Abstract

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

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Errata

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: erratum," Appl. Opt. 50, 2699-2700 (2011)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-50-17-2699

References

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    [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).
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  16. J. Fletcher, “Solution of the multigroup neutron transport equation using spherical harmonics,” Nucl. Sci. Eng. 84, 33-46 (1983).
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    [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]
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  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]

2009

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]

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).

2008

J. Bouza-Domínguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78 (2008).

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]

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]

2007

F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76, 9-14 (2007).
[CrossRef]

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]

2006

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]

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]

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]

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]

2005

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]

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]

2004

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]

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]

2003

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).

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]

2001

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]

2000

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

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]

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]

1998

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]

1996

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).

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309-325 (1996).

1995

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]

S. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539 (1995).
[CrossRef]

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532-2539 (1995).
[CrossRef]

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395-7409 (1995).
[CrossRef]

1993

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]

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]

1991

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]

1990

W. Cheong, S. Prahl, and A. Welch, “Optical properties of tissues in vitro,” IEEE J. Quantum Electron. 26, 2166-2185(1990).
[CrossRef]

1986

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]

1983

J. Fletcher, “Solution of the multigroup neutron transport equation using spherical harmonics,” Nucl. Sci. Eng. 84, 33-46 (1983).

Alcouffe, R. E.

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]

Alianelli, L.

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]

Aronson, R.

Arridge, D.

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]

Arridge, S.

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]

S. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539 (1995).
[CrossRef]

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]

Arridge, S. R.

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]

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).

S. R. Arridge, “Photon-measurement density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395-7409 (1995).
[CrossRef]

S. R. Arridge, “Diffusion tomography in dense media,” in Scattering and Inverse Scattering, R. Pike and P. Sabatier, eds. (Academic, 2002), pp. 920-936.

Backofen, R.

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]

Bacskai, B. J.

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]

Barbour, R. L.

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]

Barnett, A. H.

Bassani, M.

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]

Bérubé-Lauzière, Y.

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).

Bilz, T.

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]

Boas, D. A.

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]

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]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Bouza-Domínguez, J.

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).

J. Bouza-Domínguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78 (2008).

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]

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]

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).

Brantley, P. S.

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

Britton, C.

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]

Celorio, R. A.

Celorio, R. Martínez

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]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Cheong, W.

W. Cheong, S. Prahl, and A. Welch, “Optical properties of tissues in vitro,” IEEE J. Quantum Electron. 26, 2166-2185(1990).
[CrossRef]

Chu, M.

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]

Comsa, D. C.

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]

Custo, A.

Davis, T.

T. Davis, Direct Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2006).

de Oliveira, C. R. E.

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]

Dehghani, H.

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]

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]

Delpy, D.

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]

Delpy, S. R.

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]

Domínguez, J. Bouza

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]

Dunn, A. K.

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]

Erosha, P.

Farrell, T. J.

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]

Fletcher, J.

J. Fletcher, “Solution of the multigroup neutron transport equation using spherical harmonics,” Nucl. Sci. Eng. 84, 33-46 (1983).

Frank, M.

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]

Hebden, J.

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]

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]

Hebden, J. C.

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]

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).

Helminen, H.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76, 9-14 (2007).
[CrossRef]

Hielscher, A. H.

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]

Hillman, E. M.

Hiraoka, M.

S. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539 (1995).
[CrossRef]

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]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Issa, V.

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).

Jiang, H.

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]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

Kaipio, J. P.

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]

Khan, T.

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]

Klar, A.

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]

F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).

Klose, A.

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]

Klose, A. D.

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]

Koch, S. W. R.

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]

Kolehmainen, V.

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]

Kotiluoto, P.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76, 9-14 (2007).
[CrossRef]

Krebs, W.

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]

Kumar, A. T.

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]

Larsen, E.

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]

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]

Larsen, E. W.

F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

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).

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309-325 (1996).

Lowery, A. J.

Malin, P.

Martelli, F.

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]

Martí-López, L.

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]

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]

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]

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).

Martin, F.

F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).

