Analytical solution of the simplified spherical harmonics equations in spherical turbid media. (October 2016) - Edjlali, Ehsan; Bérubé-Lauzière, Yves
Abstract: We present for the first time an analytical solution for the simplified spherical harmonics equations (so-called SP N equations) in the case of a steady-state isotropic point source inside a spherical homogeneous absorbing and scattering medium. The SP N equations provide a reliable approximation to the radiative transfer equation for describing light transport inside turbid media. The SP N equations consist of a set of coupled partial differential equations and the eigen method is used to obtain a set of decoupled equations, each resembling the heat equation in the Laplace domain. The equations are solved for the realistic partial reflection boundary conditions accounting for the difference in refractive indices between the turbid medium and its environment (air) as occurs in practical cases of interest in biomedical optics. Specifically, we provide the complete solution methodology for the SP 3, which is readily applicable to higher orders as well, and also give results for the SP 5 . This computationally easy to obtain solution is investigated for different optical properties of the turbid medium. For validation, the solution is also compared to the analytical solution of the diffusion equation and to gold standard Monte Carlo simulation results. The SP 3 and SP 5 analytical solutions prove to be in good agreement with the Monte Carlo results. This work provides an additional tool for validating numerical solutions of the SP N equations for curved geometries. Abstract : Highlights: A methodology for analytically solving the SPN equations is proposed. It relies on diagonalization and expressing the solution in spherical coordinates. Monte Carlo simulations are incorporated to validate the analytical solutions.