Continuous-Galerkin Summation-by-Parts Discretization of the Khokhlov-Zabolotskaya-Kuznetsov Equation with Application to High-Intensity-Focused Ultrasound

Abstract

Over the last two decades, High Intensity Focused Ultrasound (HIFU) has emerged as a promising non-invasive medical approach for locally and precisely ablating tissue, offering versatile applications in tumor treatment, drug delivery, and addressing brain disorders such as essential tremor. Its advantages include targeted energy delivery with no affect on skin integrity, low system maintenance costs, minimal impact on normal tissues, and swift recovery. Despite its’ merits, HIFU remains underutilized, primarily employed in specific breast cancer and prostate cancer treatments. To expand its range of applicability, a comprehensive understanding of the interaction between the ultrasound beam and local tissues at the focal point is essential. This thesis focuses on modeling critical nonlinear effects in the thermal modulation of local tissues by numerically solving the Khokhlov- Zabolotskaya-Kuznetsov (KZK) equation—which is an excellent model for the nonlinear acoustic field arising in HIFU. Constructing high-order stable discretizations of the KZK equations poses significant challenges due to the presence of polynomial nonlinear terms and a second derivative of an integral term within in this equation. Employing a continuous Galerkin approach, an operator is formulated to approximate the integral term, facilitating the construction of a modified second derivative operator. This establishes a clear correspondence between continuous and discrete stability proofs. Additionally, a skew-symmetric splitting technique is used to discretize the nonlinear advective term. The resulting semi-discrete scheme is proven to be stable. Numerical experiments using the method of manufactured solutions demonstrate the high-order accuracy and stability of the proposed numerical method. Finally, a HIFU verification test case demonstrates the applicability of the proposed scheme to investigate HIFU.