McGhee, J. M.

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).

Morel, J. E.

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).

Nieto-Vesperinas, M.

Ntziachristos, V.

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]

Patterson, M. S.

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]

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]

Prahl, S.

W. Cheong, S. Prahl, and A. Welch, “Optical properties of tissues in vitro,” IEEE J. Quantum Electron. 26, 2166-2185(1990).
[CrossRef]

Prahl, S. A.

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).

Pyyry, J.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76, 9-14 (2007).
[CrossRef]

Raymond, S. B.

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]

Ribalta, A.

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]

Riley, J.

Ripoll, J.

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]

J. Ripoll, “Light diffusion in turbid media with biomedical application,” Ph.D. dissertation (Universidad Autónoma de Madrid, 2000).

Schweiger, M.

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]

S. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539 (1995).
[CrossRef]

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]

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]

Tarvainen, T.

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]

Tomasevic, D. I.

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309-325 (1996).

Vauhkonen, M.

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]

Vishwanath, K.

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]

Viskanta, R.

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]

Voigt, A.

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]

Wang, L.

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley-Interscience, 2007).

Welch, A.

W. Cheong, S. Prahl, and A. Welch, “Optical properties of tissues in vitro,” IEEE J. Quantum Electron. 26, 2166-2185(1990).
[CrossRef]

Wells, W. M.

Wilson, B. C.

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]

Wittig, S.

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]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

Wu, H.

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley-Interscience, 2007).

Wyman, D. R.

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]

Yasuda, S.

F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).

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]

Zaccanti, G.

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]

Zangheri, L.

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]

Appl. Opt.

Bull. Inst. Math. Acad. Sin.

F. Martin, A. Klar, E. W. Larsen, and S. Yasuda, “Approximate models of radiative transfer,” Bull. Inst. Math. Acad. Sin. 2, 409-432 (2007).

IEEE J. Quantum Electron.

W. Cheong, S. Prahl, and A. Welch, “Optical properties of tissues in vitro,” IEEE J. Quantum Electron. 26, 2166-2185(1990).
[CrossRef]

IEEE Trans. Med. Imaging

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]

J. Comput. Phys.

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]

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]

J. Cryst. Growth

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]

J. Math. Imaging Vision

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]

J. Opt. A Pure Appl. Opt.

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]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

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]

Lasers Med. Sci.

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]

Med. Phys.

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]

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]

Nucl. Sci. Eng.

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).

D. I. Tomasevic and E. W. Larsen, “The simplified P2 approximation,” Nucl. Sci. Eng. 122, 309-325 (1996).

P. S. Brantley and E. W. Larsen, “The simplified P3 approximation,” Nucl. Sci. Eng. 134, 121 (2000).

J. Fletcher, “Solution of the multigroup neutron transport equation using spherical harmonics,” Nucl. Sci. Eng. 84, 33-46 (1983).

Opt. Commun.

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]

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]

Opt. Express

Opt. Lasers Eng.

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]

Phys. Med. Biol.

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]

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]

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]

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]

S. Arridge, M. Hiraoka, and M. Schweiger, “Statistical basis for the determination of optical pathlength in tissue,” Phys. Med. Biol. 40, 1539 (1995).
[CrossRef]

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]

Phys. Rev. E

J. Bouza-Domínguez, “Light propagation at interfaces of biological media: boundary conditions,” Phys. Rev. E 78 (2008).

Proc. SPIE

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).

Prog. Nucl. Energy

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]

Radiat. Phys. Chem.

P. Kotiluoto, J. Pyyry, and H. Helminen, “Multitrans SP3 code in coupled photon-electron transport problems,” Radiat. Phys. Chem. 76, 9-14 (2007).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2003).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

S. R. Arridge, “Diffusion tomography in dense media,” in Scattering and Inverse Scattering, R. Pike and P. Sabatier, eds. (Academic, 2002), pp. 920-936.

S. A. Prahl, “Light transport in tissue,” Ph.D. dissertation (University of Texas at Austin, 1988).

L. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley-Interscience, 2007).

