Diffusion in deforming semi-permeable media: a Monte Carlo random walk approach

Abstract Diffusion processes in deforming media with semi-permeable barriers are fundamental to understanding many biological and physical systems. The presence of semi-permeable barriers causes temporal and spatial departures from normal diffusion. We present an analytic expression for diffusion in a dynamically strained cuboidal cell that demonstrates one extreme of behaviour. Unfortunately however, analytical models cannot represent the effects of dynamic strain in microscopically restrictive media with varying permeability. To address these limitations, we explore how different cyclical strain patterns affect the apparent diffusion tensor in a simplified microstructure inspired by cardiac tissue. We show how the diffusion tensor in a dynamically strained medium may be derived for a Monte Carlo Random Walk (MCRW) model. The MCRW procedure is readily adaptable to more complex geometries and to cyclic and non-cyclic strains. Computations using the MCRW method show that the effects of physiologically relevant strain ( $$\sim 20\%$$ ) are significantly attenuated by semi-permeable barriers. Strain correction methods, derived for homogeneous diffusion, are found to systematically overcorrect as barrier permeability decreases. These findings are significant for interpreting measurements, particularly in biology, where both microstructural barriers and dynamic deformation alter observed diffusion properties.