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The Quiescence Growth Model is a type of mathematical model for a time series where growth is influenced by reproduction, death and quiescence. It modifies a Malthusian growth model to include the effects of individuals in a population exiting and entering a dividing state. The resulting curvatures can be similar to a Gompertz function or simple logistic function, but instead of resource limitations directly effecting population growth, individuals are dropping out of the dividing population at some given rate which leadsto a different biological interpretation of the parameters. It is formulated as a Multi-compartment model.

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

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

where

• a is the upper asymptote, since ${\displaystyle ae^{be^{-\infty }}=ae^{0}=a}$
• b, c are negative numbers
• b sets the x displacement
• c sets the growth rate (x scaling)
• e is Euler's Number (e = 2.71828...)

The function curve can be derived from a Gompertz law of mortality, which states the rate of mortality (decay) falls exponentially with current size. Mathematically

${\displaystyle k^{r}\propto {\frac {1}{y(t)}}}$

where

• ${\displaystyle r={\frac {y'(t)}{y(t)}}}$ is the rate of growth.
• k is an arbitrary constant.

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

In the sixties A.K. Laird^[1] 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:

${\displaystyle X(t)=K\exp \left(\log \left({\frac {X(0)}{K}}\right)\exp \left(-\alpha t\right)\right)}$

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:

${\displaystyle \lim _{t\rightarrow +\infty }X(t)=K}$

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;

• 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:

${\displaystyle X^{\prime }(t)=\alpha \log \left({\frac {K}{X(t)}}\right)X(t)}$

i.e. is of the form:

${\displaystyle X^{\prime }(t)=F\left(X(t)\right)X(t),F^{\prime }(X)\leq 0}$

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:

${\displaystyle F(X)=\alpha \left(1-\left({\frac {X}{K}}\right)^{\nu }\right)\Rightarrow F(0)=\alpha <+\infty }$

whereas in the Gompertz case the proliferation rate is unbounded:

${\displaystyle \lim _{X\rightarrow 0^{+}}F(X)=\lim _{X\rightarrow 0^{+}}\alpha \log \left({\frac {K}{X}}\right)=+\infty }$

As noticed by Steel^[2] and by Wheldon^[3], 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^[4] that, including the interaction with immune system, Gompertz and other laws characterized by unbounded F(0) would preclude the possibility of immune surveillance.

The Gompertz differential equation

${\displaystyle X^{\prime }(t)=\alpha \log \left({\frac {K}{X(t)}}\right)X(t)}$

is the limiting case of the generalized logistic differential equation

${\displaystyle X^{\prime }(t)=\alpha \nu \left(1-\left({\frac {X(t)}{K}}\right)^{\frac {1}{\nu }}\right)X(t)}$

(where ${\displaystyle \nu >0}$ is a positive real number) since

${\displaystyle \lim _{\nu \rightarrow +\infty }\nu \left(1-x^{1/\nu }\right)=-\log \left(x\right)}$.

In addition, there is an inflection point in the graph of the generalized logistic function when

${\displaystyle X(t)=\left({\frac {\nu }{\nu +1}}\right)^{\nu }K}$

and one in the graph of the Gompertz function when

${\displaystyle X(t)={\frac {K}{e}}=K\cdot \lim _{\nu \rightarrow +\infty }\left({\frac {\nu }{\nu +1}}\right)^{\nu }}$.

Based on the above considerations, Wheldon^[3] 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 ${\displaystyle X_{C}}$ such that for ${\displaystyle X>X_{C}}$ the growth follows the Gompertz Law:

${\displaystyle F(X)=\max \left(a,\alpha \log \left({\frac {K}{X}}\right)\right)}$

so that:

${\displaystyle X_{C}=K\exp \left(-{\frac {a}{\alpha }}\right).}$

Here there are some numerical estimates^[3] for ${\displaystyle X_{C}}$:

• ${\displaystyle X_{C}\approx 10^{9}}$ for human tumors
• ${\displaystyle X_{C}\approx 10^{6}}$ for murine (mouse) tumors

• Gompertz distribution
• Growth curve

• Weisstein, Eric W. "Gompertz Curve". MathWorld.