Abstract

Starting from the radiative transport equation we derive the scaling relationships that enable a single Monte Carlo (MC) simulation to predict the spatially- and temporally-resolved reflectance from homogeneous semi-infinite media with arbitrary scattering and absorption coefficients. This derivation shows that a rigorous application of this single Monte Carlo (sMC) approach requires the rescaling to be done individually for each photon biography. We examine the accuracy of the sMC method when processing simulations on an individual photon basis and also demonstrate the use of adaptive binning and interpolation using non-uniform rational B-splines (NURBS) to achieve order of magnitude reductions in the relative error as compared to the use of uniform binning and linear interpolation. This improved implementation for sMC simulation serves as a fast and accurate solver to address both forward and inverse problems and is available for use at http://www.virtualphotonics.org/.

© 2011 OSA

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References

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  1. A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]
  2. A. H. Hielscher, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
    [CrossRef] [PubMed]
  3. J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Dover Publications, 2008).
  4. A. D. Kim, “Transport theory for light propagation in biological tissue,” J. Opt. Soc. Am. A 21, 820–827 (2004).
    [CrossRef]
  5. A. H. Hielscher, R. E. Alcouffe, and R. E. 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]
  6. R. C. Haskell, L. O. Svaasand, T. Tsay, T. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  7. A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
    [CrossRef] [PubMed]
  8. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  9. B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  10. R. Graaff, M. H. Koelink, F. F. M. De Mul, W. G. Zijistra, 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] [PubMed]
  11. A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
    [CrossRef] [PubMed]
  12. A. Pifferi, P. Taroni, G. Valentini, and S. Andersson-Engels, “Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model,” Appl. Opt. 37, 2774–2780 (1998).
    [CrossRef]
  13. E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041301 (2008).
    [CrossRef]
  14. G. M. Palmer and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Theory and validation on synthetic phantoms,” Appl. Opt. 45, 1062–1071 (2006).
    [CrossRef] [PubMed]
  15. G. M. Palmer, C. F. Zhu, T. M. Breslin, F. S. Xu, K. W. Gilchrist, and N. Ramanujam, “Monte Carlo-based inverse model for calculating tissue optical properties. Part I: Application to breast cancer,” Appl. Opt. 45, 1072–1078 (2006).
    [CrossRef] [PubMed]
  16. H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. 11, 1–18 (2006).
    [CrossRef]
  17. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
    [CrossRef] [PubMed]
  18. C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
    [CrossRef]
  19. I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
    [CrossRef]
  20. A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnières, and H. van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
    [CrossRef]
  21. S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  22. W. Cheong and S. A. Prahl. “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  23. L. G. Henyey and J. L. Greenstein, “Diffuse radiation of the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  24. W. Hines, C. D. Montgomery, M. D. Goldsman, and M. C. Borror, Probability and Statistics in Engineering (John Wiley & Sons Inc., 2003).
  25. J. Dougherty, R. Kohavi, and S. Mehran, “Supervised and unsupervised discretization of continuous features,” in Proceedings 12th International Conference on Machine Learning (Morgan Kaufmann, 1995).
  26. M. Boulle, “Optimal bin number for equal frequency discretization,” Intell. Data Anal. 9, 175–188 (2005).
  27. M. H. Kalos and P. A. Whitlock, Monte Carlo Methods Volume 1: Basics (Wiley-Interscience, 1995).
  28. W. Tiller and L. Piegl, The NURBS Book (Springer, 1995).
  29. F. D. Rogers, An Introduction to NURBS With Historical Perspective (Morgan Kaufmann, 2004).

2009 (1)

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

2008 (1)

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041301 (2008).
[CrossRef]

2007 (1)

I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
[CrossRef]

2006 (3)

2005 (1)

M. Boulle, “Optimal bin number for equal frequency discretization,” Intell. Data Anal. 9, 175–188 (2005).

2004 (1)

2001 (1)

1998 (3)

1997 (2)

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

1996 (1)

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

1995 (2)

A. H. Hielscher, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1990 (1)

W. Cheong and S. A. Prahl. “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1989 (1)

1983 (1)

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

1941 (1)

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

Aarnoudse, J. G.

Adam, G.

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

Alcouffe, R. E.

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

Alerstam, E.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041301 (2008).
[CrossRef]

Andersson-Engels, S.

Ayers, F. R.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

Barbour, R. E.

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

Bays, R.

Bevilacqua, F.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[CrossRef]

Borror, M. C.

W. Hines, C. D. Montgomery, M. D. Goldsman, and M. C. Borror, Probability and Statistics in Engineering (John Wiley & Sons Inc., 2003).

Boulle, M.

