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.

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

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:

:$Y(t)\; =\; \{\; K\; over\; (1\; +\; Q\; e^\{-\; alpha\; u\; (t\; -\; t\_0)\})\; ^\; \{1\; /\; u\}\; \}$

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

:$Y^\{prime\}(t)\; =\; alpha\; left(1\; -\; left(frac\{Y\}\{K\}\; ight)^\{\; u\}\; ight)Y$

with initial condition

:$Y(t\_0)\; =\; Y\_0$

provided that:

:$Q\; =\; -1\; +\; Y\_0^\{\; u\}$

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 $u\; ightarrow\; 0^+$ provided that:

:$alpha\; =\; Oleft(frac\{1\}\{\; u\}\; ight)$

In fact, for small ν it is

:$Y^\{prime\}(t)\; =\; Y\; r\; frac\{1-expleft(\; u\; lnleft(frac\{Y\}\{K\}\; ight)\; ight)\}\{\; u\}\; approx\; r\; Y\; lnleft(frac\{Y\}\{K\}\; ight)$

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.

ee also

*Ludwig von Bertalanffy

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.

ee also

*Logistic function
*Gompertz curve