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.