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.

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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016060
Classification : 35C20, 35Q92
Mots clés : Asymptotic expansion, Bloch-Torrey equation, anisotropic diffusion transmission condition, diffusion magnetic resonance imaging
Caubet, Fabien 1 ; Haddar, Houssem 2 ; li, Jing-Rebecca 2 ; Van Nguyen, Dang 2

1 Institut de Mathématiques de Toulouse, Université de Toulouse, 31062 Toulouse cedex 9, France.
2 INRIA Saclay, Équipe DéFI, CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau, France.
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     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},
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     doi = {10.1051/m2an/2016060},
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     zbl = {1378.35309},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016060/}
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Caubet, Fabien; Haddar, Houssem; li, Jing-Rebecca; Van Nguyen, 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. http://www.numdam.org/articles/10.1051/m2an/2016060/

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