Ming Miao | Applied Math, University of Waterloo
Monte Carlo Simulation of Diffusion Magnetic Resonance Imaging
The goal of this thesis is to describe, implement and analyse Monte Carlo (MC) algorithms for simulating the mechanism of di.usion magnetic resonance imaging (dMRI). As the inverse problem of mapping the sub-voxel mirco-structure remains challenging, MC methods provide an important numerical approach for creating ground-truth data. The main idea of such simulations is first generating a large sample of independent random trajectories in a prescribed geometry and then synthesize the imaging signals from the sample according to a desired imaging sequence.
The thesis starts by providing a concise introduction of the mathematical background for understanding dMRI. It then proceeds to describe the workflow and implementation of the most basic Monte Carlo method with experiments performed on simple geometries. A theoretical framework for error analysis is introduced, which to the best of the author’s knowledge, has been absent in literature. In an effort to mitigate the costly nature of MC algorithms, the fast random walk (FRW) is implemented, invented by Grebenkov. Additional mathematical justification is provided in the appendix should the reader .nd details in the original paper by Grebenkov lacking. The result suggests that the FRW algorithm only provides moderate accuracy improvement over the crute MC method in the geometry modelled after white matter fibres. Overall, both approaches are shown to be flexible for a variety of geometries and pulse sequences. The MC simulation algorithms will be used to generate data for a newly formulated inverse problem described in the last chapter.