#### Transcript Biomedical Imaging I

Biomedical Imaging 2 Class 2 – Diffuse Optical Tomography (DOT) 01/22/08 BMI2 SS08 – Class 2 “DOT Theory” Slide 1 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 2 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])) BMI2 SS08 – Class 2 “DOT Theory” Slide 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])] r dV = (dr)3 0 BMI2 SS08 – Class 2 “DOT Theory” Slide 4 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: d rs , Ωs ; rd , Ωd ri 3 T r , s r , Ω d d r s s Ω i i i • 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. BMI2 SS08 – Class 2 “DOT Theory” Slide 5 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) = T0·x, or T(x) = T0·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: – 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. BMI2 SS08 – Class 2 “DOT Theory” Slide 6 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 7 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. BMI2 SS08 – Class 2 “DOT Theory” Slide 8 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. BMI2 SS08 – Class 2 “DOT Theory” Slide 9 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?) 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? BMI2 SS08 – Class 2 “DOT Theory” Slide 10 How Photons Interact with Biological Tissue Scattered and reflected s s’ Scattered and absorbed mal, msl, g Scattered and transmitted BMI2 SS08 – Class 2 “DOT Theory” Slide 11 Quantitative Assessment of Absorption and Scattering 1. Inner surfaces are coated with a bright, white, highly reflective material (very high µs, very low µa) Detector 2. Eventually, all nonabsorbed photons are captured by one or another of the detectors [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).] 3. An upper limit on the sample material’s µa can be computed from the difference between incident and detected light levels BMI2 SS08 – Class 2 “DOT Theory” Slide 12 Quantitative Assessment of Absorption and Scattering 1. Inner surfaces are coated with a dark, matte, highly absorptive material (very high µa, very low µs) Detector 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 13 Scattering is Caused by Tissue Ultrastructure (http://omlc.ogi.edu) BMI2 SS08 – Class 2 “DOT Theory” Slide 14 Absorption is Caused by Multiple Chromophores BMI2 SS08 – Class 2 “DOT Theory” Slide 15 In NIR Region, Hb and HbO are Major Sensitive Absorber 4000 - Deoxy-hemoglobin l2 l1 HbO m al1 HbO m al 2 Hb l1 l 2 l 2 l1 Hb HbO Hb HbO extinct coeff (cm-1/mol/liter) 3500 - Oxy-hemoglobin 3000 2500 l1 l 2 l 2 l1 Hb m a Hb m HbO l1 l 2 l 2 la1 Hb HbO Hb HbO l1 = 690nm 2000 l2 = 830nm 1500 [ HbT ] [ Hb] [ HbO ] 1000 SO2 500 650 700 750 800 850 [ HbO ] [ HbT ] 900 wavelength (nm) BMI2 SS08 – Class 2 “DOT Theory” Slide 16 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 … BMI2 SS08 – Class 2 “DOT Theory” Slide 17 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 18 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) Absorption coefficient of X (cm-1) Molar extinction coefficient (cm-1M-1) BMI2 SS08 – Class 2 “DOT Theory” Slide 19 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 20 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 21 Near Infrared Diffuse Optical Imaging: Problem Definition • Find embedded tissue heterogeneity • By solving: OD ln fo f (St O2B , H bt B , StO2T , H bt T ) fi source fi fo detector StO2B, HbtB StO2T HbtT BMI2 SS08 – Class 2 “DOT Theory” Slide 22 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 23 Example: Optical brain imaging • “Partial view” or back reflection geometry Source / Detector 1 Detector 2 2-3 cm Detector 3 Scalp Bone CSF Cortex BMI2 SS08 – Class 2 “DOT Theory” Slide 24 Time-Resolved Measurements Prompt or ballistic Photons (t = d/c) I “Snake” Photons I Diffuse Photons t t0 t d t0 • 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 BMI2 SS08 – Class 2 “DOT Theory” Slide 25 Frequency-Domain Measurements I I t t0 Photon density waves t t0 I I Modulation Phase t t t0 t0 • Propagation of photon density waves (PDW): lPDW = 9 cm, cPDW = 0.06 c (* • Measure PDW modulation (or amplitude) and phase delay • RF equipment (100MHz-1GHz) • Wave strongly damped, challenging measurement (* f = 200 MHz, μa = 0.1 cm-1, μs = 10 cm-1 n = 1.37 BMI2 SS08 – Class 2 “DOT Theory” Slide 26 Theoretical Descriptions of NIR Propagation Through Tissue • Quantum Electrodynamics • Classical Electrodynamics (Maxwell’s equations) • Radiation Transport Equation 1 ¶ r , ,t S r , ,t r , ,t ma r ms r r , , t c ¶t source streaming absorption and scattering out ms r f r , ,t d 4 scattering in • Diffusion Equation 1 ¶f r ,t D r f r ,t ma r f r ,t S r ,t c ¶t – Assumes (among other things) that μs(r) >> μa(r). BMI2 SS08 – Class 2 “DOT Theory” Slide 27 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., d d 0 ¶d ¶ma m V d d 0 d 0 a V ,D 0 ma ¶d ¶D m 0 ,D 0 D d 3r , a Wa ma WD D d 3r Wa ¶d ¶ma m 0 a ,D 0 , WD ¶d ¶D m 0 ,D 0 a BMI2 SS08 – Class 2 “DOT Theory” Slide 28 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: N d Wan man WDn D n n 1 Wan ¶d ¶ma m 0 a ,D 0 V n , WD ¶d ¶D m 0 ,D 0 V n a • Perturbation equations for all source/detector combinations are combined into a system of linear equations, or matrix equation. BMI2 SS08 – Class 2 “DOT Theory” Slide 29 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: d Wa d 1 Wa11 Wa12 d 21 22 W W 2 a a M1 d WaM 2 M Wa μa WD D Wa1N WD11 Wa2N WD21 WD12 WD22 WaMN WDM 1 WDM 2 ma1 2 ma WD1N N 2N WD ma D 1 MN 2 WD D N D BMI2 SS08 – Class 2 “DOT Theory” Slide 30 Dilemma: Many different combinations of μa and μs are consistent with any given non-invasive light intensity measurement log10(Intensity) 0.01 -1 0.02 -2 0.03 0.04 -3 μa 0.05 -4 0.06 0.07 -5 0.08 -6 0.09 0.1 -7 2 4 μs 6 8 10 BMI2 SS08 – Class 2 “DOT Theory” Slide 31 Solution, Part 1: Few spatial distributions of μa and μs are consistent with many nearly simultaneous noninvasive light intensity measurement (Cavernous hemangioma) BMI2 SS08 – Class 2 “DOT Theory” Slide 32 Solution, Part 2: 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 measurements around its border, is very difficult Figuring out the difference between the spatial distributions of μa and μs in these two media is much easier BMI2 SS08 – Class 2 “DOT Theory” Slide 33 Solution, Part 2: As a practical matter, most useful method is to use a spatially homogeneous second medium (i.e., reference medium) µs = 9 cm-1 µs = 9 cm-1 µa = 0.05 cm-1 µa = 0.07 cm-1 µs = 10 cm-1 µs = 11 cm-1 µa = 0.06 cm-1 µs = 11 cm-1 µa = 0.05 cm-1 µa = 0.07 cm-1 Δµs = -1 cm-1 Δµs = -1 cm-1 Δµa = -0.01 cm-1 Δµa = 0.01 cm-1 Δµs = 1 cm-1 Δµs = 1 cm-1 Δµa = -0.01 cm-1 Δµa = 0.01 cm-1 BMI2 SS08 – Class 2 “DOT Theory” Slide 34 Solution, Part 3: 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”) BMI2 SS08 – Class 2 “DOT Theory” Slide 35 Solution, Part 3: 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 μa decreases the amount of light that leaves the medium One number (weight) for each voxel BMI2 SS08 – Class 2 “DOT Theory” Slide 36 Solution, Part 3: Linear perturbation strategy for image reconstruction Repeat process just described, for all source-detector combinations. WEIGHT matrix BMI2 SS08 – Class 2 “DOT Theory” Slide 37 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. µs = 9 cm-1 µs = 9 cm-1 µa = 0.05 cm-1 µa = 0.07 cm-1 µs = 11 cm-1 µs = 11 cm-1 µs = 10 cm-1 µa = 0.06 cm-1 R WX µa = 0.05 cm-1 µa = 0.07 cm-1 Δµs = -1 cm-1 Δµs = -1 cm-1 Δµa = -0.01 cm-1 Δµa = 0.01 cm-1 Δµs = 1 cm-1 Δµs = 1 cm-1 Δµa = -0.01 cm-1 Δµa = 0.01 cm-1 BMI2 SS08 – Class 2 “DOT Theory” Slide 38 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. 1 R WX X W R BMI2 SS08 – Class 2 “DOT Theory” Slide 39 Real-world Issue 1: Coping with noise (random error) in clinical measurement data Linear system solutions are additive: W R1 R 2 W 1 1 1 R1 W R 2 X1 X 2 W R W 1 Noise in data 1 R W 1 X E Noise image BMI2 SS08 – Class 2 “DOT Theory” Slide 40 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. BMI2 SS08 – Class 2 “DOT Theory” Slide 41 w a s m , d n ; x, y ¶ Rsm ,dn ¶ ma x ,y ms 10 cm-1, ma 0.06 cm-1 D 1/ 3 ms ma 0.0331 cm BMI2 SS08 – Class 2 “DOT Theory” Slide 42 BMI2 SS08 – Class 2 “DOT Theory” Slide 43 BMI2 SS08 – Class 2 “DOT Theory” Slide 44 BMI2 SS08 – Class 2 “DOT Theory” Slide 45 BMI2 SS08 – Class 2 “DOT Theory” Slide 46 BMI2 SS08 – Class 2 “DOT Theory” Slide 47 BMI2 SS08 – Class 2 “DOT Theory” Slide 48 BMI2 SS08 – Class 2 “DOT Theory” Slide 49 BMI2 SS08 – Class 2 “DOT Theory” Slide 50 BMI2 SS08 – Class 2 “DOT Theory” Slide 51 BMI2 SS08 – Class 2 “DOT Theory” Slide 52 BMI2 SS08 – Class 2 “DOT Theory” Slide 53 BMI2 SS08 – Class 2 “DOT Theory” Slide 54 BMI2 SS08 – Class 2 “DOT Theory” Slide 55 BMI2 SS08 – Class 2 “DOT Theory” Slide 56 w D s m , d n ; x, y ¶ Rsm ,dn ¶ Dx ,y w s s m , d n ; x, y ¶ Rsm ,dn ¶ ms x ,y ¶ Rsm ,dn ¶ Dx ,y ¶ Dx ,y ¶ ms x ,y BMI2 SS08 – Class 2 “DOT Theory” Slide 57 BMI2 SS08 – Class 2 “DOT Theory” Slide 58 BMI2 SS08 – Class 2 “DOT Theory” Slide 59 BMI2 SS08 – Class 2 “DOT Theory” Slide 60 BMI2 SS08 – Class 2 “DOT Theory” Slide 61 Physical diameter / thickness (8 cm)(10.06 cm-1) = 80.48 Optical diameter / thickness Total attenuation coefficient BMI2 SS08 – Class 2 “DOT Theory” Slide 62