Abstract

We show that the diffusion approximation (DA) to the radiative transport equation, which is commonly used in biomedical optics to describe propagation of light in tissues, contains a previously unexplored adjustable parameter. This parameter is related to the rate of exponential decay of the reduced intensity. In conventional theories, there are two distinct choices for this parameter. However, neither of these choices is optimal. When the optimal value for the parameter is used, the resulting DA becomes much more accurate near the medium boundaries, e.g., at the depth of up to a few *, where * is the transport mean free path (typically, about 1 mm in tissues). We refer to the new adjustable parameter as the reduced extinction coefficient. The proposed technique can reduce the relative error of the predicted diffuse density of the optical energy from about 30% to less than 1%. The optimized DA can still be inaccurate very close to an interface or in some other physical situations. Still, the proposed development extends the applicability range of the DA significantly. This result can be useful, for instance, in tomographic imaging of relatively shallow (up to a few * deep) layers of tissues in the reflection geometry.

© 2018 Optical Society of America

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References

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  1. D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
    [Crossref]
  2. S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [Crossref]
  3. G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imag. 12, 594–601 (2003).
    [Crossref]
  4. A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
    [Crossref]
  5. G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
    [Crossref]
  6. A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. A 68, 1045–1050 (1978).
    [Crossref]
  7. We do not support the point of view that the DA is applicable only when μs≫μa, and believe that the diffuse propagation regime sets in sufficiently far from sources and boundaries for any nonzero μs. However, the definition of the diffusion coefficient can be in this case complicated and not derivable from the P1 approximation or asymptotic analysis. In this paper, we always work in the regime when μs≫μa so that the P1 expression for the diffusion coefficient is not in question.
  8. A. D. Kim and A. Ishimaru, “Optical diffusion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
    [Crossref]
  9. R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
    [Crossref]
  10. R. Elaloufi, R. Carminanti, and J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).
    [Crossref]
  11. R. Pierrat, J.-J. Greffet, and R. Carminati, “Photon diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 23, 1106–1110 (2006).
    [Crossref]
  12. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [Crossref]
  13. A. D. Kim, “Correcting the diffusion approximation at the boundary,” J. Opt. Soc. Am. A 28, 1007–1015 (2011).
    [Crossref]
  14. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26, 1291–1300 (2009).
    [Crossref]
  15. D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 23, 1502–1504 (1998).
    [Crossref]
  16. I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
    [Crossref]
  17. R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [Crossref]
  18. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Deghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
    [Crossref]
  19. C. Chen, J. Du, and L. Pan, “Extending the diffusion approximation to the boundary using an integrated diffusion model,” AIP Adv. 5, 067115 (2015).
    [Crossref]
  20. S. Fantini, M. A. Franceschini, and E. Gratton, “Effective source term in the diffusion equation for photon transport in turbid media,” Appl. Opt. 36, 156–163 (1997).
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  22. X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
    [Crossref]
  23. E. M. Sevick-Muraca and C. L. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
    [Crossref]
  24. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  25. As is well known, external radiation incident on a scattering medium can be described in the transport theory either by introducing inhomogeneous boundary condition with nonzero ingoing intensity or by introducing surface sources and using a homogeneous half-range boundary condition.
  26. J. R. Lorenzo, Principles of Diffuse Light Propagation (World Scientific, 2012).
  27. There are two main approaches to deriving the diffusion approximation: the one presented here and asymptotic analysis, which considers the limit of the RTE when μa/μs→0. A subtle difference between the two approaches is that, in the asymptotic analysis, the Fick’s law J=−D∇u holds everywhere without restriction, and the approximation for the intensity is formulated in terms of u and ∇u, e.g., [13]. In the approach used in this paper, a more general relation (6b) is obtained, which becomes equivalent to the Fick’s law only sufficiently far from the boundaries.
  28. This statement is not strictly true if the medium has a finite depth. But it can be true with exponential precision if the depth is sufficiently large.
  29. V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).
    [Crossref]
  30. https://www.cbica.upenn.edu/vmarkel/CODES/MC/ .
  31. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  32. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  33. D. J. Durian and J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
    [Crossref]

