Gompertz Growth in Tumor-Immune Competition: Bifurcations, Multistability, and Chemotherapeutic Implications

This study investigates the nonlinear dynamics that emerge from the interactions between cancer cells and immune cells within a predator–prey model, wherein cancer cell growth obeys the Gompertz law. A bifurcation analysis allows for the identification of states of dormancy, uncontrolled growth, and multistability with and without chemotherapy. In the absence of chemotherapy, a Gompertz model predicts bistability with low (dormant) and high (active) tumor levels. However, unlike models based on the logistic equation, it shows that a stable tumor-free solution does not exist, consistent with the medical understanding about the risk of remaining disease even with successful treatment. Under chemotherapy, the model demonstrates highly complex dynamics with up to four coexisting stable steady states, stable tumor levels, and chemotherapy-induced oscillations. Parameter continuation studies show that the potency of immune recruitment, rate of cell inactivation, and drug saturation are essential in characterizing transitions among these dynamical regions. The analysis indicates that the choice of growth rate plays a significant role in determining the physiological implications for therapy, suggesting a cure for a model with a logistic growth rate, but merely tumor control for a Gompertzian scenario. Moreover, these results provide insights into optimal chemotherapy dosage to prevent problems associated with bistability and to capitalize on stable dormancy.