| Glioma is one of the most challenging types of brain tumors to be treated or controlled locally. One of the main problems is to determine which areas of the apparently normal brain contain glioma cells, as gliomas are known to infiltrate several centimetres beyond the clinically apparent lesion that is visualized on standard CT or MRI. To ensure that radiation treatment encompasses the whole tumor, including the cancerous cells not revealed by MRI, doctors treat the volume of brain that extends 2cm out from the margin of the visible tumor. This approach does not consider varying tumor-growth dynamics in different brain tissues, thus it may result in killing some healthy cells while leaving cancerous cells alive in other areas. These cells may cause recurrence of the tumor later in time which limits the effectiveness of the therapy.;In this thesis, we propose two models to define the tumor invasion margin based on the fact that glioma cells preferentially spread along nerve fibers. The first model is an anisotropic reaction-diffusion type tumor growth model that prioritizes diffusion along nerve fibers, as given by DW-MRI data. The second proposed approach computes the tumor invasion margin using a geodesic distance defined on the Riemannian manifold of brain fibers. Both mathematical models result in Partial Differential Equations (PDEs) that have to be numerically solved. Numerical methods used for solving differential equations should be chosen with great care. A part of this thesis is dedicated to discuss in detail, the numerical aspects such as stability and consistency of different finite difference methods used to solve these PDEs. We review the stability issues of several 2D methods that discretize the anisotropic diffusion equation and we propose an extension of one 2D stable method to 3D. We also analyze the stability issues of the geodesic model. In comparison, the geodesic model is numerically more stable than the anisotropic diffusion model since it results in a first-order PDE. Finally, we evaluate both models on actual DTI data from patients with glioma by comparing our predicted growth with follow-up MRI scans. Results show improvement in predicting the invasion margin when using the geodesic distance model as opposed to the 2cm conventional Euclidean distance. |
