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PREFACE

During the past 50 years, advanced developments in physics and engineering technology have played anincreasingly important role in the development of modern biomedical sciences. The present set of ten volumeson Biomedical Physics was written to cover the key areas of developments and to illustrate the new approachesas well as methods and theories employed.

Following the general structure of medical care, the new fundamental imaging methods are first coveredstarting from the most sensitive molecular approach with positron emission and single photon emissiontomography (PET and SPECT, Volume 1) via computed tomography and ultrasound (CT and US, Volume2), to magnetic resonance imaging and spectroscopy (MRI and MRSI, Volume 3), and optical molecularimaging (OMI, Volume 4). Each of these volumes is describing the important contributions to the four imagingrevolutions in biomedical imaging since the early 1970s. This part is then followed by an overview of moregeneral biomedical measurement techniques (Volume 5), modern bioinformatics covering the important areasof genomics, proteomics, lipidomics, and metabolomics (Volume 6), and molecular radiation biology andradiation safety (Volume 7). The last set of volumes cover the therapeuticly important field of biomedicalphysics treatment methods starting from Volume 8 on accelerators and radiation sources and detectors, toVolume 9 on radiation interaction with matter, radiation transport theory, and absorbed dose, followed bytreatment planning and treatment optimization. Finally, there is Volume 10 on the wide range of otherphysically based treatment and rehabilitation methods and their biological effects.

It is hoped that the present series will stay alive not least in its Internet version, so new, important areas canbe covered and incorporated as they mature and comprehensive chapters get written. It is hoped that thepresent set of volumes will be useful in biomedical research as well as education where the wide spectrum ofimaging, diagnostics, and therapeutic approaches based on biophysical processes have not been so extensivelycovered in recent years. They may allow an efficient transfer of knowledge from the wide range of methodsavailable that may become of increasing interest in your own area of expertise. I therefore hope that thisextensive reference work on biomedical physics will be of interest for the whole biomedical and applied physicscommunities for years to come.

A. BrahmeKarolinska Institute, Stockholm, Sweden

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INTRODUCTION TO VOLUME 2: X-RAY AND ULTRASOUNDIMAGING

This Volume of Comprehensive Biomedical Physics focuses on the technology and applications of the two mostwidely available categories of morphological imaging techniques, x-ray and ultrasound (US) imaging. Forsure, these two classes of methods dominate the Medical Imaging arena in terms of local availability andfacilities, the number of devices installed worldwide, and the number of exams performed routinely inimaging departments. The word ‘morphological’ used above is, indeed, misleading. When talking aboutRadiology, we suddenly start thinking about chest radiographs, dental panoramic images, fetal USs, oranything else that has to do with the ‘shapes’ of internal organs. The very first radiograph in history (thevery famous Mrs. Roentgen’s hand with ring in 1895) was so impressive that it was immediately clear whatthe main practical application of the x-ray would be after its discovery. Unlike the development of radionu-clide imaging techniques, which have always focused on organ functionality, it took almost a decade for thescientific and medical community to figure out how to get functional information from US images (intro-duced in the 1940s with the first applications in echocardiography in the ’50s and ’60s) and more than acentury to do the same with x-rays.

When Prof. Brahme and Elsevier contacted us about this project, we accepted to collaborate willingly sincewe immediately realized that it would be a great opportunity for us to improve our knowledge in our respectiveareas of research. Given our scientific interests, we were confident in our knowledge of the physics of US andx-rays, but not to the extent that we could consider ourselves true experts in the field. We have certainly learneda lot during the preparation of this Volume, and we must thank all the authors for their patience and for havingfulfilled almost all our requests, even though some of them probably went beyond what they considered to be‘acceptable.’ We spent a lot of time asking the authors to make every single sentence and every mathematicalstep in their Chapters clear. The reader will appreciate the effort devoted to ensuring that there is a logical threadrunning through the book and that the necessary integrations between the chapters were made so as toguarantee a thorough description, not only of the main topics but also of niche and advanced topics, whichwill hopefully help the book find a distinctive collocation within the wide range of US and x-ray imagingliterature. As regards this last point, we have chosen to reserve space for new ideas and new perspectives as far aspossible and to devote entire chapters to the description of important research tools.

The first Section of Volume 2 covers the technology of Medical Imaging based on x-rays. The series of articlespresented in this Section attempts not only to give the reader a clear picture of the state-of-art of current x-rayimaging physics and technology, which are primarily based on the selective attenuation of photons in tissueswith a different atomic number and density, but also of the emerging techniques which are, conversely, basedon the differences of phase shifts of the x-ray wavefront on tissues with different complex refractive indices.Although still experimental, it is of primary importance for physicists, engineers, and scientists to keepup-to-date with the current developments of such emerging techniques which may have a prominent role inthe diagnostic scenario in the near future.

The first Section is therefore structured as follows. Chapters 2.01–2.03 provide the foundations for under-standing the physical and technological aspects of planar and tomographic x-ray imaging. Due to the severalpeculiarities of each diagnostic application of x-ray imaging, Chapters 2.04–2.07 are structured in such a waythat the reader can have a clearer understanding of how each diagnostic task has led to different technologicalevolutions of imaging devices starting from the same physical principles of x-ray emission, interaction withtissues, and detection. Chapters 2.08–2.11 are, on the other hand, more focused on advanced or niche topics inradiology and emerging applications such as phase-contrast imaging, micro-CT, radiation protection, and anin-depth overview of the mathematical foundations of tomographic reconstruction in two and threedimensions.

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xiv Introduction to Volume 2: X-Ray and Ultrasound Imaging

More specifically, this is what the reader will find in each Chapter.Chapter 2.01 introduces the basic aspects of x-ray imaging for diagnostic radiology, the description of

photon interaction processes in the diagnostic energy range, and the introductory physical aspects of x-rayimage formation mechanisms (image contrast based on x-ray attenuation or phase change). It comprises athorough discussion on the radiation quality, from the most common type of source for x-ray imaging, that is,the x-ray tube, and a description of x-ray tube spectra and its shaping through filtration.

Chapter 2.02 describes the primary physical parameters of image quality and their relationship, as well asthe derived parameters obtained from them. In particular, it explains fundamental objective and subjectiveimage quality metrics such as modulation transfer function, noise power spectrum, contrast, contrast transferfunction, signal-to-noise ratio, contrast-to-noise ratio, and low-contrast detectability.

Chapter 2.03 provides an introductory overview of Computed Tomography physics and technology, with astrong focus on the technological evolution that has led to the current configuration of modern multislice CTand cone-beam CT scanners, along with the most important dose reduction techniques as well as a concisereview of common image artifacts and the possible strategies for their correction.

Chapter 2.04 opens the application-driven part of this Section. This chapter addresses the technologies andthe applications of radiology used in the field of oral (or dental) and maxillofacial imaging, describingdedicated x-ray sources for dental intraoral radiology, intraoral detectors, equipment for panoramic and forcephalometric extraoral radiology and cone-beam volumetric imaging of the head.

Chapter 2.05 focuses on mammography. The various stages of the breast imaging chain and the specificrequirements to optimize image quality and patient dose in mammography are reviewed and discussed, andthe latest information on advanced applications such as x-ray tomosynthesis, breast CT, and dual-energymammography are presented.

Chapter 2.06 introduces the reader to the principles and techniques of x-ray imaging based on photonenergy discrimination. This field of study has important applications in quantitative imaging. The theory andapplications of dual-energy and multi-energy (or spectral) imaging are presented and discussed.

Chapter 2.07 deals with quality control (QC) in x-ray imaging. This chapter is intended mainly for medicalphysics experts whose role is to organize and accomplish adequate programs of Quality Assurance within theirimaging departments. QCs for computed and digital radiography, mammography systems, dental systems,digital angiography systems, and to conclude computed tomography systems are presented. Procedures,reference values, and typical periodicity of QCs are reported for each type of equipment.

The wave nature of x-rays, which is commonly disregarded in conventional biomedical imaging applica-tions, plays a fundamental role when coherent sources of x-rays and phase-sensitive techniques are employed.This important and emerging field of application is the subject of Chapter 2.08. The five main different phase-sensitive techniques, namely propagation-based phase-contrast imaging, analyzer-based imaging, coded aper-tures phase-contrast x-ray imaging, interferometry, and grating interferometry are introduced in this Chapter,providing the basic physical principles and presenting selected biomedical applications.

The downscaling of tomographic x-ray imaging technology to the size of small laboratory animals isdiscussed in Chapter 2.09. Due to the growing role of preclinical imaging in phenotyping, drug discovery,and in providing understanding of the mechanisms of disease, we thought it important that the readerunderstand how challenging it might be to obtain high performance in terms of image quality and temporalresolution in subjects that are more than one order of magnitude smaller than humans (and with heart rates upto ten times faster!).

Chapter 2.10 deals with the important and often controversial, delicate topic of the risks associated with themedical employment of x-rays for patients and workers, which however can never be stressed enough. Besidesdebates on the issue, which are mostly about the interpretation of radio-epidemiological data (totally beyondthe scope of this Section), the physical dosimetry of ionizing radiation is a mature science and its application onRadiation Protection in diagnostic radiology is the main focus of this Chapter.

At last, for all those who do not like to look upon the reconstruction software packages of tomographicinstrumentation as just ‘black boxes’ (as it should always be for a serious medical physicist or biomedicalengineer), Chapter 2.11 provides an in-depth description of the mathematical foundations of image recon-struction from projections, a fascinating field of research that has never stopped growing since the invention ofthe CT in the 1970s. This Chapter covers the fundamentals of 2D and 3D image reconstruction algorithms, withan emphasis on the analytical methods for x-ray CT imaging, but with a useful overview of iterative methods fortransmission tomography that are continuously gaining ground on today’s CT instrumentation.

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Introduction to Volume 2: X-Ray and Ultrasound Imaging xv

The second Section of Volume 2 covers the technology of Medical Imaging based on US. US is a multifacetedtechnique which offers innumerable advantages. It can provide information regarding the location of anyacoustic discontinuity and an estimate of the reflection coefficient. It can also provide maps of the propagationspeed and maps of the attenuation coefficient. Besides, it is harmless, simple to use, does not require a speciallyequipped room, and can hence be used effectively at the bedside. Moreover, it provides a good time resolution,it can estimate the speed of moving objects and as far as equipment goes, US equipment is economical andextremely popular.

Chapter 2.12 introduces the reader to the basic principles of US propagation and to the well-known B-modeimaging technique. Chapter 2.13 describes the basic principles which govern the probe operation. The maintopic of this chapter is beamforming by multielement array transducers. Chapter 2.14 deals with the topic ofDoppler US where the Doppler shift is utilized to measure blood flow velocities and direction. Chapter 2.15presents and discusses currently available and emerging US imaging modalities ranging from the standardmodes to nonlinear imaging, quantitative imaging, and perfusion imaging quantification. Chapter 2.16introduces the reader to the basics of nonlinear acoustics and its application to medical US imaging. Thecoefficient of nonlinearity is introduced, together with a variety of equations, which develop step by step fromsimple to more sophisticated models. Chapter 2.17 describes the main medical applications of US and devotesspace to their role as bedside diagnostic completions of the physical examination of the patient. The biologicaleffects occurring in diagnostic ultrasound (DUS) represent an important field of inquiry in non-ionizingradiation biology and Chapter 2.18 is devoted to this topic. Chapters 2.19 and 2.20 are entirely devoted tothe description of two important research tools: the simulation of US fields and the US research platforms(highly flexible scanners with wide access to raw echo data).

Below, the authors themselves introduce their respective chapters to the readers by means of short abstractsand personal messages.

Marcello Demi: Chapter 2.12wants to introduce the reader to the basic principles of US propagation. Firstly,an ideal medium is considered to simplify the approach to the equations that govern US propagation in abiological medium. Secondly, the simplifying assumptions are progressively removed in order to come close tothe work conditions within which physicians usually operate. The most popular US imaging technique, thewell-known echo-pulse or B-mode imaging technique, is also introduced. The main assumptions, which are atthe basis of the reconstruction of the US image sequences, are analyzed separately and the image artifactsgenerated by such assumptions are illustrated.

Han Thijssen: In my opinion, Chapter 2.13 is meant to expose the basic physics of (medical) US to thereader. Furthermore, we extensively introduced the various methods and techniques of imaging with arraytransducers and reviewed the principles and backgrounds of image quality assessment and assurance. Recentdevelopments are introduced and explained.

Massimo Mischi: I am convinced that Chapter 2.13 provides the reader with the essence of array beamform-ing through a few selected and well-connected equations and will help them when they approach relatedproblems, before digging into hundreds of pages without any prior understanding of the topic.

Hans Torp: Chapter 2.14 deals with the topic of Doppler US. In Doppler US, the Doppler shift frommovingblood is utilized to measure blood flow velocities and direction and to extract the weak scattering from bloodfrom much stronger echoes from the vessel wall and other larger tissue structures in the human body. The twoclassical modalities of continuous-wave and pulsed-wave Doppler analysis are introduced and discussed. Thecolor Doppler and vector Doppler imaging techniques are also explored.

Massimo Mischi: Quantitative US imaging is the ultimate instrument we can provide doctors with. In thepast few years, we have experienced a tremendous growth in this field, which is paving the way for newdiagnostic options that we could not have even imagined a few years back. Providing a concise and compre-hensive overview of all the imaging modalities, both the traditional ones, such as B-mode or Doppler, as well asthe emerging ones, such as molecular imaging, elastography, and contrast-enhanced US, has therefore been avery exciting challenge. Hopefully, this work (Chapter 2.15) will set a solid basis and lead to furtherdevelopments.

Libertario Demi: Chapter 2.16 introduces the reader to the basics of nonlinear acoustics and its applicationto medical US imaging. The coefficient of nonlinearity is introduced, together with a variety of equations,which develop step by step from simple to more sophisticated models. Starting from sinusoidal plane waves inlossless nonlinear media, the models are gradually expanded to include the effect of absorption, pulsed planewaves, quasiplanar waves, diffraction, and local nonlinearity.

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xvi Introduction to Volume 2: X-Ray and Ultrasound Imaging

Gino Soldati: Chapter 2.17 describes some medical applications of US and devotes space to their role asbedside diagnostic completions of the physical examination of the patient. The chapter shows how, in thissetting, medical sonography represents a powerful tool for integrating imaging and functional data in thecritically ill. Here, some space is also reserved for unusual applications of sonography and its role in diagnosingdiseases of aerated organs such as the lung.

Douglas Miller: Chapter 2.18 Bioeffects in DUS are a little known and poorly understood biomedical topic.Although the energy radiated from US probes is non-ionizing, the interaction of US with living tissue is ascientifically interesting and complex biophysical problem. There is no distinct boundary between therapeuticand DUS, and on-screen dosimetric indexes therefore are provided on DUS machines. As with all medicalprocedures, the practitioner should be aware of the potential interactions with a biological significance andconsider the risks as well as the benefits to the patient.

Martin Verweij: Chapter 2.19 There are numerous conference proceedings and journal articles that describeindividual simulation methods for medical ultrasound fields. However, there exist very few texts that provide ageneral overview of these methods. In my opinion, what is largely missing is a combined description of thefundamentals and taxonomy of this field, preferably in the form of a review article or book chapter. The editorsgave me the opportunity to write such a text for Comprehensive Biomedical Physics, and some very capable co-authors agreed to join me in this task. Under those circumstances it was not hard to accept the invitation forwriting this book chapter. Knowing the main correspondences and distinctions between simulation methods,you can more efficiently and confidently choose the right method for solving a particular problem.

Piero Tortoli: Chapter 2.20 US research platforms have recently gained unexpected success which isongoing. In a few years, they have gone from being cumbersome pieces of equipment, with their use restrictedto the lab of origin, to portable but sophisticated systems that are distributed worldwide. Due to their nature ofbeing open platforms dedicated to research, they are implicitly dynamic, that is, they must evolve continuouslyso as to be always on the boundary between those needs that are already mature and emerging needs associatedwith brand new ‘hot’ ideas.

Daniele PanettaInstitute of Clinical Physiology (IFC-CNR), National Research Council, Pisa, Italy

Marcello DemiFondazione Toscana Gabriele Monasterio, Pisa, ItalyEVIE

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2.01 Physical Basis of x-Ray ImagingP Russo, Universita di Napoli Federico II, Napoli, Italy; INFN Sezione di Napoli, Napoli, Italy

ã 2014 Elsevier B.V. All rights reserved.

2.01.1 Introductory Concepts 22.01.1.1 Imaging Basics 22.01.1.2 Absorption, Scattering, Refraction of x-Rays 32.01.1.3 Tissue Substitute Materials 82.01.1.4 Absorption Contrast and Phase Contrast 82.01.2 Interaction Processes 142.01.2.1 Photoelectric Absorption 142.01.2.2 Rayleigh (Coherent) Scattering 152.01.2.3 Compton (Incoherent) Scattering 192.01.2.4 Mass Attenuation Coefficients and Dosimetry 232.01.2.4.1 Mass energy transfer coefficient 232.01.2.4.2 Mass energy absorption coefficient 242.01.2.4.3 Exposure and absorbed dose 252.01.3 x-Ray Tubes and Beam Quality in Diagnostic Radiology 262.01.3.1 Beam Attenuation and Beam Shape Descriptors 302.01.3.2 Effect of Varying the Kilovoltage at Fixed Beam Filtration 342.01.3.3 Effect of Varying the Added Filtration at a Fixed Kilovoltage 342.01.3.4 Effect of Varying the Tube Current and Exposure Time at Fixed Kilovoltage 352.01.3.5 Effect of Filtration by Air in the Beam Line 362.01.3.6 Beam Output at Varying Kilovoltages 362.01.3.7 Effect of Voltage Ripple and of Target Angle 372.01.3.8 Attenuation and Beam Hardening 372.01.3.9 Focal Spot Size 392.01.4 Examples of x-Ray Image Formation and Contrast Mechanisms 442.01.4.1 Attenuation Contrast (Absorption and Scattering) 442.01.4.2 Attenuation Plus Phase Contrast 46Acknowledgments 47References 47EVIE

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GlossaryBeam hardening The phenomenon of shift toward higher

energies of the spectral distribution of a polychromatic x-ray

beam, upon transmission of the beam through an

attenuating material, as a consequence of the decreasing

x-ray attenuation coefficient of materials with increasing

photon energy.

K-edge filtering A technique of x-ray beam filtration in

which the spectral intensity distribution of the x-ray beam is

heavily attenuated at energies above a threshold given by the

K-edge absorption energy, characteristic of the filter material,

via photoelectric interaction of the incident photons with

K-shell electrons of atoms in the filter material.

x-Ray absorption imaging A technique for producing

projected shadow images of opaque objects using

penetrating radiation as x-rays, revealing the internal

structure of the objects, based on the measurement with an

image detector of the spatially varying transmitted intensity

through the object determined by the varying amount of

absorption of x-rays.

x-Ray beam filtration Technique for changing the

shape of the spectral distribution of the photon beam from

an x-ray source via transmission through a thin (metal)

sheet in front of the x-ray beam. Spectral shaping occurs

by means of energy-selective x-ray absorption in the sheet,

due to the energy dependence of the absorption

coefficient of the filter material at varying x-ray

energies.

x-Ray phase-contrast imaging A technique for

producing a projected image of opaque objects using

penetrating radiation as x-rays, by measuring how the

electromagnetic waves associated with the propagation of

x-rays are phase-shifted in propagation through matter, as a

result of the spatially varying refractive index at x-ray

wavelengths.

x-Ray photon cross section A quantity with the dimension

of an area, which is proportional to the probability of

occurrence of a particular interaction process (e.g.,

photoelectric effect, Compton effect, and Rayleigh

scattering) of x-ray photons with matter.

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mprehensive Biomedical Physics http://dx.doi.org/10.1016/B978-0-444-53632-7
.00201-X 1
http://dx.doi.org/10.1016/B978-0-444-53632-7.00201-X

Source

x-Rays

R1 R2

Wavefronts

x

y z = 0 z = z 0 z = z 0+ D z

D x

z = z l

z

Object

A

B

a

b

a�

b�

c�c

d�

d

e�

e�

e

Detector

Figure 1 Geometry of x-ray planar imaging. x-Rays diverge from thesource and their direction is deviated by scattering (rays e, e´, e) or leftundeviated (rays b, b´) by transmission through the object, producing a‘shadow’ radiography of the object. An area detector with elements ofsize Dx records a signal in each detector element of size Dx, produced byabsorption of x-rays in the sensitive volume of the detector. Thetransmitted x-ray intensity is dependent on the local properties ofabsorption, scattering, and refraction in the object. In terms ofpropagation of the electromagnetic wave associated with the x-rays, theobject introduces a deformation of the incident waveform due to changeof the phase of the wave, also dependent on the local variation of theobject composition. If the source is of finite size, an image blur(geometric unsharpness) is produced at the edges of the projected objectshadow (the light-gray shaded penumbra regions of rays a and a´). Forsuitably small source size, the phenomenon of x-ray refraction can beobserved: rays at the boundary of denser object details are deviated tothe side (diverge) (e.g., rays c, c´) while at the boundary of lighter objectdetails rays are deviated (converge) toward the denser medium (e.g.,rays d, d´).

2 Physical Basis of x-Ray Imaging

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2.01.1 Introductory Concepts

2.01.1.1 Imaging Basics

x-Ray imaging is adopted in many fields, including diagnostic

radiology, in order to visualize the three-dimensional (3D)

internal structure of ‘objects,’ as biological tissues, which are

opaque to visible radiation. The short wavelengths of x-rays

adopted in diagnostic radiology (in the order of 0.01–0.1 nm)

allow one to penetrate the tissues so that a part of the incident

radiation is transmitted through the sample and can be ana-

lyzed with an area detector for the visualization purpose; the

remaining part is either transmitted while remaining unde-

tected (back- or side-scattered radiation, radiation penetrating

the detector) or absorbed in the object. The fraction of radia-

tion that interacts with the atoms in the body by releasing

energy to them produces absorbed dose of ionizing radiation.

Research into stochastic biological effects of ionizing radiation

showed that adverse effects might result from the absorption of

low doses of ionizing radiation in living matter, so that

hypotheses of a risk hazard for the human body might be

postulated, on a statistical base, for diagnostic medical imaging

procedures employing ionizing radiation producing exposure

to as low effective doses as 100 mSv or less (see Chapters 2.10

and 7.12). These hypotheses of risk factors have to be com-

pared with the beneficial effects on individuals, and on the

population, of such diagnostic procedures in the assessment of

body tissue lesions and medical therapies. In any case, any

diagnostic imaging procedure employing ionizing radiation

should aim at reducing to as low as reasonably possible levels

the radiation dose to the patient, so that it is fundamental to

consider image quality in conjunction with the associated

radiation dose, when discussing the relative merits of any

technique for x-ray imaging of biological organisms.

The purpose of this chapter is to introduce basic aspects of

the physics of x-ray imaging in terms of interaction processes,

spectral properties of the radiation generated by an x-ray tube,

and image contrast mechanism.

The various x-ray imaging techniques require a source of

continuous or pulsed radiation; an object to be imaged, at a

distance R1 from the source; and an area detector, at a distance

R2 from the object (Figure 1). This figure intends to summarize

various aspects of x-ray imaging. First, one can consider

the imaging radiation as rays (of energy� tens of keV) emanat-

ing from a point-like or extended source, traveling in straight

lines in vacuum (or air), or as traveling electromagnetic (e.m.)

waves (of wavelength in the order of 0.01–0.1 nm) and corre-

sponding wave fronts. Some of the rays are unaffected by

passage through the sample (e.g., rays b, b0 in Figure 1), while

some others are deviated (scattered) in the forward or in the

backward direction (e.g., rays e, e´). Rays that are absorbed (to

varying extent) in the different regions of the sample generate a

geometrical shadow of the object contour and of its internal

details (indicated with varying gray levels in Figure 1 on the

detector elements). In the case of a point-like source, rays

passing through any point of discontinuity of the sample (as

at its external contour or at the border of an internal detail)

may pass through it and generate a point image on the detec-

tor. However, in the case of a finite-size source (as with all x-ray

tube sources used in diagnostic radiography units) the rays

originating from various points of the source generate a

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finite-size image of any such object point, when reaching a

distant detector (as for rays a, a´). This generates a penumbra

region (e.g., the region between rays a and a´), which intro-

duces an image blur (‘geometric unsharpness’ effect) increasing

as the distance between the object and the detector increases.

In umbra and penumbra regions, the e.m. wave associated

with x-ray propagation behind the object presents a modula-

tion of its amplitude which determines intensity variations on

the detector, with less intense waves corresponding to propa-

gation through more absorbing regions in the sample. On the

other hand, at the same time, wave fronts (of spherical shape,

for a point-like source at finite distance from the object) are

distorted after traversing the sample because of change of the

wave phase. The deformation of the wave fronts corresponds to

refraction of x-rays, with deviation toward the side of less

material density. This is shown, for example, by rays (c, c´)

and (d, d´) in Figure 1 at the border regions of internal details

of higher density or lower density regions than the background

(details A and B, respectively). This refraction produces an

effect of converging or diverging rays, measurable as x-ray

intensity modulations when placing the detector at suitably

large distances R2 from the object, as indicated in Figure 1

(so-called propagation-based phase contrast imaging; Zhou

and Brahme, 2008; Bravin et al., 2013).

Figure 1

Physical Basis of x-Ray Imaging 3

The relative positions of the source, object, and detector

determine the geometry for projection radiography (with the

detector in front of the source acquiring planar view of the

transmitted x-ray beam through the object) or for scatter imag-

ing (with the detector placed at an angle, e.g., 90�, with respect

to the beam axis, acquiring Compton and Rayleigh scattered

photons in the object). Computed tomography (CT) x-ray

transmission imaging requires the acquisition of hundred

views with a linear or area detector array, by relative rotation

of the source–detector assembly around the object, in planar

geometry (Kalender, 2011); if the detector is placed laterally,

scatter CT is performed (Cesareo et al., 2002). In projection

imaging, R2ffi0 and a contact radiography is executed, for

which R1þR2ffiR1 and no image magnification takes place. If

a divergent x-ray beam is used and if R2 6¼0, then magnification

imaging occurs, with magnification factor M¼(image size/

object size)¼(R1þR2)/R1>1.

The characteristics of the radiographic image acquired

depend on many variables, including

(a) the size, shape, and angular intensity distribution of the

x-ray source;

(b) the spectral distribution of the radiation emitted by the

source (e.g., a monochromatic or a polychromatic beam,

and its spectrum of energies/wavelengths);

(c) the beam divergence angle;

(d) the dose to the irradiated object;

(e) the polarization state, coherence, and the temporal struc-

ture of the emitted radiation;

(f) the 3D absorption, scattering, and refraction properties of

the object, which may include details with a 3D spatial

frequency distribution up to a limit frequency dependent

on the size of the finest details;

(g) the distances R1 and R2;

(h) the two-dimensional (2D) spatial resolution, the intrin-

sic detection efficiency, the 2D noise properties (power

spectrum), and the temporal resolution properties of the

detector;

(i) the angle between the beam axis and the imaging plane;

(j) the energy-dependent attenuation properties of the

medium between the source and the object, and between

the object and the detector.

