Drug delivery enhanced by ultrasound: Mathematical modeling and simulation

Ultrasound enhanced drug transport is a multiphysics problem involving acoustic waves propagation, bioheat transfer, and drug transport. In this paper, we study a model for this problem that consists of a wave-type equation for acoustic pressure, a diffusion-reaction equation for bioheat transfer, and a convection-diffusion-reaction equation for drug transport. We focus in particular on the numerical analysis of such a system. We propose and derive convergence estimates for a piecewise linear finite element method (FEM) with quadrature. We prove that the FEM is second order convergent for concentration in a discrete L2-norm. Since concentration depends on the gradient of acoustic pressure, this result shows that the FEM is superconvergent. Note that one expects the FEM to be convergent of order one in the L2-norm because the optimal convergence rate for the acoustic pressure is two in the H1-norm. We include numerical results backing the theoretical findings and an application of the model to a laboratory experiment of ultrasound enhanced transdermal drug delivery.