Optimal chemotherapy in cancer treatment: state dependent Riccati equation control and extended Kalman filter

In this work, we design a nonlinear state feedback controller based on the State Dependent Riccati Equation (SDRE) technique to eliminate the tumor. One of the most interesting advantages of the SDRE is that it is possible to consider the specific conditions of patients by defining appropriate weights in the cost function and by limiting the administrated drug. Another advantage of this approach is that there are infinite ways to form the state dependent matrices. For each patient, a suitable drug regimen has been obtained using these advantages. A nonlinear model has been utilized to predict the growth of tumor. The model is a system of ODE with four state variables: normal cells, tumor cells, immune cells, and drug concentration. To use the SDRE controller, all state variables must be available for feedback. But for measuring the tumor size, the professional equipment is needed. So, it is impossible to measure the tumor size any time. We suppose that the number of normal cells could be measured in the presence of the Gaussian white noise. Therefore, we can design a state observer to estimate the immeasurable states from measurements. Extended Kalman Filter (EKF) can be used as a state observer for a nonlinear system, and in this work, we use EKF as a nonlinear state observer. Consequently, we can use the SDRE technique just by measuring the normal cell population. Numerical simulations are given to illustrate the design procedure and to show the flexibility of the method. Copyright © 2012 John Wiley & Sons, Ltd.