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George Pólya Awards for 2019

This paper starts with a question of obvious real-world significance: how can we tell whether a generic drug is “equivalent” to its brand-name counterpart? It is not enough just to check that they contain the same amount of the same active ingredient, since factors such as pill coatings or inactive ingredients could affect the rates at which the drug is absorbed and eliminated by a patient’s body. These rates cannot be measured directly, so they must somehow be inferred from measurements over time of the drug concentration in the patient’s blood. How should the concentration curves of the two drugs be compared? This leads us to the intriguing mathematical problem presented in the article. The US Food and Drug Administration (FDA) has set equivalence criteria which involve comparing three quantities: the time of the maximal concentration peak, the maximal concentration at that time, and the area under the concentration curve. But why did they select these three quantities in particular? It seems clear that some calculus will be involved in answering the question, and the authors lead us gracefully along this path. Starting with a commonly used pharmacokinetic model for the shape of the concentration curve, they eventually demonstrate that any two of the FDA’s three measured quantities are sufficient to determine the desired absorption and elimination rates; thus, the FDA criteria are mathematically sound. The journey takes us through several calculus topics, including integration, optimization, and, surprisingly, the Lambert W function. The latter provides a good opportunity for readers to think about the notion of implicitly defined functions with multiple branches, an important concept that calculus students may not realize goes beyond the familiar inverse trigonometric functions. Not every equation can be solved by mere algebra, and not every useful function has a nice closed-form expression or a button on your calculator! The authors skillfully illustrate the interplay between applied and theoretical mathematics: understanding the properties of a mathematical model may involve different tools than were required to formulate the model in the first place. This well-written and engaging article could form the basis of a very nice project for students in a calculus or modeling course, providing an opportunity to apply and explore concepts from coursework while demonstrating that calculus is useful—and used—beyond the classroom.