Fast finite element methods for focused ultrasound applications

Abstract

This thesis presents the development of a three-dimensional, time-domain, fast finite element solver for focused ultrasound (FUS) applications. FUS is a rapidly evolving medical technology used for tissue heating, ablation, and neuromodulation. In recent years, interest in FUS has grown significantly, especially for the treatment of cancers and brain diseases. Effective patient-specific treatment planning can be supported by accurate simulations. However, this poses a challenge from a computational standpoint since the ratio of the domain size to the wavelength is typically large, therefore computationally demanding. This is made even more challenging for nonlinear models such as the Westervelt model, commonly used in high-intensity focused ultrasound (HIFU) treatments, as higher harmonics need to be resolved. To address this challenge, a three-dimensional, time-domain acoustic solver for FUS applications is developed. The solver is based on the high-order finite element method and the explicit Runge-Kutta method. It is well known that low-order finite element methods suffer from pollution errors, especially for high-frequency problems. However, it is demonstrated that with high-order finite elements, the pollution error is significantly reduced. Moreover, the availability of modern open-source finite element software, such as FEniCSx, eases the implementation of high-order finite element methods. A mass-lumped finite element scheme is implemented, which, combined with the explicit Runge-Kutta method avoids the need to solve a linear system of equations. To further accelerate the solver, a fast finite element integration algorithm is employed. In particular, an algorithm with the lowest known computational cost complexity is implemented, namely the sum-factorisation algorithm. Furthermore, the algorithm is carefully implemented to ensure optimal performance. The solver is implemented for both CPU and GPU architectures. On both architectures, it is shown that the finite element integration implementations achieve a good fraction of peak hardware performance. Additionally, it is shown that the solver exhibits excellent parallel scalability. Contrary to common perceptions, it is demonstrated that a high-order finite element method is well-suited for realistic acoustic simulation for focused ultrasound applications. The novel contributions in this thesis include the development of a high-order finite element method with a fast finite element integration algorithm. The solver is validated against other acoustic wave solvers for several benchmark problems. Overall, it is shown that the developed solver is fast, accurate, scalable, and is suitable for large-scale linear and nonlinear time-domain acoustic simulations.