Estudio de la dinámica global para un modelo de Evasion-Inmune de un tumor cancerígeno
Study of the Global Dynamics for a Tumor Immune-Evasion System
Paul A. Valle1, Luis N. Coria2, and Konstantin E. Starkov1
1 Centro de Investigación y Desarrollo de Tecnología Digital (CITEDI-IPN), Instituto Politécnico Nacional, México. pvallet1200@alumno.ipn.mx, luis.coria@gmail.com
]]> 2 Instituto Tecnológico de Tijuana (ITT), México. kstarkov@ipn.mx
Article received on 10/12/2013.
Accepted on 06/10/2014.
Resumen
En este artículo se estudia la dinámica global del modelo de Evasión-Inmune presentado por Arciero, Jackson y Kirschner [1], el cual describe la interacción entre células efectoras, células cancerígenas y las citocinas IL - 2 y TGF - β en el sitio del tumor. El sistema modela distintos comportamientos, como lo son: puntos de equilibrio, órbitas periódicas y ciclos límite estables. Utilizando el método de Localización de Conjuntos Compactos Invariantes se logra definir un dominio en el espacio de estados donde se localizan todas las dinámicas que exhibe el modelo de Evasión-inmune. La localización de dicho dominio es importante debido a que proporciona información sobre la salud del individuo en corto y largo plazo. Los límites de tal dominio representan los valores mínimos y máximos de las variables de estado y se expresan mediante desigualdades algebraicas dadas por una combinación de los parámetros del sistema. Adicionalmente, mediante una función candidata de Lyapunov, se demuestra que la región de localización es un dominio positivamente invariante, lo que permite asegurar que dada cualquier condición inicial, las trayectorias del sistema no divergen. Finalmente, se presentan simulaciones numéricas y se realiza un análisis de las posibles implicaciones biológicas de los resultados obtenidos.
Palabras clave: Conjunto compacto invariante, dominio acotado positivamente invariante, Lyapunov, cáncer, sistema biológico.
Abstract
]]> In this paper we study the global dynamics for a Tumor Immune-Evasion model proposed by Arciero, Jackson and Kirschner [1], which describes the interaction between effector cells, cancer cells, and the cytokines IL -2 and TGF — β in the tumor site. This system describes different behaviors such as equilibrium points, periodic orbits, and stable limit cycles. By using the Localization of Compact Invariant Sets method, we define a domain where all the dynamics of the Immune-Evasion system are located. The localization of these sets is important because they provide information about the individual's health in the short and long term. The domain boundaries are expressed by inequalities depending on the system's parameters and represent the minimum and maximum values of the system variables. Furthermore, by taking a Lyapunov candidate function, we demonstrate that the localizing region is a positively invariant domain. This ensures that for any initial condition outside this domain, the trajectories of the system will not diverge. Finally, we present numerical simulations and realize an analysis of possible biological implications of our results.Keywords: Compact invariant set, bounded positively invariant domain, Lyapunov function, cancer, biological system.
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