The Gompertz curve or Gompertz function is a type of
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 in the natural sciences (such as physics, ...
for a
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
, named after
Benjamin Gompertz
Benjamin Gompertz (5 March 1779 – 14 July 1865) was a British self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1 ...
(1779–1865). It is a
sigmoid function
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve.
A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula:
:S(x) = \frac = \ ...
which describes growth as being slowest at the start and end of a given time period. The right-side or future value
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 one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the
simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the
generalised logistic function
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after ...
. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.
History
Benjamin Gompertz
Benjamin Gompertz (5 March 1779 – 14 July 1865) was a British self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1 ...
(1779–1865) was an actuary in London who was privately educated. He was elected a fellow of the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
in 1819. The function was first presented in his June 16, 1825 paper at the bottom of page 518. The Gompertz function reduced a significant collection of data in life tables into a single function. It is based on the assumption that the mortality rate increases exponentially as a person ages. The resulting Gompertz function is for the number of individuals living at a given age as a function of age.
Earlier work on the construction of functional models of mortality was done by the French mathematician
Abraham de Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He moved ...
(1667–1754) in the 1750s. However, de Moivre assumed that the mortality rate was constant. An extension to Gompertz's work was proposed by the English actuary and mathematician
William Matthew Makeham (1826–1891) in 1860, who added a constant background mortality rate to Gompertz’s exponentially increasing one.
Formula
where
* ''a'' is an asymptote, since
* ''b'' sets the displacement along the ''x''-axis (translates the graph to the left or right).
* ''c'' sets the growth rate (''y'' scaling)
* e is
Euler's Number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
(e = 2.71828...)
Properties
The halfway point is found by solving
for t.
The point of maximum rate of increase (
) is found by solving
for t.
The increase at
is
Derivation
The function curve can be derived from a
Gompertz law of mortality, which states the rate of absolute 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
A mobile phone, cellular phone, cell phone, cellphone, handphone, hand phone or pocket phone, sometimes shortened to simply mobile, cell, or just phone, is a portable telephone that can make and receive calls over a radio frequency link whil ...
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
* Modelling of growth of tumors
* Modelling market impact in finance and aggregated subnational loans dynamic.
*Detailing population growth in animals of prey, with regard to predator-prey relationships
*Modelling bacterial cells within a population
*Examining disease spread
Applications
Gompertz curve
Population biology is especially concerned with the Gompertz function. This function is especially useful in describing the rapid growth of a certain population of organisms while also being able to account for the eventual horizontal asymptote, once the
carrying capacity
The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as t ...
is determined (plateau cell/population number).
It is modeled as follows:
where:
*
is time
*
is the initial density of cells
*
is the plateau cell/population density
*
is the initial rate of tumor growth
This function consideration of the plateau cell number makes it useful in accurately mimicking real-life
population dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.
History
Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...
. The function also adheres to the
sigmoid function
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve.
A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula:
:S(x) = \frac = \ ...
, which is the most widely accepted convention of generally detailing a population's growth. Moreover, the function makes use of initial growth rate, which is commonly seen in populations of bacterial and cancer cells, which undergo the
log phase
Log most often refers to:
* Trunk (botany), the stem and main wooden axis of a tree, called logs when cut
** Logging, cutting down trees for logs
** Firewood, logs used for fuel
** Lumber or timber, converted from wood logs
* Logarithm, in mathem ...
and grow rapidly in numbers. Despite its popularity, the function initial rate of tumor growth is difficult to predetermine given the varying microcosms present with a patient, or varying environmental factors in the case of population biology. In cancer patients, factors such as age, diet, ethnicity, genetic pre-dispositions,
metabolism
Metabolism (, from el, μεταβολή ''metabolē'', "change") is the set of life-sustaining chemical reactions in organisms. The three main functions of metabolism are: the conversion of the energy in food to energy available to run cell ...
