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 nth 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 nth dose is desired, subtract the amount of the recurring dose (or initial dose for n = 0) from the value calculated for A(n).

Task 

1. 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 

1. 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?
2.  For the initial conditions of the problem, determine an elimination rate for which no stabilization level appears to occur.
3.  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 M0 = 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

Reference

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