Using Graphs, Equations, and

Tables to Investigate the

Elimination of Medicine from the Body

The Problem:
 

A student injured her knee in a volleyball game and her doctor prescribed an anti-inflammatory drug to reduce the swelling. She is to take two 220 mg tablets every eight hours for ten days (30 doses). Her kidneys can eliminate 60% of the drug from her body every 8 hours.  Assuming she takes the correct dosage every 8 hours, how much of the medication is actually in her body at any given time.

This is an example of generating data using a recursively defined function and, then, analyzing that data using two web-based applets: one for table data and one for graphing.

Generating and Analyzing the Data:

Step 1:         Fill out the following chart for the first two days indicating the amount of medicine in the body at the time the student takes each dose.

                     n = dosage number       mg of medicine in the body

                     n = 0 (initial dose)        440

              n = 1                           .4(440) + 440 = 616 mg

              n = 2                          

              n = 3                          

              n = 4                          

              n = 5

              n = 6                                  

Step 2:         In the space below, describe in your own words (or with mathematical symbols) how you computed the amount of medicine in the body at the beginning of each dose.

Step 3:         Notice that each successive answer was dependent on the amount of medicine left in the body after the previous dose.  This type of function is called recursive function or recursively-defined function.

                     An example of a recursive function is the table below:

                     n                   S(n)

                     n = 0           S(0) = 3

              n = 1           S(1) = 6

              n = 2           S(2) = 12

              n = 3           S(3) = 24

              n = 4           ___________

              n = 5           ___________

              n = 6           ___________

                     This function is expressed in recursive form as:

                     S(0) = 3

              S(n) = 2 S(n-1) for n = 1, 2, 3, ·

                     Which means "new answer" = 2 times "previous" answer.

                     Fill in the last 3 values of the function in the table above.

Step 4:         Write a recursive function for the volleyball medicine problem [Recall: D(0) = 440 and D(1) = .4(440) + 440 ]

Step 5:         Use your recursive function from Step 4 to generate the first four doses of the anti-inflammatory drug used by the volleyball player.

                     n = dosage number       D(n) =

              n = 1

              n = 2

              n = 3

              n = 4

Step 6:            Use the applet at    http://standards.nctm.org/document/eexamples/chap7/7.2/index.htm

                        to complete Parts a, b, and c below.  You will need to scroll down this web page to find the applet.

Part a:       Enter the following data into the applet:

                  an initial dose of 440 mg,

            an elimination rate of 60%, and

            a recurring dose of 440 mg.

                  Press Calculate button on the applet to generate a table of values for the amount of medication in the body at each period.

Part b:       What do you notice about the amount of medication in the body for the first ten day period?

Part c:       Change the initial dosage to 880 mg but leave the elimination rate and recurring dosage the same.  Describe what changes and what remains from the original dosage.

Step 7:         Access the graphing version of the applet at

                     http://standards.nctm.org/document/eexamples/chap7/7.2/part3.htm 

                     Use the axes below to graph BOTH dosage regimens from Step 6 part a and part c.

Step 8:         Now letâs see what happens after this volleyball player quits taking the medicine.  Remember that for the first 30 doses she had an initial dose of 440 mg, an elimination rate of 60%, and a recurring dose of 440 mg.  Notice that when she quits taking the medicine (on the 31^st dose), she will have a recurring dose of 0 mg.

                     Fill out the chart below for a 15 day period (15 days = 45 doses):

Step 9:         On the axes below, graph the amount of medicine in the volleyball player's body over a 15 day period beginning when she started taking the medication.

The Problem:
 

Azythromycin (Z-Pac) is an antibiotic that uses a "loading dose" of 500 mg and is followed by a daily dosage of 250 mg for 7-10 days. Its elimination rate is approximately 78% per day. How does a loading dose regimen compare to a "straight dose" (equal amounts of 250 mg) regimen?

This is an example of generating data using a recursively defined function and, then, analyzing that data using two web-based applets.

Generating and Analyzing the Data:

Step 1:         Using the applet available at

http://standards.nctm.org/document/eexamples/chap7/7.2/index.htm

                     to fill out the following table below and to compare the amounts of medicine in the body under the two dosage regimens described in the problem.

Step 2:         Use the axes below to graph the amount of medicine in the body under the two dosage regimens described in the problem.

Step 3:         According to a pharmacology study:

                     "With a regimen of 500 mg on day 1, the amount of medicine in the body remained essentially unchanged from day 2 through day 5. However, without the loading dose, 5 to 7 days were required to reach steady-state."

                     Do your results agree or disagree with this statement? Explain.