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Differential equation


Equation XX-2 Differential Equation

It is not possible to integrate this equation but by looking at low and high concentrations we can get some idea of the plasma concentration versus time curve.

Low Cp approximation to first order

At low concentrations, where Km > Cp, Km + Cp is approximately equal to Km

where the Vm/Km is a constant term and the whole equation now looks like that for first order elimination, with Vm/Km a first order elimination rate constant.

Therefore at low plasma concentrations we would expect first order kinetics. Remember, this is the usual situation for most drugs. That is Km is usually larger than the plasma concentrations that are achieved.

High Cp approximation to zero order

For some drugs, higher concentrations are achieved, that is Cp > Km, then Km + Cp is approximately equal to Cp.

and we now have zero order elimination of drug. At high plasma concentrations we have zero order or concentration independent kinetics.

Figure XX-1 Linear Plot of Cp Versus Time Showing High Cp and Low Cp - Zero Order and First Order Elimination

From the plot.

High Cp, in the zero order part, the slope is fairly constant (straight line on linear graph paper) but steeper, that is, the rate of elimination is faster than at lower concentrations. [However, the apparent rate constant is lower. This is easier to see on the semi-log graph.]

At higher concentrations the slope = -Vm. At lower concentrations we see an exponential decline in plasma concentration such as with first order elimination.

On semi-log graph paper we can see that in the zero order region the slope is more shallow, thus the rate constant is lower. The straight line at lower concentrations is indicative of first order kinetics.

Figure XX-2 Semi-Log Plot of Cp Versus Time Showing High Cp and Low Cp

The presence of saturation kinetics can be quite important when high doses of certain drugs are given, or in case of over-dose. In the case of high dose administration the effective elimination rate constant is reduced and the drug will accumulate excessively if saturation kinetics are not understood.

Figure XX-3 Linear Plot of Versus Dose Per Day

Phenytoin is an example of a drug which commonly has a Km value within or below the therapeutic range. The average Km value about 4 mg/L. The normally effective plasma concentrations for phenytoin are between 10 and 20 mg/L. Therefore it is quite possible for patients to be overdosed due to drug accumulation. At low concentration the apparent half-life is about 12 hours, whereas at higher concentration it may well be much greater than 24 hours. Dosing every 12 hours, the normal half-life, can rapidly lead to dangerous accumulation. At concentrations above 20 mg/L elimination maybe very slow in some patients. Dropping for example from 25 to 23 mg/L in 24 hours, whereas normally you would expect it to drop from 25 -> 12.5 -> 6 mg/L in 24 hours. Typical Vm values are 300 to 700 mg/day. These are the maximum amounts of drug which can be eliminated by these patients per day. Giving doses approaching these values or higher would cause dangerous accumulation of drug. Figure 90 is a plot of versus dose for example.

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This page was last modified: 12 February 2001

Copyright 2001 David W.A. Bourne

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