Coupling nonlinear electric fields and temperature to enhance drug transport: An accurate numerical tool

Abstract

The main motivation of the present work is the numerical study of a system of Partial Differential Equations that governs drug transport, through a target tissue or organ, when enhanced by the simultaneous action of an electric field and a temperature rise. The electric field, while forcing charged drug molecules through the tissue or the organ, thus creating a convection field, also leads to a rise in temperature that affects drug diffusion. The differential system is composed by a nonlinear elliptic equation, describing the potential of the electric field, and by two parabolic equations: a diffusion–reaction equation for temperature and a convection–diffusion–reaction for drug concentration. The temperature and the concentration equations are coupled with the potential equation via a reaction term and the convection and diffusion terms respectively. As the parabolic equations depend directly on the potential and its gradient, the central question is the design and mathematical study of an accurate method for the elliptic equation and its gradient. We propose a finite difference method, which is equivalent to a fully discrete piecewise linear finite element method, with superconvergent/supercloseness properties. The method is second order convergent with respect to a H1-discrete norm for the elliptic problem, and with respect to a L2-discrete norm for the two parabolic problems. The stability properties of the method are also analyzed. Numerical experiments illustrating the drug transport for different electrical protocols are also included.