Abstract

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  33. 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] [PubMed]
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    [CrossRef]
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    [CrossRef]
  36. M. Schweiger, S.R. Arridge, M. Hiraoka, and D.T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22, 1779–1792 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]

2011 (5)

G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inv. Prob.27, 075003 (2011).
[CrossRef]

A.Q. Bauer, R.E. Nothdurft, T.N. Erpelding, L.V. Wang, and J.P. Culver, “Quantitative photoacoustic imaging: correcting for heterogeneous light fluence distributions using diffuse optical tomography,” J. Biomed. Opt.16, 096016 (2011).
[CrossRef] [PubMed]

A.D. Kim and M. Moscoso, “Diffusion of polarized light,” Multiscale Model. Simul.9, 1624–1645 (2011).
[CrossRef]

T. Tarvainen, V. Kolehmainen, S.R. Arridge, and J.P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spect. Rad. Trans.1122600–2608 (2011).
[CrossRef]

A.D. Kim, “Correcting the diffusion approximation at the boundary,” J. Opt. Soc. Am. A28, 1007–1015 (2011).
[CrossRef]

2009 (2)

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys38, 149–192 (2009).
[CrossRef]

S.R. Arridge and J.C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Prob.25123010 (2009).
[CrossRef]

2006 (2)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Num. Math. Eng.65, 383–405 (2006).
[CrossRef]

B.T. Cox, S.R. Arridge, K.P. Köstli, and P.C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt.45, 1866–1875 (2006).
[CrossRef] [PubMed]

2005 (3)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J.P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt.44, 876–886 (2005).
[CrossRef] [PubMed]

A.P. Gibson, J.C. Hebden, and S.R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50R1–R43 (2005).
[CrossRef] [PubMed]

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] [PubMed]

2004 (1)

2003 (2)

A.D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A20, 92–98 (2003).
[CrossRef]

J. Heino, S.R. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E68, 031908 (2003).
[CrossRef]

2001 (1)

S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements,” Astron. Astrophys.380, 776–788 (2001).
[CrossRef]

2000 (1)

1999 (5)

R. Aronson and N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A16, 1066–1071 (1999).
[CrossRef]

J. Ripoll and M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse-diffuse interfaces,” J. Opt. Soc. Am. A16, 1947–1957 (1999).
[CrossRef]

V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
[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 Media9, 489–499 (1999).
[CrossRef]

S.R. Arridge, “Optical tomography in medical imaging,” Inv. Prob.15, R41–R93 (1999).
[CrossRef]

1998 (1)

G. Kanschat, “A robust finite element discretization for radiative transfer problems with scattering,” East West J. Num. Math.6, 265–272 (1998).

1997 (2)

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] [PubMed]

M. Schweiger and S.R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys.24, 895–902 (1997).
[CrossRef] [PubMed]

1995 (4)

K.D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys.22, 691–702 (1995).
[CrossRef] [PubMed]

G.C. Pomraning and B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusion theory,” Ann. Nucl. Energy22, 787–817 (1995).
[CrossRef]

R. Aronson, “Boundary conditions for the diffusion of light,” J. Opt. Soc. Am. A12, 2532–2539 (1995).
[CrossRef]

M. Schweiger, S.R. Arridge, M. Hiraoka, and D.T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22, 1779–1792 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (3)

L.-H. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A10, 1746–1752 (1993).
[CrossRef]

S.R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–299 (1993).
[CrossRef] [PubMed]

R. Aronson, “Extrapolation distance for diffusion of light,” Proc. SPIE1888, 297–304 (1993).
[CrossRef]

1988 (1)

1975 (1)

G.J. Habetler and B. J. Matkowsky, “Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation,” J. Math. Phys.16, 846–854 (1975).
[CrossRef]

1974 (1)

E.W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys.15, 75–81 (1974).
[CrossRef]

1941 (1)

L.G. Henyey and J.L. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

Aronson, R.

Arridge, S.R.