In principle, all of the above conditions affect the quality of

the x-ray image. Quite generally, one could state that the ‘ideal’

imaging conditions are obtained when using a point-like,

intense x-ray source of monochromatic radiation, delivering

low radiation dose to the sample, at practically short imaging

distances from a detector having a spatial resolution permitting

the visibility of the finest details in the irradiated object and

high detection efficiency, and with negligible noise. In practical

situations in biomedical imaging, an unpolarized, incoherent,

finite-sized, and polyenergetic source of x-rays is adopted

(commonly given by the radiation beam emanating from the

focal spot of an x-ray tube); the detector has a finite pixel size in

the order of 0.1 mm, and a quantum detection efficiency and

an energy absorption efficiency less than 100%. Moreover, the

source has a size in the order of 1 mm, the distance R1þR2 is in

the order of 1 m, and R2ffi0. Owing to fluctuations in the

number of x-ray photons impinging on each detector element

around its mean level (related to the statistics of x-ray photon

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generation), which introduce ‘noise’ in the image, the signal-

to-noise ratio (SNR) for detection has a value dependent on the

intensity of the x-ray field and, thus, on the radiation dose to

the object. Owing to nonideal detector characteristics, the SNR

intrinsic to the incident x-ray field on the area of a detector

element is decreased in the output signal of that element, so

that a degradation of SNR characteristic is introduced, and a

dose higher than in the ideal case must be adopted for imaging.

2.01.1.2 Absorption, Scattering, Refraction of x-Rays

In the diagnostic energy range (10–150 keV), the atomic total

interaction cross section stot (cm2 per atom) can be expressed

as the sum over the individual photon interaction cross sec-

tions of the relevant interaction processes in this range (i.e.,

excluding pair and triplet production and photonuclear inter-

actions which occur at higher energies):

stot ¼ sR þ sC þ tPE [1]

where sR is the cross section for Rayleigh (coherent) scattering,

sC that for Compton (incoherent) scattering, and tPE is the

cross section for photoelectric absorption. In this range, for

photon energies E away from the atomic binding energies

where absorption threshold regions are present, both the

photoabsorption and coherent scattering interactions inmatter

can be described by using complex atomic scattering factors (or

form factors), f¼ f1þ if2. The atomic photoabsorption cross sec-

tion, tPE(E), is related to the imaginary part of the atomic

complex form factor and can be obtained from the values of

f2 using the relation

tPE ¼ 2hcreE

f2 ¼ 2relf2 [2]

where re is the classical electron radius¼(e2/4pe0mec2)¼

2.819380�10�15 m, h the Planck’s constant¼6.62606957

�10�34 J s, c the velocity of light in vacuum¼2.99792458�108 m s�1, E¼hv is the photon energy, and

l¼hc/E the corresponding wavelength, with l (nm)¼1.2398520/E (keV). The atomic coherent scattering cross section

is related to the real part f1 of the complex form factor. Using the

complex form factor f (electrons per atom), one can derive

expressions for the material refractive index and for scattering

and absorption coefficients. The (semiempirical) atomic scatter-

ing factors are based upon photoabsorption measurements of

elements in their elemental state, modeling condensed matter as

a collection of noninteracting atoms: this is true for energies

sufficiently far from absorption thresholds.

The atomic photoelectric cross section tPE is related to the

corresponding cross section per unit mass, or photoelectric mass

attenuation coefficient ma/r (cm2g�1), through

mar¼ NA

AtPE [3]

where NA¼Avogadro’s number (¼6.02214129�1023 mol�1)and A¼atomic mass (number of grams per mole of material).

With Z the atomic number (number of electrons per atom

of an element), in the diagnostic energy range, the cross section

tPE for photoelectric absorption varies approximately as Z4,

while the atomic cross section sC for Compton scattering varies

as Z1 and the atomic cross section sR for Rayleigh scattering

VIER

Photon energy (keV)

m/r

(cm

2 g-1

)

mR/r mtr/r

tPE/r

mC/r

m/r

Carbon (Z = 6)

1

104

103

102

101

100

10-1

10-2

10 100

Figure 2 Mass attenuation coefficients and mass energy transfercoefficient for carbon (graphite) are plotted in the diagnostic energyrange. Data are calculated with the code XMuDat (Nowotny, 1998) withinteraction cross section data from Boone and Chavez (1996).

Photon energy (keV)

Phosphorus (Z = 15)

1 10

104

103

102

101

100

10-1

10-2

10-3

100

m/r

(cm

2 g-1

)

mR/r

mtr/r

tPE/r

mC/r

m/r

Figure 3 Mass attenuation coefficients and mass energy transfercoefficient for phosphorus are plotted in the diagnostic energy range. TheK-edge is at 2.15 keV. Data are calculated with the code XMuDat(Nowotny, 1998) with interaction cross section data from Boone andChavez (1996).


VIER

varies approximately as Z2. For all elements but hydrogen (for

which Z/A¼1), the value of Z/A is between 0.4 and 0.5, with

low-Z elements having Z/Affi0.5. Hence, one can assume that

NA/A∝Z�1: this gives the mass attenuation coefficients a Z

dependence of one power of Z less than for the atomic atten-

uation coefficients.

The atomic cross section for Compton scattering sC is

related to the corresponding cross section per unit mass, or

Compton mass attenuation coefficient mC/r (cm2g�1), through

mCr¼ NA

AsC [4]

while the Rayleigh mass attenuation coefficient mR/r (cm2g�1), isdefined as

mRr¼ NA

AsR [5]

The total mass attenuation coefficient m/r (cm2g�1) can be

obtained as the sum of the mass attenuation coefficients for

each interaction process; without considering pair and triplet

production and neglecting photonuclear interactions, it is

given by

mr

� �¼ mR

r

� �þ mC

r

� �þ ma

r

� �[6]

where mR/r indicates the contribution of the Rayleigh scatter-

ing, mC/r indicates the contribution of the Compton scattering,

and ma/r that of the photoelectric effect. For compounds and

mixtures, the mass attenuation coefficient (m/r)comp can be

expressed as the weighted average of the mass attenuation

coefficients (m/r)i of their elemental constituents, with weights

given by their weight fractions wi :

mr

� �comp

¼Xi

wimr

� �i

[7]

For a compound of chemical formula (X1)a1(X2)a2� � �(Xn)an,

one has

wi ¼ aiAiXi

aiAi

[8]

where Ai is the atomic weight of element Xi in the composition.

The linear attenuation coefficient m (cm�1) for a compound

material of mass density rcomp is m¼(m/r)comprcomp.

The linear attenuation coefficient m (cm�1) for an elemental

material of mass density r, is given by

m ¼ rmr

� �¼ r

marþ mR

rþ mC

r

� �[9]

can be expressed in terms of atomic cross sections as

m ¼ rmarþ mR

rþ mC

r

0@

1A

¼ rNA

AtPE þ sR þ sCð Þ

¼ N tPE þ sR þ sCð Þ

[10]

where N is the number of atoms per unit volume (cm�3).While the linear attenuation coefficient m of a material varies

ELSE

linearly with the density r of the material, the mass attenuation

coefficient m/r is independent of density. The mass density r(g cm�3) can be expressed as r¼(A/NA)N, where NA/A is the

number of atoms per gram of material and the quantity NAZ/A

is the number of electrons per gram of material. The mass

attenuation coefficients (m/r, tPE/r, sR/r, sC/r) [and the mass

energy transfer coefficient (mtr/r) (defined in Section

2.01.2.4)] for some elements (C, Ca, P, I, Pb) and for soft

tissue are shown in Figures 2–6, from attenuation data of

Hubbell and Seltzer (1995) or Boone and Chavez (1996), as

tabulated by the computer code XmuDat (Nowotny, 1998).

Photon attenuation data are also available via the computer

code XCOM by Berger and Hubbell (1987).

For the above-mentioned elements, Figure 7 shows the

ratio of Compton to photoelectric mass attenuation coeffi-

cients, indicating that for low-Z elements Compton scattering

may dominate over photoabsorption at energies of interest for

radiography. On the other hand, for high-Z materials photo-

electric interaction is dominant. Similarly, the coefficients for

soft tissue are plotted versus photon energy in Figure 8, while

Figure 9 shows that Compton scattering in soft tissue is dom-

inant at energies above ffi30 keV where the highest output of a

100-kVp spectrum is contained. For water (Figure 10), the

Figure 2

Figure 3

Photon energy (keV)

m/r

(cm

2 g-1

)

mR/r

mtr/r

tPE/rmC/r

m/r

Calcium (Z = 20)

1

104

103

102

101

100

10-1

10-2

10 100

Figure 4 Mass attenuation coefficients and mass energy transfercoefficient for calcium are plotted in the diagnostic energy range. Notethe K-edge at 4.0 keV. Data are calculated with the code XMuDat(Nowotny, 1998) with interaction cross section data from Boone andChavez (1996).

Photon energy (keV)

m/r

(cm

2 g-1

)

mR/r mtr/r

tPE/rmC/r

m/r

Iodine (Z = 53)

1

104

103

102

101

100

10-1

10-2

10 100

Figure 5 Mass attenuation coefficients and mass energy transfercoefficient for iodine are plotted in the diagnostic energy range. Note theK-edge at 33.2 keV and the L-edges at about 5 keV. Iodinatedcompounds are used as injected contrast agents in some contrast-enhanced radiological diagnostic procedures in order to enhance thelabeled tissue contrast with respect to surrounding tissues, due to thelarge increase in attenuation coefficient around the K-edge of iodine. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996).

Photon energy (keV)

m/r

(cm

2 g-1

)

mR/r

mtr/r

tPE/r

mC/r

m/r Lead (Z = 82)

1

104

103

102

101

100

10-1

10-2

10-3

10 100

Figure 6 Mass attenuation coefficients and mass energy transfercoefficient for lead are plotted in the diagnostic energy range. Note theK-edge at 88.0 keV, the L-edges between 13.0 and 15.9 keV, and theM-edges between 2.48 and 3.85 keV. Data are calculated with the codeXMuDat (Nowotny, 1998) with interaction cross section data from Booneand Chavez (1996).

Photon energy (keV)

Rat

io (m

C/r

)/(t

PE/r

)

Compton to photoelectric ratio

C

Ca

P

Pb

I

10

102

101

100

10-1

10-2

100

Figure 7 Ratio of Compton to photoelectric mass attenuationcoefficients in the diagnostic energy range, for some low-Z and high-Zelements (C, P, Ca, I, Pb). The horizontal line corresponds to mC/r¼tPE/r : the zone above this line indicates prevalence of Compton scattering,and below the line photoelectric interaction dominates over scattering.

Photon energy (keV)

mC/r tPE/r

mR/r

mtr/r

m/r

1 10

104

103

102

101

100

10-1

10-2

10-3

100

Soft tissue (ICRU-44)

m/r

(cm

2 g-1

)

Figure 8 Mass attenuation coefficients and mass energy transfercoefficient for soft tissue are plotted in the diagnostic energy range. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996).

Photon energy (keV)

Rat

io (m

C/r

) / (t

PE/r

)


Nor

mal

ized

sp

ectr

al in

tens

ity

Soft tissue (ICRU-44)

100 kVp x-ray spectrum

10

102

101

100

10–1

10–2

10010–2

10–1

100

101

102

Figure 9 Ratio of Compton to photoelectric mass attenuationcoefficients in the diagnostic energy range (thick line), for soft tissue. Thehorizontal line corresponds to mC/r¼tPE/r. Also shown is the energyspectrum at 100 kVp from an x-ray tube (thin line), normalized to 100%at 34 keV where the maximum of the bremsstrahlung output occurs: thecomparison of the two trends shows that in general radiography, in mostof the energy spectrum, soft tissue scattering through Comptoninteractions is dominant over photoelectric absorption, apart from alow-energy tail in the 15–30 keV range.


ELSEVIE

R

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Photon energy (keV)

(mC/r

) / (t

PE/r

)

Water


10

102

101

100

10-1

10-2

100

Figure 10 Ratio of Compton to photoelectric mass attenuationcoefficients in the diagnostic energy range, for water.


Compton mass attenuation coefficient equals the photoelectric

mass attenuation coefficient at about 28 keV, and at 100 keV it

is about 60 times greater.

In what follows, the relation between index of refraction

and attenuation coefficient will be illustrated.

For low x-ray energies or forward scattering (small scatter-

ing angle), one can assume that atoms in a liquid or solid

material scatter the incident radiation as electric dipoles, so

that x-ray interaction with the material can be described using

optical constants like the complex refractive index.

Let us consider atoms as damped, isotropic harmonic oscil-

lators driven by an incident e.m. wave (for simplicity assumed

linearly polarized) exerting on each bound atomic electron of

charge e and mass me an oscillating driving force �eE0eiot atangular frequency o¼2pn¼2pc/l. The equation of motion in

a direction x normal to the propagation direction z, for an

electron having a natural frequency of oscillation o0¼2pn0(resonance frequency), can be written as

med2x

dt2þmeg

dx

dtþmeo2

0x ¼ �eE0eiot [11]

where g is the damping coefficient for that electronic state,

representing a dissipation constant associated with each natu-

ral frequency of oscillation of the atom, so that for each mode,

i, of oscillation we have the three quantities o0i, gi, and f0i, with

f0i being the oscillator strength factor (see Feynman et al.,

1964). The solution of this equation is

x tð Þ ¼ e=m

o2 � o20 � igo

E0eiot [12]

from which the module of the induced dipole moment

p¼�ex is

p oð Þ ¼ e2=me

�o2 þ o20 þ igo

E0 [13]

and the atomic polarizability a(o) can be defined as

a oð Þ ¼ e2=e0me

�o2 þ o20 þ igo

[14]

with e0 being the permittivity of vacuum. Then, in order to take

into account the action exerted by the incident e.m. wave on

each atomic electron and all modes of atomic oscillation char-

acterized by the values of (o0i, gi, fi), one takes the sum over

ELSE

those modes, thus arriving at the expression for the atomic

polarizability

a oð Þ ¼ e2

e0me

Xi

f0i

o20i � o2

� �þ igio[15]

It can be shown that this model leads to an expression for

the complex index of refraction which, for frequencies o�o0i,

can be written as

n ¼ 1� Ne2

2e0me

Xi

f0i

o20i � o2

� �2 þ g2i o2

264

375 o2 � o2

0i

� �þ igio� �

¼ 1�Nrel2o2

2p

Xi

f0i

o20i � o2

� �2 þ g2i o2

264

375 o2 � o2

0i

� �þ igio� �

[16]

where N is the number of atoms of the element per unit

volume, and where the sum is extended over all electronic

states of the atom.

Under the assumption of weak damping and for oscillation

frequencies o�o0i, the complex index of refraction, n, for an

elemental material can be calculated by the simplified

expression

n ¼ 1�Nrel2

2pf1 þ if2ð Þ [17]

with the atomic form factor, f, having a real part f1 and an

imaginary part if2. The index of refraction is then written as

n ¼ 1� d� ib [18]

having defined the dimensionless real numbers refractive index

decrement, d, and the absorption index (or linear absorption coef-

ficient) b as

d rel2

2pNf1 [19]

b rel2

2pNf2 [20]

For a compound or mixture of density rcomp, with Ni the

atom number density of the atomic species i, each having

atomic number Zi, a weight fraction wi, atomic weight Ai, and

complex atomic form factor fi, one has

Ni ¼ wiNA

Arcomp [21]

and the complex refractive index, in the approximation of

atoms scattering as dipoles, can then be written as

n ffi 1� rel2

2p

Xi

Nifi [22]

From this expression, the refractive index decrement of the

compound or mixture, dcomp, can be calculated as

dcomp ¼ rel2

2p

Xi

Nif1,i [23]

VIER

Figure 10

Photon energy (keV)(a)

(b)

n = 1-d-ib

d

bdel

ta o

r b

eta

Breast tissue (ICRU-44)

0

10-4

10-6

10-8

10-10

10-12

10-14

20 40 60 80 100 120 140

Photon energy (keV)

d / bRat

io d /

b

Breast tissue (ICRU-44)107

106

105

104

103

20 40 60 80 100 120 140

Figure 11 (a) The refractive index decrement, d, and the absorptionindex, b, of breast tissue (whose composition is shown in Table 1)versus photon energy in the diagnostic range. The ratio d/b is shown in(b). Data are calculated with the code XOP v2.3 (Sanchez del Rio andDejus, 2003).

Photon energy (keV)

Water, liquid

d

bdel

ta o

r b

eta

1

10-4

10-5

10-6

10-7

10-8

10-9

10-10

10-11

10 100

Figure 12 The refractive index decrement, d, and the absorption index,b, of liquid water versus photon energy in the diagnostic range. Data arecalculated with the code XOP (X-ray Oriented Programs) (Sanchez delRio and Dejus, 2003).


VIER

f1,i being the real part of the atomic form factor for forward

scattering; at the same time, the absorption coefficient, bcomp,

can be calculated as

bcomp ¼rel

2

2p

Xi

Nif2i [24]

f2,i being the coefficient of the imaginary part of the atomic

form factor for forward scattering. Away from absorption

edges, f1,iffiZi electrons per atom so that Nif1,i is the number

of electrons of the atomic species i in the compound or mixture

and SiNif1,i is the electron density re (electrons cm�3). For an

elemental material, the electron density is

re ¼ rNA

AZ [25]

so that by combining eqns [19] and [25], at energies well above

the absorption energies, the refractive index decrement is line-

arly proportional to the electron density via

d ¼ rel2

2pre [26]

For a compound (or mixture), one defines the effective

electron density (re)eff

reð Þeff ¼ rcomp

NA

AeffZeff [27]

where Zeff is the effective atomic number of the compound

(number of electrons per molecule of the compound) and Aeff

is the effective atomic weight, defined as the ratio of the molec-

ular weight of the sample divided by the total number of the

atoms in the compound. Hence, in analogy with eqn [23], for

energies significantly above the absorption edges, the refractive

index decrement is proportional to the effective electron den-

sity of the material:

dcomp ¼ rel2

2preð Þeff ¼

reh2c2

2pE2reð Þeff [28]

and in terms of mass density:

dcomp ¼ reh2c2NA

2p

� �Zeff

Aeff

� �1

E2rcomp [29]

The values of d(E) and b(E) are largely different, with dbeing much larger than b for biological tissues; they show a

generally decreasing trend as a function of energy, away from

threshold energies where discontinuities in b are present.

Figure 11(a) shows the refractive index decrement and the

absorption index for breast tissue as a function of photon

energy, in the diagnostic energy range; their ratio is plotted in

Figure 11(b), where it is seen that d is three to six orders of

magnitude higher than b in the range interesting for radiol-

ogy. The data for d and b for water are shown in Figure 12,

where in the semilog scale the 1/E2 dependence of d is

evident (eqn [28]).

ELSE

The absorption index, b, is related to the cross section for

photoelectric absorption tPE and to the mass attenuation coef-

ficient ma/r by

tPE ¼ 4plN

b [30]

mar¼ 4p

lrb [31]

The transmission T of a collimated x-ray beam through a

slab of thickness d is then given by

Figure 11

Figure 12


T ¼ exp �NtPEdð Þ ¼ exp � 4plbd

� �¼ exp �madð Þ [32]

where ma is the photoelectric attenuation coefficient (cm�1).The elemental composition of some soft tissues is shown

in Table 1; the total mass attenuation coefficient in the diag-

nostic energy range for some tissues in this table is shown in

Figure 13, while Figure 14(a) shows corresponding values of

the linear attenuation coefficient.

It is seen that in that energy range, m/r and m are a decreas-

ing function of energy, with differences between linear attenu-

ation coefficients of tissues (hence, tissue contrast) decreasing

as energy increases. In Figure 14(b), the plot of ma versus

energy for soft tissue, blood, and lung tissue shows that they

practically have the same values of attenuation coefficient in

the diagnostic range, so that for increasing the radiopacity of,

for example, blood vessels, a suitable contrast agent can be

injected, that is, a solution containing a highly attenuating

medium like iodine-containing compounds (see Figure 5).

2.01.1.3 Tissue Substitute Materials

A number of compound materials are commonly used as tissue

substitutes in radiological imaging, for quality control and for

laboratory studies: for attenuation-based imaging, the require-

ment is to show a linear attenuation coefficient similar to

that of the substituted tissue in the diagnostic energy range.

Soft tissue substitutes include water, polymethylmethacrylate

(PMMA), and, for breast tissues (schematically comprising

skin, fibroglandular, adipose tissues, and microcalcifications),

several epoxy resins like BR10, BR12, CB1, CB2, CB3, CB4,

in addition to paraffin wax, polyethylene as adipose tissue

substitutes, as well as CaCO3 or Al2O3 or Al or Au as possible

substitutes for microcalcifications. Some parameters of tissue

substitute material are reported in Table 2. Figure 15 shows

plots of the attenuation coefficient versus energy in the range of

10–150 keV, for some tissue substitutes.

PMMA is a common soft tissue substitute material for

imaging diagnostics, being cheap and easily machinable. In

attenuation-based imaging, PMMA has attenuation properties

similar to those of a soft tissue; in Figure 16(a) its linear

attenuation coefficient m is plotted versus photon energy

in the diagnostic range together with that of liquid water.

The differences in m are within �25% and þ15% between

PMMA and water in the diagnostic range, with the deviation

(mPMMA�mwater) vanishing at about 36 keV (Figure 16(b)).

In phase contrast imaging, PMMA shows analogous

good properties as a tissue substitute material: Figure 17

shows its values of d and b for energies in the diagnostic

range. In Figure 18, a comparison is shown between the values

of the refractive index decrement d for PMMA and breast tissue

in that energy range, showing a relative deviation of about 12%

between PMMA and tissue. The d/b ratio versus photon energy

is shown in Figure 19 for PMMA and, for comparison, also for

water: the two ratios assume close values at high energies,

differing by less than 1%, whereas they differ by as much as

60% at 10 keV.

Breast tissue microcalcifications (i.e., deposit of a fraction

of a millimeter in size composed of calcium phosphate

or calcium oxalate dehydrate, known as type II and type I

ELSE

microcalcifications, respectively) are important findings in

the diagnosis of breast cancer through mammography (see

Chapter 2.05). As a tissue substitute for such a tissue, fine

grains of calcium carbonate (CaCO3) can be adopted. Figure

20(a) shows values of d and b for CaCO3 in the low-energy

range up to 30 keV, typical of mammography;

microcalcifications (as simulated by CaCO3 grains) generate

contrast in an attenuation-based or phase-based image because

of the differences in their values of b and d, respectively, in the

photon energy range around 20 keV, as shown in Figure

20(b). Note that in this case, contrast is higher in terms of bvalues rather than d values.

2.01.1.4 Absorption Contrast and Phase Contrast

In general, the refractive index n(x,y,z) is a function of the

three-dimensional spatial coordinates of the point of observa-

tion in a nonhomogeneous sample (let us neglect any depen-

dence on time of the refractive index for simplicity). Hence,

both d¼d(x,y,z) and b¼b(x,y,z) depend on the spatial position

as well. With o¼2pc/l being the angular frequency, let us

consider the field at a point P(x,y,z) of an e.m. wave E¼E0exp[i(ot�oz/c)] propagating in free space along the direction

zwith amplitude E0. If the propagation between the source and

P occurs in a homogeneous medium of refractive index n, then

the field of the wave is E´¼E0 exp[i(ot�onz/c)]. By substitut-

ing the complex index of refraction n¼1�d� ib, the wave canbe written as

E0 ¼ E0 exp i ot � on

z

c

0@

1A

24

35

¼ E0 exp i ot � o 1� dð Þ zcþ iob

z

c

24

35

8<:

9=;

¼ E0 exp i ot � oz

c

0@

1A

24

35 exp iod

z

c

0@

1A exp �ob z

c

0@

1A

[33]

In eqn [33], we can see that with respect to propagation in

free space where E¼E0 exp[i(ot�oz/c)], the net effect of con-sidering the propagation in a material slab of complex index of

refraction n is a change (decrease) in amplitude E´/E0 given by

E0

E¼ exp iod

z

c

0@

1A exp �ob z

c

0@

1A

¼ exp i2pldz

0@

1A exp � 2p

lbz

0@

1A

[34]

The above expression shows the presence of a change in the

phase of the wave (given by the first factor on the right-hand

side, related to the refractive index decrement d) and a change

in its amplitude (attenuation) (given by the second factor on

the right-hand side, related to the absorption index b). In other

words, if we interpose a slab of thickness Dz of refractive indexn, between the source and the point P in free space, the result-

ing field at P is E0 exp[i2pdDz/l] exp[�2pbDz/l] and the

changes in phase (Df) and wave amplitude (A/A0) with respect

to propagation in free space are given, respectively, by

VIER

Table 1 Composition of some soft and hard tissues according to ICRU Report 44 (1989), as tabulated in the code XMuDat (Nowotny, 1998) and ascan be found also on the NIST website (http://physics.nist.gov/PhysRefData/XrayMassCoef/tab2.html)

Breast tissue (ICRU-44) Adipose tissue (ICRU-44)

Z Element Weight fraction Z Element Weight fraction

1 H (hydrogen) 0.106 1 H (hydrogen) 0.1146 C (carbon) 0.332 6 C (carbon) 0.5987 N (nitrogen) 0.030 7 N (nitrogen) 0.0078 O (oxygen) 0.527 8 O (oxygen) 0.27811 Na (sodium) 0.001 11 Na (sodium) 0.00115 P (phosphorus) 0.001 16 S (sulfur) 0.00116 S (sulfur) 0.002 17 Cl (chlorine) 0.00117 Cl (chlorine) 0.001Electrons (g�1) 3.32Eþ23 Electrons (g�1) 3.35Eþ23Calc. electrons (cm�3) 3.39Eþ23 Calc. electrons (cm�3) 3.18Eþ23Effective Z 7.07 Effective Z 6.47Mean ratio hZ/Ai 0.55196 Mean ratio hZ/Ai 0.55579Density r (g cm�3) 1.020Eþ00 Density r (g cm�3) 9.500E�01

Bone, cortical (ICRU-44) Brain, gray/white matter (ICRU-44)


1 H (hydrogen) 0.034 1 H (hydrogen) 0.1076 C (carbon) 0.155 6 C (carbon) 0.1457 N (nitrogen) 0.042 7 N (nitrogen) 0.0228 O (oxygen) 0.435 8 O (oxygen) 0.71211 Na (sodium) 0.001 11 Na (sodium) 0.00212 Mg (magnesium) 0.002 15 P (phosphorus) 0.00415 P (phosphorus) 0.103 16 S (sulfur) 0.00216 S (sulfur) 0.003 17 Cl (chlorine) 0.00320 Ca (calcium) 0.225 19 K (potassium) 0.003Electrons (g�1) 3.10Eþ23 Electrons (g�1) 3.33Eþ23Calc. electrons (cm�3) 5.95Eþ23 Calc. electrons (cm�3) 3.46Eþ23Effective Z 13.84 Effective Z 7.65Mean ratio hZ/Ai 0.51478 Mean ratio hZ/Ai 0.55239Density r (g cm�3) 1.920Eþ00 Density r (g cm�3) 1.040Eþ00

Tissue, soft (ICRU-44) Blood, whole (ICRU-44)


1 H (hydrogen) 0.102 1 H (hydrogen) 0.1026 C (carbon) 0.143 6 C (carbon) 0.117 N (nitrogen) 0.034 7 N (nitrogen) 0.0338 O (oxygen) 0.708 8 O (oxygen) 0.74511 Na (sodium) 0.002 11 Na (sodium) 0.00115 P (phosphorus) 0.003 15 P (phosphorus) 0.00116 S (sulfur) 0.003 16 S (sulfur) 0.00217 Cl (chlorine) 0.002 17 Cl (chlorine) 0.00319 K (potassium) 0.003 19 K (potassium) 0.002

26 Fe (iron) 0.001Electrons (g�1) 3.31Eþ23 Electrons (g�1) 3.31Eþ23Calc. electrons (cm�3) 3.51Eþ23 Calc. electrons (cm�3) 3.51Eþ23Effective Z 7.64 Effective Z 7.74Mean ratio hZ/Ai 0.54996 Mean ratio hZ/Ai 0.54999Density r (g cm�3) 1.060Eþ00 Density r (g cm�3) 1.060Eþ00

Lung tissue (ICRU-44) Muscle, skeletal (ICRU-44)


1 H (hydrogen) 0.103 1 H (hydrogen) 0.1026 C (carbon) 0.105 6 C (carbon) 0.1437 N (nitrogen) 0.031 7 N (nitrogen) 0.034

(Continued)


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http://physics.nist.gov/PhysRefData/XrayMassCoef/tab2.html

Photon energy (keV)

m/r

(cm

2 g-1

)

Tissues:Bone, corticalBrainAdiposeBreast

Mass attenuation coefficients

10

101

100

10-1

100

Figure 13 Total mass attenuation coefficient for some soft and hardtissues (cortical bone, brain (gray/white matter), and adipose and breasttissues) as a function of energy, in the diagnostic energy range. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996), for tissues in Table 1.


