Numerical methods for simulating diffusion in cellular media

Abstract

Diffusion imaging is a relatively recent branch of magnetic resonance imaging that produces images of human physiology through diffusion of water molecules within the body. One difficulty in calculating diffusion coefficients, particularly in the brain, is the multitude of natural barriers to water diffusion, such as cell membranes, myelin sheaths, and fiber tracts. These barriers mean that water diffusion is not a homogeneous random process. Due to the complexity of modeling these structures, a simplifying assumption made in some methods of data analysis is that there are no barriers to water diffusion. We develop tools to simulate the diffusion of water in an inhomogeneous medium, which may then be used to test the accuracy of this assumption. The inherent difficulty (and computational cost) of including barriers (e.g., cell membranes) can be lessened by employing the immersed boundary (IB) method to represent these structures without the need for complicated computational grids. The contribution of this thesis is the implementation and validation of an IB method that allows for diffusion across semi-permeable membranes. The method is tested for a square interface aligned with the computational grid by comparing it to a second numerical scheme that uses standard finite differences. We also calculate the rate of convergence for the IB method to assess the numerical accuracy. To demonstrate the flexibility of the IB method to simulate diffusion with any interface shape, we also present simulations for irregular interfaces.