Chapter 2

Background Mathematical Material

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Writing Differential Equations

Rate processes in the field of pharmacokinetics are usually limited to first order, zero order and occasionally Michaelis-Menten kinetics. Linear pharmacokinetic systems consist of first order disposition processes and bolus doses, first order or zero order absorption rate processes. These rate processes can be described mathematically.

First Order Equation First Order Rate Equation

Each first order rate process ("arrow") is described by a first order rate constant (k1) and the amount or concentration remaining to be transferred (X1).

Zero Order Equation Zero Order Rate Equation

Zero order rate processes are described by the rate constant alone. Amount or concentration to the zero power is 1.

Michaelis Menten Equation Michaelis-Menten Rate Equation

The Michaelis Menten process is somewhat more complicated with a maximum rate (velocity, Vm) and a Michaelis constant (Km) and the amount or concentration remaining.


The full differential equation for any component of a pharmacokinetic model can be constructed by adding an equation segment for each arrow in the pharmacokinetic model. The rules for each segment:

  1. Direction of the arrow
  2. Type of rate process

An example

An example of a PK model

Figure 2.7.1 An example pharmacokinetic model with zero order, first order and Michaelis Menten processes.

In Figure 2.7.1 the rate process from one to two is zero order. The process from two to three is first order and the process from two to four follows Michaelis-Menten kinetics. We can now systematically write the differential equations for each component of the model.


More examples of pharmacokinetic models and writing differential equations and even more examples (Ignore the Laplace parts).


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