Modelling anomalous diffusion using fractional Bloch-Torrey equations on approximate irregular domains

Diffusion-weighted imaging is an in vivo, non-invasive medical diagnosis technique that uses the Brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment. Recent work highlights that the diffusion-induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay. To investigate the underlying mechanism of this deviated signal decay, diffusion is re-examined through the Bloch-Torrey equation by using fractional calculus with respect to both time and space. In this study, we explore the influence of the complex geometrical structure on the diffusion process. An effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two-dimensional time-space Riesz fractional partial differential equation with Dirichlet boundary conditions. This scheme is proved to be unconditionally stable and convergent. Numerical examples are given to support our analysis. We then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time-space fractional Bloch-Torrey equations on irregular domains.