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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.
Yang, Zheyi 1 ; Mekkaoui, Imen 2 ; Hesthaven, Jan 3 ; Li, Jing-Rebecca 4
CC-BY 4.0
@article{MSIA_2023__12_1_65_0,
author = {Yang, Zheyi and Mekkaoui, Imen and Hesthaven, Jan and Li, Jing-Rebecca},
title = {Asymptotic models of the diffusion {MRI} signal accounting for geometrical deformations},
journal = {MathematicS In Action},
pages = {65--85},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {12},
number = {1},
year = {2023},
doi = {10.5802/msia.32},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/msia.32/}
}
TY - JOUR AU - Yang, Zheyi AU - Mekkaoui, Imen AU - Hesthaven, Jan AU - Li, Jing-Rebecca TI - Asymptotic models of the diffusion MRI signal accounting for geometrical deformations JO - MathematicS In Action PY - 2023 SP - 65 EP - 85 VL - 12 IS - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - http://geodesic.mathdoc.fr/articles/10.5802/msia.32/ DO - 10.5802/msia.32 LA - en ID - MSIA_2023__12_1_65_0 ER -
%0 Journal Article %A Yang, Zheyi %A Mekkaoui, Imen %A Hesthaven, Jan %A Li, Jing-Rebecca %T Asymptotic models of the diffusion MRI signal accounting for geometrical deformations %J MathematicS In Action %D 2023 %P 65-85 %V 12 %N 1 %I Société de Mathématiques Appliquées et Industrielles %U http://geodesic.mathdoc.fr/articles/10.5802/msia.32/ %R 10.5802/msia.32 %G en %F MSIA_2023__12_1_65_0
Yang, Zheyi; Mekkaoui, Imen; Hesthaven, Jan; Li, Jing-Rebecca. Asymptotic models of the diffusion MRI signal accounting for geometrical deformations. MathematicS In Action, Maths Bio, Tome 12 (2023) no. 1, pp. 65-85. doi: 10.5802/msia.32
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