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

Wave Equation Modeling via Physics-Informed Neural Networks: Models of Soft and Hard Constraints for Initial and Boundary Conditions

Therapeutic ultrasound waves are the main instruments used in many noninvasive clinical procedures. They are continuously transforming medical treatments through mechanical and thermal effects. To allow for effective and safe delivery of ultrasound waves, numerical modeling methods such as the Finite Difference Method (FDM) and the Finite Element Method (FEM) are used. However, modeling the acoustic wave equation can result in several computational complications. In this work, we study the accuracy of using Physics-Informed Neural Networks (PINNs) to solve the wave equation when applying different combinations of initial and boundary conditions (ICs and BCs) constraints. By exploiting the mesh-free nature of PINNs and their prediction speed, we specifically model the wave equation with a continuous time-dependent point source function. Four main models are designed and studied to monitor the effects of soft or hard constraints on the prediction accuracy and performance. The predicted solutions in all the models were compared to an FDM solution for prediction error estimation. The trials of this work reveal that the wave equation modeled by a PINN with soft IC and BC (soft-soft) constraints reflects the lowest prediction error among the four combinations of constraints.

An Optimized Control Approach for HIFU Tissue Ablation Using PDE Constrained Optimization Method

High-intensity focused ultrasound (HIFU) is a widely used technique capable of providing noninvasive heating and ablation for a wide range of applications. However, the major challenges lie in the determination of the position and the amount of heat deposition over a target area. In order to assure that the thermal area is confined to tumor locations, an optimization method should be employed. Sequential quadratic programming and steepest gradient method with closed-form solution have been previously used to solve this kind of problem. However, these methods are complex and computationally inefficient. The goal of this article is to solve and control the solution of inverse problems with partial differential equation (PDE) constraints. Therefore, a distinguishing challenge of this technique is the handling of large numbers of optimization variables in combination with the complexities of discretized PDEs. In our method, the objective function is formulated as the square difference between the actual thermal dose and the desired one. At each iteration of the optimization procedure, we need to develop and solve the variation problem, the adjoint problem, and the gradient of the objective function. The analytical formula for the gradient is derived and calculated based on the solution of the adjoint problem. Several factors have been taken into consideration to demonstrate the robustness and efficiency of the proposed algorithm. The simulation results for all cases indicate the robustness and the computational efficiency of our proposed method compared to the steepest gradient descent method with the closed-form solution.

Calculating the Effect of Ribs on the Focus Quality of a Therapeutic Spherical Random Phased Array

The overlaying rib cage is a major hindrance in treating liver tumors with high intensity focused ultrasound (HIFU). The problems caused are overheating of the ribs due to its high ultrasonic absorption capability and degradation of the ultrasound intensity distribution in the target plane. In this work, a correction method based on binarized apodization and geometric ray tracing approach was employed to avoid heating the ribs. A detailed calculation of the intensity distribution in the focus plane was undertaken to quantify and avoid the effect on HIFU beam generated by a 1-MHz 256-element random phased array after the ultrasonic beam passes through the rib cage. Focusing through the ribs was simulated for 18 different idealized ribs-array configurations and 10 anatomically correct ribs-array configurations, to show the effect of width of the ribs, intercostal spacing and the relative position of ribs and array on the quality of focus, and to identify the positions that are more effective for HIFU applications in the presence of ribs. Acoustic simulations showed that for a single focus without beam steering and for the same total acoustic power, the peak intensity at the target varies from a minimum of 211 W/cm² to a maximum of 293 W/cm² for a nominal acoustic input power of 15 W, whereas the side lobe level varies from 0.07 I_peak to 0.28 I_peak and the separation between the main lobe and side lobes varies from 2.5 mm to 6.3 mm, depending on the relative positioning of the array and ribs and the beam alignment. An increase in the side lobe level was observed by increasing the distance between the array and the ribs. The parameters of focus splitting and the deterioration of focus quality caused by the ultrasonic propagation through the ribs were quantified in various possible different clinical scenarios. In addition to idealized rib topology, anatomical realistic ribs were used to determine the focus quality of the HIFU beam when the beam is steered both in axial and transverse directions and when the transducer is positioned at different depths from the rib cage.