Cancer cell eradication in a 6D metastatic tumor model with time delay

In this paper we study ultimate dynamics of one time delay cancer metastatic 6D model with chemotherapy. This model describes interactions between normal and cancer cells under chemotherapy agents that occurs in two tumor sites, primary and secondary. We derive ultimate upper bounds for normal and cancer cells; ultimate upper/lower bounds for the chemical concentration in the both of tumor sites and describe the location of the attracting set. The study of equilibrium points is included. Global asymptotic tumor eradication conditions are obtained by a two-step process. Firstly, using the localization method of compact invariant sets (LMCIS) we find conditions of the tumor clearance at the primary site. Further, exploiting the cascade structure of the whole system and properties of asymptotically autonomous and competitive systems we derive tumor eradication conditions at the secondary site under the assumption that cancer cells at the primary site are asymptotically eradicated. Based on these results, cancer eradication conditions of the whole system are obtained. In particular, it is shown that the ω-limit set of each trajectory starting in the nonnegative orthant is one of equilibrium points located in the plane which is free from both primary and secondary cancer cells.