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Overview Optical Tomographic Imaging of Small Animals • Introduction X-Ray Tomography vs Optical Tomography • Model-based iterative image reconstruction Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system Andreas H. Hielscher, Ph.D. • Applications Brain Imaging Tumor Imaging Fluorescence Imaging Columbia University, New York City Dept. of Biomedical Engineering Dept. of Radiology Overview X-Ray Imaging Uses X-rays to generate shadowgrams M(ϕ,ξ). • Introduction Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system • Applications Brain Imaging Tumor Imaging Fluorescence Imaging (measurable attenuation) unknown absorption cross-section A(x,y) X-ray source • Model-based iterative image reconstruction M(ϕ,ξ) X-Ray Tomography vs Optical Tomography electromagnetic wave λ~10-10 m energy~104eV Energy propagates on straight lines through medium 1 M(ϕ,ξ) X-ray source X-Ray Tomography X-Ray Shadowgram X-Ray Tomography X-Ray Tomography Xra y so ur ce M (ϕ ,ξ) 2 Xra y so ur ce M (ϕ ,ξ ) X-Ray Tomography 2D Scan of Head unknown absorption cross-section M(ϕ,ξ) X-ray source A(x,y) =>Simple image reconstruction scheme: backprojection of M on lines of transmission. (Inverse Radon Transform) Optical Imaging Optical Shadowgram Uses near-infrared light (700< λ<900nm) A(x,y) {unknown absorption & scattering profile} light source EM - wave λ ~ 800•10-9m energy ~ 1 eV Energy does not propagate on straight line between source and detector (light is strongly scattered) 3 Optical Tomography light source Optical Tomography light source Optical Tomography Optical Tomography light source light source 4 Overview Optical Imaging Uses near-infrared light (700< λ<900nm) • Introduction X-Ray Tomography vs Optical Tomography A(x,y) {unknown absorption & scattering profile} • Model-based iterative image reconstruction light source EM - wave λ ~ 800•10-9m energy ~ 1 eV Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system • Applications Brain Imaging Tumor Imaging Fluorescence Imaging How to reconstruct cross-sectional images A(x,y) from measurement on surface? (Inverse Problem) initial guess D= 1 cm2n s Forward Model I detectors ? Theory: sources Experiment detectors sources Model-Based Iterative Image Reconstruction Forward Model, F ( ) depends on NxN unkowns measured detector readings I M,i predicted detector reading I P,i( 3D-Time-Resolved Diffusion Equation ∂U = ∂ D ∂U + ∂ D ∂U + ∂ D ∂U - cµaU + S ∂z ∂x ∂y ∂y ∂z ∂t ∂x ) with c := speed of light in medium, S = Source, and diffusion coefficient : D = c ( 3 [ µa + µs' ] ) with µ a = absorption coefficient and µ s' = reduced scattering coefficient . 5 Diffusion vs Transport Model Limits of Diffusion Model laser beam ring filled with water ∂U = ∇c/(3µ +3µ ') ∇U - cµaU + S a s ∂t discretize into N spacial variables leads to N finite-difference equations milk ∫ discretization into N spacial and A angular variables leads to N x A coupled finite-difference equations 1.5 Diffusion 1 Experiments 0.5 0 5 10 15 1 Experiments 0.8 0.6 0.4 Transport 0.2 0 slower by factor ~A 1.2 20 25 30 35 40 Transport 0 5 10 15 y [mm] Forward Model applied to Mouse Head 20 x [mm] 25 30 35 40 Model-Based Iterative Image Reconstruction sources Experiment ~ 1 cm ? Theory: initial guess D= 1 