1. BMI2 SS07 – Class 2 “DOT Theory” Slide 1 Biomedical Imaging 2 Class 2 – Diffuse Optical Tomography (DOT) 01/23/07

2. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 3 What Are We Measuring? Input (source): s(r s,Ω s ) Output (measurement): d(r s,Ω s ;r d,Ω d ) Constitutive property/ies (contrast): x(r i [,Ω i ]) Transfer function: T(r i,Ω i ) = T(x(r i [,Ω i ]))

4. BMI2 SS07 – Class 2 “DOT Theory” Slide 4 What Are We Measuring? Input (source): s(r s,Ω s ) Output (measurement): d(r s,Ω s ;r d,Ω d ) Constitutive property/ies (contrast): x(r i [,Ω i ]) Transfer function: T(r i,Ω i ) = T[x(r i [,Ω i ])] 0 r dV = (dr) 3

5. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 6 When Can We Solve for x(r)? Most generally, T is influenced by x Most tractable case: T is medium-independent –i.e., T(x) = T 0 ·x, or T(x) = T 0 ·f(x). Also sometimes doable: T is not medium-independent, but can be treated as if it were, for the purpose of computing a successive approximation sequence: –T 0  x 1  T 1  x 2  T 2  x 3 ... 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 7 x–ray CT — Tractable or Not? Because we exclude the scattered photon component from the detectors, we have T 0 =  –functions, and f(x) = f(μ) = e -μ

8. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 10 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?)

11. BMI2 SS07 – Class 2 “DOT Theory” Slide 11 How Photons Interact with Biological Tissue s s ’ Scattered and reflected Scattered and absorbed Scattered and transmitted  a,  s, g

12. BMI2 SS07 – Class 2 “DOT Theory” Slide 12 [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 µ s, very low µ a ) 2. Eventually, all non- absorbed photons are captured by one or another of the detectors 3. An upper limit on the sample material’s µ a can be computed from the difference between incident and detected light levels Quantitative Assessment of Absorption and Scattering

13. BMI2 SS07 – Class 2 “DOT Theory” Slide 13 Detector 1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µ a, very low µ s ) 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 µ a + µ s Quantitative Assessment of Absorption and Scattering

14. BMI2 SS07 – Class 2 “DOT Theory” Slide 14 Scattering is Caused by Tissue Ultrastructure (http://omlc.ogi.edu)

15. BMI2 SS07 – Class 2 “DOT Theory” Slide 15 Absorption is Caused by Multiple Chromophores

16. BMI2 SS07 – Class 2 “DOT Theory” Slide 16 In NIR Region, Hb and HbO are Major Sensitive Absorber extinct coeff (cm-1/mol/liter) wavelength (nm) 650700750800850900 500 1000 1500 2000 2500 3000 3500 4000 - Deoxy-hemoglobin - Oxy-hemoglobin 1 = 690nm 2 = 830nm

17. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 19 Why do we want to know μ a ? μ a = μ a (Hb-oxy) + μ a (Hb-deoxy) + μ a (H 2 O) + μ a (lipid) + μ a (cyt-oxidase) + μ a (myoglobin) + … μ a (X, λ) = ε(X,λ)∙[X] Concentration of X (M, mol-L -1 ) Molar extinction coefficient (cm -1 M -1 ) Absorption coefficient of X (cm -1 )

20. BMI2 SS07 – Class 2 “DOT Theory” Slide 20 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 μ a μ a = μ a (Hb-oxy) + μ a (Hb-deoxy) + μ a (H 2 O) + μ a (lipid) + μ a (cyt-oxidase) + μ a (myoglobin) + … Why do we want to know μ a ?

21. BMI2 SS07 – Class 2 “DOT Theory” Slide 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. BMI2 SS07 – Class 2 “DOT Theory” Slide 22 S t O 2 B, H b t B StO2THbtTStO2THbtT StO2THbtTStO2THbtT Near Infrared Diffuse Optical Imaging: Problem Definition Find embedded tissue heterogeneity By solving: ii source oo detector

23. BMI2 SS07 – Class 2 “DOT Theory” Slide 23 Continuous Wave (C.W.) Measurements Simplest form of OT: lowest spatial resolution, “easy” implementation, greatest penetration –Measuring transmission of constant light intensity (DC) –Simple, least expensive technology  most S-D pairs –High “frame rates” possible

24. BMI2 SS07 – Class 2 “DOT Theory” Slide 24 Example: Optical brain imaging “Partial view” or back reflection geometry Scalp Bone Cortex CSF 2-3 cm Source / Detector 1 Detector 2 Detector 3

25. BMI2 SS07 – Class 2 “DOT Theory” Slide 25 Time-Resolved Measurements Measuring the arrival time/temporal spread of short pulses (<ns) due to scattering & absorption (narrowing the “banana”) Expensive, delicate hardware (single-photon counters, fast lasers, optical reflections, delays…) Long acquisition times (low frame rates) Potentially better spatial resolution than DC measurements t I t0t0 t I t0t0 Prompt or ballistic Photons (t = d/c) d “Snake” Photons Diffuse Photons

26. BMI2 SS07 – Class 2 “DOT Theory” Slide 26 Frequency-Domain Measurements Propagation of photon density waves (PDW): PDW = 9 cm, c PDW = 0.06 c ( * Measure PDW modulation (or amplitude) and phase delay RF equipment (100MHz-1GHz) Wave strongly damped, challenging measurement t I t0t0 t I t0t0 t I t0t0 Photon density waves t I t0t0 Phase Modulation ( * f = 200 MHz, μ a = 0.1 cm -1, μ s = 10 cm -1 n = 1.37

27. BMI2 SS07 – Class 2 “DOT Theory” Slide 27 Theoretical Descriptions of NIR Propagation Through Tissue Quantum Electrodynamics Classical Electrodynamics (Maxwell’s equations) Radiation Transport Equation Diffusion Equation –Assumes (among other things) that μ s (r) >> μ a (r).

