Light Propagation Through Biological Tissue and Other Diffusive Media: Theory, Solutions, and...

Light Propagation throughBiological Tissueand Other Diffusive MediaTHEORY, SOLUTIONS, AND SOFTWARE

Bellingham, Washington USA

Light Propagation throughBiological Tissueand Other Diffusive MediaTHEORY, SOLUTIONS, AND SOFTWARE

Fabrizio MartelliSamuele Del Bianco

Andrea IsmaelliGiovanni Zaccanti

Library of Congress Cataloging-in-Publication Data Light propagation through biological tissue and other diffusive media : theory, solutions, and software / Fabrizio Martelli ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-0-8194-7658-6 1. Light--Transmission--Mathematical models. 2. Tissues--Optical properties. I. Martelli, Fabrizio, 1969- QC389.L54 2009 535'.3--dc22 2009049647 Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: +1 360.676.3290 Fax: +1 360.647.1445 Email: [email protected] Web: http://spie.org Copyright © 2010 Society of Photo-Optical Instrumentation Engineers (SPIE) All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the work and thought of the author(s). Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America.

To our families

Contents

Acknowledgements xiii

Disclaimer xv

List of Acronyms xvii

List of Symbols xix

1 Introduction 1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I THEORY 7

2 Scattering and Absorption Properties of Diffusive Media 92.1 Approach Followed in this Book . . . . . . . . . . . . . . . . . . 92.2 Optical Properties of a Turbid Medium . . . . . . . . . . . . . . . . 11

2.2.1 Absorption properties . . . . . . . . . . . . . . . . . . . . 122.2.2 Scattering properties . . . . . . . . . . . . . . . . . . . . 13

2.3 Statistical Meaning of the Optical Properties of a Turbid Medium . 182.4 Similarity Relation and Reduced Scattering Coefficient . . . . . . 192.5 Examples of Diffusive Media . . . . . . . . . . . . . . . . . . . . 22References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 The Radiative Transfer Equation and Diffusion Equation 293.1 Quantities Used to Describe Radiative Transfer . . . . . . . . . . 293.2 The Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . 313.3 The Green’s Function Method . . . . . . . . . . . . . . . . . . . 323.4 Properties of the Radiative Transfer Equation . . . . . . . . . . . 33

3.4.1 Scaling properties . . . . . . . . . . . . . . . . . . . . . . 333.4.2 Dependence on absorption . . . . . . . . . . . . . . . . . 35

3.5 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5.1 The diffusion approximation . . . . . . . . . . . . . . . . 38

3.6 Derivation of the Diffusion Equation . . . . . . . . . . . . . . . . 39

viii CONTENTS

3.7 Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 413.8 Properties of the Diffusion Equation . . . . . . . . . . . . . . . . 42

3.8.1 Scaling properties . . . . . . . . . . . . . . . . . . . . . . 423.8.2 Dependence on absorption . . . . . . . . . . . . . . . . . 42

3.9 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 433.9.1 Boundary conditions at the interface between diffusive and

non-scattering media . . . . . . . . . . . . . . . . . . . . 433.9.2 Boundary conditions at the interface between two diffusive

media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

II SOLUTIONS 55

4 Solutions of the Diffusion Equation for Homogeneous Media 574.1 Solution of the Diffusion Equation for an Infinite Medium . . . . . 574.2 Solution of the Diffusion Equation for the Slab Geometry . . . . . 624.3 Analytical Green’s Functions for Transmittance and Reflectance . 664.4 Other Solutions for the Outgoing Flux . . . . . . . . . . . . . . . 744.5 Analytical Green’s Function for the Parallelepiped . . . . . . . . . 804.6 Analytical Green’s Function for the Infinite Cylinder . . . . . . . . 814.7 Analytical Green’s Function for the Sphere . . . . . . . . . . . . 834.8 Angular Dependence of Radiance Outgoing from a Diffusive Medium 84References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Hybrid Solutions of the Radiative Transfer Equation 935.1 General Hybrid Approach to the Solutions for the Slab Geometry . 945.2 Analytical Solutions of the Time-Dependent Radiative Transfer

Equation for an Infinite Homogeneous Medium . . . . . . . . . . . 975.2.1 Almost exact time-resolved Green’s function of the radiative

transfer equation for an infinite medium with isotropicscattering . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.2 Heuristic time-resolved Green’s function of the radiativetransfer equation for an infinite medium with non-isotropicscattering . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.3 Time-resolved Green’s function of the telegrapher equationfor an infinite medium . . . . . . . . . . . . . . . . . . . 99

5.3 Comparison of the Hybrid Models Based on the Radiative TransferEquation and Telegrapher Equation with the Solution of the DiffusionEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 The Diffusion Equation for Layered Media 109

CONTENTS ix

6.1 Photon Migration through Layered Media . . . . . . . . . . . . . 1096.2 Initial and Boundary Value Problems for Parabolic Equations . . . 1106.3 Solution of the DE for a Two-Layer Cylinder . . . . . . . . . . . 1126.4 Examples of Reflectance and Transmittance of a Layered Medium 1186.5 General Property of Light Re-emitted by a Diffusive Medium . . . . 121

6.5.1 Mean time of flight in a generic layer of a homogeneouscylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.5.2 Mean time of flight in a two-layer cylinder . . . . . . . . 1236.5.3 Penetration depth in a homogeneous medium . . . . . . . 1246.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 125

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7 Solutions of the Diffusion Equation with Perturbation Theory 1317.1 Perturbation Theory in a Diffusive Medium and the Born

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.2 Perturbation Theory: Solutions for the Infinite Medium . . . . . . 135

7.2.1 Examples of perturbation for the infinite medium . . . . . . 1377.3 Perturbation Theory: Solutions for the Slab . . . . . . . . . . . . . 141

7.3.1 Examples of perturbation for the slab . . . . . . . . . . . 1487.4 Perturbation Approach for Hybrid Models . . . . . . . . . . . . . 1547.5 Perturbation Approach for the Layered Slab and for Other Geometries1557.6 Absorption Perturbation by Use of the Internal Pathlength Moments 156References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

III SOFTWARE AND ACCURACY OF SOLUTIONS 163

8 Software 1658.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.2 The Diffusion&Perturbation Program . . . . . . . . . . . . . . . 1668.3 Source Code: Solutions of the Diffusion Equation and Hybrid Models170

8.3.1 Solutions of the diffusion equation for homogeneous media 1718.3.2 Solutions of the diffusion equation for layered media . . . 1758.3.3 Hybrid models for the homogeneous infinite medium . . . . 1778.3.4 Hybrid models for the homogeneous slab . . . . . . . . . 1808.3.5 Hybrid models for the homogeneous parallelepiped . . . . 1828.3.6 General purpose subroutines and functions . . . . . . . . 182

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9 Reference Monte Carlo Results 1879.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879.2 Rules to Simulate the Trajectories and General Remarks . . . . . . 1879.3 Monte Carlo Program for the Infinite Homogeneous Medium . . . . 191

x CONTENTS

9.4 Monte Carlo Programs for the Homogeneous and the Layered Slab 1939.5 Monte Carlo Code for the Slab Containing an Inhomogeneity . . . 1959.6 Description of the Monte Carlo Results Reported in the CD-ROM . 197

9.6.1 Homogeneous infinite medium . . . . . . . . . . . . . . . . 1979.6.2 Homogeneous slab . . . . . . . . . . . . . . . . . . . . . 1989.6.3 Layered slab . . . . . . . . . . . . . . . . . . . . . . . . 1999.6.4 Perturbation due to inhomogeneities inside the homogeneous

slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

10 Comparisons of Analytical Solutions with Monte Carlo Results 20310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20310.2 Comparisons Between Monte Carlo and the Diffusion Equation:

Homogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . 20410.2.1 Infinite homogeneous medium . . . . . . . . . . . . . . . 204

10.2.1.1 Time-resolved results . . . . . . . . . . . . . . 20410.2.1.2 Continuous wave results . . . . . . . . . . . . . 209

10.2.2 Homogeneous slab . . . . . . . . . . . . . . . . . . . . . 21310.2.2.1 Time-resolved results . . . . . . . . . . . . . . 21310.2.2.2 Continuous wave results . . . . . . . . . . . . . . 217

10.3 Comparison Between Monte Carlo and the Diffusion Equation:Homogeneous Slab with an Internal Inhomogeneity . . . . . . . . . 217

10.4 Comparisons Between Monte Carlo and the Diffusion Equation:Layered Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.5 Comparisons Between Monte Carlo and Hybrid Models . . . . . . 22510.5.1 Infinite homogeneous medium . . . . . . . . . . . . . . . 22610.5.2 Slab geometry . . . . . . . . . . . . . . . . . . . . . . . 230

10.6 Outgoing Flux: Comparison between Fick and Extrapolated BoundaryPartial Current Approaches . . . . . . . . . . . . . . . . . . . . . 232

10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23410.7.1 Infinite medium . . . . . . . . . . . . . . . . . . . . . . . 23410.7.2 Homogeneous slab . . . . . . . . . . . . . . . . . . . . . . 23710.7.3 Layered slab . . . . . . . . . . . . . . . . . . . . . . . . 23810.7.4 Slab with inhomogeneities inside . . . . . . . . . . . . . 23810.7.5 Diffusive media . . . . . . . . . . . . . . . . . . . . . . . 238

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Appendices 243Appendix A: The First Simplifying Assumption of the Diffusion Appro-

ximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Appendix B: Fick’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 246Appendix C: Boundary Conditions at the Interface Between Diffusive and

Non-Scattering Media . . . . . . . . . . . . . . . . . . . . . . . . 250

CONTENTS xi

Appendix D: Boundary Conditions at the Interface Between two DiffusiveMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Appendix E: Green’s Function of the Diffusion Equation in an InfiniteHomogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . 255

Appendix F: Temporal Integration of the Time-Dependent Green’s Func-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Appendix G: Eigenfunction Expansion . . . . . . . . . . . . . . . . . . 262Appendix H: Green’s Function of the Diffusion Equation for the Homoge-

neous Cube Obtained with the Eigenfunction Method . . . . . . . 265Appendix I: Expression for the Normalizing Factor . . . . . . . . . . . 268References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Index 271

AcknowledgementsWe have to thank many people for their valuable contributions to a growing body ofknowledge from which this book originated.

We are very thankful to Piero Bruscaglioni, who introduced us to the study oflight propagation through turbid media, for his constant help and fruitful discussions.

A great amount of the research that led to this book has been contributed bythe many students who worked in our laboratory to prepare their dissertations.Their questions, the fruitful discussions with them, their work, and the new ideasthey proposed helped us very much in developing new models and methodologiesfor investigating photon migration through turbid media. Thanks a lot to EnricoBattistelli, Adriana Taddeucci, Michele Bassani, Daniele Contini, Lucia Alianelli,Silvia Carraresi, and others!

We would like also to thank our numerous colleagues all over the world for theirsuggestions and stimulating discussions. A special thanks to the colleagues involvedin the nEUROPt project, and in particular to friends Antonio Pifferi, Lorenzo Spinelli,and Alessandro Torricelli from Politecnico di Milano: We are very grateful for theirsuggestions and the fruitful cooperation.

We thank our colleague and friend Danilo Marcucci not only for his help to solvepractical problems but also for his cheerful companionship, and Ricardo Lepore,with whom we recently started a promising collaboration.

We express our gratitude to Yukio Yamada for inviting Fabrizio Martelli toJapan’s Mechanical Engineering Laboratory (MEL) with the STA fellowship pro-gram: The research activity carried out at MEL by Fabrizio Martelli has beenprecious for the studies on photon migration through layered diffusive media.

We wish to thank our friend Angelo Sassaroli from Tufts University for hisadvice and opinions on the work done. The discussions and the exchange of viewson several subjects have been of constant help with writing this book.

The research leading to these results received funding from the European Com-munity’s Seventh Framework Programme [FP7/2007-2013] under grant agreementn. HEALTH-F5-2008-201076 (nEUROPt project).

DisclaimerWhile the authors have used their best efforts in preparing this book and the enclosedCD-ROM, they make no warranties, express or implied, that the formulas of thebook and the software and results contained in the enclosed CD-ROM are freeof error, or are consistent with any standard of merchantability, or will meet therequirements for any particular application. They should not be relied upon forsolving a problem whose incorrect solution could result in injury to a person or lossof property. Use of the programs in such a manner is at the user’s own risk. Theauthors disclaim all liability for direct or consequential damages resulting from useof the programs.

List of AcronymsAcronym DescriptionCW Continuous WaveDE Diffusion EquationDLL Dynamic-Link LibraryDOT Diffuse Optical TomographyEBC Extrapolated Boundary ConditionEBPC Extrapolated Boundary Partial CurrentFWHM Full Width Half MaximumHG Henyey and GreensteinMC Monte CarloNIRS Near-Infrared SpectroscopyPCBC Partial Current Boundary ConditionRTE Radiative Transfer EquationTE Telegrapher EquationTCSPC Time-Correlated Single-Photon CountingTPSF Temporal Point Spread FunctionTR Time-ResolvedZBC Zero Boundary Condition

List of SymbolsLatin-basedNotation Descriptiona Radius of the spherea′

Extrapolated radius of the sphereA Coefficient for the extrapolated boundary conditionc Speed of light in vacuumCa Absorption cross sectionCs Scattering cross sectionCg Geometrical cross section⟨d2k

⟩Mean square distance from the source after kscattering events

dE Elementary energydΣ Elementary areadV Elementary volumedΩ Elementary solid angledP Elementary powerD Diffusion coefficientE2, E3, E4 Coefficients for the boundary condition between

two diffusive mediaf1(θ), f2(ϕ), f3(z) Probability distribution functions for the photon’s

scatteringF Cumulative probability functionF (θe) Angular dependence of the outgoing radianceg Asymmetry factorG Green’s functionh Planck’s constantI Radiance or specific intensityIn Modified Bessel function of order n~J Flux vectorJn Bessel functions of order nKn Modified Bessel function of order n〈k〉 Mean number of scattering events undergone by photonslmax Maximum length of a photon trajectory

xx List of Symbols

Latin-basedNotation Description`s = 1/µs Scattering mean free path`a = 1/µa Absorption mean free path`t = 1/µt Extinction mean free path`′

= 1/µ′s Transport mean free path

〈l〉 Mean pathlength of photons〈lR〉 Mean pathlength for the total reflectance〈lT 〉 Mean pathlength for the total transmittanceLx, Ly, Lz Dimensions for the parallelepipedL Radius of the cylinderL′

Extrapolated radius of the cylindern Refractive indexni Refractive index of the diffusive mediumne Refractive index of the external mediumnr Relative refractive indexN Particle concentrationNln Normalizing factor for the two-layer cylinder solutionp(θ) Scalar scattering functionp(z) Penetration depth of photonsP Impinging power of a light beam or detected powerPn Legendre Polynomialsq0 Source term of the diffusion equationQa = Ca/Cg Absorption efficiencyQs = Cs/Cg Scattering efficiencyQ′s = Qs (1− g) Reduced scattering efficiency

r Radius of particles~r Position of the receiver→r0 Position of the source→r2 Position of the inhomogeneity→r3 Position of the detector−→r+m,−→r−m Sources positions with the method of images for the

slab→rs Position vector of the real source→r′ Position of the sourceR ReflectanceRF Fresnel reflection coefficient for unpolarized lights Thickness of the slabs Unit vector for the direction of observation of radianceSvirt Virtual source term

List of Symbols xxi

Latin-basedNotation Descriptiont Current observation timet′

Emission time of the source〈t〉 Mean time of flight of photons〈tn〉 nth–order momentt0 Time of flight for ballistic photons〈ti〉 Mean time of flight spent inside a layer iT Transmittanceu Energy densityv Speed of light inside the mediumwmp Contribution to the TPSF of the mth trajectoryx = 2πr/λ Size parameter〈xk〉 , 〈yk〉 , 〈zk〉 Mean photon’s coordinates after k scattering eventsze = 2AD Extrapolated distance

xxii List of Symbols

Greek-basedNotation Descriptionα Angular range of the field of view of the receiverβm Positive roots of the equation Jm(βmL

′) = 0

Γ Gamma functionδΦ = Φpert − Φ0 Perturbation for the fluence rateδΦa Absorption perturbation for the fluence rateδΦD Scattering perturbation for the fluence rateδRa Absorption perturbation for the reflectanceδRD Scattering perturbation for the reflectanceδT a Absorption perturbation for the transmittanceδTD Scattering perturbation for the transmittance∂V External physical boundary of a domain Vε Source term of the radiative transfer equationθ Polar angleΘ(x) Step function (0 for x < 0 and 1 for x > 0)λ WavelengthΛ Single-scattering albedoλn Eigenvaluesµa Absorption coefficientµs Scattering coefficientµt = µa + µs Extinction coefficientµ′s = µs(1− g) Reduced scattering coefficientµeff Effective attenuation coefficientξn Eigenfunctionsρ Distance of the receiver from the pencil light beamρv Volume fraction of particlesσ Standard deviationΣ Cross section of the light beamτ Optical thicknessϕ Azimuthal angleΦ Fluence rateΦ0 Unperturbed fluence rateΦpert Perturbed fluence rateΩd Acceptance solid angle of the detection system

Chapter 1

IntroductionThe purpose of this book and the enclosed software is to provide a tool with dualfunctionality. The text summarizes the theory of light propagation through diffusivemedia, while the software serves as a computational tool giving users an interactiveapproach to understanding the link between theory and real calculations. Throughthis two-pronged approach, the authors believe that this tool will be valuable bothfor research investigations and educational purposes.

The software on the CD-ROM is designed to calculate the solutions of thediffusion equation (DE) — solutions that can be verified by comparing against theenclosed set of reference Monte Carlo (MC) results. The text describes the basictheory of photon transport along with analytical solutions of the diffusion equationfor several geometries. The book also includes a description of the software, relatedmaterials, and their use.

We will show that propagation of light through turbid media (i.e., media withscattering and absorption properties) can be accurately described with the radiativetransfer equation (RTE), a complex integro-differential equation of which analyt-ical solutions are not available for geometries of practical interest. The DE is anapproximation that can be obtained from the RTE by making some simplifyingassumptions. For the DE, analytical solutions are available in many geometries,but it is necessary to stress that these solutions are approximate. Therefore, eachapplication should be checked to verify that the accuracy of these approximations issufficient. This check can be performed by comparing the approximate solutionsagainst reference solutions of the RTE. For this purpose, the CD-ROM also includesexamples of numerical solutions of the RTE obtained with MC simulations fordifferent geometries.

Diffusive media are turbid media for which the solutions of the DE provide asufficiently accurate description of light propagation. Through these media, photonspropagate in a diffusive regime, i.e., the path followed by any photon migratingfrom the source to the detector looks like a random walk (zigzag trajectory). Thisoccurs when photons undergo a sufficiently high number of scattering events thattheir trajectories become randomized. Section 2.5 lists a number of media commonin daily life for which a diffusive regime of propagation can be assumed. This listincludes highly scattering media like biological tissues, agricultural products, wood,

1

2 Chapter 1

paper, plastic materials, sugar, salt, and milk, for which the diffusive regime can bereached even when the volume of the medium is of a few cubic centimeters. The listalso includes slightly scattering media like clouds of gas and dust in the interstellarmedium, in which case an extremely large volume is necessary.

In the book, a review is presented of the theories and the formulae based onthe DE that have been widely used for biomedical applications. These theories andformulae are the basis of the programs provided with the CD-ROM. The contentsof the book are divided into three parts. The first part (Chapters 2 and 3) presentsthe basic theory of photon transport. In Chapter 2, the general concepts and thephysical quantities necessary to describe light propagation through absorbing andscattering media are introduced. In Chapter 3, the RTE and the DE are describedand discussed. In the second part (Chapters 4, 5, 6 and 7), various solutions of theRTE are presented. Chapter 4 is devoted to solutions of the DE for homogeneousmedia. Chapter 5 is dedicated to hybrid solutions for the homogeneous slab basedon solutions of the RTE and the telegrapher equation. In Chapter 6, a solution of theDE for a two-layer medium is described. In Chapter 7, solutions of the perturbedDE when small defects are introduced into the medium are obtained with the Bornapproximation. In the third part (Chapters 8, 9 and 10), the software provided withthe CD-ROM is described, and the accuracy of the solutions presented is investigatedand discussed. In Chapter 8, the software included in the CD-ROM is described.Chapters 9 and 10 are dedicated to the validation of the solutions and of the softwaredescribed in the previous chapters. The validation is done by means of comparisonswith the results of MC simulations. In Chapter 9, a detailed description of the MCprograms employed is provided, and in Chapter 10 the results of the comparisonsare described and discussed.

In the CD-ROM, all the programs for calculating the solutions presented inthe book are provided with the results of MC simulations that can be used as astandard reference. The CD-ROM is divided into three parts: the first includes thesoftware named Diffusion&Perturbation; the second contains all the source codefor the solutions found in the book; and the third part contains the results of MCsimulations that can be used as a standard reference.

The Diffusion&Perturbation software is dedicated to the calculation and thevisualization of the solutions of the DE together with its perturbation theory fora homogeneous medium. This software package has a user-friendly interface de-veloped in Visual C and can be easily installed on any PC with a WINDOWSoperating system. With this software, the calculations of typical data types—suchas steady-state and time-resolved reflectance, transmittance and fluence rate, bothfor the case of homogeneous and inhomogeneous media—are implemented. Forinhomogeneous media, the calculations are based on the solutions of perturbationtheory of the DE.

The second part of the software provides the subroutines to estimate almostall the solutions of photon transport presented in the book: the solutions of the

Introduction 3

DE for homogeneous media, the hybrid solutions based on the RTE and on thetelegrapher equation, the solutions of the DE for a two-layer medium, and theperturbed solutions for absorbing and scattering inclusions. The source code isdistributed with a list of examples and drivers that lead the reader to use thissoftware for photon migration studies. The intent of this second part of the softwareis to involve and to stimulate the readers to build their own programs to makeinvestigations on photon migration through diffusive media. A set of subroutinesis provided that can be composed like a mosaic to perform various calculations.Detailed calculations of the physical quantities used in photon migration (e.g., flux,radiance, etc.) are carried out in various geometries of the medium. The source codehas been written using the FORTRAN language. The FORTRAN subroutines canbe converted to a dynamic-link library (DLL) interface for processes that use them.In this way, the subroutines can be employed in programs written in other languageslike C or, although it is not a straightforward procedure, in other environmentslike MATLAB.1 For a correct utilization of this second part of the CD-ROM, eventhough the software is accompanied by comments and examples of use, some basicprogramming skills are required; however, programming skills are not necessary forusing the Diffusion&Perturbation software.

The third part of the CD-ROM provides a set of results of MC simulations thatcan be used as a standard reference. The results are for three different geometries:the homogeneous infinite medium, the homogeneous slab (with and without inho-mogeneities inside), and the layered slab. All the results are presented in Excel files(*.xls).

The theories and computations presented here are relevant to near-infraredspectroscopy (NIRS) and diffuse optical tomography (DOT). NIRS and DOT areoptical techniques that use near-infrared light to probe biological tissues. Near-infrared light can penetrate deeply into tissues (some centimeters) and is sensitiveto several tissue constituents. For this reason, NIRS techniques provide importantphysiological information on living tissues. Biological tissue is a complex randommedium where light undergoes many scattering events, and its propagation is similarto a diffusion process. Although the interaction of near-infrared light with tissueis dominated by scattering effects (the distance between two subsequent scatteringevents is on the order of ≈ 100 µm), most of the physiological information is led bythe absorption of chromophores like oxy- and deoxy-hemoglobin. Since biologicaltissue behaves as a diffusive medium in the near-infrared region, many applicationsin biomedical optics require an appropriate knowledge of photon migration throughdiffusive media. Thus, in NIRS and DOT the diffusion equation is widely used topredict measurements of data types at different detectors’ sites. The main challengeof NIRS and DOT is to achieve an accurate map of the optical properties (absorptionand reduced scattering coefficients) from measurements taken at the boundary of atarget organ (inverse problem). Usually, the solution to the inverse problem in DOTor NIRS relies on accurate and possibly fast models of photon migration in tissues

4 Chapter 1

(forward problem) able to provide the light distribution inside the medium, given thegeometry, the distribution of sources, and the optical properties of the medium. Thisbook examines the forward problem, and the software provided within the enclosedCD-ROM can thus be used as a package of forward solvers for the study of photonmigration in tissue.

Thousands of papers on the diffusion of light have been published, and a reviewof this huge amount of work would require a review of several research fields, butthat is beyond the scope of this book. For these reasons, the bibliography does notexhaustively cover all of the work done on the diffusion equation but refers mainlyto publications in the field of biomedical optics and more precisely to the field ofNIRS and DOT. In order to have a more complete view of photon transport, wesuggest the reader refer to other classical books on light propagation.2–6

Although the theoretical and computational tools provided with this book andCD-ROM have their primary use in the field of biomedical optics, there are manyother applications in which they can be used. In fact, many media (agriculturalproducts, wood, food, plastic materials, paper, pharmaceutical products, etc.) haveoptical properties at visible and/or near-infrared wavelengths for which light propa-gation in the diffusive regime can be established, even in a few cubic centimetersof material. Therefore, the same techniques used to study biological tissues can beused in many practical applications to monitor industrial processes or for qualitycontrol,7–12 or in completely different fields.

In fact, the theories used in this book (i.e., the radiative transfer equation andthe diffusion equation) arise from the more general transport theory. Transporttheory concerns the transport of particles through a background medium and isused in several applications where the transported particles and the host mediumcan have very different quantities. The advent of personal computers has madeseveral numerical methods affordable to solve transport theory, and the availabilityof numerical solutions has further encouraged the use of this theory to solve practicalproblems. The list of applications is surprisingly long. Duderstadt and Martin6

summarized some of the most relevant applications of transport theory. Transporttheory has been applied to the study of

• neutron transport in nuclear reactors,

• penetration of light through the atmosphere,

• diffusion of molecules in gases,

• diffusion of holes and electrons in semiconductors, and

• photon transport through biological tissues.

Despite the different kinds of particles (neutrons, gas molecules, electrons,photons) that may be involved in the transport processes, all these phenomena can bestudied and described by using the same basic equation. When the transport processbecomes diffusive, the transport equation can be simplified through the diffusion

Introduction 5

equation. Given a physical quantity u representative of the physical process studied(for instance, the particle density), whenever u is described by the equation

∂∂tu(~r, t)− k1∇2u(~r, t) + k2u(~r, t) = 0, (1.1)

we are dealing with a diffusive process. The coefficient k1 is related to the spatialand temporal scale of the diffusive phenomenon studied, and the coefficient k2

is related to the probability that the transported particles will be absorbed. Forradiative transfer processes, k1 will be related to the transport coefficient or diffusioncoefficient of photons through the medium. For neutron transport processes, k1

will be related to the transport coefficient of neutrons through the medium. Forthe diffusion of electrons and holes in semiconductors, k1 will be related to theelectrical conductivity. For the diffusion of molecules in gases, k1 will be related tothe transport coefficient of the molecules through the gas. The above equation withk2 = 0 can be also used to describe the conduction of heat in solid isotropic materials,where u will be the temperature of the medium, and k1 will be the thermometricconductivity of the substance, i.e., a material-specific quantity depending on thethermal conductivity, the density, and the specific heat of the substance.13 Then,with the same mathematical tool, very different physical processes can be studiedwhere very different physical interpretations are required.

Theories and software presented in the book and in the CD-ROM can therefore,in principle, be re-adapted to other fields where the same basic equation is used, i.e.,where a diffusive phenomenon is involved.

We finally point out that the theories and solutions presented in this book havebeen obtained with reference to media illuminated by unpolarized light. However,it should be stressed that the solutions are also applicable to media illuminatedby polarized light, which commonly occurs when laser sources are used. In fact,multiple scattering randomly changes polarization of scattered light, so that lightdetected after a sufficiently large number of scattering events is completely de-polarized. Memory of polarization only remains near the source where photonsarrive after a small number of scattering events. It has been shown with numericalsimulations14, 15 and experiments15 that when propagation occurs in diffusive regime(i.e., when the solutions presented in this book become applicable), received photonshave lost memory of the initial state of polarization, and the results for polarizedlight become almost identical to those obtained for unpolarized light.

6 Chapter 1

References

1. The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, MATLAB 7External Interfaces (2008).

2. H. C. van de Hulst, Light Scattering by Small Particles, John Wiley & Sons,New York (1957).

3. S. Chandrasekhar, Radiative Transfer, Oxford University Press, London/Dover,New York (1960).

4. K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-WesleyPublishing Company, Massachusetts, Palo Alto, London (1967).

5. A. Ishimaru, Wave Propagation and Scattering in Random Media, Vol. 1,Academic Press, New York (1978).

6. J. J. Duderstadt and W. R. Martin, Transport Theory, John Wiley & Sons,New York (1979).

7. P. Tatzer, M. Wolf, and T. Panner, “Industrial application for inline materialsorting using hyperspectral imaging in the NIR range,” Real-Time Imaging 11,99–107 (2005).

8. M. Blanco, J. Coello, H. Iturriaga, S. Maspoch, and C. de-la Pezuela, “Near-infrared spectroscopy in the pharmaceutical industry,” Analyst 123, 135R–150R(1998).

9. Y. Roggo, P. Chalus, L. Maurer, C. Lema-Martinez, A. Edmond, and N. Jent,“A review of near infrared spectroscopy and chemometrics in pharmaceuticaltechnologies,” J. Pharm. Biomed. Anal. 44, 683–700 (2007).

10. G. Reich, “Near-infrared spectroscopy and imaging: Basic principles andpharmaceutical applications,” Appl. Opt. 57, 1109–1143 (2005).

11. M. C. Cruz and J. A. Lopes, “Quality control of pharmaceuticals with NIR:from lab to process line,” Vib. Spectrosc. 49, 204–210 (2009).

12. M. Zude (Ed.), Optical Monitoring of Fresh and Processed Agricultural Crops,(Contemporary Food Engineering Series), CRC Press, Boca Raton, Florida(2009).

13. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford UniversityPress, Ely House, London (1959).

14. P. Bruscaglioni, G. Zaccanti, and Q. Wei, “Transmission of a pulsed polarizedlight beam through thick turbid media: numerical results,” Appl. Opt. 32,6142–6150 (1993).

15. J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized lightto discriminate short-path photons in a multiply scattering medium,” Appl. Opt.31, 6535–6546 (1992).

Chapter 2

Scattering and AbsorptionProperties of Diffusive Media2.1 Approach Followed in this Book

The collection of analytical solutions reported in this book and the related softwarein the enclosed CD-ROM concern propagation of light through diffusive media.Diffusive media are turbid media where propagation occurs in the diffusive regime,i.e., propagation is dominated by multiple scattering, and photons undergo manyscattering events before being detected. The solutions describe how the energypropagates through turbid media in which light-matter interaction can be modeledwith elastic scattering and absorption. Scattering interaction deflects photons alongnew directions of propagation, but the energy, and thus the wavelength and thefrequency, of scattered photons remains unchanged. Absorption interaction causesthe disappearance of photons. Therefore, the wavelength of the radiation that outlivesabsorption and propagates through the medium remains unchanged. The turbidmedia considered in this book are random media, i.e., media where the amplitudeand phase of the waves propagating through such media fluctuate randomly in timeand space.1

The simple model used in this book to represent light-matter interaction is acrude assumption that avoids getting inside the actual complexity of the phenomenaof light-matter interaction, and it has the advantage of leading to relatively simpleanalytical solutions for photon migration valid for many real media of everyday life.Light-matter interaction is characterized by several phenomena that will be ignoredin this book since they are not going to affect the results of our investigations. Somelight-matter interaction phenomena are mentioned below.

Absorption and scattering are the main phenomena of light-matter interaction.Absorption is a phenomenon related to the absorption bands of molecules. When anelectron within a molecule is raised by photon absorption to an excited state, therelaxation to the ground state can occur following a non-radiative decay processwith the emission of heat or/and a radiative process with the emission of a photon ata different wavelength (luminescence effects). The non-radiative processes precedeand/or compete with the luminescence processes. When the electron relaxes to the

9

10 Chapter 2

ground state with the emission of one photon, the involved processes are called fluo-rescence and phosphorescence. These photons are re-emitted at a longer wavelength(their energy is smaller with respect to absorbed photons) and can continue their mi-gration through the medium. In this book, absorbed photons will be considered lostfor propagation, and these effects will not be considered. For details on fluorescenceand phosphorescence, see Ref. 2. Only the effects of linear absorption (where thestrength of absorption depends linearly on the intensity of light) will be considered,while effects like two-photon absorption (where the strength of absorption dependson the square of the light intensity) will be neglected. We stress that this effectis many orders of magnitude weaker than linear absorption and therefore is not acommon phenomenon.3 For the same reasons, the effects of multi-photon absorptionare not discussed.

Light scattering originates from the interaction of photons with structural het-erogeneities present inside material bodies at the wavelength scale. The interactionbetween a photon and a molecule results in a photon moving in a different directionand a molecule that may maintain, increase, or decrease its energy. If the energyof the scattered photon is the same as the incident photon, the photon-moleculeinteraction is denoted as elastic scattering; while if the energy of the scattered photonis lower or higher than the incident photon, the interaction is denoted as inelasticscattering. Rayleigh scattering is an example of elastic scattering, which occurswhen light propagates through gases, while Raman scattering is an example ofinelastic scattering. Light scattering can be also influenced by the movement ofscattering particles (e.g., the Doppler effect,4, 5 dynamic light scattering6, 7). Noneof these effects will be covered in this book.

In this chapter, we introduce the optical properties (absorption and scatteringcoefficients, scattering function) used to describe the interaction of light with theturbid medium, and we summarize their dependence on the physical and chemicalproperties of the medium. The statistical meaning of the optical properties is dis-cussed in Sec. 2.3. In Sec. 2.4, the reduced scattering coefficient, used to describe thescattering properties of a diffusive medium, is introduced together with a similarityrelation useful to understanding propagation in the multiple-scattering regime. InSec. 2.5, examples of diffusive media are reported.

We point out that the simple modeling assumed for the interaction of the radiationwith the turbid medium is applicable to many problems of practical interest andhas the advantage of significantly reducing both the complexity of the problemand the mathematical formalism used for describing light propagation. However,this modeling is not suitable for studying light propagation that is dominated byinelastic scattering (e.g., fluorescence, phosphorescence, etc.). For a deeper study oflight-matter interaction and of more general theories for describing light propagation,the interested reader is referred to Refs. 1, 8–16.

Scattering and Absorption Properties of Diffusive Media 11

2.2 Optical Properties of a Turbid Medium

Light-matter interaction due to absorption is described by the absorption coefficientµa, defined as the ratio between the power absorbed in the unit volume and thepower incident per unit area.

The interaction due to scattering is described by the scattering coefficient µsand by the scattering phase function p(s, s′). The scattering coefficient is defined asthe ratio between the power scattered in the unit volume and the power incident perunit area. The scattering phase function, or simply the scattering function, is definedas the probability that a photon traveling in direction s is scattered within the unitsolid angle around the direction s′. The scattering function has the dimensions ofsr−1. Since, as mentioned in Chapter 1, theories and solutions presented in this bookpertain to media illuminated by unpolarized light, the scattering properties of themedium are introduced with reference to unpolarized radiation. In this case, if weconsider isotropic scatterers (spherical particles or randomly oriented non-sphericalparticles), the scattering function only depends on the scattering angle θ, i.e., theangle between directions s and s′. So p(s, s′) = p(θ). The following normalizationfor the scattering function is thus assumed:

∫4π

p(s, s′)dΩ ′ =2π

π∫0

p(θ) sin θdθ = 1. (2.1)

When propagation is dominated by multiple scattering, a single number canbe sufficient to characterize the scattering function: It is the asymmetry factor g,defined as the average cosine of the scattering angle:

g = 〈cos θ〉=2π

π∫0

cos θp(θ) sin θdθ. (2.2)

The total power extracted by absorption or scattering by the unit volume isdescribed by the extinction coefficient µt, defined as

µt = µa + µs. (2.3)

The coefficients µa, µs, and µt are measured in mm−1. To describe the fraction ofthe total power extracted by scattering, the single-scattering albedo Λ is also used:

Λ =µsµt. (2.4)

The extinction coefficient describes the attenuation due to absorption and scat-tering. With reference to a light beam that propagates along the z direction, thefraction of power extracted by the volume element of thickness dz is, according tothe definitions, µt(z )Σdz P (z )

Σ , where Σ is the cross section of the beam, and P (z )

12 Chapter 2

is the impinging power. The variation of power passing through the volume elementis thus

dP (z) = −µt(z )P (z )dz . (2.5)

Integrating Eq. (2.5), the Lambert-Beer law is obtained:

P (z) = Pe exp[−

z∫0

µt(z′)dz′], (2.6)

where Pe = P (z = 0) is the power emitted at z = 0. The Lambert-Beer lawdescribes the exponential decay of the beam as it travels through the turbid medium.

The fraction P (z) of the emitted power that propagates through the medium isoften indicated as the ballistic component of the beam. The exponential factor inthe Lambert-Beer law is the optical thickness τ(z) (a dimensionless quantity) ofthe medium crossed by the light beam. For a homogeneous medium, it is simplyτ(z) = µtz.

2.2.1 Absorption properties

In general, a turbid medium can be described as a background medium in whichparticles with different refractive indices are suspended. Absorption comes fromthe interaction of light with molecules both of the background medium and of thedispersed particles. Each molecule has a spectrum of absorption that can quicklychange even for small variations of the wavelength due to its characteristic lines andbands of absorption. To describe absorption, it is useful to distinguish absorptiondue to molecules of the background medium µam and absorption due to dispersedparticles µap . Depending on the type and concentration of molecules, we can haveµam > µap or vice versa. Absorption caused by molecules within the backgroundmedium can be evaluated from the absorption cross section Ci (measured in mm2)and the number of molecules per unit volume Ni (mm−3) as

µam =∑i

NiCi, (2.7)

where summation is on the different types of absorbing molecules,17 or from theimaginary part of the refractive index of the background medium Im(nm) as

µam =4πIm(nm)

λ, (2.8)

where λ is the wavelength in the medium.14, 15

Absorption due to particles can be described through the absorption cross sectionCa of each particle. Ca is defined as the ratio between the power absorbed by theparticle and the incident power per unit area. For particles randomly distributedand non-spherical particles randomly oriented, if the concentration of particles is


sufficiently low, the absorption of each particle is not affected by its neighboringparticles, and the power absorbed per unit volume is obtained by adding the powerabsorbed by each particle. With reference to spherical particles, µap can be writtenas

µap = N

∫ ∞0

Ca (r) f (r) dr, (2.9)

where N is the particle concentration (mm−3), and Nf (r) dr is the concentrationof particles with a radius in the interval (r, r + dr). The efficiency of a particle toabsorb radiation can be also expressed by the absorption efficiency Qa = Ca/Cg,where Cg = πr2 is the geometric cross section of the particle.

2.2.2 Scattering properties

Elastic light scattering originates from the heterogeneity of the refractive indexinside the medium.15 It is also possible to distinguish the effect of molecules ofthe background medium from the effect of dispersed particles. Scattering frommolecules of the background medium originates from density fluctuations; in fact,even in a homogeneous medium (e.g., gases, pure water), the number of moleculesinside a given volume element fluctuates in time, although the average number ofmolecules is constant. These fluctuations give rise to fluctuations in the refractiveindex and then to scattering. For media in which propagation is dominated by multi-ple scattering, the effect of the background is usually negligible, so we only considerthe effect of particles. Even for scattering, if the concentration of randomly orientedparticles is sufficiently low (independent scattering approximation), interferenceeffects can be disregarded, and the power scattered by the unit volume is simplyobtained by adding the power scattered by each particle. The scattering coefficientand the scattering function for the volume element can thus be obtained as

µs = N

∫ ∞0

Cs (r) f (r) dr (2.10)

and

p (θ) =N∫∞

0 p (θ, r)Cs (r) f (r) drN∫∞

0 Cs (r) f (r) dr, (2.11)

where Cs (r) and p (θ, r) are respectively the scattering cross section and the scat-tering function of particles with radius r. Cs (r) is defined as the ratio between thepower scattered by the particle and the incident power per unit area. The efficiencyof a particle to scatter radiation can be expressed by the scattering efficiency:

Qs = Cs/Cg. (2.12)

For sufficiently low concentrations of particles for which the independent scatter-ing approximation is applicable, the scattering coefficient is therefore proportionalto the particle concentration N ; thus, µs increases proportionally to the volume

14 Chapter 2

fraction ρv of particles. Conversely, the scattering function for the volume elementis obtained as the average of the scattering function of different particles and isindependent of the particle concentration.

For a dispersion of identical particles (monodispersion), the scattering functionfor the volume element is identical to the scattering function for the single particle,and µs can be simply expressed as a function of ρv and Qs. With N = ρv/(4

3πr3),

we obtain from Eqs. (2.10) and (2.12)

µs = NQsCg =34ρvQsr. (2.13)

As mentioned before, absorption is mainly related to the types of molecules inthe background medium and in the particles. Therefore, the absorption propertiesof the turbid medium mainly depend on its chemical composition. Instead, scat-tering originates from the different refractive index of particles with respect to thebackground medium, and it mainly depends on the microphysical properties of theparticles. In particular, the scattering properties depend on the relative refractiveindex nr of the particles with respect to the background medium, on the size andshape of the particles, and on the wavelength of the radiation. More precisely, thedependence on size and wavelength is through the size parameter x, which for spher-ical particles is defined as x = 2πr/λ, where λ is the wavelength in the backgroundmedium.

Small particles, also referred to as Rayleigh scatterers, are particles with r/λ ≤0.05 (or x ≤ π/10).14 These particles have a small efficiency to scatter light. Theirscattering efficiency, and thus their scattering coefficient, strongly depends on thesize parameter. Given that Qs ∝ x4, as the wavelength increases, µs decreasesas λ−4. However, their scattering function does not depend on wavelength and issimply given by

p (θ) =3

16π(1 + cos2 θ

). (2.14)

The scattering function for Rayleigh scatterers is thus symmetric with respect toθ = π/2 and has the asymmetry factor g = 0 even if it is not exactly isotropic.

For large particles, i.e., when r λ (or x 1), the scattering efficiencyis almost independent of the size parameter, and the scattering function stronglydepends on it. For these particles, Qs ∼= 2 (which is the extinction paradox in lightscattering).1, 11, 14 For the scattering function, the value in the forward direction, i.e.,p (θ = 0), is proportional to x, and thus, as the size parameter increases, the scat-tering function becomes more forward peaked. However, the total power scatteredin the forward direction (i.e., with θ < π/2) remains more or less unchanged, andthus the asymmetry factor remains almost constant, with a value that depends on therelative refractive index nr.14

Examples of results for the scattering properties of spherical non-absorbingparticles obtained with Mie theory18 are reported in Figs. 2.1–2.4. The results havebeen generated with a code based on the BHMIE subroutine reported by Bohren and


Huffman.15 Figures 2.1 and 2.2 show the scattering efficiency and the asymmetryfactor, respectively, as a function of the size parameter for different values of therelative refractive index nr of the spheres. Figures 2.3 and 2.4 show how the

10-1 100 101 1020

1

2

3

4

5

6

Qs

Size Parameter

nr = 1.04 nr = 1.1 nr = 1.2 nr = 1.33 nr = 1.5 nr = 2

10-1 100 101 1020

1

2

3

Qs

Size Parameter

nr = 0.96 nr = 0.9 nr = 0.8 nr = 0.7 nr = 0.6 nr = 0.5

Figure 2.1 Scattering efficiency versus the size parameter for different values of nr.

10-1 100 101 1020.0

0.2

0.4

0.6

0.8

1.0

1.2

Asy

mm

etry

Fac

tor

Size Parameter

nr = 1.04 nr = 1.1 nr = 1.2 nr = 1.33 nr = 1.5 nr = 2

10-1 100 101 1020.0

0.2

0.4

0.6

0.8

1.0

1.2

Asy

mm

etry

Fac

tor

Size Parameter

nr = 0.96 nr = 0.9 nr = 0.8 nr = 0.7 nr = 0.6 nr = 0.5

Figure 2.2 Asymmetry factor versus the size parameter for different values of nr.

scattering coefficient for a fixed volume fraction of monodispersed spheres dependson r and λ. The results pertain to ρv = 0.01 (i.e., spheres occupy 1% of the volume)and have been obtained using the independent scattering approximation. Figure 2.3reports the results as a function of r for λ = 0.6328 µm and different values of nr.There is a strong dependence on r: According to Eq. (2.13) and to the dependence ofQs on the size parameter, µs increases proportionally to r3 for r λ, whereas µsdecreases as r−1 for r λ. A maximum is reached for r ≈ λ. Figure 2.4 reportsµs as a function of λ for different values of r and for nr = 1.04 and 1.33. There is astrong dependence also on wavelength: In particular, µs decreases as λ−4 for values

16 Chapter 2

λ r, whereas it is almost independent of λ for λ r.

10-2 10-1 100 10110-2

10-1

100

101

102

103

s (m

m-1

)

r ( m)

nr = 1.04 nr = 1.1 nr = 1.2 nr = 1.33 nr = 1.5 nr = 2

10-2 10-1 100 10110-2

10-1

100

101

102

s (m

m-1

)r ( m)

nr = 0.96 nr = 0.9 nr = 0.8 nr = 0.7 nr = 0.6 nr = 0.5

Figure 2.3 Scattering coefficient versus the radius for monodispersions of spheres withvolume fraction 0.01 for different values of nr. The wavelength in the background mediumis 0.6328 µm.

10-1 100 10110-2

10-1

100

101

102

s (m

m-1

)

Wavelength ( m)

r = 0.1 m r = 0.5 m r = 1 m r = 5 m

nr = 1.04

10-1 100 10110-1

100

101

102

103

s (m

m-1

)

Wavelength ( m)

r = 0.1 m r = 0.5 m r = 1 m r = 5 m

nr = 1.33

Figure 2.4 Scattering coefficient versus the wavelength in the background medium formonodispersions of spheres with volume fraction 0.01. The different curves correspondto different values of the radius of the spheres for nr = 1.04 and 1.33.

We point out that the scattering and absorption coefficients and the scatteringfunction previously defined do not depend on the direction of the incident unpolar-ized light. This is true for spherical particles for which, due to the symmetry, thereare not preferred directions; however, for non-spherical particles, this is true onlyif particles are randomly oriented. Otherwise, both the scattering coefficient andthe scattering function can be dependent on the direction of the incident light. Thisis the case, for instance, for cylindrical scatterers oriented in the same direction,for which the optical properties also depend on the direction of the incident light


with respect to the axis of the cylinders. In this case, the optical properties must bere-evaluated each time the incident angle changes.

As long as the independent scattering approximation is applicable, i.e., whenthe inter-particle separation is large enough to make particles independent scatterers,the optical properties of turbid media are simply related to the optical properties ofsingle particles that can be evaluated with the Mie theory for spherical particles15, 18

or with more complicated theories for non-spherical particles (for example, seeMishchenko et al.16). For practical applications, it may be useful to know howfar the independent scattering approximation can be applied. From theoreticalinvestigations, Mishchenko et al.16 suggested a simple approximated condition:For particles significantly smaller than the wavelength of the radiation, the inter-particle distance should be larger than one wavelength in the medium. For largeparticles (x > 5), the mutual distance should be larger than three times their radius.Therefore, the volume occupied by a particle should be larger than λ3 for smallparticles and larger than (3r)3 for large particles. For small particles, the volumefraction should be smaller than ρvmax = 4

3π( rλ)3. As an example, for λ = 0.6328µm, ρvmax ∼= 0.002 for r = 0.05 µm (x ∼= 0.5) and ρvmax ∼= 0.01 for r = 0.085µm. For large particles, ρvmax = 4π

81∼= 0.155. Therefore, the effect of dependent

scattering is expected to be larger for small particles. This has been confirmedby experiments.19, 20 Measurements of the scattering coefficient on small particlesshowed that with respect to the value expected from Eq. (2.13), attenuation decreasessharply as ρv is increased above 0.01. For large particles, variations are significantlysmaller and in the opposite direction, i.e., attenuation increases.19

In many experiments and practical applications in which propagation of lightdominated by multiple scattering is involved, the volume fraction can be significantlyhigher than 0.01, and the independent scattering approximation is not applicable. Forsuch media, the optical properties cannot be estimated with the simple rules basedon the optical properties of single particles. In particular, the simple proportionalityof the scattering coefficient with the particle concentration can be lost. Nevertheless,the Lambert-Beer law [Eq. (2.6)] may remain valid to describe the attenuation ofthe light beam through the medium. Thus, it remains possible to use the absorptioncoefficient to characterize the interaction due to absorption, and to use the scatteringcoefficient and the scattering function to characterize the interaction due to scattering.For example, experimental results have been reported for the optical properties ofdense scattering media with a volume fraction of particles of about 0.23.20

The definitions of absorption and scattering coefficients and other physicalparameters adopted in this book are also usually valid for media so densely packedthat suspended particles and background media can be confused. This is the case forbiological tissues, paper, many food products and drugs, and other media. However,if the scattering coefficient becomes so high that the scattering mean free path`s = 1/µs is comparable with the wavelength (see next section), due to the wave-like nature of photons, interference effects occur that may completely change the

18 Chapter 2

regime of light propagation. If k`s . 1 , with k = 2πλ (this is possible for particles

with very high refractive index and x ≈ 1), the electric field cannot even performone oscillation before the wave is scattered again, and interference effects can beso strong that transition to the Anderson localization of light can occur.21 This isbeyond the scope of this book since, for the most part, natural media have `s λ.

2.3 Statistical Meaning of the Optical Properties of a TurbidMedium

The optical properties of turbid media have a simple statistical meaning.22, 23 Thescattering function p(s, s′) is defined as the probability that a photon traveling indirection s is scattered within the unit solid angle around the direction s′. Withthe assumption of unpolarized radiation and of isotropic scatterers (spherical orrandomly oriented non-spherical particles), the scattering function only depends onthe scattering angle θ. The probability distribution functions f1 (θ) and f2 (ϕ), forthe angles θ and ϕ that specify the direction s′, are therefore simply given by

f1(θ) = 2πp(θ) sin θ (2.15)

andf2(ϕ) =

12π. (2.16)

The Lambert-Beer law [Eq. (2.6)] can be used to obtain the probability distri-bution function for the pathlength followed by photons before interacting with theturbid medium. In fact, the ratio between the power µt(z)P (z)dz extracted by thevolume element of thickness dz at depth z and the emitted power Pe can be seenas the probability that a photon emitted by the source will be scattered or absorbedwithin (z, z + dz). For the probability distribution function f3 (z), we thereforeobtain

f3 (z) = µt(z) exp[−

z∫0

µt(z′)dz′]. (2.17)

For a homogeneous medium, f3 (z) becomes

f3 (z) = µt exp(−µtz). (2.18)

The probability distribution function f3 (z) can be used to evaluate the meanfree path `t traveled by photons before being scattered or absorbed. With referenceto a homogeneous infinitely extended medium we have

`t =∫ ∞

0zf3 (z) dz =

∫ ∞0

zµt exp(−µtz)dz =1µt. (2.19)

If we consider a non-absorbing medium, we obtain for the scattering mean freepath traveled between two subsequent scattering events `s = 1/µs, and µs can bedefined as the number of scattering events undergone by a photon per unit length.For a non-scattering medium, the absorption mean free path, i.e., the mean pathtraveled by photons before being absorbed, is `a = 1/µa.


2.4 Similarity Relation and Reduced Scattering Coefficient

For a non-absorbing turbid medium in which the interaction of light can be describedwith the scattering coefficient and the scattering function, the propagation of photonscan be represented as a random walk in which they frequently change direction dueto scattering. The rules that govern photon migration are given by the probabilitydistribution functions for θ, ϕ, and z, which were introduced in the previous section.These functions are the starting point for developing MC routines to simulate lightpropagation (see Chapter 9). They can be also used to obtain some exact analyticrelations for photon migration. With reference to photons emitted in an infinitenon-absorbing homogeneous medium at z = 0 along the z-axis, if we indicate with(xk, yk, zk) the coordinates of the point in which the kth scattering event occurs, ithas been shown22 that

〈xk〉 = 〈yk〉 = 0, 〈zk〉 =1µs

k−1∑i=0

gi =1µs

1− gk

1− g(2.20)

and ⟨d2k

⟩=⟨x2k + y2

k + z2k

⟩=

2µ2s

k − (k + 1)g + gk+1

(1− g)2 . (2.21)

Since −1 ≤ g ≤ 1, for large values of k the mean value of zk reaches the value

limk→∞

〈zk〉 =1µs

11− g

=1µ′s, (2.22)

and the mean value of the square distance from the source after k scattering eventsbecomes ⟨

d2k

⟩ ∼= 2µ2s

k(1− g)(1− g)2 =

2µ′2s

k(1− g) , (2.23)

whereµ′s = µs(1− g) (2.24)

is the reduced scattering or transport coefficient of the medium. The quantity

`′

=1µ′s

(2.25)

is the transport mean free path. Equation (2.22) shows that for a homogeneousnon-absorbing medium the transport mean free path represents the mean distancetraveled by photons along the initial direction of propagation before they haveeffectively “forgotten” their original direction of motion.

Equation (2.23) suggests a similarity relation for propagation in two mediahaving different single-scattering properties—µs1 , p1(θ) and µs2 , p2(θ)—but withµs1(1− g1) = µs2(1− g2), i.e., with the same reduced scattering coefficient: Themean distance from the source after k1 scattering events in medium 1 is almost

20 Chapter 2

the same as for photons that have undergone k2 scattering events in medium 2provided k1(1− g1) = k2(1− g2). This similarity relation can be used to evaluatethe number of scattering events necessary to randomize the direction of propagationof photons propagating through a medium with non-isotropic scattering (g 6= 0). Forthe medium with isotropic scattering (g = 0), one scattering event is sufficient torandomize propagation; for non-isotropic scattering, we expect that k = 1/(1− g)scattering events are necessary. The corresponding mean pathlength is k/µs = 1/µ

′s.

These results suggest that when propagation is dominated by multiple scattering,the most important parameter to describe the scattering properties of the medium isµ′s. This is confirmed by the results obtained when the diffusion approximation

becomes applicable: In Sec. 3.6, we will show that photons undergoing manyscattering events migrate in a diffusive regime that is fully characterized by thediffusion coefficient D = 1/(3µ

′s).

It may be interesting to study how µ′s depends on the properties of scattering

particles. For this purpose, it is useful to introduce the reduced scattering efficiencyQ′s for the single particle as

Q′s = Qs (1− g) , (2.26)

from which the reduced scattering coefficient can be obtained as

µ′s = N

∫ ∞0

Q′s (r)Cg (r) f (r) dr. (2.27)

10-1 100 101 10210-4

10-3

10-2

10-1

100

101

Qs'

Size Parameter

nr = 1.04 nr = 1.1 nr = 1.2 nr = 1.33 nr = 1.5 nr = 2

10-1 100 101 10210-4

10-3

10-2

10-1

100

Qs'

Size Parameter

nr = 0.96 nr = 0.9 nr = 0.8 nr = 0.7 nr = 0.6 nr = 0.5

Figure 2.5 Reduced scattering efficiency versus the size parameter for different values ofnr.

Examples of results for Q′s as a function of the size parameter are reported in

Fig. 2.5 for different values of the relative refractive index of the spheres. Q′s isstrongly affected by nr, even if small variations are observed on the shape of thecurves. Figures 2.6 and 2.7 report µ

′s as a function of r and of λ, respectively, for


monodispersions of spheres with volume fraction ρv = 0.01. Figure 2.6 pertainsto λ = 0.6328 µm and to the same values of nr as in Fig. 2.5. µ

′s increases

proportionally to r3 for r λ, reaches a maximum for r ≈ λ, and decreases as r−1

for r λ. Also, for µ′s the shape of the curves is not much influenced by nr. In

Fig. 2.7, the results are reported for different values of r for nr = 1.04 and 1.33.

10-2 10-1 100 10110-3

10-2

10-1

100

101

102

103

s' (m

m-1

)

r ( m)

nr = 1.04 nr = 1.1 nr = 1.2 nr = 1.33 nr = 1.5 nr = 2

10-2 10-1 100 10110-3

10-2

10-1

100

101

102

103

s' (m

m-1

)

r ( m)

nr = 0.96 nr = 0.9 nr = 0.8 nr = 0.7 nr = 0.6 nr = 0.5

Figure 2.6 Reduced scattering coefficient versus the radius for monodispersions of sphereswith volume fraction 0.01 for different values of nr. The wavelength in the backgroundmedium is 0.6328 µm.

10-1 100 10110-3

10-2

10-1

100

s' (m

m-1

)

Wavelength ( m)

r = 0.1 m r = 0.5 m r = 1 m r = 5 m

nr = 1.04

10-1 100 10110-2

10-1

100

101

102

s' (m

m-1

)

Wavelength ( m)

r = 0.1 m r = 0.5 m r = 1 m r = 5 m

nr = 1.33

Figure 2.7 Examples of reduced scattering coefficient versus the wavelength in the back-ground medium for monodispersions of spheres with volume fraction 0.01. The differentcurves correspond to different values of the radius of the spheres for nr = 1.04 and 1.33.The values of the radius are indicated in the legend.

Only the case r λ has a strong dependence of µ′s on λ (µ

′s proportional to λ−4).

In the range of visible and near-infrared wavelengths, appreciable variations of µ′s

are expected only for small scatterers (r < 0.5µm).

22 Chapter 2

2.5 Examples of Diffusive Media

In this section, we provide some examples of diffusive media in daily life. In order todiscuss diffusive media, we need a definition of the diffusive regime of propagation.As will be shown in Chapter 3, light propagation through turbid media can bedescribed by the RTE. For many turbid materials, propagation can be described by asimpler but more approximate equation, the diffusion equation, for which relativelysimple analytical solutions are available. Turbid media in which propagation canbe described with good accuracy by the DE are named diffusive media. As willbe shown in subsequent chapters, the conditions necessary to establish a diffusiveregime of propagation for photons migrating from the source to the receiver dependboth on the optical properties of the turbid medium and on the geometry. In Sec. 10.7,these conditions will be indicated for establishing a diffusive regime of propagation:1) µa/µ

′s . 0.1, 2) the volume of turbid medium & (10`

′)3, and 3) photons should

have traveled paths with length & 4`′, and the corresponding number of scattering

events should be & 4/(1− g). When these conditions are satisfied, the largest partof radiation detected at distances & 2/µ

′s is due to diffused photons, and solutions

of the RTE based on the diffusion approximation can be therefore properly used forthe analysis of experimental results.

At visible and/or near-infrared wavelengths, the condition µa/µ′s . 0.1 is

satisfied for many turbid materials, and a diffusive regime of propagation cantherefore be established, provided their volume is sufficiently wide. By means ofexamples, we report here typical values for the optical properties of some turbidmedia observable in daily life.

Solid materials are often strongly scattering media and have a reduced scatteringcoefficient ≈ 1 mm−1. This is the case of biological tissues and agricultural prod-ucts. Measurements of the optical properties of biological tissue show a significantintersubject variability and a dependence on the spectral range considered. Typicalvalues of µ

′s reported at near-infrared wavelengths range between∼ 0.5 and 1 mm−1

for muscle;24, 25 ∼ 1 and 1.6 mm−1 for subcutaneous adipose tissue;24, 26 ∼ 0.5 and2.5 mm−1 for breast;27–29 ∼ 1 and 1.6 mm−1 for bone;30 ∼ 0.3 and 2 mm−1 forforehead;31, 32 ∼ 0.8 and 4 mm−1 for brain (grey and white matter);33–35 and so on.For 650 nm < λ < 1000 nm, the absorption coefficient is . 0.04 mm−1.24, 25, 27–32

There are also tissues like tendon36 and tooth37 in which scattering is due toapproximately cylindrical particles (dental tubules and collagen fibers, respectively)with a pronounced alignment. For these media, the scattering properties stronglydepend on the photon’s direction. As an example, in Ref. 36 it has been shown thatfor a bovine achilles tendon the reduced scattering coefficient changes about oneorder of magnitude (from about 0.25 to about 2.5 mm−1) when measured along theparallel or the perpendicular direction with respect to the alignment of the fibers.Another example of a scattering medium with aligned fibers is wood. For wood,significantly different results have been reported for µ

′s measured along the parallel

or the perpendicular direction (µ′s ' 2 and 20 mm−1).38, 39 For these media, theories


presented in this book are not applicable.Scattering properties of many agricultural products are in the range indicated

for biological tissues. Spectral measurements of reduced scattering coefficients inthe near infrared showed values between 1.5 and 2.5 mm−1 for apples, pears, andpeaches; between 1 and 1.5 mm−1 for kiwifruits; and between 0.5 and 1 mm−1 forcarrots and tomatoes; with absorption coefficients < 0.05 mm−1.40–42

Other examples of solid, highly scattering materials with low absorption atvisible and/or near-infrared wavelengths are plastic materials20, 43 and many othersobservable in daily life, like sugar, salt, wheat flour, cheese, paper, and manypharmaceutical products. For sugar, wheat flour, and corn meal in particular,measurements44 showed that the reduced scattering coefficient at near-infraredwavelengths spans in the range 1-70 mm−1. For cane sugar it is around 1 mm−1,white sugar is around 3 mm−1, icing sugar is around 70 mm−1, corn meal is around5 mm−1, and wheat flour is around 30 mm−1. For 650 nm < λ < 1000 nm, theabsorption coefficient remains . 0.04 mm−1. For pharmaceutical tablets, a veryhigh value of µ

′s (∼ 50 mm−1) has been reported.45

Examples of liquids with high scattering and low absorption in which prop-agation in the diffusive regime can be established are white paints, milk and fatemulsions, etc. A fat emulsion like Intralipid, a soybean oil emulsion in water thatis used clinically as an intravenously administrated nutrient and is frequently usedas a tissue phantom for calibrating NIRS and DOT instrumentation, has a reducedscattering coefficient between 1 and 2 mm−1 for 400 < λ < 900 nm when theconcentration of fat is 1%.46–48

Propagation through gases and aerosols can also occur in the diffusive regime,even if the reduced scattering coefficient can be very low. Examples of aerosolswith relatively high scattering are fogs and clouds. Dense fogs with visibility <50 m have a scattering coefficient & 0.08 m−1 [the practical relation betweenthe horizontal visibility R and µs is R = 3.912/µs (λ = 0.55µm)]49 and µ

′s &

0.01 m−1. Solutions based on the diffusion equation can thus be used to describepropagation of photons that traveled paths longer than some hundreds of meters.The clear atmosphere is also a turbid medium, but it is not sufficiently extendedfor establishing a diffusive regime of propagation. Its scattering coefficient at sealevel ranges from about 0.04 km−1 at 400 nm to 0.004 km−1 at 700 nm, and thecorresponding values for the vertical optical thickness are 0.35 and 0.035.50 Butpropagation in the diffusive regime can be reached even in media with a smallerscattering coefficient, provided their extension is sufficiently large. This may occur,for instance, on an extremely large scale for clouds of gas and dust in the interstellarmedium.

24 Chapter 2

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5. S. Wojtkiewicz, A. Liebert, H. Rix, N. Zołek, and R. Maniewski, “Laser-dopplerspectrum decomposition applied for the estimation of speed distribution ofparticles moving in a multiple scattering medium,” Phys. Med. Biol. 54, 679–697 (2009).

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13. E. P. Zege, A. I. Ivanov, and I. L. Katsev, Image Transfer Through a ScatteringMedium, Springer-Verlag, New York (1991).

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19. A. Ishimaru and Y. Kuga, “Attenuation constant of a coherent field in a densedistribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).

20. G. Zaccanti, S. D. Bianco, and F. Martelli, “Measurements of optical propertiesof high density media,” Appl. Opt. 42, 4023–4030 (2003).

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22. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relation-ships for the statistical moments of scattering point coordinates for photonmigration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).

23. L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging, JohnWiley & Sons, New York (2007).

24. A. Kienle and T. Glanzmann, “In vivo determination of the optical propertiesof muscle with time-resolved reflectance using a layered model,” Phys. Med.Biol. 44, 2689–2702 (1999).

