Radiative transfer equation for predicting light propagation in biological media: comparison of a modified finite volume method, the Monte Carlo technique, and an exact analytical solution

Abstract

We examine the accuracy of a modified finite volume method compared to analytical and Monte Carlo solutions for solving the radiative transfer equation. The model is used for predicting light propagation within a two-dimensional absorbing and highly forward-scattering medium such as biological tissue subjected to a collimated light beam. Numerical simulations for the spatially resolved reflectance and transmittance are presented considering refractive index mismatch with Fresnel reflection at the interface, homogeneous and two-layered media. Time-dependent as well as steady-state cases are considered. In the steady state, it is found that the modified finite volume method is in good agreement with the other two methods. The relative differences between the solutions are found to decrease with spatial mesh refinement applied for the modified finite volume method obtaining <2.4%. In the time domain, the fourth-order Runge-Kutta method is used for the time semi-discretization of the radiative transfer equation. An agreement among the modified finite volume method, Runge-Kutta method, and Monte Carlo solutions are shown, but with relative differences higher than in the steady state.

Analytical solutions of the radiative transport equation for turbid and fluorescent layered media

“…Commonly, the RTE is solved using numerical methods, e.g., with the finite volume method 4 or with Monte Carlo simulations 5 , due to the lack of analytical solutions. Recently, however relevant analytical solutions were obtained for the infinite scattering medium by applying the P N method and the method of rotated reference frames 6, 7 .…”

Three‐dimensional frequency‐domain optical anisotropy imaging of biological tissues with near‐infrared light

“…The reconstructions of the medium were performed with a structured mesh of 269 001 nodes (degrees of freedom) and 1536 000 tetrahedral elements. The angular space (4 π Sr) was uniformly discretized into 64 control solid angles and each angle was also subdivided into eight azimuthal and polar directions for the phase function normalization . A Gaussian Laser source is used to illuminate the western surface ( x = 0 mm) of the medium.…”

“…The angular space (4pSr) was uniformly discretized into 64 control solid angles and each angle was also subdivided into eight azimuthal and polar directions for the phase function normalization. 24 A Gaussian Laser source is used to illuminate the western surface (x = 0 mm) of the medium. The expression of the spatial Gaussian function along the y-axis and z-axis is given by:…”

Fluorescence molecular imaging based on the adjoint radiative transport equation

“…for (r r r, Ω Ω Ω) ∈ Γ − . The directional reflection coefficient ρ is given by Snell-Descartes laws assuming that the refractive index of the outside medium (air) is unity [10,76,79]. The specular reflection Ω sp Ω sp Ω sp = Ω Ω Ω − 2(Ω Ω Ω • n n n) n n n is defined as the direction from which a laser beam must hit the surface.…”