The U versus time is a mirror image of the amount of drug in the body V*Cp versus time. As we lose drug from the body, it will appear in urine. We can derive an equation for U versus time by integrating the differential equation.
(Equation
V-1, see above)
Differential equation
Since Cp = Cp0 * e- kel * t and V * Cp0 = DOSE
dU = DOSE* kel * e- kel * t * dt
then
U = DOSE [1 - e- kel * t]
Equation V-2. Integration Equation for U
Cumulative amount excreted into urine at time = t
At t = 0; e- kel * t = 1
then U0= DOSE [1 - 1] = 0
At t = ; e- kel * t = 0
then U = DOSE [1 - 0] = DOSE
The cumulative amount excreted into urine is one way of representing this data. It is fairly qualitative and it is difficult to get quantitative results directly. There are other ways of looking at this data which can give you more information. (For example, an approximate idea of the half- life). See Figure V-1).
Figure V-1. Plot of Cp versus Time and U versus Time
A second method is the rate of excretion versus time plot. Going back to the differential equation.
Equation V-3. Rate of Change of U versus Time
Taking the logarithms of both sides gives:-
Thus by plotting ln (dU/dt) versus time you should get a straight line with a slope of - kel.
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
concentration of drug in the sample. Again, this is a rate
measurement. Since we estimate dU/dt as U/
t, an average rate of excretion over the
collection interval, the t value used with this plot is the time at
the midpoint of the collection interval.
Table V-1. Example Data Analysis of Drug in Urine Data
(hr) |
(mg) |
(mg) |
(hr) |
(mg/hr) |
(mg) |
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After giving an IV dose of 500 mg, urine is collected and assayed for drug content. Data collected is time interval versus amount (mg) excreted during the time interval. The cumulative amount excreted is readily calculated.
Figure V-2. Plot of Cumulative Amount Excreted versus Time
The plot shows U rapidly increasing at first then leveling off to
U. U
= DOSE for
this set of data. Notice that U
/2 (250 mg) is excreted in about
11/2 hours. Otherwise it is a fairly qualitative
representation of the data.
Figure V-3. Plot of Rate of Excretion and A.R.E. versus Time
Rate of Excretion (R/E) Plots
Amount Remaining to be Excreted (A.R.E.) Plots
From the equation before we have:-
Intercept = ln (DOSE * kel)
Slope = - kel
kel = 0.440 hr- 1 (t1/2 = 1.58 hr)
Consequently with this plot we can estimate kel and t1/2. A disadvantage of this type of plot is that the error present in "real" data can obscure the straight line and lead to results which lack precision. Also it can be difficult to collect frequent, accurately timed urine samples. This is especially true when the elimination half-life is small.
There is a third method of plotting the data which is commonly used. That is the amount remaining to be excreted (A.R.E.) plot. The equation describing this plot can be derived from the differential equation.
(Equation V-1)
Since, DOSE = Amount in body + Amount eliminated
Thus, DOSE = V * Cp + U = U
U,
total amount excreted
Then U
- U = V.Cp
Integrating gives:
ln (U -
U) = ln U
- kel * t
Alternate Derivation:
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).
These results are shown as red circles on the semi- log graph paper (see Figure V-4).
kel = 0.464 hr- 1
One disadvantage of this approach is that the errors are
cumulative, with collection interval, and the total error is
incorporated into the U values and therefore into each A.R.E. value. Another
problem is that total (all) urine collections are necessary. One
missed sample means errors in all the results calculated.
Copyright 2001 David W.A. Bourne