Abstract

In the field of photon migration in turbid media, different Monte Carlo methods are usually employed to solve the radiative transfer equation. We consider four different Monte Carlo methods, widely used in the field of tissue optics, that are based on four different ways to build photons’ trajectories. We provide both theoretical arguments and numerical results showing the statistical equivalence of the four methods. In the numerical results we compare the temporal point spread functions calculated by the four methods for a wide range of the optical properties in the slab and semi-infinite medium geometry. The convergence of the methods is also briefly discussed.

© 2012 Optical Society of America

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  1. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef]
  2. L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biol. 47, 131–146 (1995).
    [CrossRef]
  3. S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
    [CrossRef]
  4. R. Graaff, M. H. Koelink, F. F. M. Demul, W. G. Zijlstra, A. C. M. Dassel, and J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
    [CrossRef]
  5. G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031–2041 (1991).
    [CrossRef]
  6. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
    [CrossRef]
  7. A. Liebert, H. Wabnitz, N. Zołek, and R. Macdonald, “Monte Carlo algorithm for efficient simulation of time resolved fluorescence in layered turbid media,” Opt. Express 16, 13188–13202(2008).
    [CrossRef]
  8. D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-3-159 .
  9. J. M. Schmitt and K. Ben-Letaief, “Efficient Monte Carlo simulation of confocal microscopy in biological tissue,” J. Opt. Soc. Am. A 13, 952–960 (1996).
    [CrossRef]
  10. G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
    [CrossRef]
  11. B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
    [CrossRef]
  12. H. W. Jentink, F. F. M. de Mul, R. G. A. M. Hermsen, R. Graaff, and J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1990).
    [CrossRef]
  13. J. L. Reuss and D. Siker, “The pulse in reflectance pulse oximetry: modeling and experimental studies,” J. Clin. Monitor. Comput. 18, 289–299 (2004).
    [CrossRef]
  14. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software, Vol. PM193 of SPIE Press Monograph (SPIE, 2009).
  15. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
    [CrossRef]
  16. E. Alerstam, “Optical spectroscopy of turbid media: time-domain measurements and accelerated Monte Carlo modeling,” Ph.D. dissertation (Lund University, 2011).
  17. K. W. Calabro and I. Bigio, “On the validity of assumptions to incorporate absorption in Monte Carlo simulations,” in Proceedings of Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BSu5A.3.
  18. J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).
  19. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E 56, 1135–1141 (1997).
    [CrossRef]
  20. A. Liemert and A. Kienle, “Analytical solution of the radiative transfer equation for the infinite-space fluence,” Phys. Rev. A 83, 015804 (2011).
    [CrossRef]
  21. A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E 83, 036605 (2011).
    [CrossRef]
  22. E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
    [CrossRef]
  23. A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29, 1475–1481 (2012).
    [CrossRef]
  24. G. Zaccanti, E. Battistelli, P. Bruscaglioni, and Q. N. Wei, “Analytic relationships for the statistical moments of scattering point coordinates for photon migration in a scattering medium,” Pure Appl. Opt. 3, 897–905 (1994).
    [CrossRef]

2012 (1)

2011 (3)

A. Liemert and A. Kienle, “Analytical solution of the radiative transfer equation for the infinite-space fluence,” Phys. Rev. A 83, 015804 (2011).
[CrossRef]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E 83, 036605 (2011).
[CrossRef]

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

2008 (2)

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
[CrossRef]

A. Liebert, H. Wabnitz, N. Zołek, and R. Macdonald, “Monte Carlo algorithm for efficient simulation of time resolved fluorescence in layered turbid media,” Opt. Express 16, 13188–13202(2008).
[CrossRef]

2004 (1)

J. L. Reuss and D. Siker, “The pulse in reflectance pulse oximetry: modeling and experimental studies,” J. Clin. Monitor. Comput. 18, 289–299 (2004).
[CrossRef]

2002 (1)

1999 (2)

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef]

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

1998 (1)

1997 (1)

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

1996 (1)

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biol. 47, 131–146 (1995).
[CrossRef]

1994 (1)

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

1993 (1)

1991 (1)

1990 (1)

1989 (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

1983 (1)

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Aarnoudse, J. G.

Adam, G.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Alerstam, E.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
[CrossRef]

E. Alerstam, “Optical spectroscopy of turbid media: time-domain measurements and accelerated Monte Carlo modeling,” Ph.D. dissertation (Lund University, 2011).

Alianelli, L.

Andersson-Engels, S.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
[CrossRef]

Battistelli, E.

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

Ben-Letaief, K.

Bigio, I.

K. W. Calabro and I. Bigio, “On the validity of assumptions to incorporate absorption in Monte Carlo simulations,” in Proceedings of Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BSu5A.3.

