Generalised logistic function

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A=0, C=1, M=0.5, B=1.5, Q=ν=0.5
A=0, C=1, M=0.5, B=1.5, Q=ν=0.5

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

Contents

[edit] The Generalized Logistic Differential Equation

A particular case of Richard's function is:

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(t0) = Y0.

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 = O\left(\frac{1}{\nu}\right)

In fact, for small ν it is

Y^{\prime}(t) = Y r \frac{1-Exp\left(\nu Ln\left(\frac{Y}{K}\right) \right)}{\nu} \approx r Y Ln\left(\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.

[edit] See also

[edit] 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.

[edit] See also