Bridging Continuous and Lattice-Based Models of Two-Dimensional Diffusion: A Systematic Approach for Estimating Transition Probabilities, Grid Size and Diffusivity

Abstract

Lattice-based models have been broadly applied in mathematical and computational modeling of biological and biomedical systems for which spatial effects are important. These discrete models commonly include diffusion of mobile constituents as a key underlying mechanism. While the direct simulation of diffusion in continuous (off-lattice) domains is possible, it is computationally intensive, particularly when multiple coupled mechanisms are involved. This study presents a systematic approach for connecting continuous models of two-dimensional diffusion with internal obstacles to discrete, lattice-based (surrogate) models of diffusion. Results from continuous model simulations on a representative domain, and over many realizations, are used to develop accurate lattice-based surrogate models by exploiting internal symmetries. Probabilities determined for the lattice-based surrogate models are also connected to theoretical diffusivities for 2D random walks on a square lattice, necessitating the calibration of a spatial grid size. This approach can facilitate the inclusion of more accurate diffusive transport models of complex media within the general framework of lattice-based models that incorporate multiple coupled mechanisms.