A diffusion and delivery model of a drug across the skin with diffusivity spatially modulated is formulated and solved analytically using computer algebra. The model is developed using one-dimensional diffusion equation with a diffusivity which is a function of position in the skin; with an initial condition which is describing that the drug is initially contained inside a therapeutic patch; with a boundary condition according to which the change in concentration in the patch is minimal, such that assumption of zero flux at the patch-skin interface is valid; and with other boundary condition according to which the microcirculation in the capillaries just below the dermis carries the drug molecules away from the site at a very fast rate, maintaining the inner concentration at 0. The model is solved analytically by the method of the Laplace transform, with Bromwich integral and residue theorem. The concentration profile of the drug in the skin is expressed as an infinite series of Bessel functions. The corresponding total amount of delivered drug is expressed as an infinite series of decreasing exponentials. Also, the corresponding effective time for the therapeutic patch is determined. All computations were performed using computer algebra software, specifically Maple. The analytical results obtained are important for understanding and improving currentapplications of therapeutic patches. For future research it is interesting to consider more general models of spatial modulation of the diffusivity and the possible application of other computer algebra software such as Mathematica and Maxima.

Date Published: 22 May 2014
PDF: 14 pages
Proc. SPIE 9107, Smart Biomedical and Physiological Sensor Technology XI, 91071F (22 May 2014); doi: 10.1117/12.2049766

Published in SPIE Proceedings Vol. 9107:
Smart Biomedical and Physiological Sensor Technology XI
Brian M. Cullum; Eric S. McLamore, Editor(s)