Abstract

We introduce a generalized diffusion equation that models the propagation of photons in highly scattering domains with thin nonscattering clear layers. Classical diffusion models break down in the presence of clear layers. The model that we propose accurately accounts for the clear-layer effects and has a computational cost comparable to that of classical diffusion. It is based on modeling the propagation in the clear layer as a local tangential diffusion process. It can be justified mathematically in the limit of small mean free paths and is shown numerically to be very accurate in two- and three-dimensional idealized cases. We believe that this model can be used as an accurate forward model in optical tomography.

© 2003 Optical Society of America

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References

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  2. A. Y. Bluestone, G. S. Abdoulaev, C. Schmitz, R. L. Barbour, A. H. Hielscher, “Three-dimensional optical-tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001), optics-express.org .
    [CrossRef] [PubMed]
  3. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
    [CrossRef]
  4. V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. I. Fourier–Laplace inverse formula,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
    [CrossRef]
  5. V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Role of boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
    [CrossRef]
  6. G. J. Müller, ed., Medical Optical Tomography: Functional Imaging and Optical Technologies, IS Series Vol. IS11 (SPIE Press, Bellingham, Wash., 1993).
  7. F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modelling and Computation (SIAM, Philadelphia, Pa., 2001).
  8. “Session on Optical Tomography and Optical Imaging,” Progress in Electromagnetic Research Symposium (PIERS), July 1–5, 2002, Cambridge, Massachusetts.
  9. A. D. Klose, A. H. Hielscher, “Optical tomography using the time independent equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
    [CrossRef]
  10. R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Engineering (Springer-Verlag, Berlin, 1993), Vol. 6.
  11. E. W. Larsen, J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974).
    [CrossRef]
  12. A. H. Hielscher, R. E. Alcouffe, 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]
  13. J. Ripoll, M. Nieto-Vesperinas, S. R. Arridge, H. Deh-ghani, “Boundary conditions for light propagation in diffuse media with non-scattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
    [CrossRef]
  14. H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Deply, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
    [CrossRef]
  15. S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
    [CrossRef] [PubMed]
  16. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
    [CrossRef]
  17. G. Bal, Y. Maday, “Coupling of transport and diffusion models in linear transport theory,” M2AN Math. Model. Numer. Anal. 36, 69–86 (2002).
    [CrossRef]
  18. G. Bal, X. Warin, “Discrete ordinate methods in xy-geometry with spatially varying angular discretization,” Nucl. Sci. Eng. 127, 169–181 (1997).
  19. F. Golse, S. Jin, C. D. Levermore, “The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 1333–1369 (1999).
    [CrossRef]
  20. E. W. Larsen, J. E. Morel, W. F. Miller, “Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes,” J. Chem. Phys. 69, 283–324 (1987).
  21. M. Tidriri, “Asymptotic analysis of a coupled system of kinetic equations,” C. R. Acad. Sci. Paris Ser. I Math. t.328, 637–642 (1999).
    [CrossRef]
  22. G. Bal, “Particle transport through scattering regions with clear layers and inclusions,” J. Comput. Phys. 180, 659–685 (2002).
    [CrossRef]
  23. G. Bal, “Transport through diffusive and non-diffusive regions, embedded objects, and clear layers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677–1697 (2002).
    [CrossRef]
  24. M. Choulli, P. Stefanov, “Reconstruction of the coefficients of the stationary transport equation from boundary measurements,” Inverse Probl. 12, L19–L23 (1996).
    [CrossRef]
  25. A. Tamasan, “An inverse boundary value problem in two-dimensional transport,” Inverse Probl. 18, 209–219 (2002).
    [CrossRef]
  26. G. Bal, V. Freilikher, G. Papanicolaou, L. Ryzhik, “Wave transport along surfaces with random impedance,” Phys. Rev. B 62, 6228–6240 (2000).
    [CrossRef]
  27. J. M. Luck, T. M. Nieuwenhuizen, “Light scattering from mesoscopic objects in diffusive media,” Eur. Phys. J. B 7, 483–500 (1999).
    [CrossRef]
  28. M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
    [CrossRef] [PubMed]
  29. S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York, 2002).
  30. J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).

