The effect of covariance between the Kα and the Kβ lead peak concentrations on the uncertainty in the result of in vivo ¹⁰⁹Cd KXRF bone lead measurement

The effect of covariance between the Kα and Kβ lead peak concentrations on the uncertainty in the ¹⁰⁹Cd K x-ray fluorescence measurement of lead in bone is addressed here. It is commonly believed that this covariance arises as a result of the mutual dependence of the ratios of x-ray to coherent amplitudes on the same coherent peak amplitude. Previous work assumes statistical independence between spectral quantities of interest, crudely estimates the uncertainties in the lead peak concentrations, and suggests that the effect of covariance on the measurement uncertainty is small and can be ignored. Consequently, the current method followed by most laboratories reports the measurement uncertainty as if the fluctuations in the measured peak concentrations were independent. The robustness of such assumption, however, is undermined by existing epidemiological data. This paper assesses the magnitude of the covariance effect, using a method based on the observed significant correlations between the ratios of x-ray to coherent peak amplitudes in series of repeat phantom measurements. The revised uncertainties following this approach can exceed the uncertainties estimated by the accepted method by as much as 40%, which suggests a much stronger effect of covariance on the measurement uncertainty than previously reported.