Bifurcation analyses and hormetic effects of a discrete-time tumor model

Administering combinatorial anti-cancer therapy with radiotherapy and immune checkpoint blockade has shown considerable synergistic effect and has been considered as a promising strategy. The key question is how the timing and intensity of radiotherapy and the secondary immune response affect the tumor therapy. To address this question, we propose and extend a discrete-time tumor growth model combining radiotherapy and inhibitor of the CTLA4 pathway to describe the development of tumor micro-environment. We initially study the existence and local stability of the fixed points, and discuss the flip and Neimark–Sacker bifurcations for some special cases, which reveals that the intensity of radiotherapy significantly affects the dynamics of tumor growth. For the generalized case, our main results indicate that if the timing of radiotherapy is within a certain interval, lower intensity of radiotherapy stimulates the growth of tumor mass, while higher intensity inhibits the growth, i.e., hormetic effects occur. Moreover, proper duration of the secondary immune response, induced by suitable immunotherapy regime, is beneficial to reduce hormetic effects, and consequently to maintain the homeostatic state of tumor mass and ultimately control the development of tumor. Thus, we conclude that based on the tumor dynamics and cumulative effects of the secondary immune response and its duration, the proper choice of timing and intensity of radiotherapy can not only reduce the complexity of the system, but also play an important role in optimizing tumor therapy.