28. BMI2 SS07 – Class 2 “DOT Theory” Slide 28 Making the Problem Tractable — Perturbation Strategy For a medium of known properties x 0 (r) = {μ a 0 (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., μ a (r) = μ a 0 (r) + Δμ a (r), |Δμ a (r)| << μ a 0 (r); D (r) = D 0 (r) + ΔD (r), |ΔD(r)| << D 0 (r). –Δμ a (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.,

29. BMI2 SS07 – Class 2 “DOT Theory” Slide 29 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 μ a 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.

30. BMI2 SS07 – Class 2 “DOT Theory” Slide 30 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:

31. BMI2 SS07 – Class 2 “DOT Theory” Slide 31 Many different combinations of μ a and μ s are consistent with any given non-invasive light intensity measurement μsμs μaμa log 10 (Intensity) Dilemma:

32. BMI2 SS07 – Class 2 “DOT Theory” Slide 32 Few spatial distributions of μ a and μ s are consistent with many nearly simultaneous non- invasive light intensity measurement (Cavernous hemangioma) Solution, Part 1:

33. BMI2 SS07 – Class 2 “DOT Theory” Slide 33 Simplify mathematical problem by introducing an additional light-scattering medium into the mix The problem of deducing the spatial distributions of μ a and μ s in this medium, from light intensity measure- ments around its border, is very difficult Figuring out the difference between the spatial distribu- tions of μ a and μ s in these two media is much easier - = Solution, Part 2:

34. BMI2 SS07 – Class 2 “DOT Theory” Slide 34 As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium) µ s = 9 cm -1 µ a = 0.05 cm -1 µ s = 9 cm -1 µ a = 0.07 cm -1 µ s = 11 cm -1 µ a = 0.05 cm -1 µ s = 11 cm -1 µ a = 0.07 cm -1 µ s = 10 cm -1 µ a = 0.06 cm -1 Δµ s = -1 cm -1 Δµ a = -0.01 cm -1 Δµ s = -1 cm -1 Δµ a = 0.01 cm -1 Δµ s = 1 cm -1 Δµ a = -0.01 cm -1 Δµ s = 1 cm -1 Δµ a = 0.01 cm -1 Solution, Part 2:

35. BMI2 SS07 – Class 2 “DOT Theory” Slide 35 Linear perturbation strategy for image reconstruction µ s = 10 cm -1 µ a = 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 μ a or μ s in any volume element (“voxel”) Solution, Part 3:

36. BMI2 SS07 – Class 2 “DOT Theory” Slide 36 Linear perturbation strategy for image reconstruction Each of these shades of gray represents a different number. Let’s write them all as a row vector. Because increasing μ a decreases the amount of light that leaves the medium One number (weight) for each voxel Solution, Part 3:

37. BMI2 SS07 – Class 2 “DOT Theory” Slide 37 Linear perturbation strategy for image reconstruction Repeat process just described, for all source-detector combinations. WEIGHT matrix Solution, Part 3:

38. BMI2 SS07 – Class 2 “DOT Theory” Slide 38 Measurement perturbation (difference) is directly proportional to interior optical coefficient perturbation. Weight matrix gives us the proportionality. µ s = 9 cm -1 µ a = 0.05 cm -1 µ s = 9 cm -1 µ a = 0.07 cm -1 µ s = 11 cm -1 µ a = 0.05 cm -1 µ s = 11 cm -1 µ a = 0.07 cm -1 µ s = 10 cm -1 µ a = 0.06 cm -1 Δµ s = -1 cm -1 Δµ a = -0.01 cm -1 Δµ s = -1 cm -1 Δµ a = 0.01 cm -1 Δµ s = 1 cm -1 Δµ a = -0.01 cm -1 Δµ s = 1 cm -1 Δµ a = 0.01 cm -1 Linear perturbation strategy for image reconstruction Solution, Part 3:

39. BMI2 SS07 – Class 2 “DOT Theory” Slide 39 Reconstructing image of μ a and μ s 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. Solution, Part 3: Linear perturbation strategy for image reconstruction

40. BMI2 SS07 – Class 2 “DOT Theory” Slide 40 Coping with noise (random error) in clinical measurement data Linear system solutions are additive: Noise in data Noise image Real-world Issue 1:

41. BMI2 SS07 – Class 2 “DOT Theory” Slide 41 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: Coping with noise (random error) in clinical measurement data

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62. BMI2 SS07 – Class 2 “DOT Theory” Slide 62 (8 cm)(10.06 cm -1 ) = 80.48 Physical diameter / thickness Total attenuation coefficient Optical diameter / thickness