Answer to Question #4905 Submitted to "Ask the Experts"

Q

I want to know how the activity of potassium-40 present in the human body is calculated from weight and height ratio. Is there any formula for calculating potassium-40 activity?

A

There are several ways to estimate the potassium-40 content of the body. You can find a simple calculation in the response to Ask the Experts Question 322 in which the responder assumes an approximate value of total potassium in the body equal to 0.2 percent of the body mass. More accurate estimates of total potassium have been made. For example, a paper by Cohn and others titled "Compartmental Body Composition of Cancer Patients by Measurement of Total Body Nitrogen, Potassium, and Water" provides values of 2.26 grams/kilogram of body mass for women and 2.52 grams per kilogram of body mass for men. Once the total potassium figure has been established, the quantity of potassium-40 may be determined as outlined in the cited Question 322. A more accurate assessment of total potassium may be made, taking account of certain variables. A rather inclusive expression, available in a paper by Larsson and others accounts for height, weight, and age; two equations are given, one for men and the other for women. The equations give millimoles of total potassium per kilogram of fat-free tissue. Thus, to evaluate total potassium one also needs an estimate of the fat content of the individual. The equations, adjusted to yield millimoles of total potassium, are:

For men: TBK = (98.3 - 0.1594 x age +0.1431 x weight - 0.1848 x height)(FFM)

For women: TBK = (94.39 - 0.1735 x age +0.1169 x weight - 0.1567 x height)(FFM).

In these equations TBK stands for total-body potassium, FFM represents kilograms of fat-free mass, age is in years, weight is in kilograms, and height is in centimeters.

Thus, for a 45-year-old man, 5'10" (152.4 cm) in height, 78 kg in weight, with a 20 percent body fat, (FFM = [1 - 0.20] x weight = 0.80 x weight = 62.4 kg), we obtain:

TBK = (98.3 - 0.1594 x 45 + 0.1431 x 78 - 0.1848 x 152.4)( 62.4) = 4,625 millimoles.

Note that 4,625 millimoles represents 181 grams; if we had used the above estimate by Cohn of 2.52 g/kg, we would have obtained 197 grams as our estimated potassium content. The two equations above can be easily adjusted to yield potassium-40 content by evaluating the number of pCi of potassium-40 contained in one millimole of total potassium, and multiplying the TBK by this value. Thus,

pCi potassium-40/millimole (mmole) total potassium = (10^-3 moles)(6.023 x 10²³ atoms/mole)(0.000117 atoms potassium-40/ atom total potassium)(ln2/[(1.28x10⁹ yr)(365 day/yr)(24 hr/day)( 60 min/hr)])(1 pCi/2.22 dpm) = 32.7 pCi/mmole. Therefore, if we multiply the TBK obtained by 32.7 we will get the estimated number of pCi of potassium-40 in the individual. In the above case we would have obtained 1.51 x 10⁵ pCi.

Other approaches are also possible, but the above should be sufficient to demonstrate the process. Good luck.

George Chabot, PhD, CHP