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

T. Davis, Direct Methods for Sparse Linear Systems (Society for Industrial and Applied Mathematics, 2006).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

J. Ripoll, “Light diffusion in turbid media with biomedical application,” Ph.D. dissertation (Universidad Autónoma de Madrid, 2000).

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Figures (6)

Fig. 1
Fig. 1

Refined mesh used for the calculations. The source and the point where the values of the fluence are collected are denoted by S and D, respectively.

Fig. 2
Fig. 2

Top: time resolved curve profiles of the fluence for the Monte Carlo simulation, the DE, and order N = 3 of our model (TD- p SP 3 ). Fluence values are determined at point D appearing in Fig. 1 for a medium with an absorption coefficient μ a = 0.04 cm 1 and reduced scattering coefficient μ s = 20 cm 1 . Bottom: absolute differences of each model with respect to the Monte Carlo simulation for the same medium.

Fig. 3
Fig. 3

Top: time resolved curve profiles of the fluence for the Monte Carlo simulation, the DE, and order N = 3 of our model (TD- p SP 3 ). Fluence values are determined at point D appearing in Fig. 1 for a medium with an absorption coefficient μ a = 1 cm 1 and a reduced scattering coefficient μ s = 10 cm 1 . Bottom: absolute differences of each model with respect to the Monte Carlo simulation for the same medium.

Fig. 4
Fig. 4

Values of Δ for different values of the inclusion’s absorption coefficient in the case of the 2D square: μ a , incl. = 0.05 , 0.1, and 1 cm 1 (left to right) at 120, 220, and 600 ps (upper, middle, and lower rows).

Fig. 5
Fig. 5

Mesh of the cylinder cut at half its height. The spherical absorbing inclusion is shown in black.

Fig. 6
Fig. 6

Values of Δ for different values of the inclusion’s absorption coefficient in the case of the cylinder: μ a , incl. = 0.05 , 0.1, and 1 cm 1 (left to right) at 6 ns .

Equations (72)

Equations on this page are rendered with MathJax. Learn more.

