A mathematical study of anti-VEGF and cytotoxic therapies of cancer with optimal control

This work studies two cancer treatments, anti-angiogenic therapy and chemotherapy, with a novel mathematical model and the associated optimal control problem. The model includes tumor cells, endothelial cells, immune cells, and Vascular Endothelial Growth Factor (VEGF), where the anti-angiogenic therapy only targets VEGF and the chemotherapy kills all cells indiscriminately. The optimal control problem minimizes the tumor burden and drug toxicity over a set of time-variant drug doses. The mathematical analysis shows the existence of the positive invariant set of the model over all the therapeutic strategies, the stability of multiple steady state solutions, as well as the existence and uniqueness of the optimal control solutions. The analysis and simulations lead to several significant findings. First, all the steady states with the vanished tumor are unstable under the anti-VEGF therapy, which confirms its limited efficacy as observed in clinics. Second, the Hopf bifurcation appears in each treatment approach with a common feature: the system exhibits periodic oscillations at low drug doses and transitions to a stable coexistence state at higher drug doses. Third, the optimal treatment strategy involves a delicate combination of both treatment types. This strategy is particularly effective when the anti-VEGF drug has a high binding affinity to VEGF molecules, and the chemotherapy drug has a small killing rate of immune cells and large killing rates of endothelial cells and tumor cells.