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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.

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

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

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

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

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

provided that:

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

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 $nu\; rightarrow\; 0^+$ provided that:

- $alpha\; =\; Oleft(frac\{1\}\{nu\}right)$

In fact, for small ν it is

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

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.

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Last updated on Thursday August 28, 2008 at 08:19:24 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday August 28, 2008 at 08:19:24 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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