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]

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]

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]

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

[Crossref]

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]

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]

A. F. Kostko and V. A. Pavlov, “Location of the diffusing-photon source in a strongly scattering medium,” Appl. Opt. 36, 7577–7582 (1997).

[Crossref]

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).

[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]

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

[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]

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

[Crossref]

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

[Crossref]

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]

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

[Crossref]

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]

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]

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]

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

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

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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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]

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]

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).

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]

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]

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]

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

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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

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]

A. F. Kostko and V. A. Pavlov, “Location of the diffusing-photon source in a strongly scattering medium,” Appl. Opt. 36, 7577–7582 (1997).

[Crossref]

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]

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).

[Crossref]

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]

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]

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. 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. 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]

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]

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

[Crossref]

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

[Crossref]

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

[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]

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

[Crossref]

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]

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

[Crossref]

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

[Crossref]

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]

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.

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/ .

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

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

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.

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

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

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.