Shape Optimization And Asymptotic Behavior Of Solutions For High Intensity Focused Ultrasound

In recent years,with the improvement of the performance of piezoelectric materials,the de-velopment of clinical imaging and the rapid progress of electronic science,high-intensity focused ultrasound,as a non-invasive tissue ablation technology,has been widely used in clinical treatment in the medical field,which is one of the cutting-edge topics in the intersection of ultrasound technology and mathematical medicine.However,the existing technology of high-intensity focused ultrasound therapy is difficult to accurately locate the focal area,and the therapeutic effect can only be guar-anteed by increasing the time and ultrasonic intensity of excessive irradiation.This paper focuses on the shape optimization and asymptotic behavior of high-intensity focused ultrasound from the perspective of improving accuracy and controlling risk.Specifically,the main content of this thesis is as follows:In order to improve the accuracy,we modify the geometric shape of the acoustic lens through shape optimization,and consider the tracking-type cost function to match a desired pressure dis-tribution in the focal region.First,we obtain the corresponding adjoint equation through the state equation and the Lagrangian functional.Then we consider the well-posednesss of the state equation and the well-posedness of the adjoint equation.In the state equation and the adjoint equation has the certain space regularity,we calculate the shape derivative of the cost function through a variational approach.In the literature,the optimization problem subject to the constrain equation contains only-(?)operator.Due to the optimization studied in this paper,the constrain equation in the optimization problem contains not only_-(?)operator,but also(?)~2operator.The emergence of(?)~2operator brings great difficulties to our calculation,and the treatment of this operator is also one of the innovations of this paper.In order to control the risk,we study the asymptotic behavior of high intensity focus ultrasound.The key problem is to study the asymptotic behavior of the Moore-Gibson-Thompson(MGT)equa-tion,and we mainly consider the MGT nonlinear plate equation in the unbounded domain.For the asymptotic behavior of solutions for the MGT nonlinear plate equations in the unbounded domain,we first use the semigroup theory to prove the well-posedness of the MGT linear plate equation.And then we use the energy method in the Fourier space to get the decay result.In addition,we study the asymptotic expansion of the eigenvalues to analyze the optimality of the decay results.For MGT linear plate equation,the decay estimate of non-critical case is considered,while for MGT coupled with Gurtin-Pipkin thermal law,the decay estimates of critical case and non-critical case are con-sidered.We use the contraction mapping theorem in appropriately chosen spaces to prove a local existence result of the MGT nonlinear plate equation.Next,by constructing an appropriate energy norm and show that this norm remains uniformly bounded with respect to time,we prove a global existence result for small initial data.Finally,based on the decay estimates for the linear problem in the previous chapter,we prove the decay result for the nonlinear problem.In this paper,a more scientific and accurate mathematical model for shape optimization of high-intensity focused ultrasound is established by selecting appropriate constraint conditions and cost functional.The shape optimization of high-intensity focused ultrasound was studied systematically and the asymptotic behavior of the constraints was analyzed.This work provides a theoretical basis for more advanced high-intensity focused ultrasound therapy technology,which can further promote the wide application of high-intensity focused ultrasound in the medical field,and further promote the development of precision medicine.