25. P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorp-tion and scattering spectroscopy of biological tissues,” Photochem. Photobiol.Sci. 2, 124–129 (2003).

26. A. N. Bashkatov, E. A. Genina, V. I. Kochubey, and V. V. Tuchin, “Opticalproperties of the subcutaneous adipose tissue in the spectral range 400-2500nm,” Opt. Spectrosc. 99, 836–842 (2005).

27. A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S.Andersson-Engels, and R. Cubeddu, “Spectroscopic time-resolved diffusereflectance and transmittance measurements of the female breast at differentinterfiber distances,” J. Biomed. Opt. 9, 1143–1151 (2004).

28. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Noninvasiveabsorption and scattering spectroscopy of bulk diffusive media: An applicationto the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876(1999).

29. T. Durduran, R. Choe, J. P. Culver, L. Zubkov, M. J. Holboke, J. Giammarco,B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breasttissue,” Phys. Med. Biol. 47, 2847–2861 (2002).

26 Chapter 2

30. A. Pifferi, A. Torricelli, P. Taroni, A. Bassi, E. Chikoidze, E. Giambattistelli,and R. Cubeddu, “Optical biopsy of bone tissue: a step toward the diagnosis ofbone pathologies,” J. Biomed. Opt. 9, 474–480 (2004).

31. D. Comelli, A. Bassi, A. Pifferi, P. Taroni, A. Torricelli, R. Cubeddu, F. Martelli,and G. Zaccanti, “In vivo time-resolved reflectance spectroscopy of the humanforehead,” Appl. Opt. 46, 1717–1725 (2007).

32. A. Torricelli, A. Pifferi, P. Taroni, E. Giambattistelli, and R. Cubeddu, “In vivooptical characterization of human tissues from 610 to 1010 nm by time-resolvedreflectance spectroscopy,” Phys. Med. Biol. 46, 2227–2237 (2001).

33. F. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, and C. De-peursinge, “In vivo local determination of tissue optical properties: Applica-tions to human brain,” Appl. Opt. 38, 4939–4950 (1999).

34. M. Johns, C. A. Giller, D. C. German, and H. Liu, “Determination of reducedscattering coefficient of biological tissue from a needle-like probe,” Opt. Exp.13, 4828–4842 (2005).

35. A. Sassaroli, F. Martelli, Y. Tanikawa, K. Tanaka, R. Araki, Y. Onodera, andY. Yamada, “Time-resolved measurements of in vivo optical properties of pigletbrain,” Opt. Rev. 7, 420–425 (2000).

36. A. Kienle, C. Wetzel, A. Bassi, D. Comelli, P. Taroni, and A. Pifferi, “De-termination of the optical properties of anisotropic biological media using anisotropic diffusion model,” J. Biom. Opt. 12, 014026 (2007).

37. A. Kienle, F. K. Forster, R. Diebolder, and R. Hibst, “Light propagation indentin: influence of microstructure on anisotropy,” Phys. Med. Biol. 48, N7–N14 (2003).

38. A. Kienle, C. D’Andrea, F. Foschum, P. Taroni, and A. Pifferi, “Light propaga-tion in dry and wet softwood,” Opt. Express 16, 9895–9906 (2008).

39. C. D’Andrea, A. Farina, D. Comelli, A. Pifferi, P. Taroni, G. Valentini, R. Cub-eddu, L. Zoia, M. Orlandi, and A. Kienle, “Time-resolved optical spectroscopyof wood,” Appl. Spectrosc. 62, 569–574 (2008).

40. M. Zude (Ed.), Optical Monitoring of Fresh and Processed Agricultural Crops,(Contemporary Food Engineering Series), CRC Press, Boca Raton, Florida(2009).

41. B. M. Nicolaï, B. E. Verlinden, M. Desmet, S. Saevels, W. Saeys, K. Theron,R. Cubeddu, A. Pifferi, and A. Torricelli, “Time-resolved and continuous waveNIR reflectance spectroscopy to predict soluble solids content and firmness ofpear,” Postharvest Biol. Tec. 47, 68–74 (2008).


42. R. Cubeddu, C. D’Andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini,C. Dover, D. Johonson, M. Ruiz-Altisent, and C. Valero, “In vivo time-resolvedreflectance spectroscopy of the human forehead,” Appl. Opt. 40, 538–543(2001).

43. H. O. Di Rocco, D. I. Iriarte, J. A. Pomarico, H. F. Ranea-Sandoval, R. Mac-donald, and J. Voigt, “Determination of optical properties of slices of turbidmedia by diffuse CW laser light scattering profilometry,” J. Quant. Spectrosc.Ra. 105, 68–83 (2007).

44. A. Pifferi, A. Torricelli, P. Taroni, D. Comelli, A. Bassi, and R. Cubeddu,“Fully automated time domain spectrometer for the absorption and scatteringcharacterization of diffusive media,” Rev. Sci. Instrum. 78, 053103 (2007).

45. J. Johansson, S. Folestad, M. Josefson, A. Sparén, C. Abrahamsson, S. Anderson-Engels, and S. Svanberg, “Time-resolved NIR/Vis spectroscopy for analysisof solids: Pharmaceutical tablets,” Appl. Spectrosc. 56, 725–731 (2002).

46. H. van Staveren, C. Moes, J. van Marle, S. Prahl, and M. Gemert, “Lightscattering in intralipid-10% in the wavelength range of 400-1100 nm,” Appl.Opt. 30, 4507–4514 (1991).

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Chapter 3

The Radiative TransferEquation and DiffusionEquationAs noted by Ishimaru,1 two distinct theories have been developed to deal withmultiple-scattering problems. One is called the analytical theory or multiple-scattering theory and is based on Maxwell’s equations; the other is the radiativetransport theory. The analytical theory starts from Maxwell’s equations governingthe field quantities.1 This theory is mathematically rigorous since, in principle, onecan account for all the effects of multiple scattering, diffraction, and interference.However, this rather complex method has not yet produced practical models ofgeneral use to describe multiple-scattering problems.

The radiative transfer equation (RTE) is a phenomenological and heuristic theorydescribing the transport of energy through a scattering medium that lacks a rigorousmathematical formulation able to account for all the physical effects involved inlight propagation. However, it has been demonstrated that under certain simplifyingassumptions, the RTE can be derived from the electromagnetic theory of multiplescattering in discrete random media.2, 3 Despite the fact that the RTE does notrigorously represent the real physics of light propagation, it has led to useful modelsfor many practical problems.1

The RTE and the DE represent the most widely used approaches to the study ofphoton migration through highly scattering media, i.e., media where propagationis dominated by multiple scattering. In this chapter, the main framework of thesetheories is reviewed.

3.1 Quantities Used to Describe Radiative Transfer

The basic quantities used in transport theory to describe energy propagation aredefined starting from the spectral radiance, or spectral specific intensity Is(~r, s, t, ν).This is measured in Wm−2sr−1Hz−1, and is defined as the average power that atposition ~r and time t flows through the unit area oriented in the direction of the unitvector s, due to photons within a unit frequency band centered at ν, moving within

29

30 Chapter 3

the unit solid angle around s. Knowledge of the spectral specific intensity providesmore detailed information on photons that are moving inside the medium. Withreference to Fig. 3.1, the power dP that at time t flows within the solid angle dΩthrough the elementary area dΣ (oriented along the direction n) placed at ~r in thefrequency interval (ν, ν + dν) is given by

dP = Is(~r, s, t, ν)|n · s|dΣdΩdν. (3.1)

d

),,,( tsrI s

sn

ddP

d

),,,( tsrI s

sn

ddP

Figure 3.1 Relationship between the power dP and the spectral radiance Is(~r, s, t, ν) givenin Eq. (3.1).

Notice that Is(~r, s, t, ν) is related in a simple way to the energy density. In fact,the energy dE, that in the time interval dt crosses the elementary area dΣ orientedalong the direction s per unit solid angle and per unit frequency interval, occupiesthe volume dV = dΣvdt, where v is the speed of light in the medium. The energydensity per unit frequency interval and unit solid angle is therefore

dE(~r, s, t, ν)dV

=Is(~r, s, t, ν)dΣdt

dΣvdt=Is(~r, s, t, ν)

v. (3.2)

The spectral radiance Is(~r, s, t, ν) is thus proportional to the number of photonsin the unit volume with frequency ν that at time t are moving along the direction s.

Since, as mentioned in Chapter 2, we refer to media in which the frequencyof the radiation does not change during propagation, and if we consider quasi-monochromatic radiation, then we can use the term “radiance” or “specific inten-sity” I(~r, s, t) (Wm−2 sr−1) defined as the integral of Is(~r, s, t, ν) over a narrowrange of frequency.

To measure the radiance I(~r, s, t), a receiver with a small area and a small fieldof view should be placed at ~r, pointed in the direction −s.

The fluence rate (or irradiance or simply fluence) Φ(~r, t) is obtained integratingthe radiance over the entire solid angle

Φ(~r, t) =∫4π

I(~r, s, t)dΩ. (Wm−2) (3.3)

The Radiative Transfer Equation and Diffusion Equation 31

The quantity u(~r, t) = Φ(~r, t)/v (J m−3) represents the energy density at position~r at time t. The quantity u(~r, t)/(hν) (m−3), where h is Planck’s constant and hν isthe energy of a single photon, is therefore equal to the photon density. The fluenceis thus proportional to the number of photons in the unit volume regardless of theirdirection of motion.

To measure the fluence, we need a small receiver capable of collecting photonswith the same efficiency over the entire solid angle (i.e., an isotropic probe). Wecan measure a quantity proportional to the fluence by using a small sphere of highlyscattering material attached to the tip of an optical fiber.

The quantity Φ(~r, t)/(4π), i.e., the fluence divided by the whole solid angle,represents the average radiance.

Another quantity useful in describing propagation is the flux vector, defined as

~J(~r, t) =∫4π

I(~r, s, t)sdΩ , (Wm−2) (3.4)

which represents the amount and the direction of the net flux of power.All the quantities introduced to describe the energy propagation through turbid

media have been defined with reference to the more general case of a time-dependentsource. Therefore, these quantities are suitable to describe the time-resolved (TR)response when the energy emitted by the source is a function of time. In manyapplications, the medium is illuminated by Continuous Wave (CW) sources, whichare steady-state sources emitting a constant power. CW response for the radiance,the fluence, and the flux do not depend on time and can be simply indicated asI(~r, s), Φ(~r), and ~J(~r). These symbols will be used in equations and solutionsreferred to CW sources.

Radiation impinging on a volume element of a turbid medium is partially ab-sorbed and scattered. If I(~r, s, t) is the incident radiance, then, from the definitionof absorption and scattering properties, the fraction of power extracted in the volumeelement of thickness ds oriented along s is given by µtI(~r, s, t)ds. Similarly, thefraction of power re-emitted within the solid angle dΩ′ around the direction s′ due toscattering is µsI(~r, s, t)p(s, s′)dsdΩ′. Furthermore, the fraction of the total powerabsorbed in the unit volume is µaΦ(~r, t).

3.2 The Radiative Transfer Equation

The RTE1, 4–7 is an integro-differential equation that represents the balance of energy(principle of conservation of energy) for light propagation through a volume elementof an absorbing and scattering medium. The RTE can be obtained by balancing thevarious mechanisms by which the radiance at a given wavelength, I(~r, s, t), canincrease or decrease inside an arbitrary volume of the medium. In the more general

32 Chapter 3

case of time-dependent sources, the RTE is written as

∂v∂tI(~r, s, t) +∇ · [sI(~r, s, t)] + µtI(~r, s, t)

= µs∫4π

p(s, s′)I(~r, s′, t)dΩ′ + ε(~r, s, t),(3.5)

where v is the speed of light inside the medium, ε(~r, s, t) is the source term that isthe power emitted at time t per unit volume and unit solid angle along s, and dΩ′ isthe elementary solid angle in the direction s′. Considering the volume element dVidentified by the position vector ~r, the terms of Eq. (3.5) represent the following:

• ∂v∂tI(~r, s, t)dV dΩdt represents the total temporal change of energy that ispropagating along s within dV , dΩ, and dt. This quantity is proportional tothe total temporal variation of the number of photons moving along s withindV , dΩ, and dt.

• ∇ · [sI(~r, s, t)] dV dΩdt = s · [∇I(~r, s, t)] dV dΩdt represents the net flux ofenergy that is propagating along s through the volume dV , within dΩ, and dt.The term s · [∇I(~r, s, t)] is often indicated as ∂I(~r, s, t)

∂s .

• µtI(~r, s, t)dV dΩdt represents the fraction of energy that is propagating alongs within dV , dΩ, and dt extracted by scattering and absorption phenomena.

• µs∫4π

p(s, s′)I(~r, s′, t)dΩ′dV dΩdt represents the energy coming from any

direction s′ that, within dV , dΩ, and dt, is scattered in direction s.

• ε(~r, s, t)dV dΩdt represents the energy generated along s in dΩ and dt bysources inside dV .

For steady-state sources, the RTE is1

∇ · [sI(~r, s)] + µtI(~r, s) = µs

∫4π

p(s, s′)I(~r, s′)dΩ′ + ε(~r, s). (3.6)

3.3 The Green’s Function Method

The Green’s function method is a powerful approach for solving radiative transferproblems with complex source distributions and complex boundary conditions.6, 8, 9

It is convenient to express the solution of the RTE with arbitrary sources andboundary conditions as a superposition of the solutions of “basilar” problemsemploying the formalism of the Green’s function. The superposition principle canbe applied since the RTE is a linear equation.

If the source term of Eq. (3.5) is assumed to be a Dirac delta function of unitarystrength, i.e.,

ε(~r, s, t) = δ3(~r −→r′ )δ(s− s′)δ(t− t′), (3.7)


the TR solution of Eq. (3.5) represents the Green’s function of the problem. Sincethe actual source can be expressed as a superposition of elementary Dirac deltasources, the solution of the RTE can be represented as a superposition of solutionsfor elementary sources. This approach leads to the Green’s function method.6, 8, 9

The generic solution of Eq. (3.5) can be represented by the superposition integral,where the Green’s function is multiplied by the actual spatial, angular, and temporaldistribution of the source and integrated over the whole spatial, angular, and temporaldomain

I(~r, s, t) =∫V

∫4π

∞∫−∞

I(~r,→r′ , s, s′,t, t′) ε(

→r′ , s′, t′)dV ′dΩ′dt′, (3.8)

where I(~r,→r′ , s, s′,t, t′) indicates the Green’s function of Eq. (3.5), i.e., the solution

of Eq. (3.5) when the source term is Eq. (3.7), and where I(~r, s, t) is the solutionfor the actual source term ε(

→r′ , s′, t′). For the sake of simplicity, the symbol I will

be used to denote either the Green’s function of the RTE or the solution of the RTEwith arbitrary sources and boundary conditions. The variables

→r′ , s′, t′ represent

the source’s variables, while ~r, s, t represent the variables of the field point. In thisbook, the time-dependent Green’s function will be also denoted as Temporal PointSpread Function (TPSF).

From Eq. (3.8), the CW Green’s function I(~r,→r′ , s, s′) [i.e., the solution of

Eq. (3.6) for a steady-state source of unitary strength ε(~r, s) = δ3(~r−→r′ )δ(s− s′)]

is

I(~r,→r′ , s, s′)

=∫V

∫4π

∞∫−∞

I(~r,→r′′, s, s′′,t, t′′)δ3(

→r′′ −

→r′ )δ(s′′ − s′)dV ′′dΩ′′dt′′

=∞∫−∞

I(~r,→r′ , s, s′,t, t′′)dt′′

=∞∫0

I(~r,→r′ ,s, s′,t)dt.

(3.9)

3.4 Properties of the Radiative Transfer Equation

3.4.1 Scaling properties

If I(~r, s, t) is the Green’s function of Eq. (3.5), where (for the sake of simplicity) thevariables of the source are not shown, and a source is implicitly assumed at ~r′ = 0,s′ = s0, t′ = 0 for a homogeneous medium characterized by µt and p(s, s′), thefunction I(~r, s, t) is defined as

I(~r, s, t) =(µtµt

)3I(~r, s, t), (3.10)

34 Chapter 3

with~r = ~r

(µtµt

), t = t

(µtµt

), (3.11)

is the Green’s function for a homogeneous medium having the extinction coefficientµt, the same scattering function p(s, s′), and the same single-scattering albedoΛ = µs/µt (see Fig. 3.2 for a representation of the scaled domain). This propertyis demonstrated by observing that, after introducing in the RTE the dimensionlessvariables ~r = µt~r and t = µtv t, [that is equivalent to considering 1/µt as the unitto measure lengths and 1/(µtv) as the unit to measure time] the following equationis obtained:

∂∂tI(~r, s, t) +∇ ·

[sI(~r, s, t)

]+ I(~r, s, t)

= Λ∫4π

p(s, s′)I(~r, s′, t)dΩ′ + vµ3t δ

3(~r)δ(s)δ(t).(3.12)

To obtain Eq. (3.12), the following properties have been used:9 δ(αt) = 1|α|δ(t) and

δ3(α~r) = 1|α|3 δ

3(~r). In two turbid media having different µt, but the same scatteringfunction and single-scattering albedo, photon migration is governed by the samehomogeneous equation, and the difference is only in the source term. Furthermore,substituting the scaled distances and time of Eq. (3.11) in the RTE written for amedium with extinction coefficient µt and assuming µs/µt =µs/µt, the followingequation is obtained:

∂v∂tI(~r, s, t) +∇ ·

[sI(~r, s, t)

]+ µtI(~r, s, t)

= µs∫4π

p(s, s′)I(~r, s′, t)dΩ′ +(µtµt

)3δ3(~r)δ(s)δ(t),

(3.13)

from which I can be represented as the Green’s function I multiplied by(µtµt

)3.

This scaling property means that the energy flowing within the time interval(t, t+ ∆t) through the surface element ∆Σ, normal to the direction s, at the point~r of a medium with extinction coefficient µt, is the same energy that, within thescaled time interval (t, t+ ∆t)(µt/µt), passes through the scaled surface element∆Σ(µt/µt)2 at the point ~r(µt/µt) for a medium with extinction coefficient µt.Thus, the scale coefficient

(µt/µt

)3 in Eq. (3.10) takes into account the energyredistribution inside the medium determined by a variation of µt. This property,also known as the similarity principle,10 is very useful since it enables us to use theresults obtained in a certain geometry for any scaled geometry, providing Λ andp(s, s′) are kept unchanged. On the basis of this principle, when studying radiativetransfer problems, it is possible to choose the most convenient scale for laboratoryexperiments instead of making experiments in their actual scale, in which it is oftendifficult to carry out measurements.


t

2/ tt

ttrr /

ttvtl /

sourcedetector

t

r

vtl

tt

),,( tsrI),,( tsrI

t

2/ tt

ttrr /

ttvtl /

sourcedetector

t

r

vtl

tt

),,( tsrI),,( tsrI

Figure 3.2 Schematic of two scaled geometries for two turbid media with different extinctioncoefficients and the same scattering function and the same albedo.

3.4.2 Dependence on absorption

For media with homogeneous absorption [µa(~r) = constant], if I(~r, s, t, µa = 0) isthe solution of Eq. (3.5) for a non-absorbing medium with the source term describedby a temporal Dirac delta function ε(~r, s, t) = S(~r, s)δ(t), then

I(~r, s, t, µa) = I(~r, s, t, µa = 0) exp(−µavt) (3.14)

is the solution of the same equation when the absorption coefficient of the mediumis µa. The property (3.14) can be simply demonstrated by substituting Eq. (3.14) inthe RTE. This property is a generalization of the Lambert-Beer law:11, 12 Regardlessof the location of a source point, radiation received at time t after traveling alongpaths of length l = vt is attenuated by a factor exp(−µal). This property allows usto reduce the study of radiative transfer problems in which µa is independent of ~r inthe case of a non-absorbing medium and to generalize the solution by simply usingEq. (3.14).

From Eq. (3.9) and Eq. (3.14), the solution for the steady-state source S(~r, s)can be expressed as

I(~r, s, µa) =

∞∫0

I(~r, s, t, µa)dt =

∞∫0

I(~r, s, t, µa = 0) exp(−µavt)dt. (3.15)

Equation (3.15) states that in a medium with constant absorption and with stationarysources, the received power is related to the time-dependent solution for the non-absorbing medium, I(~r, s, t, µa = 0), by means of the Laplace transform.9, 13 Inprinciple, measurements carried out with a steady-state source for different valuesof µa can be used for reconstructing the temporal response.

The above property for the radiance can be extended to the power measured bya detector of area ΣR and with an acceptance solid angle ΩR. In fact, denoting with

36 Chapter 3

q the unit vector normal to ΣR, the Green’s function for the detected power P (t, µa)can be calculated, using Eq. (3.14), as

P (t, µa) =∫

ΣR

dΣ∫

ΩR

I(~r, s, t, µa) |s · q| dΩ = P (t, µa = 0) exp(−µavt),

(3.16)and the Green’s function for the steady-state source P (µa) can be consequentlycalculated as

P (µa) =

∞∫0

P (t, µa = 0) exp(−µavt)dt . (3.17)

Using Eq. (3.17), it is possible to obtain a useful expression for the mean time offlight 〈t〉 spent by photons that migrate from the source to the detector. Accordingto the definition, 〈t〉 is obtained as

〈t〉 (µa) =

∞∫0

tP (t, µa)dt

∞∫0

P (t, µa)dt= − 1

vP (µa)∂P (µa)∂µa

= − ∂

v∂µalnP (µa). (3.18)

The mean pathlength followed by photons inside the medium is obtained as 〈l〉 =v 〈t〉. Using Eq. (3.18), it is possible to obtain accurate measurements for the meantime of flight (or for the mean pathlength), even from measurements of relativeattenuation with steady-state sources. The expression of Eq. (3.18) can be furthergeneralized for the nth–order moment, 〈tn〉, as

〈tn〉 (µa) = (−1)n1

vnP (µa)∂nP (µa)∂nµa

. (3.19)

The property of Eq. (3.14) can be further generalized to obtain the depen-dence of the radiance on the absorption coefficient µai of a generic volume Viof a homogeneous or non-homogeneous medium. Due to the intrinsic meaningof the absorption coefficient, the probability of a photon being detected when theabsorption coefficient in the region i is µai is obtained by multiplying the proba-bility of being detected when µai = 0 by exp(−µaiviti) (vi and ti are the speed oflight and the time of flight in the volume Vi). Thinking in terms of the RTE, thisproperty can be written using the radiance detected at a certain point ~r. If I0(~r, s, t)is the solution of Eq. (3.5) for µai = 0, and with the source term described by atemporal Dirac delta function, then the solution when an absorption coefficient µaiis introduced in Vi can be represented as

I(~r, s, t, µai) = I0(~r, s, t)

t∫0

f0i(ti, t) exp(−µaiviti)dti , (3.20)


where f0i(ti, t) indicates the probability density function (normalized to 1) for thetime of flight ti. From Eq. (3.20), the expression for the mean time of flight spentinside the volume Vi, by photons contributing to I(~r, s, t, µai), is

〈ti〉 (~r, s, t, µai) =

t∫0

tif0i(ti,t) exp(−µaiviti)dtit∫0

f0i(ti,t) exp(−µaiviti)dti

= − 1viI(~r,s,t,µai)

∂I(~r,s,t,µai)∂µai

= − ∂vi∂µai

ln I(~r, s, t, µai) .

(3.21)

In all generality, the region i represents any region of the medium, and thus thisresult has no restrictions in its validity.

The same property stated above for the radiance is also valid for the power Pdetected by a receiver of area ΣR and with acceptance angle ΩR, since P is obtainedby an integral of the radiance [see Eq. (3.1)]. The mean time of flight spent byphotons received at time t inside the volume Vi can thus be obtained similarly toEq. (3.21) as

〈ti〉 (µai, t) = − 1viP (µai, t)

∂P (µai, t)∂µai

= − ∂

vi∂µailnP (µai, t). (3.22)

Similar relations can be obtained for steady-state sources. In particular, the meantime of flight spent by received photons inside the volume Vi of the medium is

〈ti〉 (µai) = − 1viP (µai)

∂P (µai)∂µai

= − ∂

vi∂µailnP (µai). (3.23)

This equation can be generalized for the nth–order moment 〈tni 〉 as

〈tni 〉 (µai) = (−1)n1

vni P (µai)∂nP (µai)∂nµai

. (3.24)

3.5 Diffusion Equation

The RTE is an integro-differential equation, and the retrieval of solutions is anextremely expensive computational process.7, 14 Solutions of the RTE are usuallybased on numerical methods15–22 like the finite element method7, 17, 23–30 or otherslike the discrete ordinates method,7, 31–33 the spherical harmonics method,7, 19, 34, 35

the finite difference method,36 the integral transport methods,7 and the path integralmethod.37 Among the numerical procedures there are also stochastic methods likethe MC, which is largely used to reconstruct solutions of the RTE.7, 38

The several numerical methods used to treat the RTE are a consequence ofthe high complexity of this equation. No general analytical (closed-form) so-lutions of the RTE are available,7 and simpler approximate models are usually

38 Chapter 3

sought.25, 26, 39–41 When propagation is dominated by multiple scattering, the mostwidely and successfully used model employs the diffusion approximation to yielda variety of solutions for both steady-state and time-dependent sources.42–45 Thediffusion equation is a parabolic-type partial-differential equation largely applied inseveral physics fields. In this section, its derivation from the RTE will be reviewedand discussed.

3.5.1 The diffusion approximation

In the more general case of time-dependent sources, the diffusion approximationconsists of two simplifying assumptions. The first one assumes the radiance inside adiffusive medium to be almost isotropic: The diffuse intensity I(~r, s, t) is approx-imated by the first two terms (isotropic and linearly anisotropic terms) of a seriesexpansion in spherical harmonics:1, 7, 46

I(~r, s, t) =1

4πΦ(~r, t) +

34π

~J(~r, t) · s. (3.25)

A spherical harmonic expansion truncated at the second term is usually denoted asthe P1 approximation.7, 25 Equation (3.25) is a good approximation for the radianceif the contribution of the higher-order spherical harmonics is negligible. This isusually true if the second term of the expansion is small with respect to the first,i.e., Φ(~r, t) 3 ~J(~r, t) · s. A complete derivation of Eq. (3.25) can be found inRef. 46, where a spherical harmonic expansion is introduced for the radiance. InAppendix A, Eq. (3.25) is introduced using an intuitive physical reasoning.

The second simplifying assumption assumes that the time variation of the diffuseflux vector ~J(~r, t) over a time range ∆t = 1/(vµ

′s) is negligible with respect to the

vector itself and can be expressed as7

1vµ′s

∣∣∣∣∣∂ ~J(~r, t)∂t

∣∣∣∣∣ ∣∣∣ ~J(~r, t)∣∣∣ . (3.26)

With Eq. (3.26), “slow” time variations of the flux are therefore assumed.When steady-state sources are considered, the diffusion approximation is simply

summarized by the expansion of the radiance in spherical harmonics, i.e.,

I(~r, s) =1

4πΦ(~r) +

34π

~J(~r) · s. (3.27)

In general, Eqs. (3.25), (3.26), and (3.27) are well fulfilled when photons haveundergone many scattering events, since scattering tends to randomize the directionof light propagation. Conversely, absorption obstructs the diffusive regime. In fact,absorption selectively extinguishes photons with long pathlengths; consequently, ina strongly absorbing medium only photons with few scattering events will not beabsorbed. Therefore, the diffusive regime of light propagation can be establishedwhen scattering effects are predominant on absorption.


Making use of these simplifying assumptions, it is possible to obtain from theRTE a simpler equation, the Diffusion Equation (DE), for which several analyticalsolutions are available.

3.6 Derivation of the Diffusion Equation

The RTE is an integro-differential equation for the radiance, while the DE is apartial-differential equation for the fluence rate. Thus, an integration procedure isrequired to obtain the DE from the RTE.

In order to obtain the diffusion equation for time-dependent sources, Eq. (3.5) isintegrated over all directions as follows:∫

4π


dΩ

=∫4π

[µs∫4π

p(s, s′)I(~r, s′, t)dΩ′ + ε(~r, s, t)]dΩ .

(3.28)

From this equation, simply exchanging the order of derivatives and integrals andthe orders of integration, the continuity equation is obtained without any need forsimplifying assumptions:

∂v∂tΦ(~r, t) +∇ · ~J(~r, t) + µaΦ(~r, t) =

∫4π

ε(~r, s, t)dΩ . (3.29)

To obtain the diffusion equation, the flux vector needs to be expressed as a functionof the fluence rate. For this purpose, the RTE [Eq. (3.5)] is multiplied by s and thenintegrated over all the directions as follows:∫

4π


sdΩ

=∫4π

[µs∫4π

p(s, s′)I(~r, s′, t)dΩ′ + ε(~r, s, t)]sdΩ .

(3.30)

From Eq. (3.30), making use of the simplifying assumptions of the diffusionapproximation [Eqs. (3.25) and (3.26)], Fick’s law is obtained (see Appendix B):

~J(~r, t) = −D[∇Φ(~r, t)− 3

∫4π

ε(~r, s, t)sdΩ], (3.31)

where D is the diffusion coefficient defined as

D =1

3(µa + µ′s), (3.32)

and µ′s = µs(1− g) is the reduced scattering coefficient. For media without sources

or with isotropic sources, Fick’s law becomes ~J(~r, t) = −D∇Φ(~r, t), which is theusual form used to represent the flux inside diffusive media.7

40 Chapter 3

The physical meaning of Fick’s law is that photons tend to migrate toward thoseregions of the medium where the photon density is smaller. Substituting Eq. (3.31)in the continuity equation [Eq. (3.29)], the time-dependent DE is obtained:

∂v∂tΦ(~r, t)−∇ ·

D[∇Φ(~r, t)− 3

∫4π

ε(~r, s, t)sdΩ]

+ µaΦ(~r, t)

=∫4π

ε(~r, s, t)dΩ,(3.33)

which can be written as[1v

∂

∂t−∇ ·

(D∇

)+ µa

]Φ(~r, t) = q0(~r, t), (3.34)

whereq0(~r, t) =

∫4π

ε(~r, s, t)dΩ− 3∇ ·[D

∫4π

ε(~r, s, t)sdΩ]

(3.35)

is the source term. For a homogeneous medium, the DE becomes(1v

∂

∂t−D∇2 + µa

)Φ(~r, t) = q0(~r, t). (3.36)

For an isotropic source, q0(~r, t) becomes

q0(~r, t) =∫4π

ε(~r, s, t)dΩ = 4πε(~r, t). (3.37)

Equation (3.34) shows that photon migration in the diffusive regime is fullydescribed by the knowledge of the absorption coefficient and of the reduced scatter-ing coefficient. Therefore, the detailed knowledge of the scattering function is notnecessary: It is sufficient to know the asymmetry factor g.

For steady-state sources, the continuity equation (obtained from the RTE withoutany need for simplifying assumptions) becomes

∇ · ~J(~r) + µaΦ(~r) =∫4π

ε(~r, s)dΩ . (3.38)

Since the term µaΦ(~r) represents the power absorbed in the unit volume, Eq. (3.38)has an immediate rigorous physical interpretation: The net flux emerging from theunit volume is equal to the generated power minus the absorbed power.

Fick’s law for steady-state sources is

~J(~r) = −D[∇Φ(~r)− 3

∫4π

ε(~r, s)sdΩ], (3.39)

and the DE for a homogeneous medium with isotropic sources is


(−D∇2 + µa

)Φ(~r) = q0(~r),

q0(~r) =∫4π

ε(~r, s)dΩ = 4πε(~r). (3.40)

We point out that the DE can be also obtained following a significantly differentapproach, starting from different simplifying assumptions. For steady-state sources,it can be shown (see Ref. 47) that expressing the flux through a generic elementaryarea in an infinite homogeneous medium as the sum of contributions due to radiationscattered in each volume element, Fick’s law can be obtained making only theassumption that the fluence is a slowly varying function of position: It is sufficient toconsider the Taylor series expansion of the fluence requiring a negligible contributionfrom the terms subsequent to the second derivatives. For time-dependent sources, thefurther assumption that the fractional change of the fluence is small during the time

required for radiation to travel about three reduced mean free paths[ ∣∣∣ 1

Φ∂Φ(~r,t)∂t

∣∣∣vµ′s

3

]is also required.47

3.7 Diffusion Coefficient

In the last few years, a debate about the dependence of the diffusion coefficient onabsorption has arisen.48–51 It is not our purpose to get inside the details of this debatesince its implications are of minor interest for the aim of this book. Undoubtedly,when the DE is derived from the RTE in complete generality, as it has been shownin the previous section, the diffusion coefficient shows a dependence on absorptionin accordance with Eq. (3.32). In this book, a different approach to the problem isproposed. It can be summarized in two steps:

I. The DE is derived for the non-absorbing medium (lossless diffusion equation),and the DE with a diffusion coefficient independent of absorption is obtained:

(1v

∂

∂t−D∇2

)Φ(~r, t) = q0(~r, t) (3.41)

withD = 1/

(3µ′s

). (3.42)

II. The dependence on absorption is assumed in accordance with the RTE, i.e.,in accordance with Eq. (3.14) for homogeneous media and with Eq. (3.20)for inhomogeneous media. This procedure is equivalent to adding the termµaΦ(~r, t, µa) to the lossless DE. The meaning of this procedure can be clar-ified. The lossless DE is an approximate equation of balance for Φ fornon-absorbing media. It is stressed that for the intensity coming from alldirections, the leakage of power within the unit volume due to absorption

42 Chapter 3

is, according to the RTE, µaΦ(~r, t, µa). As a consequence, adding the termµaΦ(~r, t, µa) to the lossless DE allows us to describe the effect of absorptionusing the same rules derived from the RTE.

It can be argued that the equation obtained with this procedure is slightly dif-ferent from the DE as it is known from the literature, since a diffusion coefficientindependent of µa is assumed. This is actually true. But this further assumptionallows us to treat the dependence on absorption in accordance with the RTE withoutthe additional intrinsic approximations of the diffusion approximation. Thus, tostudy photon migration in an absorbing medium, no further approximations withrespect to the non-absorbing case are introduced. Therefore, from now onward, wewill assume that the diffusion coefficient is defined by Eq. (3.42). From a practicalpoint of view, since the DE is applicable with good accuracy only when µa µ

′s,

the solutions of the DE expressed with D = 1/(3µ′s

)or with D = 1/

[3(µa + µ

′s)]

remain almost indistinguishable. Results of experiments49, 52 have shown that, foran infinite medium and for high absorption values up to µa/µ

′s = 1/3, the DE with

the diffusion coefficient independent of absorption [Eq. (3.42)] provides a betterdescription of measurements.

3.8 Properties of the Diffusion Equation

The DE shows some general properties like the RTE.

3.8.1 Scaling properties

Following the same procedure used to obtain Eq. (3.10), it is possible to show that ifΦ(~r, t) is the solution of the DE with a Dirac delta source q0(~r, t) = δ3(~r) δ(t) for ahomogeneous diffusive medium characterized by µ

′s and µa, then Φ(~r, t), defined as

Φ(~r, t) =

(µ′s

µ′s

)3

Φ(~r, t) (3.43)

with

~r = ~rµ′s

µ′s

, t = tµ′s

µ′s

, (3.44)

is the solution for a homogeneous medium with the transport coefficient µ′s and the

absorption coefficient µa, provided that µa/µa = µ′s/µ

′s.

3.8.2 Dependence on absorption

The postulated independence of the diffusion coefficient from absorption keepsthe actual physical meaning to the absorption coefficient. This means that all theproperties related to the dependence on absorption of RTE solutions (see Sec. 3.4.2)


can be extended also to any DE solution. In particular, for a homogeneous non-absorbing medium, if Φ(~r, t, µa = 0) is the solution of Eq. (3.36) for a source termq0(~r, t) = S(~r)δ(t), then

Φ(~r, t, µa) = Φ(~r, t, µa = 0) exp(−µavt) (3.45)

is the solution when the absorption coefficient of the medium (independent of ~r) is µa.Making use of Fick’s law and of the first assumption of the diffusion approximation,this property can be also extended to the flux and the radiance. Consequently, theproperty is also valid for the detected power. Finally, the properties stated for theRTE about the dependence on absorption [Eqs. (3.20) and (3.24)] remain validwithin the DE.

It is important to point out that if the diffusion coefficient were that of Eq. (3.32),i.e., D = 1/

[3(µa + µ

′s)], the absorption coefficient in diffusion theory would not

keep the actual physical meaning derived from the RTE, and all the above-mentionedproperties would not be exactly valid for the DE.

3.9 Boundary Conditions

The DE is a partial-differential equation, and any general solution will containunknown constants that need to be determined by using proper boundary conditionsoriginated from the physics of the problem investigated. The first boundary conditionto be mentioned is the behavior at infinity of any intensity concerning a radiativeprocess. It is trivial to require that any intensity vanishes at infinity. The boundaryconditions are much more complex when the diffusive medium has finite boundaries.Two main situations will be considered: diffusive/non-scattering interfaces anddiffusive/diffusive interfaces.

3.9.1 Boundary conditions at the interface between diffusive andnon-scattering media

For a diffusive medium bounded by a convex or flat surface Σ at the interface witha non-scattering region, the exact boundary condition for the radiance I(~r, s, t) isthat there should be no diffuse light entering the medium from the external non-scattering region at the interface Σ. The diffuse intensity I(~r, s, t) (~r ∈ Σ) enteringtowards any direction s pointing toward the diffusive medium can only originatefrom reflections at the boundary. With reference to Fig. 3.3, if I(~r, s′, t) is theradiance along s′, the mirror image direction (with respect to Σ) to s results in

I(~r, s, t) = RF (s′ · q)I(~r, s′, t), (3.46)

44 Chapter 3

where q is the unit vector normal to Σ, and RF ( s′ · q) is the Fresnel reflectioncoefficient for unpolarized light for the light emerging from the diffusive medium:53

RF (s′ · q) =

12

(ni cos θ−ne cos θrni cos θ+ne cos θr

)2+ 1

2

(ne cos θ−ni cos θrne cos θ+ni cos θr

)2for 0 < θ < θc

1 for θ > θc = arcsin(ne/ni).(3.47)

In Eq. (3.47), ni is the refractive index of the diffusive medium, ne is the refractiveindex of the external medium, θ is the angle of incidence (cos θ = s′ · q), and θrθr = arcsin [(ni/ne) sin θ] is the angle of refraction. The ratio ni/ne = nr isdenoted as the relative refractive index of the medium.

Diffusive mediumExternal medium

s

q's u

),,( tsrI

r

ni ne

r


s

q's u

),,( tsrI

r

ni ne

r

Figure 3.3 Schematic of a diffusive/non-scattering interface and the notation used.

With the simple angular distribution assumed for I(~r, s, t) [Eq. (3.25)], thebehavior of Eq. (3.46) cannot be satisfied exactly, and an approximate boundarycondition must be considered. It is assumed that the condition of Eq. (3.46) is trueon average for all directions s inwardly directed,1 and the following relation isobtained:

−∫

s·q<0

I(~r, s, t)(s · q)dΩ =∫

s·q>0

RF (s · q)I(~r, s, t)(s · q)dΩ. (3.48)

This is a boundary condition for the total radiation coming from the boundarysurface.

Making use of the angular distribution for the radiance obtained from thediffusion approximation [Eq. (3.25)], and calculating the integrals of Eq. (3.48), the


boundary condition for the fluence rate can be written as (see Appendix C for thederivation) [

Φ(~r, t)− 2A~J(~r, t) · q] ∣∣∣∣~r∈Σ

= 0 (3.49)

with

A =1 + 3

π/2∫0

RF (cos θ) cos2 θ sin θdθ

1− 2π/2∫0

RF (cos θ) cos θ sin θdθ

. (3.50)

The coefficient A depends on the relative refractive index nr = ni/ne. Figure 3.4shows how A depends on nr for values of refractive index mismatch of practicalinterest. A = 1 if nr = 1, and A > 1 if nr 6= 1.

0.5 1.0 1.5 2.01

2

3

4

5

6

7

8

Coe

ffici

ent A

nr

Figure 3.4 Dependence of the coefficient A on the relative refractive index.

This boundary condition is denoted as the Partial Current Boundary Condition(PCBC) and represents the most accurate boundary condition for photon diffusion ata finite boundary.1, 54–60 The PCBC states a simple relationship between the outgoingflux ~J(~r, t) · q and the fluence Φ(~r, t) on the interface of the medium:[

~J(~r, t) · q] ∣∣∣∣~r∈Σ

=Φ(~r, t)

2A

∣∣∣∣~r∈Σ

. (3.51)

The PCBC involves both the flux and the fluence. It is possible to obtain anequation in which only the fluence is involved making use of Fick’s law [Eq. (3.25)]and assuming that sources on the boundary have an isotropic emission. With theseassumptions, the PCBC can be expressed as[

Φ(~r, t) + 2AD∂

∂qΦ(~r, t)

] ∣∣∣∣~r∈Σ

= 0. (3.52)

46 Chapter 3

This boundary condition is also known as the Robin boundary condition.25, 61, 62

According to this condition, the derivative of Φ(~r, t) along the direction normalto the boundary is proportional to Φ(~r, t) itself. An extrapolated decrement ofΦ(~r, t) inside the non-scattering region is obtained if the derivative of Φ(~r, t) isassumed to remain constant in the non-scattering region to the value on the boundary.The distance from the geometrical boundary at which Φ(~r, t) is extrapolated to zerois denoted as the extrapolated distance ze and results in

ze = 2AD. (3.53)

The boundary condition that assumes Φ = 0 on the surface at the extrapolateddistance ze is denoted as the extrapolated boundary condition (EBC).57, 58, 60 Wewill assume this boundary condition for deriving almost all the analytical solutionsreported in the next chapters. A simpler, but more approximated, boundary conditionthat has been used in the literature to obtain analytical solutions of the DE is the so-called zero boundary condition (ZBC).42, 43 The ZBC simply assumes Φ(~r, t) = 0at the physical boundary of the medium. These boundary conditions are also validfor steady-state sources, as can be seen in the procedure followed in Appendix C.

It is worth making a final remark about the behavior of the flux vector at theinterface Σ. From the continuity equation [Eq. (3.29)], it is possible to demonstrate ageneral condition of continuity of the component of the flux normal to the interfaceΣ. The derivation is straightforward through Gauss’s theorem. This boundarycondition is also valid both for time-dependent sources and for steady-state sources,and it is consistent with the RTE.

3.9.2 Boundary conditions at the interface between two diffusivemedia

When there is no refractive index mismatch at a diffusive/diffusive interface, both thefluence and the normal component of the flux must be continuous on the boundary.Conversely, when Fresnel reflections occur, as a general statement, the continuity ofthe normal component of the flux vector still holds at the interface, while reflectionsdetermine a discontinuity of the fluence rate. In this section, these properties will beshown making an energy balance at the interface of the two media.

For medium 1 [refractive index n1, radiance I1(~r, s, t), and flux ~J1(~r, t)] andmedium 2 [refractive index n2, radiance I2(~r, s, t), and flux ~J2(~r, t)], in contact ata boundary Σ (see Fig. 3.5), Eq. (3.48) needs to be rewritten. In this case, the energybalance at the interface has to take into account both the reflected light from Σ andthe transmitted light through Σ of the two diffusive media. Following the procedureused by Faris,63 a couple of equations can be written:

∫s·q>0

I1(~r, s, t)(s · q)dΩ = −∫

s·q<0

RF1(−s · q)I1(~r, s, t)(s · q)dΩ

+∫

s·q>0

[1−RF2(s · q)] I2(~r, s, t)( s · q)ds (3.54)


Diffusive medium 1

Diffusive medium 2

rq

),,(1 tsrIs

Refractive index n1

Refractive index n2

Diffusive medium 1

Diffusive medium 2

rq

),,(1 tsrIs

Refractive index n1

Refractive index n2

Figure 3.5 Schematic of a diffusive/diffusive interface and notation used.

and

−∫

s·q<0

I2(~r, s, t)(s · q)dΩ =∫

s·q>0

RF2(s · q)I2(~r, s, t)(s · q)dΩ

−∫

s·q<0

[1−RF1(−s · q)] I1(~r, s, t)(s · q)dΩ,(3.55)

where q is the unit vector normal to Σ, inwardly directed to medium 1; and RF1

and RF2 are the Fresnel reflection coefficients from medium 1 to medium 2 andvice versa, respectively. Making use of the diffusion approximation [Eq. (3.25)] andadding Eqs. (3.54) and (3.55) (see Appendix D for the derivation), the continuity ofthe normal component of the flux is obtained:[

~J1(~r, t) · q]∣∣∣∣~r∈Σ

=[~J2(~r, t) · q

]∣∣∣∣~r∈Σ

. (3.56)

This result, although obtained with the diffusion approximation, coincides with thegeneral condition of continuity of the normal component of the flux at the interface(mentioned in the previous section) that can be derived using the continuity equation.Subtracting Eq. (3.55) from Eq. (3.54) (see Appendix D), the following boundarycondition for the fluence is obtained:

Φ2(~r, t)∣∣∣∣~r∈Σ

−(n2

n1

)2

Φ1(~r, t)∣∣∣∣~r∈Σ

= 2Jq(~r, t)∣∣∣∣~r∈Σ

(E3 + E4)(1− E2)

, (3.57)

where

E2 = 2

π/2∫0

RF2(cos θ) cos θ sin θdθ , (3.58)

48 Chapter 3

E3 = 3

π/2∫0

RF1(cos θ) cos2 θ sin θdθ , (3.59)

and

E4 = 3

π/2∫0

RF2(cos θ) cos2 θ sin θdθ . (3.60)

Equations (3.56) and (3.57) represent the boundary conditions at a diffusive/diffusiveinterface. When the two media have the same refractive index, the fluence at theinterface is also continuous since RF1 = RF2 = 0.

Ripoll and Nieto-Vesperinas64 have shown that for values of the ratio n2/n1

sufficiently close to 1, e.g., 0.75 < n2/n1 < 1.25, and for values of the opticalproperties typical of biological tissues, the right-hand term of Eq. (3.57) has an effecton the boundary conditions lower than 7% and can be in many cases neglected,leading to the simplified boundary condition

Φ2(~r, t)∣∣∣∣~r∈Σ

−(n2

n1

)2

Φ1(~r, t)∣∣∣∣~r∈Σ

= 0 . (3.61)

The above boundary condition will be used in Chapter 6 to obtain a solution of theDE for layered media.


References


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9. G. Arfken and H. Weber, Mathematical Methods for Physicists, AcademicPress, San Diego, London (2001).


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25. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15,R41–R93 (1999).

26. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuseoptical imaging,” Phys. Med. Biol. 50, R1–R43 (2000).

27. R. L. Barbour, H. L. Graber, Y. Pei, S. Zhong, and C. H. Schmitz, “Opticaltomographic imaging of dynamic features of dense-scattering media,” J. Opt.Soc. Am. 18, 3018–3036 (2001).

28. S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transferwith finite elements. I. Basic method and tests,” Astron. Astrophys. 380, 776–788 (2001).

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30. L. H. Liu and P. f. Hsu, “Analysis of transient radiative transfer in semitranspar-ent graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 105, 357–376(2007).


31. A. Trivedi, S. Basu, and K. Mitra, “Temporal analysis of reflected opticalsignals for short pulse laser interaction with nonhomogeneous tissue phantoms,”J. Quant. Spectrosc. Radiat. Transf. 93, 337–348 (2005).

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36. A. H. Hielscher, R. E. Raymond, E. Alcouffe, and R. L. Barbour, “Comparisonof finite-difference transport and diffusion calculations for photon migrationin homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302(1998).

37. L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, “Time-dependent photon migration using path integrals,” Phys. Rev. E 51, 6134–6141(1995).

38. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy ofthe diffusion equation to describe photon migration through an infinite medium:Numerical and experimental investigation,” Phys. Med. Biol. 45, 1359–1373(2000).

39. A. A. Kokhanovsky, “Small-angle approximations of the radiative transfertheory,” J. Phys. D: Appl. Phys. 30, 2837–2840 (1997).

40. A. A. Kokhanovsky, “Analytical solutions of multiple light scattering problems:a review,” Meas. Sci. Technol. 13, 233–240 (2002).

41. G. V. Efimov, N. V. Bryzhevoi, W. von Waldenfels, and R. Wehrse, “Solutionof the radiative transfer equation in the separable approximation,” J. Quant.Spectrosc. Radiat. Transf. 94, 291–309 (2005).

42. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance andtransmittance for the noninvasive measurement of tissue optical properties,”Appl. Opt. 28, 2331–2336 (1989).

43. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determi-nation of optical pathlengths in tissue: temporal and frequency analysis,” Phys.Med. Biol. 37, 1531–1560 (1992).

52 Chapter 3

44. S. R. Arridge, “Photon measurements density functions. Part I: Analyticalforms,” Appl. Opt. 34, 7395–7409 (1995).

45. M. R. Ostermeyer and S. L. Jacques, “Perturbation theory for diffuse lighttransport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255–261 (1997).


47. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, Addison-Wesley,Reading (1966).

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49. M. Bassani, F. Martelli, D. Contini, and G. Zaccanti, “Independence of thediffusion coefficient on absorption: experimental and numerical evidence,” Opt.Lett. 22, 853–855 (1997).

50. R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbingmedium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).

51. R. Graaff and J. J. Ten Bosch, “Diffusion coefficient in photon diffusion theory,”Opt. Lett. 25, 43–45 (2000).

52. T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, “Expression of theoptical diffusion coefficient in high-absorption turbid media,” Phys. Med. Biol.42, 2541–2549 (1997).

53. M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cam-bridge (1980).


55. M. Keijzer, W. M. Star, and P. R. M. Storchi, “Optical diffusion in layeredmedia,” Appl. Opt. 27, 1820–1824 (1988).

56. I. Freund, “Surface reflections and boundary conditions for diffusive photontransport,” Phys. Rev. A 45, 8854–8858 (1992).

57. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, andB. J. Tromberg, “Boundary conditions for the diffusion equation in radiativetransfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).

58. A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influenceof boundary conditions on the accuracy of diffusion theory in time-resolvedreflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975(1995).


59. A. Kienle and M. S. Patterson, “Improved solution of the steady-state and thetime-resolved diffusion equations for reflectance from a semi-infinite turbidmedium,” J. Opt. Soc. Am. A 14, 246–524 (1997).

60. D. Contini, F. Martelli, and G. Zaccanti, “Photon migration through a turbidslab described by a model based on diffusion approximation. I. Theory,” Appl.Opt. 36, 4587–4599 (1997).

61. D. Daners, “Robin boundary value problems on arbitrary domain,” Trans. Amer.Math. Soc. 352, 4207–4236 (2000).

62. J. Bouza-Domínguez, “Light propagation at interfaces of biological media:Boundary conditions,” Phys. Rev. E 78, 031926 (2008).

63. G. W. Faris, “Diffusion equation boundary conditions for the interface betweenturbid media: A comment,” J. Opt. Soc. Am. A 19, 519–520 (2002).

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Chapter 4

Solutions of the DiffusionEquation for HomogeneousMediaIn this chapter, the Green’s functions of the diffusion equation for homogeneousgeometries like the infinite medium, the infinite slab, the parallelepiped, the infi-nite cylinder, and the sphere are presented. Green’s functions of the DE for otherhomogeneous geometries like the finite cylinder are available.1, 2 The solutions ofthe DE are approximate solutions of the RTE. Information on the accuracy of thesolutions can be obtained from comparisons with the results of MC simulations thatwill be presented in Chapter 10.

In this chapter, figures with examples of solutions for the infinite medium and theslab will be also presented. All the figures can be reproduced using the software pro-vided with the enclosed CD-ROM. In particular, the results for fluence, reflectance,and transmittance can be easily obtained using the Diffusion&Perturbation code.Those for flux, radiance, total reflectance, and total transmittance can be obtainedusing the enclosed subroutines.

4.1 Solution of the Diffusion Equation for an Infinite Medium

The infinite medium is a relevant geometry for photon migration studies because itis also a reference for photon migration in other geometries. In an infinite medium,photon migration occurs in the most general conditions without restrictions dueto boundary effects. For this reason, when approximate theories like the diffusionequation are used in this geometry, the effects of the approximations show theirintrinsic consequences and can be studied in all generality. Moreover, several solu-tions in more complex geometries use the solution for the infinite geometry.2–5 Thus,the infinite medium represents a fundamental test necessary for the comprehensionof photon transport in any type of turbid medium.

Let us consider an infinite homogeneous medium characterized by absorptioncoefficient µa, reduced scattering coefficient µ

′s, and diffusion coefficient D =

1/(3µ′s). Given a spatial and temporal isotropic Dirac delta source of unitary strength

57

58 Chapter 4

in the originq0(~r, t) = δ3(~r)δ(t), (4.1)

the diffusion equation for the fluence rate can be written as(∂

v∂t−D∇2 + µa

)Φ(~r, t) = q0(~r, t). (4.2)

Following the procedure described in Appendix E, the time-dependent Green’sfunction for t > 0 is

Φ(~r, t) =v

(4πDvt)3/2exp

(− r2

4Dvt− µavt

), (4.3)

with r = |~r| as the distance from the source.It is important to stress that Eq. (4.3) also predicts energy for t < t0 = r/v,

which is the ballistic time necessary to propagate from the source to the receiver,following a straight path. Therefore, the simplifying assumptions necessary to obtainthe DE imply an infinite speed for the propagation of energy so that a local variationin photon density instantaneously spreads over the medium. This aspect of the DEis physically incorrect since it violates the causality principle. The violation of thecausality principle also originates from the requirement that the two assumptions ofthe DE are instantaneously valid after the light source is injected inside the medium,whilst they actually become valid only after a detected photon has undergoneseveral scattering events. In fact, the diffusion approximation correctly describesthe asymptotical behavior of light propagation, i.e., when the elapsed time is muchlonger than the ballistic time.

The Green’s function of Eq. (4.3) can be calculated with the following subrou-tines provided on the enclosed CD-ROM:

• DE_Fluence_Infinite_TR

• DE_Fluence_Perturbation_Infinite_TR

Through the use of Fick’s law [Eq. (3.31)], the time-dependent Green’s functionfor the flux for any t > 0 results in

~J(~r, t) =r

16 (πDv)3/2 t5/2exp

(− r2

4Dvt− µavt

)r, (4.4)

and the Green’s functions for the radiance, using Eqs. (3.25), (4.3) and (4.4), can bewritten as

I(~r, s, t) =1

(4π)5/2 (Dvt)3/2

[v +

3r2t

(r · s)]

exp(− r2

4Dvt− µavt

). (4.5)

The Green’s function for the flux can be calculated with these subroutines:

Solutions of the Diffusion Equation for Homogeneous Media 59

• Homogeneous_Flux_Infinite

• Homogeneous_Flux_Perturbation_Infinite

The Green’s function for the radiance can be calculated with the following subrou-tines:

• Homogeneous_Radiance_Infinite

• Homogeneous_Radiance_Perturbation_Infinite

In Fig. 4.1, the TR expressions for the fluence, flux, and radiance of Eqs. (4.3),(4.4), and (4.5) are plotted versus time for an infinite medium with µa = 0, µ

′s = 1

mm−1, and refractive index n = 1. The results for the fluence and flux are shownfor r = 1, 5, 10 and 20 mm. The results for the radiance are shown for r = 1 and10 mm and for different values of the angle α between the directions s and r. Thefigure shows that the fluence, and thus the energy density, decreases as r increases.A different behavior can be noticed at early and late times for the flux: At earlytimes, the flux decreases as r increases and vice versa at late times. This means thatat late times the energy gradient is larger for the larger distances. We point out thatat r = 1 mm the radiance takes negative values for large values of α and early times.

Integrating Eqs. (4.3), (4.4), and (4.5) over the whole time range (see Appen-dices E and F), the following expressions for the Green’s functions for steady-statesources Φ(~r), ~J(~r), and I(~r, s) are obtained:

Φ(~r) =1

4πrDexp (−µeffr) , (4.6)

~J(~r) =1

4πr

(1r

+ µeff

)exp (−µeffr) r, (4.7)

I(~r, s) =1

42π2rD

[1 + 3

(D

r+ µeffD

)(r · s)

]exp (−µeffr) , (4.8)

whereµeff =

√µa/D =

√3µaµ

′s (4.9)

is the effective attenuation coefficient.It is important to stress that the analytical expression for the CW flux [Eq. (4.7)]

is the exact solution of the RTE for the case of a non-absorbing medium (seeAppendix E).

The Green’s functions for CW fluence, flux, and radiance can be calculated withthese subroutines:

• DE_Fluence_Perturbation_Infinite_CW


60 Chapter 4


Alternatively, the CW Green’s function can be calculated integrating the TR Green’sfunction using the Trap subroutine provided with the enclosed CD-ROM.

0 500 1000 1500 2000 250010-8

10-7

10-6

10-5

10-4

10-3

10-2

Flue

nce

(mm

-2ps

-1)

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

0 500 1000 1500 2000 250010-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Flux

(mm

-2ps

-1)

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

0 20 40 60 80 10010-5

10-4

10-3

10-2

r = 1 mm

Rad

ianc

e (m

m-2

ps-1

sr-1

)

Time (ps)

0 degrees 45 degrees 90 degrees 135 degrees 180 degrees

0 500 1000 150010-7

10-6

10-5

r = 10 mm

Rad

ianc

e (m

m-2

ps-1

sr-1

)

Time (ps)

0 degrees 45 degrees 90 degrees 135 degrees 180 degrees

Figure 4.1 Time-dependent fluence, flux, and radiance for an infinite medium with µa = 0,µ

′s = 1 mm−1, and n = 1. The results for the fluence and the flux are shown for several

distances r. The results for the radiance are shown for several values of the angle betweenr and s for r = 1 and 10 mm.

In Fig. 4.2, the CW fluence, flux, and radiance of Eqs. (4.6), (4.7), and (4.8) areplotted for a medium with µ

′s = 1 mm−1. Fluence and flux are plotted versus r for

µa = 0, 0.005, 0.01, 0.05, and 0.1 mm−1. The radiance is plotted versus α for µa =0 and r = 0.5, 1, 2, and 5 mm. The figure shows how the CW fluence and the CWflux decrease when the absorption coefficient or the distance increase. For the CWradiance, the figure shows an almost isotropic behavior, as expected in the diffusiveregime, for large values of r (r ≥ 5 mm).

Finally, the expression for the mean pathlength traveled by photons detected at a


distance r from the source 〈l(r)〉 is:

〈l(r)〉 = − ∂

∂µaln Φ(~r) =

r

21√µaD

=r

2

√3µ′sµa

. (4.10)

For a non-absorbing medium, the expression of the mean pathlength diverges.

0 5 10 15 2010-5

10-4

10-3

10-2

10-1

100

101

Flue

nce

(mm

-2)

r (mm)

a = 0 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

0 5 10 15 2010-5

10-4

10-3

10-2

10-1

100

101

Flux

(mm

-2)

r (mm)

a = 0 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

0 30 60 90 120 150 180-0.06

-0.03

0.00

0.03

0.06

0.09

0.12

Rad

ianc

e (m

m-2

sr-1

)

Angle (degrees)

r = 0.5 mm r = 1 mm

0 30 60 90 120 150 1800.000

0.005

0.010

0.015

Rad

ianc

e (m

m-2

sr-1

)

Angle (degrees)

r = 2 mm r = 5 mm

Figure 4.2 Fluence, flux, and radiance for steady-state sources in an infinite homogeneousmedium with µ

′s = 1 mm−1. The fluence and flux are plotted versus r for several values of

µa. The radiance is plotted versus the angle between s and r for some values of r for µa =0.

Physically, this means that a photon inside an infinite non-absorbing medium travelsfor an infinite time. Similarly, the second-order moment 〈l2(r)〉 can be calculated as

〈l2(r)〉 =1

Φ(~r)∂2

∂2µaΦ(~r) =

r

41√µaD

1 + µeffr

µa= 〈l(r)〉

1 + µeffr

2µa. (4.11)

Averaging 〈l(r)〉 [given by Eq. (4.10)], multiplied by the weighting factor Φ(~r)over the whole medium, the following expression for the total mean pathlength 〈l〉

62 Chapter 4

is obtained:

〈l〉 =

+∞∫0

〈l(r)〉Φ(~r)r2dr

+∞∫0

Φ(~r)r2dr

=1µa. (4.12)

The total mean pathlength traveled by photons in a scattering and absorbing infinitemedium before being absorbed is therefore equal to the mean free path followed inan infinite non-scattering medium with the same absorption [Eq. (2.19)]. However,while trajectories in the non-scattering medium are straight lines, in the scatteringmedium they are chaotic broken lines in accordance with the scattering properties.It is interesting to know the mean number of scattering events 〈k〉 undergone byphotons before being absorbed in an infinite medium. Since the mean free pathfollowed between two subsequent scattering events is given by `s = 1/µs, it resultsin

〈k〉 =〈l〉`s

= µs〈l〉 =µsµa. (4.13)

The information on 〈k〉 can be useful in evaluating the average square distance fromthe source at which photons migrating through the medium are absorbed. UsingEq. (2.23) for the average distance at which photons are located after k scatteringevents, and using Eq. (4.13), we obtain

〈d2〈k〉〉 ∼=

2µ′2s〈k〉(1− g) =

6µ2eff

. (4.14)

The same result can be also obtained averaging the square distance r2 over thewhole medium with the probability density function µaΦ(r) [µaΦ(r)4πr2dr isthe probability to have absorption of photons within the spherical shell of radiusr, r + dr].6

4.2 Solution of the Diffusion Equation for the Slab Geometry

The problem of light propagation through random media bounded by parallel planeshas been a subject of interest for decades because many physical systems are likelyto be represented in this way.1, 7–11 The slab is used for applications of the radiativetransfer in the atmosphere7, 9, 12–15 and in biological tissues.2–4, 10, 11, 16–23 For thisreason, the solutions for the slab are widely used in photon migration studies.

To obtain the solutions reported in this section and in Sec. 4.3, we follow themethod of images (or mirror images)24 that is commonly used for the slab.2–5, 17 Themethod makes use of boundary conditions (ZBC2, 3 or EBC4, 5, 17, 25) that assume thefluence equal to 0 on the physical boundary of the diffusive medium (ZBC) or onthe extrapolated surface at distance 2AD [Eq. (3.53)] from the physical boundary(EBC). For regular, bounded geometries like the slab and the parallelepiped, themethod of images allows us to reconstruct the solution for the fluence inside the


medium as a superposition of solutions for the infinite medium. The flux exiting thediffusive medium (i.e., in the slab geometry the reflectance and the transmittance) isobtained by applying Fick’s law at the boundary of the medium [see Eqs. (4.23) and(4.24)]. The method of images leads relatively simple explicit analytical solutionsexpressed as summations of infinite terms. A very few of these terms is sufficientfor the convergence of the solution in almost all applications of practical interest.

Solutions of the DE have been also obtained following other approaches. Inparticular, solutions for the semi-infinite medium and slab have been obtainedusing the PCBC expressed as the Robin boundary condition,1, 8, 25–30 a boundarycondition less approximated with respect to the ZBC or the EBC. However, thesesolutions have been implemented for the time domain in integral form using theLaplace transformation method, and their use is significantly more complicated.For the CW plane wave incident upon a slab, a simple closed-form solution basedon the Robin boundary condition is also available.8, 27–30 Other hybrid solutions,especially devoted to obtaining improved solutions for the outgoing flux, have beenalso proposed. These solutions will be presented and discussed in Sec. 4.4.

The geometry together with a description of notations used is shown in Fig. 4.3.The solutions reported in this section were obtained for an isotropic point sourceof unitary strength, i.e., a spatial and temporal Dirac delta source q(~r, t) = δ3(~r −→rs )δ(t) at→rs = (0, 0, zs) with 0 ≤ zs ≤ s and s geometrical thickness of the slab.According to the EBC (Sec. 3.9.1), the fluence is assumed equal to zero at two

z

xy

z = 0

z = s

z = - z e

z = s + z e E x t r a p o l a t e db o u n d a r y

E x t r a p o l a t e db o u n d a r y

r e a l s o u r c ez 0 + = z s

P o s i t i v e s o u r c e sN e g a t i v e s o u r c e s

q

q

z 1 - = 2 s + 2 z e - z s

s

z 1 + = 2 s + 4 z e + z s

z 0 - = - 2 z e - z s

Figure 4.3 Model of the slab geometry and image sources.

