A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues

Abstract : Diffusion magnetic resonance imaging (dMRI) is a non-invasive imaging technique that gives a measure of the diffusion characteristics of water in biological tissues, notably, in the brain. The hindrances that the microscopic cellular structure poses to water diffusion are statistically aggregated into the measurable macroscopic dMRI signal. Inferring the microscopic structure of the tissue from the dMRI signal allows one to detect pathological regions and to monitor functional properties of the brain. For this purpose, one needs a clearer understanding of the relation between the tissue microstructure and the dMRI signal. This requires novel numerical tools capable of simulating the dMRI signal arising from complex microscopic geometrical models of tissues. We propose such a numerical method based on linear finite elements that allows for a more accurate description of complex geometries. The finite elements discretization is coupled to the adaptive Runge-Kutta Chebyshev time stepping method. This method, which leads to the second order convergence in both time and space, is implemented on FeniCS C++ platform. We also use the mesh generator Salome to work efficiently with multiple-compartment and periodic geometries. Four applications of the method for studying the dMRI signal inside multi-compartment models are considered. In the first application, we investigate the long-time asymptotic behavior of the dMRI signal and show the convergence of the apparent diffusion coefficient to the effective diffusion tensor computed by homogenization. The second application aims to numerically verify that a two-compartment model of cells accurately approximates the three-compartment model, in which the interior cellular compartment and the extracellular space are separated by a finite thickness membrane compartment. The third application consists in validating the macroscopic Karger model of dMRI signals that takes into account compartmental exchange. The last application focuses on the dMRI signal arising from isolated neurons. We propose an efficient one-dimensional model for accurately computing the dMRI signal inside neurite networks in which the neurites may have different radii. We also test the validity of a semi-analytical expression for the dMRI signal arising from neurite networks.
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Dang Van Nguyen. A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues. Mathematical Physics [math-ph]. Ecole Polytechnique X, 2014. English. ⟨pastel-00957750⟩

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