Solving analytically the simplified spherical harmonics equations in cylindrical turbid media.

Abstract

A methodology is presented for analytically solving simplified spherical harmonics equations (SPN) in a finite homogeneous absorbing and scattering cylindrical medium. The SPN equations are a reliable approximation to the radiative transfer equation for describing light propagation inside turbid media. The equations consist of a set of coupled partial differential equations (PDEs). The analytical solution developed here is for a steady-state isotropic point source located at an arbitrary point inside a cylindrical turbid medium. Partial-reflection boundary conditions are considered, as they realistically model the refractive index mismatch between a turbid medium and its surroundings (air), as occurs in practice in biomedical optics. The eigen method is used to decouple the set of SPN PDEs. The methodology is applied to the SP3, which has proved to be sufficiently accurate in practice, but it is readily generalizable to higher orders. The solution is compared with the analytical solution of the diffusion equation as well as to gold standard Monte Carlo simulations for validation, against which it shows good agreement. This work is important, as it provides an additional tool for validating numerical solutions of SPN equations for curved geometries, namely, cylindrical shapes, which are often used in practice.