Fractional Diffusion with Geometric Constraints: Application to Signal Decay in Magnetic Resonance Imaging (MRI)

Ervin K. Lenzi,Haroldo V. Ribeiro,Marcelo K. Lenzi,Richard L. Magin,Luiz R. Evangelista

doi:10.3390/math10030389

Abstract

We investigate diffusion in three dimensions on a comb-like structure in which the particles move freely in a plane, but, out of this plane, are constrained to move only in the perpendicular direction. This model is an extension of the two-dimensional version of the comb model, which allows diffusion along the backbone when the particles are not in the branches. We also consider memory effects, which may be handled with different fractional derivative operators involving singular and non-singular kernels. We find exact solutions for the particle distributions in this model that display normal and anomalous diffusion regimes when the mean-squared displacement is determined. As an application, we use this model to fit the anisotropic diffusion of water along and across the axons in the optic nerve using magnetic resonance imaging. The results for the observed diffusion times (8 to 30 milliseconds) show an anomalous diffusion both along and across the fibers.

Highlights

Mathematical models of diffusion are needed to interpet experimental measurements designed to probe the micro-structure of heterogeneous materials [1,2,3]
The mean square displacement exhibits a linear time dependence, i.e., (r − hr i)2 ∼ t, where r is the position coordinate and t the time, which is typical of Markovian processes
We investigate the diffusion process of a system on a backbone structure, such as the one illustrated in Figure 1, where the particles at z = 0 may diffuse on the xy-plane, and, at z 6= 0 the particles only diffuse along the z-direction

Summary

Introduction

Mathematical models of diffusion are needed to interpet experimental measurements designed to probe the micro-structure of heterogeneous materials [1,2,3]. It is possible to find behaviors such as (r − hr i)2 ∼ lnγ t, which are related to ultraslow diffusion [13] Motivated by these anomalous regimes, a comb-like structure has been proposed as the main ingredient of a mathematical diffusive model to investigate anomalous diffusion in percolation clusters with topological bias [14,15]. The choice for the kernels may be related to structure of the media where the diffusion proceeds with barriers, traps, and tortuosity, among others, which may be connected to random walk with a long-tailed distribution for the waiting time distribution.

The Mathematical Problem and Experimental Data

Discussion and Conclusions