Calculating bone-lead measurement variance.

The technique of (109)Cd-based X-ray fluorescence (XRF) measurements of lead in bone is well established. A paper by some XRF researchers [Gordon CL, et al. The Reproducibility of (109)Cd-based X-ray Fluorescence Measurements of Bone Lead. Environ Health Perspect 102:690-694 (1994)] presented the currently practiced method for calculating the variance of an in vivo measurement once a calibration line has been established. This paper corrects typographical errors in the method published by those authors; presents a crude estimate of the measurement error that can be acquired without computational peak fitting programs; and draws attention to the measurement error attributable to covariance, an important feature in the construct of the currently accepted method that is flawed under certain circumstances.

htp://e/rpnetl.niehs.nih.gov/doc/2000/108p383-386todd/abstrac.tnl The in vivo measurement of lead in human bone using 109Cd-based fluorescence of the K-shell X-rays of lead (KXRF) is a wellestablished technique that has been widely applied to studies of the human health effects of lead and has been reviewed, most recently, by Todd and Chettle (1) in a technical manner and by Hu et al. (2) in a conceptual manner. This paper addresses the method for calculating the measurement uncertainty in a bone-lead measurement given in a 1994 paper by Gordon et al. (3).
In 109Cd-based KXRF, the 88.034 keV 7-rays from 109Cd are used to fluoresce the K-shell X-rays of lead (in increasing energy, those with Siegbahn notation: Ka2, Ka1, K41, KP3, and KP2). The 109Cd y-rays can also elastically scatter off of the calcium and phosphorus (and, to a lesser extent, oxygen) atoms in bone and inelastically scatter off all of the elements in the sample undergoing measurement (principally the bone, soft tissue, and skin). The photons are recorded by a spectroscopy system that yields an energy distribution of the recorded photons that is then fitted using a nonlinear least-squares technique with a mathematical function to extract the amplitudes of the X-ray and elastic scatter peaks. The ratio of the X-ray-toelastic peaks is the response of the system and is regressed, for each X-ray peak under analysis, against the lead concentration of the calibration standards to produce a calibration line.
The in vivo signal from a subject is measured for each lead X-ray to be analyzed and is compared to the established calibration line to obtain one or more estimates of the subject's bone-lead level. The individual Xray estimates are then combined, usually in an inverse-variance weighted manner, to produce the result.
The remainder of this paper addresses methods for the mathematical treatment of the measurement uncertainty; corrects typographical errors in the published method of Gordon et al. (3); presents a crude estimate of the measurement error that can be acquired without computational peak-fitting programs; and addresses the measurement error attributable to covariance.

Materials and Methods
Gordon et al. (3) published a study of the reproducibility of109Cd-based X-ray fluorescence (XRF) measurements of bone lead. Their paper contained an "Appendix" wherein they gave a near-complete description of the mathematical method by which they calculated the variance of an in vivo bone-lead measurement. In brief, they made multiple measurements of a series of plasterof-paris phantoms doped with a range of lead concentrations. The spectrum of scattered radiation showed characteristic peaks from the emission of lead K X-rays that varied in size depending, in part, on the lead concentration of the phantom. Gordon et al. used four of the lead K X-rays for analysis: those with Siegbahn (International Union of Pure and Applied Chemistry notation in parentheses) Kul (K-L3), Ku2 (K-L2), Kil (K-M3), and K,3 (K-M2). For clarity and ease of comparison, I will use the notation of Gordon et al.: xi denotes the amplitude of each X-ray peak, coh denotes the coherent peak amplitude, and RJ denotes the ratio of the two peak amplitudes. Peak amplitudes and SDs are extracted from the spectra by applying a nonlinear least-squares technique. A calibration line is constructed for the ratio of the X-ray-to-coherent peak amplitudes against lead concentration. The ratio is used because it is independent, to a good approximation, of two important factors that affect in vivo and phantom measurements; namely, source-to-skin distance and overlying tissue thickness. Each calibration line is calculated using least-squares regression. I perform weighted least-squares regression and I suspect that Gordon et al. did also, although they did not state what method they used. However, the method of least-squares regression is irrelevant to the arguments of this paper.
Regression gives estimates of the calibration line slope (mi), the slope's variance (a2A), the intercept (C), the intercept's variance (aci), and the covariance between the slope and intercept (Gcimi). The X-ray-tocoherent ratios from an in vivo measurement can be converted, using the calibration lines, into estimates of the in vivo lead concentration (Pbi). A matrix correction term accounts for the difference between phantom (plaster-of-paris) and human (bone) matrices, and the estimates of the in vivo bone-lead concentration are combined into an inverse-variance weighted mean to give a single estimate (Pb,,). The inverse-variance weighted estimate has a variance that is denoted cypb P' For each of the lead X-rays in use  Gordon 4) can be simplified into a form that can be obtained at the time of measurement ("online") and with slightly less computational effort than the estimate of Gordon et al. A cruder estimate of measurement error can be obtained by using the fact that the variance of the X-ray peak amplitude (or area) dominates the variance of the ratio of the X-ray to coherent peak amplitudes (or areas). The fractional error in the ratio is therefore approximately equal to the fractional error in the X-ray peak: a2 whereupon An expression that combines the variance of the XRF response and the variance of the calibration line is then given: Several spectroscopy package regions of interest give a2 allowing an online estimate of Opbito be obtained (assuming the calibration line slope is already known). An online estimate of the measurement error has been useful when physicians require rapid assessment of a patient, and it may prove useful if a target measurement error is needed for all subjects. The expression for variance that accounts for only the X-ray amplitude is indistinguishable from the expression that accounts for the variances in both the coherent and the X-ray amplitudes. Table 1  These assumptions are valid and are stated here only for completeness. Gordon et al. did not derive the expression that accounts for the covariance introduced by the mutual dependence of the ratios of X-ray-to-coherent amplitudes (R) on the same coherent peak amplitude. It may, however, be derived from the product of crude estimates of Gpb from two X-rays i and j aPbi and oPbj If the assumption about how this term was derived is correct, there is a potential problem because the final term of Equation Todd 1 is not the largest term. Using the data of Table A2 in Gordon et al. (3), Table  2 of this paper shows that the first term, (2.2./coh4), contributes > 95% of the value of the whole, whereas the final term used by Gordon et al. contributes very little to the value of the whole. Table 1. The proportional contribution of the variance in the X-ray peak to the variance in the X-ray-tocoherent peak ratio for two human subjects measured by Gordon     Conclusions I have corrected typographical errors in the published method of Gordon et al. (3) and I provided a crude estimate of measurement error that may be of some use. I propose that the correction for the mutual dependence of the X-rays on the coherent peak not be used because of the small size of the covariance correction and the variability due to rounding.