cm2n s detectors 4π µs' = (1-g) µs 1.4 sources 4π and Diffusion 1.6 2 detectors ∫ with U = Ψ(Ω') dΩ' 1.8 2.5 Intensity [au] equation of radiative transport ∂Ψ/c∂t = S - Ω ∇Ψ - (µa + µs)Ψ + Ψ(Ω') p(Ω∗Ω') dΩ' Intensity [au] approximation diffusion equation Forward Model, F ( ) depends on NxN unkowns log (Fluence [Wcm -2]) measured detector readings I M,i predicted detector reading I P,i( ) source µa=0.1 cm -1 , µs =10 cm -1 ; 14781 nodes, 24 ordinates 6 ? Theory: new guess Forward Model, F ( ) e.g. transport equation predicted detector reading I P,i( measured detector readings I M,i Analysis Scheme Φ ≈ { I M,i - I P,i( Σ ) Forward Model, F ( ) e.g. transport equation Analysis Scheme Φ ≈ { I M,i - I P,i( )}2 i ) Error Value Φ ( no Model-Based Iterative Image Reconstruction Experiment sources new guess ? Theory: new guess Forward Model, F ( ) e.g. transport equation measured detector readings I M,i predicted detector reading I P,i( Analysis Scheme Φ ≈ { I M,i - I P,i( Σ )}2 i Updating Scheme Model-Based Iterative Image Reconstruction sources Theory: detectors sources Experiment no Φ<ε detectors Φ<ε ) ) sources Error Value Φ ( Forward Model, F ( ) e.g. transport equation measured detector readings I M,i predicted detector reading I P,i( Analysis Scheme Φ ≈ { I M,i - I P,i( Σ ) )}2 i Error Value Φ ( yes ) )}2 Σ (This is just one number!) ? predicted detector reading I P,i( measured detector readings I M,i i yes sources Experiment detectors D= 1 cm2n s sources initial guess Model-Based Iterative Image Reconstruction detectors ? Theory: detectors sources Experiment sources Model-Based Iterative Image Reconstruction Φ<ε ) no Error Value Φ ( Φ<ε ) no Updating Scheme 7 Experiment ? Theory: detectors sources ? new guess Model-Based Iterative Image Reconstruction sources Theory: detectors sources Experiment new guess Forward Model, F ( ) e.g. transport equation measured detector readings I M,i predicted detector reading I P,i( Analysis Scheme Φ ≈ { I M,i - I P,i( Σ Forward Model, F ( ) e.g. transport equation measured detector readings I M,i ) )}2 Σ ) )}2 i Error Value Φ ( yes predicted detector reading I P,i( Analysis Scheme Φ ≈ { I M,i - I P,i( i final sources Model-Based Iterative Image Reconstruction ) Error Value Φ ( final yes Φ<ε Iteration Example Φ<ε ) no Updating Scheme Iterative Reconstruction Initial Guess: D = 1.0 cm 2ns -1 Detector Source 2nd Iteration 8th Iteration D [cm/ns 2] D [cm/ns 2] 8 cm Intensity 0 predictions Time Steps (Δt = .05 ns) 50 7 0 measurements 7 7 predictions 0 Time Steps 50 homogeneous initial guess (D = 1 cm 2ns-1) 0.5 0.5 measurements 24th Iteration 1.5 1.5 0 0 Time Steps 50 0 0 Time Steps 50 iteratively change properties of medium until measurements and predictions agree 4 cm 8 Image Reconstruction as an Optimization Problem Data Analysis Scheme Find image for which error value is smallest ! objective error function Φ(D,µa) Measurement Data Y Predicted data U (Ysdt - Usdt (µa,D))2 Φ(µa,D) = 2σ2sdt s d t Contour plot of Φ(D,µa) ΣΣΣ Objective Function µa D Gradient Path Conjugate Gradient Path = χ2 Error Function Goal : Find minimum of Φ(µa,D) Employ minimization technique that uses information about gradient dΦ(µa,D) . d(µa,D) each image = 40x40 unknowns Gradient Calculation Divided Difference Gradient Calculation Divided