Abstract
Cancer development is a dynamic and continuously evolving process, with the emergence of drug-resistant cancer cells being one of the primary reasons for the failure of traditional treatments. Adaptive therapy, as an emerging cancer treatment strategy, is increasingly being applied in oncology. In this study, we incorporate pharmacokinetics into a cancer evolutionary game theory model and propose an optimal control problem constrained by maximum drug concentration and maximum tumor burden. Firstly, we demonstrate the existence of an optimal control for this problem. Secondly, using Pontryagin's minimum principle, we formulated the structure of the optimal control to design an optimal adaptive therapy strategy. Finally, through numerical simulations, we compare the optimal adaptive therapy strategy with other adaptive therapies and traditional treatments, and further develop personalized treatment plans for different patient groups. The results demonstrate that the optimized adaptive treatment strategy effectively preserves a high survival rate of healthy cells during treatment. By maintaining drug-sensitive and drug-resistant cell populations in a state of low-level competition, this approach prevents the proliferation of drug-resistant cells, reduces the tumor burden on patients, and extends overall survival.