ma047 - The Exponential-Decay Law Applied to Medical...

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The Exponential-Decay Law Applied to Medical Dosages Gerald M. Armstrong, Brigham Young University, Provo, UT 84602 Calvin P. Midgley, Condell Memorial Hospital, Libertyville, IL 60048 Mathematics Teacher, February 1987, Volume 80, Number 2, pp. 110–113. Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). More than 200 books, videos, software, posters, and research reports are available through NCTM’S publication program. Individual members receive a 20% reduction off the list price. For more information on membership in the NCTM, please call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright May 1991 by the National Council of Teachers of Mathematics. All rights reserved. M any drugs are used up by the human body at rates described in terms of exponential decay. Two simple exponential formulas are given in this article. One describes how a therapeutic level of a drug is achieved in terms of its half-life and other parameters. The other describes how this level is maintained. If the body eliminates a drug so that the rate at which the amount of the drug is changing is proportional to the amount of the drug present, this relationship can be stated in the notation of calculus as Students of calculus learn that this equation has a solution of the form If H is the half-life of such a drug—the time required for half of it to be used up—and P is the initial dose given, then with is the amount left in the system after t units of time. (The half-life H is written and called half-life in medical literature.) b t 1 y 2 k 5 2 ln 2 H , Pe kt , f s t d 5 Ce kt . f 9 s t d 5 kf s t d .
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Therapeutic Level and Dosage Assume now that a drug of half-life H is administered to a patient. If P units of the drug are administered at times ..., uniform time intervals of length the total amount T of the drug in the body at time is easily found. At time the first P units are given. Then at time the second P units are administered, and the amount left from the first dose is giving a total of in the system. At time the total amount in the system is with the terms representing, respectively, the amounts left from the third, second, and first doses. To continue, the amount T of the drug in the system at the time one time unit after the last dose is given, is Since this expression is the sum of d terms of a geometric sequence with the first term and constant ratio (1) This formula is useful in different ways. As given, T can be obtained in terms of the dose P , the length of the time intervals, the total number d
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