Profs. Brooks and DiMarzio Northeastern University Spring 2004

1. ECEU692Subsurface ImagingCourse NotesPart 12: Imaging with Light (4):Diffusive Optical Tomography Profs. Brooks and DiMarzio Northeastern University Spring 2004 Chuck DiMarzio, Northeastern University

2. Topic Outline • Goal: “Find the Matrix Elements” • A Bit of Radiometry • Terminology and Units • Radiative Transport • Approximation to Radiative Transport Equation • Diffusion Approximation • Wave Solution • Generating the Diffusive Waves • Examples • Adding Ultrasound • Solving for the Matrix Elements Chuck DiMarzio, Northeastern University

3. P P t t The Matrix Elements DC AC Amplitude AC Phase Chuck DiMarzio, Northeastern University

4. Radiometric Quantities Chuck DiMarzio, Northeastern University

5. Radiometry and Photometry M, Flux/Proj. Area Notes: Spectral x=dx/dn or dx/dl: Add subscript n or w, divide units by Hz or mm. F, Flux Radiant Flux Watts Luminous Flux Lumens Radiant Exitance Watts/m2 Luminous Exitance Lumens/m2=Lux 1 W is 683 L at 555 nm. Radiance Watts/m2/sr Luminance Lumens/m2/sr 1 Lambert= (1L/cm2/sr)/p I, Flux/W L,Flux/AW Radiant Intensity Watts/sr Luminous Intensity Lumens/sr E, Flux/Area Rcd. Irradiance Watts/m2 Illuminance Lumens/m2=Lux 1 ftLambert= (1L/ft2/sr)/p 1mLambert= (1L/m2/sr)/p 1 Ft Candle=1L/ft2 1 Candela=1cd=1L/sr Chuck DiMarzio, Northeastern University

6. What Is Radiative Transport? • The Radiative Transport Equation L+dL dW dW L ds Chuck DiMarzio, Northeastern University

7. Solutions to RTE • Monte-Carlo • Low Scattering • High Scattering • Diffusion Approximation • Usually Valid in Tissue, Except... • Certain Tissue Types • Certain Imaging Modalities (eg. Confocal, OCT) • Close to Source or to Rapid Changes in Parameters Chuck DiMarzio, Northeastern University

8. Resolution Limits (M-C) Tissue Parameters ma = 0.03 /cm ms = 200 /cm g = 0.95 d = 1 cm • Approach • Monte-Carlo • Reciprocity • Fourier Transform • Parameters • Depth 1 cm. • Thickness 2 cm. • Transillumination MTF 125 150 200-ps Gate Spatial Frequency, /cm Dunn, Andrew, and Charles A. DiMarzio, “Efficient Computation of Time--Resolved Transfer Functions for Imaging in Turbid Media,” Journal of the Optical Society of America A 13, No. 1, January 1996. Pp. 65--70. Chuck DiMarzio, Northeastern University

9. Photon Diffusion Approximation • The Radiative Transport Equation • Taylor Series: f is Fluence Rate, J is Flux • Result Chuck DiMarzio, Northeastern University

10. Fluence Rate? • Another Radiometric Quantity • Fluence is Energy/Area • Fluence Rate is Energy/Area/Time • =Power/Area • Units Like E or M, but Different Meaning • Relation to Absorbed Power/Volume • A=fma • Used to Determine f in Monte-Carlo Chuck DiMarzio, Northeastern University

11. ¶ F ( ) Ñ · D Ñ F - - a F = 0 Dispersion Equation • The Diffusion Equation • Wave Solution ¶ t k • k2 Im =0 Re Chuck DiMarzio, Northeastern University

12. Dispersion Results Chuck DiMarzio, Northeastern University

13. Spherical Waves Chuck DiMarzio, Northeastern University

14. 8 10 Light (Real) 6 10 -1 4 10 DPDW Sound ), Wavenumber, m (Imag) (Imag) 2 10 p (Real) 0 k/(2 10 -2 10 -4 10 0 5 10 15 20 10 10 10 10 10 f, Frequency, Hz. Different Types of Waves 1mm 1mm 1m 1km 10059_1 Chuck DiMarzio, Northeastern University

15. Physical Reason for Dispersion Imaginary part of k increases with frequency Easy to understand in terms of multiple paths. m100574a.m Chuck DiMarzio, Northeastern University

16. 20 Photon Tracks 20,000 Photon Tracks Pabs=0.1 Pext=0.3 Watch the Photons Migrate! • Received Photons 90 80 70 60 Photons in Box 50 40 30 20 10 0 0 20 40 60 80 100 Time Step Chuck DiMarzio, Northeastern University

17. How Diffuisve Waves Begin? Tissue Extrapolated Boundary • Generation • From Light Wave • Wave Behavior • Absorption • Reflection • Refraction • Diffraction • Interference • Scattering Detector Image Source Image Source Effective Source Input Chuck DiMarzio, Northeastern University

18. Noise Issues Noise proportional to square root of DC signal. m100574a.m Chuck DiMarzio, Northeastern University

19. DOT Instrumentation at MGH Imaging Center • TECHNOLOGY • Near-infrared light • Fiber optics • Computed Tomography • ADVANTAGES • Optical contrast • Portable - bedside, ambulance • Continuous • Inexpensive • DISADVANTAGES • Resolution • Depth penetration From David A. Boas - MGH NMR Center Chuck DiMarzio, Northeastern University

20. Detectors Sources Functional Imaging of a Neonate 6 cm Mid-line 4 cm Passive movement of right arm Passive movement of right arm At Rest Data Set I - 98-05-14 From David A. Boas - MGH NMR Center Chuck DiMarzio, Northeastern University

21. 0 -1 Z axis -2 -3 -4 6 -5 5 4 6 3 5 2 4 3 1 2 1 0 0 X axis Y axis 0.15 0 -1 0.1 0.05 -2 0 0 0.14 -3 0.05 0.12 0.04 -1 -1 -4 Reconstruction with Reflection only (Top Sources) 0.1 -5 0 0.03 -2 -2 0 2 4 6 0.08 -3 -3 0.06 0.02 0.04 -4 -4 Reflection and Transmission (All Sources) 0.01 0.02 -5 -5 0 1 2 3 4 5 6 0 0 1 2 3 4 5 6 Keeping the Matrix Rank Up Source z y=4 Detector Object x • DiMarzio, et. al., Presented at Photonics West, Jan 1999 Chuck DiMarzio, Northeastern University

22. API Virtual Source Ultrasound Beam Optical Source Optical Source Optical Source Optical Receiver Optical Receiver Optical Receiver Ultrasound Focal Point Light from Source to Receiver Light from Source to Receiver through Ultrasound Focus All Light from Source Fiber Chuck DiMarzio, Northeastern University

23. Solving the Wave Equation (1) Chuck DiMarzio, Northeastern University

24. Solving the Wave Equation (2) Chuck DiMarzio, Northeastern University

25. The First Born Approximation Chuck DiMarzio, Northeastern University

26. Why Do We Want a Model? • Applications • Forward Model • Will it work? • Inverse Algorithms • How Much Does k Change? • ie. Is there a Tumor? • And Where? • Understanding • What is k? • See Panel to Right. Chuck DiMarzio, Northeastern University