64 Chapter 4

extrapolated flat surfaces outside the turbid medium at the extrapolated distanceze = 2AD from the physical boundaries of the slab. By the method of images,24

the correct boundary conditions are obtained by using, in addition to the real sourcein →rs , an infinite number of pairs of positive and negative sources in an infinitediffusive medium having the same optical properties of the slab. The locations

−→r+m

of the positive sources and−→r−m of the negative sources are shown in Fig. 4.3. The

real source is that at z+0 ; all the other sources are image sources. In this way, the

fluence inside the slab is composed by terms of fluence from an infinite medium:The positive terms from the sources at

−→r+m and the negative ones from the sources at

−→r−m. The sources are placed along the z-axis at

z+m = 2m(s+ 2ze) + zsz−m = 2m(s+ 2ze)− 2ze − zsm = 0,±1,±2, .........±∞ .

(4.15)

Adding the contributions of all of the source pairs, the Green’s function for the TRfluence rate at ~r = (x, y, z) results in

Φ(~r, t) = v

(4πDvt)3/2exp

(− ρ2

4Dvt − µavt)

×m=+∞∑m=−∞

exp

[− (z−z+m)2

4Dvt

]− exp

[− (z−z−m)2

4Dvt

],

(4.16)

with ρ =√x2 + y2 and 0 ≤ z ≤ s.

The flux and the radiance, making use of Fick’s law [Eq. (3.31)] and of the firstassumption of the diffusion approximation [Eq. (3.25)], can be calculated as

~J(~r, t) = −D∇Φ(~r, t) (4.17)

and

I(~r, s, t) =1

4πΦ(~r, t)− 3

4πD∇Φ(~r, t) · s. (4.18)

The expressions of Eqs. (4.16)–(4.18) can be calculated with the followingsubroutines provided with the enclosed CD-ROM:

• DE_Fluence_Perturbation_Slab_TR (for the fluence)

• Homogeneous_Fluence_Slab (for the fluence)

• Homogeneous_Flux_Slab (for the flux)

• Homogeneous_Radiance_Slab (for the radiance)


The Green’s function for the TR reflectance R(ρ, t), i.e., the power crossing thesurface at z = 0, per unit area, at distance ρ from the z-axis with any exit angle, canbe evaluated as

R(ρ, t) =∫

s·q>0

[1−RF (s · q)] I(ρ, z = 0, s, t)(s · q)dΩ

= ~J(ρ, z = 0, t) · q ,(4.19)

where q is the unit vector normal to the boundary surface outwardly directed, andRF is the Fresnel reflection coefficient for unpolarized light. Making use of Fick’slaw results in

R(ρ, t) = D∂

∂zΦ(ρ, z = 0, t). (4.20)

With a similar procedure, it is possible to evaluate the TR transmittance T (ρ, t):

T (ρ, t) = −D ∂

∂zΦ(ρ, z = s, t). (4.21)

The functions R(ρ, t) and T (ρ, t) of Eqs. (4.20) and (4.21), since they corre-spond to a unitary energy point-like source, also represent the probability per unittime and unit area that a photon, emitted at →rs and t = 0, exits at time t and atdistance ρ from the z-axis.

The reflectance and the transmittance can be obtained by performing the deriva-tive of Eqs. (4.20) and (4.21) with a numerical evaluation. This can be done usingthe subroutine

• Homogeneous_Flux_Slab

provided with the enclosed CD-ROM. Alternatively, explicit analytical expressionsfor R(ρ, t) and T (ρ, t) can be obtained from the same equation, as will be shown inthe next section.

The Green’s function for CW sources, i.e., the solution for an isotropic pointsource of unitary strength [q(~r) = δ3(~r −→rs )] is

Φ(~r) = 14πD

m=+∞∑m=−∞

exp[−µeff

√ρ2+(z−z+m)2

]√ρ2+(z−z+m)2

−exp[−µeff

√ρ2+(z−z−m)2

]√ρ2+(z−z−m)2

.

(4.22)

Using Fick’s law, the Green’s function for the CW reflectance R(ρ) for steady-state sources results in

R(ρ) = D∂

∂zΦ(ρ, z = 0), (4.23)

and for the CW transmittance

T (ρ) = −D ∂

∂zΦ(ρ, z = s). (4.24)

66 Chapter 4

The CW Green’s functions for the flux, radiance, reflectance, and transmittancecan be calculated integrating the corresponding TR Green’s functions using, forinstance, the Trap subroutine provided with the enclosed CD-ROM. Alternatively,the fluence can be calculated with the subroutine

• DE_Fluence_Perturbation_Slab_CW

provided with the enclosed CD-ROM and the reflectance and the transmittance withthe software that implements the analytical expressions reported in Sec. 4.3.

The whole procedure described here provides the Green’s function for a slabilluminated by an isotropic light source placed in zs. When a pencil light beamis normally impinging the medium along the z-axis, the solutions can be derivedwith the superposition integral [Eq. (3.8)] of the above Green’s functions with thereal source. The pencil beam inside the medium can be represented as a line ofpoint-like isotropic sources with decreasing intensity, according to the Lambert-Beerlaw. Alternatively, the real source term can be substituted by a single isotropic pointsource located at ~r = (0, 0, zs). The coordinate zs is usually obtained by imposingthat the line of isotropic point sources and the single point source have the same firstmoment.20, 31, 32 This assumption results in

zs = 1/(µa + µ′s). (4.25)

In this book, we argue that the physical meaning of the absorption coefficientin diffusion and transport theory should be equivalent [Eqs. (3.14) and (3.20)];therefore, the coordinate zs is assumed as

zs = 1/µ′s. (4.26)

With this assumption, the DE solutions maintain the same dependence on theabsorption coefficient of the RTE solutions. From a practical point of view, since theDE is applicable with good accuracy only when µa µ

′s, the solutions of the DE

expressed with zs = 1/(µa + µ′s) or with zs = 1/µ

′s are almost indistinguishable.

We stress that all the figures of this chapter for the slab pertain to to an isotropicsource at zs = 1/µ

′s.

The approximation of a semi-infinite medium has often been used to describeproblems that involve highly diffusive media. The approximation is valid for slabssufficiently thick so that the probability that the radiation will escape from thesurface opposite the source is negligible. In this case, the boundary conditions arefulfilled, retaining only the first of the dipole sources (that with m = 0) necessaryfor the slab.

4.3 Analytical Green’s Functions for Transmittance andReflectance

In this section, the analytical expressions for the reflectance and the transmittanceof an infinite slab obtained with Fick’s law are reported both for time-dependentsources and for steady-state sources.


The analytical expressions for the TR reflectance and transmittance, making useof Eqs. (4.16), (4.20), and (4.21), respectively, result in

R(ρ, t) = −exp

(− ρ2

4Dvt−µavt

)2(4πDv)3/2t5/2

×m=+∞∑m=−∞

[z3m exp

(− z23m

4Dvt

)− z4m exp

(− z24m

4Dvt

)],

(4.27)

T (ρ, t) =exp

(− ρ2

4Dvt−µavt

)2(4πDv)3/2t5/2

×m=+∞∑m=−∞

[z1m exp

(− z21m

4Dvt

)− z2m exp

(− z22m

4Dvt

)],

(4.28)

with z1m = (1− 2m)s− 4mze − zsz2m = (1− 2m)s− (4m− 2)ze + zsz3m = −2ms− 4mze − zsz4m = −2ms− (4m− 2)ze + zs ,

(4.29)

and ρ =√x2 + y2. Equations (4.27) and (4.28) are infinite series and should be

truncated for practical applications. Since the distance from the boundaries [onwhich R(ρ, t) and T (ρ, t) are evaluated] of the sources that correspond to the indexm increases when m increases, the contribution of sources corresponding to highvalues of m is expected to be significant only for large values of ρ and/or t.

The expressions of Eqs. (4.27) and (4.28) can be calculated with the followingsubroutines provided on the enclosed CD-ROM:

• DE_Reflectance_Perturbation_Slab_TR

• DE_Transmittance_Perturbation_Slab_TR

In Fig. 4.4, the TR reflectance and the TR transmittance from Eqs. (4.27) and(4.28) are plotted versus time for a slab with s = 40 mm, µa = 0, µ

′s = 1 mm−1,

zs = 1/µ′s = 1 mm, and refractive index ni = 1.4. The reflectance is shown for

ρ = 5 and 20 mm and the transmittance for ρ = 0 and 40 mm. Both are plottedfor the relative refractive index nr equal to 1 and 1.4. nr is the ratio between therefractive indexes inside (ni) and outside (ne) the diffusive medium, respectively.The differences between the curves of Fig. 4.4 corresponding to different values ofnr can be explained by the redistribution, due to reflection, of photons inside theslab. Reflection removes a consistent fraction of photons that, in the case of nr = 1,would exit near the source at short times toward longer distances. We note that withnr = 1.4, more than 50% of photons are reflected if an isotropic distribution for theradiance is assumed. These photons continue to migrate through the slab, and thusthe probability of exiting at longer distances and longer times increases as is shownin Fig. 4.4.

68 Chapter 4

0 1000 2000 300010-10

10-9

10-8

10-7

10-6

10-5

10-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 5 mm; nr = 1 = 20 mm; nr = 1 = 5 mm; nr = 1.4 = 20 mm; nr = 1.4

0 1000 2000 3000 4000 500010-12

10-11

10-10

10-9

10-8

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)

= 0 mm; nr = 1 = 40 mm; nr = 1 = 0 mm; nr = 1.4 = 40 mm; nr = 1.4

Figure 4.4 Time-dependent reflectance and transmittance from a homogeneous slab withs = 40 mm , µa = 0 , µ

′s = 1 mm−1, zs = 1 mm, and ni = 1.4. The reflectance is shown for

ρ = 5 and 20 mm and the transmittance for ρ = 0 and 40 mm. Both are plotted for nr = 1and 1.4.

By integrating Eqs. (4.27) and (4.28) over the entire exit surface, the total TRreflectance R(t) and the total TR transmittance T (t) are obtained:

R(t) =+∞∫0

R(ρ, t)2πρdρ = − exp(−µavt

)2(4πDv)1/2t3/2

×m=+∞∑m=−∞

[z3m exp

(− z23m

4Dvt

)− z4m exp

(− z24m

4Dvt

)] (4.30)

T (t) =+∞∫0

T (ρ, t)2πρdρ =exp(−µavt

)2(4πDv)1/2t3/2

×m=+∞∑m=−∞

[z1m exp

(− z21m

4Dvt

)− z2m exp

(− z22m

4Dvt

)].

(4.31)

The functions R(t) and T (t) represent the Green’s functions for an infinitely ex-tended receiver. For the reciprocity principle7, 13 the functions R(t) and T (t) can beused to describe the time-resolved reflectance and transmittance when an infinitelywide beam, having constant radiance, impinges perpendicularly on the surface ofthe slab.

The Green’s functions R(t) and T (t) can be calculated making use of thefollowing subroutines provided on the enclosed CD-ROM:

• DE_Total_Reflectance_Slab_TR

• DE_Total_Transmittance_Slab_TR

In Fig. 4.5, the expressions forR(t) and T (t) of Eqs. (4.30) and (4.31) are plottedversus time for a slab with s = 40 mm, µa = 0 , µ

′s = 1 mm−1, zs = 1/µ

′s = 1


mm, ni = 1.4, and nr = 1.4. For late times, the two curves tend to the same valueand thus photons have the same probability to escape from the two sides of the slab.This is because at late times the photon density inside the slab tends to have a spatialdistribution almost symmetric with respect to the middle of the slab. In contrast, atearly times R(t) and T (t) are significantly different because of the large amount ofphotons contributing to R(t) that escape from the slab at z = 0, after few scatteringevents close to the source.

0 2500 5000 7500 1000010-7

10-6

10-5

10-4

10-3

10-2

R, T

(ps-1

)

Time (ps)

Total Reflectance Total Transmittance

Figure 4.5 Time-dependent total reflectance and total transmittance from a slab with s =

40 mm, µa = 0, with µ′s = 1 mm−1, zs = 1 mm, ni = 1.4, and nr = 1.4.

With a similar integration, the expressions for the average TR reflectance andtransmittance received by a ring delimited by ρ1 and ρ2 can be obtained:

〈R(ρ1, ρ2, t)〉 =

ρ2∫ρ1

R(ρ,t)2πρdρ

π(ρ22−ρ21)= − 1

π3/24(ρ22−ρ21)(Dv)1/2t3/2

×[

exp(− ρ21

4Dvt

)− exp

(− ρ22

4Dvt

)]exp(−µavt)

×m=+∞∑m=−∞

[z3m exp

(− z23m

4Dvt

)− z4m exp

(− z24m

4Dvt

)] (4.32)

〈T (ρ1, ρ2, t)〉 =

ρ2∫ρ1

T (ρ,t)2πρdρ

π(ρ22−ρ21)= 1

π3/24(ρ22−ρ21)(Dv)1/2t3/2

×[

exp(− ρ21

4Dvt

)− exp

(− ρ22

4Dvt

)]exp(−µavt)

×m=+∞∑m=−∞

[z1m exp

(− z21m

4Dvt

)− z2m exp

(− z22m

4Dvt

)] (4.33)

These expressions are important for analyzing experimental data since a real receiveris characterized by a finite area and an integral over this area is required for a correctevaluation of the detected light.

70 Chapter 4

The functions 〈R(ρ1, ρ2, t)〉 and 〈T (ρ1, ρ2, t)〉 can be calculated making use ofthe following subroutines, respectively, provided with the enclosed CD-ROM:

• DE_Reflectance_Slab_Receiver_TR

• DE_Transmittance_Slab_Receiver_TR

The effect of the finite area of the receiver is shown in Fig. 4.6. The expressionof Eq. (4.32) for the TR reflectance is plotted versus time for a ring delimited byρ1 = 4 mm and ρ2 = 6 mm and for a ring delimited by ρ1 = 19 mm and ρ2 = 21mm and compared with the reflectance obtained for ρ = 4, 5 and 6 mm and for ρ =19, 20, and 21 mm. A slab with s = 40 mm, µa = 0, µ

′s = 1 mm−1, zs = 1/µ

′s =

1 mm, ni = 1.4, and nr = 1.4 has been considered. The comparison shows thatthe response for the receiver of finite area is almost identical to the response for areceiver element at a distance ρ = (ρ1 + ρ2)/2. This result indicates that for manypractical applications, even for a receiver with a large area, the response can beaccurately evaluated using Eqs. (4.27) and (4.28) referring to elementary receiverelements.

0 250 500 750 100010-7

10-6

10-5

10-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

4 mm < < 6 mm = 4 mm = 5 mm = 6 mm

0 250 500 750 100010-10

10-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

19 mm < < 21 mm = 19 mm = 20 mm = 21 mm

Figure 4.6 Effect of the finite area of the receiver. The average TR reflectance through aring with ρ1 = 4 mm and ρ2 = 6 mm and through a ring with ρ1 = 19 mm and ρ2 = 21 mmare compared with the reflectance obtained for ρ = 4, 5, and 6 mm and for ρ = 19, 20, and21 mm. A slab with s = 40 mm, µa = 0, µ

′s = 1 mm−1, zs = 1 mm, ni = 1.4, and nr = 1.4

has been considered.

The Green’s functions for the steady-state source can be obtained by integratingthe TR Green’s functions previously reported. Details on the integrals used canbe found in Appendix F. By integrating Eqs. (4.27) and (4.28) over the whole time


range, the following expressions are obtained:

R(ρ) =+∞∫0

R(ρ, t)dt

= − 14π

m=+∞∑m=−∞

(z3m

(ρ2 + z2

3m

)−3/2 ×

1 +

[µa(ρ2+z23m)

D

]1/2

× exp

−[µa(ρ2+z23m)

D

]1/2− z4m

(ρ2 + z2

4m

)−3/2

×

1 +

[µa(ρ2+z24m)

D

]1/2× exp

−[µa(ρ2+z24m)

D

]1/2)

(4.34)

T (ρ) =+∞∫0

T (ρ, t)dt

= 14π

m=+∞∑m=−∞

(z1m

(ρ2 + z2

1m

)−3/2 ×

1 +

[µa(ρ2+z21m)

D

]1/2

× exp

−[µa(ρ2+z21m)

D

]1/2− z2m

(ρ2 + z2

2m

)−3/2

×

1 +

[µa(ρ2+z22m)

D

]1/2× exp

−[µa(ρ2+z22m)

D

]1/2)

(4.35)

The quantity R(ρ)2πρdρ (T (ρ)2πρdρ) represents the probability that a photonemitted by the source exits from the surface element 2πρdρ at z = 0 (z = s) and atdistance ρ, regardless of the length of its trajectory.

The expression for the CW reflectance and transmittance can be calculated withthe following subroutines provided with the enclosed CD-ROM:

• DE_Reflectance_Perturbation_Slab_CW

• DE_Transmittance_Perturbation_Slab_CW

In Fig. 4.7, R(ρ) and T (ρ) are plotted versus the distance ρ for a slab withs = 40 mm, µ

′s = 1 mm−1, and zs = 1/µ

′s = 1 mm. The results are shown

for µa = 0 and 0.02 mm−1 and for nr = 1 and 1.4. The differences among thecurves of Fig. 4.7 corresponding to different values of nr can be explained by theredistribution, due to reflection, of photons inside the slab. Inside the medium,Fresnel reflections trap a consistent fraction of photons that, in the case of nr = 1,would exit near the source. Reflected photons continue to migrate through the slab,and thus the probability of exiting at longer distances from the source increases.However, absorption contrasts the effect of boundary reflections since it attenuatesreflected light that is characterized by a longer pathlength.

By integrating Eqs. (4.30) and (4.31) over the whole time range, the followingexpressions for the total CW reflectance R and the total CW transmittance T , i.e.,

72 Chapter 4

0 10 20 30 4010-6

10-5

10-4

10-3

10-2

10-1

R, T

(mm

-2)

(mm)

Reflectance, nr = 1 Reflectance, nr = 1.4 Transmittance, nr = 1 Transmittance, nr = 1.4

a = 0 mm-1

0 10 20 30 4010-1010-910-810-710-610-510-410-310-210-1

R, T

(mm

-2)

(mm)

Reflectance, nr = 1 Reflectance, nr = 1.4 Transmittance, nr = 1 Transmittance, nr = 1.4

a = 0.02 mm-1

Figure 4.7 Reflectance and transmittance for CW sources from a slab with s = 40 mm, µ′s =

1 mm−1, and zs = 1 mm. The results are shown for two values of the absorption coefficient(µa = 0 and 0.02 mm−1) and for two values of nr = 1 and 1.4.

the fraction of the incident energy reflected and transmitted, can be obtained:

R =+∞∫0

R(t)dt = −12

m=+∞∑m=−∞

[sgn(z3m) exp (−µeff |z3m|)

−sgn(z4m) exp (−µeff |z4m|)],

(4.36)

and

T =+∞∫0

T (t)dt = 12

m=+∞∑m=−∞

[sgn(z1m) exp (−µeff |z1m|)

−sgn(z2m) exp (−µeff |z2m|)],

(4.37)

where sgn(x) = − 1 if x < 0 and sgn(x) = 1 if x > 0. The quantity 1− (R+ T )is the fraction of energy absorbed by the medium. For a non-absorbing medium, wehaveR+T = 1. Therefore, the solutions of the DE presented in this section preservethe law of conservation of energy inside the medium; i.e., all the energy propagatingthough the medium with µa = 0 is going to be re-emitted. For a non-absorbingsemi-infinite medium, we have R = 1.

Equations (4.36) and (4.37) can be calculated with the following subroutinesprovided with the enclosed CD-ROM:

• DE_Total_Reflectance_Slab_CW

• DE_Total_Transmittance_Slab_CW

In Fig. 4.8, the CW total reflectance and transmittance of a slab with s = 40 mm,µ′s = 1 mm−1, zs = 1/µ

′s = 1 mm, and nr = 1.4 are plotted versus the absorption

coefficient of the medium. The figure also reports the mean pathlengths 〈lR〉 and


〈lT 〉 for photons contributing to the total reflectance and transmittance, respectively.〈lR〉 and 〈lT 〉 have been evaluated numerically applying Eq. (3.18) to Eqs. (4.36)and (4.37).

10-4 10-3 10-2 10-1 10010-5

10-4

10-3

10-2

10-1

100

R, T

a (mm-1)

Total Reflectance Total Transmittance

10-4 10-3 10-2 10-1 100100

101

102

103

<lR

>, <

l T> (m

m)

a (mm-1)

<lR> <lT>

Figure 4.8 CW total reflectance, total transmittance, and mean pathlength for the total re-ceived photons for a slab with s = 40 mm, µ

′s = 1 mm−1, zs = 1 mm, and nr = 1.4. The

data are plotted versus the absorption coefficient of the medium.

Integrating Eqs. (4.32) and (4.33) over time, we obtain the average CW re-flectance and transmittance received by a ring delimited by ρ1 and ρ2:

〈R(ρ1, ρ2)〉 =+∞∫0

〈R(ρ1, ρ2, t)〉 dt = − 1π3/24(ρ22−ρ21)D1/2

×m=+∞∑m=−∞

z3m

√4πD

ρ21+z23mexp

[− 2√

µa(ρ21+z23m)4D

]− z4m

√4πD

ρ21+z24m

× exp[− 2√

µa(ρ22+z23m)4D

]− z3m

√4πD

ρ22+z23mexp

[− 2√

µa(ρ21+z24m)4D

]+ z4m

√4πD

ρ22+z24mexp

[− 2√

µa(ρ22+z24m)4D

] (4.38)

〈T (ρ1, ρ2)〉 =+∞∫0

〈T (ρ1, ρ2, t)〉 dt = 1π3/24(ρ22−ρ21)D1/2

×m=+∞∑m=−∞

z1m

√4πD

ρ21+z21mexp

[− 2√

µa(ρ21+z21m)4D

]− z2m

√4πD

ρ21+z22m

× exp[− 2√

µa(ρ21+z22m)4D

]− z1m

√4πD

ρ22+z21mexp

[− 2√

µa(ρ22+z21m)4D

]+z2m

√4πD

ρ22+z22mexp

[− 2√

µa(ρ22+z22m)4D

] (4.39)

These expressions for 〈R(ρ1, ρ2)〉 and 〈T (ρ1, ρ2)〉 can be calculated makinguse of the following subroutines provided with the enclosed CD-ROM:

74 Chapter 4

• DE_Reflectance_Slab_Receiver_CW

• DE_Transmittance_Slab_Receiver_CW

Finally, the solutions are reported for the CW mean pathlengths 〈l(ρ)〉R and〈l(ρ)〉T , followed by photons before exiting from the slab at distance ρ from thez-axis, at z = 0 and z = s, respectively:

〈l(ρ)〉R = − ∂∂µa

lnR(ρ) = − 18πDR(ρ)

×m=+∞∑m=−∞

(z3m

(ρ2 + z2

3m

)−1/2 exp−[µa(ρ2+z23m)

D

]1/2−z4m

(ρ2 + z2

4m

)−1/2 exp−[µa(ρ2+z24m)

D

]1/2 ) (4.40)

〈l(ρ)〉T = − ∂∂µa

lnT (ρ) = 18πDT (ρ)

×m=+∞∑m=−∞

(z1m

(ρ2 + z2

1m

)−1/2 exp−[µa(ρ2+z21m)

D

]1/2−z2m

(ρ2 + z2

2m

)−1/2 exp−[µa(ρ2+z22m)

D

]1/2 ) (4.41)

4.4 Other Solutions for the Outgoing Flux

In Secs. 4.2 and 4.3, expressions for the flux outgoing from the slab (reflectanceand transmittance) have been obtained from the solution for the fluence based onthe EBC making use of Fick’s law. As mentioned in Sec. 4.2, other approacheshave been proposed that basically assume the outgoing flux to be a combination offluence and flux17, 25, 33 or simply proportional to the fluence on the boundary.33, 34

The first approach calculates the outgoing flux as the integral of the radiancetransmitted at the interface. As an example, the TR reflectance from the slab can bewritten as

R(ρ, t) =∫2π

[1

4πΦ(ρ, z = 0, t) + 3D4π

∂Φ(ρ,z=0,t)∂z cos θ

]×[1−RF (cos θ)] cos θdΩ,

(4.42)

where the radiance has been expressed using Eq. (3.25) and Fick’s law [Eq. (3.31)],and θ is the angle relative to the normal to the boundary. Integrating Eq. (4.42), anexpression significantly different with respect to the solution obtained using onlyFick’s law is obtained:17

R(ρ, t) = aΦ(ρ, z = 0, t) + bD∂Φ(ρ, z = 0, t)

∂z, (4.43)

where a and b are numerical coefficients depending on the refractive index mismatch.As an example, for ni/ne = 1.4, a = 0.118 and b = 0.306.17 The reflectance is


therefore expressed as a combination of the fluence and of the reflectance obtained inSec. 4.3 [Eq. (4.27)]. Equation (4.43) is more complicated with respect to Eq. (4.27),since both the fluence and the flux are involved, and the calculation is therefore moretime consuming. Also this approach, similarly to the one used to obtain Eq. (4.27),is based on the EBC and Fick’s law. Discrepancies between the two solutions comefrom the fact that both Eq. (3.25) and Fick’s law are approximated, and it is difficultto predict which is more accurate. Comparisons with MC results reported in Ref. 17have shown that Eq. (4.43) gives a more accurate solution.

The second approach is an hybrid approach, being based both on the EBC and onthe PCBC [it has been also named EBPC33 (extrapolated boundary partial current)].To obtain the outgoing flux Jout(r, t), the PCBC, the simple expression of Eq. (3.51),is applied to the solution for the fluence obtained using the EBC. The resultingexpression is

Jout(r, t) =Φ(r, t)

2A, (4.44)

which can be applied to any point of the boundary. Since the PCBC is expected to beless approximated than Fick’s law, we expect that Eq. (4.44) provides more accurateresults than expressions from Sec. 4.3 and Eq. (4.43). In fact, the discontinuity in theoptical properties may determine strong variations of the flux near the boundary. Asa consequence, the first [Eq. (3.25)] and second assumption [Eq. (3.26)] necessaryto obtain the DE from the RTE can be compromised, and therefore Fick’s law maybe more approximated at the boundary.

We have therefore three different expressions for the outgoing flux: The firstassumes Jout(r, t) equal to the flux obtained applying Fick’s law at the fluence on theboundary; and for the slab leads to the solutions reported in Sec. 4.3 for reflectanceand transmittance. The second expression calculates Jout(r, t) as a combination offluence and flux (evaluated with Fick’s law). Finally, the third assumes Jout(r, t)proportional to the fluence. Undoubtedly, the third solution is the simplest to use,since the same expression is applicable to any point of the boundary. As an example,for the slab, the same expression [Eq. (4.44)] can be used to evaluate both the TRreflectance and the TR transmittance. Therefore, it may be useful to have explicitexpressions for the slab based on Eq. (4.44). The Green’s functions for reflectanceand transmittance result as

R(ρ, t) =Φ(ρ, z = 0, t)

2A(4.45)

and

T (ρ, t) =Φ(ρ, z = s, t)

2A(4.46)

in the time domain, and

R(ρ) =Φ(ρ, z = 0)

2A(4.47)

76 Chapter 4

and

T (ρ) =Φ(ρ, z = s)

2A(4.48)

in the CW domain, with Φ(ρ, z, t) and Φ(ρ, z) given by Eqs. (4.16) and (4.22),respectively.

The expressions for the average reflectance and transmittance result in

〈R(ρ1, ρ2, t)〉 =〈Φ(ρ1, ρ2, z = 0, t)〉

2A(4.49)

and

〈T (ρ1, ρ2, t)〉 =〈Φ(ρ1, ρ2, z = s, t)〉

2A(4.50)


〈R(ρ1, ρ2)〉 =〈Φ(ρ1, ρ2, z = 0)〉

2A(4.51)

and

〈T (ρ1, ρ2)〉 =〈Φ(ρ1, ρ2, z = s)〉

2A(4.52)

in the CW domain, with

〈Φ(ρ1, ρ2, z, t)〉 =

ρ2∫ρ1

Φ(ρ,z,t)2πρdρ

π(ρ22−ρ21)= v

(4πDvt)1/21

π(ρ22−ρ21)

× exp(− µavt

)[exp

(− ρ21

4Dvt

)− exp

(− ρ22

4Dvt

)]×m=+∞∑m=−∞

exp

[− (z−z+m)2

4Dvt

]− exp

[− (z−z−m)2

4Dvt

] (4.53)

and

〈Φ(ρ1, ρ2, z)〉 =

ρ2∫ρ1

Φ(ρ,z)2πρdρ

π(ρ22−ρ21)

= 12π√µaD(ρ22−ρ21)

m=+∞∑m=−∞

exp

[−µeff

√ρ2

1 + (z − z+m)2

]− exp

[−µeff

√ρ2

2 + (z − z+m)2

]+ exp

[−µeff

√ρ2

2 + (z − z−m)2

]− exp

[−µeff

√ρ2

1 + (z − z−m)2

].

(4.54)

The expressions for the total reflectance and transmittance are

R(t) =Φ(z = 0, t)

2A(4.55)


and

T (t) =Φ(z = s, t)

2A(4.56)


R =Φ(z = 0)

2A(4.57)

and

T =Φ(z = s)

2A(4.58)

in the CW domain, with

Φ(z, t) =∞∫0

Φ(ρ, z, t)2πρdρ = v

(4πDvt)1/2exp

(− µavt

)×m=+∞∑m=−∞

exp

[− (z−z+m)2

4Dvt

]− exp

[− (z−z−m)2

4Dvt

] (4.59)

and

Φ(z) =∞∫0

Φ(z, t)dt = 12√µaD

×m=+∞∑m=−∞

[exp (−µeff |z − z+

m|)− exp (−µeff |z − z−m|)].

(4.60)

We stress that, as for the solutions obtained using Fick’s law, for a non-absorbingmedium the above expressions verify the condition R+ T = 1.

Equations (4.45)–(4.60) can be calculated using the following subroutines pro-vided with the enclosed CD-ROM:

• DE_Fluence_Perturbation_Slab_TR


which evaluate the TR and the CW fluence inside the slab and

• DE_Fluence_Slab_Receiver_TR

• DE_Fluence_Slab_Receiver_CW

• DE_Total_Fluence_Slab_TR

• DE_Total_Fluence_Slab_CW

which evaluate respectively Eqs. (4.53), (4.54), (4.59), and (4.60).To conclude, it should be pointed out that in spite of significant differences

among the expressions obtained following the three different approaches to evaluatethe outgoing flux (compare, for instance, Eqs. (4.27), (4.43), and (4.45) that repre-sent the three solutions for the TR reflectance), they produce results that in many

78 Chapter 4

0 50 100 1501E-6

1E-5

1E-4

1E-3

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DEFick DEEBPC

nr = 1 = 2.5 mm

0 50 100 1501E-6

1E-5

1E-4

1E-3

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DEFick DEEBPC

nr = 1.4 = 2.5 mm

0 100 200 3001E-7

1E-6

1E-5

1E-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DEFick DEEBPC

nr = 1 = 5 mm

0 100 200 3001E-7

1E-6

1E-5

1E-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DEFick DEEBPC

nr = 1.4 = 5 mm

0 200 400 6001E-8

1E-7

1E-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DEFick DEEBPC

nr = 1 = 10 mm

0 200 400 6001E-8

1E-7

1E-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DEFick DEEBPC

nr = 1.4 = 10 mm

Figure 4.9 TR reflectance from a slab with s = 40 mm, µ′s = 1 mm−1, µa = 0, zs = 1 mm,

and ni = 1.4: Comparison between results obtained using Fick and EBPC approaches bothfor the case of matched (nr = 1) and mismatched (nr = 1.4) refractive indexes.


situations are almost indistinguishable. Examples of comparisons are reported inFigs. 4.9 and 4.10. Results are shown only for the Fick and the EBPC approaches.Since the solution obtained with the other method is a linear combination of fluxand fluence, the corresponding results are in between those obtained with Fick andEBPC approaches. Figure 4.9 shows the comparison for the TR reflectance at ρ =2.5, 5, and 10 mm for a slab with s = 40 mm, µ

′s = 1 mm−1, µa = 0, zs = 1 mm,

and ni = 1.4. Results are shown both for matched (nr = 1) and mismatched (nr =1.4) refractive indexes. Figure 4.10 shows the CW reflectance of the same slab ofFig. 4.9 but with µa = 0.01 mm−1. Differences are appreciable at early times andat short source-receiver distances. Significantly larger differences are observed inpresence of refractive index mismatch. Comparisons with MC results (exampleswill be shown in Sec. 10.6) have shown that the EBPC solution provides a betterdescription of the TR outgoing flux, especially at short distances from the source.In particular, the shape of the temporal response is better described. Therefore,the solution is preferable for inversion procedures devoted to retrieving the opticalproperties of the medium from TR measurements, especially when measurementsare carried out at short distances or on small volumes of diffusive media.34 As tothe CW reflectance and transmittance, it is difficult to draw a general conclusion,while for the total reflectance (which is strongly influenced by photons exiting verynear the source) and for the total transmittance, solutions of Sec. 4.3 are in betteragreement with MC results.

0 5 10 15 201E-5

1E-4

1E-3

0.01

0.1

Ref

lect

ance

(mm

-2)

(mm)

DEFick DEEBPC

nr = 1

0 5 10 15 201E-5

1E-4

1E-3

0.01

0.1

Ref

lect

ance

(mm

-2)

(mm)

DEFick DEEBPC

nr = 1.4

Figure 4.10 CW reflectance from a slab with s = 40 mm, µ′s = 1 mm−1, µa = 0.01 mm−1,

zs = 1 mm, and ni = 1.4: Comparison between results obtained using Fick and EBPCapproaches both for the case of matched (nr = 1) and mismatched (nr = 1.4) refractiveindexes.

Finally, we point out that from Eqs. (4.27) and (4.45) the ratio r1 = RDEFick/RDEEBPC is independent of the source-receiver distance ρ and limt→+∞ r1 =1.The same property holds for r2 = TDEFick/TDEEBPC . A better comprehensionof these properties can be achieved looking at the relation existing between the

80 Chapter 4

solution for the fluence obtained with the EBC and the solution for the fluenceobtained with the PCBC. In fact, for the time domain, it is possible to evaluatethe ratio r3 = (ΦPCBC/ΦEBC)|external boundary on the external boundary betweenthe expression of the fluence obtained with the PCBC expressed as the Robinboundary condition (see Ref. 25, 26) and the expression of the fluence obtainedwith the EBC [Eq. (4.16)]. r3 is independent of the source-receiver distance andlimt→+∞ r3 =1.26, 34 Moreover, r3 for typical values of the optical properties ofbiological tissue differs from 1 by a negligible amount if the early times are excluded.Therefore, the differences between PCBC and EBC are confined to the initial part ofthe temporal range. In Refs. 26 and 34 the ratio r4 = RPCBC/RDEFick was alsostudied showing that limt→+∞ r4 =1, while for early times, when the time variationof the photon flux is very strong, r4 mostly differs from 1 significantly more than r3.Thus, for long times all the different expressions used to describe the outgoing fluxtend to give the same description against time, while for early times the use of Fick’slaw with the EBC shows a different behavior with respect to the other expressionsobtained with the PCBC and the EBPC.

4.5 Analytical Green’s Function for the Parallelepiped

We consider a diffusive parallelepiped of dimensions Lx, Ly, and Lz with a spatialand temporal isotropic Dirac delta source with a unitary strength at→rs = (xs, ys, zs),as shown in Fig. 4.11. The procedure used in Sec. 4.2 for the slab, based on themethod of images and on the extrapolated boundary condition, is extended to threedimensions to obtain the Green’s function for the parallelepiped.5 Thus, insteadof an infinite line of positive and negative point sources, a lattice of positive andnegative point sources is used, which satisfies the EBC at the six extrapolated planes

),,( ssss zyxr

yLxL

zL

),,( zyxr

xy

z ),,( ssss zyxr

yLxL

zL

),,( zyxr

xy

z

Figure 4.11 Graphic of the diffusive parallelepiped.


(x = −ze, y = −ze, z = −ze, x = Lx + ze, y = Ly + ze, and z = Lz + ze). Thetime-dependent Green’s function can thus be expressed as5

Φ(~r, t) = v

(4πDvt)3/2exp

(− µavt

) l=+∞∑l=−∞

m=+∞∑m=−∞

n=+∞∑n=−∞

×

exp[− (x−x1l)

2+(y−y1m)2+(z−z1n)2

4Dvt

]− exp

[− (x−x1l)

2+(y−y1m)2+(z−z2n)2

4Dvt

]− exp

[− (x−x1l)

2+(y−y2m)2+(z−z1n)2

4Dvt

]+ exp

[− (x−x1l)

2+(y−y2m)2+(z−z2n)2

4Dvt

]− exp

[− (x−x2l)

2+(y−y1m)2+(z−z1n)2

4Dvt

]+ exp

[− (x−x2l)

2+(y−y1m)2+(z−z2n)2

4Dvt

]+ exp

[− (x−x2l)

2+(y−y2m)2+(z−z1n)2

4Dvt

]− exp

[− (x−x2l)

2+(y−y2m)2+(z−z2n)2

4Dvt

],

(4.61)

where

x1l = 2lLx + 4lze + xsx2l = 2lLx + (4l − 2)ze − xsy1m = 2mLy + 4mze + ysy2m = 2mLy + (4m− 2)ze − ysz1n = 2nLz + 4nze + zsz2n = 2nLz + (4n− 2)ze − zs .

(4.62)

Equation (4.61) can be calculated using the following subroutine provided withthe enclosed CD-ROM:

• Homogeneous_Fluence_Parallelepiped

An analogous expression can be obtained for steady-state sources integratingEq. (4.61) or using the above procedure with steady-state isotropic Dirac deltasources. The outgoing flux, as discussed in Sec. 4.4, can be obtained applying Fick’slaw on the physical boundary of the medium or, more conveniently, using the PCBC[Eq. (3.51)].

4.6 Analytical Green’s Function for the Infinite Cylinder

Making use of the EBC, and following the procedure described in Refs. 1, 2, 35, and36, the expression of the Green’s function for an infinite cylinder is given by thefollowing equation:

Φ(~r,→rs , t) = vexp[− (z−zs)2

4Dvt−µavt

]2πL

′2√πDvt

m=+∞∑m=−∞

cos(mα)∑βm

exp(−β2mDvt)

×Jm(βmr)Jm(βmrs)

[J′m(βmL

′)]2

.

(4.63)

82 Chapter 4

For an explanation of the symbols, we refer to Fig. 4.12. →rs = (rs, 0, zs) is the posi-tion of the source using the cylindrical coordinates, ~r = (r, α, z) is the observationpoint, α is the angular separation of source and detector (i.e., the angle between→rs and ~r projected on a plane at z = constant), L is the radius of the cylinder, andL′

= L + 2AD is the radius of the extrapolated boundary. The real numbers βmare positive roots of the equation Jm(βmL

′) = 0, where Jm is the Bessel function

of integer order m, and J′m is its derivative with respect to the radius.

'L

Lsr

r

'L

Lsr

r

Figure 4.12 Projection on a plane with a constant value of z of the diffusive cylinder (greyregion) together with its extrapolated boundary (broken circle).


• DE_Fluence_Cylinder_TR

This subroutine was obtained from a former routine36 developed by AngeloSassaroli of Tufts University. Also for the cylinder, the outgoing flux, as discussedin Sec. 4.4, can be obtained applying Fick’s law on the physical boundary of themedium or, more conveniently, using the PCBC [Eq. (3.51)].

Equation (4.63), being obtained with the eigenfunction method (see Sec. 6.2),has a fast convergence at late times but a slow convergence at early times. For thisreason, due to convergence problems related to the number and to the accuracyof the calculated eigenvalues, it is also possible to have negative values for thetemporal response at early times. With 100 Bessel functions Jm and 100 zerosβm, we obtained good convergence of the temporal profiles at photon flight timeslonger than ballistic for a cylinder with radius L = 50 mm, µa = 0, and µ

′s = 1

mm−1, source at rs = L − 1/µ′s, receiver at r = L, and α = 25 deg. Only at a


close source-receiver distance of 5 or 10 mm or for a larger radius of the cylinderis a larger number of Bessel functions and zeros required for good convergenceat early times. Roughly, it is possible to state that the number of Bessel functionsrequired for the convergence of the solution at early times is proportional to theproduct µ

′sL. It is important to stress that the number of Bessel functions required

for the convergence of the solution quickly decreases at late times.Finally, we stress that using the solution for the infinite cylinder and the method

of images it is possible to obtain the solution of the DE for the finite cylinder. Theprocedure, similarly to that used for the slab in Sec. 4.2, consists of reconstructing thefluence as a superposition of solutions for the infinite cylinder. The correct boundaryconditions are obtained by using, in addition to the real source, an infinite numberof positive and negative sources in an infinite cylinder having the same opticalproperties of the finite cylinder. The locations

−→r+m of the positive sources and

−→r−m of

the negative sources are the same shown for the slab [Fig. 4.3 and Eq. (4.15)]. Thesubroutine to evaluate the TR fluence for the finite cylinder can be easily obtainedby changing a few lines in the subroutine DE_Fluence_Perturbation_Slab_TR.

4.7 Analytical Green’s Function for the Sphere

Making use of the EBC, and following the procedure described in Refs. 1, 2, and36, the expression of the Green’s function for the sphere is given by the followingequation:

Φ(~r,→rs , t) = v exp(−µavt)2πa′2

√rrs

m=+∞∑m=0

(2m+ 1)Pm(cosα)∑

βm+1/2

exp(−β2m+1/2Dvt)

×Jm+1/2(βm+1/2r)Jm+1/2(βm+1/2rs)

[J′m+1/2

(βm+1/2a′ )]2

,

(4.64)where a is the radius of the sphere,→rs = (rs, φs, ψs) is the position of the source(spherical coordinates), →r = (r, φ, ψ) is the observation point, α is the sourcedetector angle, and a

′= a + 2AD is the extrapolated radius, which depends on

the refractive index mismatch between the medium and the surroundings. βm+1/2

are the positive roots of the equation Jm+1/2(βm+1/2a′) = 0, where Jm+1/2 is the

Bessel function of half-integer order m+ 1/2, and J′

m+1/2 is its derivative. Pm isthe Legendre polynomial of order m.


• DE_Fluence_Sphere_TR

This subroutine was obtained from a former routine36 developed by Angelo Sassaroliof Tufts University. With 100 Bessel functions Jm and 100 zeros βm+1/2, weobtained good convergence of the temporal profiles at photon flight times longerthan ballistic for a sphere with radius a = 50 mm, µa = 0, and µ

′s = 1 mm−1;

84 Chapter 4

source at rs = a − 1/µ′s; receiver at r = a; and α = 25 deg. Only at a close

source-receiver distance of 5 or 10 mm or for a larger radius of the sphere is alarger number of Bessel functions and zeros required for good convergence at earlytimes. Comments on the convergence of the solution for the cylinder reported in theprevious section are basically applicable also to the sphere: a faster convergence isobtained for smaller values of µ

′sa, and the number of Bessel functions required for

the convergence of the solution quickly decreases at late times.

4.8 Angular Dependence of Radiance Outgoing from aDiffusive Medium

Measurements of diffused light devoted to characterize diffusive media are com-monly based on measurements of diffused light emerging from the medium. Thediffused light is commonly collected using optical fibers or directly with detectorshaving a limited numerical aperture. Therefore, the quantity actually measured perunit area is the outgoing radiance accepted by the detection system, which can bewritten as

P (t) =∫Ωd

dΩIe(~r, se, t)se · q , (4.65)

where Ωd is the acceptance solid angle of the detection system, Ie(~r, se, t) is theradiance on the external boundary of the medium, and q is the outwardly directednormal. Therefore, knowing the angular distribution of the outgoing radiance is keyto understanding if the solutions reported in previous sections for the total outgoingflux are suitable to describe the actually measured quantity.


isq

es

u),,( tsrI ii

r

ni ne

e

),,( tsrI eeDiffusive medium

External medium

isq

es

u),,( tsrI ii

r

ni ne

e

),,( tsrI ee

Figure 4.13 Schematic of light emerging from a diffusive non-scattering interface.

In this section, an analytical expression based on the diffusion approximationand on the PCBC is obtained for the angular dependence of the radiance outgoing


from a diffusive medium bounded by a non-scattering region. According to thediffusion approximation, the radiance is assumed to be almost isotropic, and it isapproximated by the first two terms of a series expansion in spherical harmonics[Eq. (3.25)].

With reference to Fig. 4.13, we denote with the subscripts i and e the quantities,respectively, inside and outside the diffusive medium. Following the procedure ofRef. 34, the radiance on the external surface, expressed as the fraction of the internalradiance transmitted in the external medium, can be written as8

Ie(~r, se, t) =(neni

)2 [1−RF (θi)

]Ii(~r, si, t) , (4.66)

where RF (θi) is the Fresnel reflection coefficient for unpolarized light [Eq. (3.47)],se is related to si by Snell’s law, and the term (neni )

2 accounts for the refraction ofthe solid angle. Making use of Eq. (3.25), Ii(~r, si, t) can be written as

Ii(~r, si, t) =1

4πΦi(~r, t) +

34π

(Jiq cos θi + Jiu cosϕi sin θi) , (4.67)

where Jiq and Jiu are the normal and the tangential component of the flux, and θiand ϕi are the polar and azimuthal angle, respectively (u · si = cosϕi sin θi, andq · si = cos θi). Substituting this expression into Eq. (4.66) and averaging over theazimuthal angle ϕe (ϕe = ϕi), we obtain

Ie(~r, θe, t) =(neni

)2 [1−RF (θi)

][ 14π

Φi(~r, t) +3

4πJiq cos θi

], (4.68)

where Ie(~r, θe, t) is the radiance outgoing along se averaged over ϕe. Making useof the PCBC at the diffuse non-scattering interface [Eq. (3.51)], the expression forIe(~r, θe, t) can be simplified as

Ie(~r, θe, t) =(neni

)2 Φi(~r, t)4π

[1−RF (θi)](

1 +3

2Acos θi

). (4.69)

Substituting this expression into Eq. (4.65), we obtain for the measurable quantity

P (t) =∫

Ωd

dΩ(neni

)2Φi(~r,t)

4π

1−RF

[arcsin(neni sin θe)

]×

1 + 32A cos


]se · q .

(4.70)

Since integration over the solid semi-angle 2π leads to the total outgoing flux[Eq. (4.44)], i.e., Φ(r,t)

2A , Eq. (4.69) can be rewritten as

Ie(~r, θe, t) =Φi(~r, t)

2AF (θe) (4.71)

86 Chapter 4

withF (θe) = 1

4π

(neni

)2 1−RF


]×

2A+ 3 cos[arcsin(neni sin θe)

] (4.72)

and ∫2π

dΩF (θe)se · q = 1 . (4.73)

Therefore, the angular dependence of the outgoing radiance (averaged over theazimuthal angle) is completely separated from the spatial and temporal dependenceand is fully described by F (θe). This function is independent of the geometryconsidered for the boundary and represents the distribution function for the directionof the outgoing radiation. F (θe) only depends on the relative refractive indexnr = ni/ne. Therefore, the measurable quantity can be written as

P (t) =Φi(~r, t)

2A

∫Ωd

dΩF (θe)se · q . (4.74)

To show the dependence of F (θe) on the relative refractive index in Fig. 4.14,the angular distribution is plotted for several values of nr varying in the range 0.5-2.The figure shows that F (θe) strongly depends on nr. Furthermore, the angulardistribution is quite different from that for a Lambertian surface8, 37 (consideringLambert’s cosine law, i.e., a surface emitting a radiance independent of the direction,for which F (θe) = 1/π), especially for nr < 1.

0 15 30 45 60 75 900.0

0.3

0.6

0.9

1.2

1.5

1.8

F(e)

e (degrees)

ni/ne = 0.5 ni/ne = 0.6 ni/ne = 0.7 ni/ne = 0.8 ni/ne = 0.9 ni/ne = 1.0 Lambertian

0 15 30 45 60 75 900.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

F(e)

e (degrees)

ni/ne = 2.0 ni/ne = 1.6 ni/ne = 1.4 ni/ne = 1.2 ni/ne = 1.1 ni/ne = 1.0 Lambertian

Figure 4.14 Angular distribution of the outgoing radiance: Dependence on the relative re-fractive index.

It could be expected that the angular distribution of light predicted by Eq. (3.25)is a rough approximation of the actual distribution, since near the boundary, due tothe discontinuous variation of the optical properties, strong variations of the flux


are likely, and thus the simplifying assumptions of the diffusion approximation[Eqs. (3.25) and (3.26)] may fail. However, comparisons with the results of exper-iments and MC simulations34, 38 have shown that the angular distribution for theradiance predicted by the diffusion approximation has a surprisingly large range ofvalidity.

Equations (4.71) and (4.74) are relevant for practical applications, since theystate that both TR and CW-multidistance measurements of outgoing radiation areaffected by the acceptance angle of the detection system by a factor independent bothon time and on source-receiver distance. This factor can be calculated integratingF (θe) on the numerical aperture of the detection system. However, for manyapplications the knowledge of this factor is not necessary, since measurements areavailable only in relative units. As long as the angular dependence of the outgoingradiance can be represented by Eq. (4.71), the solutions for the total outgoing flux(like the ones reported in Secs. 4.2, 4.3, and 4.4 for the slab, multiplied by a scalefactor) can be used to analyze TR or CW-multidistance measurements withoutadding further approximations. Therefore, it is important to know the accuracy ofEq. (4.71). As mentioned before, it has been shown both with MC simulations34

and with experiments38 that the angular distribution for the radiance foreseen by thediffusion approximation has a surprisingly large range of validity. An example ofMC results is shown in Fig. 4.15 for the TR reflectance from a semi-infinite mediumplotted for three different angular ranges α of the field of view of the receiver: α1 =0 deg − 30 deg, α2 = 30 deg − 60 deg, and α3 = 60 deg − 90 deg. Each curve hasbeen normalized to the whole area subtended from the same curve and the resultscorrespond to µ

′s = 1 mm−1, µa = 0.01 mm−1, g = 0.9, ni = 1.4, nr = 1.4, and

ρ = 2.5 and 20 mm. The three curves are almost indistinguishable in the logarithmicscale chosen in the figure. The figure also shows the ratios of the normalized TRreflectance Rα1/Rα2 and Rα1/Rα3 at ρ = 2.5 and 20 mm. The figure confirms theresults expected from the DE, i.e., Eqs. (4.70) and (4.71), from which the shape oftemporal distribution of exiting photons is expected to be independent of the angularreceiving range of the receiver. Even at ρ = 2.5 mm, with the exception of earlyphotons, the temporal profile of the exiting photons normalized to the whole areaonly weakly depends on α.

An expression similar to Eq. (4.68), but valid only for nr = 1, has been obtainedby Firbank et al.39 by using the hybrid radiosity-diffusion theory. Vera and Durian38

also presented a similar expression for the total CW transmittance.It should be pointed out that Eq. (4.72) has been obtained by using Snell’s law.

Therefore, it is applicable only if the external surface of the diffusive medium issmooth. In the case of a rough surface, the scattering on the surface can significantlyaffect the angular distribution.

88 Chapter 4

0 200 400 60010-5

10-4

10-3

10-2

10-1

Ref

lect

ance

(Arb

itrar

y U

nits

)

Time (ps)

1 = 0 deg - 30 deg 2 = 30 deg - 60 deg 3 = 60 deg - 90 deg

= 2.5 mm

0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 01 0 - 7

1 0 - 6

1 0 - 5

1 0 - 4

1 0 - 3

Refle

ctanc

e(Arb

itrary

Units)

T i m e ( p s )

α1 = 0 d e g - 3 0 d e gα2 = 3 0 d e g - 6 0 d e gα3 = 6 0 d e g - 9 0 d e g

ρ = 2 0 m m

0 200 400 6000.5

1.0

1.5

1 = 0 deg - 30 deg

2 = 30 deg - 60 deg

3 = 60 deg - 90 deg

RR

RR

Time (ps)

R1/ R

2

R1/ R

3

= 2.5 mm

0 500 1000 1500 2000 25000.5

1.0

1.5

1 = 0 deg - 30 deg

2 = 30 deg - 60 deg

3 = 60 deg - 90 deg

RR

RR

Time (ps)

R1/ R

2

R1/ R

3

= 20 mm

Figure 4.15 TR reflectance (MC results) from a semi-infinite medium reported for threeangular ranges α of the field of view of the receiver: α1 = 0 deg − 30 deg, α2 = 30 deg −60 deg, and α3 = 60 deg− 90 deg. The curves are normalized to the whole area subtendedand correspond to µ

′s = 1 mm−1, µa = 0.01 mm−1, g = 0.9, ni = 1.4, nr = 1.4, and ρ =

2.5 and 20 mm. In the figure, the ratios of the TR reflectance Rα1/Rα2 and Rα1/Rα3 arealso shown.


References





5. A. Kienle, “Light diffusion through a turbid parallelepiped,” J. Opt. Soc. Am. A22, 1883–1888 (2005).

6. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, Addison-Wesley,Reading (1966).



9. Y. S. Chou, “Approximate method for radiative transfer in scattering absorbingplane-parallel media,” Appl. Opt. 17, 364–373 (1978).

10. V. Tuchin, Tissue Optics, SPIE Press, Bellingham, WA (2000).


12. B. H. Armstrong and R. W. Nicholls, Emission, Absorption and Transfer ofRadiation in Heated Atmospheres, Pergamon Press, New York (1972).


14. A. I. Lyapustin, “Radiative transfer code SHARM for atmospheric and terres-trial applications,” Appl. Opt. 36, 7764–7772 (2005).

15. V. Nardino, F. Martelli, P. Bruscaglioni, G. Zaccanti, S. Del Bianco, D. Guzzi,P. Marcoionni, and I. Pippi, “McCART: Monte Carlo code for atmosphericradiative transfer,” IEEE Trans. Geosci. Remote Sens. 46, 1740–1752 (2008).

16. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering andabsorption of turbid materials determined from reflection measurements. 1.Theory,” Appl. Opt. 22, 2456–2462 (1983).

90 Chapter 4


18. J. A. Wahr, K. K. Tremper, S. Samra, and D. T. Delpy, “Near-infrared spec-troscopy - theory and applications,” J. Cardiothor. Vasc. An. 10, 406–418(1996).

19. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, and B. C. Wilson,“Spatially-resolved absolute diffuse reflectance measurements for non-invasivedetermination of the optical scattering and absorption coefficients of biologicaltissue,” Appl. Opt. 35, 2304–2314 (1996).

20. T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model forspatially resolved, steady-state diffuse reflectance for the noninvasive determi-nation of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).

21. S. T. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, andY. Hefetz, “Experimental tests of a simple diffusion model for the estimationof scattering and absorption coefficients of turbid media from time-resolveddiffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).

22. A. Torricelli, A. Pifferi, P. Taroni, E. Giambattistelli, and R. Cubeddu, “In vivooptical characterization of human tissues from 610 to 1010 nm by time-resolvedreflectance spectroscopy,” Phys. Med. Biol. 46, 2227–2237 (2001).

23. F. Martelli, S. Del Bianco, and G. Zaccanti, “Effect of the refractive indexmismatch on light propagation through diffusive layered media,” Phys. Rev. E.70, 011907 (2004).

24. E. Zauderer, Partial Differential Equations of Applied Mathematics, JohnWiley & Sons, New York (1989).

25. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, andB. J. Tromberg, “Boundary conditions for the diffusion equation in radiativetransfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).

26. A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, “The influenceof boundary conditions on the accuracy of diffusion theory in time-resolvedreflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975(1995).

27. N. Garcia, A. Z. Genack, and A. A. Lisyansky, “Measurement of the transportmean free path of diffusing photons,” Phys. Rev. B 46, 475–479 (1992).

28. D. J. Durian, “Influence of boundary reflection and refraction on diffusivephoton transport,” Phys. Rev. E 50, 857–866 (1994).


29. J. G. Rivas, R. Sprik, C. M. Soukoulis, K. Busch, and A. Lagendijk, “Op-tical transmission through strong scattering and highly polydisperse media,”Europhys. Lett. 48, 22–28 (1999).

30. J. G. Rivas, R. Sprik, and A. Lagendijk, “Static and dynamic transport of lightclose to the anderson localization transition,” Phys. Rev. E 63, 046613 (2001).

31. D. R. Wyman, M. S. Patterson, and B. C. Wilson, “Similarity relations foranisotropic scattering in Monte Carlo simulations of deeply penetrating neutralparticles,” J. Comput. Phys. 81, 137–150 (1989).

32. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3approximation,” J. Opt. Soc. Am. A 18, 584–599 (2001).

33. D. C. Comsa, T. J. Farrel, and M. S. Patterson, “Quantification of biolumines-cence images of point source objects using diffusion theory models,” Phys.Med. Biol. 51, 3733–3746 (2006).

34. F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the lightemerging from a diffusive medium: angular dependence and flux at the externalboundary,” Phys. Med. Biol. 44, 1257–1275 (1999).

35. A. Sassaroli, F. Martelli, and Y. Yamada, “Study on the propagation of ultra-short pulse light in cylindrical optical phantoms,” Phys. Med. Biol. 44, 2747–2763 (1999).

36. A. Sassaroli, F. Martelli, G. Zaccanti, and Y. Yamada, “Performance of fittingprocedures in curved geometry for retrieval of the optical properties of tissuefrom time-resolved measurements,” Appl. Opt. 40, 185–197 (2001).


38. M. U. Vera and D. J. Durian, “Angular distribution of diffusely transmittedlight,” Phys. Rev. E 53, 3215–3224 (1996).

39. M. Firbank, S. R. Arridge, M. Schweiger, and D. T. Delpy, “An investigationof light transport through scattering bodies with non-scattering regions,” Phys.Med. Biol. 41, 767–783 (1996).

Chapter 5

Hybrid Solutions of theRadiative Transfer EquationIn order to solve the RTE without simplifying assumptions, we have to resortto numerical methods. These methods convert the integral form of the transportequation into a system of algebraic equations that can be treated numerically byusing a computer. The finite element method, the discrete ordinates method, or thespherical harmonics method are examples of numerical procedures for reconstructingsolutions of the RTE.1–7 Among the numerical procedures, there are also statisticalmethods like the MC method,1, 8 which offers a way of simulating the physicalpropagation of photons in scattering media. With the MC method, the physicalquantities that describe photon migration, such as the flux and the fluence rate,are calculated after running a large number of photon trajectories. The MC can beconsidered a numerical method to solve the RTE even though the equation itself is notimplemented in the code as it is usually done in other numerical methods. Besidesthe approximate solutions obtained making use of the DE, very few analyticalsolutions are available for studying light propagation through highly scatteringmedia.

Approximate analytical solutions subjected to the same limitations of those ofthe DE are also obtained with a random walk method.9–12

Solutions based on the DE are affected by the intrinsic approximations ofthis theory that show different consequences for steady-state and time-dependentsources. For steady-state sources, solutions based on the DE cannot describe photonsdetected at short distances.8, 13 For time-dependent sources, DE solutions provide apoor description of early received photons since the DE is well established only forphotons undergoing many scattering events.14

The limitations of the DE can present a serious problem in modeling pho-ton migration, especially when small volumes of diffusive media are considered.To overcome these limitations, several improved solutions of the DE have beenproposed.13, 15–26 For this reason, the availability of analytical solutions of the RTEnot subjected to the intrinsic limitations of the DE is desirable.

In this chapter, starting from analytical solutions of the RTE for the infinitemedium more accurate with respect to the DE, a hybrid heuristic approach is

93

94 Chapter 5

implemented to derive solutions for the slab geometry. The proposed model makesuse of:

• The method of images together with the EBC in order to obtain the Green’sfunction for the fluence inside the slab as superposition of the Green’s func-tions for a series of positive and negative point-like isotropic sources in ahomogeneous infinite medium;

• The PCBC or Fick’s law to obtain the diffuse reflectance and transmittance.

For the Green’s function of the infinite medium, the idea is to use some RTEsolutions that are more accurate than the DE solution, being confident that a betterdescription of light propagation at small distances and short times could be obtained.The solutions of the RTE for the infinite medium that we report for the hybrid modelare:

• The solution of the time-dependent RTE obtained by Paasschens;27

• The solution of the telegrapher equation (TE) (an equation less approximatethan the DE) obtained by Durian and Rudnick.28

Examples of comparisons reported in Ref. 29 and in Chapter 10 show thatsignificantly more accurate results with respect to the DE can be obtained using thehybrid model. Since the hybrid solutions are in any case obtained making use of ap-proximations (EBC, PCBC, and Fick’s law arise from the diffusion approximation)their accuracy should be tested before using them in specific applications instead ofthe well-tested solutions of the DE.

5.1 General Hybrid Approach to the Solutions for the SlabGeometry

In this section, a hybrid approach is introduced to obtain solutions for the slabgeometry. When solutions of the RTE are searched for geometries like the slab,the main problem to solve for obtaining the Green’s function is the modeling ofthe boundary condition. In order to overcome this problem, the same boundarycondition used for the DE (i.e., the EBC) is here heuristically employed for anykind of model derived from the RTE. Remember that the EBC assumes the fluenceequal to zero on an extrapolated boundary at distance ze = 2AD [Eq. (3.53)] fromthe geometrical boundary. The Green’s function for the slab geometry Φslab(~r, t)is obtained by using the method of images. For a spatial and temporal Dirac deltasource q(~r, t) = δ3(~r−→rs )δ(t) with→rs = (0, 0, zs) [see Fig. 4.3], the locations

−→r+m

and−→r−m of the positive and negative sources are the same used for the solution of

the DE in Sec. 4.2. Thus, following a procedure similar to the one used for the DE,

Hybrid Solutions of the Radiative Transfer Equation 95

the solution for a generic model ΦMODEL_slab(~r, t) for an infinite slab is obtainedadding the solutions for pairs of sources placed in an infinite medium (see Fig. 4.3):

ΦMODEL_slab(~r, t) =m=+∞∑m=−∞

[ΦMODEL_i

( ∣∣∣~r −−→r+m

∣∣∣ , t)

−ΦMODEL_i

( ∣∣∣~r −−→r−m∣∣∣ , t)] ,(5.1)

where [see Eq. (4.15)]−→r+m = [0, 0, 2m(s+ 2ze) + zs]

−→r−m = [0, 0, 2m(s+ 2ze)− 2ze − zs] ,

(5.2)

and ΦMODEL_i is any exact or approximate solution of the RTE for the infinitemedium. The solutions for ΦMODEL_i that will be used in this book are Eqs. (4.3),(5.9), and (5.16) that are respectively the solutions of the DE, of the RTE, and of theTE.

In the enclosed CD-ROM, the subroutine

• Homogeneous_Fluence_Slab

is provided to calculate the fluence of Eq. (5.1). For ΦMODEL_i, the solutions ofRTE, TE, and DE can be used.

By also extending the use of the boundary conditions obtained from the DEto the calculation of the flux and radiance [Eqs. (3.31) and (3.25)], expressions for~JMODEL_slab(~r, t) and IMODEL_slab(~r, s, t) can be obtained for a generic modelby using Fick’s law:

~JMODEL_slab(~r, t) = −D∇ΦMODEL_slab(~r, t) (5.3)

IMODEL_slab(~r, s, t) =1

4πΦMODEL_slab(~r, t)+

34π

~JMODEL_slab(~r, t) · s. (5.4)

These expressions can be calculated with the following subroutines provided in theenclosed CD-ROM:

• Homogeneous_Gradient_Slab


• Homogeneous_Radiance_Slab

96 Chapter 5

The diffuse reflectance and the diffuse transmittance for the slab geometry canbe evaluated as

RMODEL(ρ, t) = ~JMODEL_slab(ρ, z = 0, t) · (−k)

= ΦMODEL_slab(ρ,z=0,t)2A

(5.5)

andTMODEL(ρ, t) = ~JMODEL_slab(ρ, z = s, t) · (k)

= ΦMODEL_slab(ρ,z=s,t)2A ,

(5.6)

where the EBPC has been used (see Sec. 4.4). Alternatively, the outgoing flux canbe calculated using Fick’s law.