2015 (1)

C. Chen, J. Du, and L. Pan, “Extending the diffusion approximation to the boundary using an integrated diffusion model,” AIP Adv. 5, 067115 (2015).
[Crossref]

2011 (1)

2009 (3)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26, 1291–1300 (2009).
[Crossref]

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
[Crossref]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

2008 (1)

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

2006 (1)

2004 (1)

V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).
[Crossref]

2003 (2)

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imag. 12, 594–601 (2003).
[Crossref]

R. Elaloufi, R. Carminanti, and J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).
[Crossref]

2001 (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

2000 (1)

1999 (2)

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
[Crossref]

1998 (2)

1997 (3)

1995 (1)

1994 (2)

1992 (1)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[Crossref]

1978 (1)

A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. A 68, 1045–1050 (1978).
[Crossref]

Abdoulaev, G. S.

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imag. 12, 594–601 (2003).
[Crossref]

Aronson, R.

Arridge, S. R.

Bal, G.

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
[Crossref]

Boas, D. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Burch, C. L.

Cariou, J.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Carminanti, R.

Carminati, R.

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chandrasekhar, S.

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

Chen, C.

C. Chen, J. Du, and L. Pan, “Extending the diffusion approximation to the boundary using an integrated diffusion model,” AIP Adv. 5, 067115 (2015).
[Crossref]

Corngold, N.

Deghani, H.

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Du, J.

C. Chen, J. Du, and L. Pan, “Extending the diffusion approximation to the boundary using an integrated diffusion model,” AIP Adv. 5, 067115 (2015).
[Crossref]

Durian, D. J.

Elaloufi, R.

Fantini, S.

Feng, T. C.

Franceschini, M. A.

Freund, I.

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[Crossref]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Gratton, E.

Greffet, J.-J.

Guern, Y.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Haskell, R. C.

Hielscher, A. H.

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imag. 12, 594–601 (2003).
[Crossref]

Intes, X.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Ishimaru, A.

A. D. Kim and A. Ishimaru, “Optical diffusion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
[Crossref]

A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. A 68, 1045–1050 (1978).
[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Joshi, A.

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Kim, A. D.

Kostko, A. F.

Le Jeune, B.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Lorenzo, J. R.

J. R. Lorenzo, Principles of Diffuse Light Propagation (World Scientific, 2012).

Lotrian, J.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Machida, M.

Markel, V. A.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26, 1291–1300 (2009).
[Crossref]

V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).
[Crossref]

McAdams, M. S.

McGhee, J.

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Nieto-Vesperinas, M.

Pan, L.

C. Chen, J. Du, and L. Pan, “Extending the diffusion approximation to the boundary using an integrated diffusion model,” AIP Adv. 5, 067115 (2015).
[Crossref]

Panasyuk, G. Y.

Pavlov, V. A.

Pellen, F.

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Pierrat, R.

Rasmussen, J. C.

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

Ripoll, J.

Rudnick, J.

Schotland, J. C.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26, 1291–1300 (2009).
[Crossref]

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).
[Crossref]

Sevick-Muraca, E. M.

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

E. M. Sevick-Muraca and C. L. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
[Crossref]

Svaasand, L. O.

Tsay, T. T.

Wareing, T. A.

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

Zhang, Q.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

AIP Adv. (1)

C. Chen, J. Du, and L. Pan, “Extending the diffusion approximation to the boundary using an integrated diffusion model,” AIP Adv. 5, 067115 (2015).
[Crossref]

Appl. Opt. (3)

IEEE Signal Process. Mag. (1)

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57–75 (2001).
[Crossref]

Inverse Probl. (2)

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[Crossref]

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001 (2009).
[Crossref]

J. Electron. Imag. (1)

G. S. Abdoulaev and A. H. Hielscher, “Three-dimensional optical tomography with the equation of radiative transfer,” J. Electron. Imag. 12, 594–601 (2003).
[Crossref]

J. Opt. Soc. Am. A (10)

A. Ishimaru, “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. A 68, 1045–1050 (1978).
[Crossref]

J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, and H. Deghani, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[Crossref]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[Crossref]

