# XX Non-linear pharmacokinetic models

## Objectives:

**=> To draw the scheme and write the differential equations for
compartmental pharmacokinetic models with non-linear metabolism elimination**
**=> To understand the process of parallel pathways as it applies with one
or more non-linear pathways**

**=> To define and use the parameters Vm and Km**

**=> To design and calculate appropriate dosage regimens when non-linear
pharmacokinetics apply**

All of the rate processes discussed so far in this course, except for the
infusion process, follow first order kinetics. In particular the elimination
process has been assumed to follow first order kinetics. However occasionally
it is observed that the elimination of a drug appears to be zero order at high
concentrations and first order at low concentrations. That is 'concentration'
or 'dose' dependent kinetics are observed. That is at high doses, which produce
higher plasma concentrations, zero order kinetics are observed, whereas at
lower doses the kinetics are linear, that is first order.

This occurs especially with drugs which are extensively metabolized. A typical
characteristic of enzymatic reactions and active transport is a limitation on
the capacity of the process. There is only so much enzyme present in the liver,
and therefore there is a maximum rate at which metabolism can occur. A further
limitation in the rate of metabolism can be the limited availability of a co-substance
or co-factor
required in the enzymatic process. This might be a limit in the amount of
glucuronide or glycine, for example.

Most of our knowledge of enzyme kinetics is derived from *in vitro*
studies where substrate, enzyme, and co-factor
concentrations are carefully controlled. Many factors are involved *in
vivo* so that each cannot be easily isolated in detail. However, the basic
principles of enzyme kinetics have application in pharmacokinetics.

Dose dependent pharmacokinetics can often be described by Michaelis-Menten
kinetics with the RATE of elimination approaching some maximum rate, Vm.

**Equation 86**

with Km a Michaelis-Menten
constant. Km is the concentration at which the rate of metabolism is 1/2 the
maximum rate, Vm

This page was last modified: 12 February 2001
Copyright 2001 David W.A. Bourne