An insight into tumor dormancy equilibrium via the analysis of its domain of attraction

The trajectories of the dynamic system which regulates the competition between the populations of malignant cells and immune cells may tend to an asymptotically stable equilibrium in which the sizes of these populations do not vary, which is called tumor dormancy. Especially for lower steady-state sizes of the population of malignant cells, this equilibrium represents a desirable clinical condition since the tumor growth is blocked. In this context, it is of mandatory importance to analyze the robustness of this clinical favorable state of health in the face of perturbations. To this end, the paper presents an optimization technique to determine whether an assigned rectangular region, which surrounds an asymptotically stable equilibrium point of a quadratic systems, is included into the domain of attraction of the equilibrium itself. The biological relevance of the application of this technique to the analysis of tumor growth dynamics is shown on the basis of a recent quadratic model of the tumor–immune system competition dynamics. Indeed the application of the proposed methodology allows to ensure that a given safety region, determined on the basis of clinical considerations, belongs to the domain of attraction of the tumor blocked equilibrium; therefore for the set of perturbed initial conditions which belong to such region, the convergence to the healthy steady state is guaranteed. The proposed methodology can also provide an optimal strategy for cancer treatment.