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There are various approaches to control a mathematical dynamic of cancer, each of which is suitable for a special goal. Optimal control is considered as an applicable method to calculate the minimum necessary drug delivery in such systems.In this paper, a mathematical dynamic of cancer is proposed considering tumor cells, natural killer cells, CD8+T cells, circulating lymphocytes, IL-2 cytokine and Regulatory T cells as the system states, and chemotherapy, IL-2 and activated CD8+T cells injection rate as the control signals. After verifying the proposed mathematical model, the importance of the drug delivery timing and the effect of cancer cells initial condition are discussed. Afterwards, an optimal control is designed by defining a proper cost function with the goal of minimizing the number of tumor cells, and two immunotherapy drug amounts during treatment CONCLUSIONS: Results show that inappropriate injection of immunotherapy time schedule and the number of initial conditions of cancer cells might result in chemoimmunotherapy failure and auxiliary treatment must be prescribed to decrease tumor size before any treatment takes place. The obtained optimal control signals show that with lower amount of drug delivery and a suitable drug injection time schedule, tumor cells can be eliminated while a fixed immunotherapy time schedule protocol fails with larger amount of drug injection. This conclusion can be utilized with the aim of personalizing drug delivery and designing more accurate clinical trials based on the improved model simulations in order to save cost and time.

Different control strategies have been proposed for drug delivery in chemotherapy during recent years. These control algorithms are designed based on dynamic models of various orders. The order of the model depends on the number of effects considered in the model. In a recent model, the effect of obesity on the tumour progression and optimal control strategy in chemotherapy have been investigated in a fifth‐order state‐space model. However, the optimal controller is open loop and not robust to the common uncertainties of such biological system. Here, the sliding surface is obtained by the optimal trajectory and by considering uncertainties of some parameters, the robust‐sliding control law is formulated in a way to slid on the optimal surface. Then, a sliding mode controller is designed to determine the drug dose rate such that the system follows the optimal desired trajectory. The stability of the control system is proved and the simulation results indicate that three states track the trajectory and the remaining two states satisfy the constraints.

Numerous models have investigated cancer behavior by considering different factors in chemotherapy. The subject of a controller design approach for these models in order to find the best rate of drug injection during the course of treatment has recently attracted much attention. The rate of drug injections is very important in chemotherapy, as it not only causes cancer cells to die, but also kills healthy cells. On the other hand, by modeling tumor growth behavior in each patient, different parameters for the dynamics of the system should be considered. In this study, optimal control signals were obtained for the most recent model of chemotherapy, using the steepest descent method. The logic of the solution was biologically compared to the experimental results. Then an adaptive controller, considering this path as the desired optimal trajectory, directs the system towards it. The global stability of the closed-loop system is achieved by means of the Lyapunov stability theory and Barbalat lemma. It is worth noting that some of the system parameters are estimated using an online recursive estimation method. Simulation results indicate the performance and effectiveness of the designed controller. The estimation parameters are also verified using experimental data from available studies.