【正文】 contributions DH carried out puter simulations. JG participated in design of the study. All authors read and approved the final manuscript. Acknowledgements This work was supported by NIH Grant HL56413. References 1. Bosch FX, Ribes J, Borras J: Epidemiology of primary liver cancer. Semin Liver Dis 1999, 19:27185. PubMed Abstract 2. Curley SA, Cusack JC Jr, Tanabe KK, Stoelzing O, Ellis LM: Advances in the treatment of liver tumors. Curr Probl Surg 2022, 39:449571. PubMed Abstract | Publisher Full Text 3. Haemmerich D, Staelin ST, Tungjitkusolmun S, Lee FT Jr, Mahvi DM, Webster JG:Hepatic bipolar radiofrequency ablation between separated multiprong electrodes. IEEE Trans Biomed Eng 2022, 48:114552. PubMed Abstract | Publisher Full Text 4. Tungjitkusolmun S, Woo EJ, Cao H, Tsai JZ, Vorperian VR, Webster JG: Finite element analyses of uniform current density electrodes for radiofrequency cardiac ablation. IEEE Trans Biomed Eng 2022, 47:3240. PubMed Abstract | Publisher Full Text 5. Panescu D, Whayne JG, Fleischman SD, Mirotznik MS, Swanson DK, Webster JG:Threedimensional finite element analysis of current density and temperature distributions during radiofrequency ablation. IEEE Trans Biomed Eng 1995, 42:879890. PubMed Abstract | Publisher Full Text 25 6. Tungjitkusolmun S, Staelin ST, Haemmerich D, Tsai JZ, Cao H, Webster JG, Lee FT, Mahvi DM, Vorperian VR: Threedimensional finiteelement analyses for radiofrequency hepatic tumor ablation. IEEE Trans Biomed Eng 2022, 49:39. PubMed Abstract | Publisher Full Text 7. Jain MK, Wolf PD: Temperaturecontrolled and constantpower radiofrequency ablation: what affects lesion growth? IEEE Trans Biomed Eng 1999, 46:14051412. PubMed Abstract | Publisher Full Text 8. Chang IA, Nguyen UD: Thermal modeling of lesion growth with radiofrequency ablation devices. Biomed Eng Online 2022, 3:27. PubMed Abstract | BioMed Central Full Text |PubMed Central Full Text 9. Chang I: Finite element analysis of hepatic radiofrequency ablation probes using temperaturedependent electrical conductivity. Biomed Eng Online 2022, 2:12. PubMed Abstract | BioMed Central Full Text |PubMed Central Full Text 10. Liu Z, Lobo SM, Humphries S, Horkan C, Solazzo SA, HinesPeralta AU, Lenkinski RE, Goldberg SN: Radiofrequency tumor ablation: insight into improved efficacy using puter modeling. AJR Am J Roentgenol 2022, 184:134752. PubMed Abstract | Publisher Full Text 11. Gopalakrishnan J: A mathematical model for irrigated epicardial radiofrequency ablation. Ann Biomed Eng 2022, 30:88493. PubMed Abstract | Publisher Full Text 12. Graham SJ, Chen L, Leitch M, Peters RD, Bronskill MJ, Foster FS, Henkelman RM, Plewes DB: Quantifying tissue damage due to focused ultrasound heating observed by MRI. Magn Reson Med 1999, 41:3218. PubMed Abstract | Publisher Full Text 26 13. Chato JC: Heat transfer to blood vessels. J Biomech Eng 1980, 102:1108. PubMed Abstract 14. Pennes HH: Analysis of tissue and arterial blood tempera。t have exact data of the tip temperature over time. We empirically chose parameters for the PI controller to minimize overshoot and obtain similar temporal behavior as in the invivo experiments. 19 The parameters we used for the PI controller were Kp = , Ki = . Fig. 4 shows the results of the closedloop control system simulation. The target tip temperature of 100176。 the C++ program then calls ABAQUS and waits until the next step is pleted. We implemented a PI controller in our control algorithm. Closed loop system simulation Since the FEM model takes several hours to plete, using the FEM model to optimize control parameters was not feasible. Therefore we used a closed loop simulation to adjust parameters of the controller, and then used these parameters in the FEM model. Before implementing the controller, we analyzed the dynamic system (. the FEM model) to be controlled. This system consisted of the ablation electrode, tissue, and dispersive electrode. The input variable of the system was the voltage applied to the electrode. The output variable was the temperature measured at the tip of the electrode. Initially, we determined the step response by applying a constant voltage of 25 V for 180 s. We then approximated the transfer function of this system by a timediscrete transfer function of the following form: We used the control system simulation software ANA (Freeware, Dept. of Control Engineering, Tech. Univ. Vienna/Austria) to analyze the control system. This software allowed us to approximate the system from its step response by a recursive least square algorithm and gave the parameters a0, a1, a2, b1 and b2 of (2). Fig. 2 shows the step responses of the original dynamic system (. the FEM model) and of the approximation 17 according to (2). The parameters used for the approximation in (2) were: a0 = , a1 = , a2 = , b1 = , b2 = . The sampling time of the approximated system was 10 s, since that was also the sampling time used later in the digital PI controller. This ensured a more accurate simulation model of the closedloop control system. Step response of the original dynamic system (FEM model) and of the approximation. Once we had identified an approximation of the dynamic system (FEM model), we designed a feedback control system. There are different ways of controlling the tip temperature such as PID control, adaptive control, neural work prediction control and fuzzy logic control. We chose the relatively simple PI controller for our control system. shows the plete closedloop control system. Ts is the desired set tip temperature and Tt is the current tip temperature. The input of the PI controller e = Ts Tt. The output of the PI controller u (corresponds to the applied voltage) is fed into the dynamic system. 18 Closed loo