Blumetti, C.

Boas, D.

Bruscaglioni, P.

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

Calabro, K. W.

K. W. Calabro and I. Bigio, “On the validity of assumptions to incorporate absorption in Monte Carlo simulations,” in Proceedings of Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BSu5A.3.

Contini, D.

Culver, J.

Dassel, A. C. M.

de Mul, F. F. M.

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software, Vol. PM193 of SPIE Press Monograph (SPIE, 2009).

Demul, F. F. M.

Duderstadt, J. J.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

Dunn, A.

Farina, B.

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Graaff, R.

Greve, J.

Guo, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Hanlon, E. B.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Hermsen, R. G. A. M.

Ismaelli, A.

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
[CrossRef]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software, Vol. PM193 of SPIE Press Monograph (SPIE, 2009).

Itzkan, I.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biol. 47, 131–146 (1995).
[CrossRef]

Jentink, H. W.

Kienle, A.

A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29, 1475–1481 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E 83, 036605 (2011).
[CrossRef]

A. Liemert and A. Kienle, “Analytical solution of the radiative transfer equation for the infinite-space fluence,” Phys. Rev. A 83, 015804 (2011).
[CrossRef]

Koelink, M. H.

Liebert, A.

Liemert, A.

A. Liemert and A. Kienle, “Light transport in three-dimensional semi-infinite scattering media,” J. Opt. Soc. Am. A 29, 1475–1481 (2012).
[CrossRef]

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E 83, 036605 (2011).
[CrossRef]

A. Liemert and A. Kienle, “Analytical solution of the radiative transfer equation for the infinite-space fluence,” Phys. Rev. A 83, 015804 (2011).
[CrossRef]

Macdonald, R.

Marchesini, R.

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

Martelli, F.

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
[CrossRef]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software, Vol. PM193 of SPIE Press Monograph (SPIE, 2009).

Martin, W. R.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

Paasschens, J. C. J.

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

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Perelman, L. T.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Pignoli, E.

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

Qiu, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Reuss, J. L.

J. L. Reuss and D. Siker, “The pulse in reflectance pulse oximetry: modeling and experimental studies,” J. Clin. Monitor. Comput. 18, 289–299 (2004).
[CrossRef]

Saponaro, S.

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

Sassaroli, A.

Schmitt, J. M.

Siker, D.

J. L. Reuss and D. Siker, “The pulse in reflectance pulse oximetry: modeling and experimental studies,” J. Clin. Monitor. Comput. 18, 289–299 (2004).
[CrossRef]

Stott, J.

Svensson, T.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
[CrossRef]

Tomatis, S.

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

Turzhitsky, V.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Vitkin, E.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Wabnitz, H.

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biol. 47, 131–146 (1995).
[CrossRef]

Wang, L. V.

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef]

Wei, Q. N.

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

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

Yao, G.

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef]

Zaccanti, G.

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, and G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
[CrossRef]

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

G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031–2041 (1991).
[CrossRef]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software, Vol. PM193 of SPIE Press Monograph (SPIE, 2009).

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biol. 47, 131–146 (1995).
[CrossRef]

Zijlstra, W. G.

Zolek, N.

Appl. Opt. (4)

Comput. Meth. Prog. Biol. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Biol. 47, 131–146 (1995).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, and D. R. Wyman, “Monte-Carlo modeling of light-propagation in highly scattering tissues: 1. model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1168 (1989).
[CrossRef]

J. Biomed. Opt. (1)

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041304 (2008).
[CrossRef]

J. Clin. Monitor. Comput. (1)

J. L. Reuss and D. Siker, “The pulse in reflectance pulse oximetry: modeling and experimental studies,” J. Clin. Monitor. Comput. 18, 289–299 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

Med. Phys. (1)

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef]

Nat. Commun. (1)

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587 (2011).
[CrossRef]

Opt. Express (2)

Phys. Med. Biol. (2)

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307–2320 (1999).
[CrossRef]

B. Farina, S. Saponaro, E. Pignoli, S. Tomatis, and R. Marchesini, “Monte Carlo simulation of light fluence in tissue in a cylindrical diffusing fibre geometry,” Phys. Med. Biol. 44, 1–11 (1999).
[CrossRef]

Phys. Rev. A (1)

A. Liemert and A. Kienle, “Analytical solution of the radiative transfer equation for the infinite-space fluence,” Phys. Rev. A 83, 015804 (2011).
[CrossRef]

Phys. Rev. E (2)

A. Liemert and A. Kienle, “Analytical Green’s function of the radiative transfer radiance for the infinite medium,” Phys. Rev. E 83, 036605 (2011).
[CrossRef]

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

Pure Appl. Opt. (1)

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

Other (4)

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation through Biological Tissue and Other Diffusive Media: Theory, Solutions, and Software, Vol. PM193 of SPIE Press Monograph (SPIE, 2009).

