1 / 50

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.

jewell
Télécharger la présentation

Probing the Prostate with Diffusive Photons (2):

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related