A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging

Journal Article · Tue Apr 15 00:00:00 EDT 2014 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2014.01.009· OSTI ID:22314860

Nguyen, Dang Van ^[1]; Li, Jing-Rebecca ^[1]; Grebenkov, Denis ^[2]; Le Bihan, Denis ^[3]

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1. INRIA Saclay, Equipe DEFI, CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex (France)
2. Laboratoire de Physique de la Matiere Condensee, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex (France)
3. NeuroSpin, Bat145, Point Courrier 156, CEA Saclay Center, 91191 Gif-sur-Yvette Cedex (France)

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.

• OSTI ID: 22314860
• Journal Information: Journal of Computational Physics, Vol. 263; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANIMAL TISSUES
BIOLOGICAL MODELS
BOUNDARY CONDITIONS
DIFFUSION
FINITE ELEMENT METHOD
INTERFACES
MAGNETIC FIELDS
MAGNETIZATION
NMR IMAGING
OSCILLATIONS
PARTIAL DIFFERENTIAL EQUATIONS
PERIODICITY
PERMEABILITY
PROTONS
STEADY-STATE CONDITIONS
TENSORS
TRANSFORMATIONS
WATER