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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:
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where
It has five parameters:
The equation can also be written:
where
Including both
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
Generalised logistic differential equation
A particular case of the generalised logistic function is:
which is the solution of the so-called Richards' differential equation (RDE):
with initial condition
where
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
In fact, for small ν it is
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