Asymptotic models of the diffusion MRI signal accounting for geometrical deformations
Zheyi Yang1 ; Imen Mekkaoui2 ; Jan Hesthaven3 ; Jing-Rebecca Li4
1 INRIA Saclay, Equipe IDEFIX, CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
2 Laboratoire LMSSA, University Dr. Tahar Moulay, Saida, Algeria
3 Chair of CMSS, EPFL, Lausanne, Switzerland
4 INRIA Saclay, Equipe IDEFIX, 1 Rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France
MathematicS In Action, Tome 12 (2023) no. 1, pp. 65-85.

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses can be modeled by the Bloch-Torrey partial differential equation (PDE). The associated diffusion MRI signal is the spatial integral of the solution of the Bloch-Torrey PDE. In addition to the signal, the time-dependent apparent diffusion coefficient (ADC) can be obtained from the solution of another partial differential equation, called the HADC model, which was obtained using homogenization techniques.

In this paper, we analyze the Bloch-Torrey PDE and the HADC model in the context of geometrical deformations starting from a canonical configuration. To be more concrete, we focused on two analytically defined deformations: bending and twisting. We derived asymptotic models of the diffusion MRI signal and the ADC where the asymptotic parameter indicates the extent of the geometrical deformation. We compute numerically the first three terms of the asymptotic models and illustrate the effects of the deformations by comparing the diffusion MRI signal and the ADC from the canonical configuration with those of the deformed configuration.

The purpose of this work is to relate the diffusion MRI signal more directly with tissue geometrical parameters.

Publié le :
DOI : 10.5802/msia.32
Mots clés : Bloch-Torrey PDE, diffusion magnetic resonance imaging, finite elements, simulation, apparent diffusion coefficient
Zheyi Yang 1 ; Imen Mekkaoui 2 ; Jan Hesthaven 3 ; Jing-Rebecca Li 4

1 INRIA Saclay, Equipe IDEFIX, CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
2 Laboratoire LMSSA, University Dr. Tahar Moulay, Saida, Algeria
3 Chair of CMSS, EPFL, Lausanne, Switzerland
4 INRIA Saclay, Equipe IDEFIX, 1 Rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Zheyi Yang; Imen Mekkaoui; Jan Hesthaven; Jing-Rebecca Li. Asymptotic models of the diffusion MRI signal accounting for geometrical deformations. MathematicS In Action, Tome 12 (2023) no. 1, pp. 65-85. doi : 10.5802/msia.32. https://msia.centre-mersenne.org/articles/10.5802/msia.32/

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