Analytical solutions to the simplified spherical harmonics equations using eigen decompositions

Abstract

We develop a modified method to simplify the analytical solutions to the simplified spherical harmonics equations (SP(N)). The scheme decouples the SP(N) partial differential equations into independent equations using eigen decompositions and calculates the Green's function of the photon migrations based on the eigenvectors and eigenvalues. In contrast to the established solutions that are based on the original coupled equations, the proposed derivation is theoretically concise and universally extendable to other regular geometries. We validate the proposed method in comparison with Monte-Carlo simulations for an infinite scattering medium and a circular geometry as an example of the boundary value problems.

Coupling between radiative transport and diffusion approximation for enhanced near-field photon-migration modeling based on transient photon kinetics

“…In these hybrid models, the transport theory is in general described by the radiative transport equation (RTE), with its solution found either by using the discrete-ordinates method (S N ) 2,3 or solving its high-order spherical harmonics (P N ∕SP N , N ≥ 3) approximation. 4,5 The physical interdependence between the two subdomains causes neither of the two equations to be solved independently. By formulating the interface conditions, e.g., the continuity of the photon-density and its high-order derivatives (flux and so on) on the crossover interface, one straightforward scheme is to deal with a coupled set of the RTE and the diffusion equation (DE), abbreviated as the coupled RTE-DE model.…”

“…By formulating the interface conditions, e.g., the continuity of the photon-density and its high-order derivatives (flux and so on) on the crossover interface, one straightforward scheme is to deal with a coupled set of the RTE and the diffusion equation (DE), abbreviated as the coupled RTE-DE model. [2][3][4][5] However, solving large-scale simultaneous equations might be mathematically intractable and numerically unstable, in particular for three-dimensional scenarios. In contrast, another roundabout but low-efficiency way takes an iteratively approximating strategy to solve the RTE and DE alternately until convergence of the successive approximations is reached.…”

“…Journal of Biomedical Optics 050501-3 May 2016 • Vol. 21 (5) tetrahedral elements for the spatial discretization, with their number depending on R T for a nearly constant element dimension.…”