Abstract

Monte-Carlo simulation is an important tool in the field of biomedical optics, but suffers from significant computational expense. In this paper, we present the multicanonical Monte-Carlo (MMC) method for improving the efficiency of classical Monte Carlo simulations of light propagation in biological media. The MMC is an adaptive importance sampling technique that iteratively equilibrates at the optimal importance distribution with little (if any) a priori knowledge of how to choose and bias the importance proposal distribution. We illustrate the efficiency of this method by evaluating the probability density function (pdf) for the radial distance of photons exiting from a semi-infinite homogeneous tissue as well as the pdf for the maximum penetration depth of photons propagating in an inhomogeneous tissue. The results agree very well with diffusion theory as well as classical Monte-Carlo simulations. A six to sevenfold improvement in computational time is achieved by the MMC algorithm in calculating pdf values as low as 10-8. This result suggests that the MMC method can be useful in efficiently studying numerous applications of light propagation in complex biological media where the remitted photon yield is low.

© 2005 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  6. A. Ishimaru Wave Propagation and Scattering in Random Media ( Academic Press, Inc., San Diego 1978 ).
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    [Crossref] [PubMed]
  8. D. Yevick , “ Multicanonical communication system modeling - application to PMD statistics ,” IEEE Photon. Tech. Lett.   14 , 1512 – 1514 ( 2002 ).
    [Crossref]
  9. R. HolzlÖhner and C. R. Menyuk , “ Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems ,” Opt. Lett.   28 , 1894 – 1896 ( 2003 ).
    [Crossref] [PubMed]
  10. A. Bilenca and G. Eisenstein , “ Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier ,” J. Quantum Electron.   41 , 36 – 44 ( 2005 ).
    [Crossref]
  11. D. P. Landau and K. Binder , A Guide to Monte Carlo Simulations in Statistical Physics ( Cambridge, MA/New York , 2000 ).
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    [Crossref]
  13. N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
    [Crossref]
  14. T. J. Farrell , M. S. Patterson , and B. Wilson , “ A diffusion theory model of spatially resolved, steady-state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys.   19 , 879 – 888 ( 1992 ).
    [Crossref] [PubMed]
  15. G. Alexandrakis , T. J. Farrell , and M. S. Patterson , “ Accuracy of the Diffusion Approximation in Determining the Optical Properties of a Two-Layer Turbid Medium ,” Appl. Opt.   37 , 7401 – 7409 ( 1998 ).
    [Crossref]

2005 (1)

A. Bilenca and G. Eisenstein , “ Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier ,” J. Quantum Electron.   41 , 36 – 44 ( 2005 ).
[Crossref]

2003 (2)

C. Andrieu , N. De Freitas , A. Doucet , and M. I. Jordan , “ An introduction to MCMC for machine learning ,” Machine Learning   50 , 5 – 43 ( 2003 ).
[Crossref]

R. HolzlÖhner and C. R. Menyuk , “ Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems ,” Opt. Lett.   28 , 1894 – 1896 ( 2003 ).
[Crossref] [PubMed]

2002 (2)

1999 (1)

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

1998 (1)

1996 (1)

1995 (1)

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

1992 (2)

B. A. Berg and T. Neuhaus , “ Multicanonical ensemble: A new approach to simulate first-order phase transitions ”, Phys. Rev. Lett.   68 , 9 – 12 ( 1992 ).
[Crossref] [PubMed]

T. J. Farrell , M. S. Patterson , and B. Wilson , “ A diffusion theory model of spatially resolved, steady-state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys.   19 , 879 – 888 ( 1992 ).
[Crossref] [PubMed]

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 – 1167 ( 1989 ).
[Crossref] [PubMed]

1953 (1)

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

Alexandrakis, G.

Andrieu, C.

C. Andrieu , N. De Freitas , A. Doucet , and M. I. Jordan , “ An introduction to MCMC for machine learning ,” Machine Learning   50 , 5 – 43 ( 2003 ).
[Crossref]

Ben-Letaief, K.

Berg, B. A.

B. A. Berg and T. Neuhaus , “ Multicanonical ensemble: A new approach to simulate first-order phase transitions ”, Phys. Rev. Lett.   68 , 9 – 12 ( 1992 ).
[Crossref] [PubMed]

Bilenca, A.

