Budman H, Dayan J, Shitzer A
Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa.
IEEE Trans Biomed Eng. 1991 Nov;38(11):1141-53. doi: 10.1109/10.99078.
A study of a controlled cryosurgical process is presented. This study is based on the energy equations describing the probe response and the phase change occurring in the medium. First-order nonlinear differential equations (state equations) are obtained by applying the integral-solution method. In order to obtain maximal cell destruction, it is desired to control a specific cooling rate at the solid-liquid interface. This cooling rate defines the desired trajectories of the state variables through the state equations. In order to satisfy the cooling rate condition on the freezing front, a closed-loop is designed to control the probe temperature program. A simple analysis of the system stability employed linearization at several points along the desired trajectories. Ranges of stability were obtained for a system containing a proportional-integral controller. It was demonstrated that these stability ranges depend mainly on the selected sampling time of the discrete control loop and that the phase-change process does not significantly affect the stability results. A complete study of the nonlinear equations was performed by a computer simulation program which enables the selection of the final values of the controller parameters, in order to minimize the error and to ensure stability. In addition, the simulation program gives information about the effects of the A/D and D/A converters accuracy on the performance of the control loop. An A/D converter accuracy of 12 bits was found necessary in order to reduce the oscillations in probe temperature to acceptable values. The simulation also yields a complete calculated temperature field in the tissue during the controlled process. From these simulated results it can be seen that oscillations of +/- 0.5 degrees C in the desired probe temperature do not significantly affect the desired cooling rate at the freezing front. An initial overshoot of 1.5 degrees C in the desired probe temperature was obtained both experimentally and theoretically from the simulation. When this initial overshoot occurs at the beginning of the freezing process, it causes an error in freezing front velocity and consequently in ice-front position. From the numerical simulation, it can be deduced that the cooling rate obtained at the front deviates from the desired value by approximately 1%. The probe-temperature error increases at two instants: a) during the super-cooling effect and the subsequent sudden crystallization, and b) when the probe temperature is below -80 degrees C and unstable boiling of the cooling medium causes oscillations.
本文介绍了一项关于可控冷冻手术过程的研究。该研究基于描述探针响应以及介质中发生的相变的能量方程。通过应用积分求解法获得了一阶非线性微分方程（状态方程）。为了实现最大程度的细胞破坏，需要控制固液界面处的特定冷却速率。该冷却速率通过状态方程定义了状态变量的期望轨迹。为了满足冻结前沿的冷却速率条件，设计了一个闭环来控制探针温度程序。通过在期望轨迹上的几个点进行线性化，对系统稳定性进行了简单分析。对于包含比例积分控制器的系统，获得了稳定范围。结果表明，这些稳定范围主要取决于离散控制回路的选定采样时间，并且相变过程对稳定性结果没有显著影响。通过计算机模拟程序对非线性方程进行了全面研究，该程序能够选择控制器参数的最终值，以最小化误差并确保稳定性。此外，模拟程序还给出了有关A/D和D/A转换器精度对控制回路性能影响的信息。发现需要12位的A/D转换器精度，以便将探针温度的振荡降低到可接受的值。模拟还得出了受控过程中组织内完整的计算温度场。从这些模拟结果可以看出，期望探针温度中±0.5摄氏度的振荡对冻结前沿的期望冷却速率没有显著影响。从模拟中通过实验和理论都获得了期望探针温度1.5摄氏度的初始超调。当这种初始超调出现在冻结过程开始时，会导致冻结前沿速度出现误差，进而导致冰前沿位置出现误差。从数值模拟可以推断，前沿处获得的冷却速率与期望值偏差约1%。探针温度误差在两个时刻会增加：a）在过冷效应及随后的突然结晶期间，以及b）当探针温度低于-80摄氏度且冷却介质的不稳定沸腾导致振荡时。