(b) Photon energy (keV)10 100

m (c

m-1

)m

(cm

-1)

Tissues:Bone, corticalBrainAdiposeBreast

Tissues (ICRU-44):SoftBloodLung

Linear attenuation coefficients

Linear attenuation coefficients

10

102

101

100

10-1

10-1

100

100

Figure 14 (a) Linear mass attenuation coefficient for some soft andhard tissues (cortical bone, brain (gray/white matter), and adipose andbreast tissues) as a function of energy, in the diagnostic energy range.(b) The almost coincident values of the linear attenuation coefficient for asoft tissue, blood, and lung tissue. Data are calculated with the codeXMuDat (Nowotny, 1998) with interaction cross section data from Booneand Chavez (1996).

Lung tissue (ICRU-44) Muscle, skeletal (ICRU-44)


8 O (oxygen) 0.749 8 O (oxygen) 0.7111 Na (sodium) 0.002 11 Na (sodium) 0.00115 P (phosphorus) 0.002 15 P (phosphorus) 0.00216 S (sulfur) 0.003 16 S (sulfur) 0.00317 Cl (chlorine) 0.003 17 Cl (chlorine) 0.00119 K (potassium) 0.002 19 K (potassium) 0.004Electrons (g�1) 3.32Eþ23 Electrons (g�1) 3.31Eþ23Calc. electrons (cm�3) 3.48Eþ23 Calc. electrons (cm�3) 3.48Eþ23Effective Z 7.66 Effective Z 7.63Mean ratio hZ/Ai 0.55048 Mean ratio hZ/Ai 0.55000Density r (g cm�3) 1.050Eþ00 Density r (g cm�3) 1.050Eþ00

Table 1 (Continued)


EVIER

Df ¼ � 2pldDz [35]

A

A0¼ exp � 2p

lbDz

� �¼ exp � m

2Dz

� (36)

The intensity I of the wave field is related to the square of

the amplitude A, so that

I

I0¼ exp �mDzð Þ [37]

where

m ¼ 4plb [38]

For a homogeneous material (spatially constant d and b),eqn [37] (known as the Lambert–Beer exponential attenuation

law) can be written as

� lnI

I0

� �¼ 4p

lbDz [39]

while eqn [35] gives the rate of change of the phase

(rad cm�1) as

�dfdz¼ 2p

ld ¼ relre [40]

ELS

We note that for a homogeneous material slab of given dand b, assumed constant in the slab of thickness Dz, at a givenl, eqns [35]–[38] can be combined to give

Df ¼ 1

2

dbln

I

I0

� �[41]

thus showing that in this hypothesis, the phase change Df can

be recovered from an attenuation measurement (I/I0) and

from prior knowledge of the ratio d/b for that material. When

Figure 13

Figure 14

Table 2 Composition of some tissue substitute materials, as calculated with the code XMuDat (Nowotny, 1998)

PMMA (C5H8O2)n BR12a


1 H (hydrogen) 0.080541 1 H (hydrogen) 0.0966 C (carbon) 0.599846 6 C (carbon) 0.7048 O (oxygen) 0.319613 7 N (nitrogen) 0.019

8 O (oxygen) 0.16917 Cl (chlorine) 0.00220 Ca (calcium) 0.009

Electrons (g�1) 3.25Eþ23 Electrons (g�1) 3.267Eþ23Calc. electrons (cm�3) 3.87Eþ23 Calc. electrons (cm�3) 3.17Eþ23Effective Z 6.56 Effective Z 6.73Density r (g cm�3) 1.19Eþ00 Density r (g cm�3) 9.70E�01

Calcium carbonate (CaCO3) Paraffin wax (C25H52)


6 C (carbon) 0.120003 1 H (hydrogen) 0.1486058 O (oxygen) 0.479554 6 C (carbon) 0.85139520 Ca (calcium) 0.400443Electrons (g�1) 3.01Eþ23 Electrons (g�1) 3.45Eþ23Calc. electrons (cm�3) 8.42Eþ23 Calc. electrons (cm�3) 3.21Eþ23Effective Z 15.62 Effective Z 5.51Density r (g cm�3) 2.80Eþ00 Density r (g cm�3) 9.30E�01

Alumina (Al2O3) Polyethylene (C2H4)n


8 O (oxygen) 0.470749 1 H (hydrogen) 0.14371613 Al (aluminum) 0.529251 6 C (carbon) 0.856284Electrons (g�1) 2.95Eþ23 Electrons (g�1) 3.43Eþ23Calc. electrons (cm�3) 1.17Eþ24 Calc. electrons (cm�3) 3.19Eþ23Effective Z 11.28 Effective Z 5.53Density r (g cm�3) 3.97Eþ00 Density r (g cm�3) 9.30E�01

Air, dry Water, liquid (H2O)


6 C (carbon) 0.000124 1 H (hydrogen) 0.1118987 N (nitrogen) 0.755268 8 O (oxygen) 0.8881028 O (oxygen) 0.23178118 Ar (argon) 0.012827Electrons (g�1) 3.01Eþ23 Electrons (g�1) 3.34Eþ23Calc. electrons (cm�3) 3.62Eþ20 Calc. electrons (cm�3) 3.34Eþ23Effective Z 7.77 Effective Z 7.51Density r (g cm�3) 1.20E�03 Density r (g cm�3) 1.00Eþ00

Data for dry air and for liquid water are also included.aData from White (1978).


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the material is not homogeneous, this simple result is no

longer valid and the measurements of phase change and inten-

sity attenuation are disentangled. From eqns [25] and [38], the

ratio d/b for a material at a given l and the corresponding

energy E can be expressed as

db¼ 2rel

mre ¼

2rehc

m Eð ÞEre [42]

In Figure 21, the rate of phase shift�df/dz for breast tissueand for PMMA (also known as Plexiglas, or lucite), a common

soft tissue substitute material, is shown in the diagnostic

range. In this figure, the relative phase shift versus photon

energy E follows the 1/E trend implied by eqn [40]; more-

over, the closeness of the two curves reflects the similarity of

the values of the effective electron density of breast tissue and

PMMA (3.39�1023 and 3.87�1023cm�3, respectively, see

Tables 1 and 2). For example, at the energy of 17.5 keV of

the Ka characteristic line of Mo of mammography x-ray tube,

a 50-mm compressed breast thickness introduces a phase

advance of about 1000p rad with respect to propagation in

50-mm air.

Photon energy (keV)

PEBR12PMMA

CaCO3

Al2O3Li

near

att

enua

tion

coef

ficie

nt

(cm

-1)

1010-1

100

101

100

Figure 15 Linear attenuation coefficient of some common tissuesubstitute materials (Table 2) in the diagnostic energy range. Data arecalculated with the code XMuDat (Nowotny, 1998) with interaction crosssection data from Boone and Chavez (1996). For BR12, data arecalculated with the XCOM database of NIST.

Photon energy (keV)


(b)

PMMAWater

36 keV

mPMMA / mwater

Rat

io o

f att

enua

tion

coef

ficie

nts

Line

ar a

tten

uatio

n co

effic

ient

, m

(cm

-1)

00.7

0.8

0.9

1.0

1.1

1.2

10-1

100

101

10 100

20 40 60 80 100 120 140

Figure 16 (a) Linear attenuation coefficient m ofpolymethylmethacrylate (PMMA) and of liquid water in the diagnosticrange. (b) The ratio mPMMA/mwater versus energy: it equals 1 at about36 keV. The data are from the XCOM database of NIST.

PMMA

Photon energy (keV)

n = 1 - d- ib

d

bDel

ta o

r b

eta

110-11

10-9

10-7

10-5

10 100

Figure 17 The refractive index decrement, d, and the absorption indexb, of PMMA versus photon energy in the diagnostic range. Data arecalculated with the code XOP (Sanchez del Rio and Dejus, 2003).

PMMA

Breast tissue(ICRU-44)

Photon energy (keV)

Ref

ract

ive

ind

ex d

ecre

men

t d

1010-8

10-7

10-6

10-5

d

100

Figure 18 The refractive index decrement, d, of PMMA and ICRU-44breast tissue versus photon energy in the diagnostic range. Data arecalculated with the code XOP (Sanchez del Rio and Dejus, 2003).

Photon energy (keV)

PMMAWater

d / b

Rat

io d

elta

/ b

eta

00

500

1000

1500

2000

2500

20 40 60 80 100 120 140

Figure 19 The ratio of the refractive index decrement, d, to theabsorption index, b, for PMMA and water versus photon energy in thediagnostic range. Data are calculated with the code XOP (Sanchez del Rioand Dejus, 2003).


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If the refractive index n of a material varies only along the

direction z, then the phase f of the propagating e.m. wave,

relative to free-space propagation, changes along z by a quan-

tity Df¼f(x,y,z0;l)�f(x,y, z0þDz;l) after traveling a distanceDz, given by

Df ¼ � 2pl

ðz0þDz0

d zð Þdz [43]

At the same time, the intensity of the e.m. wave is

reduced as

I

I0¼ exp

ðz0þDz0

� 4plb zð Þ

�dz

�

¼ exp

ðz0þDz0

�mð Þdz � [44]

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19


(b) Photon energy (keV)

CaCO3

d

d -CaCO3

b-CaCO3

d -breast tissue

b-breast tissue

d,b

d or

b

b

1 10

10

10-3

10-4

10-5

10-6

10-7

10-8

10-9

10-5

10-6

10-7

10-8

10-9

10-10

15 20 25 30

Figure 20 (a) The refractive index decrement, d, and absorptioncoefficient, b, of CaCO3 used as a substitute material formicrocalcifications in breast tissue versus photon energy in the range upto 30 keV typical of mammography. The discontinuity due to the K-edgeof Ca (4.04 keV) is visible. (b) The comparison of d and b values forCaCO3 and ICRU-44 breast tissue shows the origin of absorption andphase contrast of microcalcifications in the energy range typical ofmammography. Data are calculated with the code XOP (Sanchez del Rioand Dejus, 2003).

PMMABreast tissue(ICRU-44)

10

101

102

Photon energy (keV)

Rat

e of

pha

se s

hift

,-d

f/d

z (r

ad m

m-1

)

100

Figure 21 The rate of change of phase �df/dz of breast tissue andof PMMA versus photon energy in the diagnostic range. Data arecalculated with the code XOP (Sanchez del Rio and Dejus, 2003). Materialdata are in Tables 1 and 2.


ELSE

In general, if d ¼d(x,y,z;l) and b¼b(x,y,z;l), then the pro-

jected phase f ¼f(x,y;z,l) and the projected linear attenuation

coefficient m¼m(x,y;z,l) along the direction z are given by

f x; y; z; lð Þ ¼ � 2pl

ðz�1

d x; y; z0; l

� dz0

[45]

m x; y; z; lð Þ ¼ � 4pl

ðz�1

b x; y; z0; l

� dz0

[46]

Given the relation (see eqn [28]) between the (local) refrac-

tive index decrement of a material and its (local) electron

density re(x,y,z), eqn [44] can also be written as

f x; y; z; lð Þ ¼ �relðz�1

re x; y; z0

� dz0

[47]

For a compound or mixture, one has

f x; y; z; lð Þ ¼ �relðz�1

reð Þeff x; y; z0

� dz0

[48]

showing that the projected phase map is essentially a map of

the projected electron density re(x,y)¼Ðre(x,y,z)dz in the sam-

ple, at a given wavelength.

In the limit of 2D projection radiography, the above formal-

ism provides the description of the interaction of an x-ray wave

with a sample (e.g., a biological tissue) as introducing phase

shift and amplitude attenuation effects in the propagation of the

e.m. wave associated with an x-ray beam. For an absorbing and

scattering thick object illuminated by a given incident wave

field, extending from z¼0 to z¼Dz and containing spatial

inhomogeneities in d (i.e., in electron density) and in m(i.e., in atomic number Z and in mass density r, via photoelec-tric and Compton interactions) over a range of spatial frequen-

cies u (cm�1), the transmitted field T(x,y;zl) at distance zl>Dz ina plane normal to the projection direction z depends through

volume integrals on the contribution at each point (x,y;zl) of all

absorption and scattering events in the whole object volume.

However, if we limit our analysis to the case of a thin object,

then one can consider line integrals of phase f and linear

attenuation coefficient m along the projection direction z, as

given by eqns [45] and [46]. The expression for the transmitted

field at distance zl on a plane perpendicular to the propagation

direction z can be simplified to give the projected transmission

function T(x,y;zl) as (Wu and Liu, 2004):

T x; y; zlð Þ ¼ A x; y; zlð Þeif x;y;zlð Þ [49]

T x; y; zlð Þ ¼ exp if x; y; zlð Þ � m x; y; zlð Þ2

�[50]

Here, f(x,y;zl) and m(x,y;zl) are the projected phase and

projected attenuation coefficients, respectively. In eqn [49], A

(x,y;zl) is the x-ray-transmitted amplitude at the plane z¼zl,

and its intensity A2(x,y;zl) gives the attenuation-based image (or

attenuation image) of conventional projection radiography.

Two-dimensional spatial variations in the transmitted intensity

A2(x,y;zl) depend on the 3D spatial variation of the linear

absorption coefficient b(x,y,z) in the object (as given by eqn

[46]), so that image contrast in the attenuation image is

based on the subject contrast produced by variations in

the projected attenuation coefficient (so-called attenuation

contrast). On the other hand, the projected phase f(x,y;zl) of

the complex-transmitted field at the plane z¼zl represents a

phase image whose contrast arises from spatial variations in the

phase shift produced by x-ray propagation inside the object

(so-called phase contrast), as given by eqn [45]. Contrast in

the projected phase map f(x,y;zl) is related to the spatial

VIER

Figure 20

Figure 21


inhomogeneities of the (effective) electron density in the

object, as given by eqns [49] and [50]. The linear attenuation

coefficient m for a material scales linearly with the density of

the material and the mass density increases with increasing

atomic number Z for most naturally occurring solid or liquid

elements (i.e., excluding gases). In the diagnostic energy range,

it contains contributions to attenuation arising from photo-

electric interactions and from elastic and inelastic scattering

interactions (eqn [1]). In this range, the photoelectric inter-

action cross section tPE depends on the atomic number and

on the photon energy E approximately as Z4/E3 (see Section

2.01.2.1); correspondingly, the Rayleigh interaction cross sec-

tion sR varies as Z2/E2 while the Compton cross section sCvaries as Z and is approximately independent of E. Hence, eqns

[49] and [50] show that attenuation contrast and phase con-

trast convey information on the spatial distribution of the

(effective) atomic number and of electronic density in the

object. At a given photon energy E and the corresponding

wavelength l, the condition of x-ray thin objects for the deri-

vation of eqns [49] and [50] imposes constraints on the x-ray

transmission T and on the size of the finest detail which can be

imaged at that energy: in the diagnostic range, it has been

indicated that ‘human body parts can be treated as thin objects

for resolutions as high as 10 mm’ (Wu and Liu, 2003).

In projection radiography, 2D maps of object properties

like d(x,y) or m(x,y), integrated over the projection direction z,

can be derived in principle by measurements of the phase

change maps (phase imaging) and by intensity attenuation

maps (absorption-based imaging).

2.01.2 Interaction Processes

The interaction processes of x-ray photons with atoms and elec-

trons in matter in the diagnostic energy range are described in

this section, including the absorption and scattering processes.

Absorption of x-rays occurs through the photoelectric effect,

whereas inelastic and elastic photon scattering occur via the

Compton effect and the Rayleigh scattering effect, respectively.

SE

Atomic number, Z

Ca

Na

S

W

YL

YK

I

00.0

Fluo

resc

ence

yie

ld, Y

k

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50 60 70 80 90 100110

Figure 22 Fluorescence yield for K-shell (YK) and for L-shell (YL)electrons for elements with atomic number between 3 and 110 (YK) andbetween 3 and 100 (YL) (data from Hubbel et al., 1994).

2.01.2.1 Photoelectric Absorption

In the (internal) photoelectric effect, a photon of suitably high

energy E¼hn interacts with a tightly bound atomic electron

(i.e., an orbital electron of the inner atomic shells K, L, M, . . .,

with binding energy Eb), transferring a part of its energy and of

its momentum (q¼hn/c) to the electron, which receives a

kinetic energy Ek and a momentum pe and is ejected from the

atom. The remaining part of photon energy and momentum is

given to the atom, which recoils with kinetic energy Ek,a and a

momentum pa: in the interaction, the incident photon van-

ishes (i.e., it is totally absorbed). This process can take place

with the given bound electron only if hn>Eb and also hn ffi Eb,

or in other words, the total absorption of the incident photon

occurs only in the presence of the binding atom and for pho-

ton energies close to (but higher than) the given electron

binding energy, in order to conserve both energy and momen-

tum in the process: indeed, momentum conservation is reached

only upon considering as nonnegligible the atom recoil

momentum pa. This implies that in the kinematics of the

EL

photoelectric interaction, the direction of the photoelectron

makes an angle different from zero with the direction of the

incident photon, according to an angular distribution which

(by momentum and energy conservation) is dependent on the

photon energy. As long as the photon energy is increasingly

greater than the binding energy (EK or EL or EM) for a given

bound electron (belonging to the shell K, or L, or M, respec-

tively), the interaction becomes less probable with that elec-

tron; for photon energies below a given binding energy,

photoelectric interaction with a lower orbit electron may

occur. Owing to its largely higher rest mass with respect to

that of the bound electron, the atom recoils with a negligible

kinetic energy with respect to the kinetic energy acquired by

the (photo)electron, so that the fraction of the incident photon

energy given to the atom is equal to the potential energy Eb with

which the electron is bound to the atom. Under this descrip-

tion, conservation of energy for this effect can be written as

hn ¼ Ek þ Eb [51]

In the photoelectric effect, in which the ejection of the

electron from an atomic shell leaves a corresponding vacancy

in that shell, promptly filled by a higher shell electron, the

atomic excitation corresponding to the absorption of an

amount of energy given by Eb is followed by a de-excitation

process in which at least a part of this energy can be released

by the emission of a fluorescence (‘characteristic’) x-ray. The

energy of this emitted photon is given by the difference

between the binding energies of the two potential energy levels

between which the electronic transition occurs. This mecha-

nism of atomic de-excitation is relevant for vacancies that are

produced in a K-shell or in an L-shell, during a photoelectric

interaction. The probability of atomic emission of a fluores-

cence x-ray is called the fluorescence yield Y : in particular, if the

emission is determined by a vacancy in the K-shell, it is indi-

cated by YK, and it is YL in the case of a vacancy in the L-shell

(see also Section 2.01.2.4). Given the total number of photo-

electric interaction events in the whole atom, a fraction PKoccurs with the K-shell electrons, and a fraction PL occurs

with L-shell electrons: for values of Z less than 20, PK is 0.9 or

higher, and it is about 0.8 for a high-Z element as tungsten

(Z¼74). Figure 22 shows the fluorescence yield for shells K

and L for varying atomic number Z. It can be seen that the

curve for YL is largely below that for YK; for low-Z elements up

VIER

Figure 22

Photoelectron polar angle q (deg)with respect to the photon direction

20 keV100 keV

00

2

4

6

30 60 90 120 150 180

Pol

ar a

ngle

dep

end

ent

term

of d

s/dW

Figure 23 The term in square bracket in eqn [52] in the text is plottedversus the angle the photoelectron makes with the direction of theincoming photon, at 20 and 100 keV kinetic energy.


to Z¼20, YK is less than 0.2. Hence, an alternative de-

excitation process must exist so that the atom can release the

fraction of the binding energy not emitted by x-ray fluores-

cence. This process is the Auger effect, in which the excess

atomic energy is transferred to one or several orbital electrons

as kinetic energy, large enough to eject those electrons from the

atom: then, such Auger electrons release their kinetic energy

locally in the surrounding medium. The mechanism of non-

radiative de-excitation through emission of the Auger electron

(s) is also alternative (with probability 1�Y) to the radiative

emission of fluorescent photons for K-shell or L-shell vacancy,

so that radiative and nonradiative de-excitation can coexist in

the same photoelectric interaction event or be totally alterna-

tive to each other.

The differential angular distribution (cm2 sr�1) of the emit-

ted photoelectron in the nonrelativistic regime of the photo-

electric interaction of a photon with energy hn with an atom of

atomic number Z can be expressed with the following formula

(Heitler, 1954):

dsdO¼ 4

ffiffiffi2p

r2eZ5

1374

� �mec

2

hn

� �7=2sin 2y cos 2’

1� b cos yð Þ4" #

[52]

where y is the angle between the direction of the incident

photon and the ejected electrons, ’ is the azimuth angle of

the photoelectron with respect to the x-ray polarization vector,

mec2 is the photoelectron rest mass energy, and b is the ratio of

the photoelectron velocity and the speed of light c in vacuum.

As regards the dependence of ’ on a plane normal to

the direction of propagation of the incoming photon, it fol-

lows from eqn [52] that ds/dO∝cos2’, a condition that is at

the basis of methods for x-ray polarimetry of nonrelativistic

photoelectrons.

As regards the dependence on y for a given ’, ds/dO∝ffi sin2y/(1�bcosy)4¼ f(y). The above formula gives ds/dO¼0 for y¼0� and for y¼180�; thus the above semi-positive

function has an absolute maximum for a photoelectron angle

ymax, depending on the b value and hence on the kinetic

energy of the photoelectron. In particular, owing to the bpower dependence in the denominator of the angular term

in eqn [52], ymax is shifted toward narrower angles as the

kinetic energy of the photoelectron increases. By taking the

two terms in parentheses in eqn [52] as constant in a given

medium and for a given photon energy, the angular depen-

dence on y expressed by f(y) in the last term in square

brackets of eqn [52] is shown in Figure 23 at photoelectron

energies of 20 and 100 keV. At such low kinetic energies (and

corresponding low b values), the term (1�bcosy) is close to

unity so that f(y)∝ sin2y and the photoelectron is preferen-

tially ejected in the direction of the vector electric field of the

incident wave.

By integrating the differential cross section per atom over all

photoelectron emission angles y, one obtains the atomic cross

section for the photoelectric effect tPE (cm2 per atom), whose

dependence on the photon energy hn and on the atomic num-

ber Z in the diagnostic energy range can be approximated by

the following expression:

tPE∝Z4

hnð Þ3 [53]

ELSE

and then, the photoelectric mass attenuation coefficient ma/r(cm2 g�1) (eqn [3]) in the diagnostic range has the following

dependence on Z and on photon energy:

mar∝

Z3

hnð Þ3 [54]

The photoelectric mass attenuation coefficient ma/r of var-

ious elements versus photon energy in the diagnostic range is

plotted in Figures 2–6, while Figures 8 and 13 show the plots

for some soft biological tissues; the plot of ma/r for water is in

Figure 25(b). All these plots indicate that at low energy in low

effective Z materials like soft tissues and water, the photoelec-

tric effect (via the hn3 dependence) dominates over scattering

interactions, with photoelectric mass attenuation coefficient

equal to Compton mass attenuation coefficient at about

30 keV for water (see also Figure 25(c)).

VIER

2.01.2.2 Rayleigh (Coherent) Scattering

In Rayleigh (coherent) scattering, a photon is scattered by

bound electrons without production of excitation or ioniza-

tion of the atom; the interaction involves the entire atom

rather than only its bound electrons so that it can be consid-

ered as a process of elastic scattering since the photon loses

only a negligible fraction of its energy (Hubbell, 1999). Lord

Rayleigh studied this type of scattering of light by air mole-

cules in the last decades of nineteenth century; the term

‘coherent’ reflects the cooperative process of all atomic elec-

trons in producing destructive and constructive interference

effects. In amorphous materials like biological tissues, this

gives an oscillating behavior of the angular distribution of

the Rayleigh cross section.

For sufficiently low frequencies o of light (and correspond-

ing low energies of photons hv¼ho/2p), elastic scattering can

be described by the scattering cross section

ss ¼ 8pr2e3

o4

o2 � o20ð Þ2

[55]

where o0 is the natural frequency of oscillation of the bound

electron which scatters the incoming radiation. In the limit of

scatter from an unbound (‘free’) electron – for which o0!0

and the low frequency v�v0, but with v such that it can be still

Figure 23


considered low – one obtains the formula for the cross section

sT for classical Thomson scattering from an electron:

ssffisT¼(8/3)pre2¼0.6652448�10�24 cm2 independent of

energy. In the other limit of vv0, and for all atomic natural

frequencies in the atom energy levels for which this relation

holds, scattering occurs by the atom as a whole and atomic

electrons contribute ‘coherently’ to this so-called Rayleigh scatter-

ing process, for which ss�o4. If light scattering occurs coherently

from a whole atomic plane in a crystal, this is called Bragg

scattering.

In all these processes of elastic scattering, no energy is

transferred to the scatter center (electron or atom or molecule),

the atom recoil occurring only to the extent of momentum

conservation; upon interaction, the incident radiation is devi-

ated from its original direction, for any scatter event, according

to the geometry shown in Figure 24 in terms of the scattering

angle 0�y<180� (where 0� is forward scattering) and of the

azimuthal angle 0�f< 360�.In a given material, the probability density function of

scattering at any given angle y, p(y), is a function of energy

and can be highly peaked, while p(f) is constant over f. Thepeak of the p(y) distribution tends to shift to lower scattering

angles as the photon energy increases (see Figure 28).

