Numerical Simulation of the Evolution of Focusing Shock Wave in Extracorporeal Shock Wave Lithotripsy by Using Space-Time Conservation Element and Solution Element Scheme

Extracorporeal shock wave lithotripsy (ESWL) is the most common treatment of kidney stone disease. By firing shock waves at the stone, it can be broken down into small fragments. Although the treatment is non-invasive, both short- and long-term side effects occur. Lithotripsy has been the subject of ongoing research. Various solutions designed to maximize stone comminution and minimize tissue damage have been proposed over the years. However, the particulars of the comminution mechanism are still undetermined. Different types of lithotripters have been approved for clinical use and are classified by the type of shock wave source they utilize, such as, electro-hydraulic lithotripter, piezoelectric lithotripter, electromagnetic lithotripter, etc. .... In this paper, we will focus on piezoelectric lithotripter. A new numerical model is used to simulate the evolution of underwater shock wave in piezoelectric ESWL. In this model, space-time conservation element and solution element (CE/SE) method is applied to solve the conservation law form of the axisymmetric Euler equations by using the unstructured grids. The CE/SE method which is originally proposed by Chang in 1995 is different in both concept and methodology from well-established traditional numerical methods (such as finite difference method, finite element method, finite volume method etc.). The CE/SE method has many nontraditional features. Firstly, it is conceptually simple and robust, neither Riemann solver nor technique based on characteristics are involved. Secondly, space and time are unified and treated on the same footing, and by the introduction of conservation element and solution element, both local and global flux conservations in space and time in stead of in space only are enforced. Thirdly, all flow variables and their spatial derivatives are considered as individual unknowns to be solved for simultaneously at each grid point, its accuracy is higher than well-established traditional numerical methods in the same grids. The computational results are showed that the dynamic focus in piezoelectric ESWL is different from geometric focus in general and the cavitations are inevitable for underwater focusing of shock wave. Both the correct location of the factual focus and the cavitation play very important roles in clinical ESWL.