We point out that Eqs. (5.3) and (5.4) can also be used to obtain the flux and theradiance in the infinite medium, provided the expression ΦMODEL_slab is substi-tuted by ΦMODEL_i. For the calculation of all the corresponding expressions, thefollowing subroutines are provided on the CD-ROM:

• Homogeneous_Gradient_Infinite



This approach leads to hybrid expressions for the fluence, flux, radiance, re-flectance, and transmittance of the slab based on the solutions of the RTE and ofthe TE for the infinite medium that will be reported in the next sections. Sincethese hybrid solutions are based on solutions for the infinite medium that are moreaccurate than the corresponding solutions of the DE, they are likely to be moreaccurate to describe propagation near the source and at short times.

We stress a feature of Eq. (5.1): When solutions of the RTE that account forthe peak due to ballistic photons are used for ΦMODEL_i, this formula showssingularities due to the ballistic peaks at times t+m =

∣∣∣~r −−→r+m

∣∣∣ /v for the positive

sources and t−m =∣∣∣~r −−→r−m∣∣∣ /v for the negative sources. These effects arise from

having assumed the delta source. To avoid these singularities for the RTE and TEmodels in the subroutines provided with the CD-ROM, we force ΦMODEL_i = 0for t < [r+ (0.1/µ

′s)]/v, with r distance between source and receiver, which from a

practical point of view means to disregard ballistic photons. However, discontinuitiesremain and should be carefully treated when using this theory. For many applications(for example, the development of inversion procedures), these singularities may be aproblem.

The hybrid approach has been presented with reference to an infinitely extendedslab. We note that the same procedure can be used to derive hybrid solutions in other


geometries, e.g., the parallelepiped.30 For the homogeneous parallelepiped followingthe same procedure used for Eq. (4.61), we obtain the solution (see Fig. 4.11)

ΦMODEL_parallelepiped(~r, t) =n=+∞∑n=−∞

m=+∞∑m=−∞

l=+∞∑l=−∞

[+ΦMODEL_i

( ∣∣~r − ~r+1lmn

∣∣ , t)− ΦMODEL_i

( ∣∣~r − ~r−2lmn∣∣ , t)+ΦMODEL_i

( ∣∣~r − ~r+3lmn



( ∣∣~r − ~r+5lmn



( ∣∣~r − ~r+7lmn


( ∣∣~r − ~r−8lmn∣∣ , t)] ,(5.7)

where

~r+1lmn = (x1l, y1m, z1n) ,~r−2lmn = (x2l, y2m, z2n) ,~r+

3lmn = (x2l, y2m, z1n) ,~r−4lmn = (x1l, y1m, z2n) ,~r+

5lmn = (x2l, y1m, z2n) ,~r−6lmn = (x1l, y2m, z1n) ,~r+

7lmn = (x1l, y2m, z2n) ,~r−8lmn = (x2l, y1m, z1n) ,

with

x1l = 2lLx + 4lze + xs,x2l = 2lLx + (4l − 2)ze − xs,y1m = 2mLy + 4mze + ys,y2m = 2mLy + (4m− 2)ze − ys,z1n = 2nLz + 4nze + zs,z2n = 2nLz + (4n− 2)ze − zs,

(5.8)

and ΦMODEL_i can be the solution of the RTE, TE, and DE for the infinite medium.All the CW Green’s functions for the hybrid models can be calculated integrating

the corresponding TR Green’s functions using the Trap subroutine provided on theCD-ROM.

5.2 Analytical Solutions of the Time-Dependent RadiativeTransfer Equation for an Infinite Homogeneous Medium

Two solutions of the RTE for the infinite homogeneous medium ΦMODEL_i arepresented. The first, ΦRTE_i, is based on an almost exact solution of the RTE fora medium with isotropic scattering. The second, ΦTE_i, is a solution of the TE.28

These solutions satisfy the causality principle, i.e., the solution is equal to zero fort < r/v. In the next sections, the time-dependent Green’s functions ΦRTE_i andΦTE_i obtained by Paasschens27 and by Durian and Rudnick28 are presented.

98 Chapter 5

5.2.1 Almost exact time-resolved Green’s function of the radiativetransfer equation for an infinite medium with isotropicscattering

Paasschens27 proposed an almost exact solution of the time-dependent RTE for theinfinite medium with isotropic scattering, i.e., with scatterers having an isotropicscattering function p(θ) = 1/(4π). Paasschens, using a path-integral method, gaveexplicit time-dependent solutions of the RTE for two and four dimensions (2D and4D, respectively). At first, the Fourier transform of the solutions was computed,then through the inverse Fourier transform, and the time-dependent solutions wereobtained analytically. Unfortunately, for three dimensions (3D) the inverse Fouriertransform cannot be performed analytically, and the exact solution is availableonly in the Fourier transform domain. The analytical 3D solution was obtained byPaasschens interpolating between the two exact time-dependent solutions found for2D and 4D.

For a non-absorbing medium with isotropic scattering, the Green’s function forthe time-dependent fluence rate obtained by Paasschens,27 Φisotropic

RTE_i (r, t), can bewritten as

ΦisotropicRTE_i (~r, t) ' v exp(−µsvt)

4πr2δ(vt− r) + Θ(vt− r)

v

[1−(rvt

)2] 18(

4πvt3µs

) 32

×Gµsvt

[1− ( rvt)

2] 3

4

exp(−µsvt),

(5.9)

where r is the distance from the source, Θ(x) is the step function (0 for x < 0 and 1for x > 0), and the function G is

G(x) = 8(3x)−3/2∞∑N=1

Γ(34N + 3

2)Γ(3

4N)xN

N !, (5.10)

which can be approximated as

G(x) ' ex√

1 + 2.026 x−1 . (5.11)

In Eq. (5.9), the first term represents the contribution of the unscattered radiation(ballistic peak), the N th–term of the series of Eq. (5.10) gives the contribution ofthe N th–scattering event, and Γ is the gamma function. At early times (small valuesof x), the first expression of G [Eq. (5.10)] is more accurate, while for late timesEq. (5.10) and Eq. (5.11) tend to give the same values. The differences betweenthe two expressions of G remain within 1.6%. To check the accuracy of Eq. (5.9),Paasschens compared the results with a numerical evaluation of the inverse Fouriertransform and demonstrated that discrepancies are on the order of 2% outside theballistic peak.27


The effect of absorption is introduced, according to the RTE, by multiplyingthe solution by exp(−µavt). We observe that, in agreement with the causalityprinciple, the solution is zero for t < r/v. We also note another feature of Eq. (5.9):At t = r/v, in addition to the singularity due to unscattered radiation, there isa logarithmic singularity due to radiation that has undergone a single forwardscattering event. This singularity, explicitly mentioned in the work of Paasschens,27

has been denoted as a tail of the ballistic peak.

5.2.2 Heuristic time-resolved Green’s function of the radiativetransfer equation for an infinite medium with non-isotropicscattering

When an infinite medium with a non-isotropic scattering function is considered,Eq. (5.9) cannot be rigorously used. Following a heuristic approach, which isjustified by the similarity relations introduced in Sec. 2.4, the Green’s function for aninfinite homogeneous medium can be rewritten for the case of anisotropic scattering(g 6= 0) using µ

′s instead of µs for the scattered component as follows:

ΦRTE_i(r, t) ' v exp[−(µs+µa)vt]4πr2

δ(vt− r) + Θ(vt− r)v

[1−(rvt

)2] 18(

4πvt

3µ′s

) 32

×Gµ′svt[1− r2

(vt)2

] 34

exp[−(µ′s + µa

)vt].

(5.12)

In Eq. (5.12), the effect of absorption has been included according to the generalproperty of the RTE [Eq. (3.14)], multiplying the solution for the non-absorbingmedium by exp(−µavt). Since µ

′s = µs(1− g), for isotropic scattering Eq. (5.12)

is identical to Eq. (5.9). Equation (5.12) can be heuristically used to representthe Green’s function of the medium for anisotropic scattering and can be used toimplement a hybrid model for the slab geometry assuming ΦMODEL_i = ΦRTE_i.

The expression of Eq. (5.12) can be calculated with the following subroutineprovided with the enclosed CD-ROM:

• RTE_Fluence_Infinite_TR

5.2.3 Time-resolved Green’s function of the telegrapher equation foran infinite medium

The TE, like the DE, is derived by using an approximation of the RTE. The TE in aregion with no sources is a second-order differential equation for the fluence of thetype

α∂2

v2∂t2Φ(~r, t) + β

∂

v∂tΦ(~r, t)− 1

3∇2Φ(~r, t) + γΦ(~r, t) = 0 , (5.13)

where α, β, and γ are constants that assume different values, depending on thederivation used to obtain the TE. Some derivations of the TE use the so-called

100 Chapter 5

P1 approximation2, 31 [Eq. (3.25)]. In this case, the TE is obtained from the RTEfollowing a procedure similar to the one used for the DE but using only the P1

approximation2 (without Fick’s law). Other alternative derivations can be usedto obtain the TE.28, 32 In particular, in this section we report the TE that Durianand Rudnick28 obtained by extending the two-stream theory33 from one to threedimensions making use of a physically motivated ansatz. The equation they obtainedis written as[

13v2

∂2

∂t2+

(2µa + 3µ

′s

)3v

∂

∂t− 1

3∇2 + µa

(µa3

+ µ′s

)]Φ(~r, t) = 0 . (5.14)

A feature of the TE is that at short times the second temporal derivative is dominant,and the wave equation is recovered. Thus, this term accounts for the ballisticphotons migrating at short times. At long times, when the second temporal derivativebecomes negligible, the DE is recovered. Equation (5.14) assumes, according to theRTE, that Φ(~r, t, µa) = Φ(~r, t, µa = 0) exp(−µavt).

We observe that for steady-state sources and µa = 0, the TE proposed by Durianand Rudnick28 reduces exactly to the DE. Thus, for steady-state sources and non-absorbing media, the Green’s function of the TE coincides with the Green’s functionof the DE. According to the expression obtained by Durian and Rudnick,28 the TRGreen’s function for an infinite homogeneous medium ΦTE_i(~r, t) is written fort > r/v as

ΦTE_i(~r, t) = −vµ′sexp(− 3

2µ′svt)

4πr

(32 + 1

µ′sv∂∂t

)× ∂∂r

[I0

(32µ′s

√(vt)2 − r2

)Θ(vt− r)

]exp(−µavt),

(5.15)

where I0(x) is the modified Bessel function of order zero, and Θ is the step functionpreviously defined. Also, this solution, in accordance to the causality principle, iszero for t < r/v. In Eq. (5.15), the derivatives of the step function lead both to adelta function term and to a delta function term derivative at t = r/v. Away fromthese delta functions (i.e., for t > r/v) the Green’s function can be represented as28

ΦTE_i(~r, t) = Θ(vt− r) v(

32µ′s

)24π√

(vt)2−r2exp

(−3

2µ′svt)

exp(−µavt)

×[I1

(32µ′s

√(vt)2 − r2

)+ vt√

(vt)2−r2I2

(32µ′s

√(vt)2 − r2

)],

(5.16)

where I1 and I2 are the modified Bessel functions of order one and two. The aboveGreen’s functions can be used to implement a hybrid model for the slab geometry,assuming for ΦMODEL_i the expressions ΦTE_i of Eq. (5.15) or of Eq. (5.16). Whenthe argument x of the modified Bessel function In(x) is much larger than n, thisapproximate expression is used:34–36

In(x) ≈ 1√2πx

exp(x) . (5.17)


The solution ΦTE_i of Eq. (5.16) can be calculated with the following subroutineprovided on the enclosed CD-ROM:

• TE_Fluence_Infinite_TR

5.3 Comparison of the Hybrid Models Based on the RadiativeTransfer Equation and Telegrapher Equation with theSolution of the Diffusion Equation

The characteristics of the hybrid models based on RTE and TE show differenceswith respect to the solution of the DE, especially at early times and short distancesfrom the source. In this section, a concise overview of results is presented where thehybrid solutions are compared with the solution of the DE. All the figures presented

0 50 10010-4

10-3

10-2

10-1r = 1 mm

Flue

nce

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 15010-5

10-4

10-3

10-2r = 2 mm

Flue

nce

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-6

10-5

10-4

10-3

Flue

nce

(mm

-2ps

-1)

Time (ps)

DE RTE TE

r = 5 mm

0 50 100 150 20010-8

10-7

10-6

10-5

10-4

r = 10 mm

Flue

nce

(mm

-2ps

-1)

Time (ps)

DE RTE TE

Figure 5.1 Comparison among the TR fluence from RTE, TE, and DE for an infinite mediumwith µa = 0, µ

′s = 1 mm−1, and n = 1.4. The results are shown for r = 1, 2, 5, and 10 mm.

The vertical dotted line indicates the ballistic time t0.

in this section have been generated by using the subroutines for hybrid models

102 Chapter 5

provided with the enclosed CD-ROM.The first comparison concerns the infinite medium. Figure 5.1 shows the com-

parison among the TR fluence from the RTE [Eq. (5.12)], TE [Eq. (5.16)], and DE[Eq. (4.3)] for a medium with µa = 0, µ

′s = 1 mm−1, and refractive index n = 1.4.

The figure shows that the main differences between the models are at early times.The solutions of the RTE and the TE, according to the causality principle, predict noenergy for t < r/v. Conversely, ΦDE_i > 0 for any t > 0.

In Figs. 5.2 and 5.3, comparisons are shown for the TR reflectance from two

0 50 100 150 20010-6

10-5

10-4

10-3

10-2

= 1 mm

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-6

10-5

10-4

10-3

= 2 mmRef

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-7

10-6

10-5

10-4

= 5 mm

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-9

10-8

10-7

10-6

= 10 mmRef

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

Figure 5.2 Comparison among the solutions of RTE, TE, and DE for the TR reflectancefrom a slab with s = 40 mm, µa = 0 , µ

′s = 1 mm−1, zs = 1 mm, n = 1.4, and nr = 1. The

results are shown for ρ = 1, 2, 5, and 10 mm. The vertical dotted line indicates the ballistictime t0.

slabs with s = 40 and 4 mm, respectively, and µa = 0, µ′s = 1 mm−1, zs = 1/µ

′s =

1 mm, ni = 1.4, and nr = 1. Results are shown for different values of the distanceρ of the receiver from the z-axis. For the 40-mm-thick slab, differences between themodels are mainly at early times. As mentioned before, the solution of the DE is 6= 0


0 50 100 15010-6

10-5

10-4

10-3

10-2

= 1 mm

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 15010-6

10-5

10-4

10-3

= 2 mmRef

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-7

10-6

10-5

10-4

= 5 mm

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-9

10-8

10-7

10-6

= 10 mm

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

Figure 5.3 Comparison among the solutions of RTE, TE, and DE for the TR reflectancefrom a slab with s = 4 mm, µa = 0 , µ

′s = 1 mm−1, zs = 1 mm, n = 1.4, and nr = 1. The

results are shown for ρ = 1, 2, 5, and 10 mm. The vertical dotted line indicates the ballistictime t0.

for t < t0 = r0/v (r0 is the distance from the real source at zs = 1/µ′s), and the

solutions of the RTE and the TE show a discontinuity near t0 related to the differenttimes at which the receiver begins to receive photons from the positive source atz+

0 = zs and from the corresponding negative image source at z−0 = −zs − 2ze.This couple of sources is the only one from which photons can be received withinthe temporal range considered in Fig. 5.2.

For the slab with s = 4 mm, the main differences are at early times, but in thiscase significant differences also remain at late times: In particular, the asymptoticbehavior at long times of the TE solution is significantly different with respect tothe RTE and DE solutions. This different behavior comes from the high number ofimage sources (about 20 dipoles for s = 4 mm!) that give a significant contributionto the solution also in the short time interval displayed in the figure. The effects of

104 Chapter 5

the quick temporal succession of lighted sources placed at a short distance from thereceiver affect the reflectance so that the asymptotic behavior is influenced by thebehavior at early times of the Green’s function for the infinite medium.

The discontinuity in the RTE and TE responses near t0 observed for s = 40mm is also visible for s = 4 mm. Discontinuities are also expected each time thereceiver begins to receive photons from a new image source. However, thanks tothe precaution mentioned in Sec. 5.1, for which the response is forced to 0 for

t < [r+(0.1/µ′s)]

v (r distance between source and field point), they remain small andnot visible in the logarithmic scale used to display the results.

0 50 100 15010-6

10-5

10-4

10-3

= 0 mms = 4 mm

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 50 100 150 20010-7

10-6

10-5

10-4

= 5 mms = 4 mmTr

ansm

ittan

ce (m

m-2

ps-1

)

Time (ps)

DE RTE TE

0 200 400 600 800 100010-12

10-11

10-10

10-9

10-8

= 0 mms = 40 mm

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

0 500 1000 150010-13

10-12

10-11

10-10

= 40 mms = 40 mm

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)

DE RTE TE

Figure 5.4 Comparison among the solutions of RTE, TE, and DE for the TR transmittancefrom slabs with s = 4 and 40 mm having µa = 0, µ

′s = 1 mm−1, zs = 1 mm, n = 1.4, and

nr = 1.

In Fig. 5.4, a comparison is shown among the different models for the time-dependent transmittance of two slabs with thicknesses of 4 and 40 mm with thesame optical properties of the previous figures. The behavior we observe among thedifferent models is similar to that seen for the reflectance in Figs. 5.2 and 5.3.


References



3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuseoptical imaging,” Phys. Med. Biol. 50, R1–R43 (2000).

4. R. L. Barbour, H. L. Graber, Y. Pei, S. Zhong, and C. H. Schmitz, “Opticaltomographic imaging of dynamic features of dense-scattering media,” J. Opt.Soc. Am. 18, 3018–3036 (2001).

5. V. A. Markel, “Modified spherical harmonics method for solving the radiativetransport equation,” Wave Random Media 14, L13–L19 (2004).

6. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equationin rotated reference frames,” J. Phys. A: Math. Gen. 39, 115–137 (2006).

7. T. Feng, P. Edström, and M. Gulliksson, “Levenberg-Marquardt methods forparameter estimation problems in the radiative transfer equation,” Inverse Probl.23, 879–891 (2007).


9. A. H. Gandjbakhche, G. H. Weiss, R. F. Bonner, and R. Nossal, “Photon path-length distributions for transmission through optically turbid slabs,” Phys. Rev.E 48, 810–818 (1993).

10. S. Havlin, R. Nossal, B. Trus, and G. H. Weiss, “Photon migration in disorderedmedia,” Phys. Rev. A 45, 7511–7519 (1992).

11. A. H. Gandjbakhche and G. H. Weiss, “Random walk and diffusion-like modelof photon migration in turbid media,” In: Progress in Optics, Elsevier Science,Amsterdam, Vol. 34, 333–402 (1995).

12. G. H. Weiss, “Statistical properties of the penetration of photons into a semi-infinite turbid medium: A random-walk analysis,” Appl. Opt. 37, 3558–3563(1998).


106 Chapter 5

14. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco,and G. Zaccanti, “Time-resolved reflectance at null source-detector separation:Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett.95, 078101 (2005).

15. A. H. Hielscher, R. E. Raymond, E. Alcouffe, and R. L. Barbour, “Comparisonof finite-difference transport and diffusion calculations for photon migrationin homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302(1998).

16. V. Venugopalan, J. S. You, and B. J. Tromberg, “Radiative transport in thediffusion approximation: An extention for highly absorbing media and smallsource-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).

17. R. Graaff and K. Rinzema, “Practical improvements on photon diffusion theory:application to isotropic scattering,” Phys. Med. Biol. 46, 3043–3050 (2001).

18. E. L. Hull and T. H. Foster, “Steady-state reflectance spectroscopy in the P3approximation,” J. Opt. Soc. Am. A 18, 584–599 (2001).

19. P. N. Reinersman and K. L. Carder, “Hybrid numerical method for solution ofthe radiative transfer equation in one, two, or three dimensions,” Appl. Opt. 43,2734–2743 (2004).

20. A. Garofalakis, G. Zacharakis, G. Filippidis, E. Sanidas, D. D. Tsiftsis, V. Ntzi-achristos, T. G. Papazoglou, and J. Ripoll, “Characterization of the reducedscattering coefficient for optically thin samples: theory and experiments,” J.Opt. A: Pure Appl. Opt. 6, 725–735 (2004).

21. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybridradiative-transfer-diffusion model for optical tomography,” Appl. Opt. 44, 876–886 (2005).

22. J. S. You, C. K. Hayakawa, and V. Venugopalan, “Frequency domain photonmigration in the δ-P1 approximation: Analysis of ballistic, transport, anddiffuse regime,” Phys. Rev. E 72, 021903 (2005).

23. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. Kaipio,“Coupled radiative transfer equation and diffusion approximation model forphoton migration in turbid medium with low-scattering and non-scatteringregions,” Phys. Med. Biol. 50, 4913–4930 (2005).

24. F. Martelli, A. Sassaroli, A. Pifferi, A. Torricelli, L. Spinelli, and G. Zaccanti,“Heuristic Green’s function of the time dependent radiative transfer equationfor a semi-infinite medium,” Opt. Express 15, 18168–18175 (2007).

25. Z. Yuan, X. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,”Phys. Med. Biol. 54, 67–90 (2009).


26. M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in bi-ological tissue using three-dimensional frequency-domain simplified sphericalharmonics equations,” Phys. Med. Biol. 54, 2493–2509 (2009).

27. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,”Phys. Rev. E 56, 1135–1141 (1997).

28. D. J. Durian and J. Rudnick, “Photon migration at short times and distancesand in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).

29. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of thetime-dependent diffusion equation for a three-layer medium: application tostudy photon migration through a simplified adult head model,” Phys. Med.Biol. 52, 2827–2843 (2007).


31. A. Y. Polishchuk, S. Gutman, M. Lax, and R. R. Alfano, “Photon-densitymodes beyond the diffusion approximation: scalar wave-diffusion equation,” J.Opt. Soc. Am. A 14, 230–234 (1997).

32. B. J. Hoenders and R. Graaff, “Telegrapher’s equation for light derived fromtranport equation,” Opt. Comm. 255, 184–195 (2005).

33. C. F. Bohren, “Multiple scattering of light and some of its observable conse-quences,” Am. J. Phys. 55, 524–533 (1987).

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Chapter 6

The Diffusion Equation forLayered MediaThe method of images (or mirror images) has been largely used to derive analyticalsolutions of the diffusion equation for various geometries.1–4 An alternative approachto the method of images is the eigenfunction method,5–9 which can only be appliedfor finite-bounded geometries. From here onward, the expression eigenfunctionmethod will be used to denote the method based on an eigenfunction expansionto solve the initial and boundary value problem for parabolic equations. In thischapter, after reviewing some of the literature devoted to photon migration throughdiffusive layered media, the eigenfunction method will be applied to solve the DE ina bounded, layered medium. In particular, the solution for a two-layer medium willbe described. The corresponding subroutine is reported in the enclosed CD-ROM.Following the same approach, a solution for a three-layer medium has been alsoobtained.10

6.1 Photon Migration through Layered Media

In recent years there has been an enhanced interest in studying the problem of lightpropagation in layered diffusive media. One reason for this is its applications in thefield of tissue optics.11, 12 In fact, many diffusive media, and in particular severalbiological tissues, can be described by a sequence of diffusive layers having differentoptical properties.11 For this reason, several approaches have been proposed to studyphoton migration through layered diffusive media.

Some authors have investigated the limitations of homogeneous models forphoton migration when employed in physical systems with a layered architecture,like biological tissues.13–15 Several studies have been carried out either with steady-state sources,16–24 or for modulated sources19, 25–29 by using different methods, suchas the analytical solution of DE, random walk analysis, MC method and finiteelement method. Some studies have also been carried out for time-dependentsources following various approaches.18, 24, 30–37 The particular interest in theseinvestigations is focused mainly on finding correct and efficient methods to calculatethe Green’s function of a system and to implement fast and reliable inversion

109

110 Chapter 6

procedures. Dayan et al.18 found approximate expressions of the Green’s functionfor the case of a slab on top of a semi-infinite medium. In the work of Kienle etal.,31 the time-dependent formula for the Green’s function was obtained by usingnumerical computation of the inverse space-transverse and inverse time Fouriertransform. Models for N-layered media, based on the earlier work of Kienle et al.,31

were also developed.24, 33 Tualle et al.32 used an extension of the method of imagesto calculate the Green’s function for a layered medium. Barnet36 described a methodfor solving the time-dependent DE in a stratified finite cylinder having an arbitrarynumber of layers. The fluence rate in the medium is represented by a transversemodel representation where the series expansion consists of radial eigenfunctionsand of longitudinal functions. The radial eigenfunctions are obtained as a solutionof the Laplacian in the disc of the cylinder, i.e., as a J0 Bessel function, while thelongitudinal component is obtained by using a finite difference method. Sikora etal.37 presented two methods for solving the DE in multilayered geometries: One isanalytic for a three-layer sphere, and the other is a 3D boundary element method forgeneral layered geometries.

Even though the eigenfunction method is well known in the scientific literature,the method of images has been preferentially used during recent years, especially forhomogeneous geometries like the slab.1–4 One reason for this is that the method ofimages for some geometries can be more easily used than the eigenfunction method.Moreover, the method of images also shows a very fast convergence of the solutionsat early times so that the number of terms needed is very low when solutions areexpressed by an infinite series. The eigenfunction method or theory of Fourier’sseries has been largely used in the past by Carslaw and Jaeger6 to solve the DEfor the conduction of heat in solids for geometries like the parallelepiped, the slab,the cylinder, and the sphere. More recent works that have used the eigenfunctionmethod in layered geometries are those of Takatani and Graham16 and Schmitt etal.17 concerning the DE for steady-state sources.

In this chapter, the eigenfunction method is reframed for the case of time-dependent sources and layered geometries, following the ideas proposed in some re-cent works dedicated to the layered parallelepiped38 and to the layered cylinder.10, 39

6.2 Initial and Boundary Value Problems for ParabolicEquations

The Green’s function method is applied to solve the initial and boundary valueproblem of the DE for a bounded domain V . The initial and boundary value problemis a way of providing the boundary conditions for the diffusion equation: The fluenceis assigned in a domain V at time t = 0 and also at boundary of V (∂V ) for t >0. Asolution of the problem is then obtained with an eigenfunction expansion.

We first consider the DE in a homogeneous bounded finite region V with a

The Diffusion Equation for Layered Media 111

source term q0(~r, t) = S0(~r)δ(t). Therefore,(1v

∂

∂t−D∇2+µa

)Φ(~r, t) = 0 for ~r ∈ V, and t > 0. (6.1)

The boundary conditions for a solution of Eq. (6.1) are simplified assuming Φ(~r, t)is equal to zero on an extrapolated boundary, ∂VE , outside the physical boundary∂V (see Sec. 3.9.1 for details on the extrapolated boundary condition):

Φ(~r, t) = 0 for ~r ∈ ∂VE , and t > 0 . (6.2)

In addition, the initial distribution of Φ(~r, t) inside V must be considered, so

Φ(~r, t = 0) = vS0(~r) for ~r ∈ V. (6.3)

The initial and boundary value problem for the DE is summarized by Eqs. (6.1),(6.2), and (6.3). To solve this problem, an expansion for the solution Φ(~r, t) isassumed as

Φ(~r, t) =+∞∑n=1

ηn(t)ξn(~r) , (6.4)

where ξn(~r) is a complete set of orthonormal eigenfunctions with correspondingeigenvalues λn and solutions of the Helmholtz equation, i.e., a partial differentialequation of the type

∇2ξn(~r) + kξn(~r) = 0. (6.5)

For the eigenvalue problem investigated here, we obtain (see Appendix G for moredetails on the procedure used)

∇2ξn(~r) +D−1(λnv − µa

)ξn(~r) = 0 for ~r ∈ V

ξn(~r) = 0 for ~r ∈ ∂VE .(6.6)

It is possible to show that the function ηn(t) has to be in the form (see Appendix G)

ηn(t) = ηn0 exp(−λnt), with ηn0 =∫V

S0(~r)ξ∗n(~r)d~r. (6.7)

The Green’s function of a parabolic equation can then be represented as

G(~r,→r′ , t, t′) =

+∞∑n=1

exp [−λn(t− t′)] ξn(~r)ξ∗n(→r′ ). (6.8)

This basic form of the Green’s function for homogeneous media will be used torepresent the solution of the DE in a bounded, layered medium.

As an example the procedure to obtain the Green’s function for a homogeneousdiffusive cube will be shown in Appendix H.

112 Chapter 6

s1

s2

a2, D2, n2

a1, D1, n1

y

x

z

V1

V2

O

ne

zs

R ( , t)L

r ),( tr

T ( , t)

s1

s2

a2, D2, n2

a1, D1, n1

y

x

z

V1

V2

O

ne

zs

R ( , t)L

r ),( tr

T ( , t)

Figure 6.1 Graphic of a two-layer medium and symbols used.

6.3 Solution of the DE for a Two-Layer Cylinder

In this section, the procedure to obtain a solution of the DE with the eigenfunctionmethod in a two-layer cylinder (see Fig. 6.1) will be summarized. Figure 6.1 showsthe medium V composed of two regions: V = V1 ∪ V2. In the figure, s1 ands2 are the thickness, µa1 and µa2 are the absorption coefficients, D1 and D2 arethe diffusion coefficients, and n1 and n2 are the refractive indices of the first andsecond layer, respectively. ne is the refractive index of the surrounding mediumthat can be different at the two sides of the cylinder z < 0 and z > s1 + s2. Theradius of the cylinder is denoted by L. The origin of the reference system is chosenalong the main axis of the cylinder on the external surface; therefore, the physicalboundaries of the cylinder belong to the planes z = 0 and z = s1 + s2 and to thecylindrical surface ρ = L. An isotropic point source is at →rs = (0, 0, zs), i.e.,q (~r, t) = δ(~r −→rs )δ(t). The DE for the inhomogeneous medium is written as

1v

∂

∂t−∇[D(~r)∇] + µa

Φ (~r, t) = δ(~r −→rs )δ(t). (6.9)

The initial and boundary value problem in both the layers is then formulatedintroducing two functions for the fluence, Φ1 (~r, t) for the first layer and Φ2 (~r, t)for the second layer, as follows:

(∂

v1∂t−D1∇2 + µa1

)Φ1 (~r, t) = 0 t > 0, 0 ≤ z < s1(

∂v2∂t−D2∇2 + µa2

)Φ2 (~r, t) = 0 t > 0, s1 ≤ z ≤ s1 + s2 ,

(6.10)

and the initial and boundary value conditions are

Φ1(ρ = L, z, t) = Φ1(ρ, z = −z1e, t) = 0 , (6.11)

Φ2(ρ = L, z, t) = Φ2(ρ, z = s1 + s2 + z2e, t) = 0 , (6.12)


and

Φ1(~r, t = 0) = v1δ(~r −→rs ) , (6.13)

where z1e and z2e are the extrapolated distances at the surfaces z = 0 and z = s1+s2,respectively (for their definition, see section 3.9.1), and the source is assumed in thefirst layer. We used the EBC at the surfaces z = 0 and z = s1 + s2 and the ZBC(see Sec. 3.9.1) on the lateral surface of the cylinder. The fluence rate is representedin each layer with an eigenfunction expansion as Eq. (6.4) and Eq. (6.7):

Φ1(~r, t) =

∞∑l,n=1

αlnξ1ln(~r) exp(−λlnt) t > 0, 0 ≤ z < s1

Φ2(~r, t) =∞∑

l,n=1

αlnξ2ln(~r) exp(−λlnt) t > 0, s1 ≤ z ≤ s1 + s2 ,

(6.14)where the same temporal evolution of Φ1 and Φ2 has been assumed (same coeffi-cients αln and λln) because the boundary condition at the interface of the layersrequires this condition. We also stress that the series depends on two indexes (l andn) because the problem investigated depends on the variables ρ and z.

To account for the effect of a discontinuous change in the refractive index at theinterface z = s1, proper boundary conditions need to be introduced. In the moregeneral case of n1 6=n2, according to the boundary condition at a diffusive-diffusiveinterface described in Sec. 3.9.2, the continuity of the fluence at z = s1 does nothold while the continuity of the flux holds. The boundary condition at z = s1 canthus be considered by using Eqs. (3.61) and (3.56) as

Φ1(ρ, z = s1, t) = (n1/n2)2Φ2(ρ, z = s1, t)D1

∂∂zΦ1(ρ, z = s1, t) = D2

∂∂zΦ2(ρ, z = s1, t).

(6.15)

It is worth noting why Φ1 and Φ2 in Eq. (6.14) have been assumed with the sametemporal evolution: The temporal evolution of Φ1 and Φ2 must be coincident ifwe want the condition of Eq. (6.15) fulfilled for any value of t. We also point outthat although Eq. (3.61) is approximate (less than 7% for 0.75 < n2

n1< 1.25), the

reflectance and transmittance of a two-layer medium is described with excellentaccuracy, even for n1/n2 = 1.4 (see the comparisons with the results of MCsimulations of Chapter 10).

To solve this problem, the Laplacian in cylindrical coordinates is considered:∇2 =1

ρ∂∂ρ

(ρ ∂∂ρ

)+ 1

ρ2∂2

∂θ2+ ∂2

∂z2. This choice is convenient because, due to the

symmetry of the problem, only two independent variables are necessary. Thischoice significantly reduces the computation time. Therefore,∇2 can be written as∇2 =1

ρ∂∂ρ

(ρ ∂∂ρ

)+ ∂2

∂z2. In order to solve the Helmholtz equation [Eq. (6.6)] in the

two layers, the separation of the variables method is used, and a solution is found in

114 Chapter 6

the form of ξln(~r) = ξρl(ρ)ξzln(z). Therefore,∇2ξ1ln(~r) +K2

1lnξ1ln(~r) = 0 0 ≤ z < s1

∇2ξ2ln(~r) +K22lnξ2ln(~r) = 0 s1 ≤ z ≤ s1 + s2

K21ln = (λ1ln/v1 − µa1)/D1

K22ln = (λ2ln/v2 − µa2)/D2 ,

(6.16)

and then 1ρ∂∂ρ

(ρ ∂∂ρ

)ξρl(ρ) = −K2

ρlξρl(ρ) 0 ≤ ρ ≤ L∂2

∂z2ξz1ln(z) = −K2

z1lnξz1ln(z) 0 < z ≤ s1∂2

∂z2ξz2ln(z) = −K2

z2lnξz2ln(z) s1 ≤ z ≤ s1 + s2 .

(6.17)

A complete orthogonal set of eigenfunctions is given by the expression:

ξln(~r) =ξ1ln = J0(Kρlρ)a1n sin(Kz1lnz + γ1ln) 0 < z ≤ s1

ξ2ln = J0(Kρlρ)a2n sin(Kz2lnz + γ2ln) s1 ≤ z ≤ s1 + s2,(6.18)

where J0 is the Bessel function of the first kind of order zero, and a1n and a2n

are coefficients to be determined. It is also required that the functions ξ1ln(~r) andξ2ln(~r) satisfy the conditions of Eqs. (6.11) and (6.12); therefore, the coefficientsKρl have to be roots of the equation

J0(KρlL) = 0 . (6.19)

From Eqs. (6.16) and (6.17), we have that the conditionsK2

1ln = K2ρl +K2

z1ln

K22ln = K2

ρl +K2z2ln

(6.20)

must be satisfied. Finally, Eq. (6.18) satisfies the boundary conditions of Eqs. (6.11)and (6.12) if γ1ln and γ2ln are chosen as

γ1ln = Kz1ln z1e

γ2ln = −Kz2ln(s1 + s2 + z2e).(6.21)

The initial condition of Eq. (6.13) is used to determine αln together with a con-dition necessary to have a set of orthonormal eigenfunctions because the definitionof Eq. (6.13) does not assure orthonormalization by itself. Thus, we have

αln =

v1

∫V1e

f(~r)ξ∗1ln(~r)d~r + v2

∫V2e

f(~r)ξ∗2ln(~r)d~r

v1

∫V1e

ξ1ln(~r)ξ∗1ln(~r)d~r + v2

∫V2e

ξ2ln(~r)ξ∗2ln(~r)d~r(6.22)

and

f(~r) = Φ(~r, t = 0) = v1δ(~r −→rs ) =∞∑

l,n=1

αlnξln(~r), (6.23)


where we have used the definition of the scalar product in the space of the continuousfunctions in the region Ve of the extrapolated cylinder [ξ∗ln(~r) is the complex conju-gate of ξln(~r)]. The presence of v1 and v2 in Eq. (6.22) accounts for the differentenergy density in the two layers due to the different refractive index. The choiceof the coefficients a1n and a2n, together with the boundary conditions, assure thatξln(~r) multiplied by αln determines a set of real and orthonormal functions. Thisset of eigenfunctions can generate all the solutions of the initial and boundary valueproblem considered in Eqs. (6.11)–(6.13). Substituting Eq. (6.18) in Eq. (6.22), wehave

αln = v21a∗1n sin∗(Kz1lnzs + γ1ln)/N2

ln , (6.24)

withN2ln = v1

∫V1e

ξ1ln(~r)ξ∗1ln(~r)d~r + v2

∫V2e

ξ2ln(~r)ξ∗2ln(~r)d~r . (6.25)

Finally, we are able to write the solution of our initial and boundary valueproblem as

Φ(~r, t) =

∞∑l,n=1

v21J0(Kρlρ)|a1n|2 sin(Kz1lnz + γ1ln)

× sin∗(Kz1lnzs + γ1ln) exp[−(K21lnD1 + µa1)v1t]/N2

ln

0 ≤ z < s1

∞∑l,n=1

v21J0(Kρlρ)a∗1na2n sin(Kz2lnz + γ2ln)


ln

s1 ≤ z ≤ s1 + s2.

(6.26)

The coefficientsK1ln andK2ln (K21ln = K2

ρl+K2z1ln,K2

2ln = K2ρl+K

2z2ln) are

obtained imposing the boundary conditions. The boundary conditions of Eq. (6.15)applied to ξ1ln and ξ2ln yield the linear system of equations for a1n and a2n:a1n sin[Kz1ln(s1 + z1e)] + a2n sin[Kz2ln(s2 + z2e)](n1/n2)2 = 0a1nD1Kz1ln cos[Kz1ln(s1 + z1e)]− an2D2Kz2ln cos[Kz2ln(s2 + z2e)] = 0 .

(6.27)This system admits non-trivial solutions (a1n, a2n 6= 0) if, and only if, the de-terminant vanishes. Therefore, the transcendental equation for the longitudinaleigenvalues is obtained:

1D1Kz1ln

tan[Kz1ln(s1 + z1e)] = − 1D2Kz2ln

tan[Kz2ln(s2 + z2e)](n1/n2)2.

(6.28)The postulated temporal dependence of the fluence inside the two layers [see

Eq. (6.14)] determines the following relationship between K21ln and K2

2ln:

K22ln =

n2

n1

D1

D2K2

1ln +(n2/n1)µa1 − µa2

D2+K2

ρl

(n2/n1)D1 −D2

D2. (6.29)

116 Chapter 6

This relation is necessary to solve Eq. (6.28).The roots of the system of equations constituted by Eqs. (6.28) and (6.29) can be

found by standard numerical methods like the bisection and the Newton-Raphsonmethod.40 The radial eigenvalues K2

ρl can be calculated only once for the unitaryradius L = 1 and kept in a look-up table. Then, the radial eigenvalues for acertain radius L can be obtained simply as K2

ρl(L) = K2ρl|L=1/L. For each radial

eigenvalue K2ρl, a set of values for Kz1ln, Kz2ln is obtained.

There is the possibility that either K2z1ln or K2

z2ln, or both, are negative; there-fore, Kz1ln and Kz2ln are imaginary numbers.39 Details about the imaginary rootsof Eq. (6.28) can be found in Ref. 39. These roots are fundamental to obtainingthe solution of the DE for the problem analyzed here. In fact, whenever imaginaryroots exist, they might yield the lowest eigenvalues, and, in particular, the mini-mum eigenvalue.39 In general, three types of roots are allowed for Eq. (6.28): Kz1ln

real and Kz2ln real, Kz1ln imaginary and Kz2ln real, and Kz1ln real and Kz2ln

imaginary.The coefficients a1n and a2n, according to Eq. (6.27), are not uniquely deter-

mined; however, their ratio is determined by the boundary condition for the fluence[Eq. (6.15)]. Assuming a1n = 1, we can rewrite Eq. (6.26) as

Φ(~r, t) =

∞∑l,n=1

v21J0(Kρlρ) sin(Kz1lnz + γ1ln)


ln

0 ≤ z < s1

∞∑l,n=1

v21J0(Kρlρ)b2ln sin(Kz2lnz + γ2ln)


ln

s1 ≤ z ≤ s1 + s2,

(6.30)

with

b2ln = a2n =sin(Kz1lns1 + γ1ln)sin(Kz2lns1 + γ2ln)

(n2/n1)2 = −sin[Kz1ln(s1 + z1e)]sin[Kz2ln(s2 + z2e)]

(n2/n1)2.

(6.31)The expression for the factor N2

ln can be found in Appendix I.Equation (6.30) represents the Green’s function for the cylinder of Fig. 6.1 when

the source term is placed in the first layer along the axis of the cylinder. In casezs belongs to the second layer, the expression for the Green’s function changes,and a new expression for αln is obtained according to Eqs. (6.22) and (6.23) whilethe eigenvalues λln and the coefficients Kρl, Kz1ln, and Kz2ln remain unchanged.


When the source is placed in the second layer (zs > s1), the expression of Φ(~r, t) is

Φ(~r, t) =

∞∑l,n=1

v22J0(Kρlρ) sin(Kz1lnz + γ1ln)b∗2ln


ln

0 ≤ z < s1

∞∑l,n=1

v22J0(Kρlρ)b2ln sin(Kz2lnz + γ2ln)b∗2ln


ln

s1 ≤ z ≤ s1 + s2.

(6.32)

The expressions of Eqs. (6.30) and (6.32) can be calculated with the followingsubroutine provided on the CD-ROM:

• DE_Fluence_Layered_Slab_TR

For some details on this subroutine, see Chapter 8. The CW Green’s function canbe calculated by integrating the TR Green’s function using the Trap subroutineprovided on the CD-ROM.

It is important to stress that the Green’s function represented by Eqs. (6.30)and (6.32) verifies the modified reciprocity relation described by Aronson:41 If→r′ denotes a point in medium 1 and ~r a point in medium 2, then G(

→r′ , ~r, t′, t) =

(n2/n1)2G(~r,→r′ , t′, t).

The solutions obtained pertain to a source placed along the axis of the cylinder.If the source is not on the axis, we lose the rotation symmetry around z, andthe eigenfunctions of the system cannot be represented any longer by Eq. (6.18).Therefore, we need to introduce in the eigenfunctions a component for the rotationaround z in order to represent all the points in the x-y plane.

Finally, according to the discussion of Sec. 4.4, from the expression of the fluence[Eqs. (6.30) and (6.32)] we can calculate the reflectance and the transmittance byusing the hybrid EBPC approach:

R(ρ, t) = Φ(ρ,z=0,t)2A

T (ρ, t) = Φ(ρ,z=s1+s2,t)2A .

(6.33)

Alternatively, the outgoing flux can be calculated using Fick’s law.Note that changing the position of the Dirac-δ source from one layer to the other

is equivalent to exchanging the expressions for reflectance and transmittance (seeFig. 6.1).

For modeling the source when a pencil light beam propagating along the z-axis(Fig. 6.1) is normally impinging the medium, see to the discussion of Sec. 4.2.

The eigenfunction method applied here to obtain a solution for a two-layer cylin-der can also be applied to obtain a solution for a three-layer cylinder.10 Moreover,

118 Chapter 6

the same method can also be applied to obtain a solution for a two and three-layerparallelepiped.38

The solution obtained can be considered a weak solution or a generalizedsolution in the sense that the solution and its derivative are not continuous in thewhole domain of the medium. In fact, Φ(~r, t) and its derivatives can be discontinuousat the boundary between the two layers when different optical properties are presentin the two layers, i.e., when a parabolic equation with discontinuous coefficientsis considered. How to solve parabolic equations in regions with discontinuouscoefficients remains a complex mathematical problem that is a subject of interest forresearchers in applied sciences, mathematics,42 and physics.

6.4 Examples of Reflectance and Transmittance of a LayeredMedium

In this section, the solutions of Eq. (6.33) for reflectance and transmittance in atwo-layer medium are plotted versus time for several characteristics of the medium.The intent of the figures is to show how the different optical properties of the layersaffect photon migration. All the figures of this section can be reproduced usingthe subroutines provided with the enclosed CD-ROM. The eigenfunction methodhas a fast convergence of the solution at late times but a slow convergence at earlytimes. For this reason, due to convergence problems related to the number and tothe accuracy of the calculated eigenvalues, it is also possible to have negative valuesfor the temporal response at early times.

In Figs. 6.2 and 6.3, a two-layer cylinder with a layered architecture of thereduced scattering coefficient is considered. In Fig. 6.2, the TR reflectance at ρ =22 mm is plotted for µ

′s1 = 1 mm−1, zs = 1/µ

′s = 1 mm, µa1 = µa2 = 0, L =

100 mm, s2 = 100 mm, n1 = n2 = 1.4, and ne = 1 for several values of µ′s2 and

s1. The figure shows that the variations of µ′s2 strongly affect the TR reflectance,

especially for the smaller values of s1. The TR reflectance and transmittance areplotted in Fig. 6.3 for a two-layer cylinder with L = 100 mm, µa1 = µa2 = 0,µ′s1 = 1 mm−1, zs = 1/µ

′s = 1 mm, n1 = n2 = 1.4, and ne = 1. The reflectance

is displayed for s1 = 4 mm, s2 = 8 mm, and ρ = 18 mm; and the transmittancefor s1 = 16 mm, s2 = 4 mm, and ρ = 0 mm. These figures show the effect of alayered architecture of the reduced scattering coefficient on photon migration. Byinterchanging the reduced scattering coefficient of the layers, a significant changeof the reflectance and of the transmittance is observed, since the effective opticalthickness of the medium is modified when the scattering inside the layers changes.

In Fig. 6.4, a cylinder is considered with a layered architecture of the absorptioncoefficient. The cylinder has L = 100 mm, s1 = 4 mm, s2 = 100 mm, µ

′s1 = µ

′s2 =

1 mm−1, zs = 1/µ′s = 1 mm, n1 = n2 = 1.4, and ne = 1. The TR reflectance is

plotted at ρ = 22 mm for µa1 = 0.01 mm−1 and µa2 = 0 mm−1 (solid curves), andµa1 = 0 and µa2 = 0.01 (dashed curves).

In Fig. 6.5, the TR reflectance and transmittance are plotted for a cylinder with


0 1000 2000 3000 4000 500010-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

s2' = 0.5 mm-1

s2' = 2 mm-1

s1 = 2 mms2 = 100 mm

0 1000 2000 3000 4000 500010-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

s2' = 0.5 mm-1

s2' = 2 mm-1

s1 = 4 mms2 = 100 mm

0 1000 2000 3000 4000 500010-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

s2' = 0.5 mm-1

s2' = 2 mm-1

s1 = 8 mms2 = 100 mm

0 1000 2000 3000 4000 500010-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

s2' = 0.5 mm-1

s2' = 2 mm-1

s1 = 16 mms2 = 100 mm

Figure 6.2 Reflectance at ρ = 22 mm for a two-layer cylinder with µ′s1 = 1 mm−1, µa1 =

µa2 = 0, L = 100 mm, s1 = 2, 4, 8, and 16 mm, s2 = 100 mm, µ′s2 = 0.5 mm−1 (solid line)

and 2 mm−1 (dashed line), zs = 1 mm, ne = 1, and n1 = n2 = 1.4.

0 500 1000 1500 200010-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

s2' = 0.5 mm-1

s2' = 2 mm-1

s1 = 4 mms2 = 8 mm

0 1000 2000 3000 400010-9

10-8

10-7

10-6

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)

s2' = 0.5 mm-1

s2' = 2 mm-1

s1 = 16 mms2 = 4 mm

Figure 6.3 Reflectance and transmittance of a two-layer cylinder with L = 100 mm, µa1 =

µa2 = 0 mm−1, µ′s1 = 1 mm−1, zs = 1 mm, ne = 1, and n1 = n2 = 1.4. For the reflectance,

s1 = 4 mm, s2 = 8 mm, and ρ = 18 mm. For the transmittance, s1 = 16 mm, s2 = 4 mm,and ρ = 0 mm. The solid curves correspond to µ

′s2 = 0.5 mm−1, and the dashed curves

correspond to µ′s2 = 2 mm−1.

120 Chapter 6

0 1000 2000 3000 4000 500010-10

10-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

a1 = 0.01 mm-1; a2 = 0 mm-1

a1 = 0 mm-1; a2 = 0.01 mm-1

Figure 6.4 Reflectance of a two-layer cylinder with L = 100 mm, s1 = 4 mm, s2 = 100 mm,µa1 = 0.01 mm−1 and µa2 = 0 (solid curves), and µa1 = 0 and µa2 = 0.01 mm−1 (dashedcurves), µ

′s1 = µ

′s2 = 1 mm−1, zs = 1 mm, ne = 1, n1 = n2 = 1.4, and ρ = 22 mm.

a layered architecture of the refractive index. The reflectance is plotted at ρ = 22and 40 mm for a cylinder with L = 100 mm, s1 = 8 mm, s2 = 100 mm, µa1 =µa2 = 0.01 mm−1, µ

′s1 = µ

′s2 = 1 mm−1, zs = 1 mm, and ne = 1. These refractive

indexes for the two layers have been considered: n1 = 1, n2 = 1.4 and n1 = 1.4,n2 = 1. The TR transmittance at ρ = 0 is plotted for a cylinder with L = 100 mm,s1 = 8 mm, s2 = 12 mm, µa1 = µa2 = 0 and 0.01 mm−1, µ

′s1 = µ

′s2 = 1 mm−1,

zs = 1 mm, and for the same values of n1, n2, and ne used for the reflectance.

0 1000 2000 300010-1310-1210-1110-1010-910-810-710-610-510-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

n1 = 1.4; n2 = 1; = 22 mm n1 = 1; n2 = 1.4; = 22 mm n1 = 1.4; n2 = 1; = 40 mm n1 = 1; n2 = 1.4; = 40 mm

0 1000 2000 300010-1310-1210-1110-1010-910-810-710-610-510-4

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)

a1 = a2 = 0 mm-1; n1 = 1.4; n2 = 1

a1 = a2 = 0 mm-1; n1 = 1; n2 = 1.4

a1 = a2 = 0.01 mm-1; n1 = 1.4; n2 = 1

a1 = a2 = 0.01 mm-1; n1 = 1; n2 = 1.4

Figure 6.5 Effect of a layered architecture of the refractive index on the time-dependentreflectance and transmittance. The reflectance is plotted at ρ = 22 and 40 mm for a two-layer cylinder with L = 100 mm, s1 = 8 mm, s2 = 100 mm, µa1 = µa2 = 0.01 mm−1,µ

′s1 = µ

′s2 = 1mm−1, zs = 1 mm, ne = 1. The transmittance at ρ = 0 is plotted for a

two-layer cylinder with L = 100 mm, s1 = 8 mm, s2 = 12 mm, µa1 = µa2 = 0 and 0.01mm−1, µ

′s1 = µ

′s2 = 1 mm−1, zs = 1 mm, and ne = 1.


6.5 General Property of Light Re-emitted by a Diffusive Medium

Light re-emitted by a diffusive medium characterized by optical properties µa andµ′s brings information on the diffusive nature of the medium. Relevant quantities

are:

• The mean time of flight 〈ti〉 (µa, µ′s, ρ, t) spent inside a diffusive layer i of

the medium by photons received at time t and distance ρ;

• The TR probability p(µa, µ′s, ρ, z, t) for reflectance measurements that pho-

tons emerging from the medium with a time of flight t and at distance ρ fromthe light beam have reached at least a certain depth z.

〈ti〉 (µa, µ′s, ρ, t) brings information on the time of flight distribution at the different

depths of the medium, and p(µa, µ′s, ρ, z, t) brings information on the depth at

which photons penetrate into the medium before being re-emitted. The solution ofthe DE for the layered cylinder reported in Sec. 6.3 and the solution of the DE forthe homogeneous slab reported in Chapter 4 can be used to obtain the followingproperty: In a homogeneous medium, both 〈ti〉 (µa, µ

′s, ρ, t) and p(µa, µ

′s, ρ, z, t)

do not depend on ρ and µa. They only depend on the scattering properties of themedium.

We present the above property in three steps. In the first step, the mean timeof flight spent inside a generic layer of a homogeneous cylinder is shown to beindependent both of ρ and µa. In the second step, the mean time of flight spent inthe layers of a two-layer cylinder is shown to be independent of ρ, provided thatµ′s1 = µ

′s2 and n1 = n2. In the third step, the TR probability p(µa, µ

′s, ρ, z, t) for

the reflectance from a homogeneous slab is shown to be independent of ρ and µa.

6.5.1 Mean time of flight in a generic layer of a homogeneouscylinder

The information on the mean time of flight 〈t〉 (ρ, z,4z, t) that photons received atdistance ρ spent inside a generic layer of thickness4z at depth z of a homogeneouscylinder (see Fig. 6.6) can be obtained using Eq. (6.33) together with Eq. (3.22). Ahomogeneous cylinder with thickness s and radiusL is considered. This homogeneouscylinder will be represented as an equivalent two-layer cylinder with thicknesses s1

and s2 with s1 + s2 = s. For reflectance and transmittance measurements, the meantime of flight 〈t1〉 (ρ, s1, t) spent inside the layer of thickness s1 can be obtained as

〈t1〉 (ρ, s1, t) = − 1v1

∂ lnR(ρ, L, s1, s2, t)∂µa1

(6.34)

for reflectance measurements, and

〈t1〉 (ρ, s1, t) = − 1v1

∂ lnT (ρ, L, s1, s2, t)∂µa1

(6.35)

122 Chapter 6

z

z

s z

R( , s, t)

z

z

s z

R( , s, t)

Figure 6.6 Example of a photon trajectory in which the length of the path followed inside alayer of thickness ∆z at depth z has been pointed out.

for transmittance measurements. The expression for 〈t〉 (ρ, z,4z, t) can be obtainedas

〈t〉 (ρ, z,4z, t) = 〈t1〉 (ρ, s1 = z +4z, t)− 〈t1〉 (ρ, s1 = z, t) . (6.36)

For a homogeneous cylinder, the radial component of the solution for the fluenceΦ(~r, t) [Eq. (6.30) or Eq. (6.32)] can be factorized with respect to the longitudinalcomponent since the longitudinal eigenvalues are independent of the radial eigenval-ues, as can be verified by a check of Eqs. (6.28) and (6.29) beingK2

1ln = K2ρl+K

2z1ln

and K22ln = K2

ρl + K2z2ln. In fact, for a medium with D1 = D2 and n1 = n2,

Eqs. (6.28) and (6.29) can be expressed astan[Kz1ln(s1+z1e)]

Kz1ln= − tan[Kz2ln(s2+z2e)]

Kz2lnK2z2ln = K2

z1ln + µa1 − µa2;(6.37)

therefore, Kz1ln and Kz2ln are independent of Kρl and can be written as Kz1ln =Kz1n and Kz2ln = Kz2n. Thus, Φ(~r, t) can be represented as follows:

Φ(~r, t) =

v21

∞∑l=1

J0(Kρlρ) exp(−K2ρlD1v1t)/N2

l

×∞∑n=1

sin(Kz1nz + γ1n) sin∗(Kz1nzs + γ1n)

× exp[−(K2z1nD1 + µa1)v1t]/N2

n

0 ≤ z < s1

v21

∞∑l=1

J0(Kρlρ) exp(−K2ρlD2v2t)/N2

l

×∞∑n=1

b2n sin(Kz2nz + γ2n) sin∗(Kz1nzs + γ1n)

× exp[−(K22nD2 + µa2)v2t]/N2

n

s1 ≤ z ≤ s1 + s2 ,

(6.38)

withNl andNn normalizing factors for the radial and longitudinal eigenfunctions, re-spectively. According to Eq. (6.38), the expressions of reflectance and transmittance


can be expressed asR(ρ, t) = AR(ρ, L, µ

′s1 = µ

′s2 = µ

′s, v1 = v2 = v, t)

×BR(s1, s2, µa1 = µa2 = µa, µ′s1 = µ

′s2 = µ

′s, v1 = v2 = v, t)

T (ρ, t) = AT (ρ, L, µ′s1 = µ

′s2 = µ

′s, v1 = v2 = v, t)

×BT (s1, s2, µa1 = µa2 = µa, µ′s1 = µ

′s2 = µ

′s, v1 = v2 = v, t) .

(6.39)

From Eqs. (6.34), (6.35), and (6.39) it follows that 〈t1〉 (ρ, s1, t) is independentof ρ. Therefore, 〈t〉 (ρ, z,∆z, t) [Eq. (6.36)] is also independent of ρ and will beindicated as 〈t〉 (z,4z, t). With further investigations, it is possible to show that〈t〉 (z,4z, t) is also independent of µa.

6.5.2 Mean time of flight in a two-layer cylinder

The case analyzed here can be considered a generalization of the results of Sec. 6.5.1for a slab with a layered architecture of absorption. In this section, the mean time offlight spent by received photons inside the layers of a two-layer cylinder, 〈t1〉 (ρ, t)and 〈t2〉 (ρ, t), is analyzed instead of the mean time of flight 〈t〉 (ρ, z,4z, t) thatreceived photons have spent inside a generic layer of thickness4z at depth z. Werefer to a two-layer cylinder (see Fig. 6.1) with µ

′s1 = µ

′s2 = µ

′s and n1 = n2 = n.

Similar to the homogeneous case, the radial component of the solution can befactorized with respect to the longitudinal component, and the fluence Φ(~r, t) canbe represented as Eq. (6.38). Therefore, the TR reflectance and transmittance can berepresented as

R(ρ, t) = AR(ρ, L, µ′s, v, t)BR(s1, s2, µa1, µa2, µ

′s, v, t) ,

T (ρ, t) = AT (ρ, L, µ′s, v, t)BT (s1, s2, µa1, µa2, µ

′s, v, t) .

(6.40)

For reflectance and transmittance measurements the mean time of flight results as

〈ti〉 (ρ, t) = − ∂vi∂µai

lnR(µai, t) i = 1, 2 ,〈ti〉 (ρ, t) = − ∂

vi∂µailnT (µai, t) i = 1, 2 .

(6.41)

From Eqs. (6.40) and (6.41), we find that 〈ti〉 (ρ, t) is independent of ρ. In Fig. 6.7,the property is shown by plotting 〈ti〉 (ρ, t) for the reflectance for several values ofρ in a two-layer cylinder with L = 100 mm, s1 = 8 mm, s2 = 100 mm, µa1 = 0,µa2 = 0.01 mm−1, µ

′s1 = µ

′s2 = 1 mm−1, zs = 1/µ

′s = 1 mm, n1 = n2 = 1.4, and

ne = 1. As expected, the curves of 〈t1〉 (ρ, t) and 〈t2〉 (ρ, t) prove to be independentof ρ.

In a previous work, this property was demonstrated for a medium laterallyinfinitely extended35 that can be considered as a particular case of the general oneanalyzed here. The same property can be also obtained using the solution for athree-layer cylinder.10

124 Chapter 6

0 2000 4000 60000

1x103

2x103

3x103

<t1><t2>

<t1>( =0 mm) <t1>( =20 mm) <t1>( =40 mm)

<t2>( =0 mm) <t2>( =20 mm) <t2>( =40 mm)

mea

n tim

e of

flig

ht (p

s)

time (ps)

Figure 6.7 TR Mean time of flight spent by received photons inside the layers of a two-layer cylinder with L = 100 mm, s1 = 8 mm, s2 = 100 mm, µa1 = 0, µa2 = 0.01 mm−1,µ

′s1 = µ

′s2 = 1 mm−1, zs = 1 mm, n1 = n2 = 1.4, and ne = 1.

If 〈t1〉 and 〈t2〉 are independent of ρ, it is possible to also show, using thesolution for a three-layer cylinder,10 that the mean time of flight 〈t〉 (ρ, z,4z, t) thatreceived photons have spent inside a generic layer of thickness4z at depth z in amedium with a layered architecture of absorption is independent of ρ.

These results show that the independence of 〈ti〉 (ρ, t) of ρ is valid forhomogeneous media and for media with a layered architecture in the absorptioncoefficient. When µ

′s1 6= µ

′s2 or n1 6= n2, the mean time of flight 〈ti〉 (ρ, t) depends

in general on ρ.35 However, it has been shown in Ref. 35 that for values of µ′s and µa

of interest for tissue optics [0.5 < (µ′s1/µ

′s2) < 2 and µa1 and µa2 < 0.05 mm−1]

〈ti〉 (ρ, t) is almost independent of ρ. Moreover, insignificant effects arise from thedifferences between n1 and n2 that may occur when photon migration takes place intissue (see Fig. 10.21). Thus, in biological tissues the TR mean time of flight spentinside a layer of the medium is almost independent of ρ.

6.5.3 Penetration depth in a homogeneous medium

The TR probability p(ρ, z, t) that photons emerging from a diffusive slab withthickness s (see Fig. 6.6) with a time of flight t at distance ρ from the light beamhave reached at least the depth z is equal to the fraction of photons that are lost,due to the finite thickness of the medium, when a slab of thickness z is consideredinstead of the actual slab with thickness s:

p(ρ, z, t) =R(ρ, s, t)−R(ρ, z, t)

R(ρ, s, t)for 0 < z < s. (6.42)

For a homogeneous slab, p(ρ, z, t) is independent both of ρ and of µa. Thisproperty can be obtained both with the expression of Eq. (4.27) for the reflectanceof the homogeneous slab or the expression of Eqs. (6.30), (6.32), and (6.33) for the


two-layer cylinder. For both a homogeneous slab and a homogeneous cylinder, theTR reflectance R(ρ, s, t) depends on the optical properties of the medium (µ

′s , µa,

refractive index), on the thickness of the slab s, and on the source-receiver distanceρ, but its expression can be represented in all generality as follows:

R(ρ, z, t) = B(ρ, µ′s, t)C(s, µ

′s, t) exp(−µavt) . (6.43)

From the form of Eq. (6.43), it is straightforward to verify that the TR probability isindependent both of ρ and of µa and thus can be indicated as a function p(z, t) of zand t only.

Substituting in Eq. (6.42), the expression for the CW reflectance instead of theTR reflectance, we obtain the CW probability, p(ρ, z), which depends both on ρ andµa. Details on this point can be found in Ref. 43.

6.5.4 Conclusion

Both the TR probability and the TR mean time of flight or the TR mean pathlengthare a characteristic of the scattering properties of the diffusive slab. Thus, for TRdiffuse reflectance measurements, the mean penetration depth of photons detectedat a certain time does not depend on either the source-detector distance or theabsorption coefficient of the scattering medium, but it does increase with the arrivaltime of the photons. The consequence of these results is that for time-dependentmeasurements the only possibility of increasing the fraction of detected photonsthat deeply penetrate into the medium with respect to those that remain confined atlower depths is to examine photons with a long time of flight.43–45 Experiments havealso confirmed that longer-lived photons penetrate deeper into the medium.46, 47

Conversely, for CW sources both the penetration depth and the mean pathlengthtraveled by received photons inside a layer of the medium strongly depend on ρ andon the absorption coefficient of the medium. When steady-state sources are used,the fraction of photons that deeply penetrate inside the medium can be increased,augmenting the distance ρ from the light beam.

126 Chapter 6

References





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10. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of thetime-dependent diffusion equation for a three-layer medium: application tostudy photon migration through a simplified adult head model,” Phys. Med.Biol. 52, 2827–2843 (2007).



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15. R. J. Hunter, M. S. Patterson, T. J. Farrell, and J. E. Hayward, “Haemoglobinoxygenation of a two-layer tissue-simulating phantom from time-resolvedreflectance: Effect of top layer thickness,” J. Opt. Soc. Am. A 47, 193–208(2002).

16. S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectancefrom a two-layer tissue model,” IEEE Trans. Biomed. Eng. 26, 656–664 (1979).

