Control of the cryosurgical process in nonideal materials

Abstract

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.