2002 (6)

V. A. Markel, J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Role of boundary conditions,” J. Opt. Soc. Am. A 19, 558–566 (2002).
[CrossRef]

A. D. Klose, A. H. Hielscher, “Optical tomography using the time independent equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

G. Bal, Y. Maday, “Coupling of transport and diffusion models in linear transport theory,” M2AN Math. Model. Numer. Anal. 36, 69–86 (2002).
[CrossRef]

G. Bal, “Particle transport through scattering regions with clear layers and inclusions,” J. Comput. Phys. 180, 659–685 (2002).
[CrossRef]

G. Bal, “Transport through diffusive and non-diffusive regions, embedded objects, and clear layers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677–1697 (2002).
[CrossRef]

A. Tamasan, “An inverse boundary value problem in two-dimensional transport,” Inverse Probl. 18, 209–219 (2002).
[CrossRef]

2001 (3)

2000 (3)

1999 (4)

J. M. Luck, T. M. Nieuwenhuizen, “Light scattering from mesoscopic objects in diffusive media,” Eur. Phys. J. B 7, 483–500 (1999).
[CrossRef]

M. Tidriri, “Asymptotic analysis of a coupled system of kinetic equations,” C. R. Acad. Sci. Paris Ser. I Math. t.328, 637–642 (1999).
[CrossRef]

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

F. Golse, S. Jin, C. D. Levermore, “The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 1333–1369 (1999).
[CrossRef]

1998 (2)

A. H. Hielscher, R. E. Alcouffe, 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]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

1997 (2)

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

G. Bal, X. Warin, “Discrete ordinate methods in xy-geometry with spatially varying angular discretization,” Nucl. Sci. Eng. 127, 169–181 (1997).

1996 (2)

M. Choulli, P. Stefanov, “Reconstruction of the coefficients of the stationary transport equation from boundary measurements,” Inverse Probl. 12, L19–L23 (1996).
[CrossRef]

M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

1987 (1)

E. W. Larsen, J. E. Morel, W. F. Miller, “Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes,” J. Chem. Phys. 69, 283–324 (1987).

1974 (1)

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

Abdoulaev, G. S.

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, 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]

Arridge, S. A.

M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Arridge, S. R.

Bal, G.

G. Bal, Y. Maday, “Coupling of transport and diffusion models in linear transport theory,” M2AN Math. Model. Numer. Anal. 36, 69–86 (2002).
[CrossRef]

G. Bal, “Particle transport through scattering regions with clear layers and inclusions,” J. Comput. Phys. 180, 659–685 (2002).
[CrossRef]

G. Bal, “Transport through diffusive and non-diffusive regions, embedded objects, and clear layers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677–1697 (2002).
[CrossRef]

G. Bal, V. Freilikher, G. Papanicolaou, L. Ryzhik, “Wave transport along surfaces with random impedance,” Phys. Rev. B 62, 6228–6240 (2000).
[CrossRef]

G. Bal, X. Warin, “Discrete ordinate methods in xy-geometry with spatially varying angular discretization,” Nucl. Sci. Eng. 127, 169–181 (1997).

Barbour, R. L.

A. Y. Bluestone, G. S. Abdoulaev, C. Schmitz, R. L. Barbour, A. H. Hielscher, “Three-dimensional optical-tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001), optics-express.org .
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, 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]

Bevilacqua, F.

Bluestone, A. Y.

Brenner, S. C.

S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York, 2002).

Choulli, M.

M. Choulli, P. Stefanov, “Reconstruction of the coefficients of the stationary transport equation from boundary measurements,” Inverse Probl. 12, L19–L23 (1996).
[CrossRef]

Dautray, R.

R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Engineering (Springer-Verlag, Berlin, 1993), Vol. 6.

Dehghani, H.

Deh-ghani, H.

Delpy, D. T.

M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Deply, D. T.

Dorn, O.

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

Dunn, A. K.

Firbank, M.

M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Freilikher, V.

G. Bal, V. Freilikher, G. Papanicolaou, L. Ryzhik, “Wave transport along surfaces with random impedance,” Phys. Rev. B 62, 6228–6240 (2000).
[CrossRef]

Gelbard, E. M.

J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).

Golse, F.

F. Golse, S. Jin, C. D. Levermore, “The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 1333–1369 (1999).
[CrossRef]

Hayakawa, C. K.

Hebden, J. C.

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

Hielscher, A. H.

