Definitions

# Generalised logistic function

The generalized logistic curve or function, also known as Richards' curve is a widely-used and flexible sigmoid function for growth modelling, extending the well-known logistic curve.

$Y = A + \left\{ K over \left(1 + Q e^\left\{-B \left(t - M\right)\right\}\right) ^ \left\{1 / nu\right\} \right\}$

where Y = weight, height, size etc., and t = time.

It has five parameters:

• A: the lower asymptote;
• K: the upper asymptote minus A. If A=0 then K is called the carrying capacity;
• B: the growth rate;
• ν>0 : affects near which asymptote maximum growth occurs.
• Q: depends on the value Y(0)
• M: the time of maximum growth if Q=ν

## The Generalized Logistic Differential Equation

A particular case of Richard's function is:

$Y\left(t\right) = \left\{ K over \left(1 + Q e^\left\{- alpha nu \left(t - t_0\right)\right\}\right) ^ \left\{1 / nu\right\} \right\}$

which is the solution of the so called Richard's differential equation (RDE):

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

with initial condition

$Y\left(t_0\right) = Y_0$

provided that:

$Q = -1 + Y_0^\left\{nu\right\}$

The classical logistic differential equation is a particular case of the above equation, with ν =1, whereas Gompertz curve may be recovered as well in the limit $nu rightarrow 0^+$ provided that:

$alpha = Oleft\left(frac\left\{1\right\}\left\{nu\right\}right\right)$

In fact, for small ν it is

$Y^\left\{prime\right\}\left(t\right) = Y r frac\left\{1-expleft\left(nu lnleft\left(frac\left\{Y\right\}\left\{K\right\}right\right) right\right)\right\}\left\{nu\right\} approx r Y lnleft\left(frac\left\{Y\right\}\left\{K\right\}right\right)$

The RDE suits to model many growth phenomena, including the growth of tumors. Concerning its applications in oncology, its main biological features are similar to those of Logistic curve model.