The human skin has a complicated structure with many multi-scale, biophysical effects impacting the propagation of skin-injected substances, such as partitioning, metabolic reactions, adsorption and elimination. An extended version of Fickโs second law governing the process of the compound diffusion in various skin layer is employed in the current work by considering the conservation of mass of the substance and the metabolic reaction of the substance in viable skin. Additionally, a model assuming linear coupling between the substance concentrations that are bound and unbound with blood was developed. Using such a model, a set of coupled partial differential equations are derived as the governing equations for the 3D dynamic skin pharmacokinetics model. To approximate the solution of these equations, a Meshless Method is developed employing localized collocation of inverse multi-quadric radial basis functions (RBF) to ensure a smooth, accurate, well-conditioned solution of the global field variable in space, while simultaneously implementing a forward-time marching scheme to address the time-transience of the solution. To validate the Localized RBF Collocation Meshless Model (LRCMM), the 2D and 3D cases of verapamil diffusion in viable skin was investigated. A benchmark, given in the literature employs empirically derived values for the governing equation parameters, providing a point of comparison for the LRCMM. In conjunction with the benchmark, the numerical analysis simulated the verapamil skin diffusion process during 4 hours of continuous injection with an input concentration of 43 mg/ml, followed by 4 hours of diffusion without further injection. The 2D and 3D LRCMM solutions compare well versus the benchmark, demonstrating the ability of the LRCMM to accurately model the system behavior. Thus, results from this study will be further implemented in future compound permeation studies, where advantages of the LRCMM will be leveraged for the optimization of various pharmacokinetic parameters for transdermal drug delivery.
This is a joint work with Anthony Khoury, Vladimir V. Golubev, and Alain J. Kassa