T. Tarvainen, V. Kolehmainen, S.R. Arridge, and J.P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spect. Rad. Trans.1122600–2608 (2011).
[CrossRef]

S.R. Arridge and J.C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Prob.25123010 (2009).
[CrossRef]

B.T. Cox, S.R. Arridge, K.P. Köstli, and P.C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt.45, 1866–1875 (2006).
[CrossRef] [PubMed]

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] [PubMed]

A.P. Gibson, J.C. Hebden, and S.R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50R1–R43 (2005).
[CrossRef] [PubMed]

J. Heino, S.R. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E68, 031908 (2003).
[CrossRef]

S.R. Arridge, “Optical tomography in medical imaging,” Inv. Prob.15, R41–R93 (1999).
[CrossRef]

V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
[CrossRef]

M. Schweiger and S.R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys.24, 895–902 (1997).
[CrossRef] [PubMed]

M. Schweiger, S.R. Arridge, M. Hiraoka, and D.T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22, 1779–1792 (1995).
[CrossRef] [PubMed]

S.R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–299 (1993).
[CrossRef] [PubMed]

Bal, G.

G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inv. Prob.27, 075003 (2011).
[CrossRef]

Bauer, A.Q.

A.Q. Bauer, R.E. Nothdurft, T.N. Erpelding, L.V. Wang, and J.P. Culver, “Quantitative photoacoustic imaging: correcting for heterogeneous light fluence distributions using diffuse optical tomography,” J. Biomed. Opt.16, 096016 (2011).
[CrossRef] [PubMed]

Beard, P.C.

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 Media9, 489–499 (1999).
[CrossRef]

Corngold, N.

Cox, B.T.

Culver, J.P.

A.Q. Bauer, R.E. Nothdurft, T.N. Erpelding, L.V. Wang, and J.P. Culver, “Quantitative photoacoustic imaging: correcting for heterogeneous light fluence distributions using diffuse optical tomography,” J. Biomed. Opt.16, 096016 (2011).
[CrossRef] [PubMed]

Delpy, D. T.

S.R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–299 (1993).
[CrossRef] [PubMed]

Delpy, D.T.

M. Schweiger, S.R. Arridge, M. Hiraoka, and D.T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22, 1779–1792 (1995).
[CrossRef] [PubMed]

Erpelding, T.N.

A.Q. Bauer, R.E. Nothdurft, T.N. Erpelding, L.V. Wang, and J.P. Culver, “Quantitative photoacoustic imaging: correcting for heterogeneous light fluence distributions using diffuse optical tomography,” J. Biomed. Opt.16, 096016 (2011).
[CrossRef] [PubMed]

Fantini, S.

Feng, T.-C.

Franceschini, M. A.

Ganapol, B. D.

G.C. Pomraning and B. D. Ganapol, “Asymptotically consistent reflection boundary conditions for diffusion theory,” Ann. Nucl. Energy22, 787–817 (1995).
[CrossRef]

Gao, H.

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys38, 149–192 (2009).
[CrossRef]

Gibson, A.P.

A.P. Gibson, J.C. Hebden, and S.R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50R1–R43 (2005).
[CrossRef] [PubMed]

Gratton, E.

Greenstein, J.L.

L.G. Henyey and J.L. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

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 Media9, 489–499 (1999).
[CrossRef]

Habetler, G.J.

G.J. Habetler and B. J. Matkowsky, “Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation,” J. Math. Phys.16, 846–854 (1975).
[CrossRef]

Haskell, R. C.

Hebden, J.C.

A.P. Gibson, J.C. Hebden, and S.R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50R1–R43 (2005).
[CrossRef] [PubMed]

Heino, J.

J. Heino, S.R. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E68, 031908 (2003).
[CrossRef]

Henyey, L.G.

L.G. Henyey and J.L. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

Hiraoka, M.

