BioModels

An insight into tumor dormancy equilibrium via the analysis of its domain of
attraction
A. Merola, C. Cosentino *, F. Amato
School of Computer and Biomedical Engineering, Universita` degli Studi Magna Græcia di Catanzaro, Campus ‘‘Salvatore Venuta’’, 88100 Catanzaro, Italy

A B S T R A C T
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 quadraticmodel of the tumor–immune system
competition dynamics. Indeed the application of the proposedmethodology 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.