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### Open Access Journals

MBE

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.

MBE

In this work, a hyperchaotic system was used as a model for chronotherapy. We applied a periodic perturbation to a variable, varying the period and amplitude of forcing. The system, five-dimensional, has until three positive Lyapunov exponents. As a result, we get small periodical windows, but it was possible to get large areas of hyperchaos of two positive Lyapunov exponents from a chaotic behavior. In this chronotherapy model, chaos could be considered as a dynamical disease, and therapy goal must be to restore the hyperchaotic state.

MBE

In this work, the Räossler system is used as a model for chrono-
therapy. We applied a periodic perturbation to the y variable to take the
Rössler system from a chaotic behavior to a simple periodic one, varying the
period and amplitude of forcing. Two types of chaos were considered, spiral
and funnel chaos. As a result, the periodical windows reduced their areas as the
funnel chaos character increased in the system. Funnel chaos, in this chrono-
therapy model, could be considered as a later state of a dynamical disease,
more irregular and difficult to suppress.

MBE

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 signicantly smaller in the tumor tissues and both the mitosis rate and aggressiveness index decrease with Stage III patients’ age. These quantitative results correspond to observed behavior in clinical and genetics studies.
MBE

In this work, we propose a mesoscopic model for tumor growth to improve our understanding of the origin of the heterogeneity of tumor cells. In this sense, this stochastic formalism allows us to not only to reproduce but also explain the experimental results presented by Brú. A significant aspect found by the model is related to the predicted values for $\beta$ growth exponent, which capture a basic characteristic of the critical surface growth dynamics. According to the model, the value for growth exponent is between 0,25 and 0,5, which includes the value proposed by Kadar-Parisi-Zhang universality class (0,33) and the value proposed by Brú (0,375) related to the molecular beam epitaxy (MBE) universality class. This result suggests that the tumor dynamics are too complex to be associated to a particular universality class.

MBE

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. This fractal dimension
is distorted by the effects of external fluctuations. The model predicts a phenomenon
which indicates stochastic resonance when the multiplicative and the
additive noise are correlated.

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