Introduction

The in vivo measurement of lead in human bone using ¹⁰⁹Cd-based fluorescence of the K-shell X-rays of lead (KXRF) is a well-established 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 -rays from ¹⁰⁹Cd are used to fluoresce the K-shell X-rays of lead (in increasing energy, those with Siegbahn notation: K₂, K₁, Kß₁, Kß₃, and Kß₂). The ¹⁰⁹Cd -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-to-elastic 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 X-ray 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 of ¹⁰⁹Cd-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 plaster-of-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) K₁ (K-L₃), K₂ (K-L₂), Kß₁ (K-M₃), and Kß₃ (K-M₂). For clarity and ease of comparison, I will use the notation of Gordon et al.: x_i denotes the amplitude of each X-ray peak, coh denotes the coherent peak amplitude, and R_i 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 (m_i), the slope's variance (²_mi), the intercept (C_i), the intercept's variance (²_Ci), and the covariance between the slope and intercept (²_Cimi). The X-ray-to-coherent ratios from an in vivo measurement can be converted, using the calibration lines, into estimates of the in vivo lead concentration (Pb_i). 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_µ). The inverse-variance weighted estimate has a variance that is denoted

For each of the lead X-rays in use

[Gordon 1]

where 1.46 is "the ratio of coherent scattering cross-sections of bone mineral to hydrated plaster of paris at 88 keV and 160°" (3) and

[Gordon 2]

The variance of the ratio, is given by

[Gordon 3]

An expression for "a crude underestimate of the measurement variance " (3), which ignores the calibration line uncertainties, is given by

[Gordon 4]

Gordon et al. then gave two expressions for the inverse-variance weighted mean (Pb_µ) and the variance thereof () derived from the individual lead X-ray peak estimates:

[Gordon 5]

which has a variance of

[Gordon 6]

An expression that combines the variance of the XRF response and the variance of the calibration line is then given:

[Gordon 7]

This formula for is incorrect in the fourth term inside the square bracket; this term should be

giving a corrected equation of

[Gordon 7 (corrected)]

The typographical error in Equation 7 of Gordon et al. is propagated through their Equations 8 and 9. Gordon Equation 8 gave the variance of the lead concentration obtained from an inverse-variance weighted mean of the estimates from each of the lead X-rays considered. The final term inside the bracket of Gordon Equation 8,

should be

giving a corrected version of Gordon Equation 8 of

[Gordon 8 (corrected)]

Similarly, the final term of Gordon Equation 9,

should be

resulting in the corrected version of Gordon Equation 9:

[Gordon 9 (corrected)]

Gordon et al. (3) then pointed out that "each of the terms (x_i/coh) has a mutual dependence on coh, the coherent scatter amplitude," and that this dependence can be accounted for by adding a further term to Equation Gordon 9 (corrected):

Discussion

Crude estimates of measurement error. The crude estimate of measurement error given by Gordon et al. (Equation 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:

whereupon

Several spectroscopy package regions of interest give ²_xi, allowing an online estimate of _Pbi to 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 illustrates this using the data of Gordon et al.

The error arising from covariance. Equation Gordon 7 warrants further examination because it contains two assumptions that are not explicitly stated by Gordon et al. Equation Gordon 7 is derived from a generalized treatment of error propagation that can be represented in matrix form (4):

where V_y is the variance in y = f (x₁, x₂, x₃), cov is the covariance, and var is the variance.

Upon multiplication:

When I revert from generalized notation to the notation of Gordon et al. and write out the summation terms, the result is

Using

and

we obtain

This equation has all of the terms in Gordon's (corrected) Equation 7 but also contains two additional terms that involve the covariance between the response R_i and the calibration line slope () and intercept (). The unstated assumptions of Gordon et al. are that both of these covariances are zero:

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_i) on the same coherent peak amplitude. It may, however, be derived from the product of crude estimates of from two X-rays i and j: and

whereupon

If we then use

we obtain

[Todd 1]

It may be that Gordon et al. then ignored all of the terms except the last one in the bracket (the only one that contains the product of the X-ray peak amplitudes) to obtain:

The square root of the expression gives the term given by Gordon et al. if the weighting factors (w_i, w_j) are added and the X-ray product is evaluated for all pairs of X-rays

In the notation of Gordon et al., w_i = and w_j = , making Gordon's actual expression

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_xi²_xj/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.

Irrespective of the assumption about how Gordon et al. derived the covariance term, the quantitative values for the increase in error arising from the covariance term reported by Gordon et al. cannot be reproduced, probably because of rounding errors. Table A2 of Gordon et al. gives the measurement uncertainty resulting from accounting for the covariance as 0.009 (3.417 - 3.408) µg Pb/g bone mineral for human subject B. For Gordon et al. human subject C, the measurement uncertainty due to the covariance term is 0.00006 (100.0-99.998% of 2.972). My calculation of the contribution of the covariance term directly from the peak amplitudes given by Gordon et al., but presumably rounded, is 0.021 and 0.002 µg Pb/g bone mineral for subjects B and C, respectively. For Gordon et al.'s high-lead subject (B), my calculation of the error increase resulting from adding the covariance term is greater than that calculated by Gordon et al. by a factor of 2.3. For the low-lead subject (C), my calculations yield an increase in error 35 times of that of Gordon et al. For both subjects, the covariance correction is still small. If the differences are indeed a result of rounding errors, I question the use of making a correction to a bone-lead measurement error that can vary so much solely as a result of rounding.

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.