Modeling the diffusion-erosion crossover dynamics in drug release

A computational model is proposed to investigate drug delivery systems in which erosion and diffusion mechanisms are participating in the drug release process. Our approach allowed us to analytically estimate the crossover point between those mechanisms through the value of the parameter $b$ ($b_c = 1$) and the scaling behavior of parameter $\tau$ on the Weibull function, $\exp[-(t/\tau)^b]$, used to adjust drug release data in pharmaceutical literature. Numerical investigations on the size dependence of the characteristic release time $\tau$ found it to satisfy either linear or quadratic scaling relations on either erosive or diffusive regimes. Along the crossover the characteristic time scales with the average coefficient observed on the extreme regimes ({\it i.e.}, $\tau \sim L^{3/2}$), and we show that this result can be derived analytically by assuming an Arrhenius relation for the diffusion coefficient inside the capsule. Based on these relations a phenomenological expression for the characteristic release in terms of size $L$ and erosion rate $\kappa$ is proposed, which can be useful for predicting the crossover erosion rate $\kappa_c$. We applied this relation to the experimental literature data for the release of acetaminophen immersed in a wax matrix and found them to be consistent with our numerical results.