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A partition of unity finite element method for computational diffusion MRI
KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).ORCID iD: 0000-0002-3213-0040
KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).ORCID iD: 0000-0002-1695-8809
KTH, School of Electrical Engineering and Computer Science (EECS), Computational Science and Technology (CST).ORCID iD: 0000-0003-4256-0463
INRIA Saclay-Equipe DEFI, CMAP, Ecole Polytechnique Route de Saclay, 91128, Palaiseau Cedex, France.
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 271-290Article in journal (Refereed) Published
Abstract [en]

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 approaches have been taken to solve the Bloch–Torrey equation but an efficient high-performance computing framework is still missing. In this paper, 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 are based on the partition of unity concept which allows for a discontinuous solution across interfaces conforming with the mesh with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as an operator splitting for the exterior boundary conditions. The method is straightforward to implement and it is available in FEniCS for moderate-scale simulations and in FEniCS-HPC for large-scale simulations. The order of accuracy of the resulting method is validated in numerical tests 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 and we performed simulations for a realistic model of a neuron to show that the method can be used for complex geometries.

Place, publisher, year, edition, pages
Elsevier, 2018. Vol. 375, p. 271-290
Keywords [en]
Computational diffusion MRI, Bloch–Torrey equation, Partition of unity finite element method, Interface conditions, Weak pseudo-periodic conditions, FEniCS/FEniCS-HPC
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics; Biological Physics; Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-234286DOI: 10.1016/j.jcp.2018.08.039ISI: 000450907600014Scopus ID: 2-s2.0-85054048672OAI: oai:DiVA.org:kth-234286DiVA, id: diva2:1245780
Funder
Swedish Energy Agency, P40435-1
Note

QC 20180906

Available from: 2018-09-06 Created: 2018-09-06 Last updated: 2019-02-15Bibliographically approved

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Nguyen, Van DangJansson, JohanHoffman, Johan

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