A. D. Klose, A. H. Hielscher, “Optical tomography using the time independent equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

A. Y. Bluestone, G. S. Abdoulaev, C. Schmitz, R. L. Barbour, A. H. Hielscher, “Three-dimensional optical-tomography of hemodynamics in the human head,” Opt. Express 9, 272–286 (2001), optics-express.org .
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, 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]

Jin, S.

F. Golse, S. Jin, C. D. Levermore, “The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 1333–1369 (1999).
[CrossRef]

Keller, J. B.

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

Klose, A. D.

A. D. Klose, A. H. Hielscher, “Optical tomography using the time independent equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

Larsen, E. W.

E. W. Larsen, J. E. Morel, W. F. Miller, “Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes,” J. Chem. Phys. 69, 283–324 (1987).

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

Levermore, C. D.

F. Golse, S. Jin, C. D. Levermore, “The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 1333–1369 (1999).
[CrossRef]

Lions, J.-L.

R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Engineering (Springer-Verlag, Berlin, 1993), Vol. 6.

Luck, J. M.

J. M. Luck, T. M. Nieuwenhuizen, “Light scattering from mesoscopic objects in diffusive media,” Eur. Phys. J. B 7, 483–500 (1999).
[CrossRef]

Maday, Y.

G. Bal, Y. Maday, “Coupling of transport and diffusion models in linear transport theory,” M2AN Math. Model. Numer. Anal. 36, 69–86 (2002).
[CrossRef]

Markel, V. A.

Miller, W. F.

E. W. Larsen, J. E. Morel, W. F. Miller, “Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes,” J. Chem. Phys. 69, 283–324 (1987).

Morel, J. E.

E. W. Larsen, J. E. Morel, W. F. Miller, “Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes,” J. Chem. Phys. 69, 283–324 (1987).

Natterer, F.

F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modelling and Computation (SIAM, Philadelphia, Pa., 2001).

Nieto-Vesperinas, M.

Nieuwenhuizen, T. M.

J. M. Luck, T. M. Nieuwenhuizen, “Light scattering from mesoscopic objects in diffusive media,” Eur. Phys. J. B 7, 483–500 (1999).
[CrossRef]

Papanicolaou, G.

G. Bal, V. Freilikher, G. Papanicolaou, L. Ryzhik, “Wave transport along surfaces with random impedance,” Phys. Rev. B 62, 6228–6240 (2000).
[CrossRef]

Ripoll, J.

Ryzhik, L.

G. Bal, V. Freilikher, G. Papanicolaou, L. Ryzhik, “Wave transport along surfaces with random impedance,” Phys. Rev. B 62, 6228–6240 (2000).
[CrossRef]

Schmitz, C.

Schotland, J. C.

Schweiger, M.

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Deply, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[CrossRef]

M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Scott, L. R.

S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York, 2002).

Spanier, J.

Stefanov, P.

M. Choulli, P. Stefanov, “Reconstruction of the coefficients of the stationary transport equation from boundary measurements,” Inverse Probl. 12, L19–L23 (1996).
[CrossRef]

Tamasan, A.

A. Tamasan, “An inverse boundary value problem in two-dimensional transport,” Inverse Probl. 18, 209–219 (2002).
[CrossRef]

Tidriri, M.

M. Tidriri, “Asymptotic analysis of a coupled system of kinetic equations,” C. R. Acad. Sci. Paris Ser. I Math. t.328, 637–642 (1999).
[CrossRef]

Tromberg, B. J.

Venugopalan, V.

Warin, X.

G. Bal, X. Warin, “Discrete ordinate methods in xy-geometry with spatially varying angular discretization,” Nucl. Sci. Eng. 127, 169–181 (1997).

Wübbeling, F.

F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modelling and Computation (SIAM, Philadelphia, Pa., 2001).

You, J. S.

C. R. Acad. Sci. Paris Ser. I Math. (1)

M. Tidriri, “Asymptotic analysis of a coupled system of kinetic equations,” C. R. Acad. Sci. Paris Ser. I Math. t.328, 637–642 (1999).
[CrossRef]

Eur. Phys. J. B (1)

J. M. Luck, T. M. Nieuwenhuizen, “Light scattering from mesoscopic objects in diffusive media,” Eur. Phys. J. B 7, 483–500 (1999).
[CrossRef]

Inverse Probl. (4)

M. Choulli, P. Stefanov, “Reconstruction of the coefficients of the stationary transport equation from boundary measurements,” Inverse Probl. 12, L19–L23 (1996).
[CrossRef]

A. Tamasan, “An inverse boundary value problem in two-dimensional transport,” Inverse Probl. 18, 209–219 (2002).
[CrossRef]

O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14, 1107–1130 (1998).
[CrossRef]

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

J. Chem. Phys. (1)

E. W. Larsen, J. E. Morel, W. F. Miller, “Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes,” J. Chem. Phys. 69, 283–324 (1987).

