Numerical simulation of the distributed-order time-space fractional Bloch-Torrey equation with variable coefficients

Abstract

The purpose of this research is to establish the generalised fractional Bloch-Torrey equation for better simulating anomalous diffusion in heterogeneous biological tissues. The introduction of the distributed-order time fractional derivative allows for an improved interpretation of the complex diffusion behaviours with multi-scale effects. The use of variable coefficients in the model increases its applicability for describing the spatial heterogeneity evident in the cellular structures. The proposed distributed-order time and space fractional Bloch-Torrey equation is discretised in time and space by the L2-1σ formula and the finite element method, respectively. The stability and convergence analyses of these numerical methods are provided. To further improve the computational efficiency, a reduced-order extrapolation scheme is developed. We verify the effectiveness of the proposed methods by numerical examples. Moreover, the coupled fractional dynamic system solution behaviour is explored on a human brain-like domain divided into the white matter and grey matter regions. Compared with the model having constant coefficients, solution behaviours suggest that variable diffusion coefficients offer an effective way to differentiate the distinct diffusion phenomena evolving in different tissue micro-environments. Furthermore, to evaluate the impacts of the weight function in the distributed-order operator, we choose three types of beta distributions with the same mean but different values of the variance. The results indicate that the larger value of the variance leads to a more remarkable fluctuation and a slower decay of the transverse magnetisation. This generalised distributed-order fractional model may provide further insights into capturing anomalous diffusion in heterogeneous media.