Approaches for modelling and reconstruction in optical tomography in the presence of anisotropies

Abstract

In this thesis, models for light propagation and solution methods for the inverse problem in medical optical tomography (OT) in the presence of anisotropies are developed. Light propagation is modelled using the anisotropic diffusion equation (DE) in the frequency domain. The derivation of the diffusion equation in an anisotropic case is sketched, and the relevant boundary and source conditions presented. The numerical solution is obtained using the finite element (FE) method. The numerical work is done in two dimensional space in order to facilitate the testing of the novel methods for solving the inverse problem. To verify the light propagation model, the 2D FE solution is compared to the boundary element method solution to the DE and to a Monte Carlo simulation.

The main emphasis is on the solution of the inverse problem in OT in the presence of anisotropies. The anisotropic inverse problem is non-unique, and hence simultaneous reconstruction of both the anisotropic diffusion tensor and the absorption coefficient is not feasible without substantial prior knowledge. The goal in this work is to reconstruct the spatial distribution of the optical absorption coefficient and overcome the disturbing effect of the background anisotropies. At the same time, the prior knowledge available on the anisotropies is assumed to be rather limited. A few different approaches for estimating the absorption coefficient is presented. Firstly, an attempt to use a conventional isotropic reconstruction scheme is considered. In this case, the obtained estimates suffer from relative large artefacts. In the second approach the structure of the anisotropy is assumed known, and the spatially constant strength is reconstructed simultaneously with the absorption. The quality of the absorption estimate degrades quickly with the accuracy of the underlying anisotropy structure. To help this, statistical inversion methods are employed. Statistical methods provide means to model the prior information on unknown parameters through probability distributions. The anisotropy parameters are modelled using relatively loose Gaussian priors. By using Gaussian priors the posterior probability distribution of the absorption on condition of the measurements also assumes a Gaussian form. Hence the conditional mean estimates and covariances can be derived in a closed form without having to resort to numerical integration. The statistical approach is used to derive a modelling error approach, where the effect of anisotropies is treated as a modelling error and included into estimation. Implementing the statistical inversion methods enables recovering the main features in the absorption distribution. The statistical treatment of anisotropies is also applied to help the inverse problem when the boundary shape and source/measurement locations are modelled inaccurately.