17. J. M. Schmitt, G. X. Zhou, E. C. Walker, and R. T. Wall, “Multilayer model ofphoton diffusion in skin,” J. Opt. Soc. Am. A 7, 2141–2153 (1990).

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19. A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnieres, and H. van deBergh, “Noninvasive determination of the optical properties of two-layeredturbid medium,” Appl. Opt. 37, 779–791 (1998).

20. G. Alexandrakis, T. J. Farrel, and M. S. Patterson, “Accuracy of the diffusionapproximation in determining the optical properties of a two-layer turbidmedium,” Appl. Opt. 37, 7401–7409 (1998).

21. R. Nossal, J. Kiefer, G. H. Weissa, R. Bonner, H. Taitelbaum, and S. Havlin,“Photon migration in layered media,” Appl. Opt. 27, 3382–3391 (1988).

22. H. Taitelbaum, S. Havlin, and G. H. Weiss, “Theoretical analysis of diffusereflectance from a two-layer tissue model,” Appl. Opt. 28, 2245–2249 (1989).

23. C. Donner and H. W. Jensen, “Rapind simulations of steady-state spatiallyresolved reflectance and transmittance profiles of multilayered turbid materials,”J. Opt. Soc. Am. A 23, 1382–1390 (2006).

24. X. C. Wang and S. M. Wang, “Light transport modell in a N-layered mis-matched tissue,” Waves Rand. Compl. Media 16, 121–135 (2006).

25. G. Alexandrakis, T. J. Farrel, and M. S. Patterson, “Monte Carlo diffusion hy-brid model for photon migration in a two-layer turbid medium in the frequencydomain,” Appl. Opt. 39, 2235–2244 (2000).

26. T. H. Pham, T. Spott, L. O. Svaasand, and B. J. Tromberg, “Quantifying theproperties of two-layer turbid media with frequency-domain diffuse reflectance,”Appl. Opt. 39, 4733–4745 (2000).

27. J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, andM. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layereddiffusive media: theory and experiments,” J. Opt. Soc. Am. A 18, 821–830(2001).

28. G. Alexandrakis, D. R. Busch, G. W. Faris, and M. S. Patterson, “Determinationof the optical properties of two-layer turbid media by use of a frequency-domainhybrid Monte Carlo diffusion model,” Appl. Opt. 40, 3810–3821 (2001).

128 Chapter 6

29. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J.Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solveinverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26,1335–1337 (2001).

30. A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, and S. L. Jacques, “Timeresolved photon emission from layered turbid media,” Appl. Opt. 35, 719–728(1996).

31. A. Kienle, T. Glanzmann, G. Wagnières, and H. van den Bergh, “Investigationof two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37,6852–6862 (1998).

32. J. M. Tualle, J. Prat, E. Tinet, and S. Avrillier, “Real-space Green’s functioncalculation for the solution of the diffusion equation in stratified turbid media,”J. Opt. Soc. Am. A 17, 2046–2055 (2000).

33. J. M. Tualle, H. M. Nghiem, D. Ettori, R. Sablong, E. Tinet, and S. Avrillier,“Asymptotic behaviour and inverse problem in layered scattering media,” J. Opt.Soc. Am. A 21, 24–34 (2004).

34. A. Pifferi, A. Torricelli, P. Taroni, and R. Cubeddu, “Reconstruction of absorberconcentrations in a two-layer structure by use of multidistance time-resolvedreflectance spectroscopy,” Opt. Lett. 26, 1963–1965 (2001).

35. F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximatesolutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc.Am. A 19, 71–80 (2002).

36. A. Barnet, “A fast numerical method for time-resolved photon diffusion ingeneral stratified turbid media,” J. Comput. Phys. 201, 771–797 (2004).

37. J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. R. Arridge,and J. Ripoll, “Diffuse photon propagation in multilayered geometries,” Phys.Med. Biol. 51, 497–515 (2006).

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44. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. D. Bianco,and G. Zaccanti, “Time-resolved reflectance at null source-detector separation:Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett.95, 078101 (2005).


46. J. Steinbrink, H. Wabnitz, H. Obrig, A. Villringer, and H. Rinneberg, “Absorp-tion and scattering perturbations in homogeneous and layered diffusive mediaprobed by time-resolved reflectance at null source-detector separation,” Phys.Med. Biol. 46, 879–896 (2001).

47. J. Selb, J. J. Stott, M. A. Franceschini, A. G. Sorensen, and D. A. Boas,“Improved sensitivity to cerebral hemodynamics during brain activation with atime-gated optical system: analytical model and experimental validation,” J.Biomed. Opt. 10, 011013 (2005).

Chapter 7

Solutions of the DiffusionEquation with PerturbationTheoryThe study of photon migration in heterogeneous media is an important issue sincemost of the diffusive media encountered in nature are inhomogeneous, and solu-tions of the DE for such media are needed especially for imaging applications.1

Heterogeneous media are usually represented as background media with constantoptical properties and some defects in their interior. Perturbation theory is a classicalmethod for finding the solution of a differential equation as the superposition of anunperturbed solution for the background medium plus a small perturbative solutiondue to the presence of small defects. Perturbation theory has become a general andwidely used approach in almost all the fields of physics. In particular, the solutionsof the DE obtained with the Born approximation2 (see Sec. 7.1) of perturbationtheory are largely used in many applications because of their simplicity.

This chapter is dedicated to applying perturbation theory to the DE. Perturbationtheory has been widely used to represent the effects on photon migration of defectsembedded within diffusive homogeneous3–11 and inhomogeneous12–15 media. Thegeneral formalism of perturbation theory applied to the DE was first developedby several authors for time-dependent sources,4, 6 for steady-state sources3, 4 andmodulated sources.3, 5 To overcome the limitations of the Born approximation,several approaches have been proposed. Ostermeyer and Jacques5 suggested thatperturbation theory could be implemented in an iterative manner to calculate thefluence rate at higher orders of approximation than the Born approximation. One ofthe first attempts to implement high-order perturbation theory for diffusive mediawas by Boas.16 In the works of Wassermann10 and Grosenick et al.,11 the authorsfound useful solutions of higher-order perturbation theory in the time domain for theslab geometry, while Sassaroli et al.14, 15 considered the CW domain for layered andslab geometries. Spinelli et al.8 introduced a time-dependent perturbation modelbased on the Padé approximation in order to better exploit the information containedin the time-resolved responses.

131

132 Chapter 7

The study of photon migration in the presence of defects has been also carried outusing different approaches. Gandjbakhche et al.,17 working within the framework ofthe random-walk theory, derived a model that describes the effect of both scatteringand absorption inclusions.

Perturbation theory has also been applied to the more general RTE,18, 19 but thedifficulties while obtaining either a numerical or an analytical solution increase.With the MC method, it is possible to reconstruct solutions of RTE in heterogenousmedia.20–23 The drawbacks of the MC method are long computation time and thecomplex coding challenges when significant heterogeneity exists inside the medium.

Analytical solutions of the DE for heterogeneous media with a single sphericaldefect are achievable with the photon density wave method whenever the boundary isabsent24, 25 or is regular; e.g., the slab25, 26 and the sphere.27 Pustovit et al.28 proposeda semi-analytical forward solver for the diffusion equation that can describe the effectof multiple spherical inhomogeneities inside the half-space or the slab geometry.

Finally, we mention the Rytov method (see Sec. 7.6), which is a widely usedapproximation for studying the perturbative DE.2, 13, 29

In this chapter, solutions of the perturbative DE by the Born approximation arepresented. The main frame of the theory is described and is applied to calculatethe perturbation due to absorbing and scattering inclusions located inside diffusivemedia (Sec. 7.1). Explicit formulas derived from the DE will be provided for theinfinite homogeneous medium (Sec. 7.2) and for the infinite homogeneous slab(Sec. 7.3). In Secs. 7.4 and 7.5, the perturbation approach is also extended to thehybrid models of Chapter 5 and to solutions for the layered slab. Finally, in Sec. 7.6,the formalism of the internal pathlength moments is used to represent absorbingperturbations in scattering media.

In Secs. 7.2 and 7.3, figures with examples of solutions are also reported. All thefigures can be reproduced using the Diffusion&Perturbation code or, alternatively,using the subroutines provided with the enclosed CD-ROM.

7.1 Perturbation Theory in a Diffusive Medium and the BornApproximation

In this chapter, the terms defect, inhomogeneity, and inclusion will be used assynonyms to define a region of the turbid medium where the optical properties arevaried with respect to a background configuration of the medium. The unperturbedheterogeneous background medium is a diffusive medium with absorption coefficientµa(~r) and diffusion coefficient D(~r). In the background medium, a perturbationdue to some absorbing or scattering defects is described by an absorption coefficientµai(~r) = µa(~r) + δµa(~r) or by a diffusion coefficient Di(~r) = D(~r) + δD(~r),respectively. The volume occupied by the defects is denoted as Vi. A source q0(~r, t)is assumed as shown in Fig. 7.1. The perturbed fluence Φpert(~r, t) is represented asthe sum of an unperturbed part Φ0(~r, t), which represents the unperturbed fluence inthe background medium and a perturbation δΦ(~r, t) due to absorbing and scattering

Solutions of the Diffusion Equation with Perturbation Theory 133

Source

Defect

Detector

z

x

y'r

r0r

Source

Defect

Detector

z

x

y'r

r0r

Figure 7.1 Defect inside a diffusive medium.

defects, i.e.,Φpert(~r, t) = Φ0(~r, t) + δΦ(~r, t). (7.1)

Substituting the expression of Φpert(~r, t) in the DE [Eq. (3.34)] we obtain(1v∂∂t −∇ ·

[D(~r) + δD(~r)

]∇

+[µa(~r) + δµa(~r)

])×[Φ0(~r, t) + δΦ(~r, t)

]= q0(~r, t).

(7.2)

Since Φ0(~r, t) is the solution of the unperturbed equation, Eq. (7.2) can be writtenas a DE for the perturbation δΦ(~r, t):

1v∂∂t −∇ ·

[D(~r)∇

]+ µa(~r)

δΦ(~r, t)

=− δµa(~r) +∇ ·

[δD(~r)∇

]Φpert(~r, t).

(7.3)

The equation for δΦ(~r, t) is thus similar to the DE for the unperturbed fluence butwith a virtual source term Svirt, which can be written as

Svirt(~r, t) = −δµa(~r)Φpert(~r, t) +∇ ·[δD(~r)∇

]Φpert(~r, t). (7.4)

With the Green’s function method30, 31 the time-dependent perturbation δΦ(~r, t)can be expressed as

δΦ(~r, t) =∫Vi

d→r′∫ +∞

−∞dt′Svirt(

→r′ , t′)G(~r,

→r′ , t, t′) , (7.5)

where G is the Green’s function of the unperturbed medium.We introduce the approximation for which only small local variations of δµa(~r)

and δD(~r) are considered. Under this approximation, the superposition principlecan be applied assuming that the effects of absorption perturbations are independentof the effects of scattering perturbations. Then, the perturbations can be divided into

134 Chapter 7

two independent terms representing the absorption and the scattering perturbationsδΦa(~r, t) and δΦD(~r, t):

δΦ(~r, t) = δΦa(~r, t) + δΦD(~r, t). (7.6)

The key point for evaluating the perturbed fluence rate in ~r is to provide a goodapproximation for Φpert(~r, t) inside Vi. The result is then given by the superpositionintegral of Eq. (7.5).

The simplest approximation to represent Φpert is the Born approximation thatassumes δΦ Φ0 and thus Φpert(~r, t) ' Φ0(~r, t). With this assumption, usingEq. (7.5) we obtain

δΦ(~r, t) =∫Vid→r′∫ +∞−∞ dt′G(~r,

→r′ , t, t′)∇→

r′·δD(→r′ )[∇→r′

Φ0(→r′ , t′)

]−∫Vid→r′∫ +∞−∞ dt′δµa(

→r′ )Φ0(

→r′ , t′)G(~r,

→r′ , t, t′).

(7.7)Making use of the property a(~r)∇ ·~b = ∇ · a(~r)~b −~b · ∇a(~r), the first term onthe right of Eq. (7.7), which represents δΦD(~r, t), becomes

δΦD(~r, t) =∫Vid→r′∫ +∞−∞ dt′∇→

r′·[G(~r,

→r′ , t, t′)δD(

→r′ )∇→

r′Φ0(→r′ , t′)

]−∫Vid→r′∫ +∞−∞ dt′δD(

→r′ )[∇→r′

Φ0(→r′ , t′)

]·[∇→r′G(~r,

→r′ , t, t′)

].

(7.8)Using the Gauss’s theorem,30 the first integral of Eq. (7.8) can be written as∫

Vid→r′∫ +∞−∞ dt′∇→

r′·[G(~r,

→r′ , t, t′)δD(

→r′ )∇→

r′Φ0(→r′ , t′)

]=∫∂Vi

dΣ′∫ +∞−∞ dt′

[G(~r,

→r′ , t, t′)δD(

→r′ )∇→

r′Φ0(→r′ , t′)

]· n,

(7.9)

where ∂Vi is the external boundary of Vi, and n is the normal to this boundary.Assuming δD(

→r′ )|∂Vi = 0, this term becomes zero, and the perturbations can be

written asΦpert(~r, t) = Φ0(~r, t) + δΦa(~r, t) + δΦD(~r, t), (7.10)

δΦa(~r, t) = −∫Vi

d→r′∫ +∞

−∞dt′δµa(

→r′ )Φ0(

→r′ , t′)G(~r,

→r′ , t, t′) (7.11)

andδΦD(~r, t) = −

∫Vid→r′∫ +∞−∞ dt′δD(

→r′ )

×[∇→r′

Φ0(→r′ , t

′)]·[∇→r′G(~r,

→r′ , t, t′)

].

(7.12)

The expressions of the perturbations given by Eqs. (7.11) and (7.12) can becalculated both for homogeneous and for layered media using for Φ0 and G theGreen’s functions of the DE presented in previous chapters. When the sourceterm is a Dirac delta source, i.e., q0(~r, t) = δ(~r − →r0)δ(t − t0), we have that


Φ0(→r , t) = G(~r,→r0 , t, t0), and thus Eqs. (7.11) and (7.12) provide the Green’sfunctions for the perturbations.

The solution obtained with the Born approximation can be improved by solvingEq. (7.3) with the use of an iterative approach of successive approximations. Thenth–order approximation for the fluence Φpert

n (~r, t) can be obtained after n iterationsas5

Φpertn (~r, t) = Φ0(~r, t) +

∫Vi

d→r′∫ +∞

−∞dt′Svirt

[Φpertn−1(→r′ , t′)

]G(~r,

→r′ , t, t

′) .

(7.13)The first term Φpert

1 of this iterative procedure gives the Born approximation.Making use of Fick’s law, the expressions for the perturbed flux and for the

perturbed radiance inside the diffusive medium can be obtained as

~Jpert(~r, t) = −D∇Φpert(~r, t), (7.14)

andIpert(~r, s, t) =

14π

Φpert(~r, t)− 34πD∇Φpert(~r, t) · s . (7.15)

The perturbation theory presented can be also extended to the case of steady-statesources for which the following relations result:

Φpert(~r) = Φ0(~r) + δΦa(~r) + δΦD(~r), (7.16)

δΦa(~r) = −∫Vi

d→r′ δµa(

→r′ )Φ0(

→r′ )G(~r,

→r′ ), (7.17)

δΦD(~r) = −∫Vi

d→r′ δD(

→r′ )[∇→r′

Φ0(→r′ )]·[∇→r′G(~r,

→r′ )], (7.18)

~Jpert(~r) = −D∇Φpert(~r) (7.19)

andIpert(~r, s) =

14π

Φpert(~r)− 34πD∇Φpert(~r) · s . (7.20)

When absorbing and scattering defects are inside homogeneous media, like theinfinite medium and the homogeneous slab, analytical expressions for the perturbedfluence can be provided. In the next sections, analytical expressions of the Green’sfunctions for the perturbed fluence, reflectance, and transmittance will be presentedfor these geometries.

7.2 Perturbation Theory: Solutions for the Infinite Medium

Using Eqs. (7.11) and (7.12) and the Green’s function of the DE in an infinitehomogeneous medium [Eq. (4.3)], the Green’s functions for the perturbations δΦa

and δΦD can be expressed in analytical form. The notations used in this sectionare those of Fig. 7.2, where a delta source [q0(~r, t) = δ(~r − →r0)δ(t)] is placed

136 Chapter 7

at →r0 = (x0, y0, z0), the inhomogeneity at →r2 = (x2, y2, z2), and the detectorat→r3 = (x3, y3, z3). Using Eq. (7.11) and Eq. (4.3), the time-dependent Green’sfunction for the absorption perturbation results in

δΦa(→r3 , t) = − v2

(4πDv)5/2t3/2exp(−µavt)

∫Vid3→r2δµa(→r2)

(1ρ12

+ 1ρ23

)× exp

[− (ρ12+ρ23)2

4Dvt

],

(7.21)where ρ12 = |→r2 −→r0 | and ρ23 = |→r3 −→r2 |.

The expression for the time-dependent scattering perturbation using Eq. (7.12)and Eq. (4.3) results in

δΦD(→r3 , t) = − 2πv2


∫Vid3→r2δD(→r2)

(∇→r2ρ12 · ∇→r2ρ23

)×[

(ρ12+ρ23)2

ρ12ρ231

2Dvt + ρ312+ρ323ρ212ρ

223

]exp

[− (ρ12+ρ23)2

4Dvt

],

(7.22)with

∇→r2ρ12 ·∇→r2ρ23 = 1ρ12ρ23

(x2 − x0, y2 − y0, z2 − z0) ·(x2 − x3, y2 − y3, z2 − z3).

Source

Inhomogeneity

Detector

z

x

y

),,( 0000 zyxr

),,( 2222 zyxr

),,( 3333 zyxr

Source

Inhomogeneity

Detector

z

x

y

),,( 0000 zyxr

),,( 2222 zyxr

),,( 3333 zyxr

Figure 7.2 Inhomogeneity inside a homogeneous infinite diffusive medium.

For steady-state sources, using Eqs. (7.17), (7.18), and (4.6), the Green’s func-tions of the absorption and of the scattering perturbations for the CW fluence resultsin

δΦa(→r3) = − 1(4πD)2

∫Vid3→r2δµa(→r2) 1

ρ12ρ23exp [−µeff (ρ12 + ρ23)] , (7.23)

δΦD(→r3) = − 1(4πD)2


(∇→r2ρ12 · ∇→r2ρ23

)×1+µeffρ12

ρ212

1+µeffρ23ρ223

exp [−µeff (ρ12 + ρ23)] .(7.24)


The perturbed solutions of Eqs. (7.21), (7.22), (7.23), and (7.24) can be calcu-lated with the following subroutines provided with the enclosed CD-ROM:



7.2.1 Examples of perturbation for the infinite medium

The intent of this section is to provide some hints about the main characteristicsof the effects on photon migration of scattering and absorbing defects inside aninfinite medium. With this purpose, the expressions for the perturbations calculatedin the previous section are plotted and discussed. First, we will analyze the effectsof scattering and absorption perturbations for CW sources since they have a moreintuitive interpretation compared to those for time-dependent sources.

The geometry of the infinite medium is mainly useful for understanding theintrinsic characteristics of photon migration, since in this geometry light propagationonly depends on the optical properties of the medium and of the defect. All theresults pertain to unitary perturbations, i.e., perturbations for which δµaVi = 1 mm2

or δDVi = 1 mm4. Therefore, the perturbations plotted in the figures of this chapteractually represent the Jacobian of the perturbations calculated at the field point. Forsmall values of δµa, δD, and Vi, the perturbation can be considered proportional tothe volume Vi occupied by the defect; therefore, the integration on the volume Vi isactually not needed for a correct estimation of the perturbed signal (see Chapter 10for numerical examples).

The perturbations introduced by scattering or absorbing inclusions show differentfeatures. The differences are particularly marked when the medium is weaklyabsorbing. This is shown by examples in Fig. 7.3 where the shadowgrams forthe CW relative perturbations, δΦD/Φ0 and δΦa/Φ0, due to a unitary scatteringand to a unitary absorption perturbation, are plotted. A medium with µa = 0,µ′s = 1 mm−1, source at (0, 0, 0), receiver at (0, 0, 40) mm, and defect at (x2,

0, 20) mm is considered. The relative perturbations are plotted versus x2. Thefigures show that the scattering perturbation has a strength significantly lower anda shadowgram significantly sharper with respect to the corresponding diagram forthe absorption perturbation. Moreover, the diagram for δΦD shows both positiveand negative values. We point out that the full width half maximum (FWHM) of theshadowgrams represents the minimum distance necessary to distinguish two defectswith a confocal scanning of the source-receiver system. The results of Fig. 7.3 showthe possibility of distinguishing between scattering and absorption perturbationsusing relative CW measurements.

For a non-absorbing medium, as it is clear from Eqs. (7.23), (7.24), and (4.6),δΦD/Φ0 and δΦa/Φ0 result to be exactly proportional to µ

′s.

Figures. 7.4 and 7.5 show the dependence of the CW perturbations on theabsorption of the medium. The geometry and µ

′s are the same as in Fig. 7.3. In

138 Chapter 7

-100 -50 0 50 100-1x10-5

0

1x10-52x10-53x10-54x10-55x10-56x10-57x10-5

Rel

ativ

e S

catte

ring

Per

turb

atio

n

x2 (mm)-100 -50 0 50 100-3x10-2

-2x10-2

-1x10-2

0

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

x2 (mm)

Figure 7.3 Shadowgram for the CW relative perturbations δΦD/Φ0 and δΦa/Φ0 for unitaryinclusions. The figures pertain to an infinite medium with µa = 0, µ

′s = 1 mm−1. The source

is at (0, 0, 0), the receiver is at (0, 0, 40) mm, and the defect is at (x2, 0, 20) mm.

Fig. 7.4, the shadowgrams for the normalized perturbations, i.e., δΦD/|δΦD(x2 =0)| and δΦa/|δΦa(x2 = 0)|, are plotted versus x2. This figure shows the effect ofµa on the shape of the shadowgram: The width of the diagram narrows as absorptionincreases. The reduction of FWHM is stronger for δΦa.

-100 -50 0 50 100-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Nor

mal

ized

Sca

tterin

g P

ertu

rbat

ion

a = 0 mm-1

a = 0.001 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

x2 (mm)-100 -50 0 50 100

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Nor

mal

ized

Abs

orpt

ion

Per

turb

atio

n

a = 0 mm-1

a = 0.001 mm 1

a = 0.01 mm-1

a = 0.05 mm-1

x2 (mm)

Figure 7.4 Dependence of shadowgrams of Fig. 7.3 on µa of the background medium. Thefigure shows the normalized perturbations, i.e., δΦD/|δΦD(x2 = 0)| and δΦa/|δΦa(x2 = 0)|versus x2.

In Fig. 7.5, the contrast (δΦD/Φ0)|(x2=0) and (δΦa/Φ0)|(x2=0) is plotted versusthe absorption coefficient of the medium. When absorption increases, there is asignificant enhancement of the contrast for the scattering perturbation, and thecontrast increases almost linearly with µa. This is because a scattering inclusionplaced on the beam axis mainly perturbs early received photons,7 which are the leastattenuated when absorption is increased. The figure also shows that the contrast


0.00 0.02 0.04 0.06 0.08 0.100

3x10-3

6x10-3

9x10-3

Con

trast

a

Scattering Perturbation

0.00 0.02 0.04 0.06 0.08 0.10-3x10-2

-2x10-2

-1x10-2

0

Con

trast

a

Absorption Perturbation

0.00 0.02 0.04 0.06 0.08 0.100

5

10

15

20

FWH

M (m

m)

a


0.00 0.02 0.04 0.06 0.08 0.100

10

20

30

40FW

HM

(mm

)

a


Figure 7.5 Dependence of scattering and absorption perturbations on µa for a medium withµ

′s = 1 mm−1, source at (0, 0, 0), defect at (x2, 0, 20) mm, and receiver at (0, 0, 0) mm. The

contrast, i.e., (δΦD/Φ0)|(x2=0) and (δΦa/Φ0)|(x2=0) and the FWHM of the shadowgrams forthe scattering and absorption perturbations are plotted versus µa.

for the absorption perturbation remains constant when absorption increases asexpected from Eqs. (7.23) and (4.6). Figure 7.5 also shows how the FWHM of theshadowgram depends on µa. The absorption perturbation shows a stronger decreaseof the FWHM versus µa compared with that of the scattering perturbation. We stressthat for µa = 0 the FWHM of the absorbing defect is 40 mm, i.e., exactly equal tothe source-receiver distance. Differences in the FWHM of scattering and absorptionperturbations are larger for the lower values of µa.

Also for time-dependent sources, the perturbation due to scattering inclusions showssignificantly different features with respect to perturbations due to absorbing inclu-sions. In Fig. 7.6, the TR perturbation is shown for an infinite medium with µa = 0,µ′s = 1 mm−1, and refractive index n = 1.4. The source is at (0, 0, 0), the receiver

140 Chapter 7

0 2500 5000 7500 10000-6x10-9

-3x10-9

0

3x10-9

6x10-9

Sca

tterin

g an

d A

bsor

ptio

n P

ertu

rbat

ion

(mm

-2ps

-1)

Time (ps)

Scattering Perturbation x 50 Absorption Perturbation

-3x10-7

-2x10-7

-1x10-7

0

1x10-7

2x10-7

3x10-7

Fluence

Flue

nce

(mm

-2ps

-1)

0 2500 5000 7500 10000-3x10-3

-2x10-3

-1x10-3

0

1x10-3

2x10-3

3x10-3

4x10-3

Rel

ativ

e S

catte

ring

Per

turb

atio

n

Time (ps)

x2 = 0 mm x2 = 10 mm x2 = 20 mm x2 = 30 mm

0 2500 5000 7500 10000-3x10-2

-2x10-2

-1x10-2

0

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

Time (ps)

x2 = 0 mm x2 = 10 mm x2 = 20 mm x2 = 30 mm

Figure 7.6 Time-dependent perturbations for an infinite medium with µa = 0, µ′s = 1 mm−1,

n = 1.4. The source is at (0, 0, 0), the receiver is at (0, 0, 40) mm, and the defect is at (0,0, 20) mm. The absolute perturbation for x2 = 0 is shown together with the unperturbedfluence; the relative perturbations are shown for different values of x2.

is at (0, 0, 40) mm, and the defect is at (x2, 0, 20) mm. The absolute scatteringperturbation [Eq. (7.21)] and the absolute absorption perturbation [Eq. (7.22)] for adefect at x2 = 0 are plotted versus time together with the unperturbed fluence. Inorder to represent δΦD and δΦa in the same scale, δΦD has been multiplied by ascaling factor of 50. Moreover, the relative scattering and absorption perturbationsδΦD/Φ0 and δΦa/Φ0 are plotted for several values of x2. These figures show thatthe scattering and absorption perturbations have significantly different shape andintensity. We observe that for x2 = 20 mm, δΦD/Φ0 = 0 for every value of time.It is also important to notice that at x2 = 0 the relative absorption perturbationδΦa/Φ0 is independent of t. This behavior also explains why the CW contrastof the absorption perturbation of Fig. 7.5 is independent of µa. Finally, a generalproperty of the time-dependent relative perturbations is that they are independent ofthe absorption coefficient of the medium.


7.3 Perturbation Theory: Solutions for the Slab

Analytical expressions for the Green’s functions of the perturbed fluence can alsobe obtained for the homogeneous slab (see Fig. 7.7). A delta source [q0(~r, t) =δ(~r −→r0)δ(t)] is at→r0 = (x0, y0, z0), the inhomogeneity is at→r2 = (x2, y2, z2),and the detector is at→r3 = (x3, y3, z3). From Eqs. (7.11), (7.12), and (4.16), theGreen’s functions for δΦa(~r3, t) and δΦD(→r3 , t) results in

δΦa(→r3 , t) = − v2


m=+∞∑m=−∞

η=+∞∑η=−∞


×

(1

ρ+12m+ 1

ρ+23η

)exp

[− (ρ+12m+ρ+23η)

2

4Dvt

]+(

1ρ+12m

+ 1ρ−23η

)exp

[− (ρ+12m+ρ−23η)

2

4Dvt

]−(

1ρ−12m

+ 1ρ+23η

)exp

[− (ρ−12m+ρ+23η)

2

4Dvt

]−(

1ρ−12m

+ 1ρ−23η

)exp

[− (ρ−12m+ρ−23η)

2

4Dvt

](7.25)

and

δΦD(→r3 , t) = − 2πv2


m=+∞∑m=−∞

η=+∞∑η=−∞

∫Vid3~r2δD(→r2)

×

(∇→r2ρ

+12m · ∇→r2ρ

+23η

)[(ρ+12m+ρ+23η)3

ρ+12mρ+23η

12Dvt +

ρ+312m+ρ+3

23η

ρ+212m+ρ+2

23η

]× exp

[−(ρ+12m+ρ+23η)

2

4Dvt

]+(∇→r2ρ

+12m · ∇→r2ρ

−23η

)×[

(ρ+12m+ρ−23η)3

ρ+12mρ−23η

12Dvt +

ρ+312m+ρ−3

23η

ρ+212m+ρ−2

23η

]exp

[−(ρ+12m+ρ−23η)

2

4Dvt

]−(∇→r2ρ

−12m · ∇→r2ρ

+23η

)[(ρ−12m+ρ+23η)3

ρ−12mρ+23η

12Dvt +

ρ−312m+ρ+3

23η

ρ−212m+ρ+2

23η

]× exp

[−(ρ−12m+ρ+23η)

2

4Dvt

]−(∇→r2ρ

−12m · ∇→r2ρ

−23η

)×[

(ρ−12m+ρ−23η)3

ρ−12mρ−23η

12Dvt +

ρ−312m+ρ−3

23η

ρ−212m+ρ−2

23η

]exp

[−(ρ−12m+ρ−23η)

2

4Dvt

],

(7.26)

142 Chapter 7

source

inhomogeneity

detector

z

x

y),,( 0000 zyxr

),,( 3333 zyxr

R (x3, y3, z3 =0, t)

T (x3, y3, z3=s, t)detector

s

3r

),,( 2222 zyxr

source

inhomogeneity

detector

z

x

y),,( 0000 zyxr

),,( 3333 zyxr

R (x3, y3, z3 =0, t)

T (x3, y3, z3=s, t)detector

s

3r

),,( 2222 zyxr

Figure 7.7 Inhomogeneity inside a homogeneous infinite diffusive slab.

where

z′+12m = 2m(s+ 2ze) + z0;

−−→r+

0m ≡ (x0, y0, z′+12m)

z′−12m = 2m(s+ 2ze)− 2ze − z0;

−−→r−0m ≡ (x0, y0, z

′−12m)

z′+23η = 2η(s+ 2ze) + z2;

−→r+

2η ≡ (x2, y2, z′+23η)

z′−23η = 2η(s+ 2ze)− 2ze − z2;

−→r−2η ≡ (x2, y2, z

′−23η)

ρ+12m =

∣∣∣→r2 −−−→r+

0m

∣∣∣ =

√(x2 − x0)2 + (y2 − y0)2 +

(z2 − z

′+12m

)2

ρ−12m =∣∣∣→r2 −

−−→r−0m

∣∣∣ =

√(x2 − x0)2 + (y2 − y0)2 +

(z2 − z

′−12m

)2

ρ+23η =

∣∣∣→r3 −−→r+

2η

∣∣∣ =

√(x3 − x2)2 + (y3 − y2)2 +

(z3 − z

′+23η

)2

ρ−23η =∣∣∣→r3 −

−→r−2η

∣∣∣ =

√(x3 − x2)2 + (y2 − y2)2 +

(z3 − z

′+23η

)2

∇→r2ρ+12m =

→r2−−→r+0η

ρ+12m= 1

ρ+12m

(x2 − x0, y2 − y0, z2 − z

′+12m

)∇→r2ρ

−12m =

→r2−−→r−0η

ρ−12m= 1

ρ−12m

(x2 − x0, y2 − y0, z2 − z

′−12m

)∇→r2ρ

+23η =

→r2−−→r+2η

ρ+23η= 1

ρ+23η

(x2 − x3, y2 − y3, z2 − z

′+23η

)∇→r2ρ

−23η =

→r2−−→r−2η

ρ−23η= 1

ρ−23η

(x2 − x3, y2 − y3, z2 − z

′−23η

).

(7.27)

For steady-state sources, using Eq. (7.17), Eq. (7.18), and Eq. (4.22), the Green’s


functions for the absorption and the scattering perturbations for the fluence are

δΦa(→r3) = − 1(4πD)2

m=+∞∑m=−∞

η=+∞∑η=−∞

∫Vid3~r2δµa(→r2)

×

1

ρ+12mρ+23η

exp[− µeff

(ρ+

12m + ρ+23η

) ]+ 1ρ+12mρ

−23η

exp[− µeff

(ρ+

12m + ρ−23η

) ]− 1ρ−12mρ

+23η

exp[− µeff

(ρ−12m + ρ+

23η

) ]− 1ρ−12mρ

−23η

exp[− µeff

(ρ−12m + ρ−23η

) ](7.28)

and

δΦD(→r3) = − 1(4πD)2

m=+∞∑m=−∞

η=+∞∑η=−∞


×

(∇→r2ρ

+12m · ∇→r2ρ

+23η

)1+µeffρ

+12m

(ρ+12m)2

1+µeffρ+23η

(ρ+23η)2exp

[− µeff

(ρ+

12m + ρ+23η

) ]+(∇→r2ρ

+12m · ∇→r2ρ

−23η

)1+µeffρ

+12m

(ρ+12m)2

1+µeffρ−23η

(ρ−23η)2exp

[− µeff

(ρ+

12m + ρ−23η

) ]−(∇→r2ρ

−12m · ∇→r2ρ

+23η

)1+µeffρ

−12m

(ρ−12m)2

1+µeffρ+23η

(ρ+23η)2exp

[− µeff

(ρ−12m + ρ+

23η

) ]−(∇→r2ρ

−12m · ∇→r2ρ

−23η

)1+µeffρ

−12m

(ρ−12m)2

1+µeffρ−23η

(ρ−23η)2exp

[− µeff

(ρ−12m + ρ−23η

) ].

(7.29)According to Sec. 4.4, the perturbed reflectance Rpert(ρ, t) and the perturbed

transmittance T pert(ρ, t) can be evaluated using either Fick’s law or the EBPChybrid approach based both on the EBC and on the PCBC. First, we present thesolutions obtained using Fick’s law. Following the procedure used in Refs. 4, 6, and7, Rpert and T pert are calculated as

Rpert(ρ, t) = R0(ρ, t) + δRa(ρ, t) + δRD(ρ, t)

= D ∂∂z

[Φ0(ρ, z = 0, t) + δΦa(ρ, z = 0, t) + δΦD(ρ, z = 0, t)

],

(7.30)

T pert(ρ, t) = T 0(ρ, t) + δT a(ρ, t) + δTD(ρ, t)

= −D ∂∂z

[Φ0(ρ, z = s, t) + δΦa(ρ, z = s, t) + δΦD(ρ, z = s, t)

].

(7.31)

Therefore, the expressions for the perturbations are obtained as

δRa(ρ, t) = D∂

∂zδΦa(ρ, z = 0, t) , (7.32)

δRD(ρ, t) = D∂

∂zδΦD(ρ, z = 0, t) , (7.33)

144 Chapter 7

δT a(ρ, t) = −D ∂

∂zδΦa(ρ, z = s, t), (7.34)

and

δTD(ρ, t) = −D ∂

∂zδΦD(ρ, z = s, t) . (7.35)

Taking the derivatives in Eqs. (7.32)–(7.35), the perturbations for the time-dependent reflectance and transmittance, result in

δRa(ρ, t) = − v

4π(4πDvt)3/2exp(−µavt)

m=+∞∑m=−∞

η=+∞∑η=−∞


×

z+′23η

ρ+′23η

[f(ρ+

12m, ρ+′23η, t)− f(ρ−12m, ρ

+′23η, t)

]− z−′23η

ρ−′23η

[f(ρ+

12m, ρ−′23η, t)− f(ρ−12m, ρ

−′23η, t)

],

(7.36)

δRD(ρ, t) = − v

2(4πDvt)5/2exp(−µavt)

m=+∞∑m=−∞

η=+∞∑η=−∞


×

z+′

23η

[(∇→r2ρ

+12m · ∇→r2ρ

+′23η

)q(ρ+

12m, ρ+′23η, t)

−(∇→r2ρ

−12m · ∇→r2ρ

+′23η

)q(ρ−12m, ρ

+′23η, t)

]−z−′23η

[(∇→r2ρ

+12m · ∇→r2ρ

−′23η

)q(ρ+

12m, ρ−′23η, t)

−(∇→r2ρ

−12m · ∇→r2ρ

−′23η

)q(ρ−12m, ρ

−′23η, t)

]−z+

12m

[p(ρ+

12m, ρ+′23η, t) + p(ρ+

12m, ρ−′23η, t)

]+z−12m

[p(ρ−12m, ρ

+′23η, t) + p(ρ−12m, ρ

−′23η, t)

],

(7.37)

δT a(ρ, t) = − v

4π(4πDvt)3/2exp(−µavt)

m=+∞∑m=−∞

η=+∞∑η=−∞


×

z+23ηρ+23η

[f(ρ+

12m, ρ+23η, t)− f(ρ−12m, ρ

+23η, t)

]− z−23ηρ−23η

[f(ρ+

12m, ρ−23η, t)− f(ρ−12m, ρ

−23η, t)

] (7.38)


and

δTD(ρ, t) = − v

2(4πDvt)5/2exp(−µavt)

m=+∞∑m=−∞

η=+∞∑η=−∞


×

z+

23η

[(∇→r2ρ

+12m · ∇→r2ρ

+23η

)q(ρ+

12m, ρ+23η, t)

−(∇→r2ρ

−12m · ∇→r2ρ

+23η

)q(ρ−12m, ρ

+23η, t)

]−z−23η

[(∇→r2ρ

+12m · ∇→r2ρ

−23η

)q(ρ+

12m, ρ−23η, t)

−(∇→r2ρ

−12m · ∇→r2ρ

−23η

)q(ρ−12m, ρ

−23η, t)

]+z+

12m

[p(ρ+

12m, ρ+23η, t) + p(ρ+

12m, ρ−23η, t)

]−z−12m

[p(ρ−12m, ρ

+23η, t) + p(ρ−12m, ρ

−23η, t)

],

(7.39)

where

z+12m = z2 − 2m(s+ 2ze)− z0, z−12m = z2 − 2m(s+ 2ze) + 2ze + z0

z+23η = (1− 2η)s− 4ηze − z2, z−23η = (1− 2η)s− (4η − 2)ze + z2

z+′23η = 2ηs+ 4ηze + z2, z−′23η = 2ηs+ (4η − 2)ze − z2

ρ+12m =

√(x2 − x0)2 + (y2 − y0)2 +

(z+

12m

)2ρ−12m =

√(x2 − x0)2 + (y2 − y0)2 +

(z−12m

)2ρ+

23η =

√(x3 − x2)2 + (y3 − y2)2 +

(z+

23η

)2

ρ−23η =

√(x3 − x2)2 + (y3 − y2)2 +

(z−23η

)2

ρ+′23η =

√(x3 − x2)2 + (y3 − y2)2 +

(z+′

23η

)2

ρ−′23η =

√(x3 − x2)2 + (y3 − y2)2 +

(z−′23η

)2

∇→r2ρ+12m = 1

ρ+12m

(x2 − x0, y2 − y0, z

+12m

)∇→r2ρ

−12m = 1

ρ−12m

(x2 − x0, y2 − y0, z

−12m

)∇→r2ρ

+23η = 1

ρ+23m

(x2 − x3, y2 − y3, − z+

23η

)∇→r2ρ

−23η = 1

ρ−23m

(x2 − x3, y2 − y3, z

−23η

)∇→r2ρ

+′23η = 1

ρ+′23m

(x2 − x3, y2 − y3, z

+′23η

)∇→r2ρ

−′23η = 1

ρ−′23m

(x2 − x3, y2 − y3, − z−′23η

)ρ =

√(x3 − x0)2 + (y3 − y0)2, ze = 2AD,

(7.40)

146 Chapter 7

f(x, y, t) =

[1y2

+(x+ y)2

xy

12Dvt

]exp

[−(x+ y)2

4Dvt

], (7.41)

p(x, y, t) =

[(1x

+1y

)3 xy

2Dvt+(

1x3

+1y3

)]exp

[−(x+ y)2

4Dvt

](7.42)

and

q(x, y, t) =[

3xy4

+ (x+y)2(3x2−2xy+y2)x2y3

12Dvt + (x+y)4

xy21

(2Dvt)2

]× exp

[− (x+y)2

4Dvt

].

(7.43)

For steady-state sources, the perturbations for reflectance and transmittance are

δRa(ρ) = − 1(4π)2D

m=+∞∑m=−∞

η=+∞∑η=−∞


×

(z+′

23η

1+µeffρ+′23η

(ρ+′23η)3

exp[−µeff (ρ+12m+ρ+′23η)]

ρ+12m− exp[−µeff (ρ−12m+ρ+′23η)]

ρ−12m

−z−′23η

1+µeffρ−′23η

(ρ−′23η)3

exp[−µeff (ρ+12m+ρ−′23η)]

ρ+12m− exp[−µeff (ρ−12m+ρ−′23η)]

ρ−12m

),

(7.44)

δRD(ρ) = − 1(4π)2D

m=+∞∑m=−∞

η=+∞∑η=−∞


×z+

12m

[h(ρ+

12m, ρ+′23η, µeff ) + h(ρ+

12m, ρ−′23η, µeff )

]−z−12m

[h(ρ−12m, ρ

+′23η, µeff ) + h(ρ−12m, ρ

−′23η, µeff )

]+z+′

23m

[(∇→r2ρ

+12m · ∇→r2ρ

+′23η

)w(ρ+

12m, ρ+′23η, µeff )

−(∇→r2ρ

−12m · ∇→r2ρ

+′23η

)w(ρ−12m, ρ

+′23η, µeff )

]−z−′23m

[(∇→r2ρ

+12m · ∇→r2ρ

−′23η

)w(ρ+

12m, ρ−′23η, µeff )

−(∇→r2ρ

−12m · ∇→r2ρ

−′23η

)w(ρ−12m, ρ

−′23η, µeff )

],

(7.45)

δT a(ρ) = − 1(4π)2D

m=+∞∑m=−∞

η=+∞∑η=−∞


×

(z+

23η

1+µeffρ+23η

(ρ+23η)3

exp[−µeff (ρ+12m+ρ+23η)]

ρ+12m− exp[−µeff (ρ−12m+ρ+23η)]

ρ−12m

−z−23η

1+µeffρ−23η

(ρ−23η)3

exp[−µeff (ρ+12m+ρ−23η)]

ρ+12m− exp[−µeff (ρ−12m+ρ−23η)]

ρ−12m

) (7.46)


and

δTD(ρ) = − 1(4π)2D

m=+∞∑m=−∞

η=+∞∑η=−∞


×z+

12m

[h(ρ+

12m, ρ+23η, µeff ) + h(ρ+

12m, ρ−23η, µeff )

]−z−12m

[h(ρ−12m, ρ

+23η, µeff ) + h(ρ−12m, ρ

−23η, µeff )

]+z+

23m

[(∇→r2ρ

+12m · ∇→r2ρ

+23η

)w(ρ+

12m, ρ+23η, µeff )

−(∇→r2ρ

−12m · ∇→r2ρ

+23η

)w(ρ−12m, ρ

+23η, µeff )

]−z−23m

[(∇2ρ

+12m · ∇→r2ρ

−23η

)w(ρ+

12m, ρ−23η, µeff )

−(∇→r2ρ

−12m · ∇→r2ρ

−23η

)w(ρ−12m, ρ

−23η, µeff )

],

(7.47)

where

w(x, y, µeff ) =1 + µeffx

x2

3 + 3µeffy + µ2effy

2

y4exp [−µeff (x+ y)] , (7.48)

h(x, y, µeff ) =1 + µeffx

x3

1 + µeffy

y3exp [−µeff (x+ y)] , (7.49)

and µeff is given by Eq. (4.9).Following the hybrid EBPC approach, the Green’s functions for the perturbed

reflectance and transmittance are much more easily obtained. The TR perturbationsare simply given by

δRa(ρ, t) =δΦa(ρ, z = 0, t)

2A, (7.50)

δRD(ρ, t) =δΦD(ρ, z = 0, t)

2A, (7.51)

δT a(ρ, t) =δΦa(ρ, z = s, t)

2A, (7.52)

δTD(ρ, t) =δΦD(ρ, z = s, t)

2A, (7.53)

and the CW perturbations are given by

δRa(ρ) =δΦa(ρ, z = 0)

2A, (7.54)

δRD(ρ) =δΦD(ρ, z = 0)

2A, (7.55)

δT a(ρ) =δΦa(ρ, z = s)

2A, (7.56)

δTD(ρ) =δΦD(ρ, z = s)

2A, (7.57)

148 Chapter 7

with δΦa and δΦD given by Eqs. (7.25) and (7.26) for the time domain andEqs. (7.28) and (7.29) for the CW domain.

The solutions of Eqs. (7.25), (7.26), (7.28), (7.29), (7.36)–(7.39), and (7.44)–(7.47)can be calculated with the following subroutines provided on the enclosed CD-ROM:

• DE_Fluence_Perturbation_Slab_TR Eqs. (7.25), (7.26), (7.50)–(7.53)

• DE_Fluence_Perturbation_Slab_CW Eqs. (7.28), (7.29), (7.54)–(7.57)

• DE_Reflectance_Perturbation_Slab_TR Eqs. (7.36) and (7.37)

• DE_Transmittance_Perturbation_Slab_TR Eqs. (7.38) and (7.39)

• DE_Reflectance_Perturbation_Slab_CW Eqs. (7.44) and (7.45)

• DE_Transmittance_Perturbation_Slab_CW Eqs. (7.46) and (7.47)

It is worth stressing that the subroutines DE_Fluence_Perturbation_Slab_TR andDE_Fluence_Perturbation_Slab_CW can be also used to obtain the hybrid solutionsof Eqs. (7.50)–(7.57).

7.3.1 Examples of perturbation for the slab

To a certain extent, the effects of scattering and absorbing inclusions inside ahomogeneous slab are similar to those observed in Sec. 7.2.1 for the infinite medium.The boundary effects of the slab modify some aspects of photon migration, but themain characteristics observed for the infinite medium are still valid.

The interest in the slab geometry is motivated by several applications in the fieldof tissue optics.32–35 In particular, this geometry is widely used for transmittancemeasurements in optical mammography36–41 and for reflectance measurements inbrain functional imaging,42–46 where a slab with a large thickness is used to simulatea semi-infinite medium.

Even for the slab geometry, all the results presented pertain to unitary perturba-tions, i.e., perturbations for which δµaVi = 1 mm2 or δDVi = 1 mm4. Perturbationshave been calculated with expressions obtained using Fick’s law. In Fig. 7.8, theshadowgram for the CW relative perturbations for the transmittance due to unitaryscattering and absorption perturbations inside a slab is shown. It is considered aslab with µa = 0, µ

′s = 1 mm−1, s = 40 mm, and a relative refractive index with

the external nr = 1. The source is placed at (0, 0, 1) mm, the receiver is at (0, 0,40) mm, and the defect is at (x2, 0, 20) mm. The relative perturbations δTD/T 0

and δT a/T 0 are plotted versus x2. There is an evident similarity of this figure withFig. 7.3 obtained for the infinite medium. For the slab, you will notice a lowerFWHM of the shadowgram for the absorption perturbation (about one half of thatfor the infinite medium) and for the scattering perturbation. The reason for thisbehavior is related to the boundary effect that restricts the volume of the medium


-40 -20 0 20 40-1x10-4

0

1x10-4

2x10-4

3x10-4

4x10-4

5x10-4

Rel

ativ

e S

catte

ring

Per

turb

atio

n

x2 (mm)-40 -20 0 20 40-4x10-2

-3x10-2

-2x10-2

-1x10-2

0

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

x2 (mm)

Figure 7.8 Shadowgram for the CW relative perturbations δTD/T 0 and δT a/T 0 for unitaryinclusions. The figures pertain to a slab with s = 40 mm, µa = 0, µ

′s = 1 mm−1, and nr =

1. The source is placed at (0, 0, 1) mm, the receiver is at (0, 0, 40) mm, and the defect is at(x2, 0, 20) mm.

where photon migration occurs. For the scattering perturbation, δTD/T 0 = 0 forx2 = 13 mm, while for the infinite medium δΦD/Φ0 = 0 for x2 = 20 mm.

In Fig. 7.9 and Fig. 7.10, the dependence on absorption of the CW perturbationfor the transmittance of the same slab of Fig. 7.8 is shown. In Fig. 7.9, the normalizedperturbations, i.e., δTD/|δTD(x2 = 0)| and δT a/|δT a(x2 = 0)|, are plotted versusx2. The shape of these diagrams is similar to those for the infinite medium ofFig. 7.4. In Fig. 7.10, the contrast, i.e., (δTD/T 0)|(x2=0) and (δT a/T 0)|(x2=0),

-40 -20 0 20 40-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Nor

mal

ized

Sca

tterin

g P

ertu

rbat

ion

a = 0 mm-1

a = 0.001 mm-1

a = 0.01 mm-1

a = 0.025 mm-1

a = 0.05 mm-1

x2 (mm)-40 -20 0 20 40

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Nor

mal

ized

Abs

orpt

ion

Per

turb

atio

n

a = 0 mm-1

a = 0.001 mm-1

a = 0.01 mm-1

a = 0.025 mm-1

a = 0.05 mm-1

x2 (mm)

Figure 7.9 Dependence of scattering and of absorption CW perturbations for transmittanceon µa for a slab with s = 40 mm, µ

′s = 1 mm−1, and nr = 1. The source at (0, 0, 1) mm,

defect at (x2, 0, 20) mm and receiver at (0, 0, 40) mm. The figure shows the normalizedperturbations, i.e., δTD/|δTD(x2 = 0)| and δT a/|δT a(x2 = 0)| versus x2.

150 Chapter 7

0.00 0.02 0.04 0.06 0.08 0.100

2x10-3

4x10-3

6x10-3

8x10-3

1x10-2

Con

trast

a


0.00 0.02 0.04 0.06 0.08 0.10-4x10-2

-3x10-2

-2x10-2

-1x10-2

0

Con

trast

a


0.00 0.02 0.04 0.06 0.08 0.100

5

10

15

FWH

M (m

m)

a


0.00 0.02 0.04 0.06 0.08 0.100

5

10

15

20

25FW

HM

(mm

)

a


Figure 7.10 Dependence of scattering and absorption CW perturbations for transmittanceon µa for a slab with µ

′s = 1 mm−1, and nr = 1. The source is at (0, 0, 1) mm, the

defect is at (0, 0, 20) mm, and the receiver is at (0, 0, 40) mm. The contrast for scatteringand absorption perturbations [(δΦD/Φ0)|(x2=0) and (δΦa/Φ0)|(x2=0), respectively] and theFWHM of the shadow diagrams are plotted versus µa.

is plotted versus µa. For (δTD/T 0)|(x2=0), the behavior is similar to the infinitemedium (Fig. 7.5), while for (δT a/T 0)|(x2=0) there is a weak dependence on µa[for the infinite medium it was (δΦa/Φ0)|(x2=0) = constant]. The figure also showsthe FWHM of the shadowgrams for the scattering and the absorption perturbationsversus µa. We have a decrement of the FWHM as µa increases, but this effect isweaker compared with that of the infinite medium. This can be explained with theshorter mean pathlength followed by received photons inside the slab comparedwith the infinite geometry.

In Fig. 7.11, the TR perturbations for the transmittance are shown for a slabwith s = 40 mm, µa = 0, µ

′s = 1 mm−1, absolute refractive ni = 1.4, and nr = 1.

The source is at (0, 0, 1) mm, the receiver is at (0, 0, 40) mm, and the defect is at


0 500 1000 1500 2000 2500 3000 3500-2x10-10

-1x10-10

0

1x10-10

2x10-10

Sca

tterin

g an

d A

bsor

ptio

n P

ertu

rbat

ion

(mm

-2ps

-1)

Time (ps)


-6x10-9

-3x10-9

0

3x10-9

6x10-9

Transmittance

Tran

smitt

ance

(mm

-2ps

-1)

0 500 1000 1500 2000 2500 3000 3500-6x10-2

-4x10-2

-2x10-2

0

2x10-2

4x10-2

6x10-2

8x10-2

Rel

ativ

e S

catte

ring

Per

turb

atio

n

Time (ps)

x2 = 0 mm x2 = 5 mm x2 = 13 mm x2 = 20 mm

0 500 1000 1500 2000 2500 3000 3500-5x10-2

-4x10-2

-3x10-2

-2x10-2

-1x10-2

0

1x10-2

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

Time (ps)

x2 = 0 mm x2 = 5 mm x2 = 13 mm x2 = 20 mm

Figure 7.11 TR perturbations for the transmittance of a slab with s = 40 mm, µa = 0, µ′s =

1 mm−1, ni = 1.4, and nr = 1. The source is at (0, 0, 1) mm, the receiver is at (0, 0, 40)mm, and the defect is at (x2, 0, 20) mm. The first figure shows the absolute perturbationδTD multiplied for 30 and δT a for a defect at (0, 0, 20) mm together with the unperturbedtransmittance. The other figures show the relative perturbations δTD/T 0 (for x2 = 5, 13, 20multiplied by 10, 1000, and 10000, respectively) and δT a/T 0 for several values of x2.

(x2, 0, 20) mm. The absolute scattering perturbation for transmittance [Eq.(7.39)]multiplied by 30 and the absolute absorption perturbation [Eq.(7.38)] due to a defectat (0, 0, 20) mm are shown together with the unperturbed transmittance. Moreover,the relative TR scattering and absorption perturbations (δTD/T 0 and δT a/T 0,respectively) are shown for several values of x2. Also for the slab, the figures showthat the scattering and absorption perturbations have a different shape and intensity.The scattering perturbations for x2 = 5, 13, and 20 mm have been multiplied bya scaling factor of 10, 1000, and 10000, respectively, in order to represent themon the same scale. At x2 = 0, the relative absorption perturbation δT a/T 0 hasa weak dependence on t, while for the infinite medium δΦa/Φ0 was independentof t. At x2 = 13 mm (where the CW perturbation δTD is zero), δTD/T 0 showsboth positive and negative values versus time. Finally, the time-dependent relativeperturbations for the slab are also independent of the absorption coefficient of themedium.

152 Chapter 7

-40 -20 0 20 40 60 80-2x10-4

0

2x10-4

4x10-4

6x10-4

8x10-4

1x10-3 z2 = 10 mm z2 = 20 mm z2 = 30 mm

Rel

ativ

e S

catte

ring

Per

turb

atio

n

x2 (mm)

a = 0 mm-1

-40 -20 0 20 40 60 80-5x10-2

-4x10-2

-3x10-2

-2x10-2

-1x10-2

0

z2 = 10 mm z2 = 20 mm z2 = 30 mm

a = 0 mm-1

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

x2 (mm)

-40 -20 0 20 40 60 80

0

1x10-3

2x10-3 z2 = 10 mm z2 = 20 mm z2 = 30 mm

Rel

ativ

e S

catte

ring

Per

turb

atio

n

x2 (mm)

a = 0.01 mm-1

-40 -20 0 20 40 60 80-4x10-2

-3x10-2

-2x10-2

-1x10-2

0

z2 = 10 mm z2 = 20 mm z2 = 30 mma = 0.01 mm-1

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

x2 (mm)

Figure 7.12 Shadowgram for the CW relative perturbations δRD/R0 and δRa/R0 for unitaryinclusions. The figures pertain to a semi-infinite medium with µ

′s = 1 mm−1. The source is

placed at (0, 0, 1) mm, the receiver is at (40, 0, 0) mm, and the defect is at (x2, 0, z2) mm.The figures are plotted for a defect at z2 = 10, 20, 30 mm and for µa = 0 and 0.01 mm−1.

Figure 7.12 shows the shadowgrams for the CW relative perturbations for thereflectance from a semi-infinite medium with µ

′s = 1 mm−1 for several depths of

the defect. The source is placed at (0, 0, 1) mm, the receiver is at (40, 0, 0) mm, andthe defect is at (x2, 0, z2) mm. The relative perturbations δRD/R0 and δRa/R0

are plotted versus x2 for µa = 0 and µa = 0.01 mm−1. The effect of absorption isto reduce the FWHM of the shadowgrams. For shallow depths, the contrast of thescattering perturbation increases, while weaker changes are found for the absorptionperturbation. It has been verified that the shape of the shadowgrams are not stronglyaffected by the µ

′s of the medium, while the contrast is roughly proportional to µ

′s

for the scattering and the absorption perturbation.In Fig. 7.13, the TR perturbations for the reflectance are shown for a semi-infinite

medium with µa = 0, µ′s = 1 mm−1, ni = 1.4, and nr = 1. The source is at (0,


0 500 1000 1500 2000 2500 3000 3500-2x10-11

-1x10-11

0

1x10-11

2x10-11

Sca

tterin

g an

d A

bsor

ptio

n P

ertu

rbat

ion

(mm

-2ps

-1)

Time (ps)


-8x10-10

-4x10-10

0

4x10-10

8x10-10

Reflectance

Ref

lect

ance

(mm

-2ps

-1)

0 500 1000 1500 2000 2500 3000 3500-4x10-2

-2x10-2

0

2x10-2

4x10-2

6x10-2

Rel

ativ

e S

catte

ring

Per

turb

atio

n

Time (ps)

z2 = 5 mm z2 = 10 mm z2 = 20 mm z2 = 30 mm

0 500 1000 1500 2000 2500 3000 3500-4x10-2

-3x10-2

-2x10-2

-1x10-2

0

1x10-2

Rel

ativ

e A

bsor

ptio

n P

ertu

rbat

ion

Time (ps)

z2 = 5 mm z2 = 10 mm z2 = 20 mm z2 = 30 mm

Figure 7.13 Time-dependent perturbations for the reflectance of a semi-infinite medium withµa = 0, µ

′s = 1 mm−1, ni = 1.4, and nr = 1. The source is at (0, 0, 1) mm, the receiver

is at (40, 0, 0) mm, and the defect is at (20, 0, z2) mm. The first figure shows the absoluteperturbation δRD multiplied by 100 and δRa for a defect at (20, 0, 20) mm together with theunperturbed reflectance. The other figures show the relative perturbations for x2 = 20 mm,y2 = 0, and z2 = 5, 10, 20, and 30 mm. δRD/δRD for z2 = 10, 20, and 30 mm has beenmultiplied by 15, 300, and 3000, respectively.

0, 1) mm, the receiver is at (40, 0, 0) mm, and defect is at (20, 0, z2) mm. Theabsolute scattering perturbation for reflectance [Eq. (7.37)] multiplied by 100 andthe absolute absorption perturbation [Eq. (7.36)] for a defect at (20, 0, 20) mm areplotted together with the unperturbed reflectance. The figure also shows the relativescattering and absorption perturbation (δRD/R0 and δRa/R0, respectively) forseveral values of z2 for x2 = 20 mm and y2 = 0. To represent the perturbations onthe same scale, the scattering perturbations for z2 = 5, 10, 20, and 30 mm have beenmultiplied by the scaling factors 1, 15, 300, and 3000, respectively. These figuresunderscore that the scattering and absorption perturbations have a different shapeand intensity. As for the other examples seen before, the TR relative perturbationsare independent of the absorption coefficient of the medium.

154 Chapter 7

7.4 Perturbation Approach for Hybrid Models

The expressions for the perturbations δΦa(~r, t) and δΦD(~r, t) given by Eqs. (7.11)and (7.12) have been obtained for the DE. When using the hybrid models describedin Chapter 5, the validity of these expressions needs to be verified. In this section,we show that the expression for δΦa(~r, t) given by Eq. (7.11) is also valid for theRTE, and the validity of Eq. (7.11) is then extended to any hybrid model derivedfrom the RTE. Conversely, in a rigorous sense, the expression for δΦD(~r, t) givenby Eq. (7.12) has a validity limited to the DE since Eq. (7.12) cannot be derivedwithin the RTE.

Consider a heterogeneous turbid medium, with a background absorption coeffi-cient µa(~r), where some absorbing defects with absorption µai(~r) = µa(~r)+δµa(~r)are placed in a volume Vi. A source q0(~r, t) is assumed inside the medium. Theperturbed fluence, flux, and radiance [Φpert(~r, t), ~Jpert(~r, t), and Ipert(~r, s, t), re-spectively] are represented as the sum of an unperturbed part, which represents theunperturbed component in the background medium generated by the source with aperturbation due to the absorbing defects introduced into the medium:

Φpert(~r, t) =∫4π

Ipert(~r, s, t)dΩ = Φ0(~r, t) + δΦa(~r, t),

~Jpert(~r, t) =∫4π

Ipert(~r, s, t)sdΩ = ~J0(~r, t) + δ ~Ja(~r, t),

Ipert(~r, s, t) = I0(~r, s, t) + δIa(~r, s, t).

(7.58)

Substituting in the RTE [Eq. (3.5)], the expression for the radiance, we obtain

∂v∂tI

pert(~r, s, t) +∇ · [Ipert(~r, s, t)s] +[µa(~r) + δµa(~r) + µs(~r)

]×Ipert(~r, s, t) = µs(~r)

∫4π

p(s, s′)Ipert(~r, s′, t)dΩ′ + ε(~r, s, t) . (7.59)

Since I0(~r, t) is the solution of the unperturbed equation, Eq. (7.59) can be writtenas the RTE for the perturbation δIa(~r, s, t):

∂v∂tδI

a(~r, s, t) +∇ · [δIa(~r, s, t)s] + µa(~r)δIa(~r, s, t)= µs(~r)

∫4π

p(s, s′)δIa(~r, s′, t)dΩ′ − δµa(~r)Ipert(~r, s, t) . (7.60)

The equation for δIa(~r, s, t) is thus similar to the RTE for the unperturbedradiance I0(~r, s, t) but with a virtual source term Svirt, which can be written as

Svirt(~r, s, t) = −δµa(~r)Ipert(~r, s, t). (7.61)

With the Green’s function method,30, 31 the time-dependent perturbation for theradiance δIa(~r, s, t) can be expressed as

δIa(~r, s, t) =∫Vi

d→r′∫4π

dΩ′∫ +∞

−∞dt′Svirt(

→r′ , s′, t′)GI(~r,

→r′ , s, s′, t, t′) ,

(7.62)


where Svirt is the virtual source density generated by the perturbation introducedin the medium, and GI is the Green’s function of the radiance for the unperturbedmedium. The time-dependent perturbation for the fluence can thus be expressed as

δΦa(~r, t) =∫4π

δIa(~r, s, t)dΩ = −∫Vid→r′∫ +∞−∞ dt′δµa(

→r′ )Φpert(

→r′ , t′)

×G(~r,→r′ , t, t′) ,

(7.63)where G is the Green’s function of the fluence for the unperturbed medium. Thus,assuming Φpert = Φ0, i.e., the Born approximation, we obtain Eq. (7.11) also for theRTE. This result also justifies the use of Eq. (7.11) for the hybrid models developedin Chapter 5.

As for Eq. (7.12), we do not know a similar procedure that can rigorously extendthe validity of this equation to the RTE. Starting from the continuity equation, itis possible to re-obtain Eq. (7.12) only assuming Fick’s law. Thus, Eq. (7.12) is anapproximate expression within the RTE. However, its use with the hybrid modelsof Chapter 5 is expected to improve the behavior obtained with the DE. Therefore,following a hybrid heuristic approach, the expressions for the perturbations can becalculated for the hybrid solutions presented in Chapter 5, assuming the expressionsof Eqs. (7.11), (7.12), (7.14), and (7.15) are also valid for the hybrid models basedon solutions of the RTE and of the TE.

The perturbed hybrid solutions can be calculated with the following subroutinesprovided on the enclosed CD-ROM:

• RTE_Fluence_Perturbation_Infinite_TR

• TE_Fluence_Perturbation_Infinite_TR

• DE_Fluence_Perturbation_Infinite_TR_2

for the infinite medium, and

• RTE_Fluence_Perturbation_Slab_TR

• TE_Fluence_Perturbation_Slab_TR

• DE_Fluence_Perturbation_Slab_TR_2

for the slab.

