The derivation of homogenized diffusion kurtosis models for diffusion MRI

Abstract

We use homogenization theory to establish a new macroscopic model for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of biological tissue with impermeable membranes. In this model, new higher-order diffusion tensors emerge and offer more information about the structure of the biological tissues. We explicitly solve the macroscopic model to obtain an ordinary differential equation for the diffusion MRI signal that has similar structure as diffusional kurtosis imaging models. We finally present some validating numerical results on synthetic examples showing the accuracy of the model with respect to signals obtained by solving the Bloch-Torrey equation.

A second order asymptotic model for diffusion MRI in permeable media

“…Following the classical periodic homogenization technique [4,17,28], we use the two-scale asymptotic expansions for the solution 𝑀 𝜀 in Ω 𝑒 𝜀 and Ω 𝑐 𝜀 :…”

Apparent diffusion coefficient measured by diffusion MRI of moving and deforming domains

“…In this way, one can study the contribution to the signal that is attributable to various types of cells or to the extra-cellular space inside the imaging voxel. In static organs, such as the brain, modeling and simulation efforts that link the measured diffusion MRI signal with the geometric structure of the cells and the extra-cellular space include analytical works (see, for example, [17,18]) and numerical works (see, for example [19,20,21,22,23,24,25]). For the heart, we cite the works [26,27,28,29,30] in which ex-vivo diffusion MRI is presented by performing numerical simulations on a model of fiber phantom and virtual cardiac microstructure.…”