M. Schweiger, S.R. Arridge, M. Hiraoka, and D.T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22, 1779–1792 (1995).
[CrossRef] [PubMed]

S.R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–299 (1993).
[CrossRef] [PubMed]

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 Media9, 489–499 (1999).
[CrossRef]

Ishimaru, A.

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

Jacques, S. L.

Jiang, H.

K.D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys.22, 691–702 (1995).
[CrossRef] [PubMed]

Kaipio, J. P.

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Num. Math. Eng.65, 383–405 (2006).
[CrossRef]

Kaipio, J.P.

T. Tarvainen, V. Kolehmainen, S.R. Arridge, and J.P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spect. Rad. Trans.1122600–2608 (2011).
[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] [PubMed]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J.P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt.44, 876–886 (2005).
[CrossRef] [PubMed]

V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
[CrossRef]

Kanschat, G.

S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements,” Astron. Astrophys.380, 776–788 (2001).
[CrossRef]

G. Kanschat, “A robust finite element discretization for radiative transfer problems with scattering,” East West J. Num. Math.6, 265–272 (1998).

Keijzer, M.

Keller, J. B.

A.D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A20, 92–98 (2003).
[CrossRef]

E.W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys.15, 75–81 (1974).
[CrossRef]

Kim, A.D.

Kolehmainen, V.

T. Tarvainen, V. Kolehmainen, S.R. Arridge, and J.P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spect. Rad. Trans.1122600–2608 (2011).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Num. Math. Eng.65, 383–405 (2006).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J.P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt.44, 876–886 (2005).
[CrossRef] [PubMed]

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] [PubMed]

V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
[CrossRef]

Köstli, K.P.

Kryzhevoi, N.

S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements,” Astron. Astrophys.380, 776–788 (2001).
[CrossRef]

Larsen, E.W.

E.W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys.15, 75–81 (1974).
[CrossRef]

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 Media9, 489–499 (1999).
[CrossRef]

Lionheart, W.R.B

V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
[CrossRef]

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 Media9, 489–499 (1999).
[CrossRef]

Matkowsky, B. J.

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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 Media9, 489–499 (1999).
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[CrossRef]

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G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inv. Prob.27, 075003 (2011).
[CrossRef]

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S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements,” Astron. Astrophys.380, 776–788 (2001).
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[CrossRef]

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T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Num. Math. Eng.65, 383–405 (2006).
[CrossRef]

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J.P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt.44, 876–886 (2005).
[CrossRef] [PubMed]

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).
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V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
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Appl. Opt. (5)

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S. Richling, E. Meinköhn, N. Kryzhevoi, and G. Kanschat, “Radiative transfer with finite elements,” Astron. Astrophys.380, 776–788 (2001).
[CrossRef]

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T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Num. Math. Eng.65, 383–405 (2006).
[CrossRef]

Inv. Prob. (4)

V. Kolehmainen, S.R. Arridge, W.R.B Lionheart, M. Vauhkonen, and J.P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data,” Inv. Prob.15, 1375–1391 (1999).
[CrossRef]

S.R. Arridge, “Optical tomography in medical imaging,” Inv. Prob.15, R41–R93 (1999).
[CrossRef]

S.R. Arridge and J.C. Schotland, “Optical tomography: forward and inverse problems,” Inv. Prob.25123010 (2009).
[CrossRef]

G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime,” Inv. Prob.27, 075003 (2011).
[CrossRef]

J. Biomed. Opt. (1)

A.Q. Bauer, R.E. Nothdurft, T.N. Erpelding, L.V. Wang, and J.P. Culver, “Quantitative photoacoustic imaging: correcting for heterogeneous light fluence distributions using diffuse optical tomography,” J. Biomed. Opt.16, 096016 (2011).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

J. Opt. Soc. Am. A (8)

J. Quant. Spect. Rad. Trans. (1)

T. Tarvainen, V. Kolehmainen, S.R. Arridge, and J.P. Kaipio, “Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model,” J. Quant. Spect. Rad. Trans.1122600–2608 (2011).
[CrossRef]

