Time-dependent diffusion in one-dimensional disordered media decorated by permeable membranes

Motivation: Characterizing water diffusion in its long-time regime is relevant to most medical applications of diffusion MRI. However, this process is challenging to model even for a one-dimensional structure with semipermeable membranes. Goal(s): Developing methods that predict the asymptotic instantaneous diffusivity from the bulk diffusivity and the membranes’ locations and permeabilities. Approach: We studied the problem theoretically and expressed the instantaneous diffusivity as an infinite sum. An independent numerical scheme is developed. Several types of disorder in the membranes’ positions were considered. Results: Our findings show excellent agreement with simulations. Our methods provide an alternative means for studying time-dependent diffusion. Impact: This report provides an improved understanding of how tissue organization and disorder may affect water diffusivity, thus making it relevant to diffusion MR studies characterizing tissue microstructure. We also provide methods for readily estimating the instantaneous diffusivity.