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A finite element method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging
Ecole Polytechnique, France.ORCID iD: 0000-0002-3213-0040
2014 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 263, p. 283-302Article in journal (Refereed) Published
Abstract [en]

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation (PDE). In addition, steady-state Laplace PDEs can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristics of the medium in the long time limit. In spatial domains that model biological tissues at the cellular level, these two types of PDEs have to be completed with permeability conditions on the cellular interfaces. To solve these PDEs, we implemented a finite elements method that allows jumps in the solution at the cell interfaces by using double nodes. Using a transformation of the Bloch-Torrey PDE we reduced oscillations in the searched-for solution and simplified the implementation of the boundary conditions. The spatial discretization was then coupled to the adaptive explicit Runge-Kutta-Chebyshev time-stepping method. Our proposed method is second order accurate in space and second order accurate in time. We implemented this method on the FEniCS C++ platform and show time and spatial convergence results. Finally, this method is applied to study some relevant questions in diffusion MRI.

Place, publisher, year, edition, pages
Academic Press, 2014. Vol. 263, p. 283-302
Keywords [en]
Bloch–Torrey equation; Diffusion magnetic resonance imaging; Finite elements; RKC; Pseudo-periodic; Double-node; Interface problem
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-225137DOI: 10.1016/j.jcp.2014.01.009ISI: 000331716900016Scopus ID: 2-s2.0-84893491013OAI: oai:DiVA.org:kth-225137DiVA, id: diva2:1194333
Note

QC 20180403

Available from: 2018-03-31 Created: 2018-03-31 Last updated: 2018-05-07Bibliographically approved

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Nguyen, Van Dang

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CiteExportLink to record
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Citation style
  • apa
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  • de-DE
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  • nn-NB
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  • Other locale
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Output format
  • html
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  • asciidoc
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