When traversing any given material thickness, electronic

scattering events may occur for any atom (or groups of

atoms) encountered along the path of the incident radiation,

and the net effect is an angular redistribution of the incoming

beam energy, as described by the differential cross section per

unit solid angle ds(y)/dO versus the scatter angle y. In calcu-

lating the scattered energy fluence out of a given material

volume irradiated by x-ray photons in the energy range of

diagnostic radiology, the single scattering (or first order)

approximation considers the resulting contribution from only

the first scatter event, without consideration for possible suc-

cessive (second or multiple order) Rayleigh scatter processes.

In other words, in the commonly adopted single scatter

approximation, photons reaching the detector after being

Rayleigh scattered in a thickness of a given material have a

history of just one scatter event.

At x-ray energies in the diagnostic energy range, in terms of

angular distribution of the scattered radiation, Rayleigh

(coherent) scattering is highly forward peaked (yffi0). This

LSE

Incident photondirection

Scattered photondirection q

f

Figure 24 Illustration of the scattering angle y (between 0� and 180�)and the azimuthal angle f (between 0� and 360�) that define thegeometry of the scattering process.

E

implies that both primary and coherently (single) scattered

radiations reach the imaging detector at relatively close loca-

tions which depend, among other factors, on the distance

between the scatter center and the image plane. This difference,

due to scatter beam divergence, can be observed, for example,

by using a very narrow primary beam and observing at a

suitable distance from the irradiated object (Johns and Yaffe,

1983). Hence, the shorter the object-to-detector distance,

and the thinner the irradiated object is, the less the influence

(through any structured pattern) of coherent scatter on the

energy fluence distribution on the detector. This occurs at

variance with incoherent (Compton) scattered radiation,

which tends to distribute energy fluence over larger scattering

angles. On the other hand, the probability per unit path length

of being Rayleigh scattered is typically one order of magnitude

lower than for the total interaction probability, so that the

combined effects of high forward scatter and small interaction

coefficients tend to compensate each other in part to produce

a net result of a coherent scatter contributing to a significant

fraction of the total (RayleighþCompton) scatter reaching the

imaging detector, even more for low-energy x-ray imaging as

in mammography (Johns and Yaffe, 1983).

Mass linear attenuation coefficients m/r (cm2g�1) are

related to total atomic (or molecular) cross sections s (cm2

per atom or cm2 per molecule) via m/r¼s(ΝΑ/Α). The interac-tion cross sections (sR, sC/r, and tPE, and total cross section

stot) for water in the diagnostic energy range are shown in

Figure 25(a); the mass linear attenuation coefficients of water

(for which ΝΑ/Α ¼0.0334271�1024 molecules cm�3) are

shown in Figure 25(b); the corresponding percent ratio of

Rayleigh to total coefficients is shown in Figure 25(c).

The differential Thomson cross section per (free) electron

for elastic scattering (cm2 sr�1 per electron) (for unpolarized

photons) is

dsT yð ÞdO

� �elec

¼ r2e2

1þ cos 2y� �

[56]

and the (total, i.e., angle integrated) cross section for Thomson

scattering is

sT ¼ðy¼py¼0

dsT yð Þdy ¼ 8pr2e3

[57]

(sT¼0.6652448 barns per electron, 1 barn¼10�28 m2). The

differential Thomson cross section per scattering angle y is thengiven by

dsT yð Þdy

� �elec

¼ dsT yð ÞdO

dOdy¼ pr2e 1þ cos 2y

� �sin y [58]

and it is plotted in Figure 26. It is symmetrical about the

y¼90� axis and is zero at 0� and 180� with maxima at about

55� and 125�.The differential cross section per atom for coherent

(Rayleigh) scattering is given by

dsR yð ÞdO

� �atom

¼ dsT yð ÞdO

� �elec

F2 x;Zð Þ [59]

where Z is the atomic number and where

x q

2h¼ E

hcsin

y2

� �¼ 1

lsin

y2

� �[60]

VIER

Figure 24

Photon energy (keV)

Mass linear attenuation coefficients

TotalPhotoelectricComptonRayleigh

m/r

(cm

2 g-1

)

20

10-3

10-2

10-1

100

101

40

Water

60 80 100 120 140


(b)

(c)

Total

0.0334271x1024

Photoelectric

Compton

Cro

ss s

ectio

n (1

024 cm

2 p

er m

olec

ule)

molecules per cm3

Rayleigh

200

2

4

6

8

10

40

Water

60 80 100 120 140

Photon energy (keV)

Ratio of mass attenuation coefficients

Rayleigh / totalCompton / totalPhotoelectric / total

(m/r

) /

(m/r

) tot (%

)

200

20

40

60

80

100

40

Water

60 80 100 120 140

Figure 25 (a) Interaction cross section and (b) mass linear attenuationcoefficients as a function of energy in water, for Rayleigh, Compton,photoelectric, and total interaction, in the diagnostic energy range. (c)Ratio of mass linear attenuation coefficients for water. Rayleigh scatterinteractions contribute negligibly to the total attenuation coefficientexcept in the range around 20 keV for mammography, where it is in theorder of 10% of (m/r)tot. Data are calculated with the code XMuDat(Nowotny, 1998) with interaction cross section data from Boone andChavez (1996).

Scattering angle, q (deg)

Thom

son

diff

eren

tial c

ross

se

ctio

n (1

0-24

cm2 /

deg

)

00.00

0.05

0.10

0.15

0.20

dsT /dq0.25

0.30

30 60 90 120 150 180

Figure 26 Electron differential cross section for Thomson scattering,as a function of the scattering angle (see eqn [58] in the text).


ELSE

with q¼change of momentum of the photon¼2hx (for energy

E¼hv¼hc/l and scatter angle y).The term F2(x,Z) takes into account the collective effect of

the interference between the scattering from the various elec-

trons in the target atom of atomic number Z, through the

atomic form factor F(x,Z) for Rayleigh scattering, whose values

have been first tabulated by Hubbell et al. (1975). For energies

close to atomic absorption edges, anomalous coherent scatter-

ing occurs at resonant absorption energies, which could be

dealt with by modifying the formalism of the form factors as

(see, e.g., Hugtenburg et al., 2002)

F x;E;Zð Þ ¼ f0 x;Zð Þ þ f0E;Zð Þ þ if

00E;Zð Þ [61]

by introducing energy-dependent real (f0) and imaginary (f00)anomalous scatter (or dispersion) factors.

For an atom with Z electrons, the atomic form factor F(x,Z)

in eqn [59] takes into account the effect on the amplitude of

the wave scattered by an atom in a given direction, due to

superposition of the scattering fields from the different atomic

electrons, with their amplitude determined by the electronic

cross section. When the change of momentum q of the photon

in the atomic scattering event is negligible, simple addition of

the scattering intensities from each of the Z electrons occurs,

and F(ffi0,Z)ffiZ; for increasing q, F(q,Z) decreases. Equation

[61] and anomalous dispersion factors come into play when

considering that when they radiate in response to the incident

wavefield, atomic electrons cannot be considered free: in par-

ticular, K-shell electrons are tightly bound so that their scatter-

ing behavior is dependent on the energy of the incident

wave. This can be described by introducing the complex form

factor, assuming an angle-dependent factor f0(x,Z) and energy-

dependent factors f´(E,Z) and f00(E,Z). As an example, consider

the case of molybdenum (Z¼42) with the energy of the K-edge

20 keV; Figure 27(a) shows the energy-dependent terms f´(E)

and f00(E), with anomalous dispersion evident at the K-edge,

while Figure 27(b) shows the term f0(x,Z) as a function of

the scattering angle at the energy of the Ka1 characteristic line

of Mo (17.479 keV).

From eqn [58], the Rayleigh (coherent) scattering cross

section per atom is then

sR ¼ðy¼py¼0

dsT yð ÞF2 x;Zð Þ

¼ 3

8sT

ð1�1

1þ cos 2y� �

F2 x;Zð Þd cos yð Þ[62]

The Rayleigh scattering differential cross section is

approximately independent of scattering angle up to a few

kiloelectronvolts, but rapidly falls to zero with increasing

scattering angle, at higher energies. The atomic cross section

for Rayleigh scattering (cm2 per atom) is a function of Z and

photon energy E¼hn, and for high energies this dependence

is expressed by

sR � Zn

En2 < n < 2:5ð Þ [63]

For energy in the diagnostic range,

VIER

Figure 25

Figure 26


(b)

K-edge

Mo (Z = 42)

f�

f��

Ato

mic

form

fact

ors

0

86420

-2-4-6-8

20 40 60 80 100 120 140

Scattering angle (deg)

E = 17.479 keV (Ka1)

Mo (Z = 42)

Ato

mic

sca

tter

ing

fact

or, f

0

05

10

15

20

25

30

35

40

45

10 20 4030 6050 70 80 90

Figure 27 (a) Anomalous atomic scattering form factors f0 and f00 (seeeqn [61] in the text) for molybdenum, as a function of energy (data fromSasaki, 1989). (b) Atomic scattering form factor f0(x,Z) for Mo, at theenergy of Ka1 characteristic line (calculated with the code SCATFAC v1.0,data constant from Waasmaier and Kirfel, 1995).

Photon energy (keV)

Pb (Z = 82)AI (Z = 13)C (Z = 8)R

ayle

igh

scat

terin

g m

ass

atte

nuat

ion

coef

ficie

nt (c

m2

g-1)

1010-3

10-2

10-1

100

101

100

Figure 28 Mass attenuation coefficient for Rayleigh (coherent)scattering in lead, aluminum, and carbon, in the diagnostic energy range(data from NIST database XCOM).


sR ffi Z2

En[64]

and for the Rayleigh mass attenuation coefficient,

sRrffi Z

En[65]

The Rayleigh mass attenuation coefficient sR/r for carbon,

aluminum, and lead is shown in the log–log plot of Figure 28.

The power law dependence of sR on the atomic number also

explains the low cross section for Rayleigh scattering in soft

tissues, due to their low effective Z (Zffi7.5).

The interference between the electronic scattering fields

gives rise to the coherent overlapping of the scattering contri-

butions from each electron, thus originating differential coher-

ent cross sections significantly greater than the sum of the

single contributions. For example (Johns and Yaffe, 1983),

for water at 60 keV, if one does not consider the interference

phenomenon by the ten electrons of the H2O molecule, then

one just sums up the cross sections from the free oxygen and

from the two free hydrogen atoms, so obtaining

dsR yð ÞdO

� �water

ffi ZdsT yð ÞdO

� �elec

¼ 10dsT yð ÞdO

� �elec

[66]

while the measured differential Rayleigh scattering cross sec-

tion at 60 keV and at the peak scattering angle can be several

times larger. Also the free H2O molecule (independent

molecule approximation, IMA) shows a Rayleigh differential

cross section significantly greater than for the sum of the free O

and the two free H atoms (independent atom approximation,

ELSE

IAA, see later). On the other hand, in the material bulk, groups

of close atoms may be considered as cooperatively acting for

coherent scattering, and in amorphous materials (as in the case

of tissues in diagnostic radiology), where random orientation

of atomic groups is spatially averaged out, one can assume that

the differential cross section dsR(y)/dO for Rayleigh scattering

can be obtained by multiplication of the atomic cross section

(eqn [59], which includes the atomic form factors F(x,Z)) by

an angle-independent interference function I(x) dependent

only on x, this function describing the effects of interference

between electrons in different atomic groups (Poludniowski

et al., 2009):

dsR yð ÞdO

� �atom

¼ dsT yð ÞdO

� �elec

F2 x;Zð ÞI xð Þ [67]

Disregarding the extra-atomic interference processes

between different electrons represents the IAA (Poludniowski

et al., 2009); for example, for water, assuming the IAA implies

I xð Þ ¼ 2F2H x,Z ¼ 1ð Þ þ F2O x,Z ¼ 8ð Þ ffi F2O x,Z ¼ 8ð Þ [68]

For compounds (or mixtures), containing n elements of

weight fraction wi and atomic weight Ai, one has for the atomic

differential cross section (assuming IMA):

dsR yð ÞdO

0@

1A

comp

atom

¼Xni¼1

widsR yð ÞdO

0@

1A

atom, i

¼Xni¼1

wi ¼Xni¼1

widsT yð ÞdO

0@

1A

elec, i

F2comp x;Zi¼1, ..., nð Þ[69]

with the form factor for compounds given by

F2comp x;Zi¼1, ..., nð Þ ¼ dsRdO

0@

1A

comp

atom

�dsRdO

0@

1A

comp

elec

¼Xni¼1

wi

Ai

0@

1A�1Xn

i¼1

wi

AiF2 x;Zið Þ

[70]

Such form factors for water (from tables of the EPDL97

library of Cullen et al., 1997) are shown in polar plots at 5,

20, and 60 keV photon energies in Figure 29. Atomic differen-

tial cross sections for Rayleigh scattering in water (disregarding

VIER

Figure 27

Figure 28

210

240

270

300

3300q

30

60

901 2 3 4 5 6 7

120

150180

Waterform factor

TotalHO

60 keV

210

240

270

300

3300q

30

60

901 2 3 4 5 6 7

120

150180

TotalHO

20 keV

210

240

270

300

330

FF, SF: From EPDL97Normalization independentSAP v2.1 (c) Alma Mater Studiorum University of Bologna, 2010

WEIGHT FRACTION

H

O

0

q

30

0.11190

0.88810

60

901 2 3 4 5 6 7

120

150180

TotalHO

5 keV


ELSE

interatomic and intermolecular interference) at 5, 20, and

60 keV are plotted in Figure 30 (Fernandez et al., 2010,

2011), while Figure 31 shows the corresponding angular dis-

tribution of total transmitted intensity of Rayleigh scatter.

The differential cross section for elastic scattering (Thomson

scattering) of linearly polarized photons interacting with a free

electron is given by

dsT yð ÞdO

� �elec

¼ r2e cos2Y [71]

where Y is the angle between the direction of the incident

polarization vector and the direction of the polarization vector

of the scattered photon.

2.01.2.3 Compton (Incoherent) Scattering

Unlike photoelectric absorption, where the interaction of the

incident photon with an atomic bound electron cannot take

place without a recoiling atom, the scattering of a photon off

a loosely bound (virtually, free) electron is possible, neglecting

the influence of the binding atom. In this process, called

Compton effect, the incident photon with energy E¼hn col-

lides with a stationary electron. Then, it is scattered at an angle

f with respect to the direction of incidence, emerging with an

energy E´¼hn´ and giving the fraction (E�E´)/E (not all) of

its energy to the electron which is scattered at an angle y with

respect to the incident photon direction, with a kinetic energy

Ek¼(E�E´) as given by energy conservation. Taking into

account also the conservation law for the total momentum,

we obtain the kinematic relationships between f, y, E, and E´

of the Compton effect:

E0 ¼ E

1þ Emec2

1� cosfð Þ [72]

cot y ¼ 1þ E

mec2

� �tan

f2

� �[73]

Equation [72] is shown in Figure 32 in the diagnostic

energy range, as E´ versus E plot at varying photon scattering

angle f between 0� and 180�. In this plot, a low-energy behav-

ior can be distinguished from the high-energy trend at varying

f. In the region below�10 keV, E´ is equal to E at all scattering

angles: this is a form of elastic scattering described by the

Thomson scattering, whose differential cross section per elec-

tron (eqn [56]) is shown in Figure 33.

At high energy in the diagnostic range, E´¼E only for the

obvious case f¼0. As the photon scattering angle increases

from 0� to 180�, inelastic scattering occurs with increasing

kinetic energy of the Compton electron up to Ekffi(150

�94.5 keV)ffi55.5 keV at f¼180� and E¼150 keV. Ek is plot-

ted versus the incident photon energy in Figure 34, for varying

photon scattering angle.

VIER

Figure 29 Polar plots of atomic form factors for water (and forcontributions from single elements H and O), at 5 keV (right), 20 keV(center), and at 60 keV (left), plotted using the SAP code developed atthe University of Bologna (Fernandez et al., 2010, 2011). Program usedwith authors’ permission.

Figure 29

210

240

270

300

3300q

30

60

90.05 .10 .15

120

150180

Waterslab thickness = 10 cm

AtomicRayleigh scatteringcross section

TotalHO

60 keV

210

240

270

300

3300q

30

60

90.05 .10 .15

.05 .10 .15

120

150180 Total

HO

20 keV

210

240

270

300

330

FF, SF: From EPDL97Normalization: NoneSAP v2.1 (c) Alma Mater Studiorum University of Bologna, 2010

WEIGHT FRACTION

H

O

0q

30

0.11190

0.88810

60

90

120

150180 Total

HO

5 keV


ELSE

In the diagnostic range, the scattering angle y of the Comp-

ton electron is only weakly related to the photon scattering

angle f, at varying incident photon energy (Figure 35): y in the

range 0–90� is a decreasing function of f from 0� to 180�, withy¼90� for f¼0�, as well as y¼0� for f¼180�, at all incidentphoton energies.

The differential cross section (cm2sr�1 per electron) for

(unpolarized) photon scattering at an angle f, as derived by

Klein and Nishina, is given by (see Attix, 1986, Chapter 2.06)

dsC fð ÞdO

0@

1A

elec

¼ r2e2

1

1þ E

mec21� cosfð Þ

0BBBB@

1CCCCA

2

1þ E

mec21� cosfð Þ

0@

1Aþ 1

1þ E

mec21� cosfð Þ

0BBBB@

1CCCCA� sin 2f

266664

377775

¼ r2e2

E0

E

0@

1A2

E0

Eþ E

E0� sin 2f

0@

1A

[74]

The Klein–Nishina differential cross section per electron is

plotted in Figure 36 at three energies in the diagnostic range: it

is seen that at low energies it approaches the Thomson cross

section shown in Figure 33, with almost symmetrical distribu-

tion in the forward and backward directions, while at increas-

ing energy a more forward-peaked scattering distribution

develops. The total (i.e., angle integrated) cross section for

Compton scattering is the Klein–Nishina total cross section

(cm2 per electron) given by

sCð Þelec ¼ 2pðf¼pf¼0

dsC fð ÞdO

0@

1A

elec

sinfdf

¼ 2pr2e1þ k

k2

0@

1A 2 1þ kð Þ

1þ 2k� ln 1þ 2kð Þ

k

24

35

8<:

þ ln 1þ 2kð Þ2k

24

35� 1þ 3k

1þ 2kð Þ2

24

359=;

k E

mec2[75]

This function is shown in Figure 37 in the diagnostic

range, and it can be seen that at energies below �10 keV,

sC converges to the Thomson scattering cross section

sT (ffi66.5�10�26 cm2 per electron), decreasing to

ffi45�10�26 cm2 per electron at 150 keV.

VIER

Figure 30 Polar plots of atomic differential cross sections for Rayleighscattering for water (and for contributions from single elements H andO), at 5 keV (right), 20 keV (center), and at 60 keV (left), plotted using theSAP code developed at the University of Bologna (Fernandez et al., 2010,2011). Program used with permission.

Figure 30

210

240

270

300

3300q

30

60

90.05 .10 .15 .20

.0005 .0010

.00010 .00020

120

150180

Rayleighintensity

Waterslab thickness = 10 cm

60 keV

210

240

270

300

3300q

30

60

90

120

15018020 keV

210

240

270

300

330

FF, SF: From EPDL97Normalization: NoneSAP v2.1 (c) Alma Mater Studiorum University of Bologna, 2010

WEIGHT FRACTION

H

O

0q

30

0.11190

0.88810

60

90

120

1501805 keV

Figure 31 Polar plots of total transmitted intensity of Rayleigh scatterfor water, at 5 keV (right), 20 keV (center), and at 60 keV (left), plottedusing the SAP code developed at the University of Bologna (Fernandezet al., 2010, 2011). Program used with permission.

Energy of incident photon, E (keV)

Compton effect

Photon scattering angle, f

E� = E

Ene

rgy

of s

catt

ered

pho

ton,

E

� (k

eV)

0

0�

30�

60�

75�

90�

120�

180�

0

20

40

60

80

100

120

140

20 40 60 80 100 120 140

Figure 32 Energy of Compton scattered photon E´ versus incidentphoton energy E in the range 1–150 keV, for varying values of the photonscattering angle f.

Photon scattering angle, f (deg)

Thom

son

diff

eren

tial c

ross

se

ctio

n (1

0-26

cm2

sr-1

)

0

4

5

6

7

8dsT /dW

30 60 90 120 150 180

Figure 33 Thomson differential cross section per free electron (eqn[56] in the text). This is the limit cross section for the Compton effect atincident photon energies below �10 keV for all scattering angles.

Energy of incident photon, E (keV)

Photon scattering angle, f

Compton effect

Ene

rgy

of s

catt

ered

ele

ctro

n,

Ek

(keV

)

0

0

0�

30�

60�

75�

90�

120�

180�

10

20

30

40

50

60

20 40 60 80 100 120 140

Figure 34 Energy of Compton scattered electron, Ek, versus incidentphoton energy E in the range 1–150 keV, for varying values of the photonscattering angle f.


ELSEVIE

R

Figure 31

Figure 32

Figure 33

Figure 34


Ele

ctro

n sc

atte

ring

angl

e, q (

deg

)

Incident photon energy, E

0 keV

50 keV

100 keV

150 keV

00

10

20

30

40

50

60

70

80

90

20 40 60 80 100 120

Compton effect

140 160 180

Figure 35 Electron scattering angle y versus photon scattering angle ffor four incident photon energies E in the diagnostic energy range.


Diff

eren

tial c

omp

ton

cros

s se

ctio

n(1

0-26

cm2

sr-1

) per

ele

ctro

n

02

3

4

5

6

7

8

30 60 90 120 150 180

10 keV

50 keV

150 keV

Figure 36 Differential Klein–Nishina cross section per electron forCompton scattering versus photon scattering angle f, given by eqn [74],at varying incident photon energy E.

Incident photon energy (keV)

Com

pto

n cr

oss

sect

ion

(sC) e

lec

(10-2

6 cm

2 p

er e

lect

ron)

(sC)elec

(sT)elec

140

45

50

55

60

65

70

10 100

Figure 37 Klein–Nishina cross section per electron for Comptonscattering in the diagnostic range. At low energies, sC converges to theThomson scattering cross section sT.


ELSE

Then, the cross section per atom (sC)atom (cm2 per atom) is

obtained as

sCð Þatom ¼ Z sCð Þelec [76]

and the Compton mass attenuation coefficient mC/r (cm2 g�1)(eqn [4]) for an elemental material of density r can be

expressed as

mCr sCð Þatom

r¼ NA

AsCð Þatom ¼

NAZ

AsCð Þelec [77]

Equations [76] and [77] show that while the Compton

cross section per (free) electron (sC)elec is independent of Z

by definition, the Compton cross section per atom (sC)atomvaries as Z and that the cross section per unit mass, mC/r, isindependent of Z due to the slight variation of Z/A (between

0.4 and 0.5) for most elements.

The Compton mass attenuation coefficient for various ele-

ments is plotted in Figures 2–6. Using eqns [7] and [8], the

material compositions in Tables 1 and 2, and eqns [75] and

[77], the Compton mass attenuation coefficient for compound

materials can be derived; Figures 8 and 25(b) show mC/r for

soft tissue and water, respectively.

Equation [74] is valid under the assumption of a free elec-

tron at rest (i.e., with negligible initial kinetic energy). For low-

Z materials and loosely bound electrons, this assumption can

be considered a good approximation only for sufficiently high

photon energies. Electron binding corrections to the differen-

tial Klein–Nishina cross section can be applied by introducing

the incoherent scattering function S(x,Z) as a function of the

change of momentum of the photon, q¼2hx, and of the

atomic number Z :

dsC ’ð ÞdO

� �binding

elec

¼ dsC ’ð ÞdO

� �elec

S x;Zð Þ [78]

(see reviews by Hubbell, 1992, Hubbell et al., 1994, Hubbell,

1999; see also Hirayama, 2000). Binding corrections decrease

the Klein–Nishina differential cross section, to a higher extent

for lower energies in the diagnostic range and for higher Z.

The polarization state of incident x-ray photons has a role

in the description of x-ray scattering at energies in the diagnos-

tic range, also in relation to the fact that the scattering interac-

tion introduces a linear polarization in the scattered photon.

Currently used imaging detectors for diagnostic radiography

are not sensitive to the polarization state of photons; however,

the angular and energy distributions of scattered photons are

sensibly different from those of unpolarized photons, as spec-

troscopy measurements point out.

The Klein–Nishina Compton differential cross section for

linearly polarized photons interacting with a free electron is

given by

dsC ’ð ÞdO

0@

1Apol

elec

¼ r2e4

1

1þ E

mec21� cos’ð Þ

0BBBB@

1CCCCA

2

1þ E


0@

1Aþ 1

1þ E


0BBBB@

1CCCCA� 2þ 4 cos 2f

266664

377775

¼ r2e4

E0

E

0@

1A2

E0

Eþ E

E0 � 2þ 4 cos 2f

0@

1A

[79]

VIER

Figure 35

Figure 36

Figure 37


where ’ is the angle between the direction of the incident

polarization vector and the direction of the polarization vector

of the scattered photon, which can be either in the plane of the

incident polarization vector or perpendicular to it.

2.01.2.4 Mass Attenuation Coefficients and Dosimetry

2.01.2.4.1 Mass energy transfer coefficientFor indirectly ionizing radiation, like photon radiation, the

dosimetric quantity kerma (J kg�1 or Gy) is defined as the

sum of the initial kinetic energies of all charged particles

released by uncharged particles per unit mass (ICRU Report

33, 1980) in a given material. Let us first consider the case of

an elemental material of atomic number Z and density r(g cm�3).

