Dynamics of Gompertzian tumour growth under environmental fluctuations

Abstract

a b s t r a c tWe investigate the effect of correlated additive and multiplicative Gaussian white noise on the Gompertzian growth of tumours. Our results are obtained by solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study. We perform simulations to analyze various aspects, of the probability distribution, of the tumour cell populations in the transient-and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (τ ) to the steady-state distribution as a function of (i) of the correlation strength (λ) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (α). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.

Stochastic modelling of avascular tumour growth and therapy

“…These effects are evident in sensory systems at the tissue and subcellular levels [21], the pattern formation in cancer growth [22] and the effect of cell-mediated immune response to cancer [23]. Investigations concentrating on the study of the effect of correlated Gaussian noises on tumour growth and therapy have stimulated growing interest among researchers [24][25][26][27][28][29][30].…”

Dynamic evolution of PSD modelled using an Ornstein-Uhlenbeck process approach

“…where D is the noise intensity. As it regards the deterministic term h(L;θ), here we assume a Gompertz model (GM) for the crystal growth (Sahoo et al, 2010):…”

“…In this paper we propose a new formulation for the deterministic model that allows obtaining a close form of the resulting FPE. In particular, in the new formulation, the deterministic component of the model, describing the time evolution of the mean size of crystals, is obtained from a Gompertz equation (Sahoo et al, 2010). For the stochastic component of the model a geometric Brownian motion for the noise intensity of the system is assumed.…”