Random walk diffusion simulations in semi-permeable layered media with varying diffusivity

Abstract

In this paper we present random walk based solutions to diffusion in semi-permeable layered media with varying diffusivity. We propose a novel transit model for solving the interaction of random walkers with a membrane. This hybrid model is based on treating the membrane permeability and the step change in diffusion coefficient as two interactions separated by an infinitesimally small layer. By conducting an extensive analytical flux analysis, the performance of our hybrid model is compared with a commonly used membrane transit model (reference model). Numerical simulations demonstrate the limitations of the reference model in dealing with step changes in diffusivity and show the capability of the hybrid model to overcome this limitation and to offer substantial gains in computational efficiency. The suitability of both random walk transit models for the application to simulations of diffusion tensor cardiovascular magnetic resonance (DT-CMR) imaging is assessed in a histology-based domain relevant to DT-CMR. In order to demonstrate the usefulness of the new hybrid model for other possible applications, we also consider a larger range of permeabilities beyond those commonly found in biological tissues.

Cardiac q‐space trajectory imaging by motion‐compensated tensor‐valued diffusion encoding in human heart in vivo

“…Furthermore, we observed that MD increased, and FA decreased, with mean frequency of the power spectrum of q ( t ), demonstrating clear diffusion time effects in contrast to the in vivo data. This difference may appear due to the longer diffusion times used at the clinical scanner (a consequence of the relatively low gradient system performance in human MRI systems) and may be indicative of a higher permeability in myocardium compared with the plastic used in the phantom 64,65 . Here, we were limited by the size of the phantom; therefore, a preclinical scanner with high gradient performance was used.…”

Realistic numerical simulations of diffusion tensor cardiovascular magnetic resonance: The effects of perfusion and membrane permeability

“…[14][15][16] Whenever a particle interacts with a cell membrane, a probability of transit p t is computed via a transit model that represents the permeability embedded in the membrane boundary condition. 19,20 This transit model, that we refer to as the hybrid model, 20 treats the membrane permeability and the step change in diffusion as successive barrier interactions. Doing so reduces the complexity of the transit probabilities to…”

“…The aim of this work is to study the sensitivity of DT‐CMR parameters to changes in cell membrane permeability and microvascular perfusion. The simulations of the diffusion process use a new permeability model 20 that improves upon the treatment of the interface boundary condition 19 between regions of different diffusivity as compared to previous numerical simulations of diffusion tensor imaging. In a similar manner, the microvascular perfusion signal has been modeled assuming that a group of particles travel at constant velocities through a capillary network anisotropically oriented according to a Dimroth–Watson distribution 29,30 .…”

Precise Derivations of Radiative Properties of Porous Media Using Renewal Theory

“…Using the existing machinery of surplus risk theory and ruin analysis, we can then derive close upper bounds and explicit tight estimates for radiative metrics. Although random walk and Lévy processes have been extensively used for modeling diffuse processes [105]- [109] in various transport problems and porous media modeling, there are two shortcomings: (i) relevance to radiation heat transfer is scarce, as opposed to other problems such as probe diffusion and permeability modeling [110]- [113], (ii) closed-form derivations that lead to tangible variance reduction in MCRT are missing. The particular angle of surplus risk theory provides a richer apparatus to derive directly related quantities.…”