On high intensity ultrasound focusing by optimal acoustic lens design

Shape optimization of an acoustic lens is considered. The problem comes from the usage of ultrasound in medical treatment of kidney stones. Since it can be assumed that no topology changes take place and only small geometry modifications have to be taken into account, we can work with gradient type optimization methods on a parameterization of the lens shape. For this purpose, the lens is parameterized by cubic splines and as a first step, the linear wave equation with jumping coefficients is used to represent wave propagation through the entire domain consisting of the fluid region (i.e., the human tissue) and the lens region. I.e. our model consists of the equation u_tt – c²Δu = 0, where the speed of sound c is piecewise constant taking different values inside and outside of the lens. The cost functional is of tracking type to achieve an acoustic pressure distribution that exhibits a point focus at the desired location and has low amplitudes elsewhere. A sensitivity analysis is performed whose results are used in the numerical computations based on gradient methods using adjoint techniques for efficient gradient evaluation. Numerical results are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)