R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
[Crossref]

D. J. Durian and J. Rudnick, “Photon migration at short times and distances and in cases of strong absorption,” J. Opt. Soc. Am. A 14, 235–245 (1997).
[Crossref]

R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[Crossref]

R. Elaloufi, R. Carminanti, and J.-J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20, 678–685 (2003).
[Crossref]

R. Pierrat, J.-J. Greffet, and R. Carminati, “Photon diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 23, 1106–1110 (2006).
[Crossref]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “Diffusion approximation revisited,” J. Opt. Soc. Am. A 26, 1291–1300 (2009).
[Crossref]

A. D. Kim, “Correcting the diffusion approximation at the boundary,” J. Opt. Soc. Am. A 28, 1007–1015 (2011).
[Crossref]

Opt. Lett. (2)

Phys. Med. Biol. (1)

A. Joshi, J. C. Rasmussen, E. M. Sevick-Muraca, T. A. Wareing, and J. McGhee, “Radiative transport-based frequency-domain fluorescence tomography,” Phys. Med. Biol. 53, 2069–2088 (2008).
[Crossref]

Phys. Rev. A (1)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[Crossref]

Phys. Rev. E (1)

V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).
[Crossref]

Waves Random Complex Media (1)

X. Intes, B. Le Jeune, F. Pellen, Y. Guern, J. Cariou, and J. Lotrian, “Localization of the virtual point source used in the diffusion approximation to model a collimated beam source,” Waves Random Complex Media 9, 489–499 (1999).
[Crossref]

Other (9)

We do not support the point of view that the DA is applicable only when μs≫μa, and believe that the diffuse propagation regime sets in sufficiently far from sources and boundaries for any nonzero μs. However, the definition of the diffusion coefficient can be in this case complicated and not derivable from the P1 approximation or asymptotic analysis. In this paper, we always work in the regime when μs≫μa so that the P1 expression for the diffusion coefficient is not in question.

https://www.cbica.upenn.edu/vmarkel/CODES/MC/ .

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

As is well known, external radiation incident on a scattering medium can be described in the transport theory either by introducing inhomogeneous boundary condition with nonzero ingoing intensity or by introducing surface sources and using a homogeneous half-range boundary condition.

J. R. Lorenzo, Principles of Diffuse Light Propagation (World Scientific, 2012).

There are two main approaches to deriving the diffusion approximation: the one presented here and asymptotic analysis, which considers the limit of the RTE when μa/μs→0. A subtle difference between the two approaches is that, in the asymptotic analysis, the Fick’s law J=−D∇u holds everywhere without restriction, and the approximation for the intensity is formulated in terms of u and ∇u, e.g., [13]. In the approach used in this paper, a more general relation (6b) is obtained, which becomes equivalent to the Fick’s law only sufficiently far from the boundaries.

This statement is not strictly true if the medium has a finite depth. But it can be true with exponential precision if the depth is sufficiently large.

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Figures (11)

Fig. 1.
Fig. 1.

(a) Total density u; (b) total current J; and (c) the diffuse component of the current Jd as functions of the normalized depth z/* for one-dimensional propagation in the half space z>0. Parameters of the medium are μa/μs=0.03/500 and g=0.98. The curves μ¯=μt and μ¯=μ* were computed according to Eq. (26) with μ¯ as labeled and =2*/3. The curves labeled “OPT” were computed according to the same formulas but for the optimal values of μ¯ and , which are in this case μ¯opt=2.84μ*, opt=0.69*.

Fig. 2.
Fig. 2.

Same as in Figs. 1(a) and 1(b), but for g=0.8; the diffuse component of the current is not shown. The optimal parameters in this case are μ¯opt=2.58μ*, opt=0.69*.

Fig. 3.
Fig. 3.

Same as in Figs.  1(a) and 1(b), but for twice stronger absorption, μa/μs=0.06/500; the diffuse component of the current is not shown. The optimal parameters in this case are μ¯opt=2.82μ*, opt=0.69*.

