Gompertz function

A Gompertz curve or Gompertz function, named after Benjamin Gompertz, is a sigmoid function. It is a type of mathematical model for a time series, where growth is slowest at the start and end of a time period.

:$y(t)=ae^\{be^\{ct$

where
* "a" is the upper asymptote
* "c" is the growth rate
* "b, c" are negative numbers
* "e" is Euler's Number ("e" = 2.71828...)

Examples

Uses

Examples of uses for Gompertz curves include:
* Mobile phone 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. Lairdcite journal | author= Laird A. K. | title= Dynamics of tumor growth | journal= Br J of Cancer | volume= 18 |year=1964 | pages=490-502] 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:

:$X(t)\; =\; K\; expleft(\; logleft(\; frac\{X(0)\}\{K\}\; ight)\; expleft(-alpha\; t\; ight)\; ight)$

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::$lim\_\{t\; ightarrow\; +infty\}X(t)=K$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:

:$X^\{prime\}(t)\; =\; alpha\; logleft(\; frac\{K\}\{X(t)\}\; ight)\; X(t)$

i.e. is of the form:

:$X^\{prime\}(t)\; =\; Fleft(\; X(t)\; ight)\; X(t),\; F^\{prime\}(X)\; le\; 0$

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:

:$lim\_\{X\; ightarrow\; 0^\{+\}\; \}\; F(X)\; =\; lim\_\{X\; ightarrow\; 0^\{+\}\; \}\; alpha\; logleft(\; frac\{K\}\{X\}\; ight)\; =\; +infty$

As noticed by Steelcite book |last=Steel |first=G.G. |coauthors= |title= Growth Kinetics of Tumors |year=1977 |publisher=Clarendon Press |location= Oxford |isbn= |pages= |chapter= ] and by Wheldoncite book |last=Wheldon |first=T.E. |coauthors= |title= Mathematical Models in Cancer Research |year=1988 |publisher=Adam hilger |location= Bristol |isbn= |pages= |chapter= ] , 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 noticedcite journal | author= d'Onofrio A. | title= A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences | journal= Physica D | volume= 208 |year=2005 | pages=220-235] that, including the interaction with immune system, Gompertz and other laws characterized by unbounded F(0) would preclude the possibility of immune surveillance.

Gomp-Ex Law of Growth

Based on the above considerations, Wheldoncite book |last=Wheldon |first=T.E. |coauthors= |title= Mathematical Models in Cancer Research |year=1988 |publisher=Adam hilger |location= Bristol |isbn= |pages= |chapter= ] proposed a mathematical model of tumor growth, called 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 cellar population expands following the exponential law. However, there is a critical size threshold $X\_\{C\}$ such that for $X>X\_\{C\}$ the growth follows the Gompertz Law:

:$F(X)=maxleft(a,alpha\; logleft(\; frac\{K\}\{X\}\; ight)\; ight)$

so that:

:$X\_\{C\}=\; K\; expleft(-frac\{a\}\{alpha\}\; ight).$

Here there are some numerical estimatescite book |last=Wheldon |first=T.E. |coauthors= |title= Mathematical Models in Cancer Research |year=1988 |publisher=Adam hilger |location= Bristol |isbn= |pages= |chapter= ] for $X\_\{C\}$:

* $X\_\{C\}approx\; 10^9$ for human tumors
* $X\_\{C\}approx\; 10^6$ for murine tumors

ee also

* Logistic function
* Gompertz-Makeham law of mortality
* Growth curve

References