A. Bilenca and G. Eisenstein , “ Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier ,” J. Quantum Electron.   41 , 36 – 44 ( 2005 ).
[Crossref]

Binder, K.

D. P. Landau and K. Binder , A Guide to Monte Carlo Simulations in Statistical Physics ( Cambridge, MA/New York , 2000 ).

Boas, D. A.

Culver, J. P.

Doucet, A.

C. Andrieu , N. De Freitas , A. Doucet , and M. I. Jordan , “ An introduction to MCMC for machine learning ,” Machine Learning   50 , 5 – 43 ( 2003 ).
[Crossref]

Dunn, A. K.

Eisenstein, G.

A. Bilenca and G. Eisenstein , “ Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier ,” J. Quantum Electron.   41 , 36 – 44 ( 2005 ).
[Crossref]

Farrell, T. J.

G. Alexandrakis , T. J. Farrell , and M. S. Patterson , “ Accuracy of the Diffusion Approximation in Determining the Optical Properties of a Two-Layer Turbid Medium ,” Appl. Opt.   37 , 7401 – 7409 ( 1998 ).
[Crossref]

T. J. Farrell , M. S. Patterson , and B. Wilson , “ A diffusion theory model of spatially resolved, steady-state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys.   19 , 879 – 888 ( 1992 ).
[Crossref] [PubMed]

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 – 1167 ( 1989 ).
[Crossref] [PubMed]

Freitas, N. De

C. Andrieu , N. De Freitas , A. Doucet , and M. I. Jordan , “ An introduction to MCMC for machine learning ,” Machine Learning   50 , 5 – 43 ( 2003 ).
[Crossref]

HolzlÖhner, R.

Ishimaru, A.

A. Ishimaru Wave Propagation and Scattering in Random Media ( Academic Press, Inc., San Diego 1978 ).

Jacques, S. L.

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

Jordan, M. I.

C. Andrieu , N. De Freitas , A. Doucet , and M. I. Jordan , “ An introduction to MCMC for machine learning ,” Machine Learning   50 , 5 – 43 ( 2003 ).
[Crossref]

Landau, D. P.

D. P. Landau and K. Binder , A Guide to Monte Carlo Simulations in Statistical Physics ( Cambridge, MA/New York , 2000 ).

Menyuk, C. R.

Metropolis, N.

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

Neuhaus, T.

B. A. Berg and T. Neuhaus , “ Multicanonical ensemble: A new approach to simulate first-order phase transitions ”, Phys. Rev. Lett.   68 , 9 – 12 ( 1992 ).
[Crossref] [PubMed]

Patterson, M. S.

G. Alexandrakis , T. J. Farrell , and M. S. Patterson , “ Accuracy of the Diffusion Approximation in Determining the Optical Properties of a Two-Layer Turbid Medium ,” Appl. Opt.   37 , 7401 – 7409 ( 1998 ).
[Crossref]

T. J. Farrell , M. S. Patterson , and B. Wilson , “ A diffusion theory model of spatially resolved, steady-state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys.   19 , 879 – 888 ( 1992 ).
[Crossref] [PubMed]

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 – 1167 ( 1989 ).
[Crossref] [PubMed]

Rosenbluth, A.

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

Rosenbluth, M.

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

Schmitt, J. M.

Stott, J. J.

Teller, E.

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

Teller, M.

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

Wang, L.

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

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

Wilson, B.

T. J. Farrell , M. S. Patterson , and B. Wilson , “ A diffusion theory model of spatially resolved, steady-state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys.   19 , 879 – 888 ( 1992 ).
[Crossref] [PubMed]

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 – 1167 ( 1989 ).
[Crossref] [PubMed]

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 – 1167 ( 1989 ).
[Crossref] [PubMed]

Yao, G.

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

Yevick, D.

D. Yevick , “ Multicanonical communication system modeling - application to PMD statistics ,” IEEE Photon. Tech. Lett.   14 , 1512 – 1514 ( 2002 ).
[Crossref]

Zheng, L.

