Geometric and Optimization Methods for Diffusion Magnetic Resonance Imaging

Abstract

This thesis presents novel mathematical and computational methods aimed at enhancing and improving brain tissue structural imaging techniques that are based on diffusion Magnetic Resonance Imaging (dMRI). The most commonly used dMRI technique is Diffusion Tensor Imaging (DTI), which models water diffusion via a Gaussian pattern and estimates the corresponding covariance, also known as diffusion tensor. DTI forms the basis of brain structural connectivity methods like tractography and sub-cortical region parcellation, and thus provides useful markers for brain white matter integrity. Other, recently proposed dMRI techniques rely on modeling water diffusion in intra-axonal and extra-axonal spaces separately. Thereby, these so-called multi-compartment models hold the promise to provide detailed tissue microstructure information and to identify markers that may be specific to particular tissue development/diseases. In this thesis we address key mathematical challenges encountered by DTI, as well as by these newly proposed dMRI techniques, that pertain to recovering more detailed microstructure information. We begin by focusing on DTI and present novel geometrical methods to improve DTI analysis (Chapters 3, 4, and 5). In particular, (i) we utilize the mathematical theory of Optimal Mass Transport to improve brain parcellation by comparing sub-cortical regions connectivity profiles and compute their corresponding geometric ``average'' connectivity profiles, (ii) we introduce Ricci flow applied to diffusion tensor fields to enhance feature extraction, and finally (iii) we introduce a notion of discrete Ricci curvature in brain connectivity networks as a novel nodal measure to detect critical regions (nodes) of the structural brain networks. This notion of node curvature can be used to identify changes in brain network structure due to disease/development as it supplements information that can be obtained by other conventional network nodal measures. We then study multi-compartment dMRI models, and present a novel model fitting method to such tissue models (Chapter 6). Our proposed method is generic to all multicompartment models and enables for the first time dMRI-imaging in multiple fiber orientations and fiber-crossings situations. In addition to potential improvements in imaging technology, we hope that the advances presented in this work will contribute to the diagnosis and treatment of neurological disorders.