η ( r ) c t L ( r , s ^ , t ) + s ^ · r L ( r , s ^ , t ) + r ln η ( r ) · s ^ L ( r , s ^ , t ) = [ μ a ( r ) + μ s ( r ) + r · s ^ ] L ( r , s ^ , t ) + μ s ( r ) 4 π p ( r , s ^ , s ^ ) L ( r , s ^ , t ) d Ω + q ( r , s ^ , t ) ,
r · s ^ = μ d ( r ) s ^ · r ln η ( r ) ,
η c t L ( r , s ^ , t ) + s ^ · r L ( r , s ^ , t ) = [ μ a ( r ) + μ s ( r ) + r · s ^ ] L ( r , s ^ , t ) + μ s ( r ) 4 π p ( r , s ^ , s ^ ) L ( r , s ^ , t ) d Ω + q ( r , s ^ , t ) ,
η c t L r ( r , s ^ , t ) + s ^ · r L r ( r , s ^ , t ) = [ μ a ( r ) + μ s ( r ) + r · s ^ ] L r ( r , s ^ , t ) + q ( r , s ^ , t ) .
η c t L d ( r , s ^ , t ) + s ^ · r L d ( r , s ^ , t ) = [ μ a ( r ) + μ s ( r ) + r · s ^ ] L d ( r , s ^ , t ) + μ s ( r ) 4 π p ( r , s ^ , s ^ ) L d ( r , s ^ , t ) d Ω + Q ( r , t ) ,
Q ( r , t ) = μ s ( r ) 4 π p ( r , s ^ , s ^ ) L r ( r , s ^ , t ) d Ω .
p ( r , cos θ ) = j = 0 p j ( r ) P j ( cos θ ) ,
p j ( r ) = 2 2 j + 1 1 1 p ( r , cos θ ) P j ( cos θ ) dcos θ .
p HG ( r , cos θ ) = 1 g ( r ) 2 4 π [ 1 + g ( r ) 2 2 g ( r ) cos θ ] ,
p j ( r ) = g ( r ) j , j = 0 , , ,
L ( r , s ^ , t ) = B T ( r , s ^ , t ) + R F ( n ^ · s ^ ) L ( r , s ^ , t ) , r V , s ^ · n ^ < 0 ,
L r ( r , s ^ , t ) = B T ( r , s ^ , t ) , r V , s ^ · n ^ < 0 ,
L d ( r , s ^ , t ) = R F ( n ^ · s ^ ) L d ( r , s ^ , t ) , r V , s ^ · n ^ < 0.
J n = s ^ · n ^ > 0 [ 1 R F ( n ^ · s ^ ) ] ( n ^ · s ^ ) L ( r , s ^ , t ) d Ω ,
M f ( J n ) = M f { s ^ · n ^ > 0 [ 1 R F ( n ^ · s ^ ) ] ( n · s ^ ) L ( r , s ^ , t ) d Ω } .
η c t ψ ( r , s ^ , t ) + s ^ · r ψ ( r , s ^ , t ) = [ μ a ( r ) + μ s ( r ) + r · s ^ ] ψ ( r , s ^ , t ) + 4 π μ ˜ s ( r , s ^ · s ^ ) ψ ( r , s ^ , t ) d Ω + Q ( r , t ) ,
η c t ψ ( z , ϖ , t ) + ϖ z ψ ( z , ϖ , t ) = [ μ a ( z ) + μ s ( z ) + z ( s · k ) ] ψ ( z , ϖ , t ) + 1 1 μ ˜ s ( z , ϖ , ϖ ) ψ ( z , ϖ , t ) d ϖ + Q ( z , t ) 2 ,
ψ ( z , ϖ , t ) = B T ( z , ϖ , t ) + R F ( ϖ ) ψ ( z , ϖ , t ) , z S , 0 < ϖ 1 ,
η c t ψ n ( z , t ) + μ n ( z ) ψ n ( z , t ) + z [ n + 1 2 n + 1 ψ n + 1 ( z , t ) + n 2 n + 1 ψ n 1 ( z , t ) ] = Q ( z , t ) δ n , 0 , n = 0 , , ,
