Volume 51, Number 4, July-August 2017
|Page(s)||1279 - 1301|
|Published online||30 June 2017|
New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI
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.
Received: 30 March 2015
Revised: 29 June 2016
Accepted: 8 September 2016
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.
Mathematics Subject Classification: 35C20 / 35Q92
Key words: Asymptotic expansion / Bloch-Torrey equation / anisotropic diffusion transmission condition / diffusion magnetic resonance imaging
© EDP Sciences, SMAI 2017
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