An Adaptive Robust Control Strategy in a Cancer Tumor-Immune System Under Uncertainties

In this work, we propose an adaptive robust control for a second order nonlinear model of the interaction between cancer and immune cells of the body to control the growth of cancer and maintain the number of immune cells in an appropriate level. Up to now, most of the control approaches are based on minimizing the drug dosage based on an optimal control structure. However, in many cases, measuring the exact quantity of the model parameters is not possible. This is due to limitation in measuring devices, variational and undetermined characteristics of micro-environmental factors and the variable nature of parameters during the growth and treatment phases of cancer. It is of great importance to present a control strategy that can deal with these variables and unknown factors in a nonlinear model. Adaptive control is a suitable choice to achieve this goal. We assume linear uncertainties for the model parameters and employ a sliding term for updating the estimated parameters and the control signals. Moreover, due to difficulties in measuring the number of immune cells in biological experiments, an estimation technique is applied to infer this value. The convergence of the estimated number of immune cells to the actual value is shown. The stability and convergence of the number of cancer and immune cells to the specified target values are also proved using a time-varying Lyapunov function. Finally, we have shown the performance of the proposed control strategy in the context of various computational results.