M. Boulle, “Optimal bin number for equal frequency discretization,” Intell. Data Anal. 9, 175–188 (2005).

Breslin, T. M.

Chance, B.

Cheong, W.

W. Cheong and S. A. Prahl. “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Cuccia, D. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

Dassel, A. C. M.

De Mul, F. F. M.

Dognitz, N.

Dougherty, J.

J. Dougherty, R. Kohavi, and S. Mehran, “Supervised and unsupervised discretization of continuous features,” in Proceedings 12th International Conference on Machine Learning (Morgan Kaufmann, 1995).

Dunn, A. K.

Durkin, A. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

Farrell, T. J.

H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. 11, 1–18 (2006).
[CrossRef]

Feng, T.

Gelbard, E. M.

J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Dover Publications, 2008).

Gilchrist, K. W.

Goldsman, M. D.

W. Hines, C. D. Montgomery, M. D. Goldsman, and M. C. Borror, Probability and Statistics in Engineering (John Wiley & Sons Inc., 2003).

Graaff, R.

Greenstein, J. L.

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

Haskell, R. C.

Hayakawa, C. K.

I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
[CrossRef]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[CrossRef]

Henyey, L. G.

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

Hielscher, A. H.

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

A. H. Hielscher, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Hines, W.

W. Hines, C. D. Montgomery, M. D. Goldsman, and M. C. Borror, Probability and Statistics in Engineering (John Wiley & Sons Inc., 2003).

Jacques, S. L.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Kalos, M. H.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods Volume 1: Basics (Wiley-Interscience, 1995).

Kienle, A.

A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnières, and H. van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

Kim, A. D.

Koelink, M. H.

Kohavi, R.

J. Dougherty, R. Kohavi, and S. Mehran, “Supervised and unsupervised discretization of continuous features,” in Proceedings 12th International Conference on Machine Learning (Morgan Kaufmann, 1995).

McAdams, M. S.

Mehran, S.

J. Dougherty, R. Kohavi, and S. Mehran, “Supervised and unsupervised discretization of continuous features,” in Proceedings 12th International Conference on Machine Learning (Morgan Kaufmann, 1995).

Montgomery, C. D.

W. Hines, C. D. Montgomery, M. D. Goldsman, and M. C. Borror, Probability and Statistics in Engineering (John Wiley & Sons Inc., 2003).

Palmer, G. M.

Patterson, M. S.

H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. 11, 1–18 (2006).
[CrossRef]

A. Kienle, M. S. Patterson, N. Dognitz, R. Bays, G. Wagnières, and H. van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

A. Kienle and M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Piegl, L.

W. Tiller and L. Piegl, The NURBS Book (Springer, 1995).

Pifferi, A.

Prahl, S. A.

W. Cheong and S. A. Prahl. “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Ramanujam, N.

Rogers, F. D.

F. D. Rogers, An Introduction to NURBS With Historical Perspective (Morgan Kaufmann, 2004).

Seo, I.

I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
[CrossRef]

Spanier, J.

Svaasand, L. O.

Svensson, T.

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041301 (2008).
[CrossRef]

Taroni, P.

Tiller, W.

W. Tiller and L. Piegl, The NURBS Book (Springer, 1995).

Tromberg, B. J.

Tsay, T.

Valentini, G.

van Den Bergh, H.

Venugopalan, V.

I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
[CrossRef]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[CrossRef]

Wagnières, G.

Wang, L.

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Whitlock, P. A.

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods Volume 1: Basics (Wiley-Interscience, 1995).

Wilson, B. C.

Xu, F. S.

Xu, H.

H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. 11, 1–18 (2006).
[CrossRef]

You, J. S.

I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
[CrossRef]

C. K. Hayakawa, J. Spanier, F. Bevilacqua, A. K. Dunn, J. S. You, B. J. Tromberg, and V. Venugopalan, “Perturbation Monte Carlo methods to solve inverse photon migration problems in heterogeneous tissues,” Opt. Lett. 26, 1335–1337 (2001).
[CrossRef]

Zhu, C. F.

Zijistra, W. G.