Med. Phys. (4)

M. Schweiger, S.R. Arridge, M. Hiraoka, and D.T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22, 1779–1792 (1995).
[CrossRef] [PubMed]

S.R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20, 299–299 (1993).
[CrossRef] [PubMed]

K.D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys.22, 691–702 (1995).
[CrossRef] [PubMed]

M. Schweiger and S.R. Arridge, “The finite-element method for the propagation of light in scattering media: frequency domain case,” Med. Phys.24, 895–902 (1997).
[CrossRef] [PubMed]

Multiscale Model. Simul. (1)

A.D. Kim and M. Moscoso, “Diffusion of polarized light,” Multiscale Model. Simul.9, 1624–1645 (2011).
[CrossRef]

Phys. Med. Biol (1)

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).
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A.P. Gibson, J.C. Hebden, and S.R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50R1–R43 (2005).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Heino, S.R. Arridge, J. Sikora, and E. Somersalo, “Anisotropic effects in highly scattering media,” Phys. Rev. E68, 031908 (2003).
[CrossRef]

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R. Aronson, “Extrapolation distance for diffusion of light,” Proc. SPIE1888, 297–304 (1993).
[CrossRef]

Transp. Theory Stat. Phys (1)

H. Gao and H. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys38, 149–192 (2009).
[CrossRef]

Waves Random 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 Media9, 489–499 (1999).
[CrossRef]

Other (2)

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

L.V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, Hoboken, NJ, 2007).

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Figures (11)

Fig. 1
Fig. 1

Radiance at the boundary point (x,y) = (0,−20) computed using the cDA, the DA and the RTE for μs = 5 mm−1 with matched (left) and mismatched refractive indices (right).

Fig. 2
Fig. 2

Logarithm of exitance at the boundary of the domain computed using the cDA, the DA and the RTE for different values of μs with matched refractive indices at the boundary.

Fig. 3
Fig. 3

Percent relative error of fluence rate computed using the cDA (top row) and the DA (bottom row) for different values of μs with matched refractive indices at the boundary. The values are cut at 10%.

Fig. 4
Fig. 4

Percent relative error of exitance at the boundary computed using the cDA and the DA for different values of μs with matched refractive indices.

Fig. 5
Fig. 5

Logarithm of exitance at the boundary of the domain computed using the cDA, the DA and the RTE for different values of μs with mismatched refractive indices.

Fig. 6
Fig. 6

Percent relative error of fluence rate computed using the cDA (top row) and the DA (bottom row) for different values of μs with mismatched refractive indices. The values are cut at 10%.

Fig. 7
Fig. 7

Percent relative error of exitance computed using the cDA and the DA for different values of μs with mismatched refractive indices.

Fig. 8
Fig. 8

Optical parameters for the heterogeneous test case.

Fig. 9
Fig. 9

Logarithm of exitance at the boundary of the domain computed using the cDA, the DA and the RTE for the heterogeneous test case.

Fig. 10
Fig. 10

Percent relative error of fluence rate computed using the cDA (left) and the DA (right) for the heterogeneous test case. The values are cut at 10%.

Fig. 11
Fig. 11

Percent relative error of exitance for the heterogeneous test case.

Tables (3)

Tables Icon

Table 1 The mean of the relative error of fluence rate ΔΦ0 (%) and exitance ΔΓ(%) computed using the cDA and the DA for different values of μs(mm−1) and asymptotic parameter ε with matched refractive indices.

Tables Icon

Table 2 The computation times of the models for different values of μs.

Tables Icon

Table 3 The mean of the relative error of fluence rate ΔΦ0 (%) and exitance ΔΓ(%) computed using the cDA and the DA for different values of μs and asymptotic parameter ε with mismatched refractive indices.

