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The linear plot of cumulative amount excreted into urine as unchanged drug *versus* time is shown below. Notice that the value of U^{∞} is NOT EQUAL to the dose, it is somewhat less than dose, unless all of the dose is excreted into urine as unchanged drug. The remaining portion of the dose should be found as metabolites and from other routes of excretion.

The equation for the cumulative amount excreted *versus* time is shown below in Equation 5.3.1.

**Equation 5.3.1 Cumulative Amount excreted into Urine versus Time**

This translates into the plot shown in Figure 5.3.1. Notice that the total amount excreted as unchanged drug, U^{∞}, is less than the dose administered in this plot.

**Figure 5.3.1 Linear Plot of U versus Time showing Approach to U^{∞} not equal to DOSE**

Click on the figure to view the interactive graph

Cumulative amount excreted plots are basically descriptive in nature. At most, you can get a general sense of how much drug is excreted, an estimate of U

**Equation 5.3.2 Rate of Excretion of Unchanged Drug into Urine**

Since urine data is collected over an interval of time the data is represented as ΔU rather than dU. Also, since ΔU is collected over a discrete time interval the time point for this interval should be the midpoint of the interval, t_{midpoint}. Equation 5.3.2 can be replaced by Equation 5.3.3 to better represent the data collected. It may look like a strange way of plotting the data, but actually it's quite convenient to use because urine data results are collected as an amount of drug excreted during a time interval. The amount excreted is the product of the volume of urine voided and the concentration of drug in the sample. This is a rate measurement.

**Equation 5.3.3 Rate of Excretion of Unchanged Drug versus midpoint time**

Taking the ln of both sides gives Equation 5.3.4, an equation for a straight line. Plotting ln[ΔU/Δt] versus time_{midpoint} provides kel from the slope and ke from the intercept divided by the Dose

**Equation 5.3.4. Natural log of Rate of Excretion versus Time**

Note: The intercept (on semi-log graph paper) is ke • DOSE.

**Figure 5.3.2. Semi-log Plot of ΔU/Δt versus Time_{midpoint} Showing Slope = - kel**

with fe = 0.75, kel = 0.2 hr

Click on the figure to view the interactive graph

Rate of excretion plots can be very useful in the determination of the parameters such as kel, ke and fe. There can be a little more scatter that with the ARE plot, below. Analysis of 'real' data may show considerable scatter in the rate of excretion plot. Thus, positioning the straight line on a semi-log plot may be difficult. In practical terms it is difficult to get a lot of early times points unless the subjects are catheterized and even then early times may be difficult to interpret. This means that this method can be difficult to use with drugs which have short half-lives. However, a significant advantage of the rate of excretion plot is that each data point is essentially independent, especially if the bladder is fully voided for each sample. A missed sample or data points is not critical to the analysis.
NOTE: This plot is rate of excretion *versus* midpoint of the collection interval, time.

**Equation 5.3.5 Cumulative amount excreted versus time**

**Equation 5.3.6 Total Amount Excreted as Unchanged Drug, U ^{infinity}**

Equation 5.3.5 can be rewritten in terms of U^{infinity}.

**Equation 5.3.7 Cumulative amount excreted versus time**

**Equation 5.3.8 Cumulative amount excreted versus time**

Rearranging Equation 5.3.8 gives Equation 5.3.9.

**Equation 5.3.9 Amount Remaining to be Excreted as Unchanged Drug versus Time**

**Equation 5.3.10 Ln(ARE) versus time**

where

**Equation 5.3.11 U ^{∞}**

Thus:

**Equation 5.3.12 ARE versus time**

**Figure 5.3.3 Semi-log Plot of ARE versus time**

Click on the figure to view the interactive graph

Amount remaining to be excreted (ARE) plots use the U

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