Forecasting the growth of multicell tumour spheroids: implications for the dynamic growth of solid tumours

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Abstract
The growth dynamics of multicell tumour spheroids (MTS) were analysed by means of mathematical techniques derived from signal processing theory. Volume vs. time trajectories of individual spheroids were fitted with the Gompertz growth equation and the residuals (i.e. experimental volume determinations minus calculated values by fitting) were analysed by fast fourier transform and power spectrum. Residuals were not randomly distributed around calculated growth trajectories demonstrating that the Gompertz model partially approximates the growth kinetics of three‐dimensional tumour cell aggregates. Power spectra decreased with increasing frequency following a 1/fδ power‐law. Our findings suggest the existence of a source of ‘internal’ variability driving the time‐evolution of MTS growth. Based on these observations, a new stochastic Gompertzian‐like mathematical model was developed which allowed us to forecast the growth of MTS. In this model, white noise is additively superimposed to the trend described by the Gompertz growth equation and integrated to mimic the observed intrinsic variability of MTS growth. A correlation was found between the intensity of the added noise and the particular upper limit of volume size reached by each spheroid within two MTS populations obtained with two different cell lines. The dynamic forces generating the growth variability of three‐dimensional tumour cell aggregates also determine the fate of spheroid growth with a strong predictive significance. These findings suggest a new approach to measure tumour growth potential.