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.
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:
which is the solution of the so called Richard's differential equation (RDE):
with initial condition
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 provided that:
In fact, for small ν it is
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.
- 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.