New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI

Abstract

Abstract. 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.

Improved impedance conditions for a thin layer problem in a nonsmooth domain

“…Since both Δ (x) and ∇ (x) vanish for |x| < r 0 , we may exploit the behavior of at infinity to obtain an expansion of f with respect to . Indeed, using (7), we obtain for |x| > r 0 ,…”

“…Situations where two materials are glued together enter this scope as well; see, eg, Geymonat et al For electromagnetism, thin dielectric layers appear in many situations; see, for example, Haddar and Jiang for the eddy current problem in the context of copper deposits on tubes, or Zutter and Knockaert for the skin effect problem, which has strong connections with thin layers. Biological tissues often involve thin parts; see Perrussel and Poignard for a mathematical and numerical study of the electromagnetic field around and inside a biological cell, or Caubet et al for the description of the diffusion magnetic resonance imaging signal in biological tissues. Thin films are good examples as well, and various models have to be considered depending on their nature and size; see Richard et al and the references therein for falling films, or Verma et al for an electrochemical situation.…”

Asymptotic expansions and effective boundary conditions: a short review for smooth and nonsmooth geometries with thin layers

“…The aim of this section is to present some ideas to improve the approximation of the thin layer problem. The starting point is relation (21), where it is clear that the main terms are as follows…”

“…For electromagnetism, thin dielectric layers appear in many situations, see for example [38] for the eddy current problem in the context of copper deposites on tubes, or [61] for the skin effect problem, which has strong connections with thin layers. Biological tissues often involve thin parts, see [48] for a mathematical and numerical study of the electromagnetic field around and inside a biological cell, or [21] for the description of the diffusion Magnetic Resonance Imaging signal in biological tissues. Thin films are good examples as well, and various models have to be considered depending on their nature and size, see [53] and the references therein for falling films, or [56] for an electrochemical situation.…”

Shape Sensitivity Analysis for Elastic Structures with Generalized Impedance Boundary Conditions of the Wentzell Type—Application to Compliance Minimization

“…In linear elasticity, it deserves to mention the book of Ciarlet [21], where a local representation of the GIBC is proposed, and others works [26,11] in the context of thin elastic plates or shells. The construction of GIBC is also treated for example by Antoine et al [9], by Poignard [44] or by Haddar et al [31,32,18] in acoustics and electromagnetism for general three-dimensional surfaces.…”