Abstract

The Green’s function of the time dependent radiative transfer equation for the semi-infinite medium is derived for the first time by a heuristic approach based on the extrapolated boundary condition and on an almost exact solution for the infinite medium. Monte Carlo simulations performed both in the simple case of isotropic scattering and of an isotropic point-like source, and in the more realistic case of anisotropic scattering and pencil beam source, are used to validate the heuristic Green’s function. Except for the very early times, the proposed solution has an excellent accuracy (>98% for the isotropic case, and >97% for the anisotropic case) significantly better than the diffusion equation. The use of this solution could be extremely useful in the biomedical optics field where it can be directly employed in conditions where the use of the diffusion equation is limited, e.g. small volume samples, high absorption and/or low scattering media, short source-receiver distances and early times. Also it represents a first step to derive tools for other geometries (e.g. slab and slab with inhomogeneities inside) of practical interest for noninvasive spectroscopy and diffuse optical imaging. Moreover the proposed solution can be useful to several research fields where the study of a transport process is fundamental.

© 2007 Optical Society of America

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References

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  1. J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley&Sons, New York, 1979).
  2. S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  3. A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000).
    [CrossRef]
  4. T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007).
    [CrossRef]
  5. L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
    [CrossRef]
  6. 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] [PubMed]
  7. For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol. 49, N. 7 (2004).
    [PubMed]
  8. A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
    [CrossRef] [PubMed]
  9. 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] [PubMed]
  10. H. Dehghani, S. R. Arridge, and M. Schweiger, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17,1659-1670 (2000).
    [CrossRef]
  11. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
    [CrossRef] [PubMed]
  12. J. C. J. Paasschens, "Solution of the time-dependent Boltzmann equation," Phys. Rev. E 56, 1135-1141 (1997).
    [CrossRef]
  13. D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
    [CrossRef] [PubMed]
  14. E. Zauderer, Partial Differential Equations of Applied Mathematics, (John Wiley&Sons, New York, 1989) Sec. 7.5, p. 484.
  15. M. H. Lee, "Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion," Phys. Rev. Lett. 85, 2422-2425 (2000).
    [CrossRef] [PubMed]
  16. F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results," Appl. Opt. 30, 4600-4612 (1997).
    [CrossRef]
  17. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
    [CrossRef]
  18. F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005).
    [CrossRef] [PubMed]

2007

T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007).
[CrossRef]

2005

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

2004

For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol. 49, N. 7 (2004).
[PubMed]

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] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, and M. Schweiger, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17,1659-1670 (2000).
[CrossRef]

M. H. Lee, "Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion," Phys. Rev. Lett. 85, 2422-2425 (2000).
[CrossRef] [PubMed]

1999

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[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] [PubMed]

1997

1995

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

1994

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
[CrossRef]

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] [PubMed]

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] [PubMed]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, and M. Schweiger, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17,1659-1670 (2000).
[CrossRef]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[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] [PubMed]

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] [PubMed]

Battistelli, E.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
[CrossRef]

Bruscaglioni, P.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
[CrossRef]

Contini, D.

Cubeddu, R.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

Dasari, R. R.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

Dehghani, H.

Del Bianco, S.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

Edstr¨om, P.

T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007).
[CrossRef]

Feld, M. S.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

Feng, T.

T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007).
[CrossRef]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000).
[CrossRef]

Gulliksson, M.

T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007).
[CrossRef]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000).
[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] [PubMed]

Itzkan, I.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

Lee, M. H.

M. H. Lee, "Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion," Phys. Rev. Lett. 85, 2422-2425 (2000).
[CrossRef] [PubMed]

Martelli, F.

F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

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] [PubMed]

F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results," Appl. Opt. 30, 4600-4612 (1997).
[CrossRef]

D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
[CrossRef] [PubMed]

Ntziachristos, V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Paasschens, J. C. J.

J. C. J. Paasschens, "Solution of the time-dependent Boltzmann equation," Phys. Rev. E 56, 1135-1141 (1997).
[CrossRef]

Perelman, L. T.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

Pifferi, A.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

Ripoll, J.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Schweiger, M.

Spinelli, L.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

Taddeucci, A.

Torricelli, A.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

Wang, L. V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Wang, Y.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

Wei, Q. N.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
[CrossRef]

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Wu, J.

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

Zaccanti, G.