Equations (76)

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s ^ ϕ + μ a ϕ + μ s ϕ = 0
ϕ = ϕ S n 1 Θ ( s ^ s ^ ) ϕ ( r , s ^ ) d s ^ .
ϕ = ϕ 0 + 𝒭 ϕ on Γ in = { ( r , s ^ ) Ω × S n 1 , s ^ n ^ < 0 }
Γ ( r b ) = s ^ n ^ > 0 T ( s ^ n ^ ) ϕ ( r b , K ( s ^ ) ) d s ^ , r b Ω ,
ϒ ( r ) = S n 1 ϕ ( r , s ^ ) d s ^ , r Ω .
μ a = ε α , μ s = ε 1 σ .
ε s ^ ϕ + ε 2 α ϕ + σ ϕ = 0 .
ϕ = Φ + Ψ .
Φ = Φ 0 + ε Φ 1 + O ( ε 2 ) ,
( κ Φ 0 ) α Φ 0 = 0.
κ = [ n σ ( 1 g ) ] 1 ,
g = S n 1 ( s ^ s ^ ) Θ ( s ^ s ^ ) d s ^ .
μ ζ Ψ 0 + σ ¯ Ψ 0 = 0 , in  ζ > 0 ,
Ψ 0 | ζ = 0 = ϕ 0 + 𝒭 Ψ 0 Φ 0 + 𝒭 Φ 0 , on 0 < μ 1 .
μ ζ Ψ 1 + σ ¯ Ψ 1 = s ^ Ψ 0 , in ζ > 0 ,
Ψ 1 | ζ = 0 = 𝒭 Ψ 1 Φ 1 + 𝒭 Φ 1 , on 0 < μ 1 .
Ψ 0 , Ψ 1 0 , ζ .
𝒫 [ Φ 0 𝒭 Φ 0 + Φ 1 𝒭 Φ 1 ϕ 0 ] = 0.
a Φ 0 + b κ n ^ Φ 0 = f , on  Ω ,
a = 𝒫 [ 1 R ( μ ) ] ,
b = ε n 𝒫 [ μ ( 1 + R ( μ ) ) ] ,
f = 𝒫 [ ϕ 0 ( r b , μ ) ] .
μ ζ Ψ + σ ¯ Ψ = 0 , in ζ > 0 ,
Ψ ( 0 , μ ) = ϕ 0 ( r b , μ ) + R ( μ ) Ψ ( 0 , μ ) [ 1 R ( μ ) ] Φ 0 ( r b ) + ε n κ μ [ 1 + R ( μ ) ] n ^ Φ 0 ( r b ) , on 0 < μ 1 .
ϕ ( r , s ^ b ) Φ 0 ( r b ) ε n κ s ^ Φ 0 ( r b ) + Ψ ( 0 , μ ) .
cos θ ζ Ψ + σ ¯ Ψ σ ¯ π π Θ ( θ θ ) Ψ ( ζ , θ ) d θ = 0 , in  ζ > 0 ,
Ψ | ζ = 0 = ψ + 𝒭 Ψ , on π / 2 < θ < π / 2.
Θ ( θ θ ) = 1 2 π 1 g 2 1 + g 2 2 g cos ( θ θ ) .
Ψ ( ζ , θ ) = Ψ ( ζ , θ ) .
π π Θ ( θ θ ) Ψ ( ζ , θ ) d θ = 0 π [ Θ ( θ + θ ) + Θ ( θ θ ) ] Ψ ( ζ , θ ) d θ .
μ ζ Ψ + σ ¯ Ψ σ ¯ 1 1 h ( μ , μ ) Ψ ( ζ , μ ) d μ ( 1 μ 2 ) 1 / 2 = 0 , in  ζ > 0 ,
Ψ ( 0 , μ ) = ψ ( μ ) + R ( μ ) Ψ ( 0 , μ ) , on 0 < μ 1.
h ( μ , μ ) = 1 2 π 1 g 2 1 + g 2 2 g ( μ μ ( 1 μ 2 ) 1 / 2 ( 1 μ 2 ) 1 / 2 ) + 1 2 π 1 g 2 1 + g 2 2 g ( μ μ + ( 1 μ 2 ) 1 / 2 ( 1 μ 2 ) 1 / 2 ) .
λ μ V + σ ¯ V σ ¯ 1 1 h ( μ , μ ) V ( μ ) d μ ( 1 μ 2 ) 1 / 2 = 0.
1 1 f ( μ ) d μ ( 1 μ 2 ) 1 / 2 π N j = 1 N f ( μ j ) ,
