Gompertzian growth curves in parathyroid tumours: further evidence for the set‐point hypothesis

Abstract

Clinical and cell kinetic data in parathyroid tumours show that their rate of growth slows down progressively and that tumour size approaches an asymptotic value. The Gompertz equation has been widely used in oncology to model growth retardation in malignant tumours; we describe its first application to a benign tumour. In 41 patients with radiation associated hyperparathyroidism, individual solutions were derived for the Gompertz equation: Nt = Exp[A/ a(1 - Exp - at)], where A is the rate constant (years-1) for initial exponential growth and a is the rate constant (years-1) for exponential decline in A. Input data comprised three estimates of tumour age at surgery, 100%, 75% and 50% of the time since irradiation, cell number estimated from tumour weight, and current tumour growth rate, representing the difference between current cell birth rate, estimated from the prevalence of mitotic figures, and an assumed mean rate of cell loss of 5%. With 100% tumour age, geometric mean values were 2.76 for A, 0.134 for a, and 0.87 g for the growth asymptote. As assumed tumour age decreased, the rate constants increased and the growth asymptotes declined from 22% to 9% greater than the geometric mean tumour weight. Depending on assumed tumour age, the rate constants were about 15-45 times smaller than in myeloma and in testicular tumours, and the growth asymptotes about 2500 and about 60 times smaller, respectively. A and a were highly correlated (r2 = 0.993), with a slope of 20.9 and no significant intercept. Depending on assumed tumour age, the geometric mean time from the initial mutation to the first cell division ranged from 39 to 92 days, much longer than in malignant tumours. (1) The Gompertz modelling demonstrates that both the nonprogressive clinical course and the slow growth of parathyroid tumours can be accounted for by a single mutation. (2) The extremely low values for A and a, and consequent very long delay before the first cell division, support the notion that the initial mutation does not affect a growth regulatory gene, but increases growth indirectly via an increase in secretory set-point, the clone of mutant cells behaving as if they were in a hypocalcaemic environment until the plasma calcium rises to the new set-point. (3) The clinical characteristics of radiation-induced parathyroid tumours are modelled more closely if there is a substantial delay between time of irradiation and onset of tumour growth. (4) The rate constants A and a are highly correlated because the variability of tumour weight on a logarithmic scale is much lower than the variability of the rate constants.