Comportamento caótico em sistemas dinâmicos e aplicação no estudo de tumores de câncer

Abstract

A incidência de casos de câncer é cada vez maior em todo mundo (ver INCA 22), motivo pelo qual torna-se importante o estudo de modelos matemáticos capazes de descrever o crescimento e desenvolvimento de tumores. Diante disso propomos o estudo, do ponto de vista da teoria qualitativa e bifurcações das equações diferenciais ordinárias, de dois modelos tridimensionais, que descrevem o crescimento de tumores de câncer. O primeiro deles trata de um caso heterogêneo, onde a população de células saudáveis compete pelos recursos disponíveis com duas populações de células tumorais: células tumorais sensíveis e células tumorais resistentes a algum tipo quimioterápico. Através do estudo da estabilidade...

The increasing incidence of cancer throughout the world becomes important the study of mathematical models able to describe the development and growth of tumors. Here, from the point of view of the qualitative theory and bifurcations of ordinary differential equations, we propose the study of two three-dimensional models describing the growth of cancer tumors. The first is a heterogeneous case, where the population of healthy cells competes for resources with two populations of tumor cells, tumor cells sensitive and resistant tumor cells, to some kind of chemotherapy. Based on the local stability study of the equilibrium points of system we show that healthy cells and resistant tumor cells may coexist, if the parameter f which represents the negative effects that tumor cells exert on the healthy cells is less than 10−5. From a biological point of view this result suggests that for values of f smaller than 10−5, the patient can effectively live with the disease until a new treatment is indicated. The second model describes the interaction among healthy cells (host), tumor cells and effector cells of the immune system. For this case we propose a bifurcation analysis varying two of the eight parameters involved. Through the linear analysis of local stability of equilibrium points, using the Routh- Hurwitz stability... (Complete abstract click electronic access below)