O'Meara J M, Fleming D E B
Department of Physics, University of Guelph, Guelph, ON N1G 2W1, Canada.
Phys Med Biol. 2009 Apr 21;54(8):2449-61. doi: 10.1088/0031-9155/54/8/013. Epub 2009 Apr 1.
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.
为了通过体内X射线荧光测量来量化骨铅浓度，通常通过将来自Kα(1)和Kβ(1)特征的归一化X射线峰值幅度与校准体模的峰值幅度进行比较，来确定两个铅浓度估计值。在每种情况下，归一化包括获取X射线峰值幅度与光谱中相干散射光子峰幅度的比值。然后使用这两个铅浓度估计值来确定该样品的加权平均铅浓度。在计算这些测量的不确定度时，在适当情况下纳入任何协方差项很重要。在确定来自每个X射线峰的铅浓度的不确定度时，标准方法不包括X射线峰与相干散射特征之间的协方差。这些光谱特征源自两个不同的物理过程，因此这些特征之间不存在协方差。通过实验和模拟数据，我们证实，正如预期的那样，在检测到的铅X射线峰与相干散射光子信号之间未观察到协方差。这与Brito（2006年，《物理医学与生物学》51卷，6125 - 39页）最近发表的工作形成直接对比。然而，由于共同的相干归一化，在加权平均铅浓度的计算中引入了协方差。正如目前的情况，在计算加权平均铅浓度的不确定度时必须考虑到这一点。我们在此提出一种计算加权平均铅浓度的替代方法，以消除由共同相干归一化引入的协方差。应该强调的是，这种替代方法仅适用于从光谱数据计算铅浓度时不包括校准线截距的情况：当截距的来源得到充分表征且已知来自巴黎石膏校准标准中铅的微量污染时。在我们的方法中，相干归一化仅应用于一个参数，并且我们不再对相关量取加权平均值。我们提出的替代计算对平均铅浓度的计算误差基本上没有影响，表明现有的考虑这种协方差的方法是足够的。
