Biomedical Imaging II

BMI II SS06 – Class 5 “DOT Theory” Slide 1

Biomedical Imaging IIBiomedical Imaging II

Class 5 – Diffuse Optical Tomography (DOT). Part 1: Theory

03/06/06


AcknowledgmentAcknowledgment

Dr. Ronald XuDr. Ronald XuAssistant ProfessorAssistant Professor

Biomedical Engineering CenterBiomedical Engineering CenterOhio State UniversityOhio State University

Columbus, OhioColumbus, Ohio

Slides 11, 14-18, 21 and 22 in this presentation were created by Prof. Xu, and can be found in their original context at the following URL:

http://medimage.bmi.ohio-state.edu/resources/medimage_ws2005_Xu-image_workshop_2.16.05.ppt


What Are We Measuring?What Are We Measuring?

Input (source): s(rs,Ωs)

Output (measurement): d(rs,Ωs;rd,Ωd)

Constitutive property/ies (contrast): x(ri[,Ωi])

Transfer function: T(ri,Ωi)= T(x(ri[,Ωi]))


What Are We Measuring?What Are We Measuring?

Input (source): s(rs,Ωs)

Output (measurement): d(rs,Ωs;rd,Ωd)

Constitutive property/ies (contrast): x(ri[,Ωi])

Transfer function: T(ri,Ωi)= T(x(ri[,Ωi])) 0

rdV = (dr)3


More on Transfer FunctionMore on Transfer Function

• Strictly speaking, is a mathematical operator, not a function

– Maps one function into another function• Familiar examples: d/dx; multiply by x and add 2; ∫dx (i.e., indefinite integral)

– Different from a function (maps a number into another number) or a functional (maps a function into a number)

• Strictly speaking, a –function is actually a functional.

• T{s} d

– If medium is linear, then:

• i.e., overall effect of entire volume of material on the input is the summation of each volume element’s individual effects

– Nonlinearity makes problem of determining x(r) far more difficult

– We’re not home free even if medium is linear, given the dependence of T on x.

3, ; , , ,i i

s s d d i i s sd T r s d d rr Ω

r Ω r Ω r Ω


When Can We Solve for x(r)?When Can We Solve for x(r)?

• Most tractable case: W is separable

– i.e., T(x) = T0·x, or T(x) = T0·f(x).

• Also sometimes doable: W is not separable, but can be treated as if it were, for the purpose of computing a successive approximation sequence:

– T0 x1 T1 x2 T2 x3 ...

• In retrospect, it is easy to see why some types of medical imaging were successfully developed long before others, and why some produce higher–resolution images than others.


x–ray CT — Tractable or Not?x–ray CT — Tractable or Not?

Because we exclude the scattered photon component from the detectors, we have T0 = –functions, and f(x) = f(μ) = e -μ


Nuclear Imaging — Tractable or Not?Nuclear Imaging — Tractable or Not?

Besides collimation, we also have to deal with the attenuation phenomenon, which makes the problem non–separable

Successive approximation strategies have been employed with some success.


Ultrasound CT — Tractable or Not?Ultrasound CT — Tractable or Not?

Successive approximation strategy can be successfully employed when spatial variation of the acoustic impedance is weak.For highly heterogeneous (scattering) media, ultrasound CT may be possible if we can apply either the Born (i.e., negligible variation in ultrasound wave amplitude within scattering objects) or Rytov (i.e., negligible variation in ultrasound wave phase within scattering objects) approximation.


An Intractable Case An Intractable Case

The light spreads out in all directions from the point of illumination, similar to a droplet of ink in water diffusing away from its initial location.1) Is T strongly (and nonlinearly) dependent on x in this case? 2) What constitutes x?

Object (tissue) is illuminated with near infrared (NIR) light (i.e., wavelengths between 750 nm and 1.2 μm). (What is photon energy?)


How Photons Interact with Biological Tissue

s s ’

Scattered and reflected

Scattered and absorbed

Scattered and transmitted

a, s

, g


[From: J. W. Pickering, S. A. Prahl, et al., “Double-integrating-sphere system for measuring the optical properties of tissue,” Applied Optics 32(4), 399-410 (1993).]

