Dynamics of Gompertzian tumour growth under environmental fluctuations

Abstract

We 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.