Relative humidity

A=0, K=1, B=3, Q=ν=0.5, M=0

The generalised 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 function.

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

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

It has six parameters:

• A: the lower asymptote;
• K: the upper asymptote. 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

Generalised logistic differential equation

A particular case of Richard's function is:

$Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}$

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

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

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

When estimating parameters from data, it is often necessary to compute the partial derivatives of the parameters at a given data point t (see ):

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

See also

• Logistic function
• Gompertz curve
• Ludwig von Bertalanffy

Citations

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References

• Richards, F.J. 1959 A flexible growth function for empirical use. J. Exp. Bot. 10: 290-300.
• Pella JS and PK Tomlinson. 1969. A generalised stock-production model. Bull. IATTC 13: 421-496.
• Lei, Y.C. and Zhang, S.Y. 2004. Features and Partial Derivatives of Bertalanffy-Richards Growth Model in Forestry. Nonlinear Analysis: Modelling and Control, Vol 9, No. 1:65-73