# Non-Linear Pharmacokinetic Models

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### Student Objectives for this Chapter

• 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, for some drugs it is observed that the elimination of the drug appears to be zero order at high concentrations and first order at low concentrations. That is 'concentration' or 'dose' dependent kinetics are observed. At higher doses, which produce higher plasma concentrations, zero order kinetics are observed, whereas at lower doses the kinetics are linear or first order. This is more commonly seen after overdoses have been taken but for a few drugs it is observed at concentrations considered therapeutic.

This occurs 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 available 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 21.1.1 Rate of Change for a Saturable Process

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

References

• Examples of Causes of Drug Showing Dose- and Time-dependent Kinetics, Table 14-1 Shargel, L. and Yu, A.B.C. 1985 Applied Biopharmaceutics and Pharmacokinetics, 2nd ed., Appleton-Century-Crofts, Norwalk, CT p254