Merola2008 - An insight into tumor dormancy equilibrium via the analysis of its domain of attraction

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.

基于吸引域（domain of attraction）分析的肿瘤休眠平衡研究
A. Merola, C. Cosentino*, F. Amato
意大利卡坦扎罗马格纳·格雷西亚大学（Università degli Studi Magna Græcia di Catanzaro）计算机与生物医学工程学院，‘萨尔瓦托雷·韦努塔’校区，88100 卡坦扎罗，意大利

摘要
调控恶性细胞与免疫细胞群体竞争的动力学系统，其轨迹可收敛至渐近稳定平衡点（asymptotically stable equilibrium），此时两类群体的规模保持恒定，该状态即被称为肿瘤休眠（tumor dormancy）。尤其当恶性细胞群体的稳态规模较低时，该平衡点代表一种理想的临床状态，因为此时肿瘤生长被阻断。在此背景下，分析该临床有益健康状态在扰动下的鲁棒性（robustness），具有至关重要的意义。

为此，本文提出一种优化技术，用于判断指定的矩形区域（该区域围绕着二次系统（quadratic systems）的渐近稳定平衡点）是否属于该平衡点的吸引域（domain of attraction）。本文基于近期提出的肿瘤-免疫系统竞争动力学二次模型（quadratic model），验证了该技术应用于肿瘤生长动力学分析的生物学意义。事实上，通过应用本文提出的方法，可确保基于临床考量确定的指定安全区域，属于肿瘤阻滞平衡点的吸引域；因此，对于属于该区域的受扰初始条件集合，其收敛至健康稳态的过程可得到保证。本文提出的方法还可为癌症治疗提供最优策略。