Difference 1 variable: 2 forward calculations needed to get gradient 1 variable: 2 forward calculations needed to get gradient ∂f(ζx) f(ζ2)- f(ζ1) ∂ζ = ζ2 - ζ1 ∂f(ζx) f(ζ2)- f(ζ1) ∂ζ = ζ2 - ζ1 f(ζ1) f(ζx) f(ζ2) ζ1 ζx ζ2 Therefore, For problem with N unknowns one needs 2N forward calculations to find gradient. f(ζ1) f(ζx) f(ζ2) ζ1 ζx ζ2 Therefore, For problem with N unknowns one needs 2N forward calculations to find gradient. Adjoint Differentiation The evaluation of a gradient requires never more than five times the effort of one forward calculation! A. Griewank, “On Automatic Differentiation,” in Mathematical Programming, M. Iri, K. Tanabe, eds., Kluwer Academic Publishers, 1989, pp.83-107. Therefore, adjoint differentiation method is 2N/5 times faster than ”traditional” divided difference scheme! 9 For more details see: Overview G. Abdoulaev, K. Ren, A.H. Hielscher, "Optical tomography as a constrained optimization problem,” accepted for publication in Inverse Problems. K. Ren, G. Abdoulaev, G. Bal, A.H. Hielscher, "Frequency-domain optical tomography based on the equation of radiative transfer,” accepted for publication in SIAM Journal of Scientific Computing. K. Ren, G. Abdoulaev, G. Bal, A.H. Hielscher, "An algorithm for solving the equation of radiative transfer in the frequency domain," Optics Letters 29(6), pp. 578-580 (2004). G. Abdoulaev and A.H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," Journal of Electronic Imaging 12(4), pp. 594-60 (2003). A.H. Hielscher, A.D. Klose, U. Netz, J. Beuthan, "Optical tomography using the timeindependent equation of radiative transfer. Part 1: Forward model," Journal of Quantitative Spectroscopy and Radiative Transfer, Vol 72/5, pp. 691-713, 2002. A.D. Klose, A.H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 2: Inverse model," Journal of Quantitative Spectroscopy and Radiative Transfer, Vol 72/5, pp. 715-732, 2002. A.D. Klose and A.H. Hielscher, "Iterative reconstruction scheme for optical tomo-graphy based on the equation of radiative transfer," Medical Physics, vol. 26, no. 8, pp. 1698-1707, 1999. A.H. Hielscher, A.D. Klose, K.M. Hanson, "Gradient-based iterative image recon-struction scheme for time-resolved optical tomography," IEEE Transactions on Medical Imaging 18, pp. 262-271, 1999. www.bme.columbia.edu/biophotonics Optical Imaging Modalities STEADYSTATE DOMAIN 100k X-ray vs optical tomography • Model-based iterative image reconstruction Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system • Applications Brain Imaging Tumor Imaging Fluorescence Imaging Frequency vs Steady-State Domain 1 image /min data acquisition rate information content FREQUENCY DOMAIN complexity/price of system 1M TIME DOMAIN • Introduction 10 images /sec target steady-state frequency domain domain reconstruction reconstruction (ω = 0) (ω = 600 MHz) absorption coefficient µa scattering coefficient µs‘ 10 Instrument Diagram Overview optical fibers • Introduction detector channels X-ray vs optical tomography • Model-based iterative image reconstruction Basic concepts and mathematical background • Instrumentation rotating mirror coupler tissue SC SC SC SC PC DAQ General optical imaging modalities Dynamic optical tomography system • Applications LD 1 Brain Imaging Tumor Imaging Fluorescence Imaging laser diodes LD 2 LDD 1 LDD 2 Dynamic Optical Tomography System (DYNOT) PS 1 lock-in reference PS 2 Dynamic Optical Tomography System (details) Arm Detector Unit Iris & Folding Hemisphere Timing Board User Interface & Software Fiber Optics Opto-deMUX Laser Diodes & Driver Student Up to 10 full tomographic images per second! 