The Optimization of Transcostal Phased Array Refocusing Using the Semidefinite Relaxation Method

Tumors in organs partially obscured by the rib cage represent a challenge for high-intensity focused ultrasound (HIFU) therapy. The ribs distort the HIFU beams in a manner that reduces the focusing gain at the target, which could result in treatment-limiting collateral damage. In fact, skin burns are a common complication during the ablation of hepatic tumors. This problem can be addressed by employing optimal refocusing algorithms that are designed to achieve a specified focusing gain at the target while controlling the exposure to the ribs in the path of the HIFU beam. However, previously proposed optimal refocusing algorithms did not allow for the controlled transmission through the ribs. In this article, we introduce a new approach for refocusing that can more efficiently steer power toward the target while limiting the power deposition on the ribs. The approach utilizes the semidefinite relaxation (SDR) technique to approximate the original (nonconvex) optimization problem. An important advantage of the SDR-based method over previously proposed optimization methods is the control of the side lobes in the focal plane. The method also allows for specifying an acceptable level of exposure to the ribs. Simulation results using a 1-MHz spherical concave phased array focused on an inhomogeneous medium are presented to demonstrate the performance of the SDR refocusing approach. A finite-difference time-domain propagation model was used to model the propagation in the inhomogeneous tissues, including the ribs. Temperature simulations based on the inhomogeneous transient bioheat transfer equation (tBHTE) demonstrate the significance of the improvements in the focusing gain when using the limited power deposition (LPD) method. The results also demonstrate that the LPD method yields well-behaved array excitation vectors, realizable by currently existing drivers.