Corrections to "How to calculate lead concentration and concentration uncertainty in XRF in vivo bone lead analysis" by Kondrashov and Rothenberg

Uncertainty calculations for the measurement of in vivo bone lead by x-ray fluorescence

In order to quantify the bone lead concentration from an in vivo x-ray fluorescence measurement, typically two estimates of the lead concentration are determined by comparing the normalized x-ray peak amplitudes from the Kalpha(1) and Kbeta(1) features to those of the calibration phantoms. In each case, the normalization consists of taking the ratio of the x-ray peak amplitude to the amplitude of the coherently scattered photon peak in the spectrum. These two Pb concentration estimates are then used to determine the weighted mean lead concentration of that sample. In calculating the uncertainties of these measurements, it is important to include any covariance terms where appropriate. When determining the uncertainty of the lead concentrations from each x-ray peak, the standard approach does not include covariance between the x-ray peaks and the coherently scattered feature. These spectral features originate from two distinct physical processes, and therefore no covariance between these features can exist. Through experimental and simulated data, we confirm that there is no observed covariance between the detected Pb x-ray peaks and the coherently scattered photon signal, as expected. This is in direct contrast to recent work published by Brito (2006 Phys. Med. Biol. 51 6125-39). There is, however, covariance introduced in the calculation of the weighted mean lead concentration due to the common coherent normalization. This must be accounted for in calculating the uncertainty of the weighted mean lead concentration, as is currently the case. We propose here an alternative approach to calculating the weighted mean lead concentration in such a way as to eliminate the covariance introduced by the common coherent normalization. It should be emphasized that this alternative approach will only apply in situations in which the calibration line intercept is not included in the calculation of the Pb concentration from the spectral data: when the source of the intercept is well characterized and known to come from trace contamination by Pb in the plaster of Paris calibration standards. In our approach, the coherent normalization is only applied to one parameter and we no longer take a weighted mean of correlated quantities. Our proposed alternative calculation has essentially no effect on the calculated error of the mean lead concentration, indicating that the existing method of accounting for this covariance is sufficient.

The effect of covariance between the Kα and the Kβ lead peak concentrations on the uncertainty in the result ofin vivo109Cd KXRF bone lead measurement

The effect of covariance between the K alpha and K beta lead peak concentrations on the uncertainty in the (109)Cd 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.