11 Detector and Timing Boards Detector modules (lock-in detection scheme, individual gain settings 2 amplification stages) Interfacing Board Timing Board Back plane Dynamic Optical Tomography System (DYNOT) From power supply To DAQ board Dynamic Range of Measurement Dynamic Range of Measurement ~ 10-1 •0.01 W 0.1 W ~ 10-3 •0.1 W 0.01 W ~10-1 • 0.1 W 5 cm 5 cm ~ 10-5 •0.1 W ~ 10-5 •0.1 W ~10-3 •0.1 W 12 Dynamic Range of Measurement Dynamic Range of Detectors 10 3 amplification stages to bring signal within 0.5 - 5 V 1 × 106 5 cm 10-1 × 103 Signal [ V ] ~10-5 •0.1 W 10-3 10-4 10-5 10-6 10-7 10-8 0.1 W ~ 10-3 •0.1 W 10-2 10-9 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 Nominal OD value Timing Scheme 6 msec 6 msec move mirror to new fiber, switch gains target illumination (1 source) Lock In S/H 32 detectors in parallel DAQ TASK Src.1 Src. Pos.1 SETTL. TIME SAMPLE Src. 2 Src. Pos. 2 SETTL. TIME Src. Pos. 3 HOLD DATA READ SAMPLE Src. 3 SETTL. TIME HOLD DATA READ SAMPLE HOLD TIME Performance Overview DATA READ Parameter Value Modulation frequency 5-10 kHz Data acquisition rate ~150 Hz Settling time 1-2 ms Noise equivalent power 10 pW (rms) Dynamic range 1:109 (180 dB) Long term bias drifts ~1% over 30 min Background light reject ~100 dB 13 For more details see: A.H. Hielscher, A.Y. Bluestone, G.S.Abdoulaev, A.D. Klose, J. Lasker, M. Stewart, U. Netz, J. Beuthan, "Near-infrared diffuse optical tomography," Disease Markers 18(5-6), pp. 313-337 (2002). C.H. Schmitz, M. Löcker, J.M. Lasker, A.H. Hielscher, R.L. Barbour, "Instrumentation for fast functional optical tomography," Rev. of Scientific Instrumentation 73(2), pp. 429-439 (2002). C.H. Schmitz, Y. Pei, H.L. Graber, J.M. Lasker, A.H. Hielscher, R.L. Barbour, "Instrumentation for real-time dynamic optical tomography," in Photon Migration, Optical Coherence Tomography, and Microscopy, S. Andersson-Engels, M.F. Kaschke, eds., SPIE-The International Society for Optical Engineering, Proc. 4431, pp. 282-291, 2001. Overview • Introduction X-ray vs optical tomography • Model-based iterative image reconstruction Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system • Applications Brain Imaging Tumor Imaging Fluorescence Imaging www.bme.columbia.edu/biophotonics Animal Model Probe Geometry Forehead shaven 325 gm Sprague Dawley Rats Animal’s head fixed in place using stereotaxic 5.0 mm λ 4 sources 12 detectors 1.5 1.5 1.5 BP Regulate inspired [O2 ] and [CO 2 ] Blood Pressure and derived respiratory rate via Femoral catheter 1.5 Ant. Ventilated at: 40-60 breaths/min 1-1.5 cc/breath Optical probe with fixed geometry positioned in line with lambda (λ) suture line, optodes begin 2 mm anterior to λ. 1.5 Anesthesia: Urethane administered i.p. 14 Probe Location Carotid Occlusion Dorsal view posterior λ S1 S2 D9 D1 D5 D7 D6 D8 D4 D12 S3 S4 β animal’s right animal’s left anterior Carotid Occlusion 46. 35. Hb [µM] right occlusion 24. 13. 