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Probing the Prostate with Diffusive Photons (2):

Probing the Prostate with Diffusive Photons (2): Procession of Photon Diffuse Propagation in Cylinder-Applicator Geometry. Daqing (Daching) Piao, PhD Assistant Professor of Bioengineering School of Electrical and Computer Engineering Oklahoma State University, Stillwater, OK 74078-5032.

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Probing the Prostate with Diffusive Photons (2):

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  1. Probing the Prostate with Diffusive Photons (2): Procession of Photon Diffuse Propagation in Cylinder-Applicator Geometry Daqing (Daching) Piao, PhD Assistant Professor of Bioengineering School of Electrical and Computer Engineering Oklahoma State University, Stillwater, OK 74078-5032

  2. Outline • Motivation --------- Improve the model for trans-lumenal imaging • Steady-state photon diffusion in an infinite medium --------- Solution in cylindrical-coordinateand its numerical validation • Steady-state photon diffusion with a cylindrical boundary------ Analytic solutionfor trans-lumenal (convex & concave) geometry • Qualitative numerical evaluation of the analytic results • Quantitative examinations---Finite Element, Experiment, Monte Carlo

  3. Outline • Motivation --------- Improve the model for trans-lumenal imaging • Steady-state photon diffusion in an infinite medium --------- Solution in cylindrical-coordinateand its numerical validation • Steady-state photon diffusion with a cylindrical boundary------ Analytic solutionfor trans-lumenal (convex & concave) geometry • Qualitative numerical evaluation of the analytic results • Quantitative examinations---Finite Element, Experiment, Monte Carlo

  4. Diffuse Optical Imaging (Tomgoraphy) Source Detector Multiple source-detector pairs----tomography reconstruction

  5. Image reconstruction Measurement ? Forward problem Medium property ? ? Inverse problem

  6. Homogenous ? Source: Cameron Musgrove

  7. Tissue scattering dominates in NIR spectrum Photon propagation becomes essentially diffused after ~5mm for most soft tissue Source: Brian. Pogue

  8. Maxwell equation

  9. Radiative Transport Equation The equation of radiative transfer simply says that as a beam of radiation travels, it loses energy to absorption, gains energy by emission, and redistributes energy by scattering. Extinction Divergence L: Radiance Source Scattering

  10. Photon Diffusion Equation Fluence rate Current density Photon diffusion equation is a second order partial differential equation describing the time behavior of photon fluence rate distribution in a low-absorption high-scattering medium.

  11. Treatment of the Boundary Condition Dirichiliet Boundary Condition Where  vanishes on the boundary Neumann Boundary Condition Where the normal gradient of  vanishes on the boundary Robin Boundary Condition The linear combination of the type-I and type-II conditions

  12. Infinite Medium

  13. Semiinfite Medium

  14. Non-planar Interface • Literatures • S.R. Arridge, M. Cope, D.T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys Med Biol., 37(7):1531-60 (1992). • B.W. Pogue, and M.S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys Med Biol., 39(7):1157-80 (1994). • A. Sassaroli, F. Martelli, G. Zaccanti, and Y. Yamada, "Performance of fitting procedures in curved geometry for retrieval of the optical properties of tissue from time-resolved measurements," Appl. Opt. 40, 185-197 (2001).

  15. Non-planar Interface Probe

  16. Motivation Joseph M. Lasker, etc Journal of Biomedical Optics 12(5), 052001 September/October 2007 Probe

  17. Outline • Motivation --------- Improve the model for trans-lumenal imaging • Steady-state photon diffusion in an infinite medium --------- Solution in cylindrical-coordinateand its numerical validation • Steady-state photon diffusion with a cylindrical boundary------ Analytic solutionfor trans-lumenal (convex & concave) geometry • Qualitative numerical evaluation of the analytic results • Quantitative examinations---Finite Element, Experiment, Monte Carlo

  18. Boltzmann (Radiative) Transport Equation Photon Diffusion Equation Time-resolved Steady-state Point source Homogenous medium Isotropic detector The equation for Green’s function

  19. Boltzmann (Radiative) Transport Equation Photon Diffusion Equation Time-resolved Steady-state Point source Spherical coordinates Isotropic detector

  20. Point source Spherical coordinates Isotropic detector

  21. Cylindrical-coordinate solution to the steady-state photon diffusion in an infinite medium unknown

  22. Cylindrical-coordinate solution to the steady-state photon diffusion in an infinite medium The equation for the radial Green’s function : where or . Following the approach for solving Poisson equation introduced by Jaskson, John David in Classical Electrodynamics Chapter3 , we find: where and are the modified Bessel functions, and and indicate the smaller and larger radial coordinates of the source and the detector.

