Diffusion through skin in the light of a fractional derivative approach: progress and challenges

Abstract

This review is focussed on modelling the transport processes of different drugs across the intact human skin by introducing a memory formalism based on the fractional derivative approach. The fundamental assumption of the classic transport equation in the light of the Fick’s law is that the skin barrier behaves as a pseudo-homogeneous membrane and that its properties, summarized by the diffusion coefficient D,  do not vary with time and position. This assumption does not hold in the case of a highly heterogeneous system as the skin is, whose outermost layer (the stratum corneum) is comprised of a multi-layered structure of keratinocytes embedded in a lamellar matrix of hydrophobic lipids, followed by the dermis that contains a network of capillaries that connect to the systemic circulation. A possible way to overcome these difficulties resides in the introduction of mathematical models which involve fractional derivatives to describe complex systems with interactions in space and time, following the model originally developed by Caputo in order to consider the memory effects in materials. Although the introduction of fractional derivatives to model memory effects is completely phenomenological, i.e., characterized by a single parameter, i.e., the fractional derivative order $\nu ,$ a number of authors have found that this approach can provide a better comparison to experimental data and that this technique may be alternative to integer-order derivative models. In this review, we aim to summarize some our recent results, concerning the transport of different diffusing compounds of different structural complexity across the intact skin.