Monday, December 20, 2010

Mathematics and medicine: a cautionary tale

Mathematics is valuable in many areas of application, including medicine, but there are hazards to having doctors diagnose their own mathematical needs. A hilarious example was just brought to my attention; a well-cited medical paper that reinvents one of the first lessons of high school calculus (which the author goes on to name after himself):
A mathematical model for the determination of total area under glucose tolerance and other metabolic curves. by M M Tai, Diabetes Care February 1994 vol. 17 no. 2 152-154 .

Here's the abstract in its entirety:
"OBJECTIVE--To develop a mathematical model for the determination of total areas under curves from various metabolic studies. RESEARCH DESIGN AND METHODS--In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve. Validity of the model is established by comparing total areas obtained from this model to these same areas obtained from graphic method (less than +/- 0.4%). Other formulas widely applied by researchers under- or overestimated total area under a metabolic curve by a great margin. RESULTS--Tai's model proves to be able to 1) determine total area under a curve with precision; 2) calculate area with varied shapes that may or may not intercept on one or both X/Y axes; 3) estimate total area under a curve plotted against varied time intervals (abscissas), whereas other formulas only allow the same time interval; and 4) compare total areas of metabolic curves produced by different studies. CONCLUSIONS--The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision. "

Google Scholar reveals that this paper is Cited by 137, most of which appear to be un-ironic.

Incidentally, while scholars like to properly reference things, who knows if the world would be better if the author had cited a calculus textbook, and pointed out that this method had been well known for hundreds of years. Very likely the medical journal would have declined to publish it if they had realized it wasn't new. (It says something about the difficulty a medical journal faces in evaluating mathematical ideas that none of the referees recognized this.) And the citations the paper has received suggest that at least to some subsequent researchers, this elementary calculus lesson, delivered in a medical journal, filled a gap in their education and proved useful. On the other hand, if the paper had instead pointed out that there is lots of widely available software to do numerical integration, it might have been even more useful to docs who needed to find areas under curves.

One of the delights of interdisciplinary work is how fruitful it can be. This is particularly true in market design, which almost always involves work between economists and experts in other things who are  directly involved in some market.

One of the frustrations of interdisciplinary work is that it involves translation between different cultures. For example, parts of market design are fairly mathematical, or involve ideas from economics (e.g. about incentives) that may be unfamiliar to non-economists.

My work on kidney exchange has had more than its share of both the delights and the frustrations, in part because the non-economist experts involved--kidney surgeons--are so very expert at what they do. I've had the good fortune to be part of teams of market designers and surgeons who work really well together.

But the translation barrier to the rest of the medical profession is formidable, particularly because matching for kidney exchange is quite mathematical, and doctors are mostly selected for their talents in other things. This makes for great complementarities when you find the right docs, but it's always hard for the medical journals to evaluate contributions that have an element of mathematics, and things can go badly wrong when a doctor overestimates the breadth of his competency, which is an occupational hazard for people whose daily work involves giving advice to patients whose lives depend on it.

HT: Assaf Romm

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