A Macroscopic Model Including Membrane Exchange for Diffusion MRI

Abstract

Diffusion Magnetic Resonance Imaging is a promising tool to obtain useful information on the microscopic structure and has been extensively applied to biological tissues. We establish a new macroscopic model from homogenization theory for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of intermediate water exchange across biological cellular membranes. Based on a particular scaling of the permeability condition modeling cellular membranes, this macroscopic model reproduces the memory effects often observed in experiments. Explicit formulae given by homogenization for the coefficients of this model emphasize their link to the relevant physiological quantities. In addition, we explicitly solve the macroscopic model to obtain an ODE model for the dMRI signal. This ODE model is numerically easy to invert, and the inverse problem of retrieving model coefficients from synthetic dMRI signal data is considered.

Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

“…There are of course other ways to mathematically justify our homogenization procedure, in particular one could use the notion of two-scale convergence (which is strongly related to the unfolding operator, see [1] and [5]), but the unfolding operator will prove to be a very natural way to treat the jump condition involved by the permeability condition of our model problem. We show that there are only five possible limit models, including of course the model of [7]. These limit models are similar to those that have already appeared in the literature when homogenizing elliptic problems with jump condition (see [14]- [15]) using two-scale convergence.…”

“…The biological cells' membrane is modeled by a permeability condition. We use R d primarily to remain consistent with [7], where this choice is discussed from the physical and biological point of view. From the mathematical point of view, the results we obtain remain true for any Ω Lipschitz open subset of R d , with appropriate boundary conditions, such as Dirichlet or Neumann's (those boundary conditions, if not scaled, should simply be added to the homogenized problems in that case, the local cell problems remaining unchanged).…”

“…In a recent paper [7], a macroscopic model has been introduced through formal homogenization, under the assumption that the media is periodic. It efficiently reproduces experimental dMRI's signal measurements and generalizes some phenomenological models that have appeared in the medical imaging literature (see [11]).…”

“…Then, we define our model problem, which requires a careful treatment due to the presence of unbounded coefficients. In a second section, we state our main results : we present the homogenized macroscopic models we want to establish, and briefly display a few numerical results to compare them to the full model problem, emphasizing that the model introduced in [7] is indeed the most accurate one. The remaining sections are devoted to the homogenization proof.…”

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

“…σ n m = σ τ m , it is well-known that the following asymptotic transmission condition, which we denote the Isotropic Diffusion Transmission Condition (IDTC), can be imposed on the interface Γ (see, e.g., [9]):…”