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Habilitation à diriger des recherches

Modeling and simulation of diffusion

Jing-Rebecca Li 1 
1 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : The first part of this thesis concerns the formulation of numerical methods that are local in time for the solution of equations with memory. The main idea is that the solution will be updated in the Fourier domain in order to avoid evaluating time convolution integrals that have memory. This work was made possible by the development of a good quadrature\cite{Greengard2000} of the Fourier integral where a small number of points in the Fourier variable were sufficient for a good resolution of the problem in the physical space over a large time interval. First, we developed a numerical method to simulate diffusion in unbounded domains with sources and applied it to the modeling of crystal growth using the phase field model. Then, in order to extend this approach to boundary value problems, we addressed the issue of evaluating the single and double layer potentials on the boundary. Finally, we generalized the idea of replacing time convolution integrals by an efficient quadrature in the transform domain to fractional integrals and derivatives for general fractional orders and obtained a rigorous bound on the quadrature error. Then we applied this approach to a fractional wave equation. The second part of the thesis concerns the specific application of diffusion magnetic resonance imaging (dMRI) in the brain. The effect on the MRI signal of the water proton magnetization in biological tissue in the presence of magnetic field gradient pulses can be modeled by a {\it microscopic} multiple compartment Bloch-Torrey partial differential equation (PDE). This PDE can be best understood as imparting a spatially dependent frequency to diffusing particles in a heterogeneous medium. The dMRI signal is the integral of the solution of this PDE at the echo time. First, we numerically solved this PDE by coupling a standard Cartesian spatial discretization with an adaptive time discretization and studied the diffusion characteristics of a tissue model of the brain gray matter made up of cylindrical and spherical cells embedded in the extra-cellular space. Next we formulated a new {\it macroscopic} ODE model for the dMRI signal by mathematical homogenization. Then we showed by numerical simulations that this ODE model gives a good approximation of the dMRI signal of the full PDE model at relatively long but still physically realistic diffusion times relevant to dMRI in the brain. Finally, I will describe some future research directions in dMRI. The first is the experimental validation of the PDE model by imaging the ganglia (neuronal network) of the Aplysia (giant sea slug), to be conducted at the MRI center Neurospin. The second is the inclusion of blood flow in the bran micro-vessels in a new PDE model. The third is the formulation of a different ODE model valid at shorter diffusion times or in the presence of larger cells.
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Habilitation à diriger des recherches
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Submitted on : Tuesday, January 7, 2014 - 2:29:19 PM
Last modification on : Wednesday, August 24, 2022 - 9:40:06 AM
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  • HAL Id : tel-00925028, version 1



Jing-Rebecca Li. Modeling and simulation of diffusion. Modeling and Simulation. Université Paris Sud - Paris XI, 2013. ⟨tel-00925028⟩



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