Abstract
Formulas are derived for the mean and variance of the number of radioactive atoms present in a compartment (or urn). Initally,n 1 radioactive atoms andb stable atoms are placed in the urn; and subsequently,r stable atoms are added and an equal number,r, of a random mixture of stable and radioactive atoms is removed per unit time. The expected number of radioactive atoms,E(t), present at timet is, as expected,n 1 e−λt where λ=(r/Δt)/(b+r+n 1). The relative variance, σ2(t)/n 21 , vanishes to zero forr=1, atoms per unit time and for a large number ofn 1 radioactive atoms; but for a large number of bothr andn 1 atoms the relative variance is ∼e−λt, equal to the fractional retention, fort>1/λ. Thus in studies where radionuclides are injected into animals and a single compartment represents the data, if a large variance is observed it might be due to the fact that large numbers of atoms are transferred out in unit time. When a small variance is observed, this is probably due to the fact that few atoms are transferred in smaller units of time (such that λ is the same in both cases).
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Research sponsored by the Energy Research and Development Administration under contract with Union Carbide Corporation.
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Bernard, S.R. An urn model study of variability within a compartment. Bltn Mathcal Biology 39, 463–470 (1977). https://doi.org/10.1007/BF02462924
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DOI: https://doi.org/10.1007/BF02462924