Parameter Estimation in a Gompertzian Stochastic Model for Tumor Growth

Abstract

The problem of estimating parameters in the drift coefficient when a diffusion process is observed continuously requires some specific assumptions. In this paper, we consider a stochastic version of the Gompertzian model that describes in vivo tumor growth and its sensitivity to treatment with antiangiogenic drugs. An explicit likelihood function is obtained, and we discuss some properties of the maximum likelihood estimator for the intrinsic growth rate of the stochastic Gompertzian model. Furthermore, we show some simulation results on the behavior of the corresponding discrete estimator. Finally, an application is given to illustrate the estimate of the model parameters using real data.

Trend analysis and computational statistical estimation in a stochastic Rayleigh model: Simulation and application

“…[2,11,9]). For this purpose, a statistical-fit methodology was applied, but the fits achieved were unsatisfactory.…”

“…Stochastic diffusions have been widely used in diverse fields, such as stochastic financial analysis, animal or cell growth in a random environment, marketing, and the natural environment. In particular, Gompertz and lognormal stochastic diffusion processes have been studied with respect to specific theoretical aspects, and they have been successfully applied to real cases in Gutiérrez et al [12,10] and Ferrante et al [2]. In order to apply these diffusion processes to the modelling and prediction of real phenomena, it is necessary to develop results of statistical inference, firstly on the estimation of their parameters (general results on this question can be consulted in Prakasa Rao [16]).…”

Modelling the Cervical Cancer Growth Process by Stochastic Delay Differential Equations

“…It reflects the actual behavior of cancerous growth and implies a cancer clone may regress prior reaching a detectable size. Moreover, noise is not completely understood in deterministic model, hence not feasible to model deterministically (Ferrante et al 2000). To be realistic, mathematical models of a biological system should include noise or stochasticity.…”

“…Most studies used a stochastic version that based on Gompertz law to account for random fluctuations of the model parameters. They assumed that the growth deceleration factors do not change and the variability of environmental conditions only induces fluctuations in the intrinsic growth rate (Ferrante et al 2000;Lo 2007;Mazma Syahidatul Ayuni & Norhayati 2014).…”

“…Next, the Gompertzian stochastic model has been constructed and contributed in its own way to a better understanding in vivo tumor growth (Ferrante et al 2000) and cervical cancer growth (Mazma Syahidatul Ayuni & Norhayati 2014). In a real situation, the dynamical behavior of cancer cell growth rate depends on its structure at the present time and it also takes into account the structure at some previous time (Byrne 1997).…”