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The Bloch-Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch-Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers.
Caubet, Fabien 1 ; Haddar, Houssem 2 ; li, Jing-Rebecca 2 ; Nguyen, Van Dang 2
@article{M2AN_2017__51_4_1279_0,
author = {Caubet, Fabien and Haddar, Houssem and li, Jing-Rebecca and Nguyen, Van Dang},
title = {New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion {MRI}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1279--1301},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/m2an/2016060},
mrnumber = {3702413},
zbl = {1378.35309},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016060/}
}
TY - JOUR AU - Caubet, Fabien AU - Haddar, Houssem AU - li, Jing-Rebecca AU - Nguyen, Van Dang TI - New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1279 EP - 1301 VL - 51 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016060/ DO - 10.1051/m2an/2016060 LA - en ID - M2AN_2017__51_4_1279_0 ER -
%0 Journal Article %A Caubet, Fabien %A Haddar, Houssem %A li, Jing-Rebecca %A Nguyen, Van Dang %T New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1279-1301 %V 51 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016060/ %R 10.1051/m2an/2016060 %G en %F M2AN_2017__51_4_1279_0
Caubet, Fabien; Haddar, Houssem; li, Jing-Rebecca; Nguyen, Van Dang. New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1279-1301. doi: 10.1051/m2an/2016060
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