## 7.2.2 Long Term Effect

The interactive environment is used to investigate how changing parameter values affects the stabilization level of medicine in the body.

The interactive figure calculates the amount of medicine in a person's body immediately after taking a dose. In this scenario, the individual takes an initial dose of medicine followed by recurring doses, taken faithfully at fixed intervals of time. The interactive figure allows the initial dose to be different from the recurring doses.

The simulation requires three inputs:

- The initial dose—the amount of medicine given for the initial dose
- The elimination rate—the percent of medicine (given as a decimal) that the kidneys remove from the system between doses
- The recurring dose—the amount of the medicine to be given at fixed intervals

Click Calculate.

*A*(*n*) represents
the amount of medicine in the body immediately *after* taking the *n*th
recurring dose of medicine. So *A*(0) = initial dose, *A*(1) = amount
of medicine in body after first recurring dose, *A*(2) = amount
of medicine in body after second recurring dose, and so on.

This interactive figure calculates
the amount of drug in the system just *after* taking a dose of medicine.
If the amount of medicine in the body just *before* taking the *n*th
dose is desired, subtract the amount of the recurring dose (or initial dose
for *n* = 0) from the value calculated for *A*(*n*).

**Task**

- Consider again the situation introduced in the previous part:

A student strained her knee in an intramural volleyball game, and her doctor prescribed an anti-inflammatory drug to reduce the swelling. She is to take two 220-milligram tablets every 8 hours for 10 days. Her kidneys eliminate 60% of this drug from her body every 8 hours.

In this problem, the three relevant factors are the initial dose, the recurring dose taken every eight hours, and the elimination rate. Consider the three questions below. Make a conjecture for each question, then use the applet to check your work.

- If the initial dose is halved, what will happen to the stabilization level of the medicine in the body?
- If the recurring dose is halved, what will happen to the stabilization level of the medicine?
- If the elimination rate is halved, what will happen to the stabilization level of the medicine?

Use the interactive figure below to answer these questions. By trying other values for each parameter, systematically investigate the effect that changes in these parameters have on the stabilization level of medicine in the body. Keep track of the results of your investigations, and describe any patterns you see. Note that the values computed in the interactive figure give the amount of medicine in the body just after taking a dose of medicine.

**Additional Tasks**

- Suppose the elimination rate is 60% and the initial dose is 440 mg, as in the original situation. But now suppose the doctor wants to change the recurring dose so that the amount of medicine in the athlete's body levels off, that is, it reaches the stabilization level, at about 900 mg. What recurring dose should he prescribe?
- For the initial conditions of the problem, determine an elimination rate for which no stabilization level appears to occur.
- If the recurring dose is halved, determine an elimination rate that will hold the amount of medicine in the body after each recurring dose at a stabilization level of 440 mg.

**Discussion**

The interactive applet is an ideal tool for exploring the effects of
the various parameters in this relatively complex situation. Students
may not be surprised that halving the recurring dose halves the
stabilization level in the body. Indeed, they should observe that the
stabilization level is directly proportional to the recurring dose.
Conversely, changing the initial dose has very little effect on the
stabilization level, except that it changes the number of doses until
the stabilization level is reached. And halving the elimination rate
results in a doubling of the stabilization level; trying other values
should lead to the observation that the stabilization level is inversely
proportional to the elimination rate.

Looking at the mathematics of the situation may make this clearer. If *M* is the the amount of medicine in the body following a dose, *E* is the elimination rate, *R* is the recurring dosage, then the equation *ME = R*
must hold in order to maintain a stable level of medication. In this
case, the amount of medicine removed from the body during each time
interval is equal to the recurring dose. Note that the initial dose does
not figure at all in this equation. Also note that there is a direct
proportional relationship between *M* and *R* and an inverse relationship between *M* and *E*.

In
exploring the additional tasks, students may initially proceed by trial
and error. However, a more powerful approach would be to use either the
observations from the first task or the mathematical analysis above.
For example, in working through the original task, students should
discover that the initial dose is irrelevant. In the first additional
task, students can determine how the direct proportional relationship
can be used to find a recurring dose that will result in a stabilization
level of 900 mg. In the second additional task, they may note that as
the elimination rate decreases, a stabilization level takes longer to
reach. However, if a sufficiently large time frame is observed, a
stabilization level will eventually be reached.

An elimination
rate of 0 implies that none of the drug is being removed from the
system, so the drug will continue to accumulate. In considering the
equation from the previous part, note that* M*0* = R* implies that *R *= 0. If *R* is not 0, then *M* must be undefined. Students might also connect this observation to the familiar rule about division by 0.

**Doing the Investigation Using a Spreadsheet**

The interactive figure in this example illustrates calculation
features that can be implemented in spreadsheets or graphing
calculators. This section describes how this situation can be modeled
using a spreadsheet.

On a spreadsheet, each cell is identified by
the column and row in which it is located. For example, the cell at the
top left corner of the spreadsheet is designated cell A1 because it is
in column A and row 1. The given problem indicated that the athlete in
question was given two 220-mg tablets of medicine, or 440 mg, as an
initial dose. Position the cursor over cell A1 and click the mouse to
highlight the cell. Type 440 and push return. The 440-mg initial dose is
then entered in cell A1.

Every 8 hours the dosage is repeated.
Also, during each 8-hour interval, 60% of the amount in the body at the
beginning of the interval is eliminated by the kidneys. A formula can be
entered into cell A2 to calculate the amount of drug remaining in the
body after the second dose at the end of the first 8-hour interval.
Since 60% of the drug is eliminated, we need to take 40% of the value
for the previous interval, given in cell A1, and then add 440 for the
recurring dose. Click in cell A2, then type "=0.4*A1+440." (The "="
instructs the spreadsheet to calculate the value of the formula and
display that number in the cell.)

The power of a spreadsheet can
then become apparent. Click on cell A2, then hold down the shift key
and click on a cell several rows below, such as A24, thus highlighting a
column of cells. A fill down command will be available under one of the
choices in the menu bar, possibly from the Edit menu. The spreadsheet
will calculate and fill in the entries for each of the cells A3 through
A24. Highlight cell A3, and in it you should see the formula
=0.4*A2+440. Notice that this is the formula you typed into cell A2 with
one important difference: the A1 has become an A2. The spreadsheet is
set up to work recursively. That is, the expression you entered
instructed the spreadsheet to calculate the value for each cell by
multiplying the value of the cell above it by 0.4 and adding 440.

**Take Time to Reflect**

**Take Time to Reflect**

- What about this scenario did you find surprising? Do you think your students would also find it surprising?
- How could you structure this activity to be useful with a class of students

### Reference

National Research Council.

High School Mathematics atWork: Essays and Examples for the Education of All Students.Washington, D.C.: National Academy Press, 1998.