Approximate solution to the photon migration problem in a turbid medium based on the path-integral approach

Abstract

The functions characterizing the directional and spatial distribution of photons propagating through an infinite scattering medium can be expressed in the form of specific path-integrals. However, these are not solvable analytically and have to be approximated by a more suitable form, while we require the preservation of only a few relaxed constraints imposed on the photon transport. The path integral solutions, i.e. the sought distribution functions, are found in the form of simple closed-form expressions. They are compared with the standard diffusion solution and with the second-order cumulant approximation, which solves an equivalent problem. Photon path length profiles of the distribution functions obtained by the path-integral approximation and the cumulant approximation are almost identical, but the first formulae are much simpler. The numerical examples indicate that the path-integral solution could serve as an improvement of the diffusion equation solution under appropriate conditions.