Biological tissue scatters light mainly in the forward direction where the scattering phase function has a narrow peak. This peak makes it difficult to solve the radiative transport equation. However, it is just for forward-peaked scattering that the Fokker-Planck equation provides a good approximation, and it is easier to solve than the transport equation. Furthermore, the modification of the Fokker-Planck equation by Leakeas and Larsen provides an even better approximation and is also easier to solve. We demonstrate the accuracy of these two approximations by solving the problem of reflection and transmission of a plane wave normally incident on a slab composed of a uniform scattering medium.
We study light propagation in biological tissue using the radiative transport equation. The Green's function is the fundamental solution to the radiative transport equation from which all other solutions can be computed. We compute the Green's function as an expansion in plane-wave modes. We calculate these plane-wave modes numerically using the discrete-ordinate method. When scattering is sharply peaked, calculating the plane-wave modes for the transport equation is difficult. For that case we replace it with the Fokker-Planck equation since the latter gives a good approximation to the transport equation and requires less work to solve. We calculate the plane-wave modes for the Fokker-Planck equation numerically using a finite-difference approximation. The method of computing the Green's function for it is the same as for the transport equation. We demonstrate the use of the Green's function for the transport and Fokker-Planck equations by computing the point-spread function in a half-space composed of a uniform scattering and absorbing medium.
We derive Kubelka-Munk (KM) theory systematically from the radiative transport equation (RTE) by analyzing the system of equations resulting from applying the double spherical harmonics method of order one and transforming that system into one governing the positive- and negative-going fluxes. Through this derivation, we establish the theoretical basis of KM theory, identify all parameters, and determine its range of validity. Moreover, we are able to generalize KM theory to take into account general boundary sources and nonhomogeneous terms, for example. The generalized Kubelka-Munk (gKM) equations are also much more accurate at approximating the solution of the RTE. We validate this theory through comparison with numerical solutions of the RTE.
We discuss several outstanding theoretical problems in optical diffusion in random media. Specifically, we discuss which of several diffusion theories most closely approximates exact solutions of the equation of transfer. We consider a plane wave impinging upon a plane-parallel slab of a random medium as a model problem to compare the diffusion theories with a numerical solution of the equation of transfer for continuous-wave, pulsed, and photon density waves. In addition, we discuss the validity of the diffusion approximation for a variety of parameter settings to ascertain the diffusion approximation's applicability to imaging biological media.
We study the performance of Chebyshev spectral methods for time-dependent radiative transfer equations. Starting with a method for one-dimensional problems in homogeneous media, we show that the modifications needed to consider more general problems such as inhomogeneous media, polarization, and higher dimensions are straightforward. In this method, we approximate the spatial dependence of the intensity by an expansion of Chebyshev polynomials. This yields a coupled system of integro-differential equations for the expansion coefficients that depend on angle and time. Next, we approximate the integral operation on the angle variables using a Gaussian quadrature rule resulting in a coupled system of differential equations with respect to time. Using a second-order finite difference approximation, we discretize the time variable. We solve the resultant system of equations with an efficient algorithm that makes Chebyshev spectral methods competitive with other methods for radiative transfer equations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.