The dynamics of tumor growth and cells pattern morphology
Elena Izquierdo-Kulich Margarita Amigó de Quesada Carlos Manuel Pérez-Amor Magda Lopes Texeira José Manuel Nieto-Villar
Mathematical Biosciences & Engineering 2009, 6(3): 547-559 doi: 10.3934/mbe.2009.6.547
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.
keywords: pattern cell morphology. mesoscopic model fractal stochastic model tumor growth
Morphogenesis and aggressiveness of cervix carcinoma
Elena Izquierdo-Kulich Margarita Amigó de Quesada Carlos Manuel Pérez-Amor José Manuel Nieto-Villar
Mathematical Biosciences & Engineering 2011, 8(4): 987-997 doi: 10.3934/mbe.2011.8.987
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.
keywords: pattern cell morphology. Tumor growth fractal stochastic model mesoscopic model
Mesoscopic model for tumor growth
Elena Izquierdo-Kulich José Manuel Nieto-Villar
Mathematical Biosciences & Engineering 2007, 4(4): 687-698 doi: 10.3934/mbe.2007.4.687
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.
keywords: fractal stochastic model mesoscopic model tumor growth mathematical modeling.
Morphogenesis of the tumor patterns
Elena Izquierdo-Kulich José Manuel Nieto-Villar
Mathematical Biosciences & Engineering 2008, 5(2): 299-313 doi: 10.3934/mbe.2008.5.299
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.
keywords: mesoscopic model mathematical modeling. tumor growth fractal stochastic model

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