  We report an implementation of the lattice Boltzmann method (LBM) to
integrate the Bloch-Torrey equation, which describes the evolution of the
transverse magnetization vector and the fate of the signal of diffusion
magnetic resonance imaging (dMRI). Motivated by the need to interpret dMRI
experiments in biological tissues, and to offset the small time-step limitation
of classical LBM, a hybrid LBM scheme is introduced and implemented to solve
the Bloch-Torrey equation. A membrane boundary condition is presented which is
able to accurately represent the effects of thin curvilinear membranes
typically found in biological tissues. As implemented, the hybrid LBM scheme
accommodates piece-wise uniform transport, dMRI parameters, periodic and
mirroring outer boundary conditions, and finite membrane permeabilities on
non-boundary-conforming inner boundaries. By comparing with analytical
solutions of limiting cases, we demonstrate that the hybrid LBM scheme is more
accurate than the classical LBM scheme. The proposed explicit LBM scheme
maintains second-order spatial accuracy, stability, and first-order temporal
accuracy for a wide range of parameters. The parallel implementation of the
hybrid LBM code in a multi-CPU computer system, as well as on GPUs, is
straightforward and efficient. Along with offering certain advantages over
finite element or Monte Carlo schemes, the proposed hybrid LBM constitutes a
flexible scheme that can by easily adapted to model more complex interfacial
conditions and physics in heterogeneous multiphase tissue models and to
accommodate sophisticated dMRI sequences.