IfF (photons per m2) is the monoenergetic photon fluence of

energy E (J) and C(E)¼EF is the corresponding photon energy

fluence (J m�2), then the kerma, K (J kg�1) (for a monoener-

getic radiation), at a given site in the absorbing material can be

calculated by defining the mass energy transfer coefficient (mtr/r)E,Z (m2kg�1), the ratio of the linear energy transfer coefficient,

mtr (m�1) to the material density r (kg m�3), as

K C Eð Þ mtrr

� �E,Z¼ F E

mtrr

� �E,Z

[80]

If, as commonly used, the photon fluence is expressed in

photons per cm2, E in keV, the density in g cm�3, and the mass

energy transfer coefficient in cm2g�1, then the kerma K (Gy)

can be calculated as

K ¼ 1:6022� 10�13F Emtrr

� �E,Z

[81]

where the coefficient (1.6022�10�13 Gy gkeV�1) takes into

account the change from SI units of measurement. In the case

of a polyenergetic spectrum, then the photon fluence spectrum

FE(E)¼dF/dE and the energy fluence spectrum CE(E)¼dC/

dE can be introduced, respectively. Then, with the same units

and for a polyenergetic x-ray spectrum, the kerma K can be

calculated as

K ¼ 1:6022� 10�13ðE00

FE Emtr Eð Þr

�Z

dE [82]

x-Ray photons in the diagnostic energy range (which

excludes pair production and photonuclear interactions) trans-

fer energy to charged particles via photoelectric or Compton

processes (Rayleigh scattering is an elastic process which does

not contribute to energy transfer); hence, in analogy with the

total mass attenuation coefficient, the mass energy transfer

coefficient can be written as

mtrr

� �¼ mtr, C

r

� �þ mtr, a

r

� �[83]

where the first term on the right-hand side takes into account

the contribution of Compton scattering and the second term,

that of photoelectric events. If all of the energy of x-ray photons

interacting at a point in a material of mass attenuation coeffi-

cient (m/r)E,Z (or, more rigorously speaking, interacting in a

small volume centered to that point) is transferred to charged

ELSE

particles and is converted into kinetic energy of those particles,

then the kerma K is equal to C(E)(m/r). On the other hand, if

only a fraction of the energy of primary photons is converted

into kinetic energy of charged particles at the interaction site

(the remaining part being due to secondary photons that

escape that site carrying away energy hn0<hn), then the kerma

K is equal to C(E)(mtr/r). Hence, the ratio mtr/m of the mass

energy transfer coefficient mtr/r to the (total) mass attenuation

coefficient m/r is equal to the ratio of the energy transferred as

kinetic energy Ek to charged particles to the incident x-ray

energy E0, and can be written as

mtr, Cr

0@

1A ¼ fC

mCr

0@

1A,

mtr, ar

0@

1A ¼ fa

mar

0@

1A

fa ¼ EkE0

, fC ¼ E0k

E0

mtrr

0@

1A ¼ fC

mCr

0@

1Aþ fa

mar

0@

1A

[84]

where Ek and E0k in eqn [84] for the energy transfer fractions

fa and fC represent the kinetic energy of the photoelectron and

of the Compton scattered electron, respectively; the quantity

(mtr/r) depends on the photon energy E and on the atomic

number Z of the material. Plots of the mass energy transfer

coefficient for some elements and for soft tissue, in the diag-

nostic energy range, have been shown in Figures 2–6, and 8.

To illustrate the meaning of mtr,a/r (for photoelectric effect)

(Attix, 1986), let us consider the energy transfer process for

photons in the diagnostic energy range, taking the photoab-

sorption and scattering processes separately into account. In

the photoelectric effect, the fraction (Ek/E0) of the initial pho-

ton energy E0 transferred as kinetic energy Ek to the photoelec-

tron is given by

EkE0¼ E0 � EK, L

E0[85]

where EK,L is the binding energy of the atom for the electronic

shell (K or L) interested in the interaction. In the atomic

relaxation process following the ionization process, the atom

will dispose of this energy by either radiative (fluorescence)

transition(s), in which characteristic x-ray(s) are emitted (the

probability of occurrence of this event being given by YK, the

fluorescence yield for shell K, or YL, the fluorescence yield for

shell L) or, alternatively, by nonradiative transitions(s) in

which Auger electron(s) are emitted. In the first case, energy

is transported away from the interaction site by characteristic

photons, while in the second case energy is transferred to

charged particles and deposited locally. In other words, if

Auger electrons are produced in the photoelectric interaction,

then all of the initial photon energy is transferred to charged

particles and mtr,a/r¼ma/r, whereas if fluorescent x-rays are

emitted, then mtr,a/r<ma/r. The ratio mtr,a/ma is the energy

transfer fraction given by (1�hEfli)/E0 where hEfli is the aver-

age energy of fluorescence radiation. For a K-shell photoelectric

interaction (i.e., for E0 EK), the difference between these

two terms can be obtained by considering the ratio

(E0�PKYK hEfli)/E0. In this expression, Efl is the photon energy

of the characteristic x-ray emitted ( Efl¼EKa1, or Efl¼EKa2, or

VIER

Photon energy (keV)

Air, dry

Mas

s en

ergy

tra

nsfe

r co

effic

ient

(cm

2 g-1

)

1010-2

10-1

100

101

100

m tr /r

Figure 39 Mass energy transfer coefficient for dry air (near sea level),


VIE

Efl¼EKb1, etc.) and hEfli defined above is the average value of

these photon energies; PK is the fraction of all photoelectric

interactions occurring in the K-shell. Then, for E0 EK, one has

mtr, ar

� �¼ mtr

r

� �1� PKYK

EKh iE0

� �[86]

correspondingly, for EL�E0<EK, one has, with analogous

terms and definitions:

mtr, ar

� �¼ mtr

r

� �1� PLYL

ELh iE0

� �[87]

The fluorescence yield is an increasing function of the

atomic number, and for a given Z it decreases from the

K-series to the L-series. YK for elements with 3�Z�110 and

YL for elements with 3�Z�100 are shown in Figure 22.

For compounds or mixtures, the rule of the weighted aver-

age (by weight) of single components’ coefficients applies:

mtrr

� �comp

¼Xi

fimtrr

� �i

[88]

Then, for a compound or a mixture, the kerma in eqns

[80]–[82] can be written as

K C Eð Þ mtr Eð Þr

� �comp

¼ FEmtrr

� �comp

[89]

K ¼ 1:6022� 10�13FEmtrr

� �comp

[90]

K ¼ 1:6022� 10�13ðE00

FE Emtrr

� �comp

dE [91]

Equations [89] and [90] apply to a monochromatic beam

of energy E, and eqn [91] represents the kerma for a polychro-

matic beam with a spectrum extending up to a maximum

energy E0. In the above formulae, as in eqns [81] and [82],

the coefficient (1.6022�10�13 Gy gkeV�1) converts from SI

units to more common units (keV, Gy, cm2g�1) for energy E,

kerma K and energy transfer coefficient mtr/r, respectively.The mass energy transfer coefficients for water are plotted in

Figure 38, where the contributions from the photoelectric andLSE

Photon energy (keV)

Water

Compton, m tr,C /r

Photoelectric, m tr,a /r

m tr /r

m /r

Mas

s at

tenu

atio

n co

effic

ient

s (c

m2

g-1)

2010-3

10-2

10-1

100

40 60 80 100 120 140

Figure 38 Mass total attenuation coefficient, energy transfercoefficient, and energy transfer contribution from photoelectric andCompton effects for water, plotted in the diagnostic energy range. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996).

E

Compton effects are illustrated, as well as the occurrence of

a mtr/m ratio which decreases to an almost constant value

(ffi0.17) as energy increases, in the diagnostic range.

The mass energy transfer coefficient for dry air is shown in

Figure 39 in the diagnostic range, and from this curve, using

eqn [90], the kerma in air per unit fluence K/F (Gy cm2) is

plotted as a function of energy in Figure 40.

2.01.2.4.2 Mass energy absorption coefficientOnce photon energy is transferred to charged particles in the

form of kinetic energy, the average energy of all secondary

charged particles (electrons, in the diagnostic range) can either

be deposited locally at the site of interaction through colli-

sional losses (excitations and ionizations), or be lost through

bremsstrahlung radiation (which is nonnegligible for energetic

electrons and for high-Z materials along the electron paths)

or other means like fluorescence emission following electron

impact ionization. The occurrence of radiative losses in the

energy deposition process can be taken into account in

the definition of kerma, K, by considering, for the two separate

energy loss processes, a collision kerma, Kc, and a radiative

kerma, Kr, respectively:

K ¼ Kc þ Kr [92]R

Photon energy (keV)

Air, dry

K/F

Air

kerm

a p

er u

nit

pho

ton

fluen

ce (G

y cm

2 )

1001010-13

10-12

10-11

Figure 40 Air kerma per unit photon fluence for monoenergetic x-raysin the diagnostic range.

plotted in the diagnostic energy range. Data are calculated with thecode XMuDat (Nowotny, 1998) with interaction cross section data fromBoone and Chavez (1996).

Figure 38

Figure 40

Figure 39


At a given monoenergetic photon energy E, the collision

kerma Kc is related to the energy fluence C(E) according to

Kc ¼ Cmenr

� �E,Z¼ FE

menr

� �E,Z

[93]

where men/r is the mass energy absorption coefficient (m2kg�1 orcm2g�1). If, in a photon interaction, hEki is the average energyof all secondary charged particles and hEri is the average energyof all photon radiation produced by those particles, then by

introducing the fraction g of hEri to hEki(0<g<1) through the

quantity

Ekh i � Erh iEkh i ¼ 1� Erh i

Ekh i 1� g [94]

one can express the mass energy absorption coefficient in terms

of the energy transfer coefficient as

menr

� �E,Z¼ mtr

r

� �E,Z

1� gð Þ [95]

Thus, men/r is less than mtr/r and men/rffimtr/r when gffi0,

for example when bremsstrahlung production is negligible as

in low-Z materials and at low energy E. The mass energy

absorption coefficient for water is plotted in Figure 41. For

water and soft tissues, gffi0 in the diagnostic energy range,

while for tungsten (Z¼74) at 100 keV, gffi0.5.

2.01.2.4.3 Exposure and absorbed dosex-Rays produce ionization in matter (i.e., they create electron–

ion pairs), so that the dosimetry of an x-ray beam can be

performed with an ionization chamber, by measuring the

amount of electric charge of one type produced in a material

volume (typically, a few cubic centimeters of air) divided by

the mass of material contained in the volume, a quantity

termed as x-ray exposure, X. The energy deposited in air for

production of one ion pair (or electron) varies little (less

than 2%) in the diagnostic energy range; its mean value is�Wð Þair ffi 33:97 eV per electron, so that with e the electron

charge, the mean value in J C�1 is�We

� �air¼ 33:97JC�1. The

unit for exposure is the roentgen (R), with 1R¼2.580�10�4Ckg�1. The exposure X (C kg�1) at a point in a monoenergetic

(E¼hn) x-ray beam, where the energy fluence is C, is then

LSE

Photon energy (keV)

Water

men/r

m/r

Mas

s at

tenu

atio

n co

effic

ient

(cm

2 g-1

)

20

101

10-2

10-1

100

40 60 80 100 120 140

Figure 41 Mass total attenuation coefficient and energy absorptioncoefficient for liquid water in the diagnostic energy range (data from NISTdatabase XRAYCOEF with data from Hubbell and Seltzer (1995)).

E

X Ckg�1� � ¼ C

menr

� �E, air

�W

e

� ��1air

[96]

and by using eqn [93], the exposure X (C kg�1) in eqn [96] is

related to the collision kerma in air (J kg�1) via

X C kg�1� � ¼ Kcð Þair

�W

e

� ��1air

¼ Kcð Þair J kg�1� �

33:97JC�1[97]

By expressing X in roentgen, R (a historical unit of

measurement defined as the quantity of radiation which

liberates one electrostatic unit of charge per cubic centime-

ters of air at standard temperature and pressure), the rela-

tion between exposure (in R) and collision kerma in air

becomes

X ¼ 1

2:580� 10�4Kcð Þair

�W

e

0@

1A�1

air

¼ 1

2:580� 10�4Kcð Þair33:97

¼ Kcð Þair8:76426� 10�3

[98]

or, equivalently,

Kcð Þair J kg�1� � ¼ 8:76426� 10�3X Rð Þ [99]

X Rð Þ ¼ 1

8:76426� 10�3Kcð Þair ffi 114 Kcð Þair J kg�1

� �[100]

and also,

X Ckg�1� � ¼ 2:580� 10�4X Rð Þ [101]

X Rð Þ ¼ 3875:97X Ckg�1� �

[102]

With absorbed dose D in matter defined in terms of energy

imparted per unit mass (1 Gy¼1 J kg�1), under conditions

of charged particle equilibrium (CPE) in the air volume of

the chamber, the (absorbed) dose in air, Dair, at the point

of measurement, is related to the collisional air kerma Kc via

Dair ¼ Kcð Þair CPEð Þ [103]

and using eqn [97] and eqn [99], one can obtain the expression

relating dose in air and exposure:

Dair Gyð Þ ¼ 33:97X Ckg�1� �

[104]

and also,

Dair mGyð Þ ¼ 8:76426X Rð Þ [105]

In air and within the diagnostic energy range, the radiative

kerma is Krffi0 so that KffiKc and then, using eqns [92] and

[103], DairffiKair.

The dose at a point in a medium with energy absorption

coefficient (men/r)medium, with respect to the dose in air, is

given by

Dmedium ¼ Dair

menr

� medium

menr

� air

264

375 [106]

and in terms of exposure X, using eqn [104],

VIER

Figure 41


Dmedium Gyð Þ ¼ 33:97

menr

� medium

menr

� air

264

375X Ckg�1

� �[107]

and using eqn [105],

Dmedium mGyð Þ ¼ 8:76426

menr

� medium

menr

� air

264

375X Rð Þ [108]

The term in square brackets (‘F-factor,’ here expressed as

Gy kg C�1 andmGy R�1) converts the exposure in air into dose

in a medium; its value is shown for various materials in the

diagnostic range, in Figure 42.

Using eqns [90], [104], and [105], and by considering

that (mtr/r)airffi(men/r)air in the diagnostic range, the dose in a

medium is related to the air kerma Kair by

Dmedium ffi Kair

menr

0@

1A

medium

menr

0@

1A

air

26666664

37777775

¼ 1:6022� 10�13FEmenr

0@

1A

medium

[109]

As an example of the use of eqn [109], the dose in breast

tissue (ICRU-44) per million photons per cm2 calculated

with this formula is shown in Figure 43. In absorption-based

contrast imaging of the breast (mammography), low-energy

photons around 20 keV are used instead of high-energy pho-

tons as in general radiography, in order to exploit the larger

differences in the values of the linear attenuation coefficient of

breast tissues (adipose, fibroglandular, tumor, and microcalcifi-

cations; see, e.g., Figures 14(a) and 15). However, Figure 43

shows that low-energy photons give higher dose to breast tissue

than high-energy photons in the diagnostic range: the mini-

mum in Kbreast/F is between 50 and 70 keV. On the other

hand, phase-contrast imaging relies on the difference in the

SE

Photon energy (keV)

F-fa

ctor

(mG

yR

-1)

F-fa

ctor

(Gy

kgC

-1)

Bone, cortical

Water

Adipose

Breast

15

6789

10

20

30

40

50

6070

10 10019

2327313539

78

116

155

194

233271

Figure 42 F-factor for various materials, in the range 1–150 keV.The sharp edges correspond to element K-edges in the materialcomposition. Data for mass energy absorption coefficient were derivedwith the code XMuDat (Nowotny, 1998) with interaction cross sectiondata from Boone and Chavez (1996).

EL

refractive index decrement d of tissues, whose values are much

larger than b, so that there is potential of dose reduction in

breast imaging based on phase-contrast techniques in the diag-

nostic range (Lewis, 2004), for example, employing photon

energies at 40–80 keV.

2.01.3 x-Ray Tubes and Beam Quality in DiagnosticRadiology

For an x-ray tube of given anode target material and given type

of applied voltage waveform, the shape of the output photon

energy spectrum (generally referred to as x-ray beam quality)

depends on the applied tube voltage (as characterized by the

kVp value) and on the tube inherent filtration. With respect to

a pure (unfiltered) bremsstrahlung spectrum from a thick tar-

get, actual beam spectra from an x-ray tube show (i) super-

imposed characteristic (fluorescence) x-ray lines (due to

interaction, in the anode target material, of electrons acceler-

ated toward the anode with kinetic energy greater than the

electron K-shell or L-shell binding energies of the target mate-

rial, and also to photoelectric interactions of bremsstrahlung

x-rays with target-bound electrons), in addition to (ii) beam

attenuation at low energies, resulting from photoelectric (self)

absorption of bremsstrahlung radiation in the surface layers of

the target material and photoelectric interactions in the mate-

rials (cooling liquid, glass envelope, tube housing, exit win-

dow) encountered by the photon radiation before exiting the

output window of the x-ray tube. This latter component repre-

sents an intrinsic filtration of the beam produced in the thick

target and as it exits the output window, which can be

expressed, for example, as an equivalent thickness of filtration

in millimeters of Al, typically in the range of 0.5–1 mm Al (at

70–75 kV) for general radiography x-ray tubes, and down to

0.1 mm Al for a mammography tube. It should be noted that

the inherent filtration is a function of x-ray tube voltage and

waveform.

Figure 44 shows schematically the shape of a bremsstrah-

lung x-ray spectrum from a thick tungsten target, the curve

of an actual filtered bremsstrahlung spectrum, and the

VIER

Photon energy (keV)

K/F

(mG

y/1

mill

ion

pho

tons

/cm

2 )

Breast tissue (ICRU-44)

1010-1

100

101

100

Figure 43 Dose in breast tissue per fluence of 106 photons cm�2 in thediagnostic energy range. Data for mass energy absorption coefficientof breast tissue (ICRU-44) were derived with the code XMuDat (Nowotny,1998) with interaction cross section data from Boone and Chavez(1996). For this plot, see also Lewis (2004).

Figure 42

Figure 43

Table 4 Energy and relative intensity of characteristic K-lines oftungsten, molybdenum, and rhodium

Tungsten (Z¼74)a

x-Ray line Energy(keV)

Relativeintensity

Ka2 57.984 57.600


contribution from K-shell characteristic radiation lines. The

dominant contribution to characteristic radiation (for electron

kinetic energies above the K-edge of W at 69.525 keV, Table 3)

comes (in order of intensity of emission) from the Ka1line (59.321 keV, shell transition K LIII in W), from the Ka2line (57.984 keV, shell transition K LII), from the Kb3 line

(66.950 keV, shell transition K MII), and from the Kb1 line

(67.244 keV, shell transition K MIII) (Table 4). After consid-

ering photon attenuation by a target thickness of 2 mm in

escaping the target, the spectrum assumes its typical attenu-

ated shape at low energies; additional filtration in the beam

determines further attenuation of photons at energies

below �0.01 MeV. L-shell binding energies in tungsten

(LI¼12.098 keV; LII¼11.541 keV; LIII¼10.204 keV) deter-

mine the sharp transitions around 10 keV (Figure 44).

The unfiltered thick-target spectrum (differential radiant-

energy r(E) in J MeV�1 versus photon energy E¼hv in MeV)

has the decreasing linear trend of the Kramers spectrum,

given by

Photon energy (MeV)

Unfiltered bremsstrahlung

Filtered through 2 mm tungsten

Bre

mss

trah

lung

out

put

(J M

eV-1

)

0.000

2

4

6

8

10

12

0.02 0.04 0.06

K-edge at69.5250 keV

Ka1

Ka2

Kb1

Kb2

LIII edge(10.2068 keV)

LI edge (12.0998 keV)LII edge (11.5440 keV)

0.08

Figure 44 The continuous thick line shows the bremsstrahlungdifferential energy spectrum r(E) (J MeV�1) as a function of photonenergy E (MeV) for electron kinetic energy 0.08 MeV incident on a thicktungsten anode target. Also shown (not to scale) is the spectrum ofthe characteristic emission K-lines of tungsten (see Table 3). Thecontinuous thin line is the bremsstrahlung spectrum attenuated by alayer of tungsten of thickness 2 mm, simulating the expected effect oftarget self-attenuation on the shape of the unfiltered spectrum; L-edgesare seen at about 0.01 MeV. Characteristic L-lines are not shown forclarity.

Table 3 Physical data of elemental materials used in x-ray beam filtration

Element Symbol Atomicnumber, Z

Density, r(g cm�3)

Electronic dens(cm�3)

Aluminum Al 13 2.699 7.83Eþ23Barium Ba 56 3.50 8.60Eþ23Copper Cu 29 8.96 2.46Eþ24Iodine I 53 4.93 1.24Eþ24Molybdenum Mo 42 10.22 2.69Eþ24Rhodium Rh 45 12.41 3.27Eþ24Tin Sn 50 7.31 1.85Eþ24Tungsten W 74 19.30 4.68Eþ24Lead Pb 82 11.35 2.71Eþ24aExperimental data from NIST x-ray Transition Energies database http://physics.nist.gov/Phy

Data from the computer code XMuDat based on tabulations from Boone and Chavez (1996)

ELSE

r Eð Þ ¼ const:� I� t � Z � E0 � Eð Þ [110]

where I is the tube current (mA), t is the exposure time (s), and

hn0E0 (MeV) is the maximum photon energy in the spectrum.

In turn, the maximum photon energy is equal to the kinetic

energy Ek of electrons (elementary charge e) incident on the

anode target, accelerated by a tube voltage V (kV)¼Ek/e. By

integrating the triangular Kramers spectrum up to the maximum

photon energy, one obtains the total bremsstrahlung produc-

tion radiant energy R (J) in a thick anode target of atomic

number Z of an x-ray tube (see Attix, 1986, Chapter 2.08):

, as anode target, contrast medium or for shielding

ity Electrons(g�1)

K-shell binding energy(keV)a

Fluorescence K-yield

2.90Eþ23 1.560 0.03102.46Eþ23 37.452 0.90202.75Eþ23 8.980 0.43902.52Eþ23 33.167 0.88102.64Eþ23 20.000 0.75902.63Eþ23 23.222 0.80102.54Eþ23 29.200 0.85602.42Eþ23 69.525 0.97102.38Eþ23 88.006 0.9830

sRefData/XrayTrans/Html/search.html.

.

Ka1 59.321 100.000Kb3þKb1þKb5/1þKb5/2 ffi67.2 32.126Kb2/1þKb2/2þKb4/1þKb4/2þKb2/3þKb2/4

ffi69.1 8.417

Molybdenum (Z¼42)

x-Ray line Energy (keV)b Relative intensityc

Ka2 17.374 50–53Ka1 17.479 100Kb3 19.590 10Kb1 19.608 20

Rhodium (Z¼45)

x-Ray line Energy (keV)b Relative intensityc

Ka2 20.074 50–53Ka1 20.216 100Kb3 22.699 10Kb1 22.724 20

aAs reported in Attix (1986).bExperimental data from NIST x-ray Transition energies database http://physics.nist.

gov/PhysRefData/XrayTrans/Html/search.html.cFrom NPL database of x-ray adsorption edges and characteristic x-ray line energies

http://www.kayelaby.npl.co.uk/atomic_and_nuclear_physics/4_2/4_2_1.html.

VIER

Figure 44

http://physics.nist.gov/PhysRefData/XrayTrans/Html/search.html



http://www.kayelaby.npl.co.uk/atomic_and_nuclear_physics/4_2/4_2_1.html


R Eð Þ ¼ const:0 � I� t � Z � E2k

¼ const:00 � I� t � Z � V2 [111]

From eqn [111], it is seen that for an anode of given atomic

number Z, at a fixed tube current and exposure time, R goes as

the square of the x-ray tube voltage V (kVp), and that at fixed

tube voltage, R increases proportionally to the increase of

the product of the tube current I and the exposure time t. The

constant in eqn [110] is about 2 J MeV�2 mA�1 s�1. At fixedkilovoltage, by multiplying the current–time product (mAs) by

a factor k, each ordinate in the unfiltered r(E) bremsstrahlung

x-ray spectrum as well as the slope of the spectrum is multi-

plied by k. On the other hand, by keeping the mAs values

constant and changing the tube voltage, the spectrum shifts

in parallel along the energy axis, without changing its slope.

The quadratic dependence of R(E) on Ek (and on V) is strictly

valid only for the unfiltered bremsstrahlung x-ray spectrum,

and a different power law exponent can be observed in the

output spectrum, for filtered beams.

As regards the efficiency of the production of bremsstrah-

lung x-rays from a thick target x-ray tube, in which electrons

(Ek�0.15 MeV) are stopped in the target, it depends on the

ratio of radiative (dEk/rdx)r to collision (dEk/rdx)c mass stop-

ping power. Figure 45(a) shows this ratio in percent for tung-

sten in the range 10–150 keV of kinetic energy; it is seen that it

Kinetic energy (keV)

Kinetic energy (MeV)

(dE

k/rd

x)/(

dE

k/rd

x)c

(%)

(a)

(b)

W (Z = 74)

W (Z = 74)

Mo (Z = 42)

200.000

0.005

Rad

iatio

n yi

eld 0.010

0.015

0.00.02 0.04 0.06 0.08 0.10 0.12 0.14

0.5

1.0

1.5

2.0

2.5

3.0

40 60 80 100 120 140

Figure 45 (a) Radiative to collision mass stopping power versus kineticenergy Ek for the case Z¼74 (tungsten anode), from NIST databaseESTAR. (b) Electron radiation yield in tungsten and molybdenum in thekinetic energy range 10–150 keV; data after Berger and Seltzer, astabulated in the Radiological Toolbox code (Eckerman and Sjoreen, 1996).

ELSE

is between 0.2% and 3% in the energy range of interest for

radiography with x-ray tubes.

The ratio of radiative to collision stopping power is directly

proportional to the atomic number Z of the target material,

and at varying electron kinetic energy below 0.15 MeV, it can

be empirically approximated as {Z�Ek� [700þ200 log10(Ek/

3)]�1} (Attix, 1986). The ratio of radiative to total (radiative

and collision) mass stopping power {[(dEk/rdx)r]/[(dEk/rdx)rþ(dEk/rdx)c]} is a function of the current kinetic energy

of the slowing down electron; by averaging this ratio across

the energies up to the initial kinetic energy Ek, one obtains an

estimate of the radiation yield Y, that is, the fraction of the

electron initial kinetic energy Ek that is emitted as e.m. radia-

tion during the slowing down process up to thermalizing in

the material. Indeed, the total bremsstrahlung production radi-

ant energy R∝Y�Ek� I� t. The electron radiation yield (as

derived from NIST database ESTAR, http://physics.nist.gov/

PhysRefData/Star/Text/ESTAR.html) is plotted in Figure 45(b)

for W and Mo targets, showing that it is less than 1% for

electron kinetic energies lower than 100 keV. This means that

for tube voltages less than 100 kV, only less than 1% of the

electron beam energy incident on the target is emitted as x-rays

while more than 99% is deposited on the target through colli-

sions and produces the heating of the target: this anode ther-

mal power loading is to be accounted for by the anode cooling

system. In terms of power density at the tube (electronic) focal

spot, hot-cathode rotating solid-metal anode x-ray tubes for

diagnostic radiology (with electrical powers up to ffi100 kW

applied by the high-voltage generator) are capable of handling

up to ffi100 kW mm�2. In terms of apparent focal spot size

(i.e., the size of the focal spot as viewed from the central axis

of the x-ray beam), taking into account the angled anode

configuration in the so-called line-focus technology (see

Section 2.01.3.9) which allows for a reduction up to a factor

1:10 of the focal spot length and area, this means that powers

up to ffi100 kW can be managed with an (apparent) linear

focal spot size of a fraction of a millimeter. Microfocus x-ray

tubes with an apparent focal spot diameter of 10 mm or less

have typically an electron beam power of 10 W or less and

hence can handle a power density of less than 100 kW mm�2,while liquid metal jet anode x-ray tubes can handle in excess

of 1000 kW mm�2 (Larsson et al., 2013).

