Generalised logistic function

File:Generalized logistic function A0 K1 B1.5 Q0.5 ν0.5 M0.5.png

A=0, K=1, B=3, Q=ν=0.5, M=0, C=1

Effect of varying parameter A. All other parameters are 1.

Effect of varying parameter B. A = 0, all other parameters are 1.

Effect of varying parameter C. A = 0, all other parameters are 1.

Effect of varying parameter K. A = 0, all other parameters are 1.

Effect of varying parameter Q. A = 0, all other parameters are 1.

Effect of varying parameter $\nu$. A = 0, all other parameters are 1.

The generalised logistic function or curve, also known as Richards' curve, originally developed for growth modelling, is an extension of the logistic or sigmoid functions, allowing for more flexible S-shaped curves:

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

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

It has five parameters:

• $A$: the lower asymptote;
• $K$: the upper asymptote. If $A=0$ then $K$ is called the carrying capacity;
• $B$: the growth rate;
• $\nu > 0$ : affects near which asymptote maximum growth occurs.
• $Q$: is related to the value $Y(0)$
• $C$: typically takes a value of 1.

The equation can also be written:

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

where $M$ can be thought of a starting time, $t_0$ (at which $Y(t_0) = A + { K-A \over (C+1) ^ {1 / \nu} }$ )

Including both $Q$ and $M$ can be convenient:

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

this representation simplifies the setting of both a starting time and the value of Y at that time.

The logistic, with maximum growth rate at time $M$, is the case where $Q = \nu$ = 1.

Generalised logistic differential equation

A particular case of the generalised logistic function is:

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

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

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

with initial condition

$Y(t_0) = Y_0$

where

$Q = -1 + \left(\frac {K}{Y_0} \right)^{\nu}$

provided that ν > 0 and α > 0.

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

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

In fact, for small ν it is

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

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

Gradient of generalized logistic function

When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point $t$ (see ). For the case where $C = 1$,

$\begin{align} \frac{\partial Y}{\partial A} &= 1 - (1 + Qe^{-B(t-M)})^{-1/\nu}\\ \frac{\partial Y}{\partial K} &= (1 + Qe^{-B(t-M)})^{-1/\nu}\\ \frac{\partial Y}{\partial B} &= \frac{(K-A)(t-M)Qe^{-B(t-M)}}{\nu(1 + Qe^{-B(t-M)})^{\frac{1}{\nu}+1}}\\ \frac{\partial Y}{\partial \nu} &= \frac{(K-A)\ln(1 + Qe^{-B(t-M)})}{\nu^2(1 + Qe^{-B(t-M)})^{\frac{1}{\nu}}}\\ \frac{\partial Y}{\partial Q} &= -\frac{(K-A)e^{-B(t-M)}}{\nu(1 + Qe^{-B(t-M)})^{\frac{1}{\nu}+1}}\\ \frac{\partial Y}{\partial M} &= -\frac{(K-A)Be^{-B(t-M)}}{\nu(1 + Qe^{-B(t-M)})^{\frac{1}{\nu}+1}} \end{align}$

See also

• Logistic function
• Gompertz curve
• Ludwig von Bertalanffy