J. Comput. Phys. (1)

G. Bal, “Particle transport through scattering regions with clear layers and inclusions,” J. Comput. Phys. 180, 659–685 (2002).
[CrossRef]

J. Math. Phys. (1)

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

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transf. (1)

A. D. Klose, A. H. Hielscher, “Optical tomography using the time independent equation of radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 72, 715–732 (2002).
[CrossRef]

M2AN Math. Model. Numer. Anal. (1)

G. Bal, Y. Maday, “Coupling of transport and diffusion models in linear transport theory,” M2AN Math. Model. Numer. Anal. 36, 69–86 (2002).
[CrossRef]

Nucl. Sci. Eng. (1)

G. Bal, X. Warin, “Discrete ordinate methods in xy-geometry with spatially varying angular discretization,” Nucl. Sci. Eng. 127, 169–181 (1997).

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (3)

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, 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]

M. Firbank, S. A. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with non-scattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Phys. Rev. B (1)

G. Bal, V. Freilikher, G. Papanicolaou, L. Ryzhik, “Wave transport along surfaces with random impedance,” Phys. Rev. B 62, 6228–6240 (2000).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

G. Bal, “Transport through diffusive and non-diffusive regions, embedded objects, and clear layers,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677–1697 (2002).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

F. Golse, S. Jin, C. D. Levermore, “The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 36, 1333–1369 (1999).
[CrossRef]

Other (6)

R. Dautray, J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Engineering (Springer-Verlag, Berlin, 1993), Vol. 6.

G. J. Müller, ed., Medical Optical Tomography: Functional Imaging and Optical Technologies, IS Series Vol. IS11 (SPIE Press, Bellingham, Wash., 1993).

F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modelling and Computation (SIAM, Philadelphia, Pa., 2001).

“Session on Optical Tomography and Optical Imaging,” Progress in Electromagnetic Research Symposium (PIERS), July 1–5, 2002, Cambridge, Massachusetts.

S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York, 2002).

J. Spanier, E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, Reading, Mass., 1969).

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Figures (4)

Fig. 1
Fig. 1

Local geometry of the clear layer.

Fig. 2
Fig. 2

Geometry of the two-dimensional setting and cross section of the geometry of the three-dimensional setting with azimuthal symmetry.

Fig. 3
Fig. 3

Plots of the current between cells 14 (70 deg) and 36 (180 deg) at the boundary of the unit disk (two-dimensional simulation) for the Monte Carlo solution and the different diffusion models. The thickness of the clear layer in mean free path is 2, 3, 5, and 7 for the top-left, top-right, bottom-left, and bottom-right figures, respectively. Solid circles, Monte Carlo simulation; open circles, classical diffusion model; solid curves, generalized diffusion model with theoretical tangential diffusion coefficient; dashed–dotted curves, generalized diffusion model with best fit; dotted curves, generalized model with large tangential diffusion coefficient. The inset represents a magnification of the above results between angles 125 and 145.

Fig. 4
Fig. 4

Plots of the current between cells 15 (75 deg) and 32 (160 deg) at the boundary of the unit sphere (three-dimensional simulation with azimuthal symmetry) for the Monte Carlo solution and the different diffusion models. The thickness of the clear layer in mean free path is 1, 2, 4, and 6 for the top-left, top-right, bottom-left, and bottom-right figures, respectively. Solid circles, Monte Carlo simulation; open circles, classical diffusion model; solid curves, generalized diffusion model with theoretical tangential diffusion coefficient; dashed–dotted curves, generalized diffusion model; dotted curves, generalized model with large tangential diffusion coefficient. The inset represents a magnification of the above results between angles 125 and 145.

