A Gompertz curve or Gompertz function, named after Benjamin Gompertz

Benjamin Gompertz

Benjamin Gompertz was a British self educated mathematician and actuary, who became a Fellow of the Royal Society...

, is a sigmoid function

Sigmoid function

Many natural processes, including those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a detailed description is lacking, a sigmoid function is often used. A sigmoid curve is produced by a mathematical...

. It is a type of mathematical model

Mathematical model

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

 for a time series

Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

, where growth is slowest at the start and end of a time period. The right-hand or future value asymptote

Asymptote

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

 of the function is approached much more gradually by the curve than the left-hand or lower valued asymptote, in contrast to the logistic function

Logistic function

A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...

 in which both asymptotes are approached by the curve symmetrically.

Graphs of Gompertz curves, showing the effect of varying one of a,b,c while keeping the others constant

Formula

where

• a is the upper asymptote, since
• b, c are negative numbers
• b sets the x displacement
• c sets the growth rate (x scaling)
• e is Euler's Number
  E (mathematical constant)
  The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
  
   (e = 2.71828...)

Differentiation

The function curve can be derived from Gompertz's law

Gompertz-Makeham law of mortality

The Gompertz–Makeham lawstates that the death rate is the sum of an age-independent component and an age-dependent component , which increases exponentially with age...

, which states the rate of mortality (decay) falls exponentially with current size. Mathematically


where

• is the rate of growth.
• k is an arbitrary constant.

Example uses

Examples of uses for Gompertz curves include:

• Mobile phone
  Mobile phone
  A mobile phone is a device which can make and receive telephone calls over a radio link whilst moving around a wide geographic area. It does so by connecting to a cellular network provided by a mobile network operator...
  
   uptake, where costs were initially high (so uptake was slow), followed by a period of rapid growth, followed by a slowing of uptake as saturation was reached.
• Population in a confined space, as birth rates first increase and then slow as resource limits are reached.
• Modeling of growth of tumors

Growth of tumors and Gompertz curve

In the sixties A.K. Laird for the first time successfully used the Gompertz curve to fit data of growth of tumors. In fact, tumors are cellular populations growing in a confined space where the availability of nutrients is limited. Denoting the tumor size as X(t) it is useful to write the Gompertz Curve as follows:


where:

• X(0) is the tumor size at the starting observation time;
• K is the carrying capacity, i.e. the maximum size that can be reached with the available nutrients. In fact it is:

independently on X(0)>0. Note that, in absence of therapies etc.. usually it is X(0)K;

• α is a constant related to the proliferative ability of the cells.
• log refers to the natural log.

It is easy to verify that the dynamics of X(t) is governed by the Gompertz differential equation:


i.e. is of the form:


where F(X) is the instantaneous proliferation rate of the cellular population, whose decreasing nature is due to the competition for the nutrients due to the increase of the cellular population, similarly to the logistic growth rate. However, there is a fundamental difference: in the logistic case the proliferation rate for small cellular population is finite:


whereas in the Gompertz case the proliferation rate is unbounded:


As noticed by Steel and by Wheldon, the proliferation rate of the cellular population is ultimately bounded by the cell division time. Thus, this might be an evidence that the Gompertz equation is not good to model the growth of small tumors. Moreover, more recently it has been noticed that, including the interaction with immune system, Gompertz and other laws characterized by unbounded F(0) would preclude the possibility of immune surveillance.

Gompertz growth and logistic growth

The Gompertz differential equation


is the limiting case of the generalized logistic differential equation


(where is a positive real number) since
.

In addition, there is an inflection point

Inflection point

In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...

 in the graph of the generalized logistic function

Logistic function

A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...

 when


and one in the graph of the Gompertz function when
.

Gomp-ex law of growth

Based on the above considerations, Wheldon proposed a mathematical model of tumor growth, called the Gomp-Ex model, that slightly modifies the Gompertz law. In the Gomp-Ex model it is assumed that initially there is no competition for resources, so that the cellular population expands following the exponential law. However, there is a critical size threshold such that for the growth follows the Gompertz Law:


so that:


Here there are some numerical estimates for :

• for human tumors
• for murine (mouse) tumors

See also

• Logistic function
  Logistic function
  A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "S-shaped" curve of growth of some population P...
  
  
• Gompertz-Makeham law of mortality
  Gompertz-Makeham law of mortality
  The Gompertz–Makeham lawstates that the death rate is the sum of an age-independent component and an age-dependent component , which increases exponentially with age...
  
  
• Growth curve
  Growth curve
  A growth curve is an empirical model of the evolution of a quantity over time. Growth curves are widely used in biology for quantities such as population size, body height or biomass...
  
  
• Gompertz distribution

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