H. S. Tharp

Pole placement by performance criterion modification

A procedure is presented that modifies the state weighting matrix Q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed-loop eigenvalues along the real axis of the optimal system in the linear-quadratic regulator problem. It is based on the results of the algebraic Riccati equation (ARE) which establish the invariance of certain eigenspaces of the associated Hamiltonian matrix with respect to certain perturbations of Q and the degree of relative stability. >

Reduced-order modeling for hyperthermia control

The feasibility of using reduced-order modeling techniques in the design of multiple-input, multiple-output (MIMO) hyperthermia temperature controllers is analyzed. State-space thermal models based on a finite-difference expansion of the bioheat transfer equation model of a scanned focused ultrasound system are created. These models are reduced using the balanced realization technique, and an order-reduction criterion is tabulated. Results show that a drastic reduction in model dimension can be achieved. The reduced-order model is then used to design a reduced-order optimal servomechanism controller for a two-scan input, two-thermocouple-output tissue model. A full-order optimal servomechanism controller is designed, and both controllers are applied to a variety of perturbed tissue thermal models to test the robust nature of the reduced-order controller. The comparison validates the use of open-loop balanced reduced-order models.<<ETX>>

Optimal power deposition with finite-sized, planar hyperthermia applicator arrays

Effective utilization of planar applicator arrays requires an understanding of the interrelationships between the lateral dimensions of the tumor and the applicators, the power field produced by the applicators, the amount of surface cooling, the tumor tissue blood perfusion, and the normal tissue blood perfusion. These interrelationships are investigated using three-dimensional power patterns and temperature fields produced by optimizing the power amplitudes of the individual applicators located within an array of small, but finite, planar applicators. Five major conclusions are obtained and discussed.<<ETX>>

Minimum Energy Guidance for Aerodynamically Controlled Missiles

This paper examines the problem of guiding a missile to a target using minimal energy. First, an optimal controller is developed for a missile that has arbitrary control over its acceleration vector. Next, an optimal controller is sought for a missile that has a directional control constraint, which is intended to model the actual constraint present in aerodynamically controlled missiles. The optimal guidance law is shown to have a very convenient form that is directly related to the optimal guidance law without the control constraint. Simulation results are conducted to completely characterize the new guidance law.

Analytical study of temperature oscillations in living tissues

A nonlinear, time-delay differential equation is postulated as a possible structure to describe the temperature dynamics at a point inside living tissue. The presented analysis explains how this differential equation can lead to four different temperature response types that have been previously observed in vivo.<<ETX>>

Concurrent hyperthermia estimation schemes based on extended Kalman filtering and reduced-order modelling

The success of treating cancerous tissue with heat depends on the temperature elevation, the amount of tissue elevated to that temperature, and the length of time that the tissue temperature is elevated. In clinical situations the temperature of most of the treated tissue volume is unknown, because only a small number of temperature sensors can be inserted into the tissue. A state space model based on a finite difference approximation of the bioheat transfer equation (BHTE) is developed for identification purposes. A full-order extended Kalman filter (EKF) is designed to estimate both the unknown blood perfusion parameters and the temperature at unmeasured locations. Two reduced-order estimators are designed as computationally less intensive alternatives to the full-order EKF. Simulation results show that the success of the estimation scheme depends strongly on the number and location of the temperature sensors. Superior results occur when a temperature sensor exists in each unknown blood perfusion zone, and the number of sensors is at least as large as the number of unknown perfusion zones. Unacceptable results occur when there are more unknown perfusion parameters than temperature sensors, or when the sensors are placed in locations that do not sample the unknown perfusion information.

Optimal pole-placement in discrete systems

A technique for increasing the damping in a closed-loop discrete-time system by modifying a nominal linear quadratic performance criterion is developed. The technique uses the multiple solutions of the discrete-time algebraic Riccati equation to modify the nominal criterion. These criterion modifications increase the damping by moving the nominal closed-loop eigenvalues to an exact location along the radial line segment connecting the nominal eigenvalues with the origin. >

Feasibility of using neural networks to estimate minimum tumour temperature and perfusion values

We examine the ability of neural networks to estimate the tissue perfusion values present and the minimum temperature in numerically calculated (Pennes, Bioheat Transfer Equation) steady-state hyperthermia temperature fields based on a limited number of measured temperatures within this field A hierarchical system of neural networks consisting of a first layer of pattern recognizing neural networks and a second layer of hypersurface reconstructing neural networks is shown to be capable of estimating these variables within a selected error tolerance. The results indicate that estimating the minimum tumour temperature directly with the system of neural networks may be more effective than using the indirect method of numerically recreating a temperature field with perfusion estimates and then obtaining the minimum tumour temperature from the estimated temperature field. Additional results indicate that if the locations of the measured temperatures within the temperature field are selected appropriately, the hierarchical system of neural networks can tolerate a moderate level of model mismatch. This model mismatch can come from errors in modelling the tumour boundaries, the sensor locations, or the magnitude of the power deposition. This paper is not intended to assess or demonstrate clinical applicability but to be a first step in investigating the feasibility of neural networks for parameter estimation related to hyperthermia studies.

Robust control of a nonlinear time-delay system

Experiments dealing with the heating of living tissue have resulted in four completely different temperature response characteristics as a result of the application of four different constant power levels. The four response characteristics can be classified as overdamped, critically damped, underdamped, and undamped oscillations. A particular nonlinear, time-delay dynamic equation has previously been shown to exhibit the same type of temperature response characteristics as those in the experiments. Herein, a control strategy is applied to the nonlinear, time-delay equation to inhibit the oscillatory behavior. A proportional plus integral (PI) controller with antiwindup, that is restricted to be nonnegative, is shown to be an effective controller in eliminating the oscillatory responses. This effectiveness is shown by analyzing and explaining the robustness properties of this controller as applied to this nonlinear, time-delay system.

Application of multiple modelling to hyperthermia estimation: Reducing the effects of model mismatch

Multiple model estimation is a viable technique for dealing with the spatial perfusion model mismatch associated with hyperthermia dosimetry. Using multiple models, spatial discrimination can be obtained without increasing the number of unknown perfusion zones. Two multiple model estimators based on the extended Kalman filter (EKF) are designed and compared with two EKFs based on single models having greater perfusion zone segmentation. Results given here indicate that multiple modelling is advantageous when the number of thermal sensors is insufficient for convergence of single model estimators having greater perfusion zone segmentation. In situations where sufficient measured outputs exist for greater unknown perfusion parameter estimation, the multiple model estimators and the single model estimators yield equivalent results.