μ j = cos ( π 2 ( N j ) 1 2 ( N 1 ) + 2 ) , j = 1. , N .
λ μ i V ( μ i ) + ( δ + σ ¯ ) V ( μ i ) σ ¯ π N j = 1 N h ( μ , μ j ) V ( μ j ) = 0 , i = 1 , N .
λ N / 2 λ N / 2 + 1 λ 1 λ 1 λ N / 2 1 λ N / 2 .
π N i = 1 N μ i V m ( μ i ) V n ( μ i ) = 0 , m n .
π N i = 1 N μ i V n ( μ i ) V n ( μ i ) = { 1 n > 0 , + 1 n < 0.
p i = [ V 1 ( μ i ) n = 1 N / 2 y 1 n V n ( μ i ) ] μ i
m = 1 N / 2 [ V m ( μ i ) R ( μ i ) V m ( μ i ) ] y m n = [ V n ( μ i ) R ( μ i ) V n ( μ i ) ] , i = N / 2 + 1 , , N .
a ( r b ) = π N i = N / 2 + 1 N p i [ 1 R ( μ i ) ] ,
b ( r b ) = n ε π N i = N / 2 + 1 N p i [ μ i + μ i R ( μ i ) ] ,
f ( r b ) = π N i = N / 2 + 1 N p i ϕ 0 ( r b , μ i ) .
H i j = m = 1 N / 2 V m ( μ i ) [ V m ( μ j ) n = 1 N / 2 y m n V n ( μ j ) ] μ j .
Ψ ( 0 , μ i ) = π N j = N / 2 + 1 N H i j [ ϕ 0 ( r b , μ j ) [ 1 R ( μ j ) ] Φ 0 ( r b ) + ε n κ μ j [ 1 + R ( μ j ) ] n ^ Φ 0 ( r b ) ] , i = 1 , , N .
Ω κ Φ 0 dr + Ω α Φ 0 v dr + Ω a b Φ 0 v d S = Ω f b v d S ,
( K + C + D ) c = G ,
K ( p , k ) = Ω κ ϑ k ( r ) ϑ p ( r ) dr
C ( p , k ) = Ω α ϑ k ( r ) ϑ p ( r ) dr
D ( p , k ) = Ω a b ϑ k ( r ) ϑ p ( r ) d S
G ( p ) = Ω f b ϑ p ( r ) d S .
ϕ 0 ( r b , s ^ ) = s ^ n ^ , s ^ n ^ < 0.
ε = ( μ s + μ a ) 1 L .
ϕ ( r , s ^ ) 1 | S n 1 | Φ 0 ( r ) n | S n 1 | ( s ^ κ DA Φ 0 ( r ) ) ,
κ DA = ( n ( μ a + μ s ( 1 g ) ) ) 1 .
κ DA Φ 0 ( r ) + μ a Φ 0 ( r ) = 0 ,
Φ 0 ( r ) + 1 2 γ n κ DA A Φ 0 ( r ) n ^ = I s γ n ,
A = 2 / ( 1 R 0 ) 1 + | cos ( θ c ) | 3 1 | cos ( θ c ) | 2 ,
s ^ r = H s ^ i ,
H = ( 2 n ^ n ^ T + I ) ,
s ^ i = H 1 s ^ r = H s ^ r .
ϕ ( r , s ^ ) = ϕ 0 ( r , s ^ ) + R ϕ ( r , H s ^ ) , s ^ n ^ < 0 ,
R = 1 2 ( n in cos φ i n out cos φ t n in cos φ i + n out cos φ t ) 2 + 1 2 ( n in cos φ t n out cos φ i n in cos φ t + n out cos φ i ) 2 ,