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

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] [PubMed]

F. Martelli, D. Contini, A. Taddeucci, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. II. Comparison with Monte Carlo results," Appl. Opt. 30, 4600-4612 (1997).
[CrossRef]

D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
[CrossRef] [PubMed]

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
[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] [PubMed]

Appl. Opt.

Inverse Probl.

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

T. Feng, P. Edstr¨om, and M. Gulliksson, "Levenberg-Marquardt methods for parameter estimation problems in the radiative transfer equation," Inverse Probl. 23, 879-891 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Biotechnol.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, "Looking and listening to light: the evolution of whole-body photonic imaging, " Nat. Biotechnol. 23, 313-320 (2005).
[CrossRef] [PubMed]

Phys. Med. Biol.

F. Martelli, S. Del Bianco, and G. Zaccanti, "Perturbation model for light propagation through diffusive layered media," Phys. Med. Biol. 50, 2159-2166 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2000).
[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] [PubMed]

For recent results: Special issue on recent development in biomedical optics, Phys. Med. Biol. 49, N. 7 (2004).
[PubMed]

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] [PubMed]

Phys. Rev. E

L. T. Perelman, J. Wu, Y. Wang, I. Itzkan, R. R. Dasari, and M. S. Feld, "Time-dependent photon migration using path integrals," Phys. Rev. E 51, 6134-6141 (1995).
[CrossRef]

J. C. J. Paasschens, "Solution of the time-dependent Boltzmann equation," Phys. Rev. E 56, 1135-1141 (1997).
[CrossRef]

Phys. Rev. Lett.

M. H. Lee, "Fick’s Law, Green-Kubo Formula, and Heisenberg’s Equation of Motion," Phys. Rev. Lett. 85, 2422-2425 (2000).
[CrossRef] [PubMed]

A. Torricelli, A. Pifferi, L. Spinelli, R. Cubeddu, F. Martelli, S. Del Bianco, and G. Zaccanti, "Time-resolved reflectance at null source-detector separation: Improving contrast and resolution in diffuse optical imaging," Phys. Rev. Lett. 95, 078101 (2005).
[CrossRef] [PubMed]

Pure Appl. Opt.

G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, "Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium," Pure Appl. Opt. 3, 897-905 (1994).
[CrossRef]

Other

E. Zauderer, Partial Differential Equations of Applied Mathematics, (John Wiley&Sons, New York, 1989) Sec. 7.5, p. 484.

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley&Sons, New York, 1979).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the medium, symbols used and positions of real and image sources, r +and, r -, respectively.

Fig. 2.
Fig. 2.

Comparison between the results of MC, RTE and DE for the time-resolved reflectance from a non absorbing semi-infinite medium with µ′s =1 mm-1, ρ=1 and 5 mm, where an isotropic point-like source at depth z 0=1/µ′s and isotropic scattering is assumed. Figures (a) and (b) show the reflectance and figures (c) and (d) show the ratio of the MC results to those of RTE and DE.

Fig. 3.
Fig. 3.

Comparison between the results of MC, RTE and DE for the time-resolved reflectance from a non absorbing semi-infinite medium with µ′s =1 mm-1, ρ=1 and 5 mm. The MC simulations assume a pencil beam impinging the medium at z=0 and anisotropic scattering with g=0.9. Equation (4) for the RTE and Eq. (36) of Ref. [13] for the DE have been used with z 0=1/µ′s . Figures (a) and (b) show the reflectance and figures (c) and (d) show the ratio of the MC results to those of RTE and DE.

Equations (4)

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

Φ i ( r , t ) v e μ s vt 4 π r 2 δ ( r vt ) + v ( 1 ( r vt ) 2 ) 1 8 ( 4 π vt 3 μ s ) 3 2 G ( μ s vt [ 1 ( r vt ) 2 ] 3 4 ) e μ s vt Θ ( r vt ) ,
G ( x ) = 8 ( 3 x ) 3 2 N = 1 Γ ( 3 4 N + 3 2 ) Γ ( 3 4 N ) x N N ! e x 1 + 2.026 x .
{ Φ si ( x , y , z , z 0 , t ) = Φ i ( r r + , t ) Φ i ( r r , t ) , r r + = ( x 2 + y 2 + ( z z 0 ) 2 ) 1 2 , r r = ( x 2 + y 2 + ( z + z 0 + 2 z e ) 2 ) 1 2 .
{ R ( ρ , t ) = D Φ si ( x , y , z = 0 , z 0 , t ) z , ρ = x 2 + y 2 , D = 1 ( 3 μ s ) .

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