Comparison of analytical and numerical models for anomalous diffusion in the Bloch-Torrey equation

In this paper we use a numerical simulation to analyze the time evolution of the net spin magnetization in diffusion weighted MRI. We compare the integer and fractional cases by simulating the Bloch-Torrey equation for several different types of gradient pulses. This generalization of the diffusion process is based on the Continuous Time Random Walk (CTRW) model, where the order of the fractional time derivative a, and the order of the fractional space derivative β, encode the distributions of waiting time and jump length increments that underly anomalous dynamics in heterogeneous materials. We examine the time evolution of the net magnetization for a constant gradient field, for a pair of bipolar gradient pulses, for the Stejskal-Tanner pulse sequence, and for a set of oscillating gradients. We compare the numerical results with analytical expressions in particular cases, and present the overall system response for selected values of α and β. One novel observation is a slow recovery of magnetization in the case of non-integer fractional time derivatives, which seems to reflect the persistence (or memory) of the earlier states of magnetization that is encoded in the definition of the Caputo time-domain fractional derivative.