cos φ i = n ^ s ^ i ,
cos φ t = 1 ( n in n out ) 2 ( 1 ( cos φ i ) 2 )
Ω S n 1 s ^ v ( r , s ^ ) ϕ ( r , s ^ ) d s ^ dr + Ω δ S n 1 ( s ^ ϕ ( r , s ^ ) ) ( s ^ v ( r , s ^ ) ) d s ^ dr + Ω S n 1 ( s ^ n ^ ) + ϕ ( r , s ^ ) v ( r , s ^ ) d s ^ d S Ω S n 1 ( s ^ n ^ ) r ϕ ( r , H s ^ ) v ( r , s ^ ) d s ^ d S + Ω S n 1 ( μ s + μ a ) ϕ ( r , s ^ ) v ( r , s ^ ) d s ^ dr + Ω S n 1 δ ( μ s + μ a ) ϕ ( r , s ^ ) ( s ^ v ( r , s ^ ) ) d s ^ dr Ω S n 1 μ s s n 1 Θ ( s ^ s ^ ) ϕ ( r , s ^ ) d s ^ v ( r , s ^ ) d s ^ dr Ω S n 1 δ μ s s n 1 Θ ( s ^ s ^ ) ϕ ( r , s ^ ) d s ^ ( s ^ v ( r , s ^ ) ) d s ^ dr = Ω S n 1 ( s ^ n ^ ) ϕ 0 ( r , s ^ ) v ( r , s ^ ) d s ^ d S ,
( A 1 + A 2 + A 3 + A 4 ) β = b 1 υ 0 ,
A 1 ( h , s ) = Ω S n 1 s ^ υ j ( r ) υ m ( s ^ ) υ l ( s ^ ) d s ^ υ i ( r ) dr + Ω δ S n 1 ( s ^ υ i ( r ) ) ( s ^ υ j ( r ) υ l ( s ^ ) υ m ( s ^ ) d s ^ dr
A 2 ( h , s ) = Ω υ i ( r ) υ j ( r ) d S S n 1 ( s ^ n ^ ) + υ l ( s ^ ) υ m ( s ^ ) d s ^ Ω υ i ( r ) υ j ( r ) d S S n 1 ( s ^ n ^ ) r υ l ( H s ^ ) υ m ( s ^ ) d s ^
A 3 ( h , s ) = Ω ( μ s + μ a ) υ i ( r ) υ j ( r ) dr S n 1 υ l ( s ^ ) υ m ( s ^ ) d s ^ + Ω δ ( μ s + μ a ) υ i ( r ) S n 1 ( s ^ υ j ( r ) ) υ m ( s ^ ) υ l ( s ^ ) d s ^ dr
A 4 ( h , s ) = Ω μ s υ i ( r ) υ j ( r ) dr S n 1 S n 1 Θ ( s ^ s ^ ) υ l ( s ^ ) d s ^ υ m ( s ^ ) d s ^ Ω δ μ s S n 1 ( s ^ υ j ( r ) ) υ m ( s ^ ) s n 1 Θ ( s ^ s ^ ) υ l ( s ^ ) d s ^ d s ^ υ i ( r ) dr
b 1 ( h , s ) = Ω υ i ( r ) υ j ( r ) d S S n 1 ( s ^ n ^ ) υ l ( s ^ ) υ m ( s ^ ) d s ^ ,
s ^ t = n in n out s ^ i + ( cos φ t n in n out cos φ i ) n ^ .
ϕ t ( r , s ^ ) = { T ϕ ( r , K ( s ^ ) ) r Ω s ^ n ^ 0 , ϕ ( r , s ^ ) r Ω s ^ n ^ < 0 ϕ ( r , s ^ ) r Ω \ Ω ,

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