2.0 -3.0 Two Wavelengths (λ1, λ2) Reconstruction algorithm provides Δµ a for each volume element (voxel) of finite element mesh for each wavelength. left occlusion HbO 2 [µM] 15. -8.0 -30. -55. -78. For each voxel we get two equations: . λ1 λ1 λ1 -90. Δµa = ε Hb Δ[Hb] + ε HbO2 Δ[HbO2 ] 0.4 -10. Lt. -20. -34. -40. THb[µM] 12. Lt. λ 2 Δ[Hb] + ε λ 2 Δµaλ 2 = ε Hb HbO2 Δ[HbO2 ] ε := extinction coefficient (from literature) 15 Two Wavelengths Movie Δ Hb, HbO 2, THb (source 1, detector 12) Reconstruction algorithm provides Δµa for each volume element (voxel) of finite element mesh for each wavelength. posterior λ From this we can calculate changes in concentrations of oxy-hemoglobin, Δ[Hb], and dexoy-hemoglobin, Δ[HbO 2], for each voxel. source 1 detector 12 β anterior λ 2 Δµ λ1 − ε λ1 Δµ λ 2 ε HbO a HbO2 a Δ[Hb] = λ1 2 λ 2 λ 2 λ1 ε Hb ε HbO2 − ε Hb ε HbO2 ε λ1 Δµ λ 2 − ε λHb2 Δµaλ1 Δ[HbO2 ] = λ1 Hbλ 2 a 1 ε Hb ε HbO2 − ε λHb2 ε λHbO 2 Forepaw Stimulation Right Forepaw Stimulation lt. rt. 50 -27.0 µM Δ[HbO2]* *Oxyhemoglobin 16 Reconstruction Blood Volume A.Y. Bluestone, M. Stewart, B. Lei, I.S. Kass, J. Lasker, G.S. Abdoulaev, A.H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part I: Hypercapnia," Journal of Biomedical Optics 9(5), pp. 1046-1062 (2004). Cut 3 Cut 7 Cut 10 lt. rt. -0.003 For more details see: 0 0.004 ΔΤHb [mM] Overview • Introduction X-ray vs optical tomography • Model-based iterative image reconstruction A.Y. Bluestone, M. Stewart, J. Lasker, G.S. Abdoulaev, A.H. Hielscher, "Three-dimensional optical tomographic brain imaging in small animals, Part II: Unilateral Carotid Occlusion," Journal of Biomedical Optics 9(5), pp. 1063-1073 (2004). A.Y. Bluestone, Kenichi Sakamoto, A.H. Hielscher, M. Stewart, “ThreeDimensional Optical Tomographic Brain Imaging during Kainic-AcidInduced Seizures in Rats,” in Physiologu, Function, and Structure from Medical Images, A. Amini, A. Manduca, eds., SPIE-The International Society for Optical Engineering, Proc. 5746, pp. 58-66 (2005). www.bme.columbia.edu/biophotonics Tumors in Mice • Tumor is injected into mouse left kidney. • Tumor continues to grow unless treated. Basic concepts and mathematical background • Instrumentation Static Measurements Dynamic Measurements • Treatment with VEGF antagonist seeks to stop angiogenesis and reverse tumor growth. • Applications Brain Imaging Tumor Imaging Fluorescence Imaging 17 Tumors in Mice More Information: • Untreated tumors: highly vascularized Frischer -Chiweshe A, Kadenhe Frischer JS, JS, Huang Huang JZ, Serur A, KadenheKadenhe-Chiweshe A, McCrudden McCrudden KW, KW, O'Toole O'Toole K, K, Holash Holash J, J, Yancopoulos Yancopoulos GD, GD, Yamashiro Yamashiro DJ, DJ, Kandel Kandel JJ JJ "Effects "Effects of of potent potent VEGF VEGF blockade blockade on on experimental experimental Wilms Wilms tumor tumor and and its its persisting persisting vasculature" vasculature" INTERNATIONAL INTERNATIONAL JOURNAL JOURNAL OF OF ONCOLOGY ONCOLOGY 25 25 (3): (3): pp. pp. 549-553 549-553 (2004). (2004). • Treated tumors: much less vascularized • Currently: Many mice are sacrificed to get tumor data Fluorescent staining with Lectin (10 x) • Only 1 time point per mouse • We propose to use MRI and OT to study tumor size and vasculature in vivo fMRI vs Optical Tomography fMRI Spatial Resolution 0.1mm- 1mm Optical Tomography 2mm - 10mm Sensitive to Hb, HbO2, cytochrome, etc, blood volume, scattering properties Speed Hb (paramag.) 