Stability analysis and analytical solution of a nonlinear model for controlled drug release: Travelling wave fronts

Abstract

In this paper, the process of drug dissolution and release from a planar matrix is investigated based on two coupled nonlinear partial differential equations proposed by Göran Frenning in 2003. In the modelling the process drug adsorption has been disregarded, assuming concentration-independent diffusion coefficients, using perfect sink conditions, and specializing to a planar geometry, The concentration profile of the mobile, or diffusing, the resulting model is rather complex and has been investigated only numerically and only approximate solution have been possible. In this paper it is shown that an analytical solution can be obtained exactly in the form of a travelling wave front. We describe the method for finding the analytical solutions using the travelling wave coordinate when the wave is assumed to be moving at constant speed. The model system of partial differential equations is transformed into two coupled ordinary differential equations, which is analysed interms of the stability of its steady state. Analytical solutions are derived in three possible cases, giving travelling wave solutions. We then discuss a comparison between the exact solutions obtained here and the analytical short-time approximation as well as the curves obtained from the modified Higuchi formula reported by Frenning in 2003.