Math in Medicine: Dosages, Percentiles, and Growth

The multiplication I was sure I had got right

My son came home with an antibiotic and a pharmacy label that printed the dose per kilogram of body weight. I did what any parent does. I multiplied 20 by the number on the label, checked it twice, and wrote down 143 milligrams. The pharmacist handed back a different figure, 217, and when I asked why, she said the label assumes a bigger body than his and that children get dosed by surface area instead.

I spent a while assuming I had made an arithmetic error. I had not. I had followed the correct arithmetic for the wrong model. A dose that scales with weight is a proportion, and a proportion is the wrong shape for a body, because bodies do not scale uniformly. The rule is not wrong because people skip steps. It is wrong because the exponent was never chosen. Once you see the exponent, the pharmacy conversation makes sense, and so does the rest of this article.

A proportion is the easy answer

The rule on the label is one line. If an adult of 70 kilograms needs 500 milligrams, then a person weighing w kilograms needs 500 times w over 70. Divide, multiply, done. For a 20 kilogram child that is 143 milligrams. Our proportions guide is the right place to check the method, because the method is not the problem. Two weights in the same ratio give two doses in the same ratio, and that is true and correct arithmetic.

What it quietly assumes is that weight is a stand-in for something else. Somebody decided that body weight is a reasonable proxy for how much of the drug should be in the bloodstream. That is a good idea. It is just not the best available proxy, and the difference between the two is not small.

Surface area is a power, and the power is two-thirds

Here is the geometry. Pretend a body is a solid shape and make a bigger one by scaling every length by a factor of k. Areas scale by k squared. Volumes scale by k cubed. Weight is a volume, so weight scales by k cubed.

Now read it the other way round, which is the direction you actually need. If weight scales as k cubed, then k scales as the cube root of weight. And surface area scales as k squared, so it scales as the cube root of weight squared. Surface area grows as weight to the power of two-thirds. That is the whole correction. Our scale factor guide works through why area, length and volume each need their own exponent, and our powers and roots guide covers the notation.

Now run the pharmacy numbers with the exponent instead of with the proportion. A 20 kilogram child against a 70 kilogram adult is a ratio of about 0.286, and 0.286 to the two-thirds power is about 0.434. Times 500 milligrams is 217. The proportion said 143. The two rules disagree by 52 percent, on a real child, in the direction that under-treats.

Note also that the error reverses for large bodies and that a power law never settles at the right answer in between. The proportion is not an approximation that improves as you approach average size. It is simply a different function, and the two only agree at one weight.

Clinicians actually use a real formula rather than a power rule. The common shortcut is the square root of height in centimetres times weight in kilograms, divided by 3,600. Note that it takes both measurements. That is not an accident. Pure two-thirds scaling only holds if a body grows in every direction at once, and real bodies grow unevenly, which is exactly why our percentages guide is needed for the other half of this article: the ones that sound like percentages but are not.

A 60th percentile is not sixty percent

This one causes more confusion than the dosing, and it takes about ten seconds to explain. A child on the 60th percentile for height is taller than 60 percent of children. Which means 40 percent of children are taller than them. Nothing about the number says sixty percent good, and nothing about being above the middle says anything about health at all. The percentile describes a ranking, and a ranking is not a verdict.

This is also why the bell curve, standard deviations and z-scores sit underneath all of it. If you want the mechanism rather than the warning, start with how a percentile comes out of a bell curve and then come back.

The practical part of a growth chart is the slope, not the level. A child sitting on the 20th percentile but tracking along it steadily is usually a boring and reassuring chart. A child who was at the 60th and is now crossing down past the 40th has changed direction, and the number they landed on matters far less than the crossing. Growth is a derivative of the chart, not a reading of it.

When the maths is the medicine

The third place these numbers show up is outbreaks, and the shift in thinking is the same one. If a disease doubles every five days, then after ten days you have four times as many cases and after twenty you have sixteen times. Doublings, not additions. Two weeks of delay is three doublings, and three doublings is a factor of eight.

Which reframes the goal. You cannot negotiate with the growth rate; you can only push the multiplication factor below one so the curve stops bending upward. Every intervention that works is doing that, and the arithmetic is why timing matters more than effort. Waiting to act is not doing less, it is paying three doublings.

What to try this week

Pick any dose printed per kilogram and convert it the other way. Then do it again with the two-thirds power and compare. Next, find your child's height and weight percentiles and check only the direction of travel over the last three readings. Three numbers, one substitution, and you will have stopped reading a percentage as a score.

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