μ n ( z ) = μ a ( z ) + μ s ( z ) [ 1 g ( z ) n ] + μ d ( z ) ( 1 δ n , 0 )
η c | t J ( r , t ) | ( μ a ( r ) + μ s ( r ) ) | J ( r , t ) | ,
η c | t ψ n ( z , t ) | μ n ( z ) | ψ n ( z , t ) | ,
( η l n c ) 1 | ψ n ( z , t ) | | t ψ n ( z , t ) | 1 ,
ψ n ( z , t ) = 1 μ n ( z ) z [ n + 1 2 n + 1 ψ n + 1 ( z , t ) + n 2 n + 1 ψ n 1 ( z , t ) ] , n odd .
ψ N ( z , t ) = N ( 2 N + 1 ) μ N ( z ) ψ N 1 ( z , t ) z .
η c t ψ n ( z , t ) + μ n ( z ) ψ n ( z , t ) n + 1 2 n + 1 z { 1 μ n + 1 ( z ) z [ n + 2 2 n + 3 ψ n + 2 ( z , t ) + n + 1 2 n + 3 ψ n ( z , t ) ] } n 2 n + 1 z { 1 μ n 1 ( z ) z [ n 2 n 1 ψ n ( z , t ) + n 1 2 n 1 ψ n 2 ( z , t ) ] } = δ n , 0 Q ( z , t ) ,
η c t ψ n ( r , t ) + μ n ( r ) ψ n ( r , t ) n + 1 2 n + 1 · { 1 μ n + 1 ( r ) [ n + 2 2 n + 3 ψ n + 2 ( r , t ) + n + 1 2 n + 3 ψ n ( r , t ) ] } n 2 n + 1 · { 1 μ n 1 ( r ) [ n 2 n 1 ψ n ( r , t ) + n 1 2 n 1 ψ n 2 ( r , t ) ] } = δ n , 0 Q ( r , t ) ,
η c t ϕ ( r , t ) · [ D ( r ) ϕ ( r , t ) ] + μ a ( r ) ϕ ( r , t ) = Q ( r , t ) ,
Ψ ( r , t ) = T Φ ( r , t ) , Φ ( r , t ) = T 1 Ψ ( r , t ) ,
T = [ 1 2 3 8 15 16 35 0 1 3 4 15 8 35 0 0 1 5 6 35 0 0 0 1 7 ] , T 1 = [ 1 2 0 0 0 3 4 0 0 0 5 6 0 0 0 7 ] .
[ C + η c t T ] Φ ( r , t ) + D r Φ ( r , t ) = Q ( r , t ) .
[ C + if η c T ] Φ ( r , f ) + D r Φ ( r , f ) = Q ( r , f ) ,
A Φ ( r , t ) + B Φ ( r , t ) n ^ = S ( r , t ) ,
Φ h ( r , t ) = i = 1 d Φ i ( t ) u i ( r ) , u i ( r ) Ω h ,
[ K ˜ + M ˜ + Π ˜ + η T ˜ c t ] Φ ˜ ( t ) = F ˜ + Γ ˜ ,
[ K ˜ + M ˜ + Π ˜ + if η T ˜ c ] Φ ˜ ( f ) = F ˜ + Γ ˜ ,
diag ( K ˜ ) = [ K ˜ 1 , K ˜ 2 , K ˜ 3 , K ˜ 4 ] .
K ˜ k ( i , j ) = V 1 ( 4 k 1 ) μ 2 k 1 u i ( r ) u j ( r ) d V , k = 1 , , 4 , i , j = 1 , , d .
M ˜ = [ M ˜ 11 M ˜ 12 M ˜ 13 M ˜ 14 M ˜ 21 M ˜ 22 M ˜ 23 M ˜ 24 M ˜ 31 M ˜ 32 M ˜ 33 M ˜ 34 M ˜ 41 M ˜ 42 M ˜ 43 M ˜ 44 ] .
M ˜ k 1 , k 2 ( i , j ) = V C ( k 1 , k 2 ) u i ( r ) u j ( r ) d V , k 1 , k 2 = 1 , , 4 , j = 1 , , d ,
T ˜ = [ T ˜ 11 T ˜ 12 T ˜ 13 T ˜ 14 T ˜ 21 T ˜ 22 T ˜ 23 T ˜ 24 T ˜ 31 T ˜ 32 T ˜ 33 T ˜ 34 T ˜ 41 T ˜ 42 T ˜ 43 T ˜ 44 ] .
