Drug Dosage

TASK. A student strained her knee in an intramural volleyball game, and her doctor has 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 filter 60% of this drug from her body every 8 hours. How much of the drug is in her system after 24 hours?

COMMENTARY. This task can be approached in many different ways using numeric or algebraic methods, by hand, with calculators or with computers. Regardless of their approach, mathematically literate students will need to be familiar with the mathematical structure in this task. Whether the situation involves understanding effective drug dosages, population growth, bank accounts or loans, amortization, heating or cooling, filtering pollution from lakes and streams, models of learning and forgetting, or models of the economy, students should be familiar with some of the many situations which give rise to iterative processes--repeated processes in which future levels are determined by present levels. The importance of technology in dealing with tasks of this type is worth noting. Students should be able to think and work with iterative processes, and also should be sufficiently comfortable with technology to expect, as a matter of course, to consider long-term behavior and trends. All high school graduates need to be comfortable with both the structure of such tasks and the technological tools for their investigation.

Iterative models illustrate a mathematical tool that has become increasingly important in recent years: difference equations, the discrete-time analogs of differential equations. In the past, students did not study difference equations until well into their undergraduate or graduate courses. But with the aid of a spreadsheet or graphing calculator, students can handle such iterative problems with relative ease, provided that they can make the translation from a problem statement to an equation describing the iterative process, also called a recursion equation.

MATHEMATICAL ANALYSIS. One approach is to create a table of values relating the number of 8-hour periods to the amount of the drug remaining in the student's system. To do this, some assumptions must be made about how quickly the drug gets into her system after it is taken. The simplest model assumes that the drug is active immediately after the medication is taken.

In this task, the information given is the rate at which the drug is eliminated by the kidney, whereas the focus of the tasks is on how much remains. The fundamental unit of time is an 8-hour period.

If the student takes 440 mg of the anti-inflammatory drug, after 8 hours her kidneys have removed 60% of the 440 mg, leaving 40% of the dose in her system. After she has taken her second 440-mg dose, the total amount of drug in her system in mg is then: (.4)(440) + 440 = 616. After 16 hours, she has 40% of the total for 8 hours, plus another 440. After 24 hours, she has 40% of the total for 16 hours, plus another 440. These calculations are summarized in Table 1.

Table 1: Calculation of amount of drug in system
	Day Time in Hours Drug Remaining + Amount Taken = Total in System 1 0 0 + 440 = 440 1 8 176 + 440 = 616 1 16 246.40 + 440 = 686.4 2 24 274.56 + 440 = 714.56

One way to generalize this method is to use Table 1 to see patterns. First, let A_n denote the amount of drug in the system after dose n. Then the "drug remaining" column may be represented as in Table 2. The relationship in Table 2 between the "drug remaining" column and the "total in system" column may then be represented as shown in Table 3, which can easily be converted into a spreadsheet. Table 3 also suggests the recursion equation: .4A_{n ­ 1} + 440 = A_n, where A_n is the amount of drug present at the beginning of the nth 8-hour period (or nth dose).

Table 2: Calculation of amount of drug in system, first generalization
	Dose n Drug Remaining + Amount Taken = Total in System, An 1 0 + 440 = 440 A1 2 (.4)(440) + 440 = 616 A2 3 (.4)(616) + 440 = 686.4 A3 4 (.4)(686.4) + 440 = 714.56 A4

Table 3: Calculation of amount of drug in system, second generalization
	Dose n Drug Remaining + Amount Taken = Total in System, An 1 0 + 440 = A1 2 (.4)A1 + 440 = A2 3 (.4)A2 + 440 = A3 4 (.4)A3 + 440 = A4

More advanced students might begin their solutions by modeling the process with this recursion equation, expressing the relationship between A_{n ­ 1} and A_n. Such an equation is well suited for use in a programmable calculator or a spreadsheet, especially in investigating the long-term behavior of the model. Students who have studied iterative models may also solve this equation analytically to find an explicit formula for A_n.

EXTENSIONS. How much of the drug is in the student's system after 10 days? The recursion equation above can be translated into a spreadsheet to answer this. Students will notice that the peak levels, A_n, don't get bigger indefinitely and can investigate ways of determining the limiting value.

Suppose she doesn't like taking medicine, so she decides to take only one pill every 8 hours for 20 days. Does this strategy of taking half the amount of the drug for twice as long a period result in the same level of drug in her system? Many drugs have what is known as a therapeutic level. Unless the amount of drug in one's system reaches the therapeutic level, the drug is not effective. If the therapeutic level for the drug she is taking is 650 mg, how effective is her strategy of taking half the drug for twice as long?

Sometimes doctors suggest that the patient take a double dose initially, called a loading dose. A simple variation in the standard model can illustrate the effect of the loading dose on the effectiveness of the drug.

In addition to a therapeutic level, drugs also have a toxic level. If too much of the drug is in your system, you can become ill as a result. As you age, the ability of your kidneys and liver to remove the drug is reduced. Suppose that, for older patients, only 40% of the drug is removed in an 8-hour period. What dangers does this pose for older patients? How would such a patient fare using a strategy of taking only one tablet for twice as long?

Once these situations have been examined, students could explore different dosages (e.g., two 200-milligram pills); dosage intervals, (e.g., taking one 220-milligram pill every 4 hours); and clearing rates (for example, the kidneys of most healthy individuals will filter 45% to 65% of the drug from their system within 8 hours). Though it requires more sophisticated methods for solution, students could also explore different models in which the drug does not get into the bloodstream immediately.

Some medications, particularly cold capsules, contain several different drugs. The liver and kidneys remove nearly 70% of a typical decongestant in an 8-hour period but only approximately 20% of a typical antihistamine during the same period. If a student takes one decongestant and one antihistamine every 8 hours for 5 days, how does the level of decongestant compare to the level of antihistamine? Once the initial spreadsheet or calculator formulas are created, this very sophisticated question is within reach.

Often the information on the rate of filtering by the liver and kidneys is given in terms of the half-life of the drug. For example, theophylline, a common asthma drug, has a half-life of approximately 4 hours. Local pharmacists can identify the half-life of drugs that people typically take during cold and flu season. Students can rewrite the half-life of the drugs in terms of the decay rate for the interval of time between doses of the medication. Since half-life is such a commonly given measure, it is important that students be able to use this terminology as well to compare residual levels over specific time periods.