Integrating Differential Equations using Laplace Tranforms

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Laplace transforms are a convenient method of converting differential equations into integrated equations, that is, integrating the differential equation. It is similar to the use of logarithms to multiple or divide numbers. To multiple two numbers we convert each number into their respective logarithm and add. The sum is converted into the 'anti'-logarithm and the product of the original numbers is the result. For division, the logarithm of the two numbers are subtracted before obtaining the answer by taking the anti-logarithm.

Differential equations can be converted into the integrated form using Laplace transforms by following a number of straight forward steps.

  1. Write the differential equation. Using the approach presented you need to write the differential equation for the system of interest.
  2. Take the Laplace transform of each differential equation using a few transforms. We transform the equation from the t domain into the s domain. For most pharmacokinetic problems we only need the Laplace transform for a constant, a variable and a differential.
  3. Use some algebra to solve for the Laplace of the system component of interest. Often the Laplace of a component 'up-stream' will need to be solved first and substituted into the equation of interest.
  4. Finally the 'anti'-Laplace for the component is determined from tables such as that below or by using the 'finger-print' method describe later.

In step two above, only three Laplace transforms are necessary for most of the linear of pharmacokinetic systems that you might encounter. These are:

The Laplace of a constant:

Laplace of a constant

The Laplace of a variable (i.e. amount in a component of the model; e.g. X1):

Laplace of a variable

The Laplace of the differential of a variable (e.g. dX1/dt):

Laplace of a differential

Time for some examples

IV Bolus - Linear one compartment model

The IV bolus does is placed in component one at zero time. Elimination is by a single first order process described by the rate constant kel (elimination rate constant). The scheme can be describe by Figure 2.8.1

Linear one compartment PK model with one elimination
Figure 2.8.1 Linear one compartment model with one elimination process

  1. Write the differential equation for X1. There is only one component.

    Differential equation

  2. Transform the equation into the Laplace form

    Laplace of equation

  3. Rearranging and solving for L(X1). Substitute Dose for X1(0).

    Laplace of X(1)

  4. The final result can be determined from the Laplace Transform table (below - line 3 with A == Dose: a == kel).

    Integrated equation

Additional problems and answers are available.

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