Fractional Reaction–Diffusion Modeling of Drug Transport and Release in Biological Tissue

Abstract: Depot drug release and delivery to biologic tissues could be impeded by delay within a disordered matrix. In this work, a coupled time-fractional mathematical model was developed that describes the rate at which drugs are released from the depot into a 1-d tissue slab and subsequently removed as a function of both space and time. A Mittag- Leffler solution for the release kinetics of the depot, along with an Eigen function expansion describing the distribution of drug throughout the tissue slab when the source term is uniformly distributed and there are no flux gradients at the slab boundaries were used to solve for the concentration of drug in the tissue. A numerical implementation of the L1 method for solving the tissue transport equation was also presented. The numerical solutions were compared to exact analytical solutions for each of these cases. The relative root mean square errors for the coupled problem decreased from approximately 1.9 % to 0.24 % as the spatial resolution was increased and the temporal resolution were also increased. Synthetic data generated from known fractional and classical models were also used to estimate parameters from simulated release curves and concentration profiles. For this particular example, the best fit values of the fractional order of the drug transport process (alpha = 0.7135 [with a 95 % confidence interval of 𝛼 = 0.6888-0.7382]) and root mean squared error for prediction of concentration profiles on separate portions of the synthetic dataset were 0.0256 and 0.2096 for the fractional model and classical model respectively. Finally, sensitivity analyses demonstrated how changes in various model parameters affect concentrations of drug in the depot and tissue over time. The results obtained in this study show that our mathematical model can distinguish different behaviors due to changes in parameters. However, they represent only theoretical studies and therefore do not provide experimental evidence regarding the presence or nature of non-Fickian transport mechanisms during drug delivery in biological systems.

Keywords: Fractional reaction–diffusion, drug release, tissue transport, Caputo derivative, Mittag–Leffler function, parameter estimation, numerical verification.