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

• Published in: Journal of computational physics Vol. 263; pp. 283 - 302
• Main Authors: Nguyen, Dang Van, Li, Jing-Rebecca, Grebenkov, Denis, Le Bihan, Denis
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 15.04.2014; Elsevier
• Subjects: ANIMAL TISSUES; BIOLOGICAL MODELS; Bloch–Torrey equation; BOUNDARY CONDITIONS; Cellular; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computer Science; DIFFUSION; Diffusion magnetic resonance imaging; Diffusion plating; Double-node; FINITE ELEMENT METHOD; Finite elements; Interface problem; INTERFACES; MAGNETIC FIELDS; MAGNETIZATION; Mathematical analysis; Mathematical models; Modeling and Simulation; NMR IMAGING; OSCILLATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PERIODICITY; PERMEABILITY; PROTONS; Pseudo-periodic; RKC; STEADY-STATE CONDITIONS; TENSORS; TRANSFORMATIONS; WATER; Finite elements; Interface problem; Diffusion magnetic resonance imaging; Double-node; RKC; Bloch–Torrey equation; Pseudo-periodic
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.01.009

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.