A finite element method for the even-parity Radiative Transfer Equation using the P N Approximation

Abstract

The concept of using optical radiation to penetrate highly scattering media, combined with image reconstruction methods to recover optical parameters inside the media, has been a recurrent idea for over a century. However it has received great attention in the last decade due to advances both in measurement technology and in theoretical and practical understanding of the nature of the image reconstruction problem. This field has come to be known as Diffuse Optical Tomography (DOT); for recent reviews see [1-5]. The term "Diffuse" is employed since the usual conditions being investigated are where the medium is so highly scattering that its propagation is nearly completely described by a Diffusion Approximation (DA). However, it is well known that under certain conditions, the DA is no longer valid. In particular the presence of non-scattering (void) regions, such as occur in the Cerebro-Spinal Fluid (CSF) filled ventricles in the brain, represent a situation for which the DA is clearly inadequate. Under these circumstances, more advanced methods are required [6-9]. A general model of light transport in scattering media, but one that ignores polarisation and coherence effects, is the Boltzmann Equation. This equation has been extensively studied in the field of Neutron Transport [10-14] and in Radiation transfer [15, 16] where it is known as the Radiative Transfer Equation (RTE). In this paper, we describe the RTE in its second order form, and discuss the development of a finite element method for its solution. Related methods are described in [17-19]. © 2009 Springer Berlin Heidelberg.