Mathematical modeling and dynamical analysis of anti-tumor drug dose-response

Cancer is a serious threat to human health and life. Using anti-tumor drugs is one of the important ways for treating cancer. A large number of experiments have shown that the hormesis appeared in the dose-response relationship of various anti-tumor drugs. Modeling this phenomenon will contribute to finding the appropriate dose. However, few studies have used dynamical models to quantitatively explore the hormesis phenomenon in anti-tumor drug dose-response. In this study, we present a mathematical model and dynamical analysis to quantify hormesis of anti-tumor drugs and reveal the critical threshold of antibody dose. Firstly, a dynamical model is established to describe the interactions among tumor cells, natural killer cells and M2-polarized macrophages. Model parameters are fitted through the published experimental data. Secondly, the positivity of solution and bounded invariant set are given. The stability of equilibrium points is proved. Thirdly, through bifurcation analysis and numerical simulations, the hormesis phenomenon of low dose antibody promoting tumor growth and high dose antibody inhibiting tumor growth is revealed. Furthermore, we fit out the quantitative relationship of the dose-response of antibodies. Finally, the critical threshold point of antibody dose changing from promoting tumor growth to inhibiting tumor growth is obtained. These results can provide suggestions for the selection of appropriate drug dosage in the clinical treatment of cancer.