The generalized logistic equation

One of the few near-universal observations about solid tumors is that almost all decelerate, i.e. reduce their specific growth rate $(dN/dt)/N$, as they grow larger [1, 2]. Consequently Eq. 1 is often generalized to a non-linear first order ODE which incorporates growth deceleration [1, 3 - 6]:

$$dN/dt = f(N),$$ (2)

where $f(N)$ is an appropriate function. Observed tumor growth is consistent with the assumption that the tumor would reach some given limiting cell number, the host carrying capacity $K > 0$, if treatment or death did not intervene. The most commonly used example of an ODE which incorporates this assumption is the generalized logistic equation. This equation is of the replicator type [7] and has one global attractor for the region $N > 0$, at a point $K$, as follows:

$$f(N) = N F(N/K) = ( \mu N / \nu ) \biggl [ 1 - (N/K)^{\nu} \biggr ] , \quad \quad \mu > 0 .$$ (3)

Here $\nu$ is real and the value $\nu = 0$ is to be understood as a limit; taking the limit gives the Gompertz equation [8, 9]:

$$F(N/K) = - \mu \ln (N/K) .$$ (4)

Eq. 1 can be obtained from Eqs. 2 and 3 by restricting attention to the region $N<K$ and then taking the limit $\mu , \nu \to \infty$ with $\mu / \nu$ fixed at $\lambda$. One other special case is $\nu =1$, the logistic equation which Eq. 3 generalizes. Roughly speaking, $\mu / \nu$ in Eq. 3 replaces $\lambda$ and $\nu$ governs how fast the tumor approaches the limiting number $K$ [6].

One-ODE models have frequently been applied to experimental and clinical tumors, and to tissue culture models (surveys in [1, 5]). The most systematic clinical application of Eq. 3, to breast tumors, suggested an optimal value of close to 0.25 for $\nu$ [6].

In analyzing tumor treatments or experiments on cell killing by external agents, a very common assumption (often made implicitly) is that treatment modifies Eq. 2 by adding an extra term, as follows:

$$dN/dt = -\alpha c(t)N + f(N) .$$ (5)

Here $\alpha$ is a positive constant, the strength of the chemotherapeutic agent, and $c(t)$ is the agent concentration at the location of the tumor, governed by treatment schedule and pharmacodynamic effects. The direct proportionality of the killing term to $c$ and to $N$ corresponds to mass-action chemical kinetics for the reaction of a therapeutic agent and a cell.