T ˜ k 1 , k 2 ( i , j ) = V T ( k 1 , k 2 ) u i ( r ) u j ( r ) d V , k 1 , k 2 = 1 , , 4 , i , j = 1 , , d ,
F ˜ = [ F ˜ 1 , F ˜ 2 , F ˜ 3 , F ˜ 4 ] T .
F ˜ k ( i ) = V Q ( k ) u i ( r ) d V , k = 1 , , 4 , i = 1 , , d ,
Π ˜ = [ Π ˜ 11 Π ˜ 12 Π ˜ 13 Π ˜ 14 Π ˜ 21 Π ˜ 22 Π ˜ 23 Π ˜ 24 Π ˜ 31 Π ˜ 32 Π ˜ 33 Π ˜ 34 Π ˜ 41 Π ˜ 42 Π ˜ 43 Π ˜ 44 ] .
Π ˜ k 1 , k 2 ( i , j ) = V [ Θ ( k 1 , k 2 ) ( 4 k 1 1 ) μ 2 k 1 1 ] u i ( r ) u j ( r ) d σ , k 1 , k 2 = 1 , , 4 , i , j = 1 , , d ,
Γ ˜ = [ Γ ˜ 1 , Γ ˜ 2 , Γ ˜ 3 , Γ ˜ 4 ] T .
Γ ˜ k ( i ) = V [ G ( k ) ( 4 k 1 ) μ 2 k 1 ] u i ( r ) d σ , k = 1 , , 4 , i = 1 , , d ,
( ρ K ˜ + ρ M ˜ + ρ Π ˜ + 1 Δ t η c T ˜ ) Φ ˜ m + 1 + ( ( 1 ρ ) K ˜ + ( 1 ρ ) M ˜ + ( 1 ρ ) Π ˜ 1 Δ t η c T ˜ ) Φ ˜ m = ρ ( F ˜ m + 1 + Γ ˜ m + 1 ) + ( 1 ρ ) ( F ˜ m + Γ ˜ m ) ,
W ˜ Φ ˜ m + 1 = η c T ~ Φ ˜ m + ϒ ˜ m + 1 ,
D r ( 2 ) Φ ( 2 ) ( r , t ) + [ C ( 2 ) + η c t T ( 2 ) ] Φ ( 2 ) ( r , t ) = Q ( 2 ) ( t ) ,
D r ( 2 ) = [ r [ 1 3 μ 1 ( r ) r ] 0 0 r [ 1 7 μ 3 ( r ) r ] ] , C ( 2 ) = [ μ 0 ( r ) 2 3 μ 0 ( r ) 2 3 μ 0 ( r ) 4 9 μ 0 ( r ) + 5 9 μ 2 ( r ) ] , Q ( 2 ) ( t ) = [ Q ( t ) 2 3 Q ( t ) ] .
W ˜ ( 2 ) Φ ˜ ( 2 ) m + 1 = η c T ˜ ( 2 ) Φ ˜ ( 2 ) m + ϒ ˜ ( 2 ) m + 1 .
η c ψ ( z , ϖ , t ) t + z [ ( s · k ) ψ ( z , ϖ , t ) ] = [ μ a ( z ) + μ s ( z ) ] ψ ( z , ϖ , t ) + 4 π μ ˜ s ( z , ϖ , ϖ ) ψ ( z , ϖ , t ) d ϖ + Q ( z , t ) 2 .
η c ψ 0 ( z , t ) t + ψ 1 ( z , t ) z = μ a ( z ) ψ 0 ( z , t ) + Q ( z , t ) 2 ,
μ ˜ s ( z , ϖ , ϖ ) = n = 0 ( 2 n + 1 2 ) μ s ( z ) g ( z ) n P n ( ϖ ) P n ( ϖ ) ,
ψ ( z , ϖ , t ) = n = 0 ( 2 n + 1 2 ) ψ n ( z , t ) P n ( ϖ ) ,
ψ n ( z , t ) = 1 1 ψ ( z , ϖ , t ) P n ( ϖ ) d ϖ .
η c n = 0 ( 2 n + 1 2 ) t ψ n ( z , t ) 1 1 P n ( ϖ ) P n ( ϖ ) d ϖ + n = 0 ( 2 n + 1 2 ) 1 1 z [ ( s · k ) ψ n ( z , t ) ] P n ( ϖ ) P n ( ϖ ) d ϖ + n = 0 ( 2 n + 1 2 ) { μ a ( z ) + μ s ( z ) [ 1 g ( z ) n ] ψ n ( z , t ) } 1 1 P n ( ϖ ) P n ( ϖ ) d ϖ = 0 , n = 1 , , .
n = 0 ( 2 n + 1 2 ) { μ d ψ n ( z , t ) 1 1 P n ( ϖ ) P n ( ϖ ) d ϖ + z ψ n ( z , t ) 1 1 ϖ P n ( ϖ ) P n ( ϖ ) d ϖ } ,