Figure 46 shows the x-ray spectrum for 80-keV electrons

incident on a thick target tungsten-anode x-ray tube after

filtration through 0.5 mm of Al. The data have been calcul-

ated with the software code SpekCalc (Poludniowski, 2007;

Poludniowski and Evans, 2007) with 0.1 keV resolution and a

photon takeoff angle of 30� from the target, and are intended

to illustrate the effect of typical (inherent) beam filtration

and the contribution of the characteristic lines, which is less

than 0.25% of the total tube output. A model (TASMICS) for

producing X-ray tube spectra from 20 kV to 640 kV has been

introduced recently (Hernandez and Boone, 2014).

Adding thin sheets of a highly pure (>99.9%) material

(e.g., Al, Cu, Mo, Rh, Sn, and others) at the output port of

the x-ray tube, which selectively absorb (via photoelectric inter-

actions) the output photons as a function of their energy, is

the common way of roughly changing the shape of the x-ray

tube spectrum in order to reduce the spectral intensity in low-

energy bands. This practice is adopted for reducing patient

(skin) radiation dose during the diagnostic imaging exam, for

VIER

http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html

http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html

Figure 45

Photon energy (keV)

K-edge

K-edge

Eb= 69.5 keV

0.5 mm AI added filter

W anode 150 kVp12 deg anode angle

20105

106

107

40 60 80 100 120 140 160

Pho

tons

keV

−1 c

m−2

mA

−1

s−1 @

1m


(b)

(c)

Photon energy (keV)

0.5 mm AI added filter

W anode, 80 kVp12 deg anode angle

10

66 68 70 72 74

0.0

5.0x106

1.0x107

1.5x107

20 30 40 50 60 70 80

b)

No.

pho

tons

keV

−1 c

m−2

mA

−1

s−1 @

1m

Figure 46 (a) Calculated x-ray spectrum at 80 kVp tube voltage from atungsten anode tube, target angle 12�, after filtration through 0.5 mmAl (typical equivalent inherent filtration due to target self-filtration, tubehousing, cooling oil, exit window). At 1 m distance from the source, thetotal bremsstrahlung tube output is 216.5 mGy mA�1s�1, and the totalcharacteristic tube output is 0.7575 mGy mA�1s�1, so that the totaloutput is 217.3 mGy mA�1s�1. Note the characteristic Ka and Kb lines oftungsten at about 58–69 keV. Note also that no air filtration wasconsidered. (b) Details of the same spectrum around 70 keV, to point outthe hardly visible decrease in intensity at the K-edge of tungsten.The K-edge absorption in the target is more evident in the spectrumshown in a semilog plot in (c) (150 kV, 0.5 mm filter, 1 m air). Data arecalculated by the code SpekCalc (Poludniowski, 2007; Poludniowskiand Evans, 2007).

Table 5 Minimum HVL at various x-ray tube potentials required byFood and Drug Administration (FDA) regulation for all x-ray systems,except dental x-ray systems designed for use with intraoral imagereceptors, manufactured on or after June 10, 2006 (21 CFR, Ch. I, }1020.30, 4-1-12 Edition, U.S. Gov. Printing Office, available at web sitewww.gpo.gov)

Measuredoperatingpotential (kVp)

Minimum HVL(mm Al)


MinimumHVL(mm Al)

30 0.3 90 3.240 0.4 100 3.650 0.5 110 3.951 1.3 120 4.360 1.5 130 4.770 1.8 140 571 2.5 150 5.480 2.9

Table 6 Minimum HVL at various x-ray tube potentials required byFDA regulation for dental x-ray systems designed for use with intraoralimage receptors, manufactured after December 1, 1980 (21 CFR, Ch. I, }1020.30, 4-1-12 Edition, U.S. Gov. Printing Office, available at web sitewww.gpo.gov)


Minimum HVL(mm Al)


MinimumHVL(mm Al)

30 1.5 90 2.540 1.5 100 2.750 1.5 110 3.051 1.3 120 3.260 1.5 130 3.570 1.5 140 3.871 2.1 150 4.180 2.3


ELSEVIE

R

reducing radiation scatter in the subject, and/or to increase the

average energy of the beam, with respect to the unfiltered (i.e.,

with only inherent filtration) energy spectrum.

x-Ray tube spectral filtration can be expressed as the total

filtration (addedþ inherent), a parameter also used for

radiation protection. Indeed, regulatory boards require or rec-

ommend a limit for the minimum total filtration in the x-ray

beam for diagnostic purposes; for example, the ICRP (Interna-

tional Commission on Radiological Protection) recommends

that the total filtration above 70 kV should be equivalent to not

less than 2.5 mmAl. In the United States, federal regulations by

the Food and Drug Administration (FDA) (CFR, 2012), require

minimum half value layer (HVL) (mm Al) values as in Table 5

(for all x-ray systems, except dental x-ray systems designed

for use with intraoral image receptors, manufactured on or

after June 10, 2006) and in Table 6 (for dental x-ray systems

designed for use with intraoral image receptors, manufactured

after December 1, 1980). Increasing the filtration even further

could be advantageous for additional limitation of patient dose

in general radiography, without sacrificing image quality

(Behrman, 2003). Such a spectral filtration is the result of

the energy-dependent mass attenuation coefficient (m/r) of

the filter material of density r in the low-energy range where

photoelectric effect is dominant (Table 3; Figure 47). An abrupt

increase of the attenuation coefficient occurs at the threshold

energy corresponding to K-shell absorption (see values in the

last column of Table 3 and Figure 48), thus leading to an

increased photoabsorption at energies above such threshold.

Figure 46

http://www.gpo.gov

http://www.gpo.gov

Atomic number, Z

Ene

rgy

(keV

)

Cu

01

10

100

10 20 30 40 50 60 70 80 90 100

MoSn

AIBa

K-edge

K-edge electron binding energy

W

IRh

Figure 48 K-edge binding energy of elements with Z¼1 to 100. Someelements, indicated by the arrows, are commonly selected for x-raytube target, beam filtering, or contrast enhancement materials. Datafrom computer code MUCOEFF are from Boone and Chavez (1996).

Photon energy (keV)

Additional filtration: 0.5 mm Be + 0.6 m air

W anode 28 kV1 mm AI filter(Emean= 20.3 keV)

Rh anode 28 kV0 . 0 2 5 mm Rh filter(Emean= 18.21 keV)

Mo anode 28 kV0.030 mm Mo filter(Emean= 16.78 keV)

Rel

ativ

e no

. of p

hoto

ns

100.0

0.2

0.4

0.6

0.8

1.0

15 20 25 30

Rh K-edge

Mo K-edge

Figure 49 Example x-ray spectra for compressed breast imaging witha mammographic unit, calculated with the MASMIP, RASMIP, andTASMIP codes (Boone and Seibert, 1997; Boone et al., 1997) for x-raytubes with Mo, Rh, and W or W/Rh anodes, respectively, with addedfiltration of Mo, Rh, or Al. The spectra are normalized to thecorresponding peak value for 28 kV (Mo/Mo or Rh/Rh anode/filter), aswell as for the 28-kV, W-target spectrum with Al filter. K-edges of Moand Rh are indicated (see Table 3).

Photon energy (keV)

AI

CuMo

Mass attenuation coefficient

Rh

m/r

(cm

2 g−1

)

101

102

101

100

10−1

102

Figure 47 The graph shows the mass attenuation coefficient, in thediagnostic energy range, for elements aluminum, copper, molybdenum,rhodium, and tin, commonly used in the form of thin pure metal sheetsfor x-ray tube added filtration and beam shaping. Sharp transitions in thegraph show elemental K-shell absorption edges. Data from computercode MUCOEFF (Boone and Chavez, 1996) are available at web siteftp://ftp.aip.org/epaps/medical_phys/E-MPHYA-23-1997/.


ELSE

From the Lambert–Beer exponential attenuation law, the

(energy-dependent) beam attenuation introduced by inserting

a material (metal) layer of thickness Dx downstream in the

beam at the exit window of the x-ray tube is given by exp

[�(m/r)rDx], where m¼m(E). By multiplying the spectral ordi-

nates at each energy in the x-ray tube output spectrum by

the corresponding attenuation factor, the shape of the filtered

beam can be calculated.

x-Ray tube spectra employed in mammography setups

require a low-energy spectrum with photon energies in the

approximate range of 15–25 keV. Indeed, incident photons

with energy lower than �15 keV determine full absorption

and high absorbed dose in the breast tissue, while photons

with energy higher than �25 keV determine a reduction of

tissue radiographic contrast. On the other hand, the limited

penetration of 15–25 keV radiation is well met to the reduced

thickness (from a few to several centimeters) of compressed

breast tissue during a mammographic exam. Indeed, studies

with monochromatic x-ray beams produced at synchrotron

radiation facilities with photon energy in this range (providing

incident fluence rate in the order of 1010 photons cm�2s�1)(e.g., at the SYRMEP beamline dedicated to mammography at

the ELETTRA synchrotron radiation facility in Trieste, Italy)

(Dreossi et al., 2008) provided evidence that the optimal breast

tissue contrast for lesion detection is for monoenergetic photons

in that low-energy range; with this monochromatic source the

optimal photon energy for imaging can be selected also with

regard to the lowest radiation dose, depending on the com-

pressed breast thickness. In order to produce narrow spectral

width beams from x-ray tubes, molybdenum or rhodium target

anodes are commonly used (with Mo or Rh filtration), at tube

voltages approximately in the range 22–33 kV; for digital mam-

mography detectors, tungsten anode tubes operated up to 50 kV

(with Al filtration) are also employed. A comparison of three

spectra of possible use in mammography is shown in Figure 49.

VIER

2.01.3.1 Beam Attenuation and Beam Shape Descriptors

Direct measurements of x-ray photon spectra from an x-ray tube

is not commonly performed either in the lab or in the clinical

environment, since the high photon fluence rate (in the order of

109 photons cm�2s�1 per mA tube current at 1 m distance, see

Figure 46) far exceeds the count rate capabilities of spectroscopic

detectors (which is limited in practice to �105counts s�1) witha low pileup rate. A high-count rate waveform-digitization

technique by Stumbo et al. (2004) and Bottigli et al. (2006)

allows pileup events to be rejected via off-line processing of the

digitized signal from a semiconductor detector. In general, for

the purpose of reducing the count rate, strong beam collima-

tion (e.g., with pinhole of aperture 0.1 mm or less) is usually

employed in order to reduce the photon flux on the detector, but

this may determine spectral artifacts in terms of spectral defor-

mations and/or inaccuracy in the determination of the spectral

intensity, due to uncertainty in the effective aperture size, its

Figure 48

Figure 49

Figure 47

ftp://ftp.aip.org/epaps/medical_phys/E-MPHYA-23-1997/


orientation with respect to the beam central axis, and scatter

from the aperture sides. Spectral artifacts also arise from escape

peaks with detectors based on high-bandgap compound semi-

conductor substrates (e.g., CdTe or CdZnTe). Recovery of spec-

tral artifacts due to escape peaks is usually performed with a

post-processing stripping procedure (e.g.,Matsumoto et al.,

2000; Seelentag and Panzer, 1979). Other spectral artifacts

may be related to the detector response, to the presence of K-

edge(s) of the detectormaterial which alter the spectral shape, to

the presence of escape peaks by interaction with the detector

contacts, to edge effects from interaction events at the border of

the detector volume, to incomplete charge collection in the

detector, and to beam scattering from the setup environment

reaching the detector. Example experimental spectrawith aCdTe

detector are shown in Figure 50.

As a rough yet practical descriptor of x-ray beam quality, the

HVL is considered. HVL is defined as the thickness (e.g., in

millimeters) of a filter material (typically aluminum or copper)

required to reduce the x-ray exposure, X, by one-half. x-Ray

exposure is determined by measuring the electrical charges, in

terms of number of ion pairs, produced by x-ray ionization in a

given mass of air (1R¼2.58�10�4C kg�1 of air), using an

ionization chamber. Indeed, the number of ion pairs created in

the chamber air volume is directly related to the beam intensity at

each energy in the x-ray spectrum. For a given anodematerial, the

HVL is a function of tube voltage (kVp), the generator waveform,

and the shape of the x-ray spectrum, as determined by the total

(inherentþadded) filtration of the beam. The HVL is used as

a rough descriptor of the x-ray spectral shape in order to


(b)

Cou

nts

mm

-2 m

A-1

s-1

keV

-1

@ 6

0cm

1 mm AI added filtration

Mo anode30 kVp

50

500

1000

1500

10 15 20 25 30

Photon energy (keV)

Cou

nts

mm

-2 m

A-1

s-1

keV

-1

@ 6

0cm

1 mm AI added filtrationMo anode30 kVp

5100

101

102

103

10 15 20 25 30

Figure 50 Spectrum from a molybdenum anode x-ray tube (30 kVp,1 mm Al filtration) measured with a commercial CdTe detector (AMPTEKXR-100T-CdTe, 3�3 mm2) (a) in linear and (b) in log scale.

ELSE

characterize an x-ray beam, with reference to the attenuation

properties of the given x-ray beam in a given material.

Attenuation refers to the decrease of the number of x-rays

incident on a given area of a sample in passing through the

sample, and is caused by both photoabsorption and scatter

in the sample, depending on the photon energy and on

the sample thickness and composition. If the x-ray beam is

monochromatic, with energy E, then the HVL value would

completely characterize the beam energy. Indeed, the (mono-

energetic) Lambert–Beer attenuation law of a pencil beam in

a thin layer of thickness t (cm) of a material with density rand total mass attenuation coefficient m/r (cm2g�1) is given by

N

N0¼ exp � m Eð Þ

rrt

� �[112]

and in terms of logarithmic attenuation,

lnN

N0

� �¼ � m Eð Þ

rrt [113]

where N0 is the number of photons incident on the thin layer

and N the number of photons exiting from the thin layer.

This law is strictly valid for a monoenergetic collimated beam

and in narrow beam geometry (also good geometry), that is when

the setup for exposure measurements with the attenuated

and unattenuated beams is such to exclude any contribution

of radiation scattered from the attenuator material and from

the surroundings.

The attenuation law can be expressed in terms of exposure X

for an incident exposure X0:

X

X0¼ exp � m Eð Þ

rrt

� �[114]

and for logarithmic attenuation,

lnX

X0

� �¼ � m Eð Þ

rrt [115]

For attenuation by an HVL at energy E,

X

X0¼ 1

2¼ exp � m Eð Þ

rrHVL

� �[116]

HVL ¼ ln 2m Eð Þr r¼ 0:693147

m Eð Þ cm [117]

hence the HVL is uniquely determined by the beam energy E,

for a given material with known value of linear attenuation

coefficient m (cm�1) at that energy. On the other hand, x-ray

tube spectra are polychromatic, and then characterized by the

distribution of the photon fluence FΕ(E) (photons per cm2)

versus photon energy E (keV) up to the maximum beam energy

E0. Following (Boone, 2000), we have

N0 ¼ðE00

AFE

x Eð Þ dE [118]

N ¼ðE00

AFE

x Eð Þ exp �m Eð Þr

rt �

dE [119]

(where A is the irradiated area), and the x-ray attenuationN(t)/

N0 in a sheet of material of thickness t, and the mass attenua-

tion coefficient m(E)/r is expressed by

VIER

Figure 50

80 keV

40 keV

1m

per

1 m

As

ion

N/N

0

0.4

0.5

0.60.70.80.9

1 7.33


N tð ÞN0¼

ðE00

AFE

x Eð Þ exp �m Eð Þr

rt �

dEðE00

AFE

x Eð Þ dE[120]

The function x(E) (photons cm�2R�1) describes the pho-

ton fluence per unit exposure and is given by (Boone, 2000)

x Eð Þ ¼ 5:43� 1010

E men Eð Þr

� air

[121]

where (men/r)air (cm2g�1) is the mass energy absorption coef-

ficient for air and E is in kiloelectronvolt. x(E) is plotted in

Figure 51 in the diagnostic energy range; the marked energy

dependence reflects that of the attenuation coefficient of air,

the common absorption material in ionization chambers used

for exposure measurements.

The total photon fluence per exposure FΤ (photons

cm�2R�1) over the whole x-ray spectrum up to the maxi-

mum photon energy E0 can be obtained by calculating the

quantity (Boone, 2000)

FT ¼

ðE00

FE dEðE00

FE

x Eð Þ dE[122]

If the exposure X (in R) at the distance of interest from the

source is measured, the total photon fluence F can be obtained

simply as the product F¼FTX. The photon fluence per expo-

sureFΤ can be calculated via eqn [122] once the x-ray spectrum

FΕ is known (e.g., by calculation with codes like TASMIP and

SpekCalc or TASMICS); tables of FΤ are provided by Boone

(2000) for various combinations of kilovolt, filtration, andHVL.

Once the x-ray photon fluence is known at each energy in

the beam, using the conversion function provided by eqn

[121] and with known material attenuation data, eqn [120]

can then be used to provide the expected attenuation in any

(homogeneous) material volume, as well as the expected inci-

dent fluence spectrum on the sample and on the downstream

imaging detector. As an example of the above considerations,

consider the attenuation of an 80-kV x-ray beam (filtered by

2.5 mm Al and 1 m air) by a sheet of aluminum of thickness t

LSE

Photon energy, E (keV)

x(E

) (p

hoto

ns c

m-2

R-1

)

00

3x1010

2x1010

1x1010

20 40 60 80 100 120 140

Figure 51 The function x(E) (photon fluence in photons cm�2 per unitexposure in R, in air, at x-ray energy E) in the diagnostic energy range(data are calculated from eqn [121] in the text, with attenuation data forair calculated with the code XMuDat, Nowotny, 1998, with interactioncross section data from Boone and Chavez (1996)).

E

in the range 1–15 mm. Figure 52 shows in a semilogarithmic

plot the attenuationN(t)/N0 calculated by eqn [120], using the

spectral intensity for an 80-kV spectrum calculated with the

computer code SpekCalc (Poludniowski, 2007; Poludniowski

and Evans, 2007) and tables of mass attenuation coefficients

(e.g., from the XCOM database of Berger and Hubbell (1987),

freely available at the website of NIST, National Institute of

Standards and Technology in USA) or from freely available

software codes that make reference to those or other tabulated

data, as from Boone and Chavez (1996).

We note that the calculated attenuation curve for the poly-

energetic beam in Figure 52 shows a marked nonlinear trend,

with increasing curvature at increasing material thickness; on

the other hand, the monoenergetic beam does show a linear

trend in this semilog plot, as obvious from eqn [113]. The

nonlinearity in the plot of exposure attenuation versus attenu-

ator thickness reveals a phenomenon known as beam harden-

ing, whereby the progressive removal of low-energy photons

from the beam transmitted through increasing thicknesses of a

material determines a modification of the beam spectral shape

and a shift of the average (and effective) energy toward higher

energies (whence, a harder x-ray beam).

For the 80-kV beam, the HVL is graphically derived as

2.78 mm Al; the corresponding effective linear attenuation coef-

ficient meff is calculated as

meff mr

� �eff

r 0:693147

HVL cmð Þ cm�1 [123]

and is meff¼2.49 cm�1, or (m/r)eff¼2.49 cm�1/2.7gcm�3¼0.92 cm2g�1. From the m(E)/r versus E curve for alu-

minum (Figure 47), this value corresponds to a beam effective

energy of 32.3 keV (Al) for the polychromatic beam. Increasing

the added filtration of the x-ray beam will cause an increase of

the HVL and a decrease of the effective attenuation coefficient

VIER

Aluminum thickness (mm)

80 kV

32 keVHVL2

HVL1

2.5 mm AI filterE

xpos

ure

(mR

) @

Att

enua

t

00.090.1

0.2

0.3

5 10 15

0.73

Figure 52 Attenuation curves (in terms of normalized exposure as afunction of thickness) in aluminum, calculated for an 80-kVpolychromatic x-ray beam. The (first) half value layer (HVL) is derivedfrom this plot as the abscissa for attenuation¼0.5 and isHVL1¼2.78 mm. Also shown for comparison are attenuations calculatedfor monochromatic x-ray beams of energy 80, 40, and 32 keV, this lastbeing the beam effective energy at which the HVL is the same for thepolychromatic and monochromatic beams. The second HVL is alsoindicated as the additional aluminum thickness for which theattenuation¼0.25 and is HVL2¼4.04 mm, so that the homogeneitycoefficient for the 80-kV beam is 2.78/4.04¼0.69.

Figure 51

Figure 52

Tube voltage (kVp)

Total filtration: 2.5 mm AI + 1 m Air

Linear fits (R2> 0.999, P<0.0001)

Hal

f val

ue la

yer

(mm

AI)

200

1

2

3

4

5

6

7

8

9

40 60 80

HVL2

HVL1

100 120 140 160

Tube voltage (kVp) (b)

(a)

Total filtration: 2.5 mm AI + 0.25 mm Cu + 1 m Air

Parabolic fits (R2> 0.999, P<0.0001)

Hal

f val

ue la

yer

(mm

AI)

200123456789

101112

40 60 80

HVL2

HVL1

100 120 140 160

Figure 54 (a) The first and second HVL values as a function ofkilovoltage for a tungsten target x-ray tube with 12� anode angle and2.5 mm Al total filtration. The curve shows a linear trend as indicated bythe linear fits. (b) With respect to the same x-ray tube and filtration as inplot (a), the filtration has been increased by adding 0.25 mm Cu,producing an increase in beam HVL, closer values of HVL1 and HVL2, andthe appearance of a curvature in the trend of HVL versus tube voltage.Data are calculated with the code SpekCalc (Poludniowski, 2007;Poludniowski and Evans, 2007).


VIER

meff. The relationship between (m/r)eff and HVL for Al and Cu

attenuators is shown in Figure 53.

Equation [116] defines the first HVL, or HVL1, as the thick-

ness required to reduce the exposure (or the exposure rate) by a

factor 0.5; once inserted in the beam a thickness HVL1 of

attenuator, the second HVL, or HVL2, is the additional thickness

required to determine, in the same conditions, an additional

reduction of the exposure by a factor 0.5, that is a total atten-

uation by a factor 0.25. Also used are the quarter value layer

(QVL), the attenuator thickness that reduces the exposure to

one-quarter of the unattenuated exposure (i.e., HVL2¼QVL-

�HVL1), and the tenth value layer, the attenuator thickness

that reduces the unattenuated exposure to 1/10.

The condition HVL1¼HVL2 strictly holds only for mono-

energetic beams, so that the homogeneity coefficient HVL1/HVL2can be used as a rough parameter to indicate how added beam

filtration is effective in reducing the spectral width and increas-

ing the spectral homogeneity of the x-ray beam, with its value

approaching unity as the spectrum approaches monochroma-

ticity. The two values (first and second HVL) provide a more

accurate description of the (polychromatic) x-ray spectrum

than a single value; a more refined characterization of the

beam quality would be provided by indication of the higher-

order HVL, that is, those filter material thicknesses which, once

inserted consecutively in the beam path (usually at the output

port of the x-ray tube, in the collimator/filter unit), determine

corresponding further reduction of the beam exposure by a

fraction 0.5; HVLs exceeding the tenth order (HVL10) have

been proposed for this purpose.

As an example, the HVL as a function of the x-ray tube

voltage for a W anode tube with 12� target angle is shown in

Figure 54(a) (with 2.5 mm Al filtration) and Figure 54(b)

(with 2.5 mm Alþ0.25 mm Cu total filtration). It is seen that

adding filtration determines an increase of HVL, a quadratic

trend of HVL versus tube kilovoltage and an increase of the

homogeneity coefficient; the trend of this coefficient versus

tube voltage is shown in Figure 55 for the two example beam

filtrations.

The implicit functional relationship between the effec-

tive energy and the HVL expressed by eqn [123] is shown in

Figure 56, from data in Figure 47 of the total attenuation

coefficient for Al and Cu.

LSE

0.010.1

1

10

100

HVL (mm)

(m/r

) eff

(cm

2 g-1

)

CuAI

0.1 1 10

Figure 53 The relationship between effective mass attenuationcoefficient and first HVL in aluminum or copper, given by eqn [123]in the text.

Tube voltage (kVp)

2.5 mm AI + 0.25 mm Cu2.5 mm AI

12 deg W anode

Hom

ogen

eity

coe

ffici

ent

HV

L 1/H

VL 2

200.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

40 60 80 100 120 140 160

Figure 55 From the data in Figure 54, the homogeneity coefficient hasbeen calculated and shown as a function of tube voltage. Adding filtrationreduces the width of the x-ray beam energy spectrum (i.e., high HVL1/HVL2 ratio), which is even more for high kilovolt spectra.

E

Figure 53

Figure 54

Figure 55

HVL1 (mm)

First half value layer

Cu

AI

Effe

ctiv

e en

ergy

(keV

)

0.0110

20

30

40

50

60708090

100

0.1 1 10

QVL (mm)(b)

(a)

Quarter value layer

Cu

AI

Effe

ctiv

e en

ergy

(keV

)

0.1

10

20

30

40

50

60708090

100

1 10

Figure 56 The effective energy versus HVL (a) or quarter value layer (b)for Al and Cu, from data in Figure 53.

Photon energy (keV)

No.

pho

tons

keV

-1 c

m-2

mA

-1s-1

@ 1

m

200

8x106

2x1074x107

6x106

4x106

2x106

40 60 80 100 120 140

50 kVp

Filtration: 2.5 mm AI + 1 m Air

70 kVp90 kVp110 kVp130 kVp150 kVp

160

Figure 57 x-Ray spectra at 50–150 kVp in 20 kVp steps, at 1 mdistance from the focal spot of a tungsten anode tube with target angle30�, after filtration through 2.5 mm Al and considering the filtrationintroduced by 1 m air between the source and the detector. Data arecalculated with the code SpekCalc.


Fulfillment of the requirement of narrow beam geometry is

important in assuring correctness of data interpretation in

attenuation measurements; on the contrary, operation under

broad beam (or bad) geometry, with a nonnegligible scatter

component present in the measurement, produces inaccurate

estimates of beam quality, for example, HVL and effective

energy. Since inclusion of scattered radiation in the exposure

measurement increases the detected signal, for any given atten-

uator thickness attenuation tends to be lower in bad geometry

than in good geometry, with corresponding increase of the

(apparent) HVL.

ELSE

2.01.3.2 Effect of Varying the Kilovoltage at FixedBeam Filtration

Figure 57 shows the effect of varying the accelerating potential

on the shape of the x-ray spectrum from a tungsten anode x-ray

tube, at fixed beam filtration. Bremsstrahlung continuum is

evident from about 15 keV up to the maximum photon energy

corresponding to the given kilovoltage (50–150 kV), as well as

characteristic K-shell lines for electron kinetic energies above

the K-edge of tungsten. The spectra up to 90 kV show a linear

high-energy continuum resembling the unfiltered bremsstrah-

lung (Figure 44), while for higher kilovoltages a curvature of

the continuum spectrum is evident.

