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

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.
Authors:
 [1] ;  [2] ;  [1] ;  [2] ;  [3] ;  [4]
  1. INRIA Saclay, Equipe DEFI, CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex (France)
  2. (France)
  3. Laboratoire de Physique de la Matiere Condensee, CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex (France)
  4. NeuroSpin, Bat145, Point Courrier 156, CEA Saclay Center, 91191 Gif-sur-Yvette Cedex (France)
Publication Date:
OSTI Identifier:
22314860
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 263; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
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