7.5 Perturbation Approach for the Layered Slab and for OtherGeometries

In Sec. 7.1 expressions for the absorption and scattering perturbations have beenderived in all generality, applying the perturbation approach to the DE. In Secs. 7.2

156 Chapter 7

and 7.3 the perturbation approach has been applied to the homogeneous infinitemedium and to the homogeneous slab: explicit expressions have been reportedfor the perturbations δΦa and δΦD and the relevant software has been described.However, Eqs. (7.11) and (7.12) can be implemented using any other availableexpression for the unperturbed fluence Φ0 in the background medium. In particular,the enclosed CD-ROM provides the subroutines

• Layered_Fluence_Perturbation

• Layered_Fluence_Perturbation_2

to calculate the perturbations for the two-layer medium: Eqs. (7.11) and (7.12) arecalculated using the solutions for the two-layer medium [Eqs. (6.30) and (6.32)] forΦ0.

In a similar way, following the lines used for instance in Layered_Fluence_Pertu-rbation, subroutines to evaluate δΦa and δΦD for the parallelepiped, the cylinder,and the sphere can be easily developed using the subroutines Homogeneous_Fluence-_Parallelepiped, DE_Fluence_Cylinder, and DE_Fluence_Sphere to evaluate theexpressions of Eqs. (4.61), (4.63), and (4.64), respectively for Φ0.

7.6 Absorption Perturbation by Use of the Internal PathlengthMoments

The perturbation due to absorbing inclusions inside a diffusive medium can be alsorepresented by using the internal pathlength moments of the generalized temporalpoint-spread function.13–15 This alternative approach for absorbing defects hasthe advantage of representing the perturbation with a more intuitive mathematicalexpression than the standard perturbation theory. It is not our purpose to go intothe details of this approach, and we only present the general expression of theperturbation for the fluence due to a single absorbing inclusion with absorptionµai = µa + δµa in a volume Vi immersed in a background medium with absorptioncoefficient µa(~r).

For obtaining the time-dependent perturbation, we extend the result of Eq. (3.20)to the fluence. According to Eq. (3.20), if Φ0(~r, t) is the unperturbed solution, whena variation of absorption δµa is introduced in Vi, the perturbed solution of the RTEwill be expressed as

Φpert(~r, t) = Φ0(~r, t)

∞∫0

f0i(ti , t) exp(−δµa v ti) dti , (7.64)

where f0i(ti , t) is the probability distribution function for the time of flight ti spentinside the volume Vi by photons contributing to Φ0(~r, t), and v is the speed of light


in the volume Vi. The function f0i is defined as the generalized temporal point-spread function. By using the Taylor expansion for the exponential and swappingthe operation of series and integration, we obtain

Φpert(~r, t) = Φ0(~r, t)∞∫0

f0i(ti , t)[

+∞∑n=0

(−1)n δµna

n! (vti)n]dti

= Φ0(~r, t)[1 +

+∞∑n=1

(−1)n δµna

n! 〈lni 〉(δµa = 0, t)

],

(7.65)

where 〈lni 〉(δµa = 0, t) is the time-dependent nth–moment of the pathlength distri-bution inside Vi followed by photons contributing at time t when δµa = 0 to thefluence in the point ~r. The general expression of Eq. (7.65) is valid for both the RTEand the DE. The moments 〈lni 〉 usually are unknown functions. As also shown inRef. 7, the first moment 〈li〉, i.e., the mean pathlength that photons contributing tothe unperturbed fluence have followed inside Vi, can be obtained using Eq. (3.21)as7

〈li〉(δµa = 0, t) = − ∂

∂µailn Φpert(~r, t)|δµa=0 = − 1

δµa

δΦa(~r, t)Φ0(~r, t)

. (7.66)

Similar expressions can be obtained for steady-state sources. It results in

Φpert(~r) = Φ0(~r)[1 +

+∞∑n=1

(−1)n δµna

n! 〈lni 〉(δµa = 0)

](7.67)

and

〈li〉(δµa = 0) = − 1δµa

δΦa(~r)Φ0(~r)

. (7.68)

Moreover, expressions similar to Eqs. (7.65)–(7.68) can be obtained for multipledefects, as described by Sassaroli et al..13

Equation (7.68) shows that within the Born approximation the absorption pertur-bation is proportional to δµa and to 〈li〉. This fact gives the possibility of obtainingexpressions for 〈li〉(t) and 〈li〉 from the expression of the absorption perturbationδΦa.

Equations (7.65) and (7.67) have an immediate intuitive physical interpretation:The effect of an absorbing defect on photon migration mainly depends on the meanpathlength spent by photons inside the volume occupied by the defect and on itsabsorption. The dependence of the perturbed fluence on the higher-order momentsof the fluence 〈lni 〉 (n > 1) takes into account the different ways photons canvisit the volume occupied by the defect. Equations (7.65) and (7.67) allow us tomake a direct comparison between the Born and the Rytov approximations. TheBorn approximation consists of retaining only the term with n = 1 in the series ofEqs. (7.65) and (7.67). The Rytov method for steady-state sources is equivalent tolooking for a perturbed solution of the kind13, 29

Φpert(~r) = Φ0(~r) exp(−〈li〉δµa) . (7.69)

158 Chapter 7

By using the Taylor expansion for the exponential and comparing Eq. (7.69) withEq. (7.67), it is possible to verify that the Rytov method consists of the assumption〈lni 〉 = 〈li〉n.13 In this case, it is assumed that all photons spend the same time insidethe volume Vi.


References



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4. S. R. Arridge, “Photon measurements density functions. Part I: Analyticalforms,” Appl. Opt. 34, 7395–7409 (1995).

5. M. R. Ostermeyer and S. L. Jacques, “Perturbation theory for diffuse lighttransport in complex biological tissues,” J. Opt. Soc. Am. A 14, 255–261 (1997).

6. M. Morin, S. Verreault, A. Mailloux, J. Frechette, S. Chatigny, Y. Painchaud,and P. Beaudry, “Inclusion characterization in a scattering slab with time-resolved transmittance measurements: perturbation analysis,” Appl. Opt. 39,2840–2852 (2000).

7. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a pertur-bation model to predict the effect of scattering and absorbing inhomogeneitieson photon migration,” Appl. Opt. 40, 4622–4632 (2001).

8. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, and R. Cubeddu, “Experimentaltest of a perturbation model for time-resolved imaging in diffusive media,”Appl. Opt. 42, 3145–3153 (2003).

9. S. De Nicola, R. Esposito, and M. Lepore, “Perturbation model to predictthe effect of spatially varying absorptive inhomogeneities in diffusing media,”Phys. Rev. E 68, 021901 (2003).

10. B. Wassermann, “Limits of high-order perturbation theory in time-domainoptical mammography,” Phys. Rev. E 74, 031908 (2006).

11. D. Grosenick, A. Kummrow, R. Macdonald, P. M. Schlag, and H. Rinneberg,“Evaluation of higher-order time-domain perturbation theory of photon diffusionon breast-equivalent phantoms and optical mammograms,” Phys. Rev. E 76,061908 (2007).

12. F. Martelli, S. Del Bianco, and G. Zaccanti, “Perturbation model for lightpropagation through diffusive layered media,” Phys. Med. Biol. 50, 2159–2166(2005).

13. A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusionequation by use of the moments of the generalized temporal point-spreadfunction. I. Theory,” J. Opt. Soc. Am. A 23, 2105–2118 (2006).

160 Chapter 7

14. A. Sassaroli, F. Martelli, and S. Fantini, “Perturbation theory for the diffusionequation by use of the moments of the generalized temporal point-spreadfunction. II. Continuous-wave results,” J. Opt. Soc. Am. A 23, 2119–2131(2006).

15. A. Sassaroli, F. Martelli, and S. Fantini, “Higher-order perturbation theory forthe diffusion equation in heterogeneous media: application to layered and slabgeometries,” Appl. Opt. 48, D62–D73 (2009).

16. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuseoptical tomography,” Opt. Express 1, 404–413 (1997).

17. A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, and R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transil-lumination experiment,” Appl. Opt. 37, 1973–1981 (1998).

18. I. N. Polonsky and M. A. Box, “General perturbation technique for the calcula-tion of radiative effects in scattering and absorbing media,” J. Opt. Soc. Am. A19, 2281–2292 (2002).

19. M. A. Box, “Radiative perturbation theory: a review,” Environ. Modell. Softw.17, 95–106 (2002).

20. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, andG. Zaccanti, “Monte Carlo procedure for investigating light propagation andimaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).


22. P. K. Yalavarthy, K. Karlekar, H. S. Patel, R. M. Vasu, M. Pramanik, P. C.Mathias, B. Jain, and P. K. Gupta, “Experimental investigation of perturbationmonte-carlo based derivative estimation for imaging low-scattering tissue,” Opt.Express 13, 985–997 (2005).

23. J. Heiskala, P. Hiltunen, and I. Nissila, “Significance of background opticalproperties, time-resolved information and optode arrangement in diffuse opticalimaging of term neonates,” Phys. Med. Biol. 54, 535–554 (2009).

24. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Refraction of diffusephoton density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).

25. D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of dif-fuse photon density waves by spherical inhomoheneities within turbid media:Analytical solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91, 4887–4891 (1994).


26. D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Detection andcharacterization of optical inhomogeneities with diffuse photon density waves:a signal to noise analysis,” Appl. Opt. 36, 75–92 (1997).

27. P. N. den Outer, T. M. Nieuwenhuizen, and A. Lagendijk, “Location of objectsin multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).

28. V. N. Pustovit and V. A. Markel, “Propagation of diffuse light in a turbidmedium with multiple spherical inhomogeneities,” Appl. Opt. 43, 104–112(2004).

29. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three-dimensionalMonte Carlo code for photon migration through complex heterogeneous mediaincluding the adult human head,” Opt. Express 10, 159–170 (2002).







36. A. Torricelli, L. Spinelli, A. Pifferi, P. Taroni, R. Cubeddu, and G. M. Danesini,“Use of a nonlinear perturbation approach for in vivo breast lesion characteriza-tion by multiwavelength time-resolved optical mammography,” Opt. Express11, 853–867 (2003).

37. P. Taroni, A. Torricelli, L. Spinelli, A. Pifferi, F. Arpaia, G. Danesini, andR. Cubeddu, “Time-resolved optical mammography between 637 and 985 nm:clinical study on the detection and identification of breast lesions,” Phys. Med.Biol. 50, 2469–2488 (2005).

38. T. Svensson, J. Swartling, P. Taroni, A. Torricelli, P. Lindblom, C. Ingvar, andS. Andersson-Engels, “Characterization of normal breast tissue heterogeneityusing time-resolved near-infrared spectroscopy,” Phys. Med. Biol. 50, 2559–2571 (2005).

162 Chapter 7

39. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. Danesini, and R. Cubeddu,“Characterization of female breast lesions from multi-wavelength time-resolvedoptical mammography,” Phys. Med. Biol. 50, 2489–2502 (2005).

40. D. Grosenick, K. T. Moesta, H. Wabnitz, J. Mucke, C. Stroszczynski, R. Mac-donald, P. M. Schlag, and H. Rinneberg, “Time-domain optical mammography:initial clinical results on detection and characterization of breast tumors,” Appl.Opt. 42, 3170–3186 (2003).

41. D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. Möller, C. Stroszczynski,J. Stößel, B. Wassermann, P. M. Schlag, and H. Rinneberg, “Concentrationand oxygen saturation of haemoglobin of 50 breast tumours determined bytime-domain optical mammography,” Phys. Med. Biol. 49, 1165–1181 (2004).

42. H. Obrig, R. Wenzel, M. Kohl, S. Horst, P. Wobst, J. Steinbrink, F. Thomas,and A. Villinger, “Near-infrared spectroscopy: does it function in functionalactivation studies of the adult brain?,” Int. J. Phychophysiol. 35, 125–142(2000).

43. A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. Barbour, and A. Hielscher,“Three-dimentional optical tomography of hemodynamics in the human head,”Opt. Express 9, 272–286 (2001).

44. A. M. Siegel, J. P. Culver, J. B. Mandeville, and D. A. Boas, “Temporalcomparison of functional brain imaging with diffuse optical tomography andfMRI during rat forepaw stimulation,” Phys. Med. Biol. 48, 1391–1403 (2003).

45. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco,and G. Zaccanti, “Time-resolved reflectance at null source-detector separation:Improving contrast and resolution in diffuse optical imaging,” Phys. Rev. Lett.95, 078101 (2005).


Chapter 8

SoftwareThis chapter describes the software in the enclosed CD-ROM. The software requiresuse of the following units: µa and µ

′s as measured in mm−1, all distances measured

in mm, and time measured in ps. For the speed of light in vacuum, the value c =0.299792458 mm ps−1 has been used.

8.1 Introduction

The contents of the CD-ROM can be visualized by means of the enclosed index.htmlfile and comprises three main parts with the following subcategories:

• The Diffusion&Perturbation Program

– Install the Diffusion&Perturbation Program

– The Diffusion&Perturbation User Manual

– Examples of Results

• Source Code: Solutions of the Diffusion Equation and Hybrid Models

– Solutions of the Diffusion Equation for Homogeneous Media

– Solutions of the Diffusion Equation for Layered Media

– Hybrid Models for the Homogeneous Infinite Medium

– Hybrid Models for the Homogeneous Slab

– Hybrid Models for the Homogeneous Parallelepiped

– General Purpose Subroutines and Functions

• Reference Monte Carlo Results

In this chapter, the Diffusion&Perturbation program and the source code are de-scribed. The reference MC results and examples of comparisons will be describedin the last two chapters.

165

166 Chapter 8

8.2 The Diffusion&Perturbation Program

The Diffusion&Perturbation program is dedicated to the calculation and the visu-alization of the solutions of the DE and its perturbation theory for a homogeneousmedium. The program is thus a forward solver for the calculation of the solutions ofthe DE in homogeneous media. This software has a user-friendly interface, whichwas developed in a Visual C environment. The program is accompanied by a usermanual.

The Diffusion&Perturbation program may be installed by using the link "InstallDiffusion&Perturbation Program" on the index.html or by using the setup.exe filethat is in the directory \Diffusion&Perturbation\cvidistkit.Diffusion&Perturbation.The main panel of the program guides the user through the Diffusion&Perturbationsoftware. From the main panel (see Fig. 8.1), the user can proceed to:

• Set up the simulation

• Run the simulation

• Display the simulated data

• Save the simulated data

• Stop the program

Simulations are set up in a corresponding panel where all the relevant quantitiesfor the simulation can be set by the user (see Fig. 8.2). The program can performsimulations with three different geometries: infinite medium, semi-infinite medium,and slab. The output of Diffusion&Perturbation includes classical quantities likereflectance, transmittance, and fluence rate. Simulations can be either carried outin the time domain or in the CW domain. The selected output can be visualized indifferent ways. For the time domain, the program can display the temporal responseof the medium, a receiver map for a fixed value of time, or an inhomogeneity mapfor a fixed value of time. For the CW domain, two types of output can be displayedby the program: a receiver map or an inhomogeneity map. For temporal response,the TPSF is meant for the selected output (reflectance, transmittance, fluence)together with the corresponding absorption and scattering perturbation. The receivermap is obtained by varying the position of the receiver, while an inhomogeneitymap is obtained for a fixed position of the receiver by varying the position of aunitary perturbation (scattering or absorption with δDVi = 1 mm4 or δµaVi = 1mm2 ) inside the medium. In the set simulation panel, the optical and geometricalcharacteristics of the simulation are selected by the user. The range (temporal orspatial) of the quantities displayed by the program are selected in the same panel(see Fig. 8.2).

After the set simulation panel has been filled with the necessary informationit is possible to proceed with the simulation. Pressing the OK button returns the

Software 167

Figure 8.1 Main panel of the program Diffusion&Perturbation.

program to the main menu (Fig. 8.1) from which it is possible to run the simulationby pressing the RUN button. The computation time is negligible for most of thesimulations that can be performed with the program. The quantities calculated bythe program can be displayed and saved as an ASCII file using the comma separatedvalues (CSV) format. The data thereby can be further analyzed and elaborated bythe user.

The use of the program is supported by an included HELP function that can beaccessed by clicking the right button of the mouse on the enabled buttons.

Examples of results generated and displayed with the Diffusion&Perturbationprogram are shown in Figs. 8.3, 8.4, and 8.5. Figure 8.3 shows an example of TRreflectance from a homogeneous slab (s = 40 mm, µa = 0.01 mm−1, µ

′s = 1 mm−1,

zs = 1 mm, ni = 1.2 and nr = 1.2, ρ = 20 mm). Figure 8.4 shows how theperturbation on transmittance at a certain time (t = 1000 ps) changes as a functionof the position of an absorbing inhomogeneity (inhomogeneity map obtained byvarying the coordinate xi and yi of the inhomogeneity) (s = 10 mm, µa = 0.05mm−1, µ

′s = 1 mm−1, zs = 1 mm, ni = 1.2 and nr = 1.2, ρ = 0, depth of the

inhomogeneity zi = 5 mm). Figure 8.5 shows the CW fluence in an infinite mediumwith an internal absorbing inhomogeneity as a function of the position of the receiver(receiver map obtained by varying the coordinate xr and zr of the receiver) [µa =0.001 mm−1, µ

′s = 1 mm−1, ni = 1.2, inhomogeneity at (0, 0, 6) mm, and receiver

at (xr, 5, zr) mm].

168 Chapter 8

Figure 8.2 Set up panel of the program Diffusion&Perturbation.

Diffusion&Perturbation employs several subroutines based on the Green’s func-tions presented in Chapters 4 and 7. For the sake of the reader, we have summarizedthe links between the subroutines employed by the program and the Green’s func-tions presented in the book in Table 8.1. The source code of these subroutines iswritten in the FORTRAN language and used by the Diffusion&Perturbation programas DLL libraries. These subroutines evaluate the Green’s function at a single time forthe TR Green’s functions and at a single position of the receiver for the CW Green’sfunctions. The source code of the subroutines employed by the program is availablein the enclosed CD-ROM inside the directory Source_Code\Diffusion&Perturbation.More details on these subroutines will be given in the next section.

We point out that solutions for reflectance and transmittance are implemented us-ing solutions based on Fick’s law. The solutions based on the hybrid EBPC approachcan be obtained by dividing the solution provided by the Diffusion&Perturbationcode for the fluence evaluated on the external surface by 2A [A is the coefficient forthe EBC, Eq. (3.50)].

Software 169

Figure 8.3 Example of a TPSF generated with the Diffusion&Perturbation program for thereflectance from a diffusive slab.

Figure 8.4 Example of an inhomogeneity map for the transmittance generated with theDiffusion&Perturbation program for an absorption perturbation inside a diffusive slab.

170 Chapter 8

Figure 8.5 Example of a receiver map for the fluence generated with theDiffusion&Perturbation program for an absorption perturbation inside an infinite medium.

8.3 Source Code: Solutions of the Diffusion Equation andHybrid Models

In this section, all the source code provided with the CD-ROM is briefly introducedand described. The source code is divided into five main environments. All the codeis written in the FORTRAN language. The FORTRAN subroutines can be convertedas a dynamic-link library (DLL) interface for processes that use them. In this way,the subroutines can be employed in programs written in other languages like C or,although it is not a straightforward procedure, in other environments like MATLAB.1

For a correct utilization of this second part of the CD-ROM, even if the software isaccompanied by comments and examples of use, some basic programming skillsthat are not necessary for using the Diffusion&Perturbation software are requiredhere.

The name of the subroutines have been built using acronyms like DE, RTE, TE,TR, and CW and keywords like Fluence, Flux, Radiance, Reflectance, Transmittance,Perturbation, Infinite, Slab, Parallelepiped, Cylinder, Sphere, and Layered. Theintent is to help the reader identify the subroutine and the type of Green’s functionassociated with the subroutine. Thus, the name of the subroutine represents a codethat gives information on the task performed by the subroutine itself. For instance,the subroutine DE_Fluence_Perturbation_Infinite_TR calculates the TR fluence andTR perturbations for the fluence in an infinite medium obtained with the DE.

Software 171

The solutions provided by the subroutines have physical meaning only when thefield point is inside the diffusive medium or on its boundary.

8.3.1 Solutions of the diffusion equation for homogeneous media

The environment of the solutions of the DE for homogeneous media is based on thesubroutines used by the Diffusion&Perturbation program to evaluate the fluence,reflectance, transmittance, and perturbation due to scattering and absorbing inhomo-geneities. Table 8.1 shows a link between each subroutine and the correspondingequations in Chapters 4 and 7.

Table 8.1 In this table, the equations of Chapters 4 and 7 and the name of the correspondingsubroutines used by the program Diffusion&Perturbation are summarized.

Infinite MediumEquations SubroutinesEqs. (4.3), (7.21), and (7.22) DE_Fluence_Perturbation_Infinite_TREqs. (4.6), (7.23), and (7.24) DE_Fluence_Perturbation_Infinite_CW

Semi-Infinite Medium and SlabEquations SubroutinesEqs. (4.16), (7.25), and (7.26) DE_Fluence_Perturbation_Slab_TREqs. (4.22), (7.28), and (7.29) DE_Fluence_Perturbation_Slab_CWEqs. (4.27), (7.36), and (7.37) DE_Reflectance_Perturbation_Slab_TREqs. (4.28), (7.38), and (7.39) DE_Transmittance_Perturbation_Slab_TREqs. (4.34), (7.44), and (7.45) DE_Reflectance_Perturbation_Slab_CWEqs. (4.35), (7.46), and (7.47) DE_Transmittance_Perturbation_Slab_CW

Other subroutines for evaluating the response for a receiver with a finite area,the total reflectance, and the total transmittance are summarized in Table 8.2, wherea link between each subroutine and the corresponding equation of Chapter 4 is given.In the names used for the subroutines, the keyword "Receiver" will identify theresponse for a receiver with a finite area, and the keyword "Total" will identify theresponse for an infinitely extended receiver.

A brief description of each subroutine of Tables 8.1 and 8.2 is provided below:


Calculates the TR fluence per unit of emitted energy (mm−2 ps−1) for theinfinite medium, and the perturbations (mm−2 ps−1) due to both a scatteringand an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 orδµaVi = 1 mm2).

172 Chapter 8

Table 8.2 In this table, the equations of Chapter 4 and the name of the correspondingsubroutines of the software package for homogeneous media are summarized.

Slab GeometryEquations SubroutinesEq. (4.32) DE_Reflectance_Slab_Receiver_TREq. (4.38) DE_Reflectance_Slab_Receiver_CWEq. (4.33) DE_Transmittance_Slab_Receiver_TREq. (4.39) DE_Transmittance_Slab_Receiver_CWEq. (4.30) DE_Total_Reflectance_Slab_TREq. (4.36) DE_Total_Reflectance_Slab_CWEq. (4.31) DE_Total_Transmittance_Slab_TREq. (4.37) DE_Total_Transmittance_Slab_CWEq. (4.53) DE_Fluence_Slab_Receiver_TREq. (4.54) DE_Fluence_Slab_Receiver_CWEq. (4.59) DE_Total_Fluence_Slab_TREq. (4.60) DE_Total_Fluence_Slab_CW

Cylindrical GeometryEquation SubroutineEq. (4.63) DE_Fluence_Cylinder_TR

Spherical GeometryEquation SubroutineEq. (4.64) DE_Fluence_Sphere_TR


Calculates the CW fluence per unit of emitted power (mm−2) for the infinitemedium, and the perturbations (mm−2) due to both a scattering and an ab-sorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1mm2).

• DE_Fluence_Perturbation_Slab_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1) for the slab,and the perturbations (mm−2 ps−1) due to both a scattering and an absorbinginhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1 mm2).


Calculates the CW fluence per unit of emitted power (mm−2) for the slab,and the perturbations (mm−2) due to both a scattering and an absorbinginhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1 mm2).

Software 173

• DE_Reflectance_Perturbation_Slab_TR

Calculates the TR reflectance per unit of emitted energy (mm−2 ps−1) forthe slab, and the perturbations (mm−2 ps−1) due to both a scattering and anabsorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1mm2).

• DE_Transmittance_Perturbation_Slab_TR

Calculates the TR transmittance per unit of emitted energy (mm−2 ps−1) forthe slab, and the perturbations (mm−2 ps−1) due to both a scattering and anabsorbing inhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1mm2).

• DE_Reflectance_Perturbation_Slab_CW

Calculates the CW reflectance per unit of emitted power (mm−2) for theslab, and the perturbations (mm−2) due to both a scattering and an absorbinginhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1 mm2).

• DE_Transmittance_Perturbation_Slab_CW

Calculates the CW transmittance per unit of emitted power (mm−2) for theslab, and the perturbations (mm−2) due to both a scattering and an absorbinginhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1 mm2).

• DE_Reflectance_Slab_Receiver_TR

Calculates the TR average reflectance over the receiver area per unit of emittedenergy (mm−2 ps−1) for the slab.

• DE_Reflectance_Slab_Receiver_CW

Calculates the CW average reflectance over the receiver area per unit ofemitted power (mm−2) for the slab.

• DE_Transmittance_Slab_Receiver_TR

Calculates the TR average transmittance over the receiver area per unit ofemitted energy (mm−2 ps−1) for the slab.

• DE_Transmittance_Slab_Receiver_CW

Calculates the CW average transmittance over the receiver area per unit ofemitted power (mm−2) for the slab.

• DE_Total_Reflectance_Slab_TR

Calculates the TR total reflectance per unit of emitted energy (ps−1) for theslab.

174 Chapter 8

• DE_Total_Reflectance_Slab_CW

Calculates the CW total reflectance per unit of emitted power for the slab.

• DE_Total_Transmittance_Slab_TR

Calculates the TR total transmittance per unit of emitted energy (mm−2) forthe slab.

• DE_Total_Transmittance_Slab_CW

Calculates the CW total transmittance per unit of emitted power for the slab.

• DE_Fluence_Slab_Receiver_TR

Calculates the TR average fluence over the receiver area per unit of emittedenergy (mm−2 ps−1) for the slab.

• DE_Fluence_Slab_Receiver_CW

Calculates the CW average fluence over the receiver area per unit of emittedpower (mm−2) for the slab.

• DE_Total_Fluence_Slab_TR

Calculates the TR total fluence per unit of emitted energy (mm−2) for theslab.

• DE_Total_Fluence_Slab_CW

Calculates the CW total fluence per unit of emitted power for the slab.

• DE_Fluence_Cylinder_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1) for theinfinite cylinder.

• DE_Fluence_Sphere_TR

Calculates the TR fluence per unit of emitted energy (mm−2 ps−1) for thesphere.

Two drivers, Program1 and Program2, are also available to show examples ofhow to use the subroutines DE_Fluence_Cylinder_TR and DE_Fluence_Sphere_TR.The computation time of DE_Fluence_Cylinder_TR and DE_Fluence_Sphere_TRis significantly longer compared to that of the other subroutines for homogeneousgeometries. A faster version of these subroutines can be obtained using the rou-tines in Ref. 2 to calculate the Bessel functions. Our priority in making these twosubroutines was to guarantee a double precision accuracy for the calculation andmore flexibility, while minor importance was given to the computation time. Thus,with minor changes, it is possible to have faster routines like those implemented inRef. 3, which were used in fitting procedures to retrieve the optical properties of themedium.

Software 175

8.3.2 Solutions of the diffusion equation for layered media

The environment of the solutions of the DE for layered media is based on followingsubroutines:

• Root

• DE_Fluence_Layered_Slab_TR

The subroutine Root calculates the longitudinal eigenvalues for the solution of theDE for a two-layer cylinder [roots of Eq. (6.28)].

The subroutine DE_Fluence_Layered_Slab_TR calculates the TR fluence rateper unit of emitted energy (mm−2 ps−1) in the two-layer cylinder [Eqs. (6.30) and(6.32)], and it uses the longitudinal eigenvalues calculated by the subroutine Rootand the radial eigenvalues [roots of Eq. (6.19), i.e., roots of J0(KρlL) = 0] providedinside the file j0.dat. All the other subroutines of this environment make use of thesesubroutines to calculate the quantities of interest for photon migration through a two-layer medium. In particular, the quantities that can be evaluated with this softwarepackage are summarized in Table 8.3. A brief description of each subroutine isprovided below.

Table 8.3 The equations of Chapter 6 and the name of the corresponding subroutines ofthe software package for the solution of the diffusion equation for layered media are summa-rized.

Two-layer mediumQuantity calculated Subroutine usedEqs. (6.30) and (6.32) DE_Fluence_Layered_Slab_TRGradient Layered_Fluence_GradientEq. (3.31) Layered_FluxEq. (3.25) Layered_RadianceLongitudinal eigenvalues RootEqs. (7.11) and (7.12) Layered_Fluence_PerturbationEqs. (7.11) and (7.12) Layered__Fluence_Perturbation_2

• Layered_Fluence_Gradient

Numerically calculates the gradient of the TR fluence (mm−3 ps−1) for thetwo-layer cylinder.

• Layered_Flux

Calculates the TR flux (mm−2 ps−1) for the two-layer cylinder.

176 Chapter 8

• Layered_Radiance

Calculates the TR radiance (mm−2 ps−1 sr−1) for the two-layer cylinder.

• Layered_Fluence_Perturbation

Calculates the TR perturbations (mm−2ps−1) for a two-layer cylinder, due toboth a scattering and an absorbing inhomogeneity of unitary strength (δDVi =1 mm4 or δµaVi = 1 mm2). The perturbation is calculated by numericallyevaluating Eqs. (7.11) and (7.12), using the fluence of Eqs. (6.30) and (6.32)obtained with the subroutine DE_Fluence_Layered_Slab_TR.

• Layered_Fluence_Perturbation_2

This is a faster version of the Layered_Fluence_Perturbation, which wasdeveloped to calculate the perturbation for an array of values of time. Foreach call, the subroutine Layered_Fluence_Perturbation calculates the per-turbation for a single time, which makes the computation of the convolu-tion integrals of Eqs. (7.11) and (7.12) time consuming. In contrast, Lay-ered_Fluence_Perturbation_2 calculates with a single call the whole TPSF ofthe scattering and of the absorption perturbation. For this reason, the compu-tation time is shorter since the information on the temporal response is storedfor each convolution integral of Eqs. (7.11) and (7.12).

Although the subroutine Layered_Fluence_Perturbation is slower than the sub-routine Layered_Fluence_Perturbation_2, it is included for its greater flexibility,which may be preferred for some kinds of calculations.

As an example to help the reader get accustomed to this environment, we showsome lines of the enclosed driver Program3 that, using the subroutines of Table 8.3,can be used to calculate the TR reflectance from the two-layer cylinder:

C ................................C Calculation of the TR reflectanceC ................................C Maximum number of radial eigenvalues

n_kj=100C Maximum number of longitudinal eigenvalues

n_k0=200C Calculation of the longitudinal Eigenvalues

CALL Root(mua,musp,s,R,n0,AA,n_k0,n_kj,+ kappa_z0,kappa_j,I_bound,Nl)

DO i=1,NPOINTC Definition of the temporal scale

t(i)=0.d0+DBLE(i-1)*dTIME ! Time (ps)C Calculation of the fluence with the DE

Software 177

CALL DE_Fluence_Layered_Slab_TR(+ mua,musp,s,R,t(i),xs,ys,zs,xr,yr,zr,n0,AA,+ n_k0,n_kj,kappa_z0,kappa_j,I_bound,Nl,flu)

C Calculation of the reflectance (Eq. 6.33)DE_reflectance(i)=flu/2.d0/AA(1)

END DO

As is shown in the program, the initial requirement for the calculation is evalua-tion of the longitudinal eigenvalues with the subroutine Root. The radial eigenvaluesare provided inside the file j0.dat, which must be inserted in the same directorywhere the driver program is run. Two important parameters that need to be set on thedriver program are the number of radial (n_kj) and longitudinal (n_k0) eigenvaluesused by the program. The values selected for n_kj and n_k0 affect the computationtime and the level of convergence of the calculated solution. Increasing n_kj andn_k0 will increase both the computation time and the convergence of the calculationto the actual solution. The number of radial eigenvalues required for convergenceof the solution depends proportionally to the product of the radius of the cylinderwith the reduced scattering coefficient of the medium. It is possible that the numberselected for n_k0 will not be enough to provide all the imaginary longitudinal rootsfor the situation investigated. In this case, the program will provide a warning, and ahigher number for n_k0 will need to be inserted to perform the calculation. Notethat the eigenfunction method has a fast convergence of the solution at late timesbut a slow convergence at early times. For this reason, due to convergence problemsrelated to the number and to the accuracy of the calculated eigenvalues, it is alsopossible to have negative values for the temporal response at early times.

8.3.3 Hybrid models for the homogeneous infinite medium

The environment of the hybrid models for the infinite homogeneous medium isbased on the following subroutines:

• RTE_Fluence_Infinite_TR

Calculates the solution of the RTE for the TR fluence per unit of emittedenergy (mm−2 ps−1) for the infinite medium.

• TE_Fluence_Infinite_TR

Calculates the solution of the TE for the TR fluence per unit of emitted energy(mm−2 ps−1) for the infinite medium.

• DE_Fluence_Infinite_TR

Calculates the solution of the DE for the TR fluence rate per unit of emittedenergy (mm−2 ps−1) for the infinite medium.

178 Chapter 8

These subroutines calculate the Green’s function of the fluence for the RTE[Eq. (5.12)], the TE [Eq. (5.16)], and the DE [Eq. (4.2)], respectively. All the othersubroutines of this environment call these subroutines to calculate the quantitiesof interest for photon migration through a homogeneous infinite medium like flux,radiance, perturbation, etc. All the quantities that can be evaluated with this softwarepackage are summarized in Table 8.4 and are briefly described below:

Table 8.4 Software package for hybrid models for the homogeneous infinite medium: Sum-mary of equations and quantities calculated with the corresponding subroutines.

Infinite MediumQuantity calculated Subroutine usedEq. (5.12) RTE_Fluence_Infinite_TREq. (5.16) TE_Fluence_Infinite_TREq. (4.3) DE_Fluence_Infinite_TRGradient Homogeneous_Fluence_Gradient

_InfiniteEq. (3.31) Homogeneous_Flux_InfiniteEq. (3.25) Homogeneous_Radiance_InfiniteEqs. (7.11) and (7.12) Homogeneous_Fluence_Perturbation

_InfiniteGradient and Perturbation Homogeneous_Fluence_Gradient

_Perturbation_InfiniteEqs. (3.31) and (7.14) Homogeneous_Flux_Perturbation

_InfiniteEqs. (3.25) and (7.15) Homogeneous_Radiance_Perturbation

_InfiniteFluence and perturbation with RTE RTE_Fluence_Perturbation_Infinite_TRFluence and perturbation with TE TE_Fluence_Perturbation_Infinite_TRFluence and perturbation with DE DE_Fluence_Perturbation_Infinite_TR_2

• Homogeneous_Fluence_Gradient_Infinite

Numerically calculates the gradient of the TR fluence (mm−3 ps−1) for theinfinite medium.


Calculates the TR flux (mm−2 ps−1) for the infinite medium.


Calculates the TR radiance (mm−2 ps−1 sr−1) for the infinite medium.

Software 179

• Homogeneous_Fluence_Perturbation_Infinite

Calculates the perturbations of TR fluence (mm−2 ps−1) due to both a scatter-ing and an absorbing inhomogeneity of unitary strength (δDVi = 1 mm4 orδµaVi = 1 mm2) for the infinite medium.

• Homogeneous_Fluence_Gradient_Perturbation_Infinite:

Calculates the gradient of the TR fluence (mm−3 ps−1) or of the CW fluence(mm−3), and the perturbations due to both a scattering and an absorbinginhomogeneity of unitary strength (δDVi = 1 mm4 or δµaVi = 1 mm2) forthe infinite medium.


Calculates the TR flux (mm−2 ps−1) or the CW flux (mm−2), and the pertur-bations due to both a scattering and an absorbing inhomogeneity of unitarystrength (δDVi = 1 mm4 or δµaVi = 1 mm2) for the infinite medium.


Calculates the TR radiance (mm−2 ps−1 sr−1), and the perturbations due toboth a scattering and an absorbing inhomogeneity of unitary strength (δDVi =1 mm4 or δµaVi = 1 mm2) for the infinite medium.

• RTE_Fluence_Perturbation_Infinite_TR

Calculates the TR fluence (mm−2 ps−1), and the perturbations due to botha scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1mm4 or δµaVi = 1 mm2) for the infinite medium, making use of the subrou-tines RTE_Fluence_Infinite_TR and Homogeneous_Perturbation_Infinite.

• TE_Fluence_Perturbation_Infinite_TR

Calculates the TR fluence (mm−2 ps−1), and the perturbations due to botha scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1mm4 or δµaVi = 1 mm2) for the infinite medium, making use of the subrou-tines TE_Fluence_Infinite_TR and Homogeneous_Perturbation_Infinite.

• DE_Fluence_Perturbation_Infinite_TR_2

Calculates the TR fluence (mm−2 ps−1), and the perturbations due to botha scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1mm4 or δµaVi = 1 mm2) for the infinite medium, making use of the subrou-tines DE_Fluence_Infinite_TR and Homogeneous_Perturbation_Infinite.

180 Chapter 8

8.3.4 Hybrid models for the homogeneous slab

The environment of the hybrid models for the homogeneous slab is based on thesubroutine Homogeneous_Fluence_Slab_TR, which calculates the Green’s functionof the hybrid models of Eq. (5.1). This subroutine calculates the Green’s functionfor the slab using the subroutines of the Green’s functions for the infinite mediumbased on RTE, TE, and DE. The other subroutines of this environment make useof the Homogeneous_Fluence_Slab_TR subroutine to calculate the other quantitiesof interest for photon migration through a homogeneous slab. In particular, thequantities that can be evaluated with this software package are summarized in Table8.5 and are briefly described below.

• Homogeneous_Fluence_Gradient_Slab

Numerically calculates the gradient of the TR fluence (mm−3 ps−1) for theslab.


Calculates the TR flux (mm−2 ps−1) for the slab.

• Homogeneous_Radiance_Slab

Calculates the TR radiance (mm−2 ps−1 sr−1) for the slab.

• Homogeneous_Fluence_Perturbation_Slab

Calculates the perturbations of the TR fluence (mm−2 ps−1) for the slabdue to both a scattering and an absorbing inhomogeneity of unitary strength(δDVi = 1 mm4 or δµaVi = 1 mm2).

• RTE_Fluence_Perturbation_Slab_TR

Calculates the TR fluence (mm−2 ps−1), and the perturbations due to botha scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1mm4 or δµaVi = 1 mm2) for the slab using the subroutines RTE_Fluence_Infi-nite_TR and Homogeneous_Perturbation_Slab.

• TE_Fluence_Perturbation_Slab_TR

Calculates the TR fluence (mm−2 ps−1), and the perturbations due to botha scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1mm4 or δµaVi = 1 mm2) for the slab using the subroutines TE_Fluence_Infi-nite_TR and Homogeneous_Perturbation_Slab.

• DE_Fluence_Perturbation_Slab_TR_2

Calculates the TR fluence (mm−2 ps−1), and the perturbations due to botha scattering and an absorbing inhomogeneity of unitary strength (δDVi = 1mm4 or δµaVi = 1 mm2) for the slab using the subroutines DE_Fluence_Infi-nite_TR and Homogeneous_Perturbation_Slab.

Software 181

Table 8.5 Software package for hybrid models for the homogeneous slab: Summary ofequations and quantities calculated with the corresponding subroutines.

Homogeneous slabQuantity calculated Subroutine usedEq. (5.1) Homogeneous_Fluence_Slab_TRGradient Homogeneous_Fluence_Gradient_SlabEq. (3.31) Homogeneous_Flux_SlabEq. (3.25) Homogeneous_Radiance_SlabEqs. (7.11) and (7.12) Homogeneous_Fluence_Perturbation_SlabFluence and perturbation with RTE RTE_Fluence_Perturbation_Slab_TRFluence and perturbation with TE TE_Fluence_Perturbation_Slab_TRFluence and perturbation with DE DE_Fluence_Perturbation_Slab_TR_2

As an example to help the reader, below are some lines of the enclosed driverProgram4 that, using the subroutines of Tables 8.4 and 8.5, calculates the TRreflectance with the different models:

C ................................C Calculation of the TR reflectanceC ................................

CALL A_parameter(n0,ne,AA)DO i=1,NPOINT

C Definition of the temporal scalet(i)=0.d0+(i-1.d0)*1.d0 ! Time (ps)

C Calculation of the fluence with the Hybrid DECALL Homogeneous_Fluence_Slab(

+ mua,musp,s,t(i),xs,ys,zs,xr,yr,zr,+ n0,AA,DE_Fluence_Infinite_TR,flu)

C Calculation of the reflectance with the DE (Eq. 5.5)DE_reflectance(i)=flu/2.d0/AA

C Calculation of the fluence with the TECALL Homogeneous_Fluence_Slab(

+ mua,musp,s,t(i),xs,ys,zs,xr,yr,zr,+ n0,AA,TE_Fluence_Infinite_TR,flu)

C Calculation of the reflectance with the TE (Eq. 5.5)TE_reflectance(i)=flu/2.d0/AA

C Calculation of the fluence with the RTECALL Homogeneous_Fluence_Slab(

+ mua,musp,s,t(i),xs,ys,zs,xr,yr,zr,+ n0,AA,RTE_Fluence_Infinite_TR,flu)

C Calculation of the reflectance with the RTE (Eq. 5.5)RTE_reflectance(i)=flu/2.d0/AA

182 Chapter 8

END DOC ..................................

This driver shows how it is possible to switch from one hybrid model to anothersimply by changing the name of the subroutine used to calculate the fluence for theinfinite medium in the input variable of the subroutine Homogeneous_Flux_Slab.The same calculation can also be carried out for the perturbation on the TR re-flectance using the driver Program5.

8.3.5 Hybrid models for the homogeneous parallelepiped

The environment of the hybrid models for the homogeneous parallelepiped is basedon the subroutine Homogeneous_Fluence_Parallelepiped, which calculates theGreen’s function of the hybrid models of Eq. (5.7). This subroutine calculates theGreen’s function for the parallelepiped using the subroutines for the infinite mediumbased on RTE, TE, and DE. The number of terms (dipoles) that need to be retainedto have convergence in Eq. (5.7) in the temporal range of some ns is usually . 4.

8.3.6 General purpose subroutines and functions

All the subroutines described in previous sections make use of some service routinesthat are necessary to implement their calculations. These are subroutines andfunctions used by the subroutines that evaluate the solutions of the diffusion equationfor homogeneous and layered media and by the subroutines used to evaluate thehybrid models. These routines are grouped together inside the directory

\Source_Code\General_Purpose_Subroutines&Functions.

The calculations of the Bessel functions and of the Legendre polynomials areimplemented with the following subroutines: IKNA, JY01A, JYNA, JYV, and LPN.The copyright of this software belongs to Professor Jianming Jin of the University ofIllinois at Urbana-Champaign. These subroutines have been downloaded from theweb site http://jin.ece.uiuc.edu/routines/routines.html, with the kind permission ofProfessor Jianming Jin. Some more information on these subroutines can be foundin Ref. 4.

All the quantities that can be evaluated with this software package are summa-rized in Table 8.6 and are briefly described below.

• A_parameter

Numerically calculates parameter A of the extrapolated boundary condition[Eq. (3.50)].

Software 183

Table 8.6 General purpose subroutines and functions: Quantities calculated with the corre-sponding subroutines or functions.

Quantity calculated Subroutine or function usedEquation (3.50) A_parameterEquation (3.47) FresnelIntegration with the trapezoidal method TrapScalar product of two vectors ScaEquation (7.41) fEquation (7.49) hEquation (7.42) pEquation (7.43) qEquation (7.48) wModified Bessel functions In and Kn IKNABessel functions J0, J1, Y0, Y1 JY01ABessel functions Jn, Yn JYNABessel functions Jv, Yv JYVLegendre Polynomials Pn LPN

• Fresnel

Calculates the Fresnel reflection coefficient for unpolarized light given byEq. (3.47).

• Trap

Calculates the integral of a function y on the variable x using the trapezoidalmethod.

• Sca

Calculates the scalar product of two vectors.

• f

Calculates Eq. (7.41).

• h


• p


• q


184 Chapter 8

• w


• IKNA

Calculates the modified Bessel functions In and Kn of order n, and theirderivatives.

• JY01A

Calculates the Bessel functions J0, J1, Y0, and Y1 of integer order, and theirderivatives.

• JYNA

Calculates the Bessel functions Jn and Yn of integer order, and their deriva-tives.

• JYV

Calculates the Bessel functions Jv, Yv of real order, and their derivatives.

• LPN

Calculates the Legendre polynomials.

Software 185

References

1. The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, MATLAB 7External Interfaces (2008).


3. A. Sassaroli, F. Martelli, G. Zaccanti, and Y. Yamada, “Performance of fittingprocedures in curved geometry for retrieval of the optical properties of tissuefrom time-resolved measurements,” Appl. Opt. 40, 185–197 (2001).

4. S. Zhang and J. Jin, Computation of Special Functions, John Wiley & Sons,New York (1996).

Chapter 9

Reference Monte Carlo Results9.1 Introduction

Almost all the analytical solutions reported in Chapters 4–7 and the related softwaredescribed in Chapter 8 are approximate solutions of the RTE. Before using them, itis useful to know the level of their accuracy in order to establish their applicability indifferent fields of interest. For this purpose, the enclosed CD-ROM reports examplesof solutions of the RTE obtained with MC simulations. With MC simulations, theRTE can be solved without the need of simplifying assumptions and with an accuracyonly limited by the statistical fluctuations related to the stochastic nature of themethod. The MC results can therefore be used as a standard reference for comparison.This chapter is mainly devoted to describe the MC results reported in the CD-ROM(Sec. 9.6). The MC results have been obtained using several different MC programs.For a better understanding and a proper use of the MC results, a description of theMC programs we used is reported in Secs. 9.2–9.5. Examples of comparisons willbe shown in Chapter 10.

9.2 Rules to Simulate the Trajectories and General Remarks

The MC method provides a physical simulation of light propagation. For eachlaunched energy packet (photon packet or simply photon for brevity), the trajectoryis numerically generated using the probability functions that govern propagationthrough the turbid medium. A photon is received if the simulated trajectory entersthe receiver within the acceptance angle. With conventional MC simulations, theimpulse response or temporal point spread function (TPSF), i.e., the probability(per unit time and per unit area of the receiver) of receiving emitted photons, isreconstructed by classifying the received photons in a histogram on the basis of theirtime of flight. The temporal resolution is only limited by the length of the intervalschosen for the temporal histogram and has a feature similar to the one obtained witha time-resolved experimental setup in which photons are detected using the time-correlated single-photon counting (TCSPC) technique.1 As for the experiment, thesimulated response is noisy: Owing to the central limit theorem, the standard errordecreases proportionally to the inverse of the square root of the number of receivedphotons. To obtain results with the required accuracy, the simulation of a large

187

188 Chapter 9

number of trajectories may be necessary with computation times that sometimesbecome prohibitively long. Techniques of variance reduction can be sometimessuccessfully used to reduce the computation time.

The key point for MC simulations is the generation of the trajectories: Theymust be generated using the same statistical rules that govern propagation in realexperiments. The basis for generating photon trajectories are the probability dis-tribution functions describing the interaction of photons with the turbid mediumreported in Sec. 2.2.1 [Eqs. (2.15)–(2.17)]. Recall that with the assumptions ofisotropic scatterers (spherical particles or randomly oriented non-spherical particles)and unpolarized light, the probability distribution functions f1 (θ) and f2 (ϕ) forthe scattering angle θ and the azimuthal angle ϕ specifying the new direction after ascattering event, are given by

f1(θ) = 2πp(θ) sin θ (9.1)

andf2(ϕ) =

12π

, (9.2)

where p(θ) is the scattering function of the medium. The probability density functionf3 (`) for the path followed between two subsequent interactions with the turbidmedium is given by

f3 (`) = µt(`) exp[−

`∫0

µt(`′)d`′]. (9.3)

The sampling of the variables `, θ, and ϕ is obtained using the inverse distri-bution method (see Ref. 2), starting from pseudo-random numbers ξ with uniformdistribution between 0 and 1 generated by computer, and making use of the cumula-tive probability function F (x) defined as

F (x) =

x∫0

f(x′)dx′ . (9.4)

Since the probability distribution function is normalized to 1, F (x) assumes all thevalues between 0 and 1. The value of the variable x associated to ξ is obtainedposing

F [x(ξ)] = ξ , (9.5)

from whichx(ξ) = F−1(ξ) . (9.6)

In all the MC programs we used, the trajectories are simulated for the non-absorbing medium. As will be shown in subsequent sections, the results for the

Reference Monte Carlo Results 189

medium with absorption can be easily obtained from the results for the non-absorbingmedium making use of the general property for the RTE described in Sec. 3.4.2.We summarize here the rules for sampling the trajectories for the homogeneousnon-absorbing medium.

For the homogeneous non-absorbing medium, the cumulative probability func-tion for the step size between two subsequent scattering events results in

F3(`) =

`∫0

f3

(`′)d`′ =

`∫0

µs exp(−µs`′)d`′ = 1− exp(−µs`) . (9.7)

The length `(ξ) associated with the pseudo-random number ξ is obtained as

1− exp[−µs`(ξ)] = ξ , (9.8)

from which

`(ξ) = − ln(1− ξ)µs

. (9.9)

From Eq. (9.1) and (9.2), the relations for θ(ξ) and ϕ(ξ) result in

F1[θ(ξ)] = 2π

θ(ξ)∫0

p(θ′) sin θ′dθ′ = ξ (9.10)

and

F2[ϕ(ξ)] =

ϕ(ξ)∫0

12πdϕ′ = ξ . (9.11)

Equation (9.11) can be easily inverted to obtain ϕ(ξ) = 2πξ. On the contrary,Eq. (9.10) can be inverted analytically only when simplified analytical models forp(θ), like the Henyey and Greenstein (HG) model,3 are considered. For morerealistic scattering functions, like those obtained from Mie theory, Eq. (9.10) shouldbe inverted numerically. The MC programs that we used can work with scatteringfunctions generated both with the HG model and with Mie theory.

The HG scattering function, defined as

p(θ) =1

4π1− g2

(1 + g2 − 2g cos θ)3/2, (9.12)

is completely characterized by the parameter g. It is possible to show that gcorresponds to the asymmetry factor of the function, i.e., 〈cos θ〉 = g, and that〈cos2 θ〉 = 1+2g2

3 . For g = 0, the function is isotropic. For g > 0, the function

190 Chapter 9

decreases monotonously as θ increases. The corresponding cumulative probabilityfunction is

F1(θ) =

1−g2

2g

[1

1−g −1

(1+g2−2g cos θ)1/2

]if g 6= 0 ,

(1− cos θ) /2 if g = 0 .(9.13)

This function can be inverted analytically to obtain the relation for θ(ξ).To work with Mie scattering functions, our MC programs make use of a table of

scattering angles θi, for i = 1 to 8001, with

2π

θi∫0

p(θ′) sin θ′dθ′ = (i− 1)/8000 (9.14)

evaluated numerically by a program based on Mie theory. The scattering angle θ(ξ)is obtained by interpolating between the values of θi and θi+1 with i−1 ≤ 8000 ξ ≤i.

The procedure used to simulate trajectories is based on a uniform randomnumber generator within the interval (0, 1). Any actual algorithm for computerrandom number generators is characterized by a period, so that after long sequencesof random numbers, the algorithm repeats the same sequence. As a consequence,there is a possibility that the generation of trajectories can get into a loop whenthe number of simulated trajectories is large. For our MC simulations, we used thealgorithm ran2 proposed in Ref. 4 with a period greater than 2·1018. For all the MCsimulations we carried out, the sequence of random numbers used never exceededthe period of the random number generator.

As long as the photons travel within the medium, their trajectories are simulatedby iterating the aforementioned rules that associate three pseudo-random numberswith the pathlengths between two subsequent scattering events and with the newdirection of propagation after each scattering event. Trajectories are followed untiltheir length l either becomes longer than a limit value lmax fixed by the user orthey leave the turbid medium. The choice of lmax determines the temporal range onwhich the TPSF can be reconstructed.

Details of the rules used for non-homogeneous media, to include the effect ofboundaries, and on the specific programs used for different geometries are given insubsequent sections.

As mentioned before, the TPSF is obtained by classifying in a histogram thecontributions of each trajectory according to its time of flight t = l/v (v is the speedof light in the turbid medium). The CW response for a steady-state source has beenobtained by adding the contribution of all trajectories independently of the time offlight. Since trajectories have been broken when their length becomes longer thanlmax, a careful check must be done to verify that the contribution of trajectories withl > lmax is negligible.


For the validation of a MC simulator, it would be very useful to compare theresults of simulations with results obtained from exact theories. Unfortunately, thereare very few examples of exact solutions for photon migration, and they pertainto the total transmittance and total reflectance of a homogeneous slab obtained forsimplified models of the scattering function.5, 6 For the validation of the programs,since the core of a MC program is actually the generation of the trajectories, wecompared MC results, for the statistics of the position where different orders ofscattering occur, with exact analytical expressions.7 Comparisons showed an ex-cellent agreement: For all the scattering functions that we used (both from the HGmodel and from Mie theory), discrepancies were within the standard deviation ofMC results, even when a large number of trajectories (108) was simulated, and therelative error was as small as 0.01%. As will be shown in Secs. 10.2.1 and 10.5.1,further validations have been provided by comparisons of MC results for the CWflux and the TR fluence with almost exact solutions of the RTE for the infinitemedium.

In order to develop a MC code, it is sufficient to know the probability lawsfor the trajectories. These laws come from the intrinsic physical meaning of theoptical properties of the medium, i.e., the extinction coefficient and the scatteringfunction. With a MC simulation, the RTE can be solved without the need to knowthe corresponding integro-differential equation, since the equation itself is notimplemented in the MC code.

9.3 Monte Carlo Program for the Infinite Homogeneous Medium

To study propagation through the infinite homogeneous medium illuminated byan isotropic point source that emits a light pulse at time t = 0, we simulated theresponse of receivers at different distances from the source. The code we used isbasically the same described in Refs. 8 and 9. Both the TR and the CW responsehave been evaluated for the radiance, fluence, and flux.

Since the source is isotropic, the problem has spherical symmetry, i.e., theresponse is identical in each point of a sphere concentric with the light source andonly depends on the distance r from the source. Without losing any information, theresponse for an elementary receiver at distance rk can thus be obtained consideringa spherical receiver of radius rk concentric with the source. To evaluate the radiance,the solid angle has been divided into 100 intervals. To obtain similar statistics in eachangular interval, the intervals have been chosen to have the same number of receivedphotons when an isotropic radiance is assumed. The jth interval of solid angle ∆Ωj

is identified by αj , the angle averaged over ∆Ωj , between the direction s and theunit vector outwardly directed r normal to the surface. To obtain the radiance fromthe MC simulation, recall that the power flowing through the elementary surfacedΣ of the sphere within the elementary solid angle dΩ around the direction s isgiven [Eq. (3.5)] by I(r, s, t)|r · s|dΣdΩ. The TPSF for the radiance in the ith time

192 Chapter 9

interval and jth angular interval is obtained as

I (rk, αj , ti, µa = 0) =1

Ntot4πr2k∆Ωj∆ti

Nk,j,i∑l=1

1| cosβl|

, (9.15)

where Ntot is the total number of drawn trajectories, Nk,j,i is the number of trajecto-ries that intersect the sphere of radius rk within the solid angle ∆Ωj with time offlight ti −∆ti/2 ≤ t < ti + ∆ti/2, and βl is the angle between r and the directionof motion of the photon when it intersects the sphere. Precautions have been takento avoid division by a small number when βl approaches π/2. We observe thateach trajectory can give multiple contributions to the radiance at different times anddistances.

According to the definition, the TPSF for the fluence is obtained adding theresponse for the radiance over the 100 angular intervals:

Φ (rk, ti, µa = 0) =100∑j=1

I (rk, αj , ti, µa = 0) ∆Ωj . (9.16)

To obtain the TPSF for the flux, again we exploit the symmetry of the problem,for which the only non-null component of the flux is the one along the radialdirection that is obtained as

Jr (rk, ti, µa = 0) =1

Ntot4πr2k∆ti

Nk,i∑l=1

cosβl| cosβl|

, (9.17)

where Nk,i is the number of trajectories that intersect the sphere of radius rk withinthe time interval ∆ti.

Being the trajectories evaluated for the non-absorbing medium, Eqs. (9.15),(9.16), and (9.17) represent the response when µa = 0. The response for theabsorbing medium can be obtained, according to a general property of the RTE[Eq.(3.16)], by multiplying the response for the non-absorbing medium by thescaling factor exp(−µavti). According to Eq. (3.17), the CW response has beenobtained integrating the TPSF. As an example, the CW radiance is

I (rk, αj , µa) =Nt∑i=1

I (rk, αj , ti, µa = 0) exp(−µavti)∆ti , (9.18)

where Nt is the number of temporal intervals, and ti is the average time over the ith

interval.As mentioned in Sec. 9.2, due to the stochastic nature of the method, the results of

MC simulations are noisy. Similar to a real experiment, we carried out independentMC simulations to evaluate the error (10 simulations for the results reported in the


CD-ROM), all with the same number of simulated trajectories, which we used toevaluate both the average and the standard error (standard deviation of the average)for each quantity of interest.

The TPSF for the radiance, fluence, and flux obtained from MC simulations canbe directly compared with the Green’s functions provided by different analyticalmodels.

9.4 Monte Carlo Programs for the Homogeneous and theLayered Slab

The programs we used are basically that described in Refs. 10 and 11. To studypropagation through the homogeneous or the layered slab, the rules to draw thetrajectories described in Sec. 9.2 must be modified to include both the effect of theboundaries and of the non-homogeneity of the medium.

Simulations have been carried out for a pencil beam source on the surface ofthe infinite slab at z = 0 that emits a light pulse at time t = 0 along the z-axis(the z-axis is perpendicular to the surface). To include the effect of boundaries, theincidence angle θi, the refraction angle θt, and the corresponding Fresnel reflectioncoefficient for unpolarized light RF (θi) [Eq. (3.47)] are first calculated each timethe trajectory hits the boundary of the slab (or of the current layer when a layeredslab is considered). RF (θi) represents the probability that the impinging light willbe reflected. To determine whether reflection or refraction occurs, a pseudo-randomnumber ξ is generated and compared with RF (θi). If ξ ≤ RF (θi), a reflectionoccurs, and the trajectory goes on in the same layer. The direction is updatedaccording to the reflection law, and a new step size is calculated. If ξ > RF (θi), arefraction occurs, and the new direction is calculated with Snell’s law. If refractionoccurs on the external surface of the slab, the trajectory is terminated, and itscontribution to the reflectance or to the transmittance is calculated; otherwise, a newstep size is generated using the new scattering coefficient, and the trajectory goes onin the new layer.

When a layered slab is considered, Eq. (9.9), which is used to generate thestep size `, should be modified to include the effect of the non-homogeneity of themedium. According to Eq. (9.3), when the step intersects volumes with a differentscattering coefficient, the cumulative probability function for ` is

F3(`) = 1− exp(−∑i

µsiì

), (9.19)

where ì and µsi are the length of the segment and the scattering coefficient in theith–volume, respectively. Therefore, the step size ` =

∑iì is obtained from

1− exp[−∑i

µsiì (ξ)]

= ξ . (9.20)

194 Chapter 9

One obtains the new direction after a scattering event by using the scatteringfunction of the layer where the scattering occurs. Trajectories are followed untilthey leave the slab or their length l becomes longer than a limit value lmax.

To calculate the reflectance and the transmittance, trajectories leaving the slabare classified in histograms on the base of the exit point and of the time of flight.Since the source is a pencil beam perpendicular to the slab and the turbid medium isisotropic, both the reflectance and the transmittance only depend on the distance ρfrom the light beam. Therefore, the receivers we have considered are rings coaxialwith the pencil beam.

For the homogeneous slab, the TPSF for the reflectance (for the transmittancesimilar expressions have been used) in the kth spatial interval and ith time intervalhas been obtained as

R (ρk, ti, µa) =Nk,i

Ntotπ(ρ2kmax− ρ2

kmin

)∆ti

exp(−µavti) , (9.21)

where Ntot is the total number of drawn trajectories, Nk,i is the number of trajecto-ries exiting at z = 0 within ρkmin

≤ ρ < ρkmax , with direction within the acceptancesolid angle of the receiver, and time of flight ti −∆ti/2 ≤ t < ti + ∆ti/2. Theexponential factor accounts for the effect of absorption when an absorbing mediumis considered. To evaluate the error on the TPSF, for each spatial and temporalinterval, we have also calculated the variance, from which the standard deviation σhas been obtained. Since each trajectory that intersects the receiver in the ith timeinterval gives the same contribution, the standard deviation results in12

σ (ρk, ti, µa) ∼=R (ρk, ti, µa)√

Nk,i

. (9.22)

In order to obtain a relative error of 0.01, it is necessary to simulate a number oftrajectories sufficient to obtain Nk,i = 104. Equation (9.22) is applicable for largevalues of Nk,i. The CW response has been obtained by integrating the TPSF asshown for the infinite medium.

To calculate the response for the absorbing layered medium from simulationscarried out for the non-absorbing slab, for each trajectory that intersects the receiverour MC code stores the partial pathlengths followed in each layer, and the TPSF isobtained using the property described in Sec. 3.4.2. The reflectance is obtained as

R (ρk, ti, µa) =1

Ntotπ(ρ2kmax− ρ2

kmin

)∆ti

Nk,i∑m=1

NL∏n=1

exp(−µanvntm,n) , (9.23)

where µan, vn, and tm,n are, respectively, the absorption coefficient, the speedof light, and the partial time of flight of the mth–trajectory in the nth–layer, andsummation is over the Nk,i trajectories that intersect the kth–receiver with total timeof flight in the ith–time interval. NL is the number of layers.


The partial time of flight tm,n has been also used to evaluate the TR mean timeof flight spent inside the nth–layer by photons received at distance ρk and at time tias

〈tn〉 (ρk, ti, µa) =∑Nk,i

m=1 tm,n∏NLn=1 exp(−µanvntm,n)∑Nk,i

m=1

∏NLn=1 exp(−µanvntm,n)

. (9.24)

As mentioned before, the MC results reported in the CD-ROM pertain to a pencilbeam source. In the MC programs we used, the effect of the specular reflection ofthe pencil beam on the input surface is not considered (this is equivalent to assumingthe source on the internal surface). Therefore, it is possible to directly comparethe TPSF obtained from MC simulations with the corresponding Green’s functionobtained from the DE or from hybrid models in which the response is provided for aunit isotropic point source placed at depth z0 = 1/µ

′s.

9.5 Monte Carlo Code for the Slab Containing anInhomogeneity

To simulate the effect of an inhomogeneity within the homogeneous or layeredslab, a procedure similar to the one described for the layered slab can also beused: It is sufficient to include in the summation of Eq. (9.20) also the opticalproperties of the volume occupied by the inhomogeneity. The perturbation canbe obtained from the comparison of two independent simulations carried out withand without the inhomogeneity. However, the expression for the standard deviationon the TPSF [Eq. (9.22)] clearly shows that to obtain the perturbation with a goodaccuracy a prohibitively large number of trajectories, with a prohibitively longcomputation time, may be necessary, especially if the perturbation is small. Asan example, with reference to the reflectance, if R (ρk, ti, µa) is evaluated witha relative error of 1%, the relative error on δR (ρk, ti, µa) would be 20% whenδR (ρk, ti, µa) /R (ρk, ti, µa) = 0.1, and 200% when δR (ρk, ti, µa) /R (ρk, ti, µa)= 0.01. To overcome this limit, we used a procedure, known as perturbation MonteCarlo,13 that makes use of scaling relationships to reduce the variance.14 Withthis method, we obtained, with a reasonable computation time, results suitable forcomparisons with analytical solutions based on a perturbation approach.

