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Multicanonical Monte-Carlo simulations of light propagation in biological media

A. Bilenca, A. Desjardins, B. E. Bouma, and G. J. Tearney

Open Access Open Access

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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Figures (4)
Fig. 1.
Fig. 1. Multicanonical Monte-Carlo (MMC) algorithm for light transport in biological media.

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Fig. 2.
Fig. 2. MMC estimators of the diffuse reflectance pdf of a semi-infinite homogeneous random medium.

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Fig. 3.
Fig. 3. MMC and CMC estimators of the diffuse reflectance pdf of a semi-infinite homogeneous random medium.

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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.

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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 ) = { C· f Y ( k − 1 ) ( y i ) · Σ n = 1 N 1 L i ( x n ) if at least one sample x n ∈ L i C· 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 ) = { C· 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 C· 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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