  23. Cylindrical-coordinate solution to the steady-state photon diffusion in an infinite medium where

  24. Boltzmann (Radiative) Transport Equation Photon Diffusion Equation Time-resolved Steady-state Point source Spherical coordinates Isotropic detector Cylindrical coordinates

  25. Numerical Validation to the cylindrical-coordinate solution of the steady-state photon diffusion in an infinite medium Spherical Coordinates ? Cylindrical Coordinates

  26. Outline • Motivation --------- Improve the model for trans-lumenal imaging • Steady-state photon diffusion in an infinite medium --------- Solution in cylindrical-coordinateand its numerical validation • Steady-state photon diffusion with a cylindrical boundary------ Analytic solutionfor trans-lumenal (convex & concave) geometry • Qualitative numerical evaluation of the analytic results • Quantitative examinations---Finite Element, Experiment, Monte Carlo

  27. When there is a boundary ? Image source Point source Isotropic detector

  28. Steady-state photon diffusion in a medium with planner boundary Geometry introduced by Fantini et al. where

  29. Steady-state photon diffusion in a medium with convex boundary Convex boundary: a radius of the equivalent isotropic source the detector is at real boundary The photon intensity associated with the real source is: known unknown

  30. Steady-state photon diffusion in a medium with convex boundary where is to be determined by the boundary condition. Besides, the factor is introduced to simplify the determination of .

  31. Steady-state photon diffusion in a medium with convex boundary At the extrapolated boundary, and Evaluating the same order of m

  32. Steady-state photon diffusion in a medium with convex boundary For detector point which is located at the real boundary, and Contribution due to the real source Contribution due to the image source

  33. Steady-state photon diffusion in a medium with concave boundary Concave boundary: At the extrapolated boundary,

  34. Steady-state photon diffusion in a medium with concave boundary For detector point , and

  35. Comparison between the solution under convex boundary and concave boundary Convex boundary: Concave boundary: physical boundary extrapolated boundary Position of the equivalent real source

  36. Semi-infinite plane

  37. Outline • Motivation --------- Improve the model for trans-lumenal imaging • Steady-state photon diffusion in an infinite medium --------- Solution in cylindrical-coordinateand its numerical validation • Steady-state photon diffusion with a cylindrical boundary------ Analytic solutionfor trans-lumenal (convex & concave) geometry • Qualitative numerical evaluation of the analytic results • Quantitative examinations---Finite Element, Experiment, Monte Carlo

  38. Azimuth direction

  39. Longitudinal direction … …………………….Part 1 Journal of Optical Society of America, A, Vol. 27, No. 3, pp. 648-662 (2010)

  40. Outline • Motivation --------- Improve the model for trans-lumenal imaging • Steady-state photon diffusion in an infinite medium --------- Solution in cylindrical-coordinateand its numerical validation • Steady-state photon diffusion with a cylindrical boundary------ Analytic solutionfor trans-lumenal (convex & concave) geometry • Qualitative numerical evaluation of the analytic results • Quantitative examinations---Finite Element, Experiment, Monte Carlo

  41. Experiments

  42. Experiments

  43. COMSOL

  44. Results

  45. Results … …………………….Part 2 Journal of Optical Society of America, A, in preparation

  46. Numerical validation against Monte Carlo simulation

  47. Zhang A, Piao D, Bunting CF, Pogue BW, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator ---- Part I: steady-state theory,” Journal of Optical Society of America, A, accepted.

  48. Summary • The Green’s function for steady-state photon diffusion in infinite medium is expanded in cylindrical coordinates. • The solutions of steady-state photon diffusion is derived for a circular concave or convex boundary. • The analytic solutions are qualitatively evaluated. • The analytic solutions are quantitatively examined

  49. Acknowledgement • Anqi (Andrew) Zhang • Guan (Gary) Xu • Dr. Gang (Gary) Yao • Dr. Brian W. Pogue • Dr. Charles F. Bunting Oklahoma Center for the Advancement of Science and Technology (OCAST)

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