Appl. Opt. (6)

Astrophys. J. (1)

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

Comput. Methods Programs Biomed. (1)

S. L. Jacques and L. Wang, “Monte Carlo modeling of light transport in multi-layered tissue,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

W. Cheong and S. A. Prahl. “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Intell. Data Anal. (1)

M. Boulle, “Optimal bin number for equal frequency discretization,” Intell. Data Anal. 9, 175–188 (2005).

J. Biomed. Opt. (4)

E. Alerstam, S. Andersson-Engels, and T. Svensson, “White Monte Carlo for time-resolved photon migration,” J. Biomed. Opt. 13, 041301 (2008).
[CrossRef]

H. Xu, T. J. Farrell, and M. S. Patterson, “Investigation of light propagation models to determine the optical properties of tissue from interstitial frequency domain fluence measurements,” J. Biomed. Opt. 11, 1–18 (2006).
[CrossRef]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

I. Seo, J. S. You, C. K. Hayakawa, and V. Venugopalan, “Perturbation and differential Monte Carlo methods for measurement of optical properties in a layered epithelial tissue model,” J. Biomed. Opt. 12, 1–15 (2007).
[CrossRef]

J. Opt. Soc. Am. A (3)

Med. Phys. (1)

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

Opt. Lett. (1)

Phys. Med. Biol. (4)

A. Kienle and M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

A. H. Hielscher, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

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

A. Kienle and M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to the source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

Other (6)

J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Dover Publications, 2008).

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods Volume 1: Basics (Wiley-Interscience, 1995).

W. Tiller and L. Piegl, The NURBS Book (Springer, 1995).

F. D. Rogers, An Introduction to NURBS With Historical Perspective (Morgan Kaufmann, 2004).

W. Hines, C. D. Montgomery, M. D. Goldsman, and M. C. Borror, Probability and Statistics in Engineering (John Wiley & Sons Inc., 2003).

J. Dougherty, R. Kohavi, and S. Mehran, “Supervised and unsupervised discretization of continuous features,” in Proceedings 12th International Conference on Machine Learning (Morgan Kaufmann, 1995).

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Figures (7)

Fig. 1
Fig. 1

Flow chart representing the processes used to obtain the spatially and temporally resolved reflectance from the different Monte Carlo approaches examined in this study.

Fig. 2
Fig. 2

(a) Results of the fitting approach used to extrapolate points on the tail of the time-resolved reflectance R(t) for rk = 1 mm. (b) Values of Rr (rk ,tl ) for rk ∈ [5 – 100] mm obtained after binning, extrapolating and resampling.

Fig. 3
Fig. 3

Relative standard deviation of the reflectance provided by discretizing the results of a cMC simulation for μa = 0.1mm−1 and μ′s = 1mm−1 using (a) 200 × 200 EFD bins, (b) 200 × 200 EWD bins, (c) 250 × 1000 EWD bins.

Fig. 4
Fig. 4

(a) Reflectance signal obtained with EFD for a cMC simulation, (b) relative error of the reflectance provided by the sMC p method, (c) relative error of the reflectance provided by the sMC i , for μa = 0.001 mm−1 and μ s = 2 mm−1.

Fig. 5
Fig. 5

(a) Reflectance signal obtained with EFD from the cMC simulation, (b) relative error of the reflectance provided by the sMC p method, (c) relative error of the reflectance provided by the sMC i method, for μa = 0.01 mm−1 and μ s = 1 mm−1.

Fig. 6
Fig. 6

(a) Reflectance signal obtained with EFD from the cMC simulation, (b) relative error of the reflectance provided by the sMC p method, (c) relative error of the reflectance provided by the sMC i method, for μa = 0.1 mm−1 and μ s = 1.5 mm−1.

Fig. 7
Fig. 7

(a) Log10 of the relative error of the sMC i approach using NURBS interpolation on the reference values, (b) log10 of the relative error of the sMC i approach based on linear interpolation of the uniformly binned reflectance as done by Kienle and Patterson for μa = 0.1 mm−1 and μ′s = 0.5 mm−1

Equations (30)

