Diffusion-controlled drug delivery: Dealing with the Stochastic Dilemma in Lattice Monte Carlo (LMC) simulations

Lattice Monte Carlo (LMC) methods are commonly used to study bio/chemical processes involving diffusion phenomena, including drug delivery systems. In such systems, the drug is encapsulated in a porous material (e.g., a hydrogel) which releases it through two processes: (i) disintegration of the material and (ii) diffusive escape of the drug molecules. One way to control the release is to design layered materials with various diffusivity regions. Diffusion in such systems is often studied numerically using LMC methods. However, modelling inhomogeneous systems, boundary conditions and diffusion at interfaces is somewhat ambiguous in LMC simulations. In this talk, we present computational studies of 2D systems consisting in two different sets of immobile obstacles that create two media with different effective viscosities, as well as their equivalent obstacle-free 1D systems with corresponding effective diffusion coefficients. Using this toy model, we examine how interfacial diffusion is treated by the various flavours of stochastic calculi, and we demonstrate that Isothermal calculus is the correct choice as opposed to the generally employed Ito calculus. We then explore the corrections that must be considered to simplify such nonuniform systems.