E. Alerstam, “Optical spectroscopy of turbid media: time-domain measurements and accelerated Monte Carlo modeling,” Ph.D. dissertation (Lund University, 2011).

K. W. Calabro and I. Bigio, “On the validity of assumptions to incorporate absorption in Monte Carlo simulations,” in Proceedings of Biomedical Optics, OSA Technical Digest (Optical Society of America, 2012), paper BSu5A.3.

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

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Figures (5)

Fig. 1.
Fig. 1.

Trajectory Γ is shown. Each direction s^i is defined by the couple of angles (θi,ϕi) (i=1,2n), but the phase function is assumed to depend on the angle θι only. The interactions occur within the spatial intervals dli1 centered at ri (i=1,2n) and to within the solid angles dωi (not shown) surrounding the directions s^i.

Fig. 2.
Fig. 2.

Time-resolved reflectance from a slab 1 mm thick at the distances 0.079 mm (left panel) and 0.429 mm (right panel) from the input source. The optical properties of the medium are μa=1mm1, μs=1mm1; the refractive index of the medium and the surroundings is 1.4. The Henyey–Greenstein phase function with asymmetry factor g=0 is used.

Fig. 3.
Fig. 3.

Same as Fig. 2, except for the ratios of the time-resolved reflectance obtained with the AW, ASPR, and AR methods to the reflectance obtained with the mBLL method.

Fig. 4.
Fig. 4.

Same as Fig. 2, except for the ratios between the CW reflectance obtained with the AW, ASPR, and AR methods to the reflectance obtained with the mBLL method, plotted versus the source–detector distance. The error bars represent the standard deviation of the ratios.

Fig. 5.
Fig. 5.

Comparison between the time-resolved reflectances obtained with the mBLL and ASPR methods. The ratio of the standard deviation obtained with the ASPR method to the standard deviation obtained with the mBLL method is also plotted. The figure pertains to the same slab as the previous figures, and the source–detector distances are 1.02 mm (left panel) and 3.28 mm (right panel).

Equations (30)

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

1vI(r,s^,t)t+s^·I(r,s^,t)+[μa(r)+μs(r)]I(r,s^,t)=μs(r)4πp(s^·s^)I(r,s^,t)dω+S(r,s^,t).
1vI(r,s^,t)t+s^·I(r,s^,t)+μt(r)I(r,s^,t)=δ(r)δ(s^s^0)δ(t).
I(r,s^,t)=vδ(|r|vt)δ(s^s^0)exp(0lμt(ξ)dξ),
w=exp(0lμt(ξ)dξ).
p(l)=μt(l)exp(0lμt(ξ)dξ).
ln(w)μt=l.
W(Γ)=Γa(s^i,li).
W(Γ)=1.
I(r,s^,t)=vδ(|r|vt)δ(s^s^0)exp(0lμs(ξ)dξ)exp(0lμa(ξ)dξ).
W(Γ)=exp(Γμa(ξ)dξ),
w1=exp(0lsμs(ξ)dξ),w2=exp(0laμa(ξ)dξ).
W(Γ)=1.
dP(li,li+dli)=μt(li)dliexp(0liμt(ξ)dξ).
dP(Γ)=[i=0n1μt(li)dlip(θi+1)dωi+1]exp(Γμt(ξ)dξ).
W(Γ)=i=0n1a(li).
dP{[li,w][li,li+dli]×[0,a(li)]}=a(li)μt(li)dliexp(0liμt(ξ)dξ).
dP(Γ)=[i=0n1a(li)μt(li)dlip(θi+1)dωi+1]exp(Γμt(ξ)dξ).
W(Γ)=1.
dP(li,li+dli)=μs(li)dliexp(0liμs(ξ)dξ).
dP(Γ)=[i=0n1μs(li)dlip(θi+1)dωi+1]exp(Γμs(ξ)dξ),
W(Γ)=exp(Γμa(ξ)dξ).
dP{[ls,la][li,li+dli]×[li,]}=μs(li)dliexp(0liμs(ξ)dξ)exp(0liμa(ξ)dξ).
P=S(Γ)dP(Γ)W(Γ),
TPSF(ti,ti+dt)=S[Γ(ti.ti+dt)]dP(Γ)W(Γ),
TPSF(ti,ti+dt)=niNW(Γ).
p(la)=μa(la)exp(0laμa(ξ)dξ).
P(la>li)=lip(la)dla.
P(la>li)=liμa(la)exp(0laμa(ξ)dξ)dla.
y(la)=0laμa(ξ)dξ,
P(la>li)=0liμa(ξ)dξexp(y)dy=exp(0liμa(ξ)dξ).

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