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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 3 — Mar. 1, 2013
  • pp: 470–478

Truncated Fourier-series approximation of the time-domain radiative transfer equation using finite elements

Aki Pulkkinen and Tanja Tarvainen  »View Author Affiliations

JOSA A, Vol. 30, Issue 3, pp. 470-478 (2013)

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The radiative transfer equation (RTE) is widely accepted to accurately describe light transport in a medium with scattering particles, and it has been successfully applied as a light-transport model, for example, in diffuse optical tomography. Due to the computationally expensive nature of the RTE, most of these applications have been in the frequency domain. In this paper, an efficient solution method for the time-domain RTE is proposed. The method is based on solving the frequency-domain RTE at multiple modulation frequencies and using the Fourier-series representation of the radiance to obtain approximation of the time-domain solution. The approach is tested with simulations. The results show that the method can be used to obtain the solution of the time-domain RTE with good accuracy and with significantly fewer computational resources than are needed in the direct time-domain solution.

© 2013 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(030.5620) Coherence and statistical optics : Radiative transfer
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.6920) Medical optics and biotechnology : Time-resolved imaging
(170.7050) Medical optics and biotechnology : Turbid media

ToC Category:
Medical Optics and Biotechnology

Original Manuscript: October 23, 2012
Manuscript Accepted: January 9, 2013
Published: February 21, 2013

Virtual Issues
Vol. 8, Iss. 4 Virtual Journal for Biomedical Optics

Aki Pulkkinen and Tanja Tarvainen, "Truncated Fourier-series approximation of the time-domain radiative transfer equation using finite elements," J. Opt. Soc. Am. A 30, 470-478 (2013)

