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Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 49, Iss. 8 — Mar. 10, 2010
  • pp: 1414–1429

Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects

Jorge Bouza Domínguez and Yves Bérubé-Lauzière  »View Author Affiliations

Applied Optics, Vol. 49, Issue 8, pp. 1414-1429 (2010)

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We present a simplified spherical harmonics approximation for the time-domain radiative transfer equation including the source-divergence effect. This leads to a set of coupled partial differential equations (PDEs) of the parabolic type that model diffuse light propagation in biological-tissue-like media. We introduce a finite element approach for solving these PDEs, thereby obtaining the time-dependent spatial profile of the fluence. We compare the results with the diffusion equation and Monte Carlo simulations. The fluence obtained via our model is shown to reproduce well the Monte Carlo results in all cases and improves on the solution of the diffusion equation in homogeneous diffusive-defying media. Our solution also shows more sensitivity than the diffusion equation to changes in the absorption coefficient of small inclusions.

© 2010 Optical Society of America

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.6920) Medical optics and biotechnology : Time-resolved imaging

ToC Category:
Medical Optics and Biotechnology

Original Manuscript: August 14, 2009
Revised Manuscript: December 26, 2009
Manuscript Accepted: January 5, 2010
Published: March 8, 2010

Virtual Issues
Vol. 5, Iss. 7 Virtual Journal for Biomedical Optics

Jorge Bouza Domínguez and Yves Bérubé-Lauzière, "Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects," Appl. Opt. 49, 1414-1429 (2010)

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  1. L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley-Interscience, 2007).
  2. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. R. Arridge, and J. P. Kaipio, “Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions,” Phys. Med. Biol. 50, 4913-4930 (2005). [CrossRef]
  3. M. S. Patterson, B. C. Wilson, and D. R. Wyman, “The propagation of optical radiation in tissue i. Models of radiation transport and their application,” Lasers Med. Sci. 6, 155-168(1991). [CrossRef]
  4. S. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539 (1995). [CrossRef]
  5. L. Martí-López, J. Bouza-Domínguez, J. C. Hebden, S. R. Arridge, and R. A. Martínez Celorio, “Validity conditions for the radiative transfer equation,” J. Opt. Soc. Am. A 20, 2046-2056 (2003).
  6. T. Khan and H. Jiang, “A new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices,” J. Opt. A Pure Appl. Opt. 5, 137-141 (2003). [CrossRef]
  7. P. Malin, P. Erosha, and A. J. Lowery, “The photon transport equation for turbid biological media with spatially varying isotropic refractive index,” Opt. Express 13, 389-399 (2005). [CrossRef]
  8. L. Martí-López, J. Bouza Domínguez, R. Martínez Celorio, and J. Hebden, “An investigation of the ability of modified radiative transfer equations to accommodate laws of geometrical optics,” Opt. Commun. 266, 44-49 (2006). [CrossRef]
  9. L. Martí-López, J. Bouza-Domínguez, and J. C. Hebden, “Interpretation of the failure of the time-independent diffusion equation near a point source,” Opt. Commun. 242, 23-43 (2004). [CrossRef]
  10. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  11. S. W. R. Koch, W. Krebs, S. Wittig, and R. Viskanta, “Discrete ordinates quadrature schemes based on the angular interpolation of radiation intensity,” J. Quant. Spectrosc. Radiat. Transfer 53, 353-372 (1995). [CrossRef]
  12. J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462-467 (2000). [CrossRef]
  13. A. Custo, W. M. Wells III, A. H. Barnett, E. M. Hillman, and D. A. Boas, “Effective scattering coefficient of the cerebral spinal fluid in adult head models for diffuse optical imaging,” Appl. Opt. 45, 4747-4755 (2006). [CrossRef]
  14. A. T. Kumar, S. B. Raymond, A. K. Dunn, B. J. Bacskai, and D. A. Boas, “A time domain fluorescence tomography system for small animal imaging,” IEEE Trans. Med. Imaging 27, 1152-1163 (2008). [CrossRef]
  15. A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441-470 (2006). [CrossRef]
  16. J. Fletcher, “Solution of the multigroup neutron transport equation using spherical harmonics,” Nucl. Sci. Eng. 84, 33-46 (1983).
  17. C. R. E. de Oliveira, “An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering,” Prog. Nucl. Energy 18, 227-236 (1986). [CrossRef]
  18. F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).
  19. P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76, 9-14 (2007). [CrossRef]
  20. M. Frank, A. Klar, E. Larsen, and S. Yasuda, “Time-dependent simplified Pnn approximation to the equations of radiative transfer,” J. Comput. Phys. 226, 2289-305 (2007). [CrossRef]
  21. M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol. 54, 2493-2509 (2009). [CrossRef]
  22. Y. Bérubé-Lauzière, V. Issa, and J. Bouza-Domínguez, “Simplified spherical harmonics approximation of the time-dependent equation of radiative transfer for the forward problem in time-domain diffuse optical tomography,” Proc. SPIE 7174, 717403 (2009).
  23. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285-1302 (1998). [CrossRef]
  24. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).
  25. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  26. S. R. Arridge, “Diffusion tomography in dense media,” in Scattering and Inverse Scattering, R. Pike and P. Sabatier, eds. (Academic, 2002), pp. 920-936.
  27. S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).
  28. W. Cheong, S. Prahl, and A. Welch, “Optical properties of tissues in vitro,” IEEE J. Quantum Electron. 26, 2166-2185(1990). [CrossRef]
  29. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532-2539 (1995). [CrossRef]
  30. S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395-7409 (1995). [CrossRef]
  31. J. Bouza-Domínguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78 (2008).
  32. E. W. Larsen, J. E. Morel, and J. M. McGhee, “Asymptotic derivation of the multigroup P1 and simplified PN equations with anisotropic scattering,” Nucl. Sci. Eng. 123, 328-342 (1996).
  33. D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309-325 (1996).
  34. P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).
  35. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol. 45, 1359-1373 (2000). [CrossRef]
  36. L. Martí-López, J. Hebden, and J. Bouza-Domínguez, “Estimates of minimum pulse width and maximum modulation frequency for diffusion optical tomography,” Opt. Lasers Eng. 44, 1172-1184 (2006). [CrossRef]
  37. R. Backofen, T. Bilz, A. Ribalta, and A. Voigt, “SPN-approximations of internal radiation in crystal growth of optical materials,” J. Cryst. Growth 266, 264-270 (2004). [CrossRef]
  38. M. Schweiger and D. Arridge, and S. R. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Imaging Vision 3, 263-283 (1993). [CrossRef]
  39. S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299-309 (1993). [CrossRef]
  40. T. Davis, Direct Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2006).
  41. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
  42. J. Ripoll, “Light diffusion in turbid media with biomedical application,” Ph.D. dissertation (Universidad Autónoma de Madrid, 2000).
  43. D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantitative fluorescence imaging of point-like sources in small animals,” Phys. Med. Biol. 53, 5797-5814 (2008). [CrossRef]
  44. V. Ntziachristos and C. Britton, “Accuracy limits in the determination of absolute optical properties using time-resolved NIR spectroscopy,” Med. Phys. 28, 1115-1124(2001). [CrossRef]

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