0.1 - 1Hz ~50 Hz > $500.000 ~ $100.000 Portability no yes Continuous Monitoring no yes Cost Combine high spatial resolution of fMRI and high speed and sensitivity of optical tomography! Huang Huang JZ, JZ, Frischer Frischer JS, JS, Serur Serur A, A, Kadenhe Kadenhe A, A, Yokoi Yokoi A, A, McCrudden McCrudden KW, KW, New New T, T, O'Toole O'Toole K, K, Zabski Zabski S, S, Rudge Rudge JS, JS, Holash Holash J, J, Yancopoulos Yancopoulos GD, GD, Yamashiro Yamashiro DJ, DJ, Kandel Kandel JJ JJ "Regression "Regression of of established established tumors tumors and and metastases metastases by by potent potent vascular vascular endothelial blockade endothelial growth growth factor factor blockade” blockade”” PROCEEDINGS PROCEEDINGS OF OF THE THE NATIONAL NATIONAL ACADEMY ACADEMY OF OF SCIENCES SCIENCES OF OF THE THE UNITED UNITED STATES STATES OF OF AMERICA AMERICA 100 100 (13): (13): 7785-7790 7785-7790 (2003) (2003) Glade-Bender Glade-Bender J, J, Kandel Kandel JJ, JJ, Yamashiro Yamashiro DJ, DJ, "VEGF "VEGF blocking blocking therapy therapy in in the the treatment treatment of of cancer” cancer” EXPERT OPINION ON BIOLOGICAL THERAPY 3 (2): 263-276 APR EXPERT OPINION ON BIOLOGICAL THERAPY 3 (2): 263-276 APR 2003 2003 9.4 Tesla MRI (Bruker Avance 400) Micro2.5 Imaging set 35mm diameter Linearly polarized Birdcage coil Typical imaging time: 30 - 60 minutes (T1 sequence) 18 Optical Tomography Set Up Step 1 Step 2 Lower mouse into imaging head. Add matching fluid (Intralipid). Step 3 Axial Slice Optical [HbT] HbT] (M) MRI Kidney Back Muscle & Spinal Cord Illuminate with light (Image!) Tumor Total Hemoglobin Typical imaging time: 10 - 20 minutes Combine high spatial resolution of fMRI and high speed and sensitivity of optical tomography! Coronal Slice [HbT] HbT] Optical Compare Untreated vs. Treated MRI (M) Kidney Untreated [HbT] Treated [HbT] Untreated tumor has higher [HbT] than treated tumor because of higher vascularization. Tumor Untreated tumor has higher [Hb] than treated tumor because it is HbO 2 starved. Total Hemoglobin Untreated [Hb] (M) Treated [Hb] (M) 19 For more details see: Overview J. Masciotti, G. Abdoulaev, J. Hur, J. Papa, J. Bae, J. Huang, D. Yamashiro, J. Kandel, A.H. Hielscher, “Combined optical tomographic and magnetic resonance imaging of tumor bearing mice,” in Optical Tomography and Spectroscopy of Tissue VII, B. Chance, R.R. Alfano, B.J. Tromberg, M. Tamura, E.M. Sevick-Muraca, eds., SPIE-The International Society for Optical Engineering, Proc. 5693, pp. 74-81 (2005). • Introduction X-ray vs optical tomography • Model-based iterative image reconstruction Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system • Applications www.bme.columbia.edu/biophotonics Brain Imaging Tumor Imaging Molecular Fluorescence Imaging Rheumatoid Arthritis Molecular Imaging Light NIRF molecular probes targets mouse without RA transgenic mouse with RA Antigen: glucose-6-phosphate isomerase (GPI) Mahmood, Weissleder et al MGH-CMIR KRN transgene on the Non transgenic B6xNOD. (GPI) glycolytic enzynme is Antigen B6xNOD F1 backgrou the T cells and immunoglobins attack. (K/BxN) Only when GPI is expressed in synovial tissue rheumatoid