η c t ψ n ( z , t ) + { μ a ( z ) + μ d ( z ) + μ s ( z ) [ 1 g ( z ) n ] } ψ n ( z , t ) + z [ n + 1 2 n + 1 ψ n + 1 ( z , t ) + n 2 n + 1 ψ n 1 ( z , t ) ] = 0 , n = 1 , , .
η c t ψ n ( z , t ) + μ n ( z ) ψ n ( z , t ) + z [ n + 1 2 n + 1 ψ n + 1 ( z , t ) + n 2 n + 1 ψ n 1 ( z , t ) ] = Q ( z , t ) δ n , 0 , n = 0 , , ,
μ n ( z ) = μ a ( z ) + μ s ( z ) [ 1 g ( z ) n ] + μ d ( z ) ( 1 δ n , 0 )
1 1 P n ( ϖ ) P n ( ϖ ) d ϖ = 2 2 n + 1 δ n , n ,
ϖ P n ( ϖ ) = ( n + 1 2 n + 1 ) P n + 1 ( ϖ ) + ( n 2 n + 1 ) P n 1 ( ϖ ) .
diag ( D r ) = [ ( 1 3 μ 1 ) , ( 1 7 μ 3 ) , ( 1 11 μ 5 ) , ( 1 15 μ 7 ) ] ,
C ( , 1 ) = [ μ 0 ( r ) 2 3 μ 0 ( r ) 8 15 μ 0 ( r ) 16 35 μ 0 ( r ) ] , C ( , 2 ) = [ 2 3 μ 0 ( r ) 4 9 μ 0 ( r ) + 5 9 μ 2 ( r ) 16 45 μ 0 ( r ) 4 9 μ 2 ( r ) 32 105 μ 0 ( r ) + 8 21 μ 2 ( r ) ] , C ( , 3 ) = [ 8 15 μ 0 ( r ) 16 45 μ 0 ( r ) 4 9 μ 2 ( r ) 64 225 μ 0 ( r ) + 16 45 μ 2 ( r ) + 9 25 μ 4 ( r ) 128 525 μ 0 ( r ) 32 105 μ 2 ( r ) 54 175 μ 4 ( r ) ] , C ( , 4 ) = [ 16 35 μ 0 ( r ) 32 105 μ 0 ( r ) + 8 21 μ 2 ( r ) 128 525 μ 0 ( r ) 32 105 μ 2 ( r ) 54 175 μ 4 ( r ) 256 1225 μ 0 ( r ) + 64 245 μ 2 ( r ) + 324 1225 μ 4 ( r ) + 13 49 μ 6 ( r ) ] .
Q ( r , t ) = Q ( r , t ) [ 1 2 3 8 15 16 35 ] T .
A = [ 1 2 + A 1 1 8 C 1 1 16 E 1 5 128 G 1 1 8 C 2 7 24 + A 2 41 384 E 2 1 16 G 2 1 16 C 3 41 384 E 3 407 1920 A 3 233 2560 + G 3 5 128 C 4 1 16 E 4 233 2560 G 4 3023 17920 + A 4 ] ,
B = [ 1 + B 1 3 μ 1 ( r ) D 1 μ 3 ( r ) F 1 μ 5 ( r ) H 1 μ 7 ( r ) D 2 μ 1 ( r ) 1 + B 2 7 μ 3 ( r ) F 2 μ 5 ( r ) H 2 μ 7 ( r ) D 3 μ 1 ( r ) F 3 μ 3 ( r ) 1 + B 3 11 μ 5 ( r ) H 3 μ 7 ( r ) D 4 μ 1 ( r ) F 4 μ 3 ( r ) H 4 μ 5 ( r ) 1 + B 4 15 μ 7 ( r ) ] .
S = [ s ^ · n ^ < 0 B T ( r , s ^ , t ) 2 | s ^ · n ^ | d Ω s ^ · n ^ < 0 B T ( r , s ^ , t ) [ 5 | s ^ · n ^ | 3 3 | s ^ · n ^ | ] d Ω s ^ · n ^ < 0 B T ( r , s ^ , t ) [ 63 4 | s ^ · n ^ | 5 35 2 | s ^ · n ^ | 3 + 15 4 | s ^ · n ^ | ] d Ω s ^ · n ^ < 0 B T ( r , s ^ , t ) [ 429 8 | s ^ · n ^ | 7 693 8 | s ^ · n ^ | 5 + 315 8 | s ^ · n ^ | 3 35 8 | s ^ · n ^ | ] d Ω ] ,
R F , n = 0 π ( cos θ ) n R F ( cos θ ) d cos θ .

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