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R ( r 0 , t 0 ) 1 Δ A Δ t t 0 t 0 + Δ t r 0 r 0 + Δ r 𝕊 2 L ( r , Ω ^ , t ) | z = 0 ( n ^ Ω ^ ) d Ω ^ 2 π r d r d t ,
R ( r , t ) | μ s μ s , r ; μ a = 0 = μ s 3 μ s , r 3 R r ( μ s μ s , r r , μ s μ s , r t ) .
R ( r , t ) | μ s = μ s , r ; μ a 0 = R r ( r , t ) exp ( μ a c t )
1 c L ( r , Ω ^ , t ) t + L ( r , Ω ^ , t ) Ω ^ = μ s L ( r , Ω ^ , t ) + μ s 𝕊 2 p ( Ω ^ Ω ^ ) L ( r , Ω ^ , t ) d Ω
L ( r , Ω ^ , t ) 0 ,
L ( r , Ω ^ r , t ) = F ( n ^ Ω ^ i ) L ( r , Ω ^ i , t ) , r S , n ^ Ω ^ i > 0 ,
{ τ = μ s c t ρ = μ s r Λ ˜ ( ρ , Ω ^ , τ ) = L ( r , Ω ^ , t ) / L 0 ,
Λ ˜ ( ρ , Ω ^ , τ ) τ + Λ ˜ ( ρ , Ω ^ , τ ) Ω ^ = Λ ˜ ( ρ , Ω ^ , τ ) + 𝕊 2 p ( Ω ^ Ω ^ ) Λ ˜ ( ρ , Ω ^ , τ ) d Ω .
Λ ˜ ( ρ , Ω ^ , τ ) 0 , and
Λ ˜ ( ρ , Ω ^ r , τ ) = F ( n ^ Ω ^ i ) Λ ˜ ( ρ , Ω ^ i , τ ) , ρ S , n ^ Ω ^ i > 0 .
R ( r , t ) = 𝕊 2 L ( r , Ω ^ , t ) | z = 0 ( n ^ Ω ^ ) d Ω ^ .
R ˜ ( ρ , τ ) = 𝕊 2 Λ ˜ ( ρ , Ω ^ , τ ) | z = 0 ( n ^ Ω ^ ) d Ω ^ .
R ˜ ( ρ 0 , τ 0 ) = 1 Δ A ˜ Δ τ τ 0 τ 0 + Δ τ ρ 0 ρ 0 + Δ ρ 𝕊 2 Λ ˜ ( ρ , Ω ^ , τ ) | z = 0 ( n ^ Ω ^ ) d Ω ^ 2 π ρ d ρ d τ
Δ A ˜ = π ( ρ 0 + Δ ρ ) 2 π ρ 0 2 .
t 0 t 0 + Δ t r 0 r 0 + Δ r 𝕊 2 L ( r , Ω ^ , t ) d Ω ^ 2 π r d r d t = E 0 τ 0 τ 0 + Δ τ ρ 0 ρ 0 + Δ ρ 𝕊 2 Λ ˜ ( ρ , Ω ^ , τ ) d Ω ^ 2 π ρ d ρ d τ .
L 0 = R ( r 0 , t 0 ) R ˜ ( ρ 0 , τ 0 ) = N h ν Δ A ˜ Δ τ Δ A Δ t = E 0 c μ s 3 .
R ( r , t ) = c μ s 3 E 0 R ˜ ( ρ , τ ) = c μ s 3 E 0 𝕊 2 Λ ˜ ( ρ , Ω ^ , τ ) | z = 0 ( n ^ Ω ^ ) d Ω ^ .
R r ( r , t ) = c μ s , r 3 E 0 R ˜ r ( ρ , τ ) = c μ s , r 3 E 0 R ˜ r ( μ s , r r , c μ s , r t ) ,
R ( r , t ) = c μ s 3 E 0 R ˜ ( ρ , τ ) = c μ s 3 E 0 R ˜ ( μ s μ s , r ρ r , μ s μ s , r τ r )
R ˜ ( μ s μ s , r ρ r , μ s μ s , r τ r ) = R r ( μ s μ s , r r , μ s μ s , r t ) 1 c μ s , r 3 E 0
R ( r , t ) = μ s 3 μ s , r 3 R r ( μ s μ s , r r , μ s μ s , r t ) ,
N r ( r 0 , t 0 ) = Σ w i = Σ w 0
R r ( r 0 , t 0 ) = 1 Δ A Δ t t 0 t 0 + Δ t r 0 r 0 + Δ r 𝕊 2 h ν c 𝒩 r ( r 0 , t 0 , Ω ) | z = 0 ( n ^ Ω ^ ) d Ω ^ 2 π r d r d t ,
R ( r 0 , t 0 ) 1 Δ A Δ t t 0 t 0 + Δ t r 0 r 0 + Δ r 𝕊 2 h ν c 𝒩 r ( r 0 , t 0 , Ω ) | z = 0 exp ( μ a c t 0 ) ( n ^ Ω ^ ) d Ω ^ 2 π r d r d t = R r ( r 0 , t 0 ) exp ( μ a c t 0 ) .
R ( r , t ) = R r ( r , t ) exp ( μ a ct )
ɛ rel ( p , i ) ( r k , t l ) = | R cMC ( r k , t l ) R sMC ( p , i ) ( r k , t l ) | R cMC ( r k , t l ) .
σ rel ( r k , t l ) = σ cMC ( r k , t l ) R cMC ( r k , t l ) .
R ( r k , t l ) = 1 Δ A k 1 Δ t l 1 N t = 1 N w i ,
σ ( r k , t l ) = 1 N 1 [ i = 1 N w i 2 Δ A k 2 Δ t l 2 N R 2 ( r k , t l ) ] 1 / 2 .
f t ( t ) = a t 3 / 2 [ exp ( b t ) exp ( c t ) ] .

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