Infinite Series Solution of the Time-Dependent Radiative Transfer Equation in Anisotropically Scattering Media

Abstract: We solve the radiative transfer equation (RTE) in anisotropically scattering media as an infinite series. Each series term represents a distinct number of scattering events, with analytical solutions derived for zero and single scattering. Higher-order corrections are addressed through numerical calculations or approximations. The RTE solution corresponds to Monte Carlo sampling of photon trajectories with fixed start and end points. Validated against traditional Monte Carlo simulations, featuring random end points, our solution demonstrates enhanced efficiency for both anisotropic and isotropic scattering functions, significantly reducing computational time and resources. The advantage of our method over Monte Carlo simulations varies with the position of interest and the asymmetry of light scattering, but it is typically orders of magnitude faster while achieving the same level of accuracy. The exploitation of hidden symmetries further accelerates our numerical calculations, enhancing the method's overall efficiency. In addition, we extend our analysis to the first and second moments of the photon's flux, elucidating the transition between transport and diffusive regimes.