Rheology of the cytoskeleton as a fractal network

We model the cytoskeleton as a fractal network by identifying each segment with a simple Kelvin-Voigt element with a well defined equilibrium length. The final structure retains the elastic characteristics of a solid or a gel, which may support stress, without relaxing. By considering a very simple regular self-similar structure of segments in series and in parallel, in one, two, or three dimensions, we are able to express the viscoelasticity of the network as an effective generalized Kelvin-Voigt model with a power law spectrum of retardation times L similar to tau(alpha). We relate the parameter alpha with the fractal dimension of the gel. In some regimes ( 0 < alpha < 1), we recover the weak power law behaviors of the elastic and viscous moduli with the angular frequencies G' similar to G" similar to w(alpha) that occur in a variety of soft materials, including living cells. In other regimes, we find different power laws for G' and G".

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincareacute return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing delta(1). However, for higher values of delta(1) the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter zeta) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

Avaliação da caracterização de lesões em mamografia com recurso a sistemas CAD (Diagnóstico Assistido por Computador)

Mestrado em Radiações Aplicadas às Tecnologias da Saúde. Área de especialização: Imagem Digital com Radiação X.

Topological Complexity and Predictability in the Dynamics of a Tumor Growth Model with Shilnikov's Chaos

Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

Avaliação da caracterização de lesões em mamografia com recurso a sistemas CAD (Computer-Aided Diagnosis)

Os sistemas Computer-Aided Diagnosis (CAD) auxiliam a deteção e diferenciação de lesões benignas e malignas, aumentando a performance no diagnóstico do cancro da mama. As lesões da mama estão fortemente correlacionadas com a forma do contorno: lesões benignas apresentam contornos regulares, enquanto as lesões malignas tendem a apresentar contornos irregulares. Desta forma, a utilização de medidas quantitativas, como a dimensão fractal (DF), pode ajudar na caracterização dos contornos regulares ou irregulares de uma lesão. O principal objetivo deste estudo é verificar se a utilização concomitante de 2 (ou mais) medidas de DF – uma tradicionalmente utilizada, a qual foi designada por “DF de contorno”; outra proposta por nós, designada por “DF de área” – e ainda 3 medidas obtidas a partir destas, por operações de dilatação/erosão e por normalização de uma das medidas anteriores, melhoram a capacidade de caracterização de acordo com a escala BIRADS (Breast Imaging Reporting and Data System) e o tipo de lesão. As medidas de DF (DF contorno e DF área) foram calculadas através da aplicação do método box-counting, diretamente em imagens de lesões segmentadas e após a aplicação de um algoritmo de dilatação/erosão. A última medida baseia-se na diferença normalizada entre as duas medidas DF de área antes e após a aplicação do algoritmo de dilatação/erosão. Os resultados demonstram que a medida DF de contorno é uma ferramenta útil na diferenciação de lesões, de acordo com a escala BIRADS e o tipo de lesão; no entanto, em algumas situações, ocorrem alguns erros. O uso combinado desta medida com as quatro medidas propostas pode melhorar a classificação das lesões.

Symbolic dynamics and big bang bifurcation in Weibull-Gompertz-Fréchet's growth models

In this paper, motivated by the interest and relevance of the study of tumor growth models, a central point of our investigation is the study of the chaotic dynamics and the bifurcation structure of Weibull-Gompertz-Fréchet's functions: a class of continuousdefined one-dimensional maps. Using symbolic dynamics techniques and iteration theory, we established that depending on the properties of this class of functions in a neighborhood of a bifurcation point PBB, in a two-dimensional parameter space, there exists an order regarding how the infinite number of periodic orbits are born: the Sharkovsky ordering. Consequently, the corresponding symbolic sequences follow the usual unimodal kneading sequences in the topological ordered tree. We verified that under some sufficient conditions, Weibull-Gompertz-Fréchet's functions have a particular bifurcation structure: a big bang bifurcation point PBB. This fractal bifurcations structure is of the so-called "box-within-a-box" type, associated to a boxe ω1, where an infinite number of bifurcation curves issues from. This analysis is done making use of fold and flip bifurcation curves and symbolic dynamics techniques. The present paper is an original contribution in the framework of the big bang bifurcation analysis for continuous maps.