Invasion Dynamics and Therapeutic Effects in a Tumor–Immune Interaction Model

Cancer remains one of the leading causes of morbidity and mortality worldwide, posing a significant global health challenge despite advances in therapeutic interventions. The process of tumor growth is a complex interplay among cancer cells, the immune system, and healthy tissues, and this interplay depends on the specific therapeutic approach. Mathematical modeling can be a valuable tool for dissecting these processes and identifying thresholds for different outcomes.

This work revisits and generalizes the widely used tumor-immune system model of de Pillis and Radunskaya by developing a set of nonlinear ordinary differential equations that incorporate the impact of therapy. Drug-cell interactions are modeled using Michaelis-Menten Kinetics, while immune suppression due to tumor invasion is represented by a Hill-type functional response. In this study, we establish fundamental dynamical properties of the system, including the positivity and boundedness of solutions, determine all equilibrium points, and characterize the local stability of the tumor-free state. In addition, we introduce a tumor invasion number that represents the threshold conditions governing successful tumor establishment.

In the context of cancer immuno-editing, the elimination, equilibrium, and escape stages are considered, and the parameter ranges governing transitions between these states via transcritical bifurcations are identified. Simulations of the system are also conducted to verify the findings, investigate the effects of various treatments on tumor-immunotherapy dynamics, and perform a sensitivity analysis to identify the key biological parameters that affect outcomes. The proposed framework enhances the theoretical understanding of tumor-immunotherapy interactions and may contribute to the development of more effective treatment strategies.