X-ray survival as Gompertz growth in number killed

Gompertz kinetics model of fast chemical neurotransmission currents

At a chemical synapse, transmitter molecules ejected from presynaptic terminal(s) bind reversibly with postsynaptic receptors and trigger an increase in channel conductance to specific ions. This paper describes a simple but accurate predictive model for the time course of the synaptic conductance transient, based on Gompertz kinetics. In the model, two simple exponential decay terms set the rates of development and decline of transmitter action. The first, r, triggering conductance activation, is surrogate for the decelerated rate of growth of conductance, G. The second, r', responsible for Y, deactivation of the conductance, is surrogate for the decelerated rate of decline of transmitter action. Therefore, the differential equation for the net conductance change, g, triggered by the transmitter is dg/dt=g(r-r'). The solution of that equation yields the product of G(t), representing activation, and Y(t), which defines the proportional decline (deactivation) of the current. The model fits, over their full-time course, published records of macroscopic ionic current associated with fast chemical transmission. The Gompertz model is a convenient and accurate method for routine analysis and comparison of records of synaptic current and putative transmitter time course. A Gompertz fit requiring only three independent rate constants plus initial current appears indistinguishable from a Markov fit using seven rate constants.

Gompertz pharmacokinetic model for drug disposition

PURPOSE

Disposition of drugs among compartments of the body usually occurs at changing rates that are commonly modeled as sums of exponential terms with different rate constants. This paper describes an alternative. Gompertz kinetics, in which the rates can change systematically.

METHODS

Differential equations were developed and solved that fit typical examples taken from the literature. The three or four constants required for a visually satisfactory fit to data could readily be found by successive adjustment "by hand," but strategies and results are presented for computer fitting of the data.

RESULTS

In four examples, the amount remaining in the blood decreases as an exponentially declining fraction of the amount present at any moment, but the antecedent processes responsible for that amount differ as follows: (a) In simple i.v. disposition (e.g., lidocaine) concentration falls as a decelerated exponential decay. (b) Delayed i.v. disposition (e.g., hexobarbital) requires, as well, a decelerated exponential growth function. (c) In simple disposition after oral administration, the concentration in the blood initially increases at a decelerating rate. (d) In biphasic oral disposition (e.g., Li+ carbonate), the initial Gompertz growth is followed by decelerated exponential decay.

CONCLUSIONS

Gompertz kinetics provides an accurate and parsimonious mathematical model describing drug disposition.