Fig. 4.
Fig. 4.

Same as in Figs.  1(a) and 1(b) and for the same medium parameters, but analytical curves were computed according to the conventional DA. The optimal parameters in this case are μ¯opt=1.31μ*, opt=0.99*. The curves μ¯=μ* in this figure and in Fig. 1 are identical.

Fig. 5.
Fig. 5.

Total density in finite slabs of varying width [L=10* (a), L=20* (b), and L=40* (c)] for the same medium as in Fig. 1. Monte Carlo simulations versus optimized DA. The curves labeled OPT have the same parameters μ¯opt and opt, as in Fig. 1. The total width of the slab corresponds to the interval of z shown in each plot.

Fig. 6.
Fig. 6.

Angular dependence of the intensity I(z,θ) at various depths z [z=0.025* (a), z=0.425* (b), z=0.975* (c), and z=1.425* (d)] for the same medium parameters as in Fig. 1. Monte Carlo simulations (MC) compared to the DA. Panel (a) also shows the angular dependence of the intensity back-reflected at the plane z=0 (BR). Not all MC data points are shown, and some of them lie outside of the areas of the plots. The thin blue lines that connect the MC data points are drawn to guide the eye.

Fig. 7.
Fig. 7.

Same as in Fig. 1, but for off-normal incidence [tanθ=0.5 (a), tanθ=1.0 (b), and tanθ=2.0 (c)]. Here θ is the angle between the collimation direction of incident radiation and normal to the surface; θ=0 for normal incidence. However, the front of the incident radiation is still infinitely broad, and the problem is one-dimensional.

Fig. 8.
Fig. 8.

Lateral current Jx for off-normal incidence. Same medium and simulation parameters and plot labeling as in Fig. 7.

Fig. 9.
Fig. 9.

Density as a function of depth z for the same medium parameters as in Fig. 1 but two different phase functions labeled as HG (Henyey–Greenstein) and EXP (exponential). MC and OPT label Monte Carlo simulations and optimized theoretical curves computed according to Eq. (28). Simulations were performed in a finite slab of width L=10*. Optimal parameters are different for the HG and OPT curves. For HG, the parameters are the same as in Fig. 1. For EXP, μ¯opt=4.38μ*, opt=0.71*.

Fig. 10.
Fig. 10.

Radial dependence of u at different depths z [z=1* (a), z=2* (b), z=4* (c), and z=8* (d)]. Incident cylindrical pencil beam has the radius of 0.5* (diameter of 1*). Labels of different curves and data sets are the same as in Fig. 1. Note that the discontinuity of solutions at the edge of the incident beam (at ρ=0.5*) is an artifact of the DA in which the reduced (discontinuous) density is added to the diffuse (continuous) density; the exact RTE solutions are continuous, albeit the radial derivative can be large at the edge of the incident beam.

Fig. 11.
Fig. 11.

Radial dependence of (a) the reflected density u and (b) the normal component of the current Jz, evaluated at the interface z=0. Same parameters as in Fig. 10.

Equations (60)