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

Appl. Opt. (1)

Comput. Methods and Programs in Biomed. (1)

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

IEEE Photon. Tech. Lett. (1)

D. Yevick , “ Multicanonical communication system modeling - application to PMD statistics ,” IEEE Photon. Tech. Lett.   14 , 1512 – 1514 ( 2002 ).
[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 – 1167 ( 1989 ).
[Crossref] [PubMed]

J. Chem. Phys. (1)

N. Metropolis , A. Rosenbluth , M. Rosenbluth , M. Teller , and E. Teller , “ Equation of state calculations by fast computing machines ,” J. Chem. Phys.   21 , 1087 – 1092 ( 1953 ).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Quantum Electron. (1)

A. Bilenca and G. Eisenstein , “ Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier ,” J. Quantum Electron.   41 , 36 – 44 ( 2005 ).
[Crossref]

Machine Learning (1)

C. Andrieu , N. De Freitas , A. Doucet , and M. I. Jordan , “ An introduction to MCMC for machine learning ,” Machine Learning   50 , 5 – 43 ( 2003 ).
[Crossref]

Med. Phys. (1)

T. J. Farrell , M. S. Patterson , and B. Wilson , “ A diffusion theory model of spatially resolved, steady-state diffusive reflectance for the noninvasive determination of tissue optical properties in vivo ,” Med. Phys.   19 , 879 – 888 ( 1992 ).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (1)

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

Phys. Rev. Lett. (1)

B. A. Berg and T. Neuhaus , “ Multicanonical ensemble: A new approach to simulate first-order phase transitions ”, Phys. Rev. Lett.   68 , 9 – 12 ( 1992 ).
[Crossref] [PubMed]

Other (2)

A. Ishimaru Wave Propagation and Scattering in Random Media ( Academic Press, Inc., San Diego 1978 ).

D. P. Landau and K. Binder , A Guide to Monte Carlo Simulations in Statistical Physics ( Cambridge, MA/New York , 2000 ).

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Figures (4)

Fig. 1.
Fig. 1.

Multicanonical Monte-Carlo (MMC) algorithm for light transport in biological media.

Fig. 2.
Fig. 2.

MMC estimators of the diffuse reflectance pdf of a semi-infinite homogeneous random medium.

Fig. 3.
Fig. 3.

MMC and CMC estimators of the diffuse reflectance pdf of a semi-infinite homogeneous random medium.

Fig. 4.
Fig. 4.

MMC and CMC estimators of the pdf of the maximum penetration depth of light in a two-layered inhomogeneous random medium.

Equations (11)

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f Y ( y i ) dy = χ 1 L i ( x ) f X ( x ) d x
1 L i ( x ) = { 1 x L i 0 x L i
f ̂ Y CMC ( y i ) = 1 dy · N Σ n = 1 N 1 L i ( x n ) , i N
f Y ( y i ) dy = χ 1 L i ( x ) w ( x ) q ( x ) dx
f ̂ Y IS ( y i ) = 1 dy · N Σ n = 1 N 1 L i ( x n ) w ( x n ) , i N
q opt ( x ) = 1 L i ( x ) f X ( x ) χ 1 L i ( x ) f X ( x ) d x = 1 L i ( x ) f X ( x ) f Y ( y i ) dy , i N
q opt ( k ) ( x ) 1 L i ( x ) f X ( x ) f Y ( k 1 ) ( y i ) , i N
f ̂ Y ( k ) ( y i ) = { f Y ( k 1 ) ( y i ) · Σ n = 1 N 1 L i ( x n ) if at least one sample x n L i f Y ( k 1 ) ( y i ) otherwise
f Y ( y ) = 0 1 w packet · f Y , W packet no _ abs ( y , w packet ) dw packet / 0 1 w packet · f Y , W , packet no _ abs ( y , w packet ) d w packet dy
f ̂ Y , W packet no _ abs ( k ) ( y i , w packet j ) = { f Y , W packet no _ abs ( k ) ( y i w packet j ) · Σ n = 1 N 1 L ij ( x n ) if at least one sample x n L ij f Y , W packet no _ abs ( k ) ( y i w packet j ) otherwise
f ̂ Y ( k ) ( y ) = 0 1 w packet · f ̂ Y , W packet no _ abs ( k ) ( y , w packet ) dw packet / 0 1 w packet · f ̂ Y , W , packet no _ abs ( k ) ( y , w packet ) d w packet dy

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