Fractional derivatives in the transport of drugs across biological materials and human skin

The diffusion of drugs across a composite structure such as a biological membrane is a rather complex phenomenon, because of its inhomogeneous nature, yielding a diffusion rate and a drug solubility strongly dependent on the local position across the membrane itself. These problems are particularly strengthened in composite structures of a considerable thickness like, for example, the human skin, where the high heterogeneity provokes the transport through different simultaneous pathways. In this note, we propose a generalization of the diffusion model based on Fick's 2nd equation by substituting a diffusion constant by means of the memory formalism approach (diffusion with memory). In particular, we employ two different definitions of the fractional derivative, i.e., the usual Caputo fractional derivative and a new definition recently proposed by Caputo and Fabrizio. The model predictions have been compared to experimental results concerning the permeation of two different compounds through human skin in vivo, such as piroxicam, an anti-inflammatory drug, and 4-cyanophenol, a test chemical model compound. Moreover, we have also considered water penetration across human stratum corneum and the diffusion of an antiviral agent employed as model drugs across the skin of male hairless rats. In all cases, a satisfactory good agreement based on the diffusion with memory has been found. However, the model based on the new definition of fractional derivative gives a better description of the experimental data, on the basis of the residuals analysis. The use of the new definition widens the applicability of the fractional derivative to diffusion processes in highly heterogeneous systems.