Equilibrium Dynamics and Threshold Conditions in Tumor–Immune Models with Chemotherapy

Abstract: This research presents a non-linear tumor-immune-chemotherapy model that distinguishes drug sensitive from drug resistant tumors with respect to chemotherapy dynamics and immune response. The model includes Gompertzian tumor growth, immune mediated killing, differential drug sensitivity, treatment induced drug resistance, immune suppression, and drug clearance. Biologically feasible equilibrium states were established by analytical investigation along with biological conditions of threshold for invasion by sensitive or resistant tumor cell types. Chemotherapy is demonstrated to have a double-edged sword effect in this model; increasing treatment intensity will be effective at suppressing sensitive tumor cells, however it also promotes the relative proportion of resistant tumor cells. Numerically simulated results demonstrate that local stability remains around positive equilibria and resistant tumor cells can remain present within the model despite an increase in chemotherapy. Further sensitivity analysis was conducted to identify factors that influence the models behavior including tumor growth rate, immune killing rates, chemotherapy efficacy, selective pressure on resistance development, degree of immune suppression and drug clearance. Ultimately the data supports that reduction in sensitive tumor population through chemotherapy does not eliminate all cancer (resistant cells), nor prevent recurrence due to continued presence of resistant cells.

Keywords: Tumor–immune dynamics, chemotherapy, drug resistance, mathematical modeling, equilibrium analysis, threshold conditions, Gompertz growth, resistance invasion