Biomedical Imaging II

1. Biomedical Imaging II Class 5 – Diffuse Optical Tomography (DOT). Part 1: Theory 03/06/06

2. Acknowledgment Dr. Ronald Xu Assistant Professor Biomedical Engineering Center Ohio State University Columbus, 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

3. 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]))

4. 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])) r dV = (dr)3 0

5. More 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.

6. 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.

7. 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

8. 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.

9. 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.

10. 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. An Intractable Case Object (tissue) is illuminated with near infrared (NIR) light (i.e., wavelengths between 750 nm and 1.2 μm). (What is photon energy?) 1) Is T strongly (and nonlinearly) dependent on x in this case? 2) What constitutes x?

11. Scattered and reflected ’ s s Scattered and absorbed mal, msl, g Scattered and transmitted How PhotonsInteract with Biological Tissue

12. 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 Quantitative Assessment of Absorption and Scattering Detector 3. An upper limit on the sample material’s µa can be computed from the difference between incident and detected light levels [From: J. W. Pickering, S. A. Prahl, et al., “Double-integrating-sphere system for measuring the optical properties of tissue,” Applied Optics32(4), 399-410 (1993).]

13. 1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µa, very low µs) Detector Quantitative Assessment of Absorption and Scattering 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 sumµa + µs

14. Scattering is Caused by Tissue Ultrastructure (http://omlc.ogi.edu)

15. Absorption is Caused by Multiple Chromophores

16. In NIR Region, Hb and HbO are Major Sensitive Absorber 4000 - Deoxy-hemoglobin 3500 - Oxy-hemoglobin 3000 l1 = 690nm 2500 extinct coeff (cm-1/mol/liter) 2000 l2 = 830nm 1500 1000 500 650 700 750 800 850 900 wavelength (nm)

17. 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 • …

18. 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

19. Concentration of X (M, mol-L-1) Molar extinction coefficient(cm-1M-1) Absorption coefficient of X(cm-1) 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]

20. Why do we want to know μa? μa = μa(Hb-oxy) + μa(Hb-deoxy) + μa(H2O) + μa(lipid) + μa(cyt-oxidase) + μa(myoglobin) + … 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

21. 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

22. fi fo source detector StO2B, HbtB StO2B, HbtB StO2T HbtT StO2T HbtT Near Infrared Diffuse Optical Imaging: Problem Definition • Find embedded tissue heterogeneity • By solving:

23. Theoretical 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).

24. Making 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)| << μa0(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.,

25. Making 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.

26. Making the Problem Tractable — Perturbation Strategy III • Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation:

27. log10(Intensity) μa μs Dilemma: Many different combinations of μa and μs are consistent with any given non-invasive light intensity measurement

28. (Cavernous hemangioma) Solution, Part 1: Few spatial distributions of μa and μs are consistent with many nearly simultaneous non-invasive light intensity measurement

29. 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 Solution, Part 2: Simplify mathematical problem by introducing an additional light-scattering medium into the mix

30. 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 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 Solution, Part 2: As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium) µs = 10 cm-1 µa = 0.06 cm-1

31. Use a computer simulation (or a homogeneous laboratory phantom) to derive the pattern of light intensity measurements around the reference medium boundary µs = 10 cm-1 µa = 0.06 cm-1 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”) Solution, Part 3: Linear perturbation strategy for image reconstruction

32. Each of these shades of gray represents a different number. Let’s write them all as a row vector. Because increasing μadecreases the amount of light that leaves the medium One number (weight) for each voxel Solution, Part 3: Linear perturbation strategy for image reconstruction

33. WEIGHT matrix Solution, Part 3: Linear perturbation strategy for image reconstruction Repeat process just described, for all source-detector combinations.

34. 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 Solution, Part 3: Linear perturbation strategy for image reconstruction Measurement perturbation (difference) is directly proportional to interior optical coefficient perturbation. Weight matrix gives us the proportionality.

35. Solution, Part 3: Linear perturbation strategy for image reconstruction 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.

36. Noise in data Noise image Real-world Issue 1: Coping with noise (random error) in clinical measurement data Linear system solutions are additive:

37. Real-world Issue 1: Coping with noise (random error) in clinical measurement data 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.