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999). [CrossRef]
  2. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005). [CrossRef]
  3. S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009). [CrossRef]
  4. S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998). [CrossRef]
  5. A. Gibson and H. Dehghani, “Diffuse optical imaging,” Philos. Trans. R. Soc. A 367, 3055–3072 (2009). [CrossRef]
  6. H. Eda, I. Oda, Y. Ito, Y. Wada, Y. Oikawa, Y. Tsunazawa, M. Takada, Y. Tsuchiya, Y. Yamashita, M. Oda, A. Sassaroli, Y. Yamada, and M. Tamura, “Multichannel time-resolved optical tomographic imaging system,” Rev. Sci. Instrum. 70, 3595–3602 (1999). [CrossRef]
  7. F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, J. C. Hebden, and D. T. Delby, “A 32-channel time-resolved instrument for medical optical tomography,” Rev. Sci. Instrum. 71, 256–265 (2000). [CrossRef]
  8. J. Selb, D. Joseph, and D. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006). [CrossRef]
  9. D. Contini, A. Torricelli, A. Pifferi, L. Spinelli, P. Taroni, V. Quaresima, M. Ferrari, and R. Cubeddu, “Multichannel time-resolved tissue oximeter for functional imaging of the brain,” IEEE Trans. Instrum. Meas. 55, 85–90 (2006). [CrossRef]
  10. Q. Zhang, H. Soon, H. Tian, S. Fernando, Y. Ha, and N. Chen, “Pseudo-random single photon counting for time-resolved optical measurement,” Opt. Express 16, 13233–13239 (2008). [CrossRef]
  11. H. Wabnitz, M. Moeller, A. Liebert, H. Obrig, J. Steinbrink, and R. Macdonald, “Time-resolved near-infrared spectroscopy and imaging of the adult human brain,” in Oxygen Transport to Tissue XXXI, E. Takahashi and D. Bruley, eds. (Springer, 2010), Vol. 662, pp. 143–148.
  12. I. Nissilä, J. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006). [CrossRef]
  13. N. Ducros, C. D’Andrea, A. Bassi, and F. Peyrin, “Fluorescence diffuse optical tomography: time-resolved versus continuous-wave in the reflectance configuration,” IRBM 32, 243–250 (2011). [CrossRef]
  14. M. Boffety, M. Allain, A. Sentenac, M. Massonneau, and R. Carminati, “Cramer–Rao analysis of steady-state and time-domain fluorescence diffuse optical imaging,” Biomed. Opt. Express 2, 1626–1636 (2011). [CrossRef]
  15. F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002). [CrossRef]
  16. J. Selb, A. Dale, and D. Boas, “Linear 3D reconstruction of time-domain diffuse optical imaging differential data: improved depth localization and lateral resolution,” Opt. Express 15, 16400–16412 (2007). [CrossRef]
  17. K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 1.
  19. S. Chandrasekhar, Radiative Transfer (Oxford University, 1950).
  20. C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, 1988).
  21. R. T. Ackroyd, Finite Element Methods for Particle Transport: Applications to Reactor and Radiation Physics (Research Studies, 1997).
  22. O. Dorn, “A transport–backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998). [CrossRef]
  23. W. Martin, C. Yehnert, L. Lorence, and J. Duderstadt, “Phase-space finite element methods applied to the first order form of the transport equation,” Ann. Nucl. Energy 8, 633–646 (1981). [CrossRef]
  24. A. D. Kim and A. Ishimaru, “Optical diffusion of continuos-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998). [CrossRef]
  25. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. 26, 1698–1707 (1999). [CrossRef]
  26. A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga, “Polarized pulse waves in random discrete scatterers,” Appl. Opt. 40, 5495–5502 (2001). [CrossRef]
  27. J. Boulanger and A. Charette, “Reconstruction optical spectroscopy using transient radiative transfer equation and pulsed laser: a numerical study,” J. Quant. Spectrosc. Radiat. Transfer 93, 325–336 (2005). [CrossRef]
  28. F. Asllanaj and S. Fumeron, “Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation,” J. Biomed. Opt. 17, 075007 (2012). [CrossRef]
  29. M. Charest, C. Groth, and Ö. Gülder, “Solution of the equation of radiative transfer using a Newton–Krylov approach and adaptive mesh refinement,” J. Comput. Phys. 231, 3023–3040 (2012). [CrossRef]
  30. 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]
  31. A. Klose, U. Netz, J. Beuthan, and A. Hielscher, “Optical tomography using the time-independent equation of radiative transfer—Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transfer 72, 691–713 (2002). [CrossRef]
  32. A. Klose and A. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19, 387–409 (2003). [CrossRef]
  33. H. K. Kim and A. Charette, “A sensitivity function-based conjugate gradient method for optical tomography with the frequency-domain equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 104, 24–39 (2007). [CrossRef]
  34. G. Kanschat, “A robust finite element discretization for radiative transfer problems with scattering,” East-West J. Numer. Math. 6, 265–272 (1998).
  35. S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements I. Basic method and tests,” Astron. Astrophys. 380, 776–788 (2001). [CrossRef]
  36. G. Abdoulaev and A. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imaging 12, 594–601 (2003). [CrossRef]
  37. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. 44, 876–886 (2005). [CrossRef]
  38. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, S. Arridge, and J. 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]
  39. T. Tarvainen, M. Vauhkonen, and S. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 109, 2767–2778 (2008). [CrossRef]
  40. D. Gorpas, D. Yova, and K. Politopoulos, “A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging,” J. Quant. Spectrosc. Radiat. Transfer 111, 553–568 (2010). [CrossRef]
  41. P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys. 230, 7364–7383 (2011). [CrossRef]
  42. K. Peng, X. Gao, X. Qu, N. Ren, X. Chen, X. He, X. Wang, J. Liang, and J. Tian, “Graphics processing unit parallel accelerated solution of the discrete ordinates for photon transport in biological tissues,” Appl. Opt. 50, 3808–3823 (2011). [CrossRef]
  43. K. Ren, G. Abdoulaev, G. Bal, and A. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29, 578–580 (2004). [CrossRef]
  44. K. Ren, G. Bal, and A. Hielscher, “Frequency domain optical tomography based on the equation of radiative transfer,” SIAM J. Sci. Comput. 28, 1463–1489 (2006). [CrossRef]
  45. H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38, 149–192 (2009). [CrossRef]
  46. O. Balima, Y. Favennec, J. Boulanger, and A. Charette, “Optical tomography with the discontinuous Galerkin forumulation of the radiative transfer equation in frequency domain,” J. Quant. Spectrosc. Radiat. Transfer 113, 805–814 (2012). [CrossRef]
  47. R. Koch and R. Becker, “Evaluation of quadrature schemes for the discrete ordinates method,” J. Quant. Spectrosc. Radiat. Transfer 84, 423–435 (2004). [CrossRef]
  48. J. Tervo, P. Kolmonen, M. Vauhkonen, L. Heikkinen, and J. Kaipio, “A finite-element model of electron transport in radiation therapy and a related inverse problem,” Inverse Probl. 15, 1345–1361 (1999). [CrossRef]
  49. E. Boman, J. Tervo, and M. Vauhkonen, “Modelling the transport of ionizing radiation using the finite element method,” Phys. Med. Biol. 50, 265–280 (2005). [CrossRef]
  50. J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).
  51. H. Jiang, “Optical image reconstruction based on the third-order diffusion equations,” Opt. Express 4, 241–246 (1999). [CrossRef]
  52. E. Aydin, C. de Oliveira, and A. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002). [CrossRef]
  53. E. D. Aydin, “Three-dimensional photon migration through voidlike regions and channels,” Appl. Opt. 46, 8272–8277 (2007). [CrossRef]
  54. S. Wright, M. Schweiger, and S. Arridge, “Solutions to the transport equation using variable order angular basis,” Proc. SPIE 5859, 585914 (2005). [CrossRef]
  55. Z. Yuan, X.-H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54, 65–88(2009). [CrossRef]
  56. A. Jha, M. Kupinski, T. Masumura, E. Clarkson, A. Maslov, and H. Barrett, “Simulating photon-transport in uniform media using the radiative transport equation: a study using the Neumann-series approach,” J. Opt. Soc. Am. A 29, 1741–1756 (2012). [CrossRef]
  57. A. Jha, M. Kupinski, H. Barrett, E. Clarkson, and J. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A 29, 1885–1898 (2012). [CrossRef]
  58. M. Addam, A. Bouhamidi, and K. Jbilou, “A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method,” Appl. Math. Comput. 215, 4067–4079 (2010). [CrossRef]
  59. M. Addam, A. Bouhamidi, and K. Jbilou, “Signal reconstruction for the diffusion transport equation using tensorial spline galerking approximation,” Appl. Numer. Math. 62, 1089–1108 (2012). [CrossRef]
  60. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]
  61. H. Gao and H. Zhao, “A fast-forward solver of radiative transfer equation,” Transp. Theory Stat.. Phys. 38, 149–192 (2009). [CrossRef]
  62. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993). [CrossRef]
  63. S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, SPIE Institute Series, Vol.5, (SPIE, 1989), pp. 102–111.
  64. J. Heiskala, I. Nissilä, T. Neuvonen, S. Järvenpää, and E. Somersalo, “Modeling anisotropic light propagation in a realistic model of the human head,” Appl. Opt. 44, 2049–2057 (2005). [CrossRef]
  65. T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. Arridge, and J. Kaipio, “Approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inverse Probl. 26, 015005 (2010). [CrossRef]

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