A FEniCS-HPC framework for multi-compartment Bloch-Torrey models

In diffusion nuclear magnetic resonance (NMR) and diffusion magnetic resonance imaging (MRI), the multi-compartment Bloch-Torrey equation plays an important role in probing the diffusion characteristics from a nanometer scale to a macroscopic scale. The signal attenuation can be computed by solving the equation. If the volume of interest is composed by multiple compartments, interface conditions with permeability are imposed. Depending on applications, different gradient strengths can be used to capture the signal attenuation. In probing porous media, for instance, high gradient strengths are used. In diffusion MRI, since water molecules enter and exit the computational domain in realistic cases, pseudo-periodic boundary conditions are used. These conditions cause difficulties in solving the equation efficiently and many efforts have been made to develop an efficient numerical method. However, large-scale problems for supercomputers with realistic applications have not been considered yet. We propose a framework for the multi-compartment Bloch-Torrey models based on the FEniCSHPC platform, a part of the FEniCS project that allows for automated discretization, automated error control with mesh adaptivity and high performance computing. The framework runs on supercomputers with near optimal weak and strong scaling. Our work includes two parts. First, we simplify the multi-compartment Bloch-Torrey model used in diffusion MRI by proposing an approximation to the pseudo-periodic boundary conditions to derive a general form for the interface and boundary conditions. The second part includes implementation and numerical validation of our method on the FEniCS-HPC platform. This simplified model is straightforward to implement and to parallelize and shows promise in validation against more realistic models.