Thèse Présentée pour obtenir le grade de

A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues Soutenue le 7 Mars 2014 devant le jury composé de: M. Denis GREBENKOV Co-Directeur de Thès

A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging

International audienceThe complex transverse water proton magnetization subject to diffusion-encoding magneticfield gradient pulses in a heterogeneous medium can be modeled by themultiple compartment Bloch-Torrey partial differential equation (PDE).In addition, steady-state Laplace PDEs can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristicsof the medium in the long time limit.In spatial domains that model biological tissues at the cellular level, these two types of PDEs have to be completed with permeability conditions on the cellular interfaces.To solve these PDEs, we implemented a finite elements method thatallows jumps in the solution at the cell interfaces by using doublenodes. Using a transformation of the Bloch-Torrey PDE we reduced oscillations in the searched-for solution and simplified the implementationof the boundary conditions. The spatial discretizationwas then coupled to the adaptive explict Runge-Kutta-Chebychev time-stepping method. Our proposed method is second order accurate in space and second order accurate in time.We implemented this method on the FEniCSC++ platform and show time and spatial convergence results.Finally, this method is applied to study some relevant questions in diffusionMRI

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 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