We report the first 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 on skeletal muscle , 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 in a tissue model consisting of cylindrical inclusions (representing bundles of parallel myocytes), delineated by thin permeable membranes (sarcolemma) and surrounded by an extracellular domain (endomysium) . As implemented, the hybrid LBM scheme accommodates piece-wise uniform transport, dMRI parameters, periodic  outer boundary conditions, and finite membrane permeabilities on non-boundary-conforming inner boundaries. The geometry is discretized in uniform 2D or 3D lattices by locating the curved cylindrical boundaries halfway between lattice nodes. 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 with MPI is straightforward. For a domain decomposition algorithm on one million lattice nodes, the speedup is linear up to 168 MPI processes. While offering certain advantages over finite element or Monte Carlo schemes, the proposed hybrid LBM constitutes a sui generis scheme that can be easily adapted to model more complex interfacial conditions and physics in heterogeneous multiphase tissue models and to accommodate sophisticated dMRI sequences.
We report the first 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.
R
Lattice
lattice
R
on skeletal muscle
in biological tissues
R
in a tissue model consisting of cylindrical inclusions (representing bundles of parallel myocytes), delineated by thin permeable membranes (sarcolemma) and surrounded by an extracellular domain (endomysium)
. A membrane boundary condition is presented which is able to accurately represent the effects of thin curvilinear membranes typically found in biological tissues
A
and mirroring
D
The geometry is discretized in uniform 2D or 3D lattices by locating the curved cylindrical boundaries halfway between lattice nodes.
R
with MPI is straightforward. For a domain decomposition algorithm on one million lattice nodes, the speedup is linear up to 168 MPI processes. While
, as well as on GPUs, is straightforward and efficient. Along with
R
sui generis
flexible
R
be
by