Some drugs cause sweating while they are active, and this sweat can be harmlessly passed through the skin. If there is too much sweating an overdose can be dangerous. The maximum safe dose depends on the area of the skin. If the subject of the treatment is a child, then it is all too easy to give the wrong dose. The following method is used to find the rough area of a child, and indeed, also works for an adult.
A clinic will easily obtain measurements of a child's height in metres, and weight in kilogrammes. The shape of a human is complicated, but is assumed to be common to all, except that the similarity ratio might be different in the horizontal and vertical directions. In common terms, some humans are fatter than others. Imagine that an overfed child looks the same it would, were its thin version to be made of rubber such that, when blown up by a pump, only the horizontal dimensions were enlarged. Let F denote the fatness of a child, in metres, and let H be its height. Then its volume will be V = KF(FH), where K is a constant, the same for all humans. Its area is A = CFH, where C is another constant. We do not know F, V, K, or C, and we wish to find A in square metres. At this point, we make an approximation; the density d of all humans is the same. This is not bad...we observe that while swimming, the relative bouyancy of all children seems to be the same as for adults. Thus the known parameter, the mass M, is given by M = Vd = dKF(FH). This gives us a formula for the fatness:
F = [M/(dKH)]1/2
From this we find the area of the child:
A = CFH = C[MH]1/2[1/(dK)]1/2
.Although this answer has three unknown constants in it, they are all multiplied together, and so there is only one parameter. This can be found once and for all. The method becomes very practical when we take logarithms:
log A = constant + 1/2[log M + log H]...................(*)
The handbook for the paediatrician has a page in three columns; as its left-column, it shows the heights in metres, on a log scale, and as its right column, the weights in kilos, also on the same log scale. The centre column shows the area in square metres, on the same log scale. The medic finds the height and weight from the records, and places a ruler with its left end on the log height, and its right end on the log weight. Where this crosses the central axis is therefore 1/2[logH + log M], so its area in square metres can be read off from the log scale in the central column, provided that the constant in (*) has been found once and for all. This then gives, not the required dose, but the maximum safe dose, if the drug has been calibrated in terms of the safe area.
Before mathematicians invented this method, the district nurse would have had to dip the child in glycerine and weigh it accurately, thus finding the increase in weight over the bare weight of the child; next, she would have estimated the thickness of the glycerine by doing it again with a known area of skin. The area of the child would have then been found from the known density of glycerine (if the temperature is kept constant). This method would have needed very accurate scales.
I am endebted to Barbara Nowak for asking me to find out how the method in the book manages to give the area in square metres.
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© by Ray Streater, 29 Dec 2003.