The spectra in Figure 57 are for a tungsten anode tube with

12� anode angle, with 2.5 mm Al added filtration and includ-

ing air filtration in the beam. The beam effective energy at

varying kilovoltage, for two filtrations (0.5 mm Al, equivalent

to a typical inherent filtration, and 2.5 mm Al) is shown in

Figure 58(a) and 58(b), respectively; the mean energy shows

an almost linear trend at low kilovoltages, with evident curved

trend at higher tube voltages and higher filtration. This trend is

more evident by increasing even further the beam filtration

(0.25 mm Cu added to the 2.5 mm Al filter) as shown in

Figure 58(c).VIER

2.01.3.3 Effect of Varying the Added Filtration at a FixedKilovoltage

Once a given kilovoltage for a patient exposure has been

selected, the beam filtration should be such that the HVL is

greater than a minimum value, as prescribed by approved pro-

tocols and norms, in order to keep the radiation dose to the

patient as low as possible. For example, in the United States

under FDA regulations, for all x-ray systems except dental

systems designed for use in intraoral image receptors, a mini-

mum HVL of 2.5 mm Al is required for all tube voltages of

71 kVp or higher (Table 5). Figure 59 shows the modifications

of the spectral shape of the beam from a tungsten-anode x-ray

tube, at 80 kV, upon increasing in steps the added filtration

from 0.5 to 4 mm Al. The corresponding changes in the effec-

tive energy (Al) and in the HVL of the beam are shown in

Figure 60, indicating the spectral shift toward higher energies

(beam hardening).

A specific application of beam filtration is in x-ray imaging

of the compressed breast with a mammography setup. In this

case, with anode target materials as Mo or Rh often employed

in mammography tubes, filtration tends to decrease the brems-

strahlung continuum intensity at too-low and too-high ener-

gies without affecting much the intensity of target characteristic

lines, centered at photon energies (between 17 and 23 keV)

where the tissue contrast is optimal. For this purpose, K-edge

filtering is applied, namely, attenuation foils are inserted in

Figure 56

Figure 57

Photon energy (keV)

Added filtration mm AI

Pho

tons

keV

−1 c

m−2

mA

−1 s

−1

@ 1

m

100.0

1.2x107

8.0x106

1.0x107

4.0x106

6.0x106

2.0x106

20 30 40

4.03.02.01.00.5

50 60

W anode12 deg

70 80

Figure 59 x-Ray spectra at 80 kV tube voltage, with increasing amountof added filtration. Data are calculated with the code SpekCalc.

Tube voltage (kVp)

Mean energyEffective energy (AI)

Filtration: 2.5 mm AI + 1 m air

Mea

n or

effe

ctiv

e en

ergy

(keV

)

12 deg anode angle

2020

30

40

50

60

40 60 80 100 120 140 160

Tube voltage (kVp)(a)

(b)

(c)

Mean energyEffective energy (AI)

Filtration: 0.5 mm AI + 1 m airM

ean

or e

ffect

ive

ener

gy (k

eV)

12 deg anode angle

20

20

30

40

50

60

40 60 80 100 120 140 160

Tube voltage (kVp)

Mean energy

Effective energy (AI)

Parabolic fit(R2=0.99921, P<0.0001)

Total filtration: 2.5 mm AI + 0.25 mm Cu + 1 m air

Mea

n or

effe

ctiv

e en

ergy

(keV

)

2020

30

40

50

60

70

40 60 80 100 120 140 160

Figure 58 Mean energy and effective energy (in Al) for beams from anx-ray tube with tungsten anode, 12� anode angle, and filtration of 0.5 mmAl (a) or 2.5 mm Al (b), in addition to 1 m air filtration; the data pointshave been connected with a straight line for better showing the trend. (c)The filtration increased to 2.5 mm Alþ0.25 mm Cu, where the fitindicated by the continuous line shows the quadratic trend of the meanenergy versus kilovoltage. Data are calculated with the code SpekCalc.

Added filtration, mm AI0

01234

20

30

40

50

1 2 3 4

Effective energy (keV)X Mean energy (keV)

01234

50

40

30

20

80 kV

HV

L (m

m A

I)

Figure 60 Effective energy (Al), mean energy, and HVL for the x-rayspectra shown in Figure 59, showing beam hardening upon increasedadded filtration (mm Al).


ELSEVIE

R

the beam path at the tube output port, made of materials with

a K-shell threshold energy in the spectral region of interest:

for example, a molybdenum filter is used for a Mo anode

(so-called Mo/Mo anode/filter combination), or a rhodium

filter is used with a molybdenum (Mo/Rh) or with a rhodium

anode (Rh/Rh) (Figure 61(a) and 61(b)). In general radio-

graphy, or for breast imaging techniques for the uncompressed

breast, the use of K-edge filtration of tungsten spectra can be

investigated for selection of the optimal kilovoltage/filter

material for maximization of tissue contrast (e.g., Prionas

et al., 2011) (Figure 61(c)).

2.01.3.4 Effect of Varying the Tube Current and ExposureTime at Fixed Kilovoltage

At fixed tube potential and at a given filtration, the x-ray beam

quantity (the integral of the spectral intensity over the whole

photon energy range in the x-ray spectrum) varies with the tube

current I and exposure time t as predicted by eqn [111], that is,

the spectral intensity at each energy is multiplied by the value

of the product I∙t (mAs): this is shown in the example plot of

Figure 62, where an 80-kVp x-ray spectrum with 0.5 mm Al

filtration is calculated for I∙t¼50, 100, and 150 mAs. The area

under the curve gives the energy fluence at 1 m distance from

the source, for a corresponding calculated air kerma of

266 mGy (@50 mA s), 531 mGy (@100 mA s), and 797 mGy(@150 mA s).

Figure 59

Figure 58

Figure 60

Photon energy (keV)

Pho

tons

keV

−1 c

m−2

mA

−1 s

−1

@ 1

m

No air

1 m air

W anode80 kVp0.5 mm AI

00

2x106

4x106

6x106

8x106

1x107

10 20 30 40 50 60 70 80 90

Figure 63 The effect of air attenuation in the beam path between thesource and detector: spectra for an 80-kVp tube voltage, 12� anodeangle, 0.5 mm Al filtration, without or with 1 m air filtration. Data arecalculated with the code SpekCalc.

Photon energy (keV)

Filtration: 0.6 m air + 0.5 mm Be

Filtration: 0.6 m air + 0.5 mm Be

Added filtration:0 mm Rh (Emean= 15.19 keV)

0.025 mm Rh (Emean=18.21 keV)

Added filtration:

0 mm Mo(Emean= 15.23 keV)

0.030 mm Mo

(Emean= 16.78 keV)

Rh anode28 kVp

Mo anode28 kVp

Rel

ativ

e no

. pho

tons

Photon energy (keV)

Pho

tons

keV

−1 c

m−2

mA

−1

s−1

@ 1

m

Sn K-edge

0.5 mm AI +

0.5 mm AI + 1 m air W anode80 kVp

0.2 mm Sn

0

50.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e no

. pho

tons

0.0

0.2

0.4

0.6

0.8

1.0

10 15 20 25 30


(b)

(c)

5 10 15 20 25 30

0

2x107

1x107

8x106

6x106

4x106

2x106

10 20 30 40 50 60 70 80 90

Figure 61 The effect of K-edge filtration on x-ray tube spectra. (a) x-Rayspectra from a mammography x-ray tube with a molybdenum anode,operated at 28 kVp. The spectrum with no added filtration (thick line) has amean energy of about 15 keV, with significant contribution to beam fluencefrom low-energy photons. Upon filtration with a 0.030 mm molybdenumfoil (hence, the term Mo/Mo for anode/filter combination), the meanenergy increases to about 17 keV. (b) Mammography spectra obtainedwith a Rh/Rh combination at 28 kVp. (c) In general radiography, withtungsten anode tubes, K-edge filtration can be used for tissue contrastoptimization: here, an 80-kV spectrum (12� anode angle) is filtered with0.2 mm tin, with K-edge at about 29 keV (Table 3). Data are calculated withthe MASMIP spectral model (Boone et al., 1997) for Mo and Rh anodes,and with the code SpekCalc for W anode.

Photon energy (keV)

50 mAs100 mAs150 mAs

0.5 mm AI + 1 m air W anode80 kVp

00.0

5.0x108

Pho

tons

keV

−1 c

m−2

@ 1

m

1.0x109

1.5x109

2.0x109

2.5x109

10 20 30 40 50 60 70 80 90

Figure 62 The effect on the x-ray tube spectral shape from a change inthe product of tube current and exposure time (mAs) at fixed tubepotential, for an 80-kVp beam with 12� anode angle and 0.5 mm Alfiltration. Data are calculated with the code SpekCalc.


ELSEVIE

R

2.01.3.5 Effect of Filtration by Air in the Beam Line

In some spectra shown in the previous paragraphs, attenuation

by air in the beam path between x-ray source and detector was

included in the calculation.

Figure 63 shows the effect of including a 1 m air thick-

ness in the beam, at 80 kVp. Owing to air absorption (see

Figure 51), a slight decrease in spectral intensity is produced

at low energies. In this example, by considering air attenuation,

the calculated total (bremsstrahlungþcharacteristic) tube out-

put at 1 m distance decreases from 217 to 194 mGymA�1 s�1 and the mean energy goes from 36.2 to 36.7 keV,

with the HVL changing from 0.939 to 1.05 mm Al.

2.01.3.6 Beam Output at Varying Kilovoltages

Figure 64 shows how the beam output (mGy mA�1 s�1) variesat varying kilovoltage in the diagnostic energy range, for an

x-ray tube with 30� tungsten anode angle, with 2.5 mm Al

filtration (Figure 64(a)) or 2.5 mm Alþ0.25 mm Cu filtration

(Figure 64(b)). The total output is calculated by summing up

the contribution to exposure due to bremsstrahlung radiation

and that due to characteristic radiation, at 1 m of distance

from the source. The beam output shows a quadratic trend

with kVp, in agreement with eqn [111]. Visual comparison of

plots (a) and (b) in this figure shows that increased filtration

determines a higher curvature in the beam output versus tube

voltage curve.

Figure 63

Figure 61

Figure 62

Tube voltage (kVp)

Total output = (1)+(2)

Total filtration: 2.5 mm AI + 0.25 mm Cu + 1 m Air

Bremsstrahlung (1)

Parabolic fit

(R2=0.99995, P<0.0001)

Characteristic (2)

Bea

m o

utp

ut (µ

Gy

mA

−1

s−1 @

1 m

)

20

0

25

50

75

100

125

40 60 80 100 120 140 160

Tube voltage (kVp)(a)

(b)

Total output = (1)+(2)

Total filtration: 2.5 mm AI + 1 m Air

Bremsstrahlung (1)

Parabolic fit

(R2=0.9999, P<0.0001)

Characteristic (2)

Bea

m o

utp

ut (µ

Gy

mA

−1

s−1 @

1 m

)

20

0

50

100

150

200

250

40 60 80 100 120 140 160

Figure 64 The beam output (from bremsstrahlung, characteristic,and total radiation) from a tungsten-anode x-ray tube as a function oftube voltage. (a) 2.5 mm Al filtration; (b) 2.5 mm Alþ0.25 mm Cufiltration. The total output has been fitted with a quadratic trend(continuous lines), in agreement with eqn [111] in the text. Data arecalculated with the code SpekCalc.

Photon energy (keV)


No.

of p

hoto

ns m

m−2

keV

−1

per

1 m

Gy

air

kerm

a @

1m

0

4x105

6x105

2x105

20 30

100%5%10%1%

1%5%

10%100%

40 50 60

80 kVp

70 80

Figure 65 The effect of varying ripple levels on the shape of an 80-kVpx-ray spectrum, at fixed exposure. Data are calculated with the codeTASMIP (Boone and Seibert, 1997).

Photon energy (keV)


Pho

tons

keV

−1 c

m−2

mA

−1 s

−1

@ 1

m

100

4x106

3x106

2x106

1x106

20 30 40

7 deg10 deg12 deg14 deg16 deg

50 60

80 kVp

anode angle

70 80

Figure 66 The effect of varying anode target angle (from 7� to 16�) onthe shape of the spectrum from an x-ray tube, at 80 kVp and 2.5 mmAl filtration. Decreasing the target angle decreases the spectral intensityat low energies in the bremsstrahlung spectrum. The tube output at1 m decreases from 71 to 58 mGy mA�1s�1 in going from 16� to7� anode angle. Data are calculated with the code SpekCalc.


VIER

2.01.3.7 Effect of Voltage Ripple and of Target Angle

Ripple in the anode tube potential, that is, residual oscillations

in kilovoltage from the x-ray tube high-voltage generator mea-

sured in percent as the factor 100(Vmax�Vmin)/Vmax, indicates

the deviation from a constant-voltage operation of the x-ray

tube, producing a corresponding variation in tube output and

the maximum photon energies in the spectrum. While old-

fashioned single-phase x-ray generators have 100% ripple,

modern x-ray tube high-voltage generators employ high-

frequency inverter circuits that can reduce the ripple to 1% or

less, though ripple factors of several percent are associated with

triple-phase high-voltage generators. Voltage ripple can be eas-

ily introduced in calculations of empirical spectral models like

TASMIP (Boone et al., 1997).

Figure 65 shows an example of spectral changes due to

varying ripple levels, for an 80-kVp spectrum with 2.5 mm Al

filtration, operated at 1%, 5%, 10%, or 100%.

x-Ray tubes with different anode angles produce slightly

different spectra. Figure 66 shows calculated spectra at 80 kVp,

for varying target angles. A decrease (18%) in tube output and

an increase (3%) in beam average energy can be observed in the

calculated spectra, for decreasing target angle from 16� to 7�.

ELSE

2.01.3.8 Attenuation and Beam Hardening

Transmission of a polychromatic x-ray beam through an object

produces a modification of the spectral shape, with increasing

average photon energy for increasing traversed thickness.

This phenomenon of beam hardening is relevant to general

radiography since the spectrum of x-rays incident on the detec-

tor can be slightly yet sensibly changed with respect to the

incident spectrum; the effect is due to the energy-selective

attenuation of the beam introduced by the energy dependence

of the tissue attenuation coefficient. In CT imaging, beam

hardening produces an underestimation of the reconstructed

effective attenuation coefficient of tissues for inner regions in

the object (see Chapter 2.03, Section 5.1). To illustrate this

beam hardening effect with an example, Figure 67 shows

calculated spectra for an 80-kVp beam from an x-ray tube,

after transmission through a water slab of varying thickness,

from 25 to 150 mm. Upon increasing the material thickness in

the beam path, reduced transmission (i.e., decrease of the

spectral intensity at all photon energies in the beam) is evident

as well as an increase in the average energy due to weakening of

the low-energy part of the spectrum. This is also shown quan-

titatively in Figure 68(a), where the ratio of attenuated to

unattenuated beam is plotted against photon energy, and in

Figure 64

Figure 65

Figure 66

Water slab thickness (mm)0

28

32

36

40

Mea

n of

effe

ctiv

e en

ergy

(keV

)

44

48

52 80 kVp

Filtration: 2.5 mm AI + 1 m air56

25 50 75 100

Parabolic fits

Emean

Eeff (AI)

125 150

Photon energy (keV)10

(a)

(b)

0.0

0.1

0.2

0.3

Rat

io a

tten

uate

d/u

natt

enua

ted

spec

tral

inte

nsity

0.4

0.5

0.6 80 kVp

Filtration: 2.5 mm AI + 1 m air0.7

20 30 40 50

Water slabthickness

150 mm

100 mm

50 mm

25 mm

807060 90 100

Figure 68 From calculated spectra as in Figure 67, the following werederived: (a) the intensity ratio at each energy and the mean energy of thepolychromatic beam and (b) the corresponding effective energy as afunction of the thickness of the water slab through which the beam istransmitted. A quadratic least-squares fit is also shown for bothquantities. Increased mean (and effective) energy at increased slabthickness shows hardening of the polychromatic beam, due to anincreasingly efficient phenomenon of removal of low-energy photonsfrom the primary beam, as indicated by decreasing transmission at lowenergies.

Photon energy (keV)

Pho

tons

keV

−1 c

m−2

mA

−1 s

−1 @

1m

100

150

100

50

25

0

mm H2O


80 kVp

14 deganode angle

1x106

2x106

3x106

4x106

20 30 40 50 60 70 80

Figure 67 Spectral shape modification by transmission of an 80-kVpbeam (curve ‘0 mm H2O’) through a water slab of varying thickness, from25 to 150 mm. Data are calculated with the code SpekCalc.


ELSE

Figure 68(b), where the mean energy and the effective energy

(in Al) corresponding to the spectra in Figure 67 are plotted as

a function of the slab thickness; quadratic fits in this plot show

the approximate trend of this beam hardening effect, at 80 kVp

in water, for the range of thicknesses explored. Another way of

looking at beam hardening effects with polychromatic beams

in passing through thick objects is related to the behavior

previously shown in Figure 52.

When performing attenuation measurements via exposure

X(t) measurements with an ionization chamber and a homo-

geneous slab of thickness t made of a given material, the trend

of the logarithmic attenuation � ln[X(t)/X(t¼0)] as a function

of t is linear for thin objects (i.e., mt1, where m is the linear

attenuation coefficient of the material) and it deviates from

linearity at large thicknesses, with the slope of the curve

decreasing as the thickness increases. A calculation, following

eqn [115], was performed for the previous example of water

and an 80-kVp beam (Figure 69): the data points up to 15 mm

water thickness are well fitted by a linear trend (in the semilog

scale of this figure) whose slope is higher than the one

obtained by a linear fit to data points at slab thicknesses of

100 mm or larger.

In order to show the effect of such a phenomenon in digital

radiography imaging, Figure 70(a) shows the image (taken

with a CMOS flat panel detector with CsI:Tl scintillator layer)

of an aluminum alloy (Al-100) wedge in contact with the

detector, taken at 80 kVp; analogous images were recorded at

70, 60, and 50 kVp. The wedge introduces a spatially decreas-

ing absorption-based attenuation in the beam in the horizon-

tal direction in this Figure 70(a), due to decreasing absorber

thickness. The pixel values I were normalized to the corre-

sponding average value I0 in the absence of the object in the

beam, and a vertically-averaged horizontal line profile along

the direction of decreasing attenuation was drawn on images

at each kilovoltage; the horizontal axis was then scaled in order

to show Al-100 absorber thicknesses in the direction of the

beam. These profiles (Figure 70(b)), shown in a semilog plot,

are nonlinear and attain curvature at large thicknesses, thus

VIER

Water slab thickness, t (mm)

Linear fit, small t

Linear fit, large t


80 kVp

00.01

0.1

Att

enua

tion

X/X

0

1

20 40 60 80 100 120 140 160

Figure 69 For the 80-kVp x-ray spectrum and varying thickness t of awater slab, attenuation data X/X0 (calculated via eqn [114] in the text)versus t provide another graphical evidence of beam hardening: the slopeof the curve (in the semilog plot) decreases as the thickness increases. Aconstant slope over the data points would indicate absence of beamhardening, as in the case of a monoenergetic beam.

Figure 68

Figure 69

Figure 67

Photon energy (keV)

Cou

nts

mm

−2 m

A−1

s−1

keV

−1

@60

cm

5100

101

102

103

10

30 kVp

No PMMA

10 mm PMMA

20 mm PMMA

Mo anode1 mm AI

15 20 25 30 35

Figure 71 Spectra from a molybdenum anode x-ray tube (30 kVp,1 mm Al filtration) measured with a commercial CdTe detector (AMPTEKXR-100T-CdTe, 3�3 mm2) in air with and without insertion in thebeam path of a PMMA slab of thickness 10 or 20 mm. Few scatteredevents (at energies below �12 keV) and pileup events at energies above30 keV are detected. The three vertical arrows point to the value of themean energy of the corresponding spectrum (at 20.76, 21.49, and21.91 keV, respectively), increasing with increasing attenuation in thebeam.Aluminum thickness (mm)

Ave

rage

pix

el v

alue

I/I 0

00.09

0.1

0.2

0.3

0.4

0.5

0.60.70.80.9

1

(a)

(b)

1.00.80.60.40.2

10 20 30 40 50 60

50 kVp

60 kVp

70 kVp

80 kVp

70 80

Figure 70 (a) Digital radiography of a wedge made of an aluminumalloy (Al-100) at 80 kVp, with pixel values normalized to 1 at regionswhere there is no aluminum in the beam path. (b) Horizontal average lineprofile of the images at 80, 70, 60, and 50 kVp, plotted versus thecorresponding aluminum thickness in the beam. Curved trends in thissemilog plot show the effect of beam hardening.


showing the effect of beam hardening, less pronounced at low

kilovoltage (smaller spectral width) than at higher kilovoltage.

The analysis of these profiles, however, is complicated by the

presence of an energy-dependent absorption in the scintillator

layer of the detector.

Beam hardening is more pronounced for wider range of

photon energies in the incident spectrum (and is not present

for a purely monochromatic beam), so that reducing the x-ray

spectral width (e.g., by beam filtration) helps in reducing this

effect. In mammography, with spectral widths extending only

over a range of about 20 keV, a less evident beam hardening

effect is expected for soft-tissue attenuation, though breast

tissue exhibits a significant decrease in attenuation coefficient

from 10 to 30 keV (Figure 14(a)). Figure 71 shows measured

spectra from a Mo anode x-ray tube at 30 kVp, with additional

filtration of 1 mm Al, with and without insertion in the beam

path of a PMMA slab of 10 or 20 mm thickness; after traversing

20 mm of PMMA, the measured variation in the mean spectral

energy is just 1.2 keV (from 20.76 to 21.91 keV). An even

weaker effect can be observed with a Mo/Mo anode/filter

combination, for compressed breast thicknesses of several

centimeters.

ELSE

2.01.3.9 Focal Spot Size

The size, shape, and intensity distribution of the focal spot of

an x-ray tube have important effects on the quality of x-ray

images, in terms of 2D spatial resolution and level of spatial

coherence of the emitted radiation. In particular, image

blurring due to geometric unsharpness introduced by the finite

size of the x-ray source plays an important role in magnifica-

tion imaging (see, for instance, its impact in micro-CT imaging

as described in Chapter 2.09). x-Rays from an x-ray tube are

generated upon directing energetic electrons on a small portion

(typically less than a few square millimeters) of the target anode.

While the area of the surface of the target of the tube where x-rays

are emerging has an obvious interpretation as the x-ray source

area, indeed the ‘focal spot’ is definedas the area of x-ray emission

on the anode of the x-ray tube, as seen from the measuring device.

The focal spot properties are dependent on the x-ray tube voltage

and tube current, and its assessment implies careful indication of

the many details of the measurement setup. In principle, the

measurement of the focal spot can be performed on any plane

intercepting the x-ray beam, using ad hoc detectors andmeasure-

ment devices or employing the imaging receptor itself; in this last

case, the plane is coincident with the image plane.

In a radiographic system, the image receptor is an area

detector positioned at a certain distance from the x-ray source;

by considering various arbitrary positions (x, y) on the detector

surface in the x-ray field, one has from the above definition

that the apparent focal spot has properties (e.g., as size, shape,

and intensity distribution) which vary over the detector plane,

both in the x and in the y direction, these being perpendicular

to the anode–cathode axis and parallel to it, respectively. In

other words, at a given source-to-detector distance, the way in

which different portions of an extended object are irradiated by

the x-ray beam from the focal spot of an x-ray tube varies in

dependence of their relative position within the beam. This is

true also for the spectral properties of the incident beam, since

also the spectral shape of the x-ray beam varies (though in

limited quantitative terms) in dependence of the take-off

angle of bremsstrahlung and characteristic photons from the

surface of the x-ray tube. In particular, there is a slight beam

hardening effect in the cathode to anode direction, in addition

to a pronounced decrease of the angular intensity of the x-ray

VIER

Figure 71

Figure 70

Cathode Anode

336

349

362

375

388

Figure 72 Image of a flat field (i.e., a flood illumination of thedetector without any object in the beam path) of the 80-kVp beam froman x-ray tube obtained with a scintillator-based digital radiographydetector. The black-line overlay is the profile along the horizontal whiteline, showing decreased intensity of the beam from the cathode tothe anode direction (heel effect).


beam in the direction from the cathode to the anode due

to self-absorption in the target, a phenomenon known as

heel effect (Figure 72). Thus, the complex phenomenon of

x-ray generation from an x-ray tube introduces a great com-

plexity in the description of the spatially variant image quality

on the detector plane, due to the properties of the focal spot

and to the imaging geometry.

Generalized theory of geometric unsharpness in x-ray image

formation with an x-ray tube shows that the optical transfer

function, OTF(u,v), of the focal spot fully determines the so-

called field characteristics at any point (x,y) in the image plane

in terms of spatial frequencies u and v; specifically, it explains

the geometrical unsharpness effect.

As explained in the next chapter, the OTF is the Fourier

transform of the point spread function, PSF(x,y), which char-

acterizes the (spatially variant) intensity distribution of the

focal spot on the image plane as projected from a point in

the object plane; the modulus of the (complex) OTF is the

modulation transfer function, MTF(u,v). For simplicity, mea-

surement of the 2D PSF is often replaced by measurement of

1D line spread function, LSF(x) and LSF(y), as an approximate

representation of the PSF(x,y) as PSF(x,0) and PSF(0,y),

obtained, for example, by using a slit in the object plane and

orienting it along the anode–cathode direction (for focal spot

width assessment) and then perpendicular to it (focal spot

length), and then measuring the projected line intensities at a

position in the image plane. This implies that the 1D MTF(u)

and MTF(v) describe the geometric unsharpness at the mea-

surement point, in the spatial frequency domain. It can be

shown that once the PSF(x,y) is measured at the central beam

position in the image plane, the PSF at any field position can

be analytically derived from the central PSF, essentially from

geometrical considerations.

ELSE

The complexity of the geometric unsharpness determina-

tions is usually reduced by defining suitable metrics and con-

ventional rules that ease the task (and the related complexity

and cost of the measurement devices) for the full determina-

tion of the focal spot properties via the PSF(x,y). Among these,

a fundamental quantity is the size of the focal spot; as a crude

yet practical way of assigning the shape and size of the focal

spot, one assigns to it just two dimensions (focal spot length

and width, along the anode–cathode axis and along

the corresponding orthogonal direction, respectively). With

respect to the physical size of the source area on the target,

the effective focal spot length can be quite reduced (by a factor

siny) due to the introduction of angled anodes, where the

target surface is angled by 90þy� with respect to the anode–

cathode direction (line-focus principle). This implies a reduc-

tion of the focal spot length in the angular direction from the

cathode to the anode, while the focal spot width is left unaf-

fected by the target angle. As regards the imaging geometry,

focal spot measurement protocols usually fix the position on

the image plane where the measurements of the focal spot size

should be performed. By adopting a positioning scheme with

the x-ray tube anode-to-cathode axis parallel to the imaging

receptor plane (as is common in general radiography), the

cone-beam of x-ray emerging from the focal spot area on the

target (having an angular width limited by the target angle and

by the beam collimators) is characterized by a reference axis

(which bisects the beam cone angle) and by a central ray,

which is defined by the axis from the focal spot normal to

the receptor plane (Figure 73).

