Applications of neural networks to partial differential equations.

Efforts to improve hyperthermia treatments of cancer have motivated this research. Three fundamental goals that have have been defined are the identification of tissue parameter values, the prediction of tissue temperature behavior, and the control of tissue temperature behavior during hyperthermia treatments. This dissertation consists of three independent studies plying neural networks to hyperthermia. These studies examine neural network based systems and how these systems apply to the identification, prediction, and control of tissue temperature behavior during a hyperthermia treatment. The first study examines the ability of neural networks to estimate the tissue perfusion values and the minimum temperature associated with numerically calculated steady state hyperthermia temperature fields. This study utilizes a limited number of measured temperatures within this field. We show that 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 capable of estimating these variables within a selected error tolerance. Additional results indicate that if the locations of the measured temperatures within the temperature field are selected effectively, the hierarchical system of neural networks can tolerate a moderate level of model mismatch. The second study examines the feasibility of using a system of neural networks to estimate the Laplacian values at the sensor locations in a tissue sample by using the measured temperature data. By combining these neural network Laplacian estimates with the measured data, numerical values for three different components (conduction, advection, and external power) can be obtained at the sensor locations. These thermal terms can then be used in a model of the tissue to predict future temperatures. Using only measured data collected early in the treatment, we show that recursive application of this estimation process can provide accurate predictions of the temperature behavior at the sensor locations of a tissue sample for the duration of a treatment. This system was also found to be robust with respect to the addition of white noise to both the sensor measurements and the amount of power delivered to the sensor locations. The final study explores the ability of a neural network based predictive system to formulate a predictive control strategy. We show that with certain restrictions on the power deposition patterns, desired temperature trajectories within the tissue model can be achieved with a neural network based predictive control system