Cumulative Amount Excreted versus Time Plot

Equation 12.3.1 Differential Equations for This Model

Equation 12.3.2 Differential Equation for Rate of Elimination of Drug

Equation 12.3.3 Differential Equation for Rate of Change of Drug amount in the Body

These equation can integrated using Laplace transforms. Taking the Laplace of the differential equations, Equation 12.3.2, and Equation 12.3.3.

Equation 12.3.4 Laplace Transform of dX/dt and dU/dt

Solving for the Laplace of U (and U) gives:

Equation 12.3.5 Laplace of U

Using the fingerprint method the two roots are 0 and -kel. The integrated equation for U, the cumulative amount of drug in urine becomes:

Equation 12.3.6 Cumulative Amount Excreted versus Time

Equation 12.3.7 Integrated Equation for U versus Time

Equation 12.3.7 gives the cumulative amount excreted into urine at time equal to t. At time equal to zero the term e^{- kel • t} becomes equal to 1. Thus U⁰ is equal to Dose • [1 - 1] = 0

When time is equal to ∞ the term e^{- kel • t} is 0 and therefore U^∞ is equal to Dose • [1 - 0] = Dose. NOTE: This only applies when elimination is by excretion into urine as unchanged drug alone.

The cumulative amount excreted into urine is one way of representing amounts excreted into urine data. It is a qualitative plot and it is difficult to directly get accurate quantitative results. However, it is possible to obtain an approximate idea of the half-life or the total amount excreted. See Figure 5.3.1).

Figure 12.3.1 Plot of Cp versus Time and U versus Time

Click on the figure to view the Java Applet window
Java Applet as a Semi-log Plot

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Copyright 2001-3 David W. A. Bourne (david@boomer.org)