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(s^·+μt)I(r,s^)=μsA(s^,s^)I(r,s^)d2s+ϵ(r,s^).
A(s^,s^)d2s=2π11f(x)dx=1.
(s^·+μ¯)Ir(r,s^)=ϵ(r,s^),
Id(r,s^)14π[ud(r)+3s^·Jd(r)],
ud(r)=Id(r,s^)d2s,Jd(r)=s^Id(r,s^)d2s
·Jd+μaud=E(μ¯μa)ur,
13ud+μ*Jd=Q(μ¯μ*)Jr,
ur(r)=Ir(r,s^)d2s,Jr(r)=s^Ir(r,s^)d2s
·Dud+μaud=S,
S=E*·Q
n^·s^0(n^·s^)Id(r,s^)d2s|rΩ=0,
(ud2n^·Jd)|rΩ=0.
(ud3*n^·Jd)|rΩ=0,
(ud+n^·ud3n^·Q)|rΩ=0,
(ud+n^·ud)|rΩ=0.
u(r)=G(r,r)S(r)d3rG(r,rs)M0+G(r,r)r|r=rs·M1,
n^·G(r,r)r|r=rs=1G(r,rs).
M1=n^μ¯/μ*μa/μ¯μ¯μaM0.
u(r)(1+1μ¯/μ*μa/μ¯μ¯μa)G(r,rs)M0.
μ*M1=n^M0,
u(r)(1+*/)G(r,rs)M0.
E(z)=W(μ¯μa)exp(μ¯z),
Q(z)=W(μ¯μ*)exp(μ¯z)z^.
Jd(z)z+μaud(z)=W(μ¯μa)exp(μ¯z),
13ud(z)z+μ*Jd(z)=W(μ¯μ*)exp(μ¯z).
ud(0)+3*Jd(0)=0.
1Wud(z)=(A1)eμ¯z+Bekdz,
1WJd(z)=(μaμ¯A1)eμ¯z+μakdBekdz,
A=2μ¯2μ¯2kd2,kd=3μaμ*,
B=3μ¯2kd2μ¯2kd2+31+kd[μ*μ¯kd+kd2/3μ¯+kd].
1Wu(z)=Aeμ¯z+Bekdz,
1WJ(z)=μaμ¯Aeμ¯z+μakdBekdz.
A=μ¯23μ*μ¯μ¯2kd2,B=3μ*μ¯kd2μ¯2kd21+μ¯1+kd.
1Wu(z)=Aeμ¯z+B1ekdz+B2ekdz,
1WJ(z)=μaμ¯Aeμ¯z+μakd(B1ekdzB2ekdz).
B1=(1+kd)2(1+kd)2(1kd)2exp(2kdL)B,
B2=(1kd)[kd2(12μ¯+3μ*)3μ¯2(1+μ*)](μ¯2kd2)[(1+kd)2exp(2kdL)(1kd)2],
δλ(x)=λπ21(x1)2+λ2
z^·Jdz+μaud=W(μ¯μa)exp(μ¯z/cosθ),
z^3udz+μ*Jd=Ws^0(μ¯μ*)exp(μ¯z/cosθ).
ud(0)+3*Jdz(0)=0.
1Wu(z)=Aeμ¯z/cosθ+Bekdz,
1WJz(z)=μaμ¯Aeμ¯z/cosθ+μakdBekdz,
A=(31/cos2θ)μ¯2(μ¯/cosθ)2kd2,
B=3μ¯2kd2(μ¯/cosθ)2kd2kd1+kd3μ¯cosθ+kdμ¯/cosθ+kd+3μ*cosθ1+kd.
1WJx=μ¯/μ*sinθexp(μ¯z/cosθ).
A(s^,s^)=14π1g2(12gs^·s^+g2)3/2,
A(s^,s^)=β4πexp(βs^·s^)sinh(β).
ϵ(ρ,z)=w(ρ)δ(z)δ2(s,z^),w(ρ)d2ρ=W.
Ir(ρ,z)=w(ρ)eμ¯zδ2(s,z^).
E(ρ,z)=w(ρ)eμ¯z,Q(ρ,z)=z^E(ρ,z).
ud(ρ,z)=u˜d(q,z)eiq·ρd2q(2π)2
J˜dz(q,z)z+(q23μ*+μa)u˜d(q,z)=(μ¯μa)eμ¯zw˜(q),
13u˜d(q,z)z+μ*J˜dz(q,z)=(μ¯μ*)eμ¯zw˜(q).
J˜d(q,z)=iq3μ*u˜d(q,z).
u˜d(q,z)={[A(q)1]eμ¯z+B(q)eκ(q)z}w˜(q),
J˜dz(q,z)={[C(q)1]eμ¯z+κ(q)3μ*B(q)eκ(q)z}w˜(q),
A(q)=2μ¯2+q2μ¯2κ2(q),C(q)=2μaμ¯+(μ¯/μ*)q2μ¯2κ2(q),
B(q)=3μ¯2κ2(q)μ¯2kd2+31+κ(q)[μ*μ¯κ(q)+kd2/3μ¯+κ(q)],
κ(q)=kd2+q2.