Analysis and optimal control of a proliferating-quiescent Glioma model with combined therapy

Low-grade gliomas (LGGs) are slow-growing but infiltrative brain tumors characterized by long-term persistence and frequent progression. In this work, we develop and analyze a compartmental mathematical model describing LGG dynamics under combination therapy involving cytotoxic chemotherapy and targeted molecular treatment. The tumor population is structured into proliferating, quiescent, and damaged compartments, and therapeutic interventions are incorporated through bilinear control terms accounting for chemo-therapies. We first establish the well-posedness of the model by proving existence, uniqueness, positivity, and boundedness of solutions. The dynamics of untreated glioma growth are then analyzed, revealing that the tumor-free equilibrium is always unstable in the absence of therapy, leading to exponential tumor expansion. Under constant treatment regimens, we derive an explicit controlled reproduction number that governs tumor persistence or eradication. Using linearization and Lyapunov–LaSalle stability techniques, we establish necessary and sufficient conditions for both local and global asymptotic stability of the tumor-free state. Finally, we formulate an optimal control problem to identify treatment strategies that minimize tumor burden while limiting therapeutic toxicity. The existence of optimal controls is rigorously proven using classical results from optimal control theory. Our results provide analytical insight into the stabilizing role of combination therapy and offer a quantitative guidance for designing optimized treatment protocols for low-grade glioma.

کلیدواژه ها:

Bilinear models ، Reproduction number ، Glioma dynamics ، Optimality conditions ، Numerical Simulations