A finite element approach for modeling photon transport in tissue

Abstract

The use of optical radiation in medical physics is important in several fields for both treatment and diagnosis. In all cases an analytic and computable model of the propagation of radiation in tissue is essential for a meaningful interpretation of the procedures. A finite element method (FEM) for deriving photon density inside an object, and photon flux at its boundary, assuming that the photon transport model is the diffusion approximation to the radiative transfer equation, is introduced herein. Results from the model for a particular case are given: the calculation of the boundary flux as a function of time resulting from a delta-function input to a two-dimensional circle (equivalent to a line source in an infinite cylinder) with homogeneous scattering and absorption properties. This models the temporal point spread function of interest in near infrared spectroscopy and imaging. The convergence of the FEM results are demonstrated, as the resolution of the mesh is increased, to the analytical expression for the Green's function for this system. The diffusion approximation is very commonly adopted as appropriate for cases which are scattering dominated, i.e., where mu s >> mu a, and results from other workers have compared it to alternative models. In this article a high degree of agreement with a Monte Carlo method is demonstrated. The principle advantage of the FE method is its speed. It is in all ways as flexible as Monte Carlo methods and in addition can produce photon density everywhere, as well as flux on the boundary. One disadvantage is that there is no means of deriving individual photon histories.

A review on continuous wave functional near-infrared spectroscopy and imaging instrumentation and methodology

“…Numerical methods based on the local solution of the diffusion equation in a meshed space (Arridge et al, 1993;Dehghani et al, 2009a) such as finite elements method (FEM) have been among the most popular methods to define the forward model due to their inherent flexibility to model irregular spaces and the ability to include prior information from other tomographic sources. Linearized solutions of the analytical solution of the diffusion equation (Kak and Slaney, 1988;O'Leary, 1996) have been widely used to model heterogeneous highly scattering tissue due to its relatively simple implementation and its computational efficiency.…”

“…Proprietary reconstruction systems as well as many different open source software products implementing some of the previously described techniques (Arridge et al, 1993;Dehghani et al, 2009b;Fang and Boas, 2009) are available to evaluate the results obtained from NIRI systems. In the following we shortly describe the most important ones.…”

Methods in diffuse optical imaging

“…Figure 3 shows an example of the time-varying signal obtained for different source-detector separations across a slab with a heterogeneous distribution of optical parameters. The time-of-flight histogram of transmitted photons at each detector was calculated in two ways: (i) by solving the time-dependent system using a fully implicit finite differencing step in time (see [11] for details) and (ii) by solving for the first three temporal moments and deriving the time-varying solution via the Taylor expansion of the MGF. The comparison is virtually perfect despite the grossly heterogeneous nature of the example which precludes the exact specification of a Green's function.…”

Theoretical and numerical analysis on multispectral bioluminescence tomography

“…For this spectral range, scattering dominates for the photons in the tissue, and it is appropriate to use the diffusion approximation to describe the photon propagation ( [1]). Let Ω ⊂ R d be the biological medium with the boundary Γ.…”