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

Abstract

Accurate and efficient solutions of the three dimensional radiative transport equation were derived in all domains for the case of layered scattering media. Index mismatched boundary conditions based on Fresnel’s equations were implemented. Arbitrary rotationally symmetric phase functions can be applied to characterize the scattering in the turbid media. Solutions were derived for an obliquely incident beam having arbitrary spatial profiles. The derived solutions were successfully validated with Monte Carlo simulations and partly compared with analytical solutions of the diffusion equation.

Broadband Time Domain Diffuse Optical Reflectance Spectroscopy: A Review of Systems, Methods, and Applications

“…where D = 1 3µ s is the diffusion coefficient. The first Equation of (3) is the well-known diffusion equation (DE) and, together with the Fick's Law (second Equation (3)), has been widely used for analytically solving diffusion problems in a variety of geometries, such as: layered laterally infinite media, layered cylinder, sphere, parallelepiped [80][81][82][83][84][85]. The diffusion approximation was shown to work quite well in situations in which the diffusion is higher than the absorption and where the source-detector distance allows the detection of photons having undergone a number of scattering events such that the radiance L becomes almost isotropic [86].…”

“…The diffusion approximation was shown to work quite well in situations in which the diffusion is higher than the absorption and where the source-detector distance allows the detection of photons having undergone a number of scattering events such that the radiance L becomes almost isotropic [86]. Furthermore, the RTE equation has been recently solved in the time domain for a layered medium by Liemert et al [80], showing a perfect agreement with Monte Carlo simulations, which is considered the gold-standard for model testing.…”

A pseudospectral method for solution of the radiative transport equation

“…that is to say that in the absence of sources, there is no component of the radiance inwards across the boundary. The high dimensionality of the field variable φ(r,ŝ, t) ∈ R N × S N −1 , and the presence of non-local spatial and angular operators, is such that analytically solutions to the RTE have been found only for homogeneous infinite and semi-infinite geometries [6], and layered media [7].…”

Improving Localization of Deep Inclusions in Time-Resolved Diffuse Optical Tomography

“…The most accurate approach is to use the integro-differential Radiative Transfer Equation (RTE) [7]. Although RTE has some analytical solutions [8][9][10] these just hold for simple geometries and cannot be applied to more complex environments, such as an human head or breast models, without making strong assumptions. Apart from classical Monte-Carlo simulations [11,12], new numerical methods have been proposed, some of which are the one-way RTE [13] or hybrid RTE [14]; nevertheless, they are still highly time-consuming for real-time applications.…”