Detector

1. Inner surfaces are coated with a bright, white, highly reflective material (very high µs, very low µa)

2. Eventually, all non-absorbed photons are captured by one or another of the detectors

3. An upper limit on the sample material’s µa can

be computed from the difference between incident and detected light levels

Quantitative Assessment of Absorption and Scattering


Detector

1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µa, very low µs)

2. Detector receives photons that are not removed from the incident beam, by either absorption or scattering

3. So, measuring the decrease of detected light as the slice thickness increases gives an estimate of the sum µa + µs

Quantitative Assessment of Absorption and Scattering


Scattering is Caused by Tissue Ultrastructure

(http://omlc.ogi.edu)


Absorption is Caused by Multiple Chromophores


In NIR Region, Hb and HbO are Major Sensitive Absorber

extin

ct c

oeff

(cm

-1/m

ol/l

iter)

wavelength (nm)

650 700 750 800 850 900

500

1000

1500

2000

2500

3000

3500

4000- Deoxy-hemoglobin

- Oxy-hemoglobin

1 = 690nm

2 = 830nm

2 1 1 2

1 2 2 1

1 2 2 1

1 2 2 1

Hb

HbO

HbO a HbO a

Hb HbO Hb HbO

Hb a Hb a

Hb HbO Hb HbO

2

[ ] [ ] [ ]

[ ]

[ ]

HbT Hb HbO

HbOSO

HbT


What Near Infrared Light Can Measure?

• Absorption measurement– Tissue hemoglobin concentration– Tissue oxygen saturation– Cytochrome-c-oxidase concentration– Melanin concentration– Bilirubin, water, glucose, …

• Scattering measurement– Lipid concentration– Cell nucleus size– Cell membrane refractive index change– …


Why Tissue Oximetry?

• Tissue oxygenation and hemoglobin concentration are sensitive indicators of viability and tissue health.

• Many diseases have specific effects on tissue oxygen and blood supply: stroke, vascular diseases, cancers, …

• Non-invasive, real time, local measurement of tissue O2 and HbT is not commercially available


Why do we want to know μa?

μa = μa(Hb-oxy) + μa(Hb-deoxy) + μa(H2O) + μa(lipid)

+ μa(cyt-oxidase) + μa(myoglobin) + …

μa(X, λ) = ε(X,λ)∙[X]

Concentration of X (M, mol-L-1)

Molar extinction coefficient(cm-1M-1)

Absorption coefficient of X(cm-1)


Rule: To get quantitatively accurate chromophore concentrations, the number of distinct wavelengths used for optical imaging must be at least as large as the number of compounds that contribute to the overall μa

μa = μa(Hb-oxy) + μa(Hb-deoxy) + μa(H2O) + μa(lipid)

+ μa(cyt-oxidase) + μa(myoglobin) + …

Why do we want to know μa?


Why Near Infrared? Pros and Cons Compared with Other Imaging Modalities

• Advantages:– Deep penetration into biological tissue– Non-invasive – Non-radioactive– Real time functional imaging– Portable– Low cost– Tissue physiological parameters – Potential of molecular sensitivity

• Disadvantages:– Low spatial and depth resolution– Hard to quantify


StO2B, HbtB

StO2T

HbtT

StO2B, HbtB

StO2T

HbtT

Near Infrared Diffuse Optical Imaging: Problem Definition

• Find embedded tissue heterogeneity• By solving:

i

source

o

detector

2 2ln ( , , , )B B T Tot b t b

i

OD f S O H t S O H t


Theoretical Descriptions of NIR Propagation Through TissueTheoretical Descriptions of NIR Propagation Through Tissue

• Quantum Electrodynamics

• Classical Electrodynamics (Maxwell’s equations)

• Radiation Transport Equation

• Diffusion Equation

– Assumes (among other things) that μs(r) >> μa(r).

4

, ,1, , , , , ,

source streaming absorption and

scattering out

, ,

scattering in

a s

s

tS t t t

c t

f t d

rr r r r r

r r

¶

¶

,1, , ,a

tD t t S t

c t

rr r r r r

¶

¶


Making the Problem Tractable — Perturbation StrategyMaking the Problem Tractable — Perturbation Strategy

• For a medium of known properties x 0(r) = {μa0(r), D 0(r)}, we can find the

transfer function to any desired degree of accuracy: T(x 0){s} = d 0.– We will refer to the above as our reference medium.

• What if an (unknown) target medium is different from the reference medium by at most a small amount at each spatial location?

– i.e., μa(r) = μa0(r) + Δμa(r), |Δμa(r)| << μa

0(r);

D (r) = D 0(r) + ΔD (r), |ΔD(r)| << D 0(r).

– Δμa(r) = absorption coefficient perturbation, ΔD(r) = diffusion coefficient perturbation

• Then the resulting change in d is approximately a linear function of the coefficient perturbations

– i.e.,

0 0 0 0

0 0 0 0

0 3

, ,

0 3

, ,

,

,

a a

a a

a aD DV

a a DV

a a DD D

d d d d D D d r

d d d W W D d r

W d W d D

¶ ¶ ¶ ¶

¶ ¶ ¶ ¶


Making the Problem Tractable — Perturbation Strategy IIMaking the Problem Tractable — Perturbation Strategy II

• In practice, medium is divided into a finite number N of pixels (“picture element” – 2D imaging) or voxels (“volume element” – 3D imaging)

• We further assume that each element is sufficiently small that there is negligible spatial variation of μa or D within it.