, lifestyle and origin of
metastasis
Metastasis is a pathogenic agent's spread from an initial or primary site to a different or secondary site within the host's body; the term is typically used when referring to metastasis by a cancerous tumor. The newly pathological sites, then, ...
play a role in determining the tumor growth rate. The carrying capacity is also expected to change based on these factors, and so describing such phenomena is difficult.
Metabolic curve
The metabolic function is particularly concerned with accounting for the rate of metabolism within an organism. This function can be applied to monitor tumor cells; metabolic rate is dynamic and is greatly flexible, making it more precise in detailing cancer growth. The metabolic curve takes in to consideration the energy the body provides in maintaining and creating tissue. This energy can be considered as metabolism and follows a specific pattern in cellular division.
Energy conservation
Energy conservation is the effort to reduce wasteful energy consumption by using fewer energy services. This can be done by using energy more effectively (using less energy for continuous service) or changing one's behavior to use less service (f ...
can be used to model such growth, irrespective of differing masses and development times. All
taxa
In biology, a taxon (back-formation from ''taxonomy''; plural taxa) is a group of one or more populations of an organism or organisms seen by taxonomists to form a unit. Although neither is required, a taxon is usually known by a particular nam ...
share a similar growth pattern and this model, as a result, considers cellular division, the foundation of the development of a tumor.
*
= energy organism uses at rest
*
= number of cells in the given organism
*
= metabolic rate of an individual cell
*
= energy required to maintain the existing
tissue
*
= energy required to create new tissue from an individual cell
The differentiation between energy used at rest and metabolic rate work allows for the model to more precisely determine the rate of growth. The energy at rest is lower than the energy used to maintain a tissue, and together represent the energy required to maintain the existing tissue. The use of these two factors, alongside the energy required to create new tissue, comprehensively map the rate of growth, and moreover, lead in to an accurate representation of the
lag phase
250px, Growth is shown as ''L'' = log(numbers) where numbers is the number of colony forming units per ml, versus ''T'' (time.)
Bacterial growth is proliferation of bacterium into two daughter cells, in a process called binary fission. Providing ...
.
Growth of tumors
In the 1960s 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:
*
is the tumor size at the starting observation time;
*
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, whereas, in presence of therapies, it may be X(0) > K;
*
is a constant related to the proliferative ability of the cells.
*
refers to the
natural log
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
.
It can be shown that the dynamics of X(t) are governed by the Gompertz differential equation:
i.e. is of the form when broken down:
''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.
The theoretical study by Fornalski et al.
showed the biophysical basis of the Gompertz curve for cancer growth except very early phase where parabolic function is more appropriate. They found also that the Gompertz curve describes the most typical case among the broad family of the cancer dynamics’ functions.
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
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
in the graph of the generalized
logistic function
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
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 assumption that there is no competition for resources holds true in most scenarios. It can however be affected by
limiting factor
A limiting factor is a variable of a system that causes a noticeable change in output or another measure of a type of system. The limiting factor is in a pyramid shape of organisms going up from the producers to consumers and so on. A factor not l ...
s, that requires the creation of sub-factors variables.
the growth follows the Gompertz Law:
so that:
Here there are some numerical estimates
for
:
*
for human tumors
*
for
murine
The Old World rats and mice, part of the subfamily Murinae in the family Muridae, comprise at least 519 species. Members of this subfamily are called murines. In terms of species richness, this subfamily is larger than all mammal families excep ...
(mouse) tumors
Inverse Gompertz function
The Gompertz function is a one-to-one correspondence (also known as an
Bijective function
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
) and so its
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
can be explicitly expressed in traditional functional notation as a single continuous function. Given a Gompertz function of the form:
where
* ''d'' is the base horizontal asymptote, since
* ''a'' is the distance from the base to the second asymptote, since
* ''b'' sets the displacement along the ''x''-axis (translates the graph to the left or right).
* ''c'' sets the growth rate (''y'' scaling)
* ''e'' is
Euler's Number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
(''e'' = 2.71828...)
the corresponding inverse function can be expressed as: