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.

中文翻译：

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基于PDE约束优化方法的HIFU组织消融优化控制方法

高强度聚焦超声（HIFU）是一种广泛使用的技术，能够为广泛的应用提供无创加热和消融。但是，主要挑战在于确定目标区域上的热量沉积的位置和数量。为了确保将热区域限制在肿瘤位置，应采用一种优化方法。序列二次编程和最速梯度法（采用封闭形式的解法）先前已用于解决此类问题。但是，这些方法复杂且计算效率低下。本文的目的是解决和控制具有偏微分方程（PDE）约束的反问题的解决方案。所以，该技术的一个显着挑战是结合离散化PDE的复杂性来处理大量优化变量。在我们的方法中，目标函数被公式化为实际热剂量与所需热剂量之间的平方差。在优化过程的每次迭代中，我们都需要开发和解决变异问题，伴随问题以及目标函数的梯度。基于伴随问题的解，推导并计算了梯度的解析公式。已经考虑了几个因素来证明所提出算法的鲁棒性和效率。