Simulation results on three-dimensional freezing phenomena around a cryoprobe are presented in the manuscript. Both fixed-grid explicit and implicit schemes based on the effective heat capacity method have been employed to solve the system of governing equations. The study revealed that the grid size of 1 mm and less is adequate to model the freezing process in the cryosurgical applications. Based on this study, an explicit solver was employed for the detailed solution of the governing equations. A stair-step approximation was used to model the cylinder-shaped cryoprobe geometry in three-dimensional space. Results of numerical simulations have been benchmarked with the exact solution, in vitro as well as in vivo experimental studies were reported in the literature. The results of the present study showed that the temperature-dependent thermophysical properties lead to better accuracy in temperature predictions inside the biological tissue. The effect of convective heat transfer to ambient was found negligible for deeper cryoprobe placements. The necrotic region engulfed by the lethal -50°C temperature (isotherm) is located approximately midway between the cryoprobe and the solid boundary of 0°C. The maximum tumor volume treated by any cryoprobe could be obtained a priori by performing the transient analysis as carried out in the present study.

中文翻译：

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三维冻结现象的数值研究及其在冷冻手术中的潜在应用

手稿中显示了围绕冷冻探头的三维冻结现象的仿真结果。基于有效热容量法的固定网格显式和隐式方案都已用于求解控制方程组。研究表明，栅格尺寸小于等于1 mm足以模拟冷冻手术应用中的冷冻过程。在此研究的基础上，采用显式求解器求解控制方程。使用阶梯近似法对三维空间中的圆柱状冷冻探头几何形状进行建模。数值模拟的结果已经用精确的解决方案作为基准，文献中还报道了体外和体内实验研究。本研究的结果表明，温度相关的热物理性质导致生物组织内部温度预测的准确性更高。对流热传递到环境的影响被发现对于更深的低温探头放置来说可以忽略不计。致命的-50°C温度（等温线）吞没的坏死区大约位于冷冻探头和0°C固体边界之间的中间位置。通过进行本研究中进行的瞬时分析，可以先验获得任何冷冻探针治疗的最大肿瘤体积。致命的-50°C温度（等温线）吞没的坏死区大约位于冷冻探头和0°C固体边界之间的中间位置。通过进行本研究中进行的瞬时分析，可以先验获得任何冷冻探针治疗的最大肿瘤体积。致命的-50°C温度（等温线）吞没的坏死区大约位于冷冻探头和0°C固体边界之间的中间位置。通过进行本研究中进行的瞬时分析，可以先验获得任何冷冻探针治疗的最大肿瘤体积。