Gompertz curve: one construct, many systems

Gompertz's law of mortality holds that, across much of adult life, the force of mortality, the instantaneous risk of death, rises roughly exponentially with age, doubling over a roughly fixed interval. Gompertz's 1825 On the nature of the function expressive of the law of human mortality fitted this exponential-of-an-exponential form to life-table data and made it the backbone of actuarial mortality modelling. The referent is the age pattern of death rates in a population, and the same mathematical function reappears, unexpectedly, in tumour growth and in technological forecasting.[1]

Unit

The force of mortality as a function of age (an exponentially rising hazard)

Recovery means

Not applicable

Measured by

Fitted to age-specific death rates and summarised by the Gompertz slope (rate of ageing) and baseline level

In oncology the same curve, read as a growth model, describes how a tumour enlarges: fast when small, then decelerating as it grows, following the sigmoid Gompertz function toward a plateau. Laird's 1964 Dynamics of tumor growth showed that a wide range of experimental tumours fit the Gompertz equation, a result that still informs models of tumour kinetics and chemotherapy scheduling. It is mathematically the identical function Gompertz wrote for human mortality, here governing cell number over time rather than death rate over age.[2]

Unit

Tumour size (cell number or volume) as a function of time, following a Gompertz curve

Recovery means

Not applicable

Measured by

Estimated by fitting the Gompertz growth equation to serial tumour-size measurements (growth rate and plateau)

More generally the Gompertz function serves as a general-purpose sigmoid growth curve, used to model bounded growth of organisms and populations and the diffusion of technologies and products toward a saturation level. Winsor's 1932 The Gompertz curve as a growth curve set out its properties as a growth model and contrasted it with the logistic. The referent is bounded, asymmetric growth over time, the identical mathematical form as the mortality law and the tumour curve, applied to yet another system.[3]

Unit

A growing quantity approaching an upper asymptote along a Gompertz sigmoid over time

Recovery means

Not applicable

Measured by

Fitted by estimating the asymptote, displacement, and growth-rate parameters of the Gompertz equation from a time series