Tables (1)

Tables Icon

Table 1 Relative Root-Mean-Square Error (L2 Norm) between the Monte Carlo Simulations and the Various Diffusion Models for Several Thicknesses of the Clear Layera

Equations (47)

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

Ω·u(x, Ω)+σa(x)u(x, Ω)+Q(u)(x, Ω)=S(x),
in D×Sn-1,
u(x, Ω)=g(x, Ω)
onΓ-={(x, Ω)D×Sn-1 s.t. Ω·ν(x)<0}.
Q(u)(x, Ω)=σs(x)u(x, Ω)-Sn-1u(x, Ω)dμ(Ω).
S1u(x, Ω)dμ(Ω)=12π02πu(x, θ)dθ,
S2u(x, Ω)dμ(Ω)=14π02π0πu(x, θ, ϕ)sin θdθdϕ,
u(x, Ω)=U(x)-1σs(x)Ω·U(x)+smaller-orderterms,
-·D(x)U(x)+σa(x)U(x)=S(x)in D,
U(x)+nLnD(x)ν(x)·U(x)=g(x)onD,
D(x)=1nσs(x),n=2, 3.
DC={yD s.t. y=x+tν(x), with xand|t|<L}.
xE=x+Lν(x)E,xI=x-Lν(x)I.
Γ±C={(x, Ω)DC×Sn-1 s.t.±Ω·νC(x)>0},
Ω·v(x, Ω)+σa(x)v(x, Ω)+Q(v)(x, Ω)=0, in DC×Sn-1,
v(x, Ω)=g(x, Ω)onΓ-C.
R1C=RC-I,
Iu(x, Ω)=u(x+2Lν(x), Ω),xIu(x-2Lν(x), Ω),xE.
-·D(x)U(x)+σa(x)U(x)=S(x)in D\DC,
U(x)+nLnD(x)ν(x)·U(x)=g(x)onD,
U(xE)=U(xI)on,
ν(x)·D(xE)U(xE)-ν(x)·D(xI)U(xI)
=KU(x)on,
KU(x)=Γ+(xE)Ω·νC(xE)(R1CU)(xE, Ω)dμ(Ω)+Γ+(xI)Ω·νC(xI)(R1CU)(xI, Ω)dμ(Ω).
RCU(x, θ)=exp[-σat(x, θ)]U(x¯),
JI=Γ+(xI)Ω·νC(xI)(R1CU)(xI, Ω)dμ(Ω)=12π0π sin θ{exp[-σat(xI, θ)]U(x¯I)-U(xE)}dθ,
U(x¯I)=U(xE)+s(θ; xE) Us(xE)+12s2(θ; xE) 2Us2(xE)+smallerterms=U(xE)+ss2(θ; xE)2Us(xE)+smallerterms.
exp[-σat(xI, θ)]=1-σat(xI, θ)+smallerterms.
JI=-σaI(xI)U(xE)+bI(xE) Us(xE)+dI(xE) 2Us2(xE)+=-σaI(xI)U(xE)+sdI(xE) Us(xE)+,
σaI(xI)=σa 12π0πt(xI, θ)sin θdθ,
bI(xE)=12π0πs(θ; xE)sin θdθ,
dI(xE)=12π0π 12s2(θ; xE)sin θdθ.
KU(x)=sdC(x) Us(x)-σaC(x)U(x)+smallerterms.
KU(xE)=dCΔU(xE)+smallterms,
dC=dex-exC+dex-inC+din-exC,
dex-exC=12π0θ0 sin θ(R+L)2(2θ)2dθ,
dex-inC=12πθ0π/2 sin θ(R-L)2×θ-arccosR+LR-Lcos θ2dθ,
din-exC=12π0π/2 sin θ(R+L)2-θ+arccosR-LR+Lcos θ2dθ.
θ0=arccosR-LR+L.
-·D(x)U(x)+σa(x)U(x)=S(x)in D\,
U(x)+nLnD(x)ν(x)·U(x)=g(x)onD,
U(x+)=U(x-)on,
ν(x)·D(x+)U(x+)-ν(x)·D(x-)U(x-)
=dCΔU(x)on.
D(D(x)U(x)·w(x)+σa(x)U(x)w(x))dx+dCU(x)·w(x)dS(x)+D 1nLnU(x)w(x)dS(x)=DS(x)w(x)dx+D 1nLng(x)w(x)dS(x).
JT(x)=Sn-1Ω·ν(x)u(x, Ω)dμ(Ω),
JD(x)=D(x)ν(x)·U(x),

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