Diffusion in inhomogenous media

Abstract

This project considers the diffusion of water molecules through a cellular medium in which the cells are modeled by square compartments placed symmetrically in a square domain. We assume the diffusion process is governed by the 2D diffusion equations and the solution is provided by implementing the Crank-Nicolson scheme. These results are verified and illustrated to agree well with the finite element method using the Comsol Multiphysics package. The model is used to compute the values of the apparent diffusion coefficient, (ADC) which is a measure that is derived from diffusion weighted MRI data and can be used to identify, e.g., regions of ischemia in the brain. With our model, it is possible to examine how the value of the apparent diffusion coefficient is affected whenever the extracellular space is varied. We observe that the average distance that the water molecules travel in a definite time is highly dependent on the geometrical properties of the cellular media.