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Plot Types

Cumulative amount excreted versus time

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


Using a JAVA aware browser you can create your own version of Figure V-1.

Rate of excretion (R/E)

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.


Example Ampicillin almost 100% (actually closer to 90%) excreted unchanged into urine.

Table V-1. Example Data Analysis of Drug in Urine Data

Time Interval
(hr)

Amount Excreted
(mg)

Cumulative Amount Excreted
(mg)

Midpoint Time
(hr)

Rate of Excretion
(mg/hr)

A.R.E.
(mg)

0 - 0.5

103

103

397

0.25

206

0.5 -1

82

185

315

0.75

164

1 - 2

117

302

198

1.5

117

2 - 4

119

421

79

3

59.5

4 - 8

67

488

12

6

16.8

8 - 12

10

498

2

10

2.5

12 - 24

2

500

-

18

0.17

24 -

-

500

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


Using a JAVA aware browser you can create your own version of Figure V-3.

Rate of Excretion (R/E) Plots

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


We can then calculate the rate of excretion during each time interval. Then the time point is the midpoint time. If we plot U/t versus t midpoint on semi-log graph paper we have the blue squares on the graph paper. As you can see this gives a straight line plot.

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.

Amount remaining to be excreted (A.R.E.)

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.


This page was last modified: 12 February 2001

Copyright 2001 David W.A. Bourne


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