• Integral in preceding slide becomes a sum:

• Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation.

0 0 0 0

1

, ,,

a a

Nn n n n

a a Dn

na a n D nD D

d W W D

W d V W d D V¶ ¶ ¶ ¶


Making the Problem Tractable — Perturbation Strategy IIIMaking the Problem Tractable — Perturbation Strategy III

• Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation:

aa D

μd W W

D

1

2

11 12 1 11 12 11

21 22 2 21 22 22

1

1 2 1 2 2

a

aN N

a a a D D DNN Naa a a D D D

M M MN M M MNM a a a D D D

N

d W W W W W Wd W W W W W W

Dd W W W W W W D

D


Many different combinations of μa and μs are consistent with any given non-invasive light intensity measurement

2 4 6 8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1 -7

-6

-5

-4

-3

-2

-1

μs

μa

log10(Intensity)

Dilemma:Dilemma:


Few spatial distributions of μa and μs are consistent with many nearly simultaneous non-invasive light intensity measurement

(Cavernous hemangioma)

Solution, Part 1:Solution, Part 1:


Simplify mathematical problem by introducing an additional light-scattering medium into the mix

The problem of deducing the spatial distributions of μa and μs in this medium, from light intensity measure-ments around its border, is very difficult

Figuring out the difference between the spatial distribu-tions of μa and μs in these two media is much easier

-

=



As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium)

µs = 9 cm-1

µa = 0.05 cm-1

µs = 9 cm-1

µa = 0.07 cm-1

µs = 11 cm-1

µa = 0.05 cm-1

µs = 11 cm-1

µa = 0.07 cm-1

µs = 10 cm-1

µa = 0.06 cm-1

Δµs = -1 cm-1

Δµa = -0.01 cm-1

Δµs = -1 cm-1

Δµa = 0.01 cm-1

Δµs = 1 cm-1

Δµa = -0.01 cm-1

Δµs = 1 cm-1

Δµa = 0.01 cm-1



Linear perturbation strategy for image reconstruction

µs = 10 cm-1

µa = 0.06 cm-1

Use a computer simulation (or a homogeneous laboratory phantom) to derive the pattern of light intensity measurements around the reference medium boundary

Additional computer simulations determine the amount by which the detected light intensity will change, in response to a small increase (perturbation) in μa or μs in any volume element (“voxel”)




Each of these shades of gray represents a different number. Let’s write them all as a row vector.

Because increasing μa decreases the amount of light that leaves the medium

One number (weight) for each voxel




Repeat process just described, for all source-detector combinations.

WEIGHT matrix



Measurement perturbation (difference) is directly proportional to interior optical coefficient perturbation. Weight matrix gives us the proportionality.

µs = 9 cm-1

µa = 0.05 cm-1

µs = 9 cm-1

µa = 0.07 cm-1

µs = 11 cm-1

µa = 0.05 cm-1

µs = 11 cm-1

µa = 0.07 cm-1

µs = 10 cm-1

µa = 0.06 cm-1

Δµs = -1 cm-1

Δµa = -0.01 cm-1

Δµs = -1 cm-1

Δµa = 0.01 cm-1

Δµs = 1 cm-1

Δµa = -0.01 cm-1

Δµs = 1 cm-1

Δµa = 0.01 cm-1

R W X




Reconstructing image of μa and μs boils down to solving a large system of linear equations.

∆R and W are known, and we solve for the unknown ∆X

Formal mathematical term for this is inverting the weight matrix W.

1 R W X X W R




Coping with noise (random error) in clinical measurement data

Linear system solutions are additive:

1 1 11 2 1 2

1 2

W R R W R W R

X X

1 1 1

E

W R W R W

XNoise in data

Noise image

Real-world Issue 1:Real-world Issue 1:


In practice it can easily happen that E is larger than ∆X.

To suppress the impact of noise, mathematical techniques known as regularization are employed.

Real-world Issue 1:Real-world Issue 1:

Coping with noise (random error) in clinical measurement data


,

,

, ; , m ns da m n

ax y

Rw s d x y

¶

¶

-1 -110 cm , 0.06 cm

1/ 3 0.0331 cm

s a

s aD


,

,

, ; , m ns dD m n

x y

Rw s d x y

D

¶

¶

, , ,

, , ,

, ; , m n m ns d s d x ys m n

s x y x y s x y

R R Dw s d x y

D

¶ ¶ ¶

¶ ¶ ¶


(8 cm)(10.06 cm-1) = 80.48

Physical diameter / thickness

Total attenuation coefficient

Optical diameter / thickness

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