Measurement of the focal spot size can be performed along

the reference axis, which usually is made to intersect the imag-

ing receptor at the center of the detector-sensitive area. Note

that in case (as in many mammography setups) where the

anode-to-cathode axis has a tilt angle with respect to the imag-

ing plane, the central ray is not anymore along the reference

axis, and values of focal spot size from measurements per-

formed on the imaging plane along the reference axis should

be processed to take into account the tube tilt angle. As x-ray

optics, either a slit (e.g., 0.01 mm wide and a few millimeters

long) or a pinhole (e.g., 0.03 mm aperture diameter) is used in

the object plane, at a distance from the image plane in order to

exploit some image magnification (e.g., between 2� and 5�).Figure 74 shows the image of a focal spot from a mammo-

graphic x-ray tube evaluated at the central beam position, as

obtained using a pinhole with 0.025 mm aperture. The figure

shows how the length and width of the focal spot (nominal

size, 0.3 mm) are determined from the full width at half max-

imum (FWHM) value of the line profiles, with measurements

in practical agreement with nominal values. The complex

shape and intensity distribution of the focal spot in this

image are also evident in Figure 75, which refers to the focal

spot of the x-ray tube in a general radiography unit with a

nominal focal spot size of 0.6 mm.

A special case of focal spot sizemeasurements concernsmini-

focus andmicrofocus x-ray tubes. In the case of such fixed-anode

low-power tubes,multiple-aperture optics can be employed. The

shape, size, and intensity distribution of the focal spot of a 40-W

tube operated at constant voltage, with a nominal spot of

0.040 mm, determined at 35 kV with a multiple-pinhole optics,

are shown in Figure 76(a). The shape of the focal spot reveals a

VIER

Figure 72

Field of view(a) (b)

Image plane Image plane

Target angleTube tilt angle

Target angle

Central ray andreference axis

Referenceaxis

X-ray tube

q

q

Field of view

Figure 73 (a) Geometry of general radiography imaging with an x-ray tube; (b) the tube is tilted with respect to the image plane, so that the referenceaxis is not anymore normal to the image plane. Tube tilting coupled to small anode angles in tubes with biangular targets is commonly employed inmammography.

Vertical line profile

Horizontal line profile

Figure 74 Magnified (4.5�) pinhole image of the focal spot of a mammographic Mo–Mo x-ray tube, and corresponding vertical and horizontal centralline profiles. Nominal focal spot size 0.3 mm, technique factors 28 kV, 50 mAs; photostimulable imaging plate detector with 0.05 mm�0.05 mmpixel size. The full width at half maximum (FWHM) spot size was measured as 0.24 mm (width)�0.32 mm (length).


ELSEVIE

R

double-Gaussian intensity distribution with a central Gaussian

round spot with a 1D halo, which could be due to off-focal

radiation (Figure 77). The same focal spot imaged with a slit in

the anode–cathode axis and in the perpendicular direction pro-

duced LSF(x) and LSF(y) whose corresponding MTF(u) andMTF

(v) curves are shown in Figure 76(b), confirming the greater

geometrical unsharpness introduced in the direction of the spot

length for this tube.

Figure 78 shows the effect of the tube voltage on the shape

of the focal spot, at fixed tube current, for this 0.040-mm focal

spot. The focal spot size appears to vary with tube voltage in

one direction and to keep constant in the other direction, with

details of the tube electron optics and off-focal radiation gen-

eration possibly explaining the development of a focal spot

halo around a round central spot.

The finite size of focal spots has large effects on the spatial

resolution properties of the x-ray imaging system. As shown in

Figure 1, attenuation-based projection radiography with a

finite-size source introduces an image blur (geometrical

unsharpness) producing a penumbra region at the borders of

Figure 73

Figure 74

Figure 75 (a) Magnified (3�) pinhole image of the focal spot of a W anode x-ray tube, and (b) corresponding 3D representation of the intensitydistribution of the x-ray beam. Nominal focal spot size 0.6 mm, technique factors 80 kV, 50 mAs; photostimulable imaging plate detector with 0.1 mmpixel size. The FWHM spot size was measured as 0.74 mm (width)�0.75 mm (length).

Spatial frequency (mm-1)(b)

(a)

MTF

00.00.10.20.30.40.50.60.70.80.91.0

2 4 6 8 10

Focal spot lengthFocal spot width

80 kVp

12 14

0.1 mm

Figure 76 (a) Magnified (6�) image of the focal spot shown withhorizontal and vertical line profiles across its center, and (b)corresponding MTF curves evaluated in the direction perpendicular(width) and parallel (length) to the cathode–anode axis of the x-ray tube.The image of the focal spot was acquired with a multiple-pinholeoptics (coded aperture) and shows a bright round core with 0.099 mmFWHM diameter and a halo in the direction of the cathode–anode axis,which extends 0.2 mm across. The x-ray tube is a W target, fixed anodex-ray tube P/N 97007 by Oxford Instruments X-Ray Technology, Inc.(Scotts Valley, CA, USA) with a carbon fiber window and inherentfiltration of 1.8 mm Al, operated at 35 kV and 0.125 mA. The focal spotsize of 40 mm (manufacturer’s nominal value according to the IEC 336standard) was compatible with the above estimates given the differencesin the measurement methods. The MTF was derived from the modulusof the Fourier transform of the line spread function (LSF) of the focalspot, measured in projection imaging with a slit oriented along the lengthand width of the focal spot, respectively.


VIER

objects and internal details. By geometrical considerations, one

can easily derive that the image of a point object P at a distance

R1 from a uniform-intensity source of linear size F, projected

on the detector and a distance R2 from that point, is no longer a

point but has a fine size f given by

f ¼ FR2

R1¼ F M� 1ð Þ [124]

where M is the magnification factor for point P. Similarly, the

image of a sharp absorbing edge at P is blurred by a width f. If

the intensity profile of the (linear) source has a nonuniform

shape, for example, a Gaussian-like shape as in Figure 78, then

the image of a point has a Gaussian shape too, with a FWHM

value approximated by the value of f. The image of a sharp edge

would appear as a sigmoidal curve (edge spread function, ESF),

whose characteristic width is given by the FWHM of the deriv-

ative of the ESF, equal to the line spread function (LSF) of the

edge. This represents a degradation of the spatial resolution of

the imaging system (geometric unsharpness), whose amount

increases (i.e., f increases) with increasing focal spot size and

increasing magnificationM. Since x-ray tube focal spot is in the

order of a fraction of a millimeter, image magnification intro-

duces an undesirable image blur effect which is minimized by

placing the object as close as possible to the detector, so that

Mffi1 and fffi0 in eqn [124].

The finite size of focal spots has also important conse-

quences in phase-contrast x-ray imaging employing polychro-

matic beams from an x-ray tube. Most techniques for retrieving

the phase change map f(x,y,z) of objects irradiated with x-ray

beams require the use of partially coherent illumination of the

sample. Indeed, the incident waves should be at least partially

coherent, in order for interference effects to occur between e.m.

waves diffracted by the object contour or by spatial nonhomo-

geneities in its refractive index decrement d(x,y,z). Coherence at

ELSE

Figure 75

Figure 76

Lateral distance (nm)

vacuumtungsten

e− 80 keV12�

4200

4300 Dep

th in

tun

gste

n (n

m)

3100

1900

700

0

21000.0−2100−4200

Figure 77 Production of off-focal radiation. The plot shows the lateral view of 200 tracks of an 80-keV electron beam incident (with 12� angle and 1 mmlateral beam size) from vacuum on the surface of a tungsten target. While penetrating as much as a few micrometers in the tungsten target, afraction of the electrons are backscattered in the vacuum emerging backward in regions distant from the incidence beam. In an x-ray tube, theseelectrons would be directed again on the surface of the tungsten target by the high electric field between the cathode and anode, originating an off-focalradiation (simulation was performed with the code CASINO v. 2.42, Drouin et al., 2007).

Position (mm)

Focal spot length

80 kV50 kV35 kV

0.0(b)

(a)

0.0

0.2

0.4

Nor

mal

ized

inte

nsity

(a.u

.)

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0

Position (mm)

Focal spot width

80 kV50 kV35 kV

0.00.0

0.2

0.4

Nor

mal

ized

inte

nsity

(a.u

.)

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.0

Figure 78 LSF of the focal spot for the width (a) and length (b) of the0.040-mm focal spot shown in Figure 76, showing the change in thefocal spot shape and intensity upon varying the x-ray tube voltage (35,50, or 80 kV) at a fixed current of 0.125 mA. The curves are normalized tothe central height of the 80-kV curve. Magnification factor, 3.15�.


ELSE

a point in a radiation field refers both to the angular distribu-

tion of the wave field at that point (so-called spatial or lateral

coherence) and to the temporal structure of the radiation field

(referred to as temporal or longitudinal coherence), related to the

spectrum of frequencies (photon energies) of the source. High

temporal coherence implies high monochromaticity of the

source radiation. The degree of spatial coherence for a mono-

chromatic wave of wavelength l from an extended (angularly

incoherent) source, when the wavefront reaches a given point

P in space, is given by the Fourier transform of the angular

intensity distribution I(a) of the source, so it is related to its size

s and intensity distribution. Quantitatively, it can be calculated

by defining the complex quantity

lcoh ¼

ðI að Þei2pl sa daðI að Þda

[125]

Its modulus (the ‘fringe visibility’ in interference experi-

ments) has values between 0 and 1 (with value 1 assumed for

the case of monochromatic plane waves, and value 0 assumed

for the case of no visible interference effect due to lack of

coherence). lcoh is related to a quantity called lateral coherence

length Lcoh, the linear size of a region in space where the wave

field is strongly correlated, which can be calculated as

Lcoh ffi la¼ l

Dls

[126]

where a is the (small) angle subtended by the source of linear

size s (e.g., the focal spot size of the x-ray tube) from the point P

at a distance Dl from the source. Hence, using an x-ray tube as

the source of radiation, its coherence properties are highly

dependent on the size F of the focal spot; in clinical settings,

the focal spot (linear) size is in the order of 1 mm, and the

source-to-object distance, R1, is in the order of 1 m; the above

formula also holds in the case of polychromatic x-ray beams as

from an x-ray tube, and characterizing its spectrum by its

average wavelength �l, one has

Lcoh �lR1

F[127]

or, in terms of average spectral energy �E,

Lcoh ¼ hc�E

� �R1

F[128]

As a numerical example, at 50 keV average photon energy �E

(corresponding to an x-ray wavelength of 2.48�10�11m), one

VIER

Figure 77

Figure 78

Position (mm)

Nor

mal

ized

pix

el v

alue

0

A

(b)

(a)

A B C D E

B CD

E

0.94

0.96

0.98

1.00

10 20 30 40 50 60 70 80

Figure 80 (a) Contact digital radiography (i.e., with the detector planealmost coincident with the back surface of the object, R1þR2¼72 cm,magnification factor Mffi1) at 40 kV of a U-shaped PMMA frame,1.95-mm thick, on which thin (25 mm diameter) gold wires have beenwrapped up. Image regions marked with letters A–E indicate areascorresponding to PMMA, a wire, another wire, PMMA, and air,respectively. Pixel (50-mm pitch) values have been normalized to theaverage value in areas corresponding to air absorption. (b) Horizontal lineprofile along the white line indicated in the image in (a): absorption-basedintensity attenuation is evident in regions corresponding to areas A–Din the contact radiography. Note the regions, indicated by the arrows,where an increased signal is present at PMMA/air edges. The focal spotsize of the x-ray tube was 5 mm.

Focal spot size F:

0.01 mm

0.05 mm

0.1 mm

10

10−5

10−4

L coh

(mm

) 10−3

10−2

100Average photon energy (keV)

Lateral coherence length at R1= 1 m and focal spot size F

1 mm

Figure 79 Lateral coherence length Lcoh as a function of the averageenergy of the polychromatic beam from an x-ray tube, as calculated witheqn [126] using a source–object distance R1¼1 m and a varying focalspot size between 1 and 0.01 mm.


VIER
has Lcohffi2.48�10�5mm. Thus, the x-ray e.m. wave at 1 m
distance from a diagnostic x-ray tube may lose spatial coher-

ence over regions whose linear size is only about 25 nm. With

the above figures, in order to gain lateral coherence, even

impractically large source–object distances would be relatively

ineffective, since Lcoh would reach 0.5 mm only after the e.m.

wave travels a distance R1¼20 m. On the other hand, reducing

the focal spot size may increase the spatial coherence of the

radiation from x-ray tube sources.

Figure 79 shows the value of Lcoh at a distance of 1 m from

the focal spot of an x-ray tube having a varying size from

1 mm (as in a general radiography x-ray tube) to 0.1 mm

(minifocus source) down to 0.01 mm (microfocus source),

as a function of the average energy of the x-ray beam �E, as

given by eqn [128]. From Figure 79, it can be seen that

coherence length is higher at lower energy and that in the

diagnostic energy range, it is less than 1 mm. In this context,

breast imaging would potentially provide an important appli-

cation of phase-contrast imaging techniques employing par-

tially coherent sources, since in that case low-energy spectra

are used with mean energies at about 17–18 keV, and focal

spot sizes are typically as small as 0.3 mm (or 0.1 mm, as

employed in magnification mammography and in phase-

contrast mammography).ELS

E

2.01.4 Examples of x-Ray Image Formationand Contrast Mechanisms

In Section 2.01.4.1, a few examples of attenuation-contrast

and phase-contrast images are illustrated. These examples,

relying on simple physical objects of known materials, will

help the reader to better understand the practical aspects of

image formation according to different contrast mechanisms.

2.01.4.1 Attenuation Contrast (Absorption and Scattering)

Figure 80(a) shows the radiography of a sample test, acquired

in contact with a 50-mm-pitch flat panel digital detector at

80 kVp. The sample consists of a PMMA frame made out of a

Dz¼1.95-mm thick PMMA slab with highly polished edges

and surfaces. Thin gold wires (25 mm diameter) have been

wrapped up on the U-shaped PMMA frame, as thin absorbing

details.

The normalized, horizontal line profile shown in Figure

80(b) reveals that the PMMA slab introduces an average atten-

uation I/I0ffi0.945, from which the effective attenuation coef-

ficient of PMMA can be calculated as m¼�(1/Dz)ln(I/I0)ffi0.290 cm�1 corresponding to an effective beam energy

of 38 keV in PMMA. Such attenuation corresponds to an

image contrast between PMMA and air C¼(I�I0)/I0ffi5.5%,

and to an attenuation contrast between the wires and PMMA

Cffi1%. Owing to the limited spatial resolution of the detector

(about 0.16 mm FWHM, measured), the thin wires have a

width in this profile that is close to 0.16 mm, so that a signal

averaging effect due to detector undersampling produces an

apparent attenuation in the gold wire (I/I0ffi0.955 at the center

of the wire profile) which is much less than the one calculated

at 38 keV (I/I0¼0.555). By considering the rate of phase

change in PMMA, �df/dz¼35.237 rad mm�1 at 38 keV (see

Figure 21), the phase shift Df in the 1.95-mm thick PMMA

slab with respect to surrounding air can be calculated as

21.87p rad.On the other hand, an estimate of Df can be obtained

by using eqn [41]. From the measured attenuation � ln(I/I0¼0.945)ffi0.046 and from the value (¼2457.17) of the

ratio of d/b for PMMA at 38 keV (Figure 19), one obtains

Dfffi22.12p rad, in close agreement with the calculated value.

Figure 80

Figure 79


Figure 81(a) shows the phase map (i.e., the x,y distribution of

the phase advance Df introduced by the homogeneous slab

with respect to the corresponding propagation in free space):

the assumption of uniform d/b ratio in the object (i.e., assum-

ing that the object is made only of PMMA) allows one to apply

eqn [41] and to derive this image from the logarithm of the

corresponding attenuation map I/I0 shown in Figure 80(a).

Line profiles (along PMMA and air only, Figure 81(b), and

along PMMA, PMMAþwire, and air, Figure 81(c)) show that

the average phase shift (with respect to free space) is ffi70 rad

in PMMA with an additional ffi70 rad attributed to the gold

wires. Note that the image contrast between PMMA and air and

between the wires and PMMA in this reconstructed phase

image is increased with respect to the case of the attenuation

image, since correspondingly, d is about 2500 times higher

than b. The phase contrast between the gold wires and

PMMA is not the true one corresponding to phase change in

the wires, since the whole object was assumed a homogeneous

PMMA slab. Since at the average energy of 38 keV the ratio of

attenuation index between Au and PMMA isffi50:1, in terms of

PMMA attenuation thickness the 25-mm thick Au wires can be

thought of as being equivalent to 1.25 mm of PMMA, which

gives them an estimated phase change close to the value for the

20

0 20

A

B C

D

E

20

A

F G H I L

B C D E

M

0

-20

-40

-60

-80

Position (mm)40 60 80

0-20-40-60-80

-100-120-140-160

Position (mm)(c)

(b)

(a)

Pha

se s

hift

(rad

)P

hase

shi

ft (r

ad)

0 10

F G H I LM

20 30 40 50 60 70 80

Figure 81 (a) Retrieved phase map corresponding to the contactradiography in Figure 80(a), obtained using eqn [41] in the text. (b)Horizontal profile along the upper white line and (c) horizontal profile alongthe lower white line in (a). Letters (A–M) label regions indicated in the imagein (a). The large step in phase shift (about �70 rad) corresponds to thePMMA–air edge, numerically similar to the phase shift introduced by thegold wires as difference between��70 and ��140 rad.

ELSE

1.95-mm thick PMMA frame. This shows that even under the

rough approximation of homogeneity for a nonhomogeneous

sample, image contrast can be generated in phase imaging,

though the quantitative relationship between Df and the

material density r is lost (eqn [41]).

Figure 80(a) shows also other interesting features: the atten-

uation in air is not constant in regions close to the edges of the

slab, as pointed by the arrows in Figure 80(b); a slight increase

in the signal is present there, which can be attributed to the

contribution from Compton scattering in the PMMA slab.

Scattering at large angles, as represented by inelastic scatter-

ing at variance with elastic scattering, redirects incident photons

to lateral directions, and at the transition region between two

different materials (PMMA and air, in this case) this produces an

enhancement of the signal toward the side of the border region,

where the scattering attenuation mCDz is smaller, mC being the

Compton scattering attenuation coefficient at the average beam

energy. In the present case (with mC¼0.213 cm�1 at 38 keV in

the PMMA slab and mCDzffi0.0414), this side corresponds to

‘air’ regions with respect to ‘PMMA’ regions, hence the increased

signal visible in Figure 80. This redirection of rays from

the forward direction toward the less scattering side should

be accompanied by a decrease of the x-ray field intensity

on the more scattering side. Indeed, this is not apparent in

Figure 80 due to the image noise and to the thin (1.95 mm)

sample thickness, but imaging of a thicker (23.52 mm) PMMA

sample in exactly the same irradiation conditions as in that

figure does reveal this condition (Figure 82).

The large extent of the zone across the plate edge which is

seen in the radiography of Figure 82 as decreased or increased

intensity (about 4 cm across the sharp edge of the PMMA slab)

allows to attribute this effect to Compton scattering rather than

Rayleigh scattering in the slab, due to the low divergence of the

coherent scatter forward peaked intensity and the short dis-

tance (a few millimeters) between the rear slab surface and the

detector active surface. Moreover, the higher beam energy

(80 kV tube voltage rather than 40 kV as in Figure 81)

VIER

PMMA

10 mm

Air

Figure 82 Composite image showing the overlay, on the contactradiography of the sharp edge of a thick (23.52 mm) PMMA slabacquired at 80 kVp, of a horizontal profile drawn along the white line atthe center of the figure. The profile evidences the effect of scatteringin the slab, which produces a reduced intensity recorded for the slab (lefthalf of the figure) and a corresponding increased intensity recorded onthe side of air (right half of the figure). The dashed lines indicate theaverage intensity on the two sides of the slab away from the edge: thedifference between these intensity levels is due to absorption in thePMMA plate.

Figure 81

Figure 82


determines a higher contribution of Compton scattering in the

sample. This thick PMMA slab introduces an attenuation

(between the average values of the signal in the slab and

those in the air, away from the edge, indicated by the black

dashed lines in Figure 82) of I/I0ffi0.69. This attenuation

comes from absorption contrast between the PMMA plate

and the same thickness of air in the beam path. When sub-

tracting these average attenuation levels from the correspond-

ing intensity on the two sides of the edge, two half Gaussian

profiles are obtained with the same width of 26.2 mm FWHM,

which can be taken as an estimate of the width of the LSF for

scattering in the sample with the given x-ray spectrum.

2.01.4.2 Attenuation Plus Phase Contrast

The above examples of projection radiography were produced

with the detector in contact with the back object surface, that is,

with the image plane at a distance zl¼z0þDz from the source,

where Dz is the object thickness (Figure 1). Upon increasing

the distance R2 from the object to the detector (i.e., with

zl>z0þDz) while keeping the source–object distance fixed –

hence introducing magnification in the imaging technique if a

divergent x-ray source is adopted, as for x-ray tube focal spots –

one expects that Compton scattering intensity reaching the

detector reduces because of divergence of the beam in the air

gap beyond the object, and that Rayleigh scattering could

reveal some effects. Indeed, the angular distribution of the

differential cross section for Rayleigh scattering (Section

2.01.2.2) in an amorphous material like liquid water (and

biological tissues) shows an oscillating behavior with an abso-

lute maximum at a scattering angle of a few degrees (Johns and

Yaffe, 1983) and a minimum in the forward direction: at

60 keV photon energy, this maximum occurs at 3.8�, and at

20 keV it occurs at 11.3�. This oscillating behavior is related to

the interference of coherent scatter from electrons belonging to

different molecules, and it introduces a small-amplitude, low-

frequency modulation of the transmitted intensity distribution

on the image plane, at a distance R2>0 from the object.

Figure 1 schematizes the presence of refraction effects in

projection radiography, related to the distortion of x-ray wave-

fronts when traversing a nonhomogeneous object, irradiated

with monochromatic or polychromatic x-ray beams. Variation

in object thickness (i.e., in the direction of propagation z) and

the nonhomogeneity of the absorption index b in the sample

are responsible for attenuation contrast in the transmitted field

intensity and the phase shift effects introduced by the propa-

gation in the object with respect to propagation in air (or

coupling medium). On the other hand, in addition to varia-

tion in object thickness, local variations in the (x,y) spatial

distribution of the refractive index decrement d(x,y,z) due to,

for example, details with a lower or higher density than the

background density of the object produce perturbation of the

wavefronts and deviation (by a few arc seconds, in biological

tissues) in the direction of x-rays at the border of those non-

homogeneous regions. Slightly refracted waves propagate

through slightly different path lengths in the sample, thus

acquiring a phase difference Df(x,y) while propagating in the

direction z (Figure 1). This phenomenon introduces a phase

gradient (�@f/@x, �@f/@y) in the plane transverse to the

direction of propagation, whose modulus is proportional to

ELSE

the angular deviation of the x-rays undergoing refraction. The

phase differences of diffracted waves which propagate in the

region behind the object and downstream the x-ray beam

produce via interference effects a redistribution of the wave

intensity in a (transverse) image plane at a distance from the

object, which shows up as sharp (x,y) variations in the trans-

mitted intensity corresponding to the projected position of

nonhomogeneities in the object or variation in the thickness

of the object. In the mathematical description of this distribu-

tion (see Chapter 2.08), the intensity of the transmitted wave

exiting the object (eqn [50]) is ‘transported’ from the output

object plane to the image plane giving a lateral distribution T

(x,y) which contains dependences on both the projected atten-

uation profile and the Laplacian (@2f/@x2þ@2f/@y2) of the

projected phase change f(x,y) (eqn [47]).These effects can be

observed with monochromatic x-ray sources (e.g., from a syn-

chrotron radiation source) as well as with polychromatic

sources as from an x-ray tube, even in the near field (Fresnel

diffraction in the free space) (Wilkins et al., 1996). For a given

spatial resolution Dx of the imaging detector, one expects that

the extent of these effects depends, among other parameters,

on the distance R2 between the object and detector as a result of

the low refraction angles (a few arc seconds), on the average

wavelength �l, and on the spatial frequency 1/D introduced by

spatial nonuniformities of size D in the object. Wu and Liu

(2003) have introduced the parameter shear length, Lshear,

defined as

Lshear �lR1R2

R1 þ R2

1

D¼ �l

R2

M

1

D[129]

whereM is the magnification factor, for introducing a visibility

parameter for phase-contrast effect in the propagation-based

geometry. This figure of merit for phase-contrast visibility,

related to the coherence degree in the incident and

transmitted wavefields, is defined as (Wu and Liu, 2003):

LshearLcoh

¼�l R1R2

R1þR2

1D

�l R1

F

¼ R2F

R1 þ R2

1

D¼ M� 1ð ÞF

M

1

D[130]

For phase contrast of a nonhomogeneity of size D to be

visible, one should have Lshear/Lcoh1, and for Lshear/Lcoh<1,

one has a partially coherent wavefield.

Figure 83(a) shows details of the radiography (40 kV,

5 mm focal spot size, R1ffi7 cm, R1ffi65 cm, Mffi10,�l ffi 0:04592nm, �E ffi 27keV, detector pixel Dx¼50 mm) of

the sharp edge (transition from region D to region E) of

the PMMA frame whose contact radiography is shown in

Figure 80(a). In this case, Lcohffi0.6 mm and Lshearffi0.03 mm,

so that phase-contrast effect could be visible. Sharp dark and

white fringes are visible at the two sides of the PMMA edge

(as shown by the profile in Figure 83(b)), which reveal the

dependence of the recorded intensity on the second derivative

@2f/@y2 of the projected phase across the PMMA–air transi-

tion. However, the attenuation in the PMMA frame being not

negligible (as seen in the contact radiography of Figure 80(a)),

both attenuation and phase contrast are present in this radiog-

raphy. In principle, scaling and ‘subtraction’ of the contact

image from the phase-contrast image allow one to derive the

two separate information of the (projected) attenuation map

m(x,y) and phase map f(x,y) of the object. So-called phase

VIER

Distance (pixels)

0

(b)

(a)

0.74

0.76

0.78

0.80

100

Gra

y va

lue

200

A

300 400

Figure 83 (a) Detail of a magnification radiography (Mffi10) of thesharp edge of the PMMA edge shown in Figure 80(a), taken at 40 kV andwith a focal spot size of 5 mm. (b) The horizontal line profile (averagedover all lines in the image) reveals the presence of sharp variations in theintensity across the edge, as black and white fringes, which are thesignature of phase-contrast effects in this propagation-based geometry.The difference A between the average signal levels reflects the absorptionin the PMMA sample with respect to free space.


retrieval methods exist, as explained in Chapter 2.08 of this

volume, which under suitable approximations allow deriving

separately the absorption image and the phase image of the

sample, exploiting image acquisitions at two or even at just

one distance R2 from the detector.

Acknowledgments

The author wishes to thank Dr. G. Mettivier, S. Curion, M.

Quattrocchi, V. Capano for their assistance in the acquisition

of radiographies, and Prof. E. Perillo for his comments on parts

of the manuscript.SE

L
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ftp://ftp.aip.org

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http://www.esrf.fr

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http://www.gpo.gov

http://www.icr.ac.uk

http://www.kayelaby.npl.co.uk

http://www-nds.iaea.org

http://www.nist.gov

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