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Discontinuous Galerkin approximation of a nonlinear multiphysics problem arising in ultrasound-enhanced drug delivery

Femke de Wit, Vanja Nikolić

TL;DR

This work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug through the Westervelt wave equation coupled to a convection-diffusion equation modeling the drug concentration.

Abstract

Motivated by simulations of ultrasound-enhanced drug delivery, this work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug. The system under study consists of the Westervelt wave equation, accounting for the nonlinear propagation of ultrasound, coupled to a convection-diffusion equation modeling the drug concentration. In particular, drug delivery is affected by ultrasound through a pressure-dependent diffusion coefficient. The Westervelt equation is supplemented by linear absorbing boundary conditions as a means of reducing spurious reflections off the boundaries of computational domains. For spatial discretization of this multiphysics system, we employ a discontinuous Galerkin approach on simplicial meshes. Under suitable assumptions on the exact pressure and the mesh size, we first establish well-posedness, non-degeneracy, and optimal convergence rates in the energy norm for the semi-discrete pressure subproblem. The smallness of the semi-discrete pressure is then used to establish the well-posedness and convergence of the wave--convection-diffusion system under suitable regularity of the exact concentration. Finally, theoretical findings are illustrated through numerical experiments.

Discontinuous Galerkin approximation of a nonlinear multiphysics problem arising in ultrasound-enhanced drug delivery

TL;DR

This work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug through the Westervelt wave equation coupled to a convection-diffusion equation modeling the drug concentration.

Abstract

Motivated by simulations of ultrasound-enhanced drug delivery, this work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug. The system under study consists of the Westervelt wave equation, accounting for the nonlinear propagation of ultrasound, coupled to a convection-diffusion equation modeling the drug concentration. In particular, drug delivery is affected by ultrasound through a pressure-dependent diffusion coefficient. The Westervelt equation is supplemented by linear absorbing boundary conditions as a means of reducing spurious reflections off the boundaries of computational domains. For spatial discretization of this multiphysics system, we employ a discontinuous Galerkin approach on simplicial meshes. Under suitable assumptions on the exact pressure and the mesh size, we first establish well-posedness, non-degeneracy, and optimal convergence rates in the energy norm for the semi-discrete pressure subproblem. The smallness of the semi-discrete pressure is then used to establish the well-posedness and convergence of the wave--convection-diffusion system under suitable regularity of the exact concentration. Finally, theoretical findings are illustrated through numerical experiments.
Paper Structure (17 sections, 16 theorems, 131 equations, 5 figures)

This paper contains 17 sections, 16 theorems, 131 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical model: A nonlinear wave--convection-diffusion system
  3. Discontinuous Galerkin semi-discretization of the system
  4. Auxiliary results
  5. Inverse and trace inequalities
  6. Discrete embeddings
  7. Properties of the involved bilinear functionals
  8. Properties of the interpolant
  9. Discontinuous Galerkin analysis of the Westervelt equation with absorbing boundary conditions
  10. Existence of the semi-discrete acoustic solution on x
  11. Extending the existence and error bounds to x
  12. Empirical order of convergence for the acoustic pressure
  13. Discontinuous Galerkin analysis of the wave--convection-diffusion system
  14. Bounds on the approximate concentration
  15. Numerical results for the pressure-dependent concentration
  16. ...and 2 more sections

Key Result

lemma 2.1

Let the assumptions made on $\mathcal{T}_{h}$ in this section hold. The following inverse estimate holds for $1 \leq \ell,\ell' \leq \infty$ and $\phi^{h} \in V^q_{h}$, $K \in \mathcal{T}_{h}$: where the constant $C_{\textup{inv}}$ depends on the mesh regularity parameter, $d$, $q$, $\ell$, and $\ell'$. Further, the following discrete trace inequality holds for $\phi^{h} \in V_h^q$ and face $F \i

Figures (5)

  • Figure 1: Discrete errors for the pressure
  • Figure 2: Discrete errors for the concentration
  • Figure 3: Pressure waves affecting drug spread ($[[per-mode=symbol]{\pascal}]$).
  • Figure 4: Ultrasound-enhanced drug concentration at different times ($[[per-mode=symbol]{\kilo \gram \per \meter \cubed}]$).
  • Figure 5: Relative change \ref{['rel change']} between ultrasound-enhanced concentration and the concentration computed with a constant diffusion coefficient at the top boundary $y=0.01$.

Theorems & Definitions (29)

  • lemma 2.1: Inverse inequalities, see dipietro
  • lemma 2.2: Continuous trace inequality, see dipietro
  • lemma 2.3: see dipietro
  • lemma 2.4
  • proof
  • lemma 2.5
  • proof
  • lemma 2.6
  • proof
  • remark 1
  • ...and 19 more