arthritis develops Developed fluorescent markers that shine when GPI is present/ 20 Cancer Detection Fluorescence Tomography reconstruction of absorption and scattering profile µ(x,y) µ(x,y) reconstruction of fluorescence source profile S(x,y) S(x,y) light source light source Mfl Fluorescence Tomography 1) Excitation λ x Inverse Source Problem Ω ⋅ ∇Ψ (r,Ω ) + ( µ a + µ s )Ψ(r,Ω ) = S (r,Ω ) + µ s ∫ p(Ω,Ω')Ψ(r,Ω')dΩ' 2) Emission λm 4π φx φm [ W cm-2 ] [ W cm-2 ] 1) Excitation λ x ( ) Ω ⋅ ∇Ψ x + µ a x→ + µ a x→ m + µ s x Ψ x = S x + µ s x ∫ p( Ω,Ω')Ψ x ( Ω')dΩ' 4π φ x = ∫ Ψ x (Ω' )dΩ' 4π fluorophore 2) Emission λm µ a x→ m absorption of fluorophore € € η quantum yield of fluorophore ( ) Ω ⋅ ∇Ψ m + µ a m + µ s m Ψ m = 1 ηµ x→ m φ xx + µ s m ∫ p( Ω,Ω')Ψ m (Ω')dΩ' 4π a 4π 21 Model-Based Image Reconstruction Model-Based Image Reconstruction 1) Excitation λ x 1) Excitation λ x Forward Model Forward Model Experiment M Prediction P Prediction P Inverse Model 2) Emission λm φx Forward Model Experiment M Inverse Model µ a x→ m µ a x→ m € € Mouse Tomography € Model-Based Image Reconstruction 1) Excitation λ x 2) Emission λm € Forward Model Prediction P Inverse Model µ a x→ m φx Experiment M Forward Model Prediction P Experiment M Inverse Model µ a x→ m Image 22 Mouse Tomography For more details see: A.K. Klose, V. Ntziachristos, A.H. Hielscher, "The inverse source problem based on the radiative transfer equation in molecular optical imaging," J. of Computational Physics 202, pp. 323-345 (2005). 1 mm A.K. Klose, A.H. Hielscher, "Fluorescence tomography with the equation of radiative transfer for molecular imaging," Optics Letters 28(12), pp. 1019-1021 (2003). 3 mm 7 mm c [au] 5 mm 0 9 mm Summary • Introduction X-Ray Tomography vs Optical Tomography • Model-based iterative image reconstruction Basic concepts and mathematical background • Instrumentation General optical imaging modalities Dynamic optical tomography system • Applications Brain Imaging Tumor Imaging Fluorescence Imaging A.K. Klose, A.H. Hielscher, " Optical fluorescence tomography with the equation of radiative transfer for molecular imaging," in Optical Tomography and Spectroscopy of Tissue V, B. Chance, R.R. Alfano, B.J. Tromberg, M. Tamura, E.M. Sevick-Muraca, eds., SPIE-The International Society for Optical Engineering, Proc. 4955, pp. 219-225 (2003). www.bme.columbia.edu/biophotonics Acknowledgements I • Students: J. Masciotti, X. Gu, J. Hur, F. Provenzano, J. Lasker, A. Bluestone, B. Moa-Anderson • Postdoctoral Fellows: A. Klose, G. Abdoulaev, J. Papa • Collaborators: Columbia J. Kandel (Pediatrics & Surgery, Columbia) D. Yamashiro (Pediatrics & Surgery, Columbia) G. Bal (Applied Mathematics) SUNY - Downstate Mark Steward (Physiology & Pharmacology) R.L. Barbour (Pathology) C. Schmitz (NIRx Medical Technologies, Inc.) 23 Acknowledgements II More Information • National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS) (RO1 AR46255-01 PI: Hielscher) • National Institute for Biomedical Imaging and Bioengineering (NIBIB) (R01 EB001900-01 PI: Hielscher and 5 R33 CA 91807-3 PI: Ntziachristos) • National Heart, Lung, and Blood Institute (NHLBI) (SBIR 2R44-HL-61057-02) • Whitaker Foundation (#98-0244 PI: Hielscher) • Schering Research Foundation (PI: Klose) www.bme.columbia.edu/biophotonics . 24