The effect of correlated noise in a Gompertz tumor growth model

Abstract

We study the effect of noise in an avascular tumor growth model. The growth mechanism we consider is the Gompertz model. The steady state probability distributions and average population of tumor cells are analyzed within the Fokker-Planck formalism to investigate the importance of additive and multiplicative noise. We consider the effect of correlation on tumor growth for both the case of nonzero and zero correlation time. It is observed that the Gompertz model, driven by correlated noise exhibits a stochastic resonance and phase transition. This behaviour is attributed to multiplicative noise. In the case of nonzero correlation time, it is found that the correlation strength and correlation time have opposite effects on the steady state probability distribution. The Gompertz model simulations are also shown to be in qualitative agreement with another similiar non-bistable system, the logistic model.

Stochastic Resonance in a Bacterium Growth System with Time Delay and Colored Noise

“…[26][27][28][29][30][31]. Among them, the logistic growth model is usually used in many cases to describe the cell growth and, more particularly, tumor cell growth [32].…”

Dynamics of Gompertzian tumour growth under environmental fluctuations

“…In other words, the fluctuations of these external factors can influence the growth parameter a generating a multiplicative noise and at the same time can restrain the cell growth giving rise to an additive noise. Here we have implicitly assumed both the multiplicative and additive noise are correlated since they have a common origin [27,28]. Thus the coupling parameter can be interpreted as the ability of a tumour to compensate the external interference due to treatment effects via internal reactions.…”

“…To date many studies have been carried out on the analysis of the steady state of the tumour growth [25][26][27][28] under the influence of environmental fluctuations. Yet there are many biological systems in which the interesting dynamics take place before the system has had time to reach the steady state.…”

Electrotherapy, a recent mode for anticancer treatment

“…One way that gives us in a several minutes or hours the appropriate electrode configuration with the respective applied LIDC can be through mathematical modelling. In oncology, mathematical modelling are used in simulation: in the kinetics of tumour growth [55][56][57][58], in spatial and temporal growth of tumour avascular phase [59,60], in interaction and competition between tumour and immune system [60,61] among others. Richards, Weibull, Bertalanffy and Gompertz logistic models are applied to describe the kinetics of tumour growth, where the latter is the most employed [55,56].…”

“…In oncology, mathematical modelling are used in simulation: in the kinetics of tumour growth [55][56][57][58], in spatial and temporal growth of tumour avascular phase [59,60], in interaction and competition between tumour and immune system [60,61] among others. Richards, Weibull, Bertalanffy and Gompertz logistic models are applied to describe the kinetics of tumour growth, where the latter is the most employed [55,56]. In the field of electrotherapy, mathematical modelling are focussed in the explanation of different processes that are induced in the tumour after the application of LIDC, such as: pH modifications [21]; physicochemical reactions around electrodes [62]; potential dissipation in the tumour after applying LIDC [63].…”