Diffusion across semi-permeable barriers: spectral properties, efficient computation, and applications

TL;DR

This paper introduces an efficient computational method for analyzing diffusion across semi-permeable barriers in one-dimensional structures, enabling analytical and numerical calculations of diffusion properties relevant to MRI and other applications.

Contribution

It provides a novel, efficient approach to compute eigenvalues and eigenmodes of the diffusion operator in heterogeneous structures with semi-permeable barriers, covering both analytical and numerical solutions.

Findings

Effect of barriers on diffusion properties analyzed

Transition from no barriers to impermeable barriers studied

Method enables accurate computation of diffusion propagator

Abstract

We present an efficient method to compute the eigenvalues and eigenmodes of the diffusion operator $\nabla(D\nabla)$ on one-dimensional heterogeneous structures with multiple semi-permeable barriers. This method allows us to calculate the diffusion propagator and related quantities such as diffusion MRI signal or first exit time distribution analytically for regular geometries and numerically for arbitrary ones. The effect of the barriers and the transition from infinite permeability (no barriers) to zero permeability (impermeable barriers) are investigated.