Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy

TL;DR

This paper develops a shape sensitivity analysis method for optimizing the design of a focusing acoustic lens in lithotripsy, aiming to improve ultrasound targeting of kidney stones.

Contribution

It introduces a variational approach to compute the shape derivative without needing the shape derivative of the state, under certain regularity assumptions.

Findings

Derived the shape derivative using a variational approach.

Applied the method to optimize acoustic lens shape in lithotripsy.

Provided conditions for the regularity needed for the derivative.

Abstract

We are interested in shape sensitivity analysis for an optimization problem arising in medical applications of high intensity focused ultrasound. The goal is to find the optimal shape of a focusing acoustic lens so that the desired acoustic pressure at a kidney stone is achieved. Coupling of the silicone acoustic lens and nonlinearly acoustic fluid region is modeled by the Westervelt equation with nonlinear strong damping and piecewise constant coefficients. We follow the variational approach to calculating the shape derivative of the cost functional which does not require computing the shape derivative of the state variable; however assumptions of certain spatial regularity of the primal and the adjoint state are needed to obtain the derivative, in particular for its strong form according to the Delfour-Hadamard-Zol\' esio Structure Theorem.