The goal of this thesis is to describe, implement and analyse Monte Carlo (MC) algorithms for simulating the mechanism of diffusion magnetic resonance imaging (dMRI). As the inverse problem of mapping the sub-voxel micro-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 synthesizing the imaging signals according to given imaging sequences.
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 the literature. In an effort to mitigate the costly nature of MC algorithms, the geometrically adaptive fast random walk algorithm (GAFRW) is implemented, first invented by D.Grebenkov. Additional mathematical justification is provided in the appendix should the reader find details in the original paper by Grebenkov lacking. The result suggests that the GAFRW algorithm only provides moderate accuracy improvement over the crude MC method in the geometry modeled after white matter fibers. Overall, both approaches are shown to be flexible for a variety of geometries and pulse sequences
