How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment?

Although recent advances in oncology emphasized the role of microenvironment in tumor growth, the role of delays for modeling tumor growth is still uncertain. In this paper, we considered a model, describing the interactions of tumor cells with their microenvironment made of immune cells and host cells, in which we inserted, as suggested by the clinicians, two time delays, one in the interactions between tumor cells and immune cells and, one in the action of immune cells on tumor cells. We showed analytically that the singular point associated with the co-existence of the three cell populations loses its stability via a Hopf bifurcation. We analytically calculated a range of the delays over which tumor cells are inhibited by immune cells and over which a period-1 limit cycle induced by this Hopf bifurcation is observed. By using a global modeling technique, we investigated how the dynamics observed with two delays can be reproduced by a similar model without delays. The effects of these two delays were thus interpreted in terms of interactions between the cell populations.