<title>Fluctuations and noise in cancer development</title>

Applied Mathematical Sciences, 2016

The steady state analysis for the effect of non-immunogenic tumor microenvironmental factors modeled by correlated additive and multiplicative noises with zero correlation time is investigated. The underlying transition probability for the stochastic model equation satisfies the Markovian Fokker-Planck equation, and the steady state distribution ρ st (x) for the tumor growth system is derived. Based on the numerical computation, we find that the effect of non-immunogenic microenvironmental factors within the tumor site with strength θ have a diffusive effect on the steady state distribution ρ st (x), and the tumor response to the surrounding non-immunogenic microenvironmental factors effects with strength D inhibits growth on both the steady state distribution ρ st (x) and the mean x st of the tumor population. The result also indicates that the stronger the correlation strength φ, the more the tumor responded to the surrounding non-immunogenic microenvironmental factors effect.

International Journal of Dynamics and Control

In this article, we study the noise-induced dynamics of the tumor-immune system. A stochastic system is considered by perturbing white noise in a deterministic system. Positivity and boundedness of the solution are established in the stochastic system. The threshold for stochastic weak persistence in mean is derived. We also obtain parametric conditions of stochastic permanence in mean. We justify the extinction of tumor burden with the increase of environmental noise strength. Finally, numerical simulations are performed and validate the analytical findings with proper biological implications.

The widely accepted model of tumor growth assumes tumors grow exponentially at a constant rate during early tumorigenesis when populations are small. The possibility that tumors might exhibit altered slower growth dynamics or even net cell death below a critical tumor size has yet to be fully explored. Deterministic growth models are capable of describing larger populations because population variation becomes small compared with the average, but when the population being modeled is small, the inherent stochasticity of the birth and death process produces significant variation. Recent advances in high throughput data collection allow for precise and sufficiently large data sets needed to capture this variation. Therefore, we present a stochastic modeling framework to describe and test the potential for altered growth dynamics at small tumor populations.

This Book is designed to provide the stochastic models for cancer cell growth as an alternative of non-parametric approach of assessing the cancer severity. Much emphasis of modeling is focused to formulation of mathematical models with probabilistic measures on the pathophysiology and genetic properties of cancer. The study is stressed more on stochastic modeling of cancer growth as modeling with deterministic environment is far free from the reality. Bivariate stochastic model for normal and mutant cell growth was developed under the assumption of the growth and loss processes of mutant and normal cells are Poisson. Approach of Differential difference equations is adopted to obtain the probability functions and several statistical measures. Another stochastic model for mutant and normal cell growth under chemotherapy is developed by taking similar assumptions. Two stage stochastic models for a cancer cell growth with the assumption of (i) every malignant tumor will have the premal...

arXiv: Cell Behavior, 2014

Stochastic resonance induced by external factor is considering to investigate the complex dynamics of tumor. The surrounding environment and the treatment effects on the tumor growth are considered as additive and multiplicative noises in growth model. The adaptability of tumor to treatment is presented by correlation of these two noises. The Fokker-Plank equation is deduced to study the probability distribution function and mean number of tumor cells in different conditions. The mean number of tumor cells can be controlled by the correlation and intensity of noises.

Randomness is an unavoidable feature of inner cellular environment and its effects propagate to higher levels of living matter organization such as cells, tissues, and organisms. Approaching those systems experimentally to understand their dynamics is a complex task because of the plethora of compounds interacting in a web that combines intra and inter level elements such that a coordinate behavior come up. Such a characteristic points to the necessity of establishing principles that help on the description, categorization, classification, and the prediction of the behavior of biological systems. The theoretical machinery already available, or the ones to be discovered motivated by biological problems, can play an important role on that quest. Here we exemplify the applicability of theoretical tools by discussing some biological problems that we have approached mathematically: fluctuations in gene expression and cell proliferation in the context of loss of contact inhibition. We dis...

PloS one, 2014

Tumors are often heterogeneous in which tumor cells of different phenotypes have distinct properties. For scientific and clinical interests, it is of fundamental importance to understand their properties and the dynamic variations among different phenotypes, specifically under radio-and/or chemo-therapy. Currently there are two controversial models describing tumor heterogeneity, the cancer stem cell (CSC) model and the stochastic model. To clarify the controversy, we measured probabilities of different division types and transitions of cells via in situ immunofluorescence. Based on the experiment data, we constructed a model that combines the CSC with the stochastic concepts, showing the existence of both distinctive CSC subpopulations and the stochastic transitions from NSCCs to CSCs. The results showed that the dynamic variations between CSCs and non-stem cancer cells (NSCCs) can be simulated with the model. Further studies also showed that the model can be used to describe the dynamics of the two subpopulations after radiation treatment. More importantly, analysis demonstrated that the experimental detectable equilibrium CSC proportion can be achieved only when the stochastic transitions from NSCCs to CSCs occur, indicating that tumor heterogeneity may exist in a model coordinating with both the CSC and the stochastic concepts. The mathematic model based on experimental parameters may contribute to a better understanding of the tumor heterogeneity, and provide references on the dynamics of CSC subpopulation during radiotherapy.

Journal of Theoretical Biology, 2006

Tumor metastasis accounts for the majority of deaths in cancer patients. The metastatic behavior of cancer cells is promoted by mutations in many genes, including activation of oncogenes such as RAS and MYC. Here, we develop a mathematical framework to analyse the dynamics of mutations enabling cells to metastasize. We consider situations in which one mutation is necessary to confer

International Journal of Mathematical Analysis, 2020

This article is distributed under the Creative Commons by-nc-nd Attribution License.

A stochastic model for tumor growth with spontaneous mutation and proliferation is developed by assuming that mutation and loss process of normal and mutant cells are poisson with different parameters. The growth rates of normal and mutant cells are not homogenous due recessive oncogenesis hypothesis. In order to analyse the tumor growth more close to reality a density dependent stochastic model with heterogeneous growth and loss rates of mutant cells and normal cells is developed. The growth parameters of model like the expected number of normal and mutant cells at time "t", the variability of normal and mutant cells and the covariance between normal and mutant cells are derived and analysed. The sensitivity of the parameters is analysed by computing the values of tumor characteristics for different parameters. It is observed that the expected tumor size is an increasing function of growth and mutant rates. The tumor size and its variability are influenced by the initial size of the tumor. The dependence between the normal and mutant cells is increasing as time 't' increases. This model is much useful in optimal drug administration for tumor control.