Amount Remaining to be Excreted (A.R.E.) versus Time Plots

Equation 12.5.1 Differential Equation for U

Since, DOSE = Amount in body + Amount eliminated

DOSE = V • Cp + U = U^∞

U^∞,total amount excreted (ONLY Equal to Dose when there are no other routes of excretion).

Then U^∞ - U = V • Cp

Equation 12.5.2 Differential Equation for U reworked

Integrating gives:

ln (U^∞ - U) = ln U^∞ - kel • t

Alternate Derivation:

Since Dose = U^∞;

Equation 12.5.3 Cumulative Amount Excreted in terms of U^∞

U^∞ - U = U^∞ • e^{- kel • t}

ln (U^∞ - U) = ln U^∞ - kel • t

Thus by plotting ln (U^∞ - U) versus time we should get a straight line with a slope of - kel. The term (U^∞ - U) is called the amount remaining to be excreted (A.R.E.). If we subtract U from U^∞ at each time point we are calculating A.R.E. or (U^∞ - U).

Although the A.R.E. plot can be smoother than the rate of excretion plot, a straight line relies on a good estimate of U^∞. A missed sample is extremely critical to the analysis making the estimation of U^∞ impossible without some form of regression analysis.

Figure 12.5.1 Semi-log Plot of Rate of Excretion versus Time

Click on the figure to view the Java Applet window
Java Applet as a Linear Plot

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