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Past work has demonstrated the value of a random walk theory (RWT) to solve multiple-scattering problems arising in numerous contexts. This paper’s goal is to investigate the application range of the RWT using Monte Carlo simulations and extending it to anisotropic media using scaling laws. Meanwhile, this paper also reiterates rules for converting RWT formulas to real physical dimensions, and corrects some errors which appear in an earlier publication. The RWT theory, validated by the Monte Carlo simulations and combined with the scaling law, is expected to be useful to study multiple scattering and to greatly reduce the computation cost.

Multiple-scattering represents a fundamental scientific problem, with implications in a wide spectrum of practical applications ranging from biological imaging [

In particular, RWT expressions have been derived for such quantities as transmittance, reflectance, and the intensity distributions of transmitted and reflected photons. These expressions not only address the computational limitation of numerical techniques, but also provide physical insights into the underlying phenomena. They have been validated by extensive comparisons against Monte Carlo simulations, and have been found to be useful for a range of practical applications such as the estimation of signal levels and the design of imaging optics. However, RWT analytical expressions are obtained in terms of lattice units and step lengths, and careful attention must be given to their conversion to expressions involving physical units, _{s} and μ_{a}), and the anisotropy factor, g. Furthermore, it is desirable to quantitatively understand the applicable range of the RWT expressions, and to investigate the feasibility of extending these expressions to anisotropic media.

In the case of isotropic scattering, random walk expressions can be related to real macroscopic variables by equating the RWT lattice unit to the real displacement divided by the rms (root mean square) scattering length (equivalent to dividing the macroscopic scattering coefficient by

We first comment on several algebraic errors in the equations published in [

and

The correctness of these equations was confirmed by comparison against Monte Carlo simulations.

There also are two errors in the previously-published Equations (12b) and (23) [

Correspondingly, there is a typo in the captions of

Comparison of the radial distribution of transmitted photons at large g.

Second, many applications involve non-absorbing media, for which it is desirable to have a simple expression for the transmittance (and reflectance), including for the general case of anisotropic scattering. Equation (A4) in [

where _{s}_{a}_{s}^{-1}_{a}

where _{s}

Comparison of transmittance predicted by Equation (A4′) [

Lastly, in some applications, it is desirable to know the radial distribution of the transmitted photons. Example applications include the analysis of the angular distribution of the transmitted photons [

Comparison of the radial distribution of transmitted photons at small g.

To understand these questions, Monte Carlo simulations were performed to obtain the radial distribution of transmitted photons and representative results were summarized and compared to Equation (1) in

In summary, this work examined a theory based on random walks to model multiple light scattering. The theory is attractive because it provided close-form solutions to properties of great interest to practical applications. This work corrected some errors published previously, validated the theory by comparison with Monte Carlo simulations, and also quantified the theory applicable ranges. Also, it extends the theory by demonstrating that the theory, combined with a scaling law, can be applied to anisotropic media. These findings are expected to be useful to study multiple scattering and to greatly reduce the computation cost.

The authors greatly acknowledge the many technical discussions with Amir H. Gandjbakhche, George. H. Weiss, Robert F. Bonner, and Ralph Nossal from the National Institutes of Health, Bethesda, MD, USA.