The MC procedure can be summarized in two steps:

1. In the first step, trajectories are generated with a conventional MC code forthe non-absorbing slab—homogeneous or layered. For all trajectories thatintersect the receiver within the acceptance angle, the information necessary toreconstruct the length of the path both within each layer and within the volumeVi of the inhomogeneity is stored together with the number of scattering eventsundergone within Vi.

2. In the second step, the stored information is used to obtain both the unper-turbed TPSF and the perturbation. The unperturbed response is obtained as

196 Chapter 9

shown in the previous section [Eqs. (9.21) or (9.23) for the reflectance]. Theabsorption and the scattering perturbations are obtained taking into accountthat if the scattering and/or the absorption coefficients within Vi are changedfrom µs0, µa0 to µsi, µai, the contribution to the TPSF of the mth trajectorychanges from the unperturbed value wm0 to

wmp = wm0

(µsiµs0

)kmexp

[− (µsi − µs0 + µai − µa0) vtm

], (9.25)

where v, km, and tm are the speed of light, the number of collisions undergone,and the time of flight within Vi, respectively. The terms that multiply wm0 inEq. (9.25) represent the ratio between the probability of following the sametrajectory after and before the introduction of the inhomogeneity, providedthat the scattering function is not changed.14 Evaluating, for each time interval∆ti, the average and the variance of the differences wmp−wm0, it is possibleto obtain the time-resolved perturbation together with its standard deviation.Furthermore, the time of flight tm within Vi can also be used to evaluate themean time of flight spent by received photons inside the inhomogeneity.

Although Eq. (9.25) is exact, the use of this MC procedure is limited by thenoise on the results. As for absorbing perturbations, it has been shown in Ref. 14that the procedure can be used for a wide range of values, both for the absorptioncoefficient and for the volume of the inhomogeneity. In contrast, due to the widerange of values that wmp − wm0 can assume when the scattering coefficient ischanged, for scattering inhomogeneities the procedure provides reliable resultsonly for sufficiently small variations of the scattering coefficient, provided that thevolume of the inhomogeneity is not too small (small variations of the scatteringcoefficient limit the range of values for wmp − wm0, while large volumes increasethe number of involved trajectories improving the statistics).

This MC procedure has an important feature that makes it particularly suitablefor generating reference results to compare against analytical solutions based on aperturbation approach. In fact, since the effect of the inhomogeneity is evaluatedas a variation of the probability to receive a finite set of trajectories (all those thatintersect the volume of the inhomogeneity) when the inhomogeneity is introducedinside the medium, it is possible that the perturbation is statistically significanteven when it is significantly smaller than statistical fluctuations on the unperturbedresponse. It is therefore possible to obtain reliable results for small perturbations ofthe optical properties for which the analytical models are expected to be accurate.

The MC results reported in the CD-ROM have been obtained using two MCprograms that differ in how they store trajectory information. For each receivedtrajectory, the first program stores the coordinates of all the scattering points. Thesecond stores only the NL + 1 lengths within the NL layers and within Vi togetherwith the number of collisions within Vi. The first program has the advantage that,with the stored information, it is possible to evaluate the perturbation introduced by


one or more inhomogeneities in whichever position within the slab, but the largeamount of data that must be stored limits the number of simulated trajectories andthen the accuracy of the results. The second program has the advantage of requiringless memory to store the information, and thus a larger number of trajectories canbe simulated, but the perturbation can be evaluated only for the inhomogeneity forwhich the information has been stored.

9.6 Description of the Monte Carlo Results Reported in theCD-ROM

In this section, a brief description is provided for the MC results reported in theMC_Results directory of the enclosed CD-ROM. In both the enclosed softwareand in the MC code, we assumed for the speed of light in vacuum the value c =0.299792458 mm ps−1.15

All the results are presented in Excel files (*.xls). A summary of the results isalso provided in the index.html file, from which it is possible to open all the resultsfiles.

9.6.1 Homogeneous infinite medium

The medium is illuminated by a point source at r = 0 that emits a light pulse at timet = 0.

The results are summarized in the following files:

• Infinite_CW_g=0.0.xls

• Infinite_CW_g=0.9.xls

• Infinite_TR_g=0.0.xls

• Infinite_TR_g=0.9.xls

The files Infinite_CW_g=0.0.xls and Infinite_CW_g=0.9.xls report the CW re-sults for a medium with asymmetry factor g = 0.0 (isotropic scattering) and g =0.9, respectively. Each file contains three worksheets: Fluence_CW, Flux_CW, andRadiance_CW in which are reported the responses for the fluence, flux, and radiance,respectively. The results for the fluence and for the flux are reported as a function ofthe source-receiver distance r for several values of the absorption coefficient. Theresults for the radiance are reported as a function of r and of the angle with theradial direction for µa = 0.001 mm−1.

The files Infinite_TR_g=0.0.xls and Infinite_TR_g=0.9.xls report the TR resultsfor the scattering functions with g = 0.0 and g = 0.9. In each file, the worksheetsFluence_TR and Flux_TR report the TPSFs for the fluence and the flux, respectively,for several values of r. Three worksheets are devoted to the results for the TR radi-ance for r = 1, 5, and 10 mm. The TPSFs pertain to a non-absorbing medium. The

198 Chapter 9

results have been reported focusing on short times for which the larger differencesare expected when comparisons with results of approximated theories are made.

All the results are reported for µ′s = 1 mm−1 and refractive index n = 1. The

results for different values of µ′s can be easily obtained making use of the scaling

properties of the RTE reported in Sec. 3.4.1.

9.6.2 Homogeneous slab


• s=40mm,n_i=1.4,n_e=1.4,g=0.0.xls

• s=40mm,n_i=1.4,n_e=1.4,g=0.9.xls

• s=40mm,n_i=1.4,n_e=1.0,g=0.0.xls

• s=40mm,n_i=1.4,n_e=1.0,g=0.9.xls

• s=4mm,n_i=1.4,n_e=1.4,g=0.9.xls

• s=4mm,n_i=1.4,n_e=1.0,g=0.9.xls

The values for the thickness of the slab s for the refractive index inside the slabn_i, and in the external medium n_e, and for the asymmetry factor of the scatteringfunction g are indicated in the name of the file. The source is a pencil beam normallyincident onto the slab emitting a light pulse at time t = 0. The worksheets CW_Refland CW_Transm report the CW reflectance and the CW transmittance as a functionof the distance of the receiver from the light beam axis for some values of theabsorption coefficient. The acceptance solid angle of receivers is 2π (all photonsimpinging the receiver are detected). With the MC simulations, trajectories havebeen broken when their length becomes longer than a limit value lmax, and attentionmust be paid when using the CW results for a slab with s = 40 mm for smallvalues of absorption (µa = 0.001 mm−1) and large distances from the light beamaxis (ρ & 40 mm) since they could be underestimated. The worksheets TR_Refland TR_Transm report the TPSFs for the reflectance and the transmittance of thenon-absorbing slab together with the corresponding standard deviation evaluatedwith Eq. (9.22). Since Eq. (9.22) cannot be used for Nk,i = 0, in the results reportedin the CD-ROM, we have set σ = 1030 when Nk,i = 0 to underscore that the resultis not significant from a statistical point of view. The worksheet Total Refl&Transmsummarizes the results for the total reflectance and total transmittance both for theCW and the TR response.

We point out that the CW response for an arbitrary value of absorption can beobtained from the TPSF for the non-absorbing medium by using Eq. (3.17).


9.6.3 Layered slab


• Lay_Scat_Mus1=1_Mus2=0.5.xls

• Lay_Scat_Mus1=1_Mus2=2.xls

• Lay_Abs_Mua1=0_Mua2=0.01.xls

• Lay_Abs_Mua1=0.01_Mua2=0.xls

• Lay_Refr_n1=1_n2=1.4.xls

• Lay_Refr_n1=1.4_n2=1.xls

All files pertain to the TR reflectance from a two-layer slab illuminated by apencil beam. Results are reported for three receivers with acceptance solid angle= 2π at distances ρ = 10, 20, and 30 mm. Each file contains two worksheets: Theworksheet TPSF reports the TPSF for the reflectance; the worksheet 〈t_i〉(t) reportsthe TR mean time of flight spent by received photons inside the layers.

The files Lay_Scat_Mus1=1_Mus2=0.5.xls and Lay_Scat_Mus1=1_Mus2=2.xlspertain to slabs with a layered architecture of scattering: They have µ

′s2 = 0.5 and

2 mm−1, respectively. The other parameters are µ′s1 = 1 mm−1, g1 = g2 = 0,

µa1 = µa2 = 0, n1 = n2 = 1.4, ne = 1, s1 = 4 mm, and s2 = 100 mm.Slabs with a layered architecture of absorption are instead considered in the

files Lay_Abs_Mua1=0_Mua2=0.01.xls and Lay_Abs_Mua1=0.01_Mua2=0.xls:They have µa1 = 0 and µa2 = 0.01 mm−1, and µa1 = 0.01 mm−1 and µa2 =0, respectively. The other parameters are µ

′s1 = µ

′s2 = 1 mm−1, g1 = g2 = 0,

n1 = n2 = 1.4, ne = 1, s1 = 4 mm, and s2 = 100 mm.The files Lay_Refr_n1=1_n2=1.4.xls and Lay_Refr_n1=1.4_n2=1.xls pertain to

slabs with a layered architecture of refractive index: They have n1 = 1 and n2 =1.4, and n1 = 1.4 and n2 = 1, respectively. The other parameters are µ

′s1 = µ

′s2 =

1 mm−1, g1 = g2 = 0, µa1 = µa2 = 0.01 mm−1, ne = 1, s1 = 8 mm, and s2 =100 mm.

9.6.4 Perturbation due to inhomogeneities inside the homogeneousslab


• Pert_Refl_s=4000mm.xls

• Pert_Transm_s=40mm.xls

• Scan_Pert_Refl_s=4000mm.xls

• Scan_Pert_Transm_s=40mm.xls

200 Chapter 9

• Pert_Refl_s=4mm.xls

The file Pert_Refl_s=4000mm.xls reports the TR absorption and the TR scat-tering perturbations for reflectance from a slab with thickness s = 4000 mm (semi-infinite medium), µ

′s0 = 0.5 mm−1, g = 0, µa0 = 0, and ni = ne = 1.4. The

source is a pencil beam normally incident at (0, 0, 0). The receiver is circular witha radius of 2 mm and an acceptance solid angle of 2π centered at (30, 0, 0) mm.The inhomogeneity is spherical and centered at (15, 0, 15) mm. The results arereported for different values of the volume of the inhomogeneity: Vi = 1, 10, 100,500, and 1000 mm3 in the worksheets V = 1, V = 10 , V = 100, V = 500, and V =1000, respectively. Each worksheet reports both the absorption and the scatteringperturbation together with the TPSF for the unperturbed medium.

The file Pert_Transm_s=40mm.xls reports results for the transmittance of a slabwith s = 40 mm and the same optical properties of file Pert_Refl_s=4000mm.xls.Both the receiver, circular with a radius of 3 mm, and the spherical inhomogeneity,at (0, 0, 20) mm, are coaxial with the light beam. Also, the results for transmittanceare reported in different worksheets for Vi = 1, 10, 100, 500, and 1000 mm3.

The file Scan_Pert_Refl_s=4000mm.xls reports examples of perturbation for thesame medium and the same receiver of file Pert_Refl_s=4000mm.xls. Results arereported for the scanning of an inhomogeneity with Vi = 500 mm3 along the x-and the z-axis. The results for the absorption and for the scattering perturbation arereported in worksheets Absorption and Scattering, respectively. Similarly, the fileScan_Pert_Transm_s=40mm.xls reports the scanning for the same medium and thesame receiver of file Pert_Transm_s=40mm.xls.

The file Pert_Refl_s=4mm.xls reports the perturbations for reflectance from aslab with s = 4 mm, µ

′s0 = 1 mm−1, g = 0.9, µa0 = 0, and ni = ne = 1.4. The

source is at (0, 0, 0), and the receiver, circular with a radius of 0.2 mm and anacceptance solid angle of 2π, is at (3,0,0) mm. The spherical inhomogeneity withVi = 1 mm3 is at (1.5, 0, 1.5) mm.

For all the files, the absorption perturbation has been evaluated using µai suchthat Vi (µai − µa0) = 0.01 mm2 and µ

′si = µ

′s0, and the scattering perturbation using

µ′si such that Vi (Di −D0) = 0.01 mm4 [with Di = 1/(3µ

′si) and D0 = 1/(3µ

′s0)]

and µai = µa0. With these values of µai and µ′si, perturbations are sufficiently

small to be predictable with the Born approximation. The MC results are thussuitable for comparisons with the analytical expression from the perturbation theoryof Chapter 7.


References

1. W. Becker, Advanced Time-correlated Single Photon Counting Techniques,Springer, Berlin (2005).


3. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys.J. 93, 70–83 (1941).



6. R. G. Giovanelli, “Reflection by semi-infinite diffusers,” Opt. Acta 2, 153–162(1955).

7. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relation-ships for the statistical moments of scattering point coordinates for photonmigration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).

8. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media:point-source case,” Appl. Opt. 30, 2031–2041 (1991).


10. F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, “Photon migrationthrough a turbid slab described by a model based on diffusion approximation.II. Comparison with Monte Carlo results,” Appl. Opt. 36, 4600–4612 (1997).

11. F. Martelli, A. Sassaroli, Y. Yamada, and G. Zaccanti, “Analytical approximatesolutions of the time-domain diffusion equation in layered slabs,” J. Opt. Soc.Am. A 19, 71–80 (2002).

12. S. Brandt, Statistical and Computational Methods in Data Analysis, North-Holland, Amsterdam (1970).


202 Chapter 9

14. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, andG. Zaccanti, “Monte Carlo procedure for investigating light propagation andimaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).


Chapter 10

Comparisons of AnalyticalSolutions with Monte CarloResults10.1 Introduction

In this chapter, we detail comparisons between the solutions of the RTE describedin Chapters 4–7 and the reference MC results reported in the enclosed CD-ROM.All of the solutions based on the diffusion approximation used for comparisonshave been calculated using the code and the subroutines included in the CD-ROM.Comparisons with MC results thus provide a validation of the enclosed software.Furthermore, since almost all the analytical solutions presented are approximatesolutions of the RTE, comparisons also provide information on their accuracyand range of applicability. As validation of the Diffusion&Perturbation software,comparisons are shown in Sec. 10.2 for the homogeneous infinite medium, thehomogeneous slab, and the homogeneous slab with an internal inhomogeneity.Comparisons for the layered slab and the hybrid models are reported in Secs. 10.4and 10.5, respectively. For the infinite medium, both the MC and the analyticalsolutions pertain to a point-like isotropic source. For the slab, MC results pertain toa pencil beam. The corresponding analytical results, according to the discussion inSec. 4.2, pertain to an isotropic point-like source inside the medium at a distancezs = 1/µ

′s from the entrance point of the beam. All the figures of this chapter, apart

from Fig. 10.31 (the MC results used for this figure have not been reported in theCD-ROM), can be reproduced using the software and the MC results provided withthe enclosed CD-ROM.

Comparisons of analytical models with MC results are shown through the ratioMC/MODEL. The results are presented without any filtering even though someTR curves are very noisy. We preferred this representation because the noisegives information on the accuracy of the reference MC results, and because afiltering procedure can severely distort the response at early times where the largerdiscrepancies between the results from analytical models and reference results areexpected. In the figures, the error bars on the MC results have not been reported,

203

204 Chapter 10

although the standard deviations are available. Nevertheless, information of theaccuracy of MC results is provided by the noise on the curves, which is on the orderof the standard deviation. All the figures with TR results will be focused on theresponse at short times, i.e., times at which the larger differences between the resultsfrom the DE and the reference MC results are expected.

10.2 Comparisons Between Monte Carlo and the DiffusionEquation: Homogeneous Medium

In this section, we compare MC results with solutions of the DE.

10.2.1 Infinite homogeneous medium

All the results reported pertain to a point source at r = |~r| = 0 that emits a lightpulse at time t = 0 in a medium with µ

′s = 1 mm−1 and refractive index n = 1.

The TR results are shown only for the non-absorbing medium. However, as forthe ratios between DE and MC results used for comparisons, we point out that theratios are independent of µa since both MC and DE results depend on µa by thefactor exp(−µavt). Results and comparisons are shown for the medium both withan isotropic (g = 0) and anisotropic (g = 0.9) scattering function. The solutions ofthe DE for the fluence have been calculated with the Diffusion&Perturbation code.Flux and radiance have been obtained using the relevant subroutines described inSec. 4.1.

10.2.1.1 Time-resolved results

Examples of TR results are reported in Figs. 10.1–10.4. Figures 10.1 and 10.2 showexamples of TPSFs obtained from MC simulations for the fluence rate and for theflux, respectively. The corresponding results from the DE are shown in Fig. 4.1.The comparison of MC results with the solution of the DE is shown plotting theratio MC/DE versus time. To look for a general rule to establish the accuracy ofthe DE, the ratio is also plotted as a function of t/t0, where t0 = r/v is the time offlight for ballistic photons. We notice that, for the results reported in the figures, t0varies from 3.3 ps to 66.7 ps when r varies from 1 to 20 mm. Comparison betweenMC results for g = 0 and 0.9 shows almost identical results, as expected by thesimilarity relation presented in Sec. 2.4. Differences are appreciable (the responseis significantly larger for g = 0) only for early received photons (photons receivedafter a smaller number of scattering events).

Comparisons of Figs. 10.1 and 10.2 show that, apart from early received photons,both the fluence and the flux from the DE are in excellent agreement with MC results.Discrepancies also remain for early photons and large distances, for which evenearly photons are expected to have experienced a large number of scattering events.As an example, since the average number of scattering events for early photons isr/`s = rµs, about 200 scattering events are expected at r = 20 mm for the mediumwith g = 0.9. The agreement is slightly better with the results for g = 0.9, i.e., when

Comparisons of Analytical Solutions with Monte Carlo Results 205

0 200 400 600 80010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Flue

nce

(mm

-2ps

-1)

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

g = 0

0 200 400 600 80010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

g = 0.9

Flue

nce

(mm

-2ps

-1)

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

0 200 400 600 8000.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

Flue

nce

MC

/DE

Time (ps)0 200 400 600 800

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20Fl

uenc

e M

C/D

E

Time (ps)

0 5 10 15 200.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

Flue

nce

MC

/DE

t/t00 5 10 15 20

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

Flue

nce

MC

/DE

t/t0

Figure 10.1 Infinite medium with µ′s = 1 mm−1, µa = 0, and n = 1. Examples of TPSFs

for the fluence rate from MC simulations (first row) and comparisons with the DE. Compar-isons are shown plotting the ratio MC/DE as a function both of t and of t/t0 (second andthird row, respectively). Results are reported both for g = 0 and 0.9 (left and right column,respectively).

206 Chapter 10

0 200 400 600 80010-910-810-710-610-510-410-310-210-1

Flux

(mm

-2ps

-1)

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

g = 0

0 200 400 600 80010-910-810-710-610-510-410-310-210-1

Flux

(mm

-2ps

-1)

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

g = 0.9

0 200 400 600 8000.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Flux

MC

/DE

Time (ps)0 200 400 600 800

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Flux

MC

/DE

Time (ps)

0 5 10 15 200.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Flux

MC

/DE

t/t00 5 10 15 20

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Flux

MC

/DE

t/t0

Figure 10.2 Same as Fig. 10.1, but for the flux.


the scattering function is forward peaked, which usually occurs for light propagationthrough actual turbid media. For r & 5 mm and t/t0 &4, discrepancies for g = 0.9remain within ' 2%. For shorter distances, discrepancies remain within '2% fort & 4`

′/v (`

′= 1/µ

′s is the transport mean free path).

0 100 200 300 4000

1

2

3

I1

Time (ps)

r = 1 mm r = 5 mm r = 10 mm r = 20 mm

g = 0.9

0 5 10 15 200

1

2

3

I1

t/t0

0 100 200 300 4000

1

2

3

4

5

I2

Time (ps)0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

I2

t/t0

Figure 10.3 Infinite medium with µ′s = 1 mm−1, µa = 0, g = 0.9, and n = 1. The MC results

for the ratios I1 and I2 [Eqs. (10.1) and (10.2)] are reported as a function both of t and oft/t0.

Since the simplifying assumptions necessary to obtain the DE (Sec. 3.5.1) canbe summarized by the two inequalities

I1 =[3| ~J (~r, t) |

]/Φ (~r, t) 1 (10.1)

and

I2 =1vµ′

s

∣∣∣∣∣∂ ~J (~r, t)∂t

∣∣∣∣∣ / ∣∣∣ ~J (~r, t)∣∣∣ 1 , (10.2)

208 Chapter 10

it may be interesting to show how I1 and I2 change as t increases. The MC resultsfor I1 and I2 are reported in Fig. 10.3 as a function both of t and t/t0. The figurereports the results for g = 0.9. For I1, the curves for the different distances arealmost indistinguishable when plotted as a function of t/t0 and show that I1 .0.3 when t/t0 & 4. For I2, the curves remain distinguishable and show that whent/t0 & 4, apart from the shortest distance, I2 . 0.1. These values of I1 and I2can thus be indicated as limit values to fulfill the inequalities. The rapid decreaseof I1 as t increases shows how more quickly the flux decreases with respect to thefluence.

0 20 40 60 80 10010-5

10-4

10-3

10-2

10-1

Rad

ianc

e (m

m-2

ps-1

sr-1

)

Time (ps)

= 5.7 deg = 45.6 deg = 84.3 deg = 135.6 deg = 174.3 deg

r = 1 mmg = 0.9

0 200 400 600 800 100010-8

10-7

10-6

10-5

Rad

ianc

e (m

m-2

ps-1

sr-1

)

Time (ps)

= 5.7 deg = 45.6 deg = 84.3 deg = 135.6 deg = 174.3 deg

r = 10 mmg = 0.9

0 45 90 135 1800.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Rad

ianc

e M

C/D

E

(deg)

t/t0 = 2 t/t0 = 4.1 t/t0 = 8 t/t0 = 16.1

r = 1 mmg = 0.9

0 45 90 135 1800.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Rad

ianc

e M

C/D

E

(deg)

t/t0 = 2 t/t0 = 4.1 t/t0 = 8 t/t0 = 16.1

r = 10 mmg = 0.9

Figure 10.4 Infinite medium with µ′s = 1 mm−1, µa = 0, g = 0.9, and n = 1. Examples of

TPSF for the radiance from MC simulations and comparisons with the DE. The TPSFs arereported for some values of α for r = 1 and 10 mm. Comparisons are shown plotting theratio MC/DE as a function of α for some values of t/t0.

Examples of MC results for the radiance and comparisons with the DE arereported in Fig. 10.4. The TPSFs are reported for some values of the angle αbetween the directions s and r, for r = 1 and 10 mm. Comparisons are shown


plotting the ratio MC/DE as a function of α for some values of t/t0. Results arereported for g = 0.9. Also, for the radiance, we observe that results from the DE arein excellent agreement with MC results provided t/t0 & 4 for r & 5 mm and t &4`′/v for shorter distances.We point out a feature of the figures showing comparisons between DE and

MC results that is also common to figures reported in subsequent sections for othergeometries: As the comparisons reported use the ratio between MC and DE results,the information on the response for t < t0 is partially lost since the correspondingMC result is 0 while the results for DE is 6= 0.

10.2.1.2 Continuous wave results

Examples of CW MC results and comparisons with the DE solutions are reported inFigs. 10.5–10.8. Results are shown both for g = 0 and g = 0.9.

0 5 10 15 2010-710-610-510-410-310-210-1100101

Flue

nce

(mm

-2ps

-1)

r (mm)

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

g = 0

0 5 10 15 2010-710-610-510-410-310-210-1100101

Flue

nce

(mm

-2ps

-1)

r (mm)

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

g = 0.9

0 5 10 15 200.80

0.85

0.90

0.95

1.00

1.05

1.10

Flue

nce

MC

/DE

r (mm)0 5 10 15 20

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Flue

nce

MC

/DE

r (mm)

Figure 10.5 MC results for the fluence for an infinite medium with µ′s = 1 mm−1 and com-

parison with DE results. The fluence is reported as a function of r for some values of µaboth for g = 0 and 0.9.

210 Chapter 10

0 5 10 15 2010-710-610-510-410-310-210-1100101

Flux

(mm

-2)

r (mm)

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

g = 0

0 5 10 15 2010-710-610-510-410-310-210-1100101

Flux

(mm

-2)

r (mm)

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

g = 0.9

0 5 10 15 200.80

0.85

0.90

0.95

1.00

1.05

1.10

Flux

MC

/DE

r (mm)0 5 10 15 20

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Flux

MC

/DE

r (mm)

0 5 10 15 200.988

0.990

0.992

0.994

0.996

0.998

1.000

1.002

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

Flux

MC

/DE

r (mm)0 5 10 15 20

0.988

0.990

0.992

0.994

0.996

0.998

1.000

1.002

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

Flux

MC

/DE

r (mm)

Figure 10.6 Same as Fig. 10.5, but for the flux.


Figures 10.5 and 10.6 show MC results for the fluence and for the flux, respec-tively, as a function of r for some values of µa. The corresponding results fromthe DE are shown in Fig. 4.2. Also, CW results for g = 0 and 0.9 are almostidentical. Differences are only appreciable observing the ratio with DE results. Forthe fluence, discrepancies remain within 1% for µa . 0.01 mm−1 and r & 2 mm.The agreement is slightly better with the results for g = 0.9 for small values of µaand r, and vice versa for large values of µa and r. For r & 2 mm and large values ofabsorption, the DE significantly overestimates the CW response. For large valuesof absorption, the flux is also significantly overestimated by the DE, but for smallvalues agreement with MC results is almost exact at all the distances and for bothvalues of g. To underscore this result, Fig. 10.6 reports separate comparisons forsmall values of absorption (µa ≤ 0.01 mm−1). From this figure, we observe that forµa = 0.001 mm−1 discrepancies remain smaller than 0.1%. This exceptionally goodagreement is not surprising since, as we pointed out in Sec. 4.1 and Appendix E,for a non-absorbing medium, the solution of the DE for the flux, which becomesJ (r) = 1/

(4πr2

), represents the exact solution of the RTE and is independent of

the scattering properties of the medium. Unfortunately, the MC code we have used(see Sec. 9.3) cannot provide the CW response for µa = 0 (it would be necessary tofollow each trajectory for an infinitely long time). Nevertheless, the excellent agree-ment of MC results for small values of absorption with the almost exact solution ofthe DE can be considered as a further validation of the MC code we have used.

0 5 10 15 200.00.20.40.60.81.01.21.41.61.82.0

I3

r (mm)

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

g = 0

0 5 10 15 200.00.20.40.60.81.01.21.41.61.82.0

I3

r (mm)

a = 0.001 mm-1

a = 0.005 mm-1

a = 0.01 mm-1

a = 0.05 mm-1

a = 0.1 mm-1

g = 0.9

Figure 10.7 The MC results for the ratio I3 [Eq. (10.3)] are reported as a function of r for aninfinite medium with µ

′s = 1 mm−1, for some values of µa, both for g = 0 and 0.9.

Also, for CW results, we report examples of MC results to understand when wecan consider the inequality fulfilled:

I3 = 3| ~J (~r) |/Φ (~r) 1 , (10.3)

which represents the simplifying assumption necessary to obtain the DE for CWsources. Examples of results are shown in Fig. 10.7 in which I3 has been plotted

212 Chapter 10

0 45 90 135 18010-4

10-3

10-2

10-1

100

Rad

ianc

e (m

m-2

sr-1

)

(deg)

r = 0.5 mm r = 1 mm r = 2 mm r = 5 mm r = 10 mm r = 20 mm

g = 0

0 45 90 135 18010-4

10-3

10-2

10-1

100

Rad

ianc

e (m

m-2

sr-1

)

(deg)

r = 0.5 mm r = 1 mm r = 2 mm r = 5 mm r = 10 mm r = 20 mm

g = 0.9

0 45 90 135 18010-1

100

101

102

Rad

ianc

e M

C/D

E

(deg)

r = 0.5 mm r = 1 mm r = 2 mm

g = 0

0 45 90 135 18010-1

100

101

102

Rad

ianc

e M

C/D

E

(deg)

r = 0.5 mm r = 1 mm r = 2 mm

g = 0.9

0 45 90 135 1800.8

0.9

1.0

1.1

1.2

Rad

ianc

e M

C/D

E

(deg)

r = 5 mm r = 10 mm r = 20 mm

g = 0

0 45 90 135 1800.8

0.9

1.0

1.1

1.2

Rad

ianc

e M

C/D

E

(deg)

r = 5 mm r = 10 mm r = 20 mm

g = 0.9

Figure 10.8 Infinite medium with µ′s = 1 mm−1 and µa = 0.001 mm−1. Examples of MC

results and comparisons with the DE for the CW radiance. Results are reported as a functionof α for some values of r.


as a function of r for some values of µa both for g = 0 and 0.9. The figure showsthat for values of µa . 0.01 mm−1 and r & 2 mm, for which discrepancies remainwithin 1% for the fluence and the flux, I3 . 0.6.

Examples of MC results for the CW radiance and comparisons with the DE arereported in Fig. 10.8. The radiance is reported for several values of r as a functionof the angle α for a medium with µa = 0.001 mm−1 for g = 0 and 0.9. Also, forthe CW radiance, there is an excellent agreement with MC results provided r &2 mm. For short distances, there are significant discrepancies, especially near α =180 degrees, where the DE also provides negative values.


All the MC results reported pertain to a pencil beam source on the surface of theslab that emits a light pulse at time t = 0. The corresponding DE results pertain toan isotropic point source at zs = 1/µ

′s, and the outgoing flux has been calculated

with Fick’s law. Results are reported for slabs with µ′s = 1 mm−1 and internal

refractive index ni = 1.4. Two values, 1 and 1.4, have been considered for therelative refractive index nr = ni/ne (ne is the refractive index of the externalmedium). Results are shown for a scattering function with asymmetry factor g =0.9 representative of many real media.

10.2.2.1 Time-resolved results

Examples of TR results are reported in Figs. 10.9–10.11. Figure 10.9 reportscomparisons between the TPSFs for the reflectance from the DE and from MCsimulations for three values of the source-receiver distance ρ for a slab 40 mm thick.Figure 10.10 reports the results for the transmittance. Receivers considered in MCsimulations are rings coaxial with the pencil light beam source (ρ indicates theaverage radius of the ring). The DE results, obtained from Eqs. (4.32) and (4.33),pertain to receivers identical to those used for MC simulations. The ratio betweenMC and DE results is also plotted as a function of t/t0 (t0 is the ballistic time tocover the source-receiver distance).

For transmittance discrepancies between DE and MC, results are significantonly for early times and involve a very small fraction of transmitted photons. Forreflectance, there are discrepancies, especially at small distances and early times(notably for times shorter than the time of flight for ballistic photons t0), that caninvolve a significant fraction of received photons. For late times, there is an almostexact agreement when nr = 1.4; while for nr = 1, the DE underestimates the resultsof about 2%.

Figure 10.11 reports results for the total TR reflectance and the total TR trans-

214 Chapter 10

0 100 20010-7

10-6

10-5

10-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

MC nr = 1.4 MC nr = 1 DE nr = 1.4 DE nr = 1

= 5.1 mm

0 10 20 300.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

MC/DE

t/t0

MC/DE nr = 1.4 MC/DE nr = 1

= 5.1 mm

0 100 200 300 40010-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)


= 10.2 mm

0 10 20 300.8

0.9

1.0

1.1

1.2

MC/DE

t/t0


= 10.2 mm

0 200 400 60010-11

10-10

10-9

10-8

10-7

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)


= 20.6 mm

0 10 20 30 400.8

0.9

1.0

1.1

MC/DE

t/t0


= 20.6 mm

Figure 10.9 TR reflectance of a homogeneous slab with s = 40 mm, µ′s = 1 mm−1, g =

0.9, µa = 0, and ni = 1.4. Comparison between DE and MC results are shown for differentvalues of ρ and nr.


mittance of slabs with s = 40 and 4 mm. For s = 4 mm, the results are reported bothfor nr = 1 and 1.4; for s = 40 mm results are reported only for nr = 1.4. There isan excellent agreement between DE and MC results for s = 40 mm. Discrepanciesare appreciable only for early times and involve only a small fraction of reflected ortransmitted photons. For s = 4 mm, there are appreciable differences in the shapeof the curves, especially for nr = 1.4. However, as will be shown in Fig. 10.13, anexcellent agreement remains for the CW results.

0 2000 4000 600010-11

10-10

10-9

10-8

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)


= 1.5 mm

0 10 200.6

0.7

0.8

0.9

1.0

1.1

1.2

MC/DE

t/t0


= 1.5 mm

0 2000 4000 600010-12

10-11

10-10

10-9

Tran

smitt

ance

(mm

-2ps

-1)

Time (ps)


= 39.9 mm

0 10 200.6

0.7

0.8

0.9

1.0

1.1

1.2

MC/DE

t/t0


= 39.9 mm

Figure 10.10 TR transmittance of a homogeneous slab with s = 40 mm, µ′s = 1 mm−1, g =

0.9, µa = 0, and ni = 1.4. Comparison between DE and MC results are shown for differentvalues of ρ and nr.

216 Chapter 10

0 2500 5000 7500 1000010-7

10-6

10-5

10-4

10-3

10-2

10-1

R, T

(ps-1

)

Time (ps)

MC Total Reflectance MC Total Transmittance DE Total Reflectance DE Total Transmittance

s = 40 mmnr = 1.4

0 2500 5000 7500 100000.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MC/DE

Time (ps)

MC/DE Total Reflectance MC/DE Total Transmittance

s = 40 mmnr = 1.4

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 01 0 - 6

1 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

R,T(

ps-1 )

T i m e ( p s )

M C T o t a l R e f l e c t a n c eM C T o t a l T r a n s m i t t a n c eD E T o t a l R e f l e c t a n c eD E T o t a l T r a n s m i t t a n c e

s = 4 m mn r = 1 . 4

0 100 200 300 400 500 6000.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MC/DE

Time (ps)


s = 4 mmnr = 1.4

0 100 200 300 40010-6

10-5

10-4

10-3

10-2

10-1

R, T

(ps-1

)

Time (ps)


s = 4 mmnr = 1

0 100 200 300 4000.6

0.8

1.0

1.2

1.4

MC/DE

Time (ps)


s = 4 mmnr = 1

Figure 10.11 Comparison between DE and MC results for the total TR reflectance andtransmittance of homogeneous slabs with s = 40 and 4 mm, µ

′s = 1 mm−1, g = 0.9, µa =

0, and ni = 1.4. Results for s = 4 mm are shown both for nr = 1 and 1.4.


10.2.2.2 Continuous wave results

Comparisons between DE and MC results are reported in Figs. 10.12 and 10.13.Figure 10.12 reports the results for the CW reflectance and transmittance as afunction of ρ for µa = 0.01 mm−1. For transmittance, discrepancies remain within2%; for reflectance, they are within 3% only for ρ > 10 mm. Discrepancies increaseas the absorption coefficient increases.

0 10 20 30 4010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Ref

lect

ance

, Tra

nsm

ittan

ce (m

m-2

)

Distance (mm)

MC Transmittance nr = 1 MC Transmittance nr = 1.4 MC Reflectance nr = 1 MC Reflectance nr = 1.4

0 10 20 30 400.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

MC/DE

Distance (mm)

MC/DE Transmittance nr = 1 MC/DE Transmittance nr = 1.4 MC/DE Reflectance nr = 1 MC/DE Reflectance nr = 1.4

Figure 10.12 Comparison between DE and MC results for the CW reflectance and the CWtransmittance from a homogeneous slab with s = 40 mm, µ

′s = 1 mm−1, g = 0.9, and µa =

0.01 mm−1. Results are shown both for nr = 1 and 1.4.

Comparisons for the total reflectance R and the total transmittance T , i.e., thefraction of emitted photons exiting at z = 0 and z = s, respectively, are shown inFig. 10.13. The figure also shows the comparison for the mean pathlengths 〈l〉Rand 〈l〉T , followed by photons before exiting at z = 0 and z = s. The results arereported as a function of µa for s = 4 mm and nr = 1.4. The results from the DEare in excellent agreement with MC results for moderate values of absorption. Forµa < 0.01 mm−1, discrepancies on R and T are within 2%. For large values of µa,the agreement also remains good, especially for R. Even for µa = µ

′s = 1 mm−1,

discrepancies remain within 50%. The agreement is even better for nr = 1 or forlarger values of s. Therefore, the solution of the DE for R [Eq. (4.36)] is suitable fordeveloping inversion procedures to reconstruct the spectral dependence of µa andµ′s from measurements of spectral reflectance.

10.3 Comparison Between Monte Carlo and the DiffusionEquation: Homogeneous Slab with an InternalInhomogeneity

As described in Sec. 9.6, MC results have been reported for inhomogeneitiesfor which µ

′si = µ

′s0 and Viδµa = 0.01 mm2 for absorption perturbations, and

µai = µa0 and ViδD = 0.01 mm4 for scattering perturbations, with δµa = µai−µa0

218 Chapter 10

10-4 10-3 10-2 10-1 10010-3

10-2

10-1

100

R, T

a (mm-1)


10-4 10-3 10-2 10-1 1001

10

100

<lR

>, <

l T> (m

m)

a (mm-1)

MC <lR> MC <lT> DE <lR> DE <lT>

10-4 10-3 10-2 10-1 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

MC

/DE

a (mm-1)


10-4 10-3 10-2 10-1 1000.5

1.0

1.5

2.0 M

C/D

E

a (mm-1)

MC/DE <lR> MC/DE <lT>

Figure 10.13 Comparison between DE and MC results for the total reflectance, the totaltransmittance, and the mean pathlengths for a homogeneous slab with s = 4 mm, µ

′s =

1 mm−1, g = 0.9, and nr = 1.4. Results are shown as a function of µa.

and δD = Di −D0. The corresponding perturbations on reflectance and transmit-tance are sufficiently small (for all the results reported, the relative perturbationsremain . 0.005) to be evaluated with good accuracy with the analytical solutions ofperturbation theory reported in Chapter 7. For comparison, the analytical solutionshave been obtained multiplying the Jacobian of the perturbation, evaluated at thecenter of the inhomogeneity, by Viδµa or ViδD. Since the Jacobian is the integrandin the expression for the perturbation, this means that the integrand has been assumedto be constant on the volume of the perturbation. As will be shown from comparisonsin Figs. 10.14 and 10.15, this assumption is suitable for sufficiently small values ofVi. For large values of Vi, a numerical integration of the perturbation on the volumeof the inhomogeneity would be necessary for more reliable comparisons.


0 1000 2000 3000-2x10-12

-1x10-12

-8x10-13

-4x10-13

0

Ra

(mm

-2ps

-1)

Time (ps)

Vi = 10 mm3

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

DE

0 1000 2000 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ra M

C/

Ra D

E

Time (ps)

Vi = 10 mm3

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

0 1000 2000 3000-1x10-14

0

1x10-14

2x10-14

3x10-14

RD

(mm

-2ps

-1)

Time (ps)

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

DE

0 1000 2000 30000.0

1.0

2.0

3.0

RD

MC

/R

DD

E

Time (ps)

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

Figure 10.14 Comparison between DE and MC results for the TR perturbation on re-flectance. The slab has s = 4000 mm, µ

′s0 = 0.5 mm−1, µa0 = 0, ni = 1.4, and nr =

1. The receiver is at ρ = 30 mm, and the inhomogeneity is at (15, 0, 15) mm. The MCresults are reported for different values of Vi for Viδµa = 0.01 mm2 and ViδD = 0.01 mm4.

Figures 10.14 and 10.15 report examples of MC results and comparisons with theDE for the TR perturbation on reflectance and transmittance, respectively. Resultsfor reflectance pertain to a slab 4000 mm thick (semi-infinite medium) with µ

′s0 =

0.5 mm−1, g = 0, µa0 = 0, and ni = ne = 1.4. The source is at (0, 0, 0) and thereceiver, circular with a radius of 2 mm and an acceptance solid angle of 2π, at (30,0, 0) mm. The inhomogeneity is spherical centered at (15, 0, 15) mm. The MCresults are reported for Vi = 10, 100, 500, and 1000 mm3 for absorption perturbationand for Vi = 100, 500, and 1000 mm3 for scattering perturbation (for smaller valuesof Vi, results are dominated by noise). Results for transmittance pertain to a slab40 mm thick with the same optical properties considered for reflectance. Both thereceiver, circular with a radius of 3 mm, and the spherical inhomogeneity at (0, 0,20) mm, are coaxial with the light beam. The figures show a general good agreement

220 Chapter 10

0 1000 2000 3000-4x10-12

-3x10-12

-2x10-12

-1x10-12

0

Ta (m

m-2

ps-1

)

Time (ps)

Vi = 10 mm3

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

DE

0 1000 2000 30000.0

0.5

1.0

1.5

2.0

Ta MC

/Ta D

E

Time (ps)

Vi = 10 mm3

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

0 1000 2000 3000-1x10-13

0

1x10-13

2x10-13

TD (m

m-2

ps-1

)

Time (ps)

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

DE

0 500 1000 15000.0

1.0

2.0

3.0

TDM

C/

TDD

E

Time (ps)

Vi = 100 mm3

Vi = 500 mm3

Vi = 1000 mm3

Figure 10.15 Comparison between DE and MC results for the TR perturbation on transmit-tance. The slab has s = 40 mm, µ

′s0 = 0.5 mm−1, µa0 = 0, ni = 1.4, and nr = 1. The

receiver and the inhomogeneity at (0, 0, 20) mm are coaxial with the light beam. The MCresults are reported for different values of Vi for Viδµa = 0.01 mm2 and ViδD = 0.01 mm4.

between MC and DE results. Furthermore, comparisons among the MC resultsfor different values of Vi also show that the perturbation weakly depends on thevolume of the inhomogeneity. We also observe that the noise on the MC results forscattering perturbations is significantly larger than for absorbing perturbations. Forthis reason, the MC results for scattering perturbations have been averaged on a timeinterval of about 40 ps.

Figures 10.14 and 10.15 report comparisons for a fixed position of the inho-mogeneity. Comparisons for inhomogeneities in different positions are shown inFigs. 10.16 and 10.17 for reflectance and transmittance, respectively. An inhomo-geneity with Vi = 500 mm3 has been considered again with Viδµa = 0.01 mm2

and ViδD = 0.01 mm4. The geometry and the optical properties are the same forFigs. 10.14 and 10.15. Figure 10.16 reports MC results for δRa(t) and δRD(t) for


0 1 0 0 0 2 0 0 0 3 0 0 0- 3 x 1 0 - 1 2

- 2 x 1 0 - 1 2

- 1 x 1 0 - 1 2

0

δRa (

mm-2 p

s-1 )

T i m e ( p s )

x i = - 1 0 m mx i = 0 m mx i = 1 0 m mx i = 2 0 m mx i = 3 0 m mx i = 4 0 m m

0 1000 2000 30000.0

1.0

2.0

3.0

4.0

Ra M

C/

Ra D

E

Time (ps)

xi = 20 mm xi = 30 mm xi = 40 mm

0 1000 2000 3000-2x10-14

0

2x10-14

4x10-14

6x10-14

8x10-14

RD

(mm

-2ps

-1)

Time (ps)

xi = -10 mm xi = 0 mm xi = 10 mm xi = 20 mm xi = 30 mm xi = 40 mm

0 1000 2000 3000-1.0

0.0

1.0

2.0

3.0

4.0

RD

MC

/R

DD

E

Time (ps)

xi = 20 mm xi = 30 mm xi = 40 mm

Figure 10.16 MC results for δRa(t) and δRD(t) for different positions of an inhomogeneitywith Vi = 500 mm3. The geometry and the optical properties are the same as in Fig. 10.14.Results are reported for at xi = −10, 0, 10, 20, 30, and 40 mm with yi = 0 and zi = 10 mm.Comparisons with DE results are shown for xi = 20, 30, and 40 mm.

the inhomogeneity at xi = −10, 0, 10, 20, 30, and 40 mm with yi = 0 and zi =10 mm. For positions of the inhomogeneity symmetric with respect to the planex = 15 mm, i.e., the plane in the middle between the source and the receiver, theperturbations are very similar (differences are appreciable only for xi = 0 and 30mm). Therefore, comparisons with DE results are shown only for xi = 20, 30,and 40 mm. Figure 10.17 reports MC results for δT a(t) for the inhomogeneity atxi = 0, 10, 20, and 30 mm with yi = 0 and zi = 20 mm and for δTD(t) for theinhomogeneity at xi = 0, 5, 10, and 15 mm with yi = 0 and zi = 20 mm. The figurealso reports the comparison with the DE (for zi = 15 mm, the ratio δTDMC/δT

DDE is

not shown because it is very noisy).The results presented pertain to inhomogeneities at distances > 5/µ

′s from the

source and from the receiver, and to small perturbations of scattering or absorption

222 Chapter 10

0 1000 2000 3000-4x10-12

-3x10-12

-2x10-12

-1x10-12

0

Ta (m

m-2

ps-1

)

Time (ps)

xi = 0 mm xi = 10 mm xi = 20 mm xi = 30 mm

0 1000 2000 30000.0

0.5

1.0

1.5

2.0

2.5

3.0

Ta MC

/Ta D

E

Time (ps)


0 1000 2000 3000-2x10-14

0

2x10-14

4x10-14

6x10-14

8x10-14

1x10-13

TD (m

m-2

ps-1

)

Time (ps)


0 500 1000 1500-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

TDM

C/

TDD

E

Time (ps)

xi = 0 mm xi = 5 mm xi = 10 mm

Figure 10.17 MC results for δT a(t) and δTD(t) for different positions of an inhomogeneitywith Vi = 500 mm3. The geometry and the optical properties are the same as in Fig. 10.15.Results are reported for different values of xi, for yi = 0 and zi = 20 mm.

(ViδD = 0.01 mm4, Viδµa = 0.01 mm2). It is possible to show that, for shorterdistances, the agreement remains good provided that the distance of the inhomo-geneity both from the receiver and from the isotropic source considered for theDE remains larger than about 2/µ

′s. For large values of Viδµa (Vi|δµa| & 4 mm2

for µ′s0 = 0.5 mm−1) or ViδD (Vi|δD| & 100 mm4 for µ

′s0 = 0.5 mm−1), a good

agreement remains for the shape of the perturbation, but the intensity is overes-timated for absorption inhomogeneities and for scattering inhomogeneities withµ′si < µ

′s0; whereas, for scattering inhomogeneities with µ

′si > µ

′s0, the intensity is

underestimated.1


10.4 Comparisons Between Monte Carlo and the DiffusionEquation: Layered Slab

Comparisons between the solution of the DE for the layered slab and the MC resultsare reported in Figs. 10.18–10.21. All figures pertain to a two-layer slab illuminatedby a pencil beam. Results are reported for the TR reflectance at distances ρ = 10,20, and 30 mm. The figures show both the MC results and the ratio between MC andDE results. The DE outgoing flux has been evaluated using the EBPC approach.

0 1000 2000 3000 4000 500010-10

10-9

10-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 10 mm = 20 mm = 30 mm

s1' = 1 mm-1

s2' = 0.5 mm-1

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

1.2

MC/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

s1' = 1 mm-1

s2' = 0.5 mm-1

0 1000 2000 3000 4000 500010-10

10-9

10-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 10 mm = 20 mm = 30 mm

s1' = 1 mm-1

s2' = 2 mm-1

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

MC/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

s1' = 1 mm-1

s2' = 2 mm-1

Figure 10.18 TR reflectance from a slab with a layered architecture of scattering. The slabhas s1 = 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, µ

′s1 = 1 mm−1, and µa1 = µa2 =

0. MC results are reported for µ′s2 = 0.5 and 2 mm−1. Comparisons with the DE are also

shown for different values of ρ.

Figure 10.18 reports comparisons for a slab with a layered architecture ofscattering. The slab has s1 = 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, µ

′s1 =

1 mm−1, and µa1 = µa2 = 0. Results are reported for µ′s2 = 0.5 and 2 mm−1.

Figure 10.19 pertains to a slab with a layered architecture of absorption having s1 =4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, and µ

′s1 = µ

′s2 = 1 mm−1. Results

are reported for µa1 = 0 and µa2 = 0.01 mm−1, and for µa1 = 0.01 mm−1 and

224 Chapter 10

0 1000 2000 3000 4000 500010-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 10 mm = 20 mm = 30 mm

a1' = 0 mm-1

a2' = 0.01 mm-1

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

1.2

MC/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

a1' = 0 mm-1

a2' = 0.01 mm-1

0 1000 2000 3000 4000 500010-10

10-9

10-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 10 mm = 20 mm = 30 mm

a1' = 0.01 mm-1

a2' = 0 mm-1

0 1000 2000 3000 4000 50000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4MC/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

a1' = 0.01 mm-1

a2' = 0 mm-1

Figure 10.19 TR reflectance from a slab with a layered architecture of absorption. The slabhas s1 = 4 mm, s2 = 100 mm, n1 = n2 = 1.4, ne = 1, µ

′s1 = µ

′s2 = 1 mm−1. MC results

are reported for µa1 = 0 mm−1 and µa2 = 0.01 mm−1, and for µa1 = 0.01 mm−1 and µa2 =0 mm−1. Comparisons with the DE are also shown for different values of ρ.

µa2 = 0. Examples for a slab with a layered architecture of refractive index areshown in Fig. 10.20 for a slab with s1 = 8 mm, s2 = 100 mm, ne = 1, µ

′s1 = µ

′s2 =

1 mm−1, and µa1 = µa2 = 0.01 mm−1. Results are reported for n1 = 1 and n2 =1.4, and for n1 = 1.4 and n2 = 1. All the comparisons show that the solutions ofthe DE for the layered slab are in excellent agreement with the reference MC results:Discrepancies are appreciable only for early received photons and are of the sameorder of those observed for the homogeneous slab.

In Fig. 10.21, an example of comparison is shown for the TR mean times offlight 〈t1〉(t) and 〈t2〉(t) spent by received photons inside the first and the secondlayer, respectively. The results pertain to the slab with layered refractive index n1 =1 and n2 = 1.4 considered in Fig. 10.20. The MC results show that both 〈t1〉(t) and〈t2〉(t) are almost independent of the source-receiver distance ρ even when there


is a significant mismatch in the refractive index of the layers. These results arein excellent agreement with the theoretical predictions reported in Sec. 6.5 on thepenetration depth of light re-emitted by diffusive media. The figure also shows thatthe values provided by DE for 〈t1〉(t) and 〈t2〉(t) are in excellent agreement withthe reference MC results.

0 1000 2000 300010-12

10-11

10-10

10-9

10-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 10 mm = 20 mm = 30 mm

n1 = 1n2 = 1.4

0 1000 2000 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MC/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

n1 = 1n2 = 1.4

0 1000 2000 300010-12

10-11

10-10

10-9

10-8

10-7

10-6

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

= 10 mm = 20 mm = 30 mm

n1 = 1.4n2 = 1

0 1000 2000 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

MC/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

n1 = 1.4n2 = 1

Figure 10.20 TR reflectance from a slab with a layered architecture of refractive index. Theslab has s1 = 8 mm, s2 = 100 mm, µ

′s1 = µ

′s2 = 1 mm−1, µa1 = µa2 = 0.01 mm−1, and

ne = 1. MC results are reported for n1 = 1 and n2 = 1.4, and for n1 = 1.4 and n2 = 1.Comparisons with the DE are also shown for different values of ρ.

10.5 Comparisons Between Monte Carlo and Hybrid Models

In this section, comparisons are shown between the solutions of the RTE obtainedwith the hybrid models of Chapter 5 and MC results. Comparisons are shown forTR results plotting the ratios MC/RTE, MC/TE, and MC/DE. All the figures will befocused on the response at short times and small source-receiver distances, times

226 Chapter 10

0 1000 2000 30000

100

200

300

400

500

600

<t1>

(ps)

Time (ps)

= 10 mm = 20 mm = 30 mm

0 1000 2000 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

<t1>

MC

/DE

Time (ps)

= 10 mm = 20 mm = 30 mm

0 1000 2000 30000

1000

2000

3000

<t2>

(ps)

Time (ps)

= 10 mm = 20 mm = 30 mm

0 1 0 0 0 2 0 0 0 3 0 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2<t 2>

MC/DE

T i m e ( p s )

ρ = 1 0 m mρ = 2 0 m mρ = 3 0 m m

Figure 10.21 TR mean times of flight spent by received photons inside the first and thesecond layer of a slab with s1 = 8 mm, s2 = 100 mm, n1 = 1, n2 = 1.4, ne = 1, µ

′s1 = µ

′s2 =

1 mm−1, and µa1 = µa2 = 0.01 mm−1. The MC results are shown together with the ratiobetween MC and DE results.

and distances at which the larger differences between the models are expected.

10.5.1 Infinite homogeneous medium

We refer to a non-absorbing medium with µ′s = 1 mm−1 and refractive index n = 1.

Figure 10.22 reports comparisons for the fluence for r = 1, 2, and 5 mm both forg = 0 and 0.9. It is particularly interesting to compare RTE and MC results for theisotropic scattering function (g = 0) since MC results are compared with an almostexact analytical solution:2 The comparison shows that, apart from very early times,discrepancies remain within 2%, i.e., within the range of error indicated in Ref. 2 forthe analytical solution of the RTE. These results can be seen as a further validationof the MC code. For g = 0, we also observe that the DE solution is faster than theTE to recover the correct values at long times. From Fig. 10.22, it is difficult to


0 50 100 150 2000.8

1.0

1.2

1.4

1.6

1.8

Flue

nce

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 1 mmg = 0.0

0 50 100 150 2000.8

1.0

1.2

1.4

1.6

1.8

Flue

nce

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 1 mmg = 0.9

0 100 200 3000.8

1.0

1.2

1.4

Flue

nce

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 2 mmg = 0.0

0 100 200 3000.8

1.0

1.2

1.4Fl

uenc

e M

C/M

odel

Time (ps)

MC/DE MC/RTE MC/TE

r = 2 mmg = 0.9

0 100 200 300 400 5000.9

1.0

1.1

Flue

nce

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 5 mmg = 0.0

0 100 200 300 400 5000.9

1.0

1.1

Flue

nce

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 5 mmg = 0.9

Figure 10.22 Comparison of DE, TE, and RTE models for the TR fluence rate with MCresults for an infinite medium with µ

′s = 1 mm−1 and n = 1. Comparisons are shown for r =

1, 2, and 5 mm both for g = 0 and 0.9.

228 Chapter 10

0 50 100 150 2000.8

1.0

1.2

1.4

1.6

1.8

Flux

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 1 mmg = 0.0

0 50 100 150 2000.8

1.0

1.2

1.4

1.6

1.8

Flux

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 1 mmg = 0.9

0 100 200 3000.8

1.0

1.2

1.4

Flux

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 2 mmg = 0.0

0 100 200 3000.8

1.0

1.2

1.4Fl

ux M

C/M

odel

Time (ps)

MC/DE MC/RTE MC/TE

r = 2 mmg = 0.9

0 100 200 300 400 5000.9

1.0

1.1

1.2

1.3

Flux

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 5 mmg = 0.0

0 100 200 300 400 5000.9

1.0

1.1

1.2

1.3

Flux

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

r = 5 mmg = 0.9

Figure 10.23 Same as Fig. 10.22, but for the flux.e

indicate the model in better agreement with MC results when g = 0.9. However,if we also include in the comparison the response for t < t0, for which the figure


0 45 90 135 1800.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Rad

ianc

e M

C/M

odel

Angle (degrees)

MC/DE MC/RTE MC/TE

r = 1 mmt/t0 = 2.0

0 45 90 135 1800.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Rad

ianc

e M

C/M

odel

Angle (degrees)

MC/DE MC/RTE MC/TE

r = 5 mmt/t0 = 2.0

0 45 90 135 1800.8

1.0

1.2

1.4

1.6

1.8

Rad

ianc

e M

C/M

odel

Angle (degrees)

MC/DE MC/RTE MC/TE

r = 1 mmt/t0 = 4.1

0 45 90 135 1800.9

1.0

1.1

1.2R

adia

nce

MC

/Mod

el

Angle (degrees)

MC/DE MC/RTE MC/TE

r = 5 mmt/t0 = 4.0

0 45 90 135 1800.9

1.0

1.1

1.2

1.3

Rad

ianc

e M

C/M

odel

Angle (degrees)

MC/DE MC/RTE MC/TE

r = 1 mmt/t0 = 8.0

0 45 90 135 1800.9

1.0

1.1

Rad

ianc

e M

C/M

odel

Angle (degrees)

MC/DE MC/RTE MC/TE

r = 5 mmt/t0 = 8.0

Figure 10.24 Same as Fig. 10.22, but for the radiance. Comparisons with MC results forg = 0 are shown for three values of t/t0 for r = 1 and 5 mm.

230 Chapter 10

does not provide information, the RTE and the TE models become preferable. Infact, the MC, RTE, and TE responses are properly equal to zero for t < t0, whilethe solution of DE predicts the arrival of photons even before the ballistic ones.

Figure 10.23 reports comparisons for the flux in the same condition consideredfor the fluence. There is also excellent agreement between RTE and MC resultsfor g = 0. We point out that, since the RTE flux is obtained from the almostexact solution of the RTE making use of Fick’s law [Eq. (3.31) and Sec. 5.1], thecomparison with MC results provides information on the accuracy of Fick’s lawitself.

These results also show that Fick’s law, when applied to the correct values forthe fluence, provides significantly better results with respect to the solution of theDE where Ficks’s law is used twice: the first time to obtain the solution for thefluence, and the second time to obtain the flux. This is perhaps the key point tounderstanding why the hybrid model based on the RTE provides results significantlybetter than the DE, as will be shown in next section.

As seen for the fluence, for g = 0.9 it is difficult to indicate a model in betteragreement with MC results.

For the radiance, examples of comparisons are shown in Fig. 10.24 for r = 1and 5 mm for the medium with g = 0. Comparisons are shown for some values oft/t0. For the radiance, the RTE solution gives the best agreement with MC results,particularly at short distances.

10.5.2 Slab geometry

We refer to a non-absorbing medium with µ′s = 1 mm−1 and refractive index ni =

1.4. Comparisons with MC results are shown in Fig. 10.25 for the reflectance of aslab with s = 40 mm and in Fig. 10.26 for the reflectance and the transmittance ofa slab with s = 4 mm. Comparisons are shown for a scattering function with g =0.9 since it is more representative of real media. For DE, TE, and RTE models, theoutgoing flux has been evaluated using the EBPC approach. Comparisons show thatthe hybrid model based on the RTE provides results that are significantly better withrespect to the others. Apart from very early times, the RTE hybrid model resultsare in excellent agreement with MC simulations, both for s = 40 and 4 mm: Fort > 4`

′/v (`

′= 1/µ

′s), discrepancies remain smaller than about 5% even when the

source receiver distance is < 4 mm. The hybrid RTE model can be used insteadof the DE to provide an improved description of light propagation through smallvolumes of diffusive media.

As to the TE hybrid solution, as observed for the infinite medium, it is slowerthan the DE to recover the correct asymptotic value at long times. In particular, asalso shown in Figs. 5.3 and 5.4, due to the large number of image sources that give asignificant contribution even at the short times displayed in Fig. 10.26, the correctasymptotic behavior is not reached for the slab 4 mm thick.


0 50 100 150 2000.8

1.0

1.2

1.4

1.6

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

= 1.0 mm

0 100 200 300 4000.8

1.0

1.2

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

= 2.02 mm

0 200 400 600 8000.8

1.0

1.2

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

= 5.1 mm

0 3 0 0 6 0 0 9 0 0 1 2 0 00 . 9

1 . 0

1 . 1Re

flecta

nceM

C/Mod

el

T i m e ( p s )

M C / D EM C / R T EM C / T E

ρ = 1 0 . 2 m m

Figure 10.25 Comparison of DE, TE, and RTE models with MC results for the TR reflectancefrom a slab with s = 40 mm, µ

′s = 1 mm−1, g = 0.9, ni = 1.4, and nr = 1. Comparisons

are shown for four values of ρ.

The results presented pertain to nr = 1. A slightly worse agreement has beenobserved for nr = 1.4.

In Fig. 10.27, a comparison is shown for the perturbation due to an absorbingand to a scattering inhomogeneity on the TR reflectance. The results pertain to a slabwith s = 4 mm, µ

′s = 1 mm−1, g = 0.9, µa = 0, ni = 1.4, and nr = 1. The receiver

is at (3, 0, 0) mm, and the inhomogeneity with Vi = 1 mm3, Viδµa = 0.01 mm2, andViδD = 0.01 mm4 is at (1.5, 0, 1.5) mm. We stress that between source and receiver,between inhomogeneity and source, and between inhomogeneity and receiver, thereare short distances (the distance is ' 1.6 mm from the isotropic point source and '2.1 mm from the receiver) for which the DE is expected to give poor results. Alsoin this case, the RTE hybrid model provides results significantly better than othermodels.

232 Chapter 10

0 20 40 60 80 1000.5

1.0

1.5

2.0

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

= 1.0 mm

0 50 100 150 200 250 3000.6

0.8

1.0

1.2

1.4

1.6

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

3.5 mm

0 50 100 150 200 250 3000.6

0.8

1.0

1.2

1.4

1.6

Tran

smitt

ance

MC

/Mod

el

Time (ps)

MC/DE MC/RTE MC/TE

= 0.15 mm

0 50 100 150 200 250 3000.6

0.8

1.0

1.2

1.4

1.6Tr

ansm

ittan

ce M

C/M

odel

Time (ps)

MC/DE MC/RTE MC/TE

= 3.5 mm

Figure 10.26 Comparison of DE, TE, and RTE models with MC results for the TR reflectanceand transmittance from a slab with s = 4 mm, µ

′s = 1 mm−1, g = 0.9, ni = 1.4, and nr = 1.

Comparisons are shown for two values of ρ.

10.6 Outgoing Flux: Comparison between Fick andExtrapolated Boundary Partial Current Approaches

In Chapter 4, three different methods have been presented to evaluate the fluxoutgoing from a diffusive medium. The first method calculates the flux applyingFick’s law to the fluence on the boundary [Eqs. (4.20) and (4.21)]. The secondintegrates the radiance transmitted at the interface over the solid angle [Eq. (4.42)].The third, the hybrid approach named EBPC, applies the PCBC to the solution forthe fluence obtained using the EBC, and the outgoing flux is simply proportional tothe fluence [Eq. (4.44)].

The first method, used in the Diffusion&Perturbation software (Sec. 4.3), ismore widely used. All the figures in Chapters 4 and 7 and in Secs. 10.2.2 and 10.3have been generated using this method. The third method, being the simplest one


0 30 60 90 120 150-5x10-7

-4x10-7

-3x10-7

-2x10-7

-1x10-7

0

Ra

(mm

-2ps

-1)

Time (ps)

MC DE TE RTE

0 30 60 90 120 1500.0

1.0

2.0

3.0

Ra M

C/

Ra M

OD

EL

Time (ps)

MC/DE MC/TE MC/RTE

0 30 60 90 120 150-1x10-7

0

1x10-7

2x10-7

3x10-7

4x10-7

5x10-7

RD

(mm

-2ps

-1)

Time (ps)

MC DE TE RTE

0 30 60 90 120 1500.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

RD

MC

/R

DM

OD

EL

Time (ps)

MC/DE MC/TE MC/RTE

Figure 10.27 Comparison of DE, TE, and RTE with MC results for the TR perturbation onreflectance from a slab with s = 4 mm, µ

′s0 = 1 mm−1, g = 0.9, µa0 = 0, ni = 1.4, and nr =

1. The receiver is at (3, 0, 0) mm, and the inhomogeneity with Vi = 1 mm3 is at (1.5, 0, 1.5)mm.

and probably more accurate, has been suggested both for hybrid models (Sec. 5.1)and for the layered slab (Sec. 6.3) and has been used for the figures of Chapters 6and 7 and in Secs. 10.4 and 10.5.2.

