A partition of unity finite element method for computational diffusion MRI

• The Bloch-Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries. Many efforts have been made to solve the equation but there is still missing an efficient high performance computing framework. In this work, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations use extended finite elements with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as operator splitting for the exterior boundary conditions. The method appears to be straightforward to implement and it is implemented in the FEniCS for moderate-scale simulations and in the FEniCS-HPC for the large-scale simulations. The accuracy of the resulting method is validated numerically and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available. Finally, we do simulations on a complex neuron to study how the signals decay under the effect of the permeable membrane and to show that the method can be used to simulate for complex geometries that we have not done before.Highlights:The discontinuity in the magnetization across the interior interfaces of the medium is weakly imposed, allowing generalization to arbitrary order finite elements.Spin exchange across the external boundaries is implemented by weakly imposing an artificial, high permeability, condition, allowing generalization to non-matching meshes.Thus, optimal convergence with respect to the space discretization is achieved.The second-order Crank-Nicolson method is chosen for the time discretization to reduce oscillations at high gradient strengths and allows for larger time-step sizes.The method is of a high level of simplicity and suitable for parallelization.An efficient open-source code is implemented in the FEniCS and FEniCS-HPC platforms.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

NATURVETENSKAP  -- Data- och informationsvetenskap -- Datavetenskap (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences -- Computer Sciences (hsv//eng)

NATURVETENSKAP  -- Data- och informationsvetenskap -- Bioinformatik (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences -- Bioinformatics (hsv//eng)

Nyckelord

extended finite element

computational diffusion MRI

interface conditions

pseudo-periodic conditions

Bloch-Torrey equation

FEniCS

FEniCS-HPC.

Tillämpad matematik och beräkningsmatematik

Applied and Computational Mathematics

Computer Science

Datalogi

Biological Physics

Biologisk fysik

Publikations- och innehållstyp

vet (ämneskategori)

ovr (ämneskategori)