Fickian and Non-Fickian Transports in Ultrasound Enhanced Drug Delivery: Modeling and Numerical Simulation

In this paper we study a multiphysics/multidomain system of partial differential equations defined by hyperbolic and parabolic equations. We consider two wave equations and two diffusion equations defined in two different domains. The diffusion equations are considered of different types: a Fickian equation and a non-Fickian equation. In the interface continuity conditions are assumed for the wave equations, while for the diffusion equation only continuity of the mass fluxes is considered. The mathematical problem considered here can be used to model the drug transport in the cancer scenario when ultrasound is used as enhancer. In this case the two spatial domains model the healthy and cancer tissues. The drug transport through the healthy tissue is described by the Fickian diffusion equation, while the non-Fickian diffusion equation is used to model the drug transport in the cancer tissue. To break the physiological barriers increasing the drug transport, ultrasound has been proposed in different contexts. As ultrasound propagates through the target tissues as pressure waves, wave equations are used to describe the pressure wave intensity that induces an increase in the Brownian transport and in the convective transport. The stability of the partial differential system is studied and its behavior is numerically illustrated.