Analysis of growth of multicellular tumour spheroids by mathematical models.

We wished to determine the applicability of previously proposed deterministic mathematical models to description of growth of multicellular tumour spheroids. The models were placed into three general classes: empirical, functional and structural. From these classes, 17 models were applied systematically to growth curves of multicellular tumour spheroids used as paradigms of prevascular and microregional tumour growth. The spheroid growth curves were determined with uniquely high density of measurements and high precision. The theoretical growth curves obtained from the models were fitted by the weighted least-squares method to the 15 measured growth curves, each corresponding to a different cell line. The classical growth models such as von Bertalanffy, logistic and Gompertz were considered as nested within more general models. Our results demonstrate that most models fitted the data fairly well and that criteria other than statistical had to be used for final selection. The Gompertz, the autostimulation and the simple spheroid models were the most appropriate for spheroid growth in the empirical, functional and structural classes of models, respectively. We also showed that some models (e.g. logistic, von Bertalanffy) were clearly inadequate. Thus, contrary to the widely held belief, the sigmoid character of a three or more parameter growth function is not sufficient for adequate fits.