Morphogenesis and Complexity of the Tumor Patterns

Abstract

A mechanism to describe the apoptosis process at mesoscopic level through p53 is proposed in this paper. A deterministic model given by three differential equations is deduced from the mesoscopic approach, which exhibits sustained oscillations caused by a supercritical Andronov-Hopf bifurcation. Taking as hypothesis that the p53 sustained oscillation is the fundamental mechanism for apoptosis regulation; the model predicts that it is necessary a strict control of p53 to stimulated it, which is an important consideration to established new therapy strategy to fight cancer. The mathematical modeling of tumor growth allows us to describe the most important regularities of these systems. A stochastic model, based on the most important processes that take place at the level of individual cells, is proposed to predict the dynamical behavior of the expected radius of the tumor and its fractal dimension. It was found that the tumor has a characteristic fractal dimension, which contains the necessary information to predict the tumor growth until it reaches a stationary state. The mathematical modeling of tumor growth is an approach to explain the complex nature of these systems. A model that describes tumor growth was obtained by using a mesoscopic formalism and fractal dimension. This model theoretically predicts the relation between the morphology of the cell pattern and the mitosis/apoptosis quotient that helps to predict tumor growth from tumoral cells fractal dimension. The relation between the tumor macroscopic morphology and the cell pattern morphology is also determined. This could explain why the interface fractal dimension decreases with the increase of the cell pattern fractal dimension and consequently with the increase of the mitosis/apoptosis relation. Indexes to characterize tumoral cell proliferation and invasion capacities are proposed and used to predict the growth of different types of tumors. These indexes also show that the proliferation capacity is directly proportional to the invasion capacity. The proposed model assumes: i) only interface cells proliferate and invade the host, and ii) the fractal dimension of tumoral cell patterns, can reproduce the Gompertzian growth law. A mathematical model was obtained to describe the relation between the tissue morphology of cervix carcinoma and both dynamic processes of mitosis and apoptosis, and an expression to quantify the tumor aggressiveness, which in this context is associated with the tumor growth rate. The proposed model was applied to Stage III cervix carcinoma in vivo studies. In this study we found that the apoptosis rate was significantly smaller in the tumor tissues and both the mitosis rate and aggressiveness index decrease with Stage III patient’s age. These quantitative results correspond to observed behavior in clinical and genetics studies. Finally, the entropy production rate was determined for avascular tumor growth. The proposed formula relates the fractal dimension of the tumor contour with the quotient between mitosis and apoptosis rate, which can be used to characterize the degree of proliferation of tumor cells. The entropy production rate was determined for fourteen tumor cell lines as a physical function of cancer robustness. The entropy production rate is a hallmark that allows us the possibility of prognosis of tumor proliferation and invasion capacities, key factors to improve cancer therapy.