As pointed out in Sec. 4.4, the three methods lead to significantly differentexpressions for the outgoing flux, but in spite of the differences in the formalexpressions, they produce results that in many situations are almost indistinguishable(see Figs. 4.9 and 4.10). In Sec. 4.4, we also predicted that the EBPC solution wouldprovide a more accurate description, especially for TR measurements at short source-receiver distances. This can be seen from comparisons with reference MC results.Examples are shown in Figs. 10.28, 10.29, and 10.30. Comparisons are shown onlyfor the first and the third method, i.e., between the Fick and EBPC approaches.Since the solution obtained with the second method was a linear combination of

234 Chapter 10

those obtained with the Fick and EBPC approaches, the corresponding results arebetween those obtained with the other methods.

Figures 10.28 and 10.29 show results for the TR reflectance at moderate distances(ρ = 2.53, 5.12, and 10.2 mm) from the source and pertain to the case of matched(nr = 1) and mismatched (nr = 1.4) refractive indexes, respectively. Comparisonspertain to a slab with s = 40 mm, µ

′s = 1 mm−1, g = 0.9, and ni = 1.4 illuminated

by a pencil light beam. Differences between results from the Fick and EBPCapproaches are larger in the case of mismatched refractive indexes. As anticipated inSec. 4.4, there is a slightly better agreement between the MC results and the EBPCresults.

In Ref. 3, it has been shown that appreciably more accurate results could be ob-tained when the EBPC solution is used in the fitting procedure devoted to retrievingthe optical properties of the medium from TR measurements of reflectance.

Figure 10.30 shows comparisons for the CW reflectance from a slab with s =40 mm, µ

′s = 1 mm−1, µa = 0.01 mm−1, g = 0.9, and ni = 1.4. For both the RTE

hybrid model and the DE, the CW response has been calculated, integrating overtime and over the area of the receiver, the temporal response being provided by themodels (we considered receivers identical to those used for MC simulations). Thefigure shows that the results from the RTE hybrid model are in better agreement withMC results than those from the DE, but it is difficult to draw a general conclusionfrom the comparison between the Fick and EBPC approaches. On the contrary, forthe total reflectance (that is strongly influenced by photons exiting very near thesource) and for the total transmittance, it is possible to show that results obtainedwith Fick’s law are in significantly better agreement with MC results.

10.7 Conclusions

In this chapter, we have compared solutions of the RTE based on the diffusionapproximation with MC results. Since the MC results are exact solutions of the RTE(apart from the statistical noise), the presented comparisons can be therefore used todetermine the degree of accuracy of the DE solutions. In this section, we summarizethe results of these comparisons also with the aim to better specify the meaning of"diffusive media". Making use of the scaling properties of Sec. 3.4.1, comparisonscan be summarized as:

10.7.1 Infinite medium

Comparisons in Sec. 10.2.1 (Figs. 10.1 and 10.2) showed that for TR results discrep-ancies between the solution of the DE and MC simulations for g = 0.9 (representa-tive of real media) are within 2% for rµ

′s & 5 and t & 4t0 (t0 is the time of flight for

ballistic photons), and for t & 4`′/v (`

′= 1/µ

′s is the transport mean free path) for

shorter distances. Since both MC and DE have the same dependence on absorption,these results are independent of the absorption coefficient. On the contrary, for CWresults, discrepancies increase as absorption increases. In fact, photons with long


0 100 200 300 400 5000.8

0.9

1.0

1.1

1.2

1.3

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DEEBPC MC/RTEEBPC

EBPC = 2.53 mm

0 100 200 300 400 5000.8

0.9

1.0

1.1

1.2

1.3

Ref

lect

ance

MC

/Mod

el

Time (ps)

MC/DEFick MC/RTEFick

Fick = 2.53 mm

0 200 400 600 800 10000.90

0.95

1.00

1.05

1.10

Ref

lect

ance

MC

/Mod

el

Time (ps)


EBPC = 5.12 mm

0 200 400 600 800 10000.90

0.95

1.00

1.05

1.10

Ref

lect

ance

MC

/Mod

el

Time (ps)


Fick = 5.12 mm

0 5 0 0 1 0 0 0 1 5 0 00 . 9 0

0 . 9 5

1 . 0 0

1 . 0 5

1 . 1 0

Refle

ctanc

eMC/M

odel

T i m e ( p s )

M C / D E E B P CM C / R T E E B P C

E B P Cρ = 1 0 . 2 1 m m

0 500 1000 15000.90

0.95

1.00

1.05

1.10

Ref

lect

ance

MC

/Mod

el

Time (ps)


Fick = 10.21 mm

Figure 10.28 TR reflectance from a slab with s = 40 mm, µ′s = 1 mm−1, g = 0.9, ni =

1.4, and nr = 1 illuminated by a pencil light beam: Comparison of MC results with resultsobtained using the Fick and EBPC approaches. Comparisons are shown for the DE and theRTE.

236 Chapter 10

0 100 200 300 400 5000.80.91.01.11.21.31.41.51.61.71.8

Ref

lect

ance

MC

/Mod

el

Time (ps)


EBPC = 2.53 mm

0 100 200 300 400 5000.80.91.01.11.21.31.41.51.61.71.8

Ref

lect

ance

MC

/Mod

elTime (ps)


Fick = 2.53 mm

0 200 400 600 800 10000.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Ref

lect

ance

MC

/Mod

el

Time (ps)


EBPC = 5.12 mm

0 200 400 600 800 10000.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Ref

lect

ance

MC

/Mod

el

Time (ps)


Fick = 5.12 mm

0 500 1000 15000.90

0.95

1.00

1.05

1.10

1.15

Ref

lect

ance

MC

/Mod

el

Time (ps)


EBPC = 10.21 mm

0 500 1000 15000.90

0.95

1.00

1.05

1.10

1.15

Ref

lect

ance

MC

/Mod

el

Time (ps)


Fick = 10.21 mm

Figure 10.29 Same as Fig. 10.28 but for the case of the mismatched refractive index (nr =1.4).


0 10 20 30 400.7

0.8

0.9

1.0

1.1

Ref

lect

ance

MC

/Mod

el

(mm)


EBPCnr = 1

0 10 20 30 400.7

0.8

0.9

1.0

1.1

Ref

lect

ance

MC

/Mod

el

(mm)


Ficknr = 1

0 10 20 30 400.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

Ref

lect

ance

MC

/Mod

el

(mm)


EBPCnr = 1.4

0 10 20 30 400.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4R

efle

ctan

ce M

C/M

odel

(mm)


Ficknr = 1.4

Figure 10.30 CW reflectance from a slab with s = 40 mm, µ′s = 1 mm−1, µa = 0, g =

0.9, and ni = 1.4 illuminated by a pencil light beam: Comparison of MC results with resultsobtained using the Fick and EBPC approaches. Comparisons are shown for the DE and theRTE, both for the case of matched (nr = 1) and mismatched (nr = 1.4) refractive indexes.

time of flight (those better described by the DE) are more attenuated with respect toearly photons when absorption increases. For µa/µ

′s . 0.01, discrepancies remain

within 1% provided rµ′s & 2 (Figs. 10.5 and 10.6). For larger values of µa/µ

′s,

discrepancies increase as the source-receiver distance increases. For rµ′s < 20, they

remain within 10% for µa/µ′s = 0.05 and within about 20% for µa/µ

′s = 0.1.


For this geometry, disagreements in addition to those caused by the intrinsic approx-imations of the DE can be also due to the approximations in modeling, both theboundary conditions and the source (the pencil light beam is modeled with a point-like isotropic source at zs = 1/µ

′s). Disagreements observed with respect to MC

results are in general larger than those for the infinite medium, and they significantly

238 Chapter 10

depend on the refractive index mismatch. Therefore, it is more difficult to draw ageneral conclusion on the accuracy of the analytical solutions. Comparisons in Sec.10.2.2, with solutions reported in Chapter 4 for a slab with reduced optical thicknesssµ′s = 40, showed that for TR results the disagreement remains within 5% for t & 4

t0 only if the source receiver distance is larger than about 10/µ′s (Figs. 10.9 and

10.10). Disagreement is larger when there is a significant mismatch in the refractiveindex, and reflections play an important role on propagation. For shorter distances,discrepancies are larger and can involve a significant fraction of received photons.This is shown by comparisons for CW reflectance for which discrepancies of about5% are observed also for ρµ

′s = 10 and µa/µ

′s = 0.01 (Fig. 10.12).

Comparisons in Sec. 10.5.2 showed that more accurate results are obtained,especially at early times and at short distances, when the hybrid model based onthe solution of the RTE for the infinite medium is used. It has been shown that, forthe case of a matched refractive index, discrepancies remain .5% for t & 4 `

′/v

both for TR reflectance and for TR transmittance from a slab with sµ′s = 4, even for

distances ρµ′s < 4 (Figs. 10.25 and 10.26).

10.7.3 Layered slab

Comparisons in Sec. 10.4 showed that the accuracy of solutions for the layered slabis similar to that for the homogeneous slab. Comparisons have been shown forµ′s1/µ

′s2 = 0.5 and 2, n1/n2 = 1.4 and 0.71, µa1 = 0 and µa2 = 0.01 mm−1, and

µa1 = 0.01 mm−1 and µa2 = 0. Results reported in Ref. 4 showed that a similaraccuracy is also obtained for a larger mismatch in the optical properties.

10.7.4 Slab with inhomogeneities inside

For the perturbation due to inhomogeneities inside the slab, comparisons in Sec.10.3 showed good agreements both for absorption and for scattering perturbations,provided that perturbations are sufficiently small and the distance of the inhomo-geneity both from the receiver and from the source remains larger than about 2/µ

′s.

For large values of Viδµa or ViδD, a good agreement remains for the shape of theperturbation, but the intensity is overestimated for absorption inhomogeneities andfor scattering inhomogeneities with a reduced scattering coefficient smaller than thebackground; whereas, for scattering inhomogeneities with µ

′si > µ

′s0, the intensity

is underestimated.

10.7.5 Diffusive media

All the comparisons reported in the previous chapters pertain to the infinite mediumor to the infinite slab (results for the layered slab have also been reported for acylinder with a sufficiently large radius to behave like an infinite slab). When mea-surements are carried out on media with a finite volume, it may be useful to knowhow long it is possible to use results based on the diffusion approximation. Theinfluence of the finite volume of the medium in establishing the diffusive regime of


propagation can be investigated comparing the solution of the DE with MC resultsfor a finite parallelepiped. An example of comparison is shown in Fig. 10.31 (theMC results used for this comparison have not been reported in the CD-ROM). Thefigure shows the TR reflectance of a cube with 10 mm sides, µ

′s = 1 mm−1, g =

0.9, µa = 0, ni = 1.4, and nr = 1. The pencil light beam impinges at (2.5, 5,0) mm, and the receiver, circular with a radius of 3 mm, is centered at (7.5, 5, 0)mm (the origin of the reference system is at the corner of the cube). The temporalresponses obtained with the DE and with the RTE hybrid model are comparedwith MC results. The finite volume of the diffusive medium strongly influencesthe temporal response. This is shown from the comparison between solutions ofthe DE for the cube and for the infinitely extended slab 10 mm thick (curves DEand DESlab, respectively). Differences increase with the time of flight. At 400ps, the response for the slab is about one order of magnitude larger. The figurealso shows the ratio MC/DE and MC/RTE as a function of t/t0. The agreementis similar to the one observed for the reflectance from a slab 40 mm thick at thesame source-receiver distance (see Figs. 10.9 and 10.25): Disagreement remainswithin about 5% for t & 4t0. Therefore, we can also conclude that for a finite vol-ume of turbid medium, solutions of the RTE based on the diffusion approximationaccurately describe propagation for photons with a sufficiently long time of flight.For the CW response, since the fraction of photons received with time of flightt > 4t0 (those better described by the DE) decreases as the volume of the mediumdecreases, the results are a bit worse with respect to the infinite slab: For the cube ofFig. 10.31, the CW solution of the RTE is overestimated by 10% for µa = 0 and 21%for µa = 0.1 mm−1, about 1% more than for the infinite slab in the same conditions.

0 100 200 300 400 5001E-8

1E-7

1E-6

1E-5

1E-4

Ref

lect

ance

(mm

-2ps

-1)

Time (ps)

MC DE RTE DESlab

0 5 10 15 200.50.60.70.80.91.01.11.21.31.41.5

Ref

lect

ance

MC

/Mod

el

t/t0

MC/DE MC/RTE

Figure 10.31 Comparison of DE and RTE models for the TR reflectance with MC results fora cube with 10 mm sides, µ

′s = 1 mm−1, g = 0.9, µa = 0, ni = 1.4, and nr = 1. The figures

also show the DE model for the infinitely extended slab 10 mm thick (DESlab curve).

240 Chapter 10

From comparisons reported in this chapter and others that can be done withresults reported in the CD-ROM, we can conclude that solutions of the RTE basedon the diffusion approximation provide a description of photon migration that canbe suitable for many applications, provided the turbid medium has µa/µ

′s . 0.1 and

a volume & (10`′)3. In such conditions, the diffusive regime of propagation can be

established even near the source provided photons traveled trajectories with length& 4`

′inside the medium. Since, as shown in Sec. 2.4 [to travel the length `

′= 1/µ

′s,

the average number of scattering events is 1/(1− g)] we can also conclude that torandomize propagation in order to reach the diffusive regime, photons should haveundergone a number of scattering events & 4/(1− g). Therefore, about 4 scatteringevents are sufficient for a medium with Rayleigh scatterers, but about 20–40 arerequired for most natural media for which 0.8 . g . 0.9.

The conditions we have indicated as necessary for establishing the diffusiveregime of propagation can be satisfied for many of the materials of which wereported typical values for the optical properties in Sec. 2.5. For many of these solidand liquid materials µ

′s ≈ 1 mm−1 and µa/µ

′s . 0.1 at visible and/or near-infrared

wavelengths. Therefore, for such materials, the diffusive regime of propagation canalso be reached when the volume is as small as ≈ 1 cm3. For dense fogs and clouds,propagation in the diffusive regime can be established for photons that traveledpaths longer than some hundreds of meters, but the volume involved should be muchlarger. An even larger volume is required for media with lower scattering propertiesas may occur for clouds of gas and dust in the interstellar medium. Instead, theclear atmosphere (the optical thickness of which is between 0.35 and 0.035 for 400< λ < 700 nm) is not suitable for propagation in the diffusive regime since theavailable volume of the turbid medium is not sufficiently extended.


References

1. S. Carraresi, T. S. M. Shatir, F. Martelli, and G. Zaccanti, “Accuracy of a pertur-bation model to predict the effect of scattering and absorbing inhomogeneitieson photon migration,” Appl. Opt. 40, 4622–4632 (2001).

2. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys.Rev. E 56, 1135–1141 (1997).

3. F. Martelli, A. Sassaroli, G. Zaccanti, and Y. Yamada, “Properties of the lightemerging from a diffusive medium: angular dependence and flux at the externalboundary,” Phys. Med. Biol. 44, 1257–1275 (1999).

4. F. Martelli, A. Sassaroli, S. Del Bianco, and G. Zaccanti, “Solution of the time-dependent diffusion equation for a three-layer medium: application to studyphoton migration through a simplified adult head model,” Phys. Med. Biol. 52,2827–2843 (2007).

Appendix A

The First Simplifying Assumption ofthe Diffusion ApproximationFor obtaining Eq. (3.25), the specific intensity I(~r, s, t) can be expanded in sphericalharmonics retaining only the first two terms (isotropic and linearly anisotropic) (seeRef. 1 for a detailed description). In this appendix, an intuitive physical justificationof Eq. (3.25) is provided following the scheme proposed by Ishimaru.2 In the diffu-sive regime of propagation, due to multiple scattering, the radiance has an almostisotropic angular distribution and consequently can be represented as

I(~r, s, t) = αΦ(~r, t) + β ~J(~r, t) · s, (A-1)

where I(~r, s, t) is the sum of two terms: a first isotropic term and a second term thataccounts for the weak anisotropy of the specific intensity. The coefficient α can bedetermined accordingly to the definition of Φ(~r, t) as

Φ(~r, t) =∫4π

I(~r, s, t)dΩ = α4πΦ(~r, t) ⇒ α =1

4π. (A-2)

Therefore, the first term of Eq. (A-1) is the average value of I(~r, s, t) over the solidangle. The value of β is determined by using the definition of ~J(~r, t): If sJ is thedirection of ~J(~r, t), then∣∣∣ ~J(~r, t)

∣∣∣ = ~J(~r, t) · sJ =∫4π

I(~r, s, t)s · sJdΩ. (A-3)

For obtaining β, Eq. (A-1) is substituted in Eq. (A-3):∣∣∣ ~J(~r, t)∣∣∣ = 1

4π

∫4π

Φ(~r, t)s · sJdΩ + β∫4π

~J(~r, t) · s(s · sJ)dΩ

= 14πΦ(~r, t)sJ ·

∫4π

sdΩ + β∣∣∣ ~J(~r, t)

∣∣∣ ∫4π

(sJ · s)(s · sJ)dΩ

= β∣∣∣ ~J(~r, t)

∣∣∣ ∫4π

(s · sJ)2dΩ .

(A-4)

Since ∫4π

(s · sJ)2dΩ =

π∫0

cos2 θ2π sin θdθ =43π , (A-5)

we obtain ∣∣∣ ~J(~r, t)∣∣∣ = β

∣∣∣ ~J(~r, t)∣∣∣ 4

3π =⇒ β =

34π

(A-6)

andI(~r, s, t) =

14π

Φ(~r, t) +3

4π~J(~r, t) · s. (A-7)

245

Appendix B

Fick’s LawIn this appendix, the time-dependent Fick’s law is obtained following the mainframes of the procedure used by Ishimaru for steady-state sources.2 The two simpli-fying assumptions of the diffusion approximation are necessary for this calculation.Multiplying the RTE, i.e., Eq. (3.5), by s and integrating over the whole solid angle,a new equation is obtained where the different terms are listed below:

• I1 =∫4π

(1v∂I(~r,s,t)

∂t

)sdΩ = 1

v∂∂t

∫4π

I(~r, s, t)sdΩ = 1v∂∂t~J(~r, t)

• I2 =∫4π

(s · ∇I(~r, s, t)) sdΩ

• I3 =∫4π

µtI(~r, s, t)sdΩ =µt ~J(~r, t)

• I4 =∫4π

µs∫4π

p(s, s′)[

14πΦ(~r, t) + 3

4π~J(~r, t) · s′

]dΩ′sdΩ

• I5 =∫4π

ε(~r, s, t)sdΩ

For integral I2, making use of the first simplifying assumption [Eq. (3.25)], weobtain

I2 =∫4π

s · ∇

[1

4πΦ(~r, t) +

34π

~J(~r, t) · s]

sdΩ. (B-1)

The first term of Eq. (B-1) is an integral of the form∫4π

(s · ~A

)sdΩ with ~A indepen-

dent of s. With reference to Fig. B-1, assuming the x-axis of the reference systemalong ~A results in∫

4π

(s · ~A

)sdΩ =

= A2π∫0

dϕπ∫0

sin θ cos θ(

cos θ i+ sin θ cosϕ j + sin θ sinϕk)dθ

= A2ππ∫0

sin θ cos2 θidθ = −2π ~A−1∫1

cos2 θd cos θ = 4π3~A.

(B-2)

Thus, ∫4π

s · ∇

[Φ(~r, t)

4π

]sdΩ =

∇Φ(~r, t)3

. (B-3)

246

Fick’s Law 247

x

y

z

As

x

y

z

As

Figure B-1 x, y, z reference system used to make the integral I2.

sJ's

s

u

uJ

sJ's

s

u

uJ

Figure B-2 Decomposition of the flux vector for the integral I4.

Since it is possible to demonstrate that, for every vector, ~A is

∫4π

s[s · ∇

(~A · s

)]dΩ = 0 , (B-4)

248 Appendix B

the second term of Eq. (B-1), 34π

∫4π

s · ∇

[~J(~r, t) · s

]sdΩ, is equal to zero. In

fact, using the same notation of Fig. B-1 results in

s · ∇(~A · s

)= cos2 θ

∂A

∂x+ sin θ cos θ cosϕ

∂A

∂y+ sin θ cos θ sinϕ

∂A

∂z, (B-5)

and thus the integral of Eq. (B-4) is an integral of trigonometric functions of θ (overthe interval 0, π) and ϕ (over the interval 0, 2π). Each term of this integral vanishes.Consequently, I2 is

I2 =∇Φ(~r, t)

3. (B-6)

The fourth integral I4 is the integral of the function

h =µs∫4π

p(s, s′)[

14πΦ(~r, t) + 3

4π~J(~r, t) · s′

]dΩ′

= µs4πΦ(~r, t) + µs

34π

∫4π

p(s, s′)[~J(~r, t) · s′

]dΩ′.

(B-7)

In the second term of h, only the component of ~J(~r, t) along s gives a not nullcontribution to the integral. With reference to Fig. B-2, ~J(~r, t) can be written as~J(~r, t) = Juu+ Jss. According to this notation, taking into account that p(s, s′)only depends on the angle θ between s and s′ and that u · s′ = cosϕ sin θ, the secondterm of Eq. (B-7) becomes

µs3

4π

∫4π

p(s, s′)[~J(~r, t) · s′

]dΩ′ =

= µs3

4πJs∫4π

p(θ) cos θdΩ′ + µs3

4πJu∫4π

p(θ) cosϕ sin θdΩ′

= µs3

4πJsg = µs3

4π~J(~r, t) · sg.

(B-8)

Finally, h results in

h =µs4π

Φ(~r, t) + µs3g4π

~J(~r, t) · s (B-9)

andI4 =

∫4π

hsdΩ = µsg ~J(~r, t), (B-10)

since∫4π

Φ(~r, t)sdΩ = 0 and∫4π

[~J(~r, t)·s

]sdΩ = 4π

3~J(~r, t) [see Eq. (B-2)].

Adding I1, I2, I3, I4, and I5, the integral over the whole solid angle of the RTEmultiplied by s can be written as

1v

∂

∂t~J(~r, t) +

∇Φ(~r, t)3

+ (µt − gµs) ~J(~r, t) =∫4π

ε(~r, s, t)sdΩ. (B-11)

Fick’s Law 249

Applying the second simplifying assumption of the diffusion approximation[Eq. (3.27)], and then neglecting the time derivative of the photon flux, Fick’s law isobtained as

~J(~r, t) = − 1µt − gµs

[∇Φ(~r, t)

3−∫4π

ε(~r, s, t)sdΩ

]. (B-12)

If we assume an isotropic source ε(~r, t), then the standard form of Fick’s law isobtained:

~J(~r, t) = − 1µt − gµs

∇Φ(~r, t)3

= −D∇Φ(~r, t), (B-13)

with D = 13(µt−gµs) = 1

3(µa+µ′s)as the diffusion coefficient.

Appendix C

Boundary Conditions at theInterface Between Diffusive andNon-Scattering MediaWhen an interface between a diffusive and a non-scattering medium is considered(see Fig. 3.3), the boundary condition for radiative transfer can be summarized byEq. (3.48), which we rewrite as

−∫

s·q<0

I(~r, s, t)(s · q)dΩ =∫

s·q>0

RF (s · q)I(~r, s, t)(s · q)dΩ, (C-1)

with I specific intensity at the boundary Σ of the diffusive medium, RF is theFresnel reflection coefficient for unpolarized light, and q is the unit vector normal tothe boundary surface. For solving the integrals of Eq. (C-1), we make use of the firstsimplifying assumption of the diffusion approximation [Eq. (3.25)]. Writing the fluxas ~J(~r, t) = Juu+ Jq q with u unit vector tangential to the boundary surface, withs · q = cos θ and u · s = cosϕ sin θ, the first integral of Eq. (C-1) becomes∫s·q<0

I(~r, s, t)( s · q)dΩ

=∫

s·q<0

[1

4πΦ(~r, t) + 34π~J(~r, t) · s

](s · q)dΩ

= Φ(~r, t)4π 2π

π∫π/2

cos θ sin θdθ + 34πJu

∫2π

( u · s)(s · q)dΩ + 34πJq

∫2π

( s · q)2dΩ

= −Φ(~r, t)4 + 3

4π

(Ju

2π∫0

dϕπ∫

π/2

cosϕ cos θ sin2 θdθ + Jq2π∫0

dϕπ∫

π/2

cos2 θ sin θdθ

)= −Φ(~r, t)

4 + 34π

(Jq

2π3

),

(C-2)where we used

•π∫

π/2

cos θ sin θdθ = −1/2

•π∫

π/2

cos2 θ sin θdθ = 1/3

•2π∫0

π∫π/2

cosϕ cos θ sin2 θdθdϕ = 0

250

Boundary Conditions at the Interface Between Diffusive and Non-Scattering Media 251

Finally, the following answer is obtained:∫s·q<0

I(~r, s, t)(s · q)dΩ =

[−Φ(~r, t)

4+~J(~r, t) · q

2

]. (C-3)

A similar procedure is used for the second integral of Eq. (C-1):∫s·q>0

RF (s · q)I(~r, s, t)( s · q) dΩ

=∫

s·q>0

RF ( s · q)[

14πΦ(~r, t) + 3

4π~J(~r, t) · s

](s · q)dΩ

= Φ(~r, t)2

π/2∫0

RF (cos θ) cos θ sin θdθ + 34π

×[Ju∫2π

RF (cos θ)(u · s)(s · q)dΩ + Jq∫2π

RF (cos θ)(s · q)2dΩ].

(C-4)

Since∫2π

RF (cos θ)(u·s)(s·q)dΩ =2π∫0

dϕπ/2∫0

RF (cos θ) cosϕ sin2 θ cos θdθ =

0, then∫s·q>0

RF (s · q)I(~r, s, t)(s · q)dΩ

= Φ(~r, t)2

π/2∫0

RF (cos θ) cos θ sin θdθ + 32Jq

π/2∫0

RF (cos θ) cos2 θ sin θdθ.

(C-5)Putting Eq. (C-3) and Eq. (C-5) into Eq. (C-1) and denoting the coefficient A as

A =1 + 3

π/2∫0

RF (cos θ) cos2 θ sin θdθ

1− 2π/2∫0

RF (cos θ) cos θ sin θdθ

, (C-6)

the partial current boundary condition (PCBC) at the diffusive/non-scattering inter-face is obtained: [

Φ(~r, t)− 2A~J(~r, t) · q]~r∈Σ

= 0 . (C-7)

Appendix D

Boundary Conditions at theInterface Between Two DiffusiveMediaThe frame of this derivation is the one proposed by Faris.3 Given two diffusivemedia (see Fig. 3.5), denoted as medium 1 [refractive index n1, radiance I1(~r, s, t),and flux ~J1(~r, t)] and medium 2 [refractive index n2, radiance I2(~r, s, t), and flux~J2(~r, t)], bounded by a surface Σ, with q unit vector normal to Σ, directed frommedium 2 towards medium 1, the boundary condition at the interface Σ can bewritten with two equations:∫

s·q>0

I1(~r, s, t)(s · q) dΩ = −∫

s·q<0

RF1(−s · q)I1(~r, s, t)(s · q)dΩ

+∫

s·q>0

[1−RF2(s · q)] I2(~r, s, t)(s · q) dΩ(D-1)

and

−∫

s·q<0

I2(~r, s, t)(s · q)dΩ =∫

s·q>0

RF2(s · q)I2(~r, s, t)(s · q)dΩ

−∫

s·q<0

[1−RF1(−s · q)] I1(~r, s, t)(s · q)dΩ,(D-2)

where RF1(−s · q) and RF2(s · q) are the Fresnel reflection coefficients frommedium 1 to medium 2 and vice versa, respectively. Using the first assumptionof the diffusion approximation to represent the specific intensities I1(~r, s, t) andI2(~r, s, t) and decomposing the flux vectors along the directions normal and tangentto Σ, the following expressions are obtained:

I1(~r, s, t) =1

4πΦ1(~r, t) +

34π

[J1u(~r, t) u1 + J1q(~r, t)q] · s (D-3)

I2(~r, s, t) =1

4πΦ2(~r, t) +

34π

[J2u(~r, t) u2 + J2q(~r, t)q] · s, (D-4)

with u1 and u2 as the unit vectors tangent to Σ. The Fresnel reflection coefficientsRF1 and RF2 can be written as

RF1(cos θ1) =12

[(n1 cos θ1 − n2 cos θ2

n1 cos θ1 + n2 cos θ2

)2

+(n2 cos θ1 − n1 cos θ2


)2]

(D-5)

252

Boundary Conditions at the Interface Between Two Diffusive Media 253

and

RF2(cos θ2) =12

[(n2 cos θ2 − n1 cos θ1


)2

+(n1 cos θ2 − n2 cos θ1


)2],

(D-6)with θ1 and θ2 related by Snell’s law, i.e., n1 sin θ1 = n2 sin θ2.

Taking the integrals in Eqs. (D-1) and (D-2), we obtain

∫s·q>0

I1(~r, s, t)(s · q)dΩ =

[Φ1(~r, t)

4+~J1(~r, t) · q

2

], (D-7)

∫s·q>0

I2(~r, s, t)( s · q)dΩ =

[Φ2(~r, t)

4+~J2(~r, t) · q

2

], (D-8)

∫s·q<0

I1(~r, s, t)(s · q) dΩ =

[−Φ1(~r, t)

4+~J1(~r, t) · q

2

](D-9)

and ∫s·q<0

I2(~r, s, t)(s · q)dΩ =

[−Φ2(~r, t)

4+~J2(~r, t) · q

2

], (D-10)

where Φ1(~r, t) and Φ2(~r, t) are the fluence at the interface Σ in medium 1 and 2,respectively. Introducing the notation

E1 = 2

π/2∫0

RF1(θ1) cos θ1 sin θ1dθ1 , (D-11)

E2 = 2

π/2∫0

RF2(θ2) cos θ2 sin θ2 dθ2 , (D-12)

E3 = 3

π/2∫0

RF1(θ1) cos2 θ1 sin θ1dθ1 (D-13)

and

E4 = 3

π/2∫0

RF2(θ2) cos2 θ2 sin θ2 dθ2 , (D-14)

254 Appendix D

these other integral terms of Eqs. (D-1) and (D-2) can be expressed as−∫

s·q<0

RF1(− s · q)I1(~r, s, t)( s · q)dΩ = Φ1(~r, t)4 E1 − J1q

2 E3

−∫

s·q>0

RF2( s · q)I2(~r, s, t)( s · q)dΩ = −Φ2(~r, t)4 E2 − J2q

2 E4.(D-15)

Making use of the above integrals [Eqs. (D-7)–(D-10) and Eqs. (D-15)], Eqs. (D-1) and (D-2) become

Φ1(~r, t)4 + J1q

2 = Φ1(~r, t)4 E1 − J1q

2 E3 + Φ2(~r, t)4 + J2q

2

−Φ2(~r, t)4 E2 − J2q

2 E4 ,(D-16)

Φ2(~r, t)4 − J2q

2 = Φ2(~r, t)4 E2 + J2q

2 E4 + Φ1(~r, t)4 − J1q

2

−Φ1(~r, t)4 E1 + J1q

2 E3 .(D-17)

When adding and subtracting these equations, two independent boundary conditionsare obtained:

J1q = J2q = ~J(~r, t) · q = Jq (D-18)

andΦ2(~r, t)

2(1− E2) =

Φ1(~r, t)2

(1− E1) + Jq (E3 + E4) . (D-19)

The first equation states the continuity of the normal component of the photon fluxat the interface. The second equation shows that, in general, the fluence rate isdiscontinuous at the interface. Snell’s law, as noted by Aronson,4 gives

n21(1− cos2 θ1) = n2

2(1− cos2 θ2), (D-20)

from which the following equality is obtained:

n21 cos θ1d(cos θ1) = n2

2 cos θ2 d(cos θ2), (D-21)

and consequently it is possible to show that

n22 (1− E2) = n2

1 (1− E1) . (D-22)

Using this property, Eq. (D-19) can be written as

Φ2(~r, t)−(n2

n1

)2

Φ1(~r, t) = 2Jq(E3 + E4)(1− E2)

. (D-23)

Appendix E

Green’s Function of the DiffusionEquation in an InfiniteHomogeneous MediumThis appendix is dedicated to the Green’s function of the DE for an infinitehomogeneous medium. The Green’s function for the fluence is obtained for both thecase of a time-dependent source and of a steady-state source. The Green’s functionfor the photon flux is obtained for a non-absorbing medium with a steady-statesource.

Time-dependent source

Let us consider an infinite homogeneous medium with absorption coefficient µa,reduced scattering coefficient µ

′s, and diffusion coefficient D = 1/(3µ

′s). Given an

orthogonal coordinate system x, y, z, with a spatial and temporal Dirac delta sourceof unitary strength δ3(~r)δ(t) in the origin O ≡ (0, 0, 0), the DE for the fluence ratecan be written as (

∂

v∂t−D∇2 + µa

)Φ(~r, t) = δ3(~r)δ(t). (E-1)

The solution will have the form Φ(~r, t) = Φ0(~r, t) exp(−µavt) with Φ0(~r, t) thesolution for the non-absorbing medium. For solving Eq. (E-1) with µa = 0, becauseof the spherical symmetry of the problem, the separation of variables method is used,and a solution in the form Φ0(~r, t) = Φ0x(x, t)Φ0y(y, t)Φ0z(z, t) is sought. Since∇2 = ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2and δ3(~r) = δ(x) δ(y) δ(z), Eq. (E-1) for the non-absorbing

medium can be written as[∂

v∂tΦ0(~r, t)−D

(∂2

∂x2 +∂2

∂y2 +∂2

∂z2

)Φ0(~r, t)

]= δ(x)δ(y)δ(z)δ(t). (E-2)

The boundary conditions for an infinite medium require that the solution Φ0(~r, t)must be zero in the following ranges:

Φ0(~r, t) ≡ 0 for t < 0; Φ0(~r, t)→ 0 for |~r| → ∞. (E-3)

Taking the Fourier Transform (FT) of Eq. (E-2), Φ0(~ω, t) = FT [Φ0(~r, t)], definedas

Φ0(~ω, t) =

Φ0x(ω1, t)Φ0y(ω2, t)Φ0z(ω3, t) for t ≥ 0 ,0 for t < 0 ,

(E-4)

255

256 Appendix E

where

Φ0x(ω1, t) = FT [Φ0x(x, t)] =+∞∫−∞

exp(−iω1x)Φ0x(x, t)dx ,

Φ0y(ω2, t) = FT [Φ0y(y, t)] =+∞∫−∞

exp(−iω2y)Φ0y(y, t)dy ,

Φ0z(ω3, t) = FT [Φ0z(z, t)] =+∞∫−∞

exp(−iω3z)Φ0z(z, t)dz ,

(E-5)

and using the following property of the Fourier transform

dp

dξpf(ξ) FT⇒ (iω)pf(ω), (E-6)

Eq. (E-2) in the transformed domain becomes

∂

v∂tΦ0(~ω, t) +

(ω2

1 + ω22 + ω2

3

)D Φ0(~ω, t) = δ(t). (E-7)

From Eq. (E-7), this leads to

∂

v∂tΦ0(~ω, t) +

(ω2

1 + ω22 + ω2

3

)DΦ0(~ω, t) = 0 for t > 0 . (E-8)

Integrating Eq. (E-8), the solution Φ0(~ω, t) is obtained

Φ0(~ω, t) =A exp

[−(ω2

1 + ω22 + ω2

3

)Dvt

]for t ≥ 0 ,

0 for t < 0 .(E-9)

In order to determine the value of constant A, the behavior of the function Φ0(~ω, t)for t = 0 has to be evaluated. Integrating Eq. (E-7) in the temporal range (0−, 0+),the following condition is obtained: v−1

[Φ0(~ω, 0+)− Φ0(~ω, 0−)

]= 1. Since

Φ0(~ω, 0−) = 0⇒ A = v. Thus, the Fourier transform of the solution Φ0(~r, t) is

Φ0(~ω, t) =v exp

[−(ω2

1 + ω22 + ω2

3

)Dvt

]for t > 0;

0 for t < 0.(E-10)

The expression of Φ0(~r, t) is obtained by taking the inverse Fourier transform ofEq. (E-10)

Φ0(~r, t) =1

(2π)3

+∞∫−∞

+∞∫−∞

+∞∫−∞

exp(i~ω · ~r)Φ0(~ω, t)d~ω for t > 0 . (E-11)

Substituting Eq. (E-10) in Eq. (E-11) results in

Φ0(~r, t) =v

(2π)3 I1I2I3 , (E-12)

Green’s Function of the Diffusion Equation in an Infinite Homogeneous Medium 257

with

I1 =+∞∫−∞

exp(iω1x) exp(−ω2

1Dvt)dω1 ,

I2 =+∞∫−∞

exp(iω2y) exp(−ω2

2Dvt)dω2 ,

I3 =+∞∫−∞

exp(iω3z) exp(−ω2

3Dvt)dω3 .

(E-13)

The integrals in Eq. (E-13) are the Fourier transforms of a Gaussian function:

+∞∫−∞

exp(−aξ2 + iωξ)dξ =√π

aexp

(− ω2

4a

). (E-14)

This leads to

Φ0(~r, t) =

v(2π)3

(πDvt

)3/2exp

(− x2+y2+z2

4Dvt

)for t > 0,

0 for t < 0.(E-15)

Finally, the following expression is obtained for the Green’s function of the absorbinginfinite medium Φ(~r, t) for t > 0:

Φ(~r, t) =v

(4πDvt)3/2exp(− r2

4Dvt− µavt), (E-16)

where r2 = x2 + y2 + z2.

Steady-state source

The fluence inside the infinite homogeneous medium is subjected to the diffusionoperator with a spatial Dirac delta source [Eq. (3.40)]:(

−D∇2 + µa

)Φ(~r) = δ3(~r). (E-17)

The Green’s function Φ(~r) can be obtained by integrating over the entire timerange Eq. (E-16). Here, as an example, the solution is derived directly for the CWsource. The problem has spherical symmetry, then it is convenient to use sphericalcoordinates. The Laplace operator is

∇2 =∂2

∂r2+

2r

∂

∂r+

1r2 sin θ

(sin θ

∂

∂θ

)+

1r2 sin2 θ

∂2

∂ϕ2, (E-18)

and since the source term is independent of θ and ϕ, the fluence rate has to beindependent of θ and ϕ. Thus, Φ(~r) is a function only of |~r| = r. Moreover,

258 Appendix E

δ3(~r) = δ(r)/(4πr2), and noting that the radial derivative of the Laplacian can beexpressed as 1

r∂2

∂r2r, Eq. (E-17) becomes(

−D1r

∂2

∂r2r + µa

)Φ(r) =

δ(r)4πr2

. (E-19)

With the notation Ψ(r) = Φ(r) r Eq. (E-19) becomes

4πrD[− ∂2

∂r2Ψ(r) +

µaD

Ψ(r)]

= δ(r). (E-20)

For solving this equation, the homogeneous equation

− ∂2

∂r2Ψ(r) +

µaD

Ψ(r) = 0 (E-21)

is considered. Equation (E-21) is the classical harmonic oscillator equation. Ageneral solution of this equation is the function

Ψ(r) = A exp(−√µaDr)

+B exp(√µa

Dr), (E-22)

from which Φ(r) becomes

Φ(r) =A

rexp

(−√µaDr)

+B

rexp

(√µaDr). (E-23)

The constant factors need to be determined using the two boundary conditions:

• The first condition is given by the behavior at the infinite boundary where thesolution has to vanish, i.e., lim

r→∞Φ(r) = 0.

• The second condition states that the total net flux F , through a sphericalsurface of radius r with center in the source for r → 0, has to be equal to thepower emitted by the source, i.e., lim

r→0F = 1.

The first condition is fulfilled when B = 0. For the second, ~J(~r) is evaluated usingFick’s law:

~J(~r) = −D∇Φ(r) = DA

1 +√

µaD r

r2

exp

(−√µaDr

)r. (E-24)

The total net flux through a spherical surface of radius r, Σ(r), with the centerat r = 0 (source point), is given by

F =∫

Σ(r)

∣∣∣ ~J(~r)∣∣∣ r · qdΣ , (E-25)

Green’s Function of the Diffusion Equation in an Infinite Homogeneous Medium 259

with q normal to the surface and dΣ a surface element. Since q = r and dΣ =r2dΩ,Eq. (E-25) becomes

F =∫4π

∣∣∣ ~J(~r)∣∣∣ r2dΩ = 4πr2

∣∣∣ ~J(~r)∣∣∣ . (E-26)

Consequently, since limr→0

F = 1, we have that

A =1

4πD. (E-27)

Finally, the expression for the fluence is

Φ(~r) =1

4πrDexp (−µeffr) , (E-28)

with µeff =√µa/D =

√3µaµ

′s as the effective attenuation coefficient.

CW photon flux for the infinite non-absorbing medium

The expression of the Green’s function for the photon flux inside an infinite mediumilluminated by a steady-state source can be obtained from Eq. (E-28) by using Fick’slaw:

~J(~r) = −D∇Φ(r) =1

4πr

(1r

+ µeff

)exp (−µeffr) r. (E-29)

For obtaining Eq. (E-29), Fick’s law has been used twice: first to obtain Φ(r) andthen to obtain ~J(~r). Nevertheless, this expression is exact for a non-absorbingmedium. To show this property, the expression of ~J(~r) for µa = 0 will be obtainedusing only the continuity equation. For a non-absorbing medium and a steady-statedelta source, the continuity equation [Eq. (3.29)] becomes

∇ · ~J(~r) = δ3(~r). (E-30)

This equation is rigorous and is not subjected to any approximation. IntegratingEq. (E-30) over a sphere of radius r (volume V ), it is possible to determine ~J(~r).The symmetry of the problem requires that ~J(~r) = J(r)r. Using Gauss’s theorem,5∫

V

∇ · ~J(~r)dV =∫Σ

~J(~r) · rdΣ = J(r)4πr2, (E-31)

with Σ surface of the sphere. Since the integral of the source term is∫V

δ3(~r)dV = 1, (E-32)

260 Appendix E

the expression of ~J(~r) for µa = 0 is obtained:

~J(~r) =1

4πr2r. (E-33)

The importance of having derived Eq. (E-33) for µa = 0 from the continuity equationis that this expression for the flux is exact.

Appendix F

Temporal Integration of theTime-Dependent Green’s FunctionIn this appendix, some details are provided about the integrals on the whole timerange necessary to calculate the Green’s function for steady-state sources of Chap-ter 4 from the time-dependent Green’s functions. All the temporal integrationsperformed in Chapter 4 can be represented as integrals of the following type:6

+∞∫0

tν−1 exp(−Bt− γt)dt = 2

(B

γ

)ν/2Kv

(2√γB), (F-1)

with υ = −1/2 and −3/2 and where Kv(x) are the modified spherical Besselfunctions of the third kind for which

Kv+1/2(x) = K−v−1/2(x) . (F-2)

Using the analytical form of the K functions,√

π2xK1/2(x) = π

2x exp(−x) ,√π2xK3/2(x) =

(1 + x−1

)exp(−x) ,

(F-3)

the integral of Eq. (F-1) can be represented in analytical form. The integral ofEq. (F-1), although generally obtained for Re(B) > 0 and Re(γ) > 0, can be alsorepresented in the form of Eq. (F-1) for Re(B) = 0 and Re(γ) = 0.

261

Appendix G

Eigenfunction ExpansionIn this appendix, it is shown how an eigenfunction expansion can be used to solve aninitial and boundary value problem for parabolic equations. The initial and boundaryvalue problem for a parabolic equation like the lossless DE in a bounded region Vwith a source term q(~r, t) = S0(~r)δ(t) can be formulated as7, 8(

∂v∂t −D∇

2

)Φ(~r, t) = 0 for ~r ∈ V and t > 0, (G-1)

Φ(~r, t) = 0 for ~r ∈ ∂V and t > 0 (G-2)

andΦ(~r, 0) = vS0(~r) for ~r ∈ V . (G-3)

The method of separation of variables is applied to solve the problem. Then, asolution Φ(~r, t) is sought in the form of

Φ(~r, t) = η(t)ξ(~r) . (G-4)

Substituting Eq. (G-4) in (G-1) results in

ddtη(t)η(t)

=vD∇2ξ(~r)

ξ(~r)= −λ for ~r ∈ V and t > 0 . (G-5)

Since both the left and the right side of Eq. (G-5) depend on different variables,they must be constant functions independent of both ~r and t, and thus they can beassumed equal to a constant value λ. The problem of determining ξ(~r) is denoted asan eigenvalue problem. The function ξ(~r) is the solution of the Hemholtz equationin Eq. (G-5):

∇2ξ(~r) = − λ

vDξ(~r). (G-6)

For a non-zero eigenvalue λ corresponding to an eigenfunction ξ(~r) and the functionη(t), the solution of

d

dtη(t) = −λη(t) (G-7)

will have the form η0 exp(−λt). Provided a complete set of orthonormal eigen-functions ξn(~r), it is possible to represent the solution Φ(~r, t) of the DE as asuperposition of solutions ηn0 exp(−λnt)ξn(~r):

Φ(~r, t) =+∞∑n=1

ηn0 exp(−λnt)ξn(~r) . (G-8)

262

Eigenfunction Expansion 263

The initial and boundary value problem also requires, according to Eqs. (G-2)and (G-3), that

Φ(~r, t) =+∞∑n=1

ηn0 exp(−λnt)ξn(~r) = 0 for ~r ∈ ∂V and t > 0 , (G-9)

Φ(~r, 0) =+∞∑n=1

ηn0ξn(~r) = S0(~r) for ~r ∈ V . (G-10)

Multiplying Eq. (G-10) by an arbitrary eigenfunction ξk(~r) and integrating overV , making use of the orthonormality property of the eigenfunctions ξn(~r)7, 8, it ispossible to express the coefficients ηn0 as

ηn0 =∫V

S0(~r)ξ∗n(~r)dV. (G-11)

The eigenfunctions ξ(~r) must be, according to Eq. (G-9), zero at the boundary∂V :

ξn(~r) = 0 for ~r ∈ ∂V. (G-12)

Then Φ(~r, t) becomes

Φ(~r, t) =+∞∑n=1

exp(−λnt)ξn(~r)∫V

S0(→r′ )ξ∗n(

→r′ )d→r′

=∫V

S0(→r′ )[

+∞∑n=1

exp(−λn t)ξn(~r)ξ∗n(→r′ )]d→r′ .

(G-13)

It is thus possible to define the function G(~r,→r′ , t, t′)

G(~r,→r′ , t, t′) =

+∞∑n=1

exp[− λn(t− t′)

]ξn(~r)ξ∗n(

→r′ ) (G-14)

as the Green’s function of the lossless DE [Eq. (G-1)]. Equation (G-14) is thesolution of the DE when q(~r, t) = δ(~r −

→r′ )δ(t− t′).

The expression in Eq. (G-14) is referred to the lossless DE as in Eq. (G-1). Inorder to extend this approach to the DE as derived in Sec. 3.6 for a homogeneousmedium [see Eq. (3.36)] with µa 6= 0, the method of separation of variables appliedto Eq. (3.36) determines this new equation:

ddtη(t)η(t)

=vD∇2ξ(~r)

ξ(~r)− vµa = −λ for ~r ∈ V and t > 0. (G-15)

Therefore, the eigenvalue problem of Eq. (G-6) and Eq. (G-7) is rewritten asdη(t)/dt = −λη(t)∇2ξ(~r) +K2ξ(~r) = 0,

(G-16)

264 Appendix G

with K2 = (λ/v − µa)/D. The eigenfunction expansion can be formulated simi-larly to Eq. (G-8), taking into account that the new coefficient K in the Hemholtzequation [the second equation in the system (G-16)] determines eigenvalues λn andeigenfunctions ξn, depending also on the absorption coefficient µa.

A final remark about the boundary condition used: The solution has been as-sumed to vanish on the physical boundary ∂V according to the ZBC (see Sec. 3.9.1).Alternatively, we could assume the fluence to vanish at an extrapolated boundaryaccording to the EBC. The kind of boundary condition employed does not affect theprocedure used to obtain the solution.

Appendix H

Green’s Function of the DiffusionEquation for the HomogeneousCube Obtained with theEigenfunction MethodAs an example of applying the eigenfunction method, we consider a homogeneousscattering and absorbing cube having side L. The time-dependent DE for a Diracdelta source is [

1v

∂

∂t+ µa −D∇2

]Φ (~r, t) = δ(~r − ~rs)δ(t) . (H-1)

This problem is equivalent to the following initial and boundary value problem in anorthogonal x, y, z reference system:[

1v

∂

∂t+ µa −D∇2

]Φ (~r, t) = 0 for t > 0 , (H-2)

Φ(x = ±L

2 , y, z, t) = Φ(x, y = ±L2 , z, t) = 0

Φ(x, y, z = 0, t) = Φ(x, y, z = L, t) = 0(H-3)

andΦ(~r, t = 0) = vδ(~r − ~rs) , (H-4)

where we have assumed that the ZBC is valid throughout the whole boundary of thecube, and ~rs = (0, 0, zs). We search for a solution of the kind

Φ(~r, t) = ξ(~r)η(t). (H-5)

Substituting Eq. (H-5) in Eq. (H-2), we getdη(t)/dt = −λη(t)∇2ξ(~r) +K2ξ(~r) = 0 ,

(H-6)

where K2 = (λ/v − µa)/D.The physical soundness of the solution implies that λ ≥ 0; however, the

Helmholtz equation in the system of Eq. (H-6) admits roots also for K2 < 0. TheHelmholtz equation can be solved by the method of separation of variables. We willsearch for a solution:

ξ(~r) = fl(x)fm(y)fn(z) . (H-7)

265

266 Appendix H

After substitution of Eq. (H-7) in the Helmholtz equation of Eq. (H-6), we obtain1fl

d2

dx2 fl = −K2l

1fm

d2

dy2fm = −K2

m

1fn

d2

dz2fn = −K2

n ,

(H-8)

where K2 = K2l +K2

m +K2n. The three equations in the system of Eq. (H-8) are

formally identical, and we can study one of them; for example, the second equationin Eq. (H-8). We will treat separately the two cases K2

m > 0 and K2m < 0.

a) K2m > 0

For this case, the general solution of the equation is

fm(y) = Am cos(Kmy + γm) . (H-9)

The condition (H-3) is satisfied, for example, if Km = (2m−1)πL ,m = 1,2,3,. . . , Am

arbitrary, and γm = 0.

b) K2m < 0

For this case, the general solution is

fm(y) = Am exp(pmy) +Bm exp(−pmy) , (H-10)

whereAm,Bm must be determined from the boundary conditions, and the parameterpm verifies: p2

m = −K2m. However, the condition (H-3) can never be satisfied by

Eq. (H-10) unless pm = 0 and Am = −Bm, and this solution must obviously bediscarded. The same arguments apply for the function fl and fn in the system (H-8)for which we cannot have solutions for K2

l < 0 and K2n < 0. We notice that for

the function fn the constant phase term γn is γn = π/2; while for fl, it is γl = 0[condition (H-3)]. Finally, it is easy to verify that the functions

ξlmn(~r) =(

2L

)3/2

[cos(Klx) cos(Kmy) cos(Knz + π/2)] (H-11)

with Kl, Km, Kn given byKl = (2l − 1 ) π/L l = 1, 2, 3, . . .Km = (2m− 1 ) π/L m = 1, 2, 3, . . .Kn = nπ/L n = 1, 2, 3, . . .

(H-12)

constitute a complete orthonormal set of eigenfunctions for the cube. In order tofind the time-dependent solution in (H-6), we can apply the same arguments used inthe previous appendix. Finally, we are led to the following solution of our initialand boundary value problem:

Φ(~r, t) = (2/L)3∞∑

l,m,n=1

v cos(Klx) cos(Kmy) cos(Knz + π/2)

× cos(Knzs + π/2) exp[−(K2D + µa)vt] .(H-13)

Green’s Function of the Diffusion Equation for the Homogeneous Cube 267

The Green’s function for the cube can be represented in general form as

Φ(~r,→r′ , t, t′) = (2/L)3

∞∑l,m,n=1

v cos(Klx) cos(Kmy) cos(Knz + π2 )

× cos(Klx′) cos(Kmy

′) cos(Knz′ + π

2 )× exp−[K2D + µa]v(t− t′) .

(H-14)

Appendix I

Expression for the NormalizingFactorThe expressions for the normalizing factor Nln, like the calculation done in Refs. 9and 10, are obtained from Eq. (6.25) and result in

N2ln = π [LJ1(Kρl)]2

v1

[l12 −

sin(2Kz1lnl1)4Kz1ln

]+ v2 n4

2 sin2(Kz1lnl1)

n41 sin2(Kz2lnl2)

×[l22 −

sin(2Kz2lnl2)4Kz2ln

],

(I-1)


v1

[− l1

2 + sinh(2|Kz1ln|l1)4|Kz1ln|

]+v2 n4

2 sinh2(|Kz1ln|l1)

n41 sin2(Kz2lnl2)

[l22 −

sin(2Kz2lnl2)4Kz2ln

] (I-2)

and


v1

[l12 −

sin(2Kz1lnl1)4Kln1

]− v2 n4

2 sin2(Kz1lnl1)

n41 sinh2(|Kz2ln|l2)

×[l22 −

sinh(2|Kz2ln|l2)4|Kz2ln|

].

(I-3)

Expressions (I-1), (I-2) and (I-3) are valid for the cases when both Kz1ln andKz2ln are real, when Kz1ln = i|Kz1ln| and Kz2ln = |Kz2ln|, and Kz1ln = |Kz1ln|and Kz2ln = i| Kz2ln|, respectively. The coefficients l1 and l2 are so defined:l1 = s1 + z1e, l2 = s2 + z2e and J1 is the Bessel function of the first kind of orderone.

268

269

References



3. G. W. Faris, “Diffusion equation boundary conditions for the interface betweenturbid media: A comment,” J. Opt. Soc. Am. A 19, 519–520 (2002).

4. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A12, 2532–2539 (1995).


6. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products,Academic, New York, 1980 (1980).



9. F. Martelli, A. Sassaroli, S. Del Bianco, Y. Yamada, and G. Zaccanti, “Solutionof the time-dependent diffusion equation for layered diffusive media by theeigenfunction method,” Phys. Rev. E. 67, 056623 (2003).


Index

AbsorptionCoefficient, 11, 35, 42, 57, 112,

132, 194, 234Cross section, 12Efficiency, 13Interaction, 9Mean free path, 18Perturbation, 133, 176, 200, 217Properties, 12, 14

Agricultural products, 22Analytical theory, 29Anderson localization of light, 18Angular dependence of radiance, 84Asymmetry factor, 11, 40, 189, 213

Background medium, 12, 132Ballistic

Component, 12Peak, 98Photons, 96, 204Time, 58, 213

Bessel functions, 82, 100, 110, 182,261, 268

Biological tissues, 22Biomedical optics, 3Born approximation, 132Boundary conditions

Diffusive-diffusive interfaces, 46,252

Diffusive-non-scattering interfaces,43, 250

ComparisonsMC-DE for inhomogeneous me-

dia, 217, 223

MC-DE for the homogeneous medium,204, 213

MC-Hybrid models, 225, 230Conduction of heat, 5Continuity equation, 39, 40CW

Mean pathlength, 74Source, 31

Cylindrical particles, 22

Detected power, 36Diffuse optical tomography, 3Diffusion

Approximation, 38, 245Coefficient, 39, 41Equation, 37, 39Equation for layered media, 109Of electrons and holes, 5Of molecules in gases, 5

DiffusiveMedia, 22, 238Regime, 22, 38, 60, 240

Dirac delta source, 32, 63Discrete ordinates method, 37, 93

EBC, 46, 62, 94, 111, 182EBPC, 75, 96, 117, 143, 168Effective attenuation coefficient, 59Eigenfunction expansion, 110, 113,

262Elastic scattering, 10, 13Electrical conductivity, 5Electromagnetic theory of multiple scat-

tering, 29Energy density, 30, 31Extinction coefficient, 11, 34, 191

271

272 INDEX

Extrapolated distance, 46, 64, 113

Fick’s law, 39, 66, 74, 246Finite difference method, 37Finite element method, 37, 93Fluence rate, 30, 58, 64, 93, 113, 134,

166, 204, 254Fluorescence, 10Flux vector, 31FORTRAN, 3, 168Fresnel reflection coefficient, 44

Gamma function, 98Gauss’s theorem, 46, 134General property of light re-emitted

by a diffusive medium, 121Generalized solution, 118Geometric cross section, 13Geometry

Cube, 265Cylinder, 81Infinite medium, 57, 255Parallelepiped, 80Slab, 62Sphere, 83Two-layer cylinder, 112

Green’s functionDE

Cube, 265Cylinder, 81Infinite medium, 255, 257, 259Parallelepiped, 80Photon flux for the non-absorbing

medium, 259Reflectance, 66Slab, 64–66Sphere, 83Transmittance, 66Two-layer cylinder, 117

Method, 32RTE

Infinite medium, 98TE

Infinite medium, 99

Helmholtz equation, 111, 113, 265Henyey and Greenstein model, 189Highly scattering materials, 23Hybrid

Approach, 75, 94, 143, 232Solutions, 63, 93, 101, 148

Independent scattering approximation,13, 17

Inelastic scattering, 10Initial and boundary value problem,

110, 262Integral transport methods, 37Interference effects, 17Internal pathlengths moments, 156Interstellar medium, 23Intralipid, 23Irradiance, 30

Jacobian, 137, 218

Lambert’s cosine law, 86Lambert-Beer law, 12, 35Lambertian surface, 86Laplace transform, 35Legendre polynomials, 83, 182

MATLAB, 3Maxwell’s equations, 29MC

Code for the homogeneous andlayered slab, 193

Code for the infinite homogeneousmedium, 191

Code for the slab containing aninhomogeneity, 195

Method, 187Results, 203Results in the CD-ROM, 197Simulations, 87

Mean free path, 17, 18, 41, 62, 207Mean number of scattering events, 62

INDEX 273

Mean pathlength, 60Mean time of flight, 36, 37, 123Method of images, 62, 94, 110Mie theory, 14, 17, 189Modified reciprocity relation, 117Multiple scattering theory, 29

Nth–order moments, 36, 37Near-infrared spectroscopy, 3Neutron transport, 5Normalizing factor, 268

Optical propertiesBiological tissue, 22, 48Statistical meaning, 18Turbid medium, 11

Optical thickness, 12Outgoing flux

Comparison between Fick and EBPC,232

EBPC, 74, 96, 117, 143Fick’s law, 96PCBC, 45

P1 approximation, 38Padé approximation, 131Particle concentration, 13Path integral method, 37PCBC, 45, 251Pencil light beam, 66Penetration depth, 124, 225Perturbation

Hybrid models, 154Layered slab, 155MC, 195Theory, 132

Phosphorescence, 10Photon density, 31Power measured, 35Principle of conservation of energy,

31Probability distribution function, 18,

188

Properties of the DE, 42Dependence on absorption, 42Scaling properties, 42

Properties of the RTE, 33Dependence on absorption, 35Scaling properties, 33

Radiance, 30, 57, 95, 135, 176, 191,204

Radiative transfer equation, 31, 93Random

Media, 9Number generator, 190Walk method, 93Walk theory, 132

Rayleigh scatterers, 14Reduced scattering coefficient, 19, 39,

40, 57, 118, 238, 255Reduced scattering efficiency, 20Reflectance, 66, 96, 117Robin boundary condition, 46Rytov method, 132, 157

ScatteringCoefficient, 11, 13, 193Cross section, 13Efficiency, 13Interaction, 9Mean free path, 17Perturbation, 133, 166, 196, 217Phase function, 11, 13, 18Properties, 13

Second-order moment, 61Similarity relation, 19Single-scattering albedo, 11Size parameter, 14Snell’s law, 87, 253, 254Software of the CD-ROM, 165

Diffusion&Perturbation program,166

Solutions of the DE and HybridModels, 170

Solution

274 INDEX

DECylinder, 81Infinite medium, 57Parallelepiped, 80Perturbation theory, 132Slab, 62Sphere, 83Two-layer cylinder, 112

RTEInfinite medium, 98

TEInfinite medium, 99

Source codeGeneral purpose subroutines and

functions, 182Hybrid models for the homogeneous

infinite medium, 177Hybrid models for the homogeneous

parallepiped, 182Hybrid models for the homogeneous

slab, 180Solutions of the DE for homogeneous

media, 171Solutions of the DE for layered

media, 175Source term, 32, 40, 66, 111, 133, 257,

262Specific intensity, 30, 245Spectral radiance, 29Spectral specific intensity, 29Speed of light in vacuum, 165Spherical harmonics method, 37, 93Standard deviation, 194, 204Statistical meaning of the optical prop-

erties, 18

Telegrapher equation, 99Theory of Fourier’s series, 110Thermal conductivity, 5Thermometric conductivity, 5Time-correlated single-photon count-

ing, 187Total mean pathlength, 61

TPSF, 33TR response, 31Transmittance, 66, 96, 117Transport

Theory, 4Coefficient, 19Mean free path, 19, 207, 234

Turbid media, 11

Visual C, 2, 166Volume fraction, 13

Weak solution, 118

ZBC, 46

Fabrizio Martelli received his master’s degree in 1996 from the University of Florence, Italy studying photon migration through biological tissues. In 2002, he received his Ph.D. degree from the University of Electro-Communications (UEC) in Tokyo, Japan with a thesis on photon migration through homogeneous and layered diffusive media. He is currently with the Physics Department of the University of Florence, Italy. His primary research is in the modeling of light propagation through diffusive media and the development of inversion procedures to retrieve the optical properties of diffusive media from time-resolved and spatially-resolved measurements. He has studied the modeling of photon transport with numerical simulations and with analytical theories, i.e., using both Monte Carlo simulations and analytical solutions derived from the diffusion equation or from the radiative transfer equation. He has conducted many research investigations into the optical properties of biological tissues using time-resolved and spatially-resolved near-infrared spectroscopic measurements, and he has experience with the inverse problem implemented with standard nonlinear regression procedures and with analytical solutions of the diffusion equation. Samuele Del Bianco received his master’s degree in 2002 from the University of Florence, Italy with a thesis on the methodologies for measuring the optical properties of biological tissues using both homogeneous and layered models. In 2008, he received his Ph.D. degree from the University of Pisa, Italy with a thesis on radiative transfer and the inversion model of atmospheric measurements in the far infrared for cloudy conditions. He is currently a researcher at the Institute of Applied Physics “Nello Carrara” (IFAC) of the National Research Council (CNR), Italy. His primary research is in the modeling of light propagation through the atmosphere to retrieve the physical and chemical properties of atmospheric constituents. He has experience in modeling light propagation with Monte Carlo simulations and in using analytical theories derived from the radiative transfer equation. He has developed several Monte Carlo programs for light propagation through scattering media using both unpolarized and polarized light. He also has a good deal of experience in modeling light propagation through diffusive media like biological tissues. He has conducted several experimental investigations on measuring the optical properties of diffusive media and biological tissues by using time-resolved and spatially-resolved measurements, and has used standard nonlinear regression procedures to retrieve the optical properties of diffusive media with a special reference to biological tissues. Andrea Ismaelli has been with the Physics Department of the University of Florence, Italy since the early 1960s. Since the 1980s he has been developing software for model simulations, process automation, interfacing, and software engineering. In particular, he developed several Monte Carlo programs used in atmospheric and tissue optics and has a great deal of experience with Monte Carlo simulations for modeling the multiple scattering in lidar returns from

atmospheric structures. He has also analyzed the human voice, particularly the cries of infants, to study the effect of vocal tract diseases among infants. He is currently a senior analyst/programmer with broad experience in software architecture and development. As a private consultant, he coordinates software development projects and selects, installs, and maintains computer systems. Giovanni Zaccanti has been a researcher at the Physics Department of the University of Florence, Italy since 1982. At present he leads the Light Propagation through Turbid Media Laboratory. He has published more than one hundred peer-reviewed papers on experimental and on theoretical studies, beginning with problems related to atmospheric optics. Specifically, he investigated both experimentally and with Monte Carlo simulations the transmission of CW and pulsed laser beams through a turbid atmosphere and the contribution of multiple scattering to the lidar echo from clouds and fogs. Since 1990 his main activity has been devoted to tissue optics, developing numerical and analytical models for photon migration through diffusive media and methodologies for measuring optical properties of diffusive media. In particular, he has developed methodologies to calibrate the optical properties of liquid diffusive media. He has also investigated optical imaging in optical mammography and functional imaging.

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