CommentariesCalculating Bone-Lead Measurement VarianceAndrew Christian ToddMount Sinai School of Medicine, Department of Community and Preventive Medicine, New York, New York, USA

The technique of '09Cd-based X-ray fluorescence (XRF) measurements of lead in bone is wellestablished. A paper by some XRF researchers [Gordon CL, et al. The Reproducibility of 109Cdbased X-ray Fluorescence Measurements of Bone Lead. Environ Health Perspect 102:690-694(1994)] presented the currently practiced method for calculating the varie of an in vivo mea-surement once a calibration line has been establshed. This paper corrects typographical errors inthe method published by those authors; presents a crude estmte of the measurement error thatcan be acquired without computational peak fitting programs; and draws attention to the measure-ment error attributable to covariance, an important feature in the construct of the currentl accept-ed method that is flawed under certain circumstaces. Kiy work bone, lead, mesurement error,X-ray fluorescence. Environ Helth Pepctc 108:383-386 (2000). [Online 15 March 1999]htp://e/rpnetl.niehs.nih.gov/doc/2000/108p383-386todd/abstrac.tnl

The in vivo measurement of lead in humanbone using 109Cd-based fluorescence of theK-shell X-rays of lead (KXRF) is a well-established technique that has been widelyapplied to studies of the human healtheffects of lead and has been reviewed, mostrecently, by Todd and Chettle (1) in a tech-nical manner and by Hu et al. (2) in a con-ceptual manner. This paper addresses themethod for calculating the measurementuncertainty in a bone-lead measurementgiven in a 1994 paper by Gordon et al. (3).

In 109Cd-based KXRF, the 88.034 keV7-rays from 109Cd are used to fluoresce theK-shell X-rays of lead (in increasing energy,those with Siegbahn notation: Ka2, Ka1,K41, KP3, and KP2). The 109Cd y-rays canalso elastically scatter off of the calcium andphosphorus (and, to a lesser extent, oxygen)atoms in bone and inelastically scatter off allof the elements in the sample undergoingmeasurement (principally the bone, soft tis-sue, and skin). The photons are recorded bya spectroscopy system that yields an energydistribution of the recorded photons that isthen fitted using a nonlinear least-squarestechnique with a mathematical function toextract the amplitudes of the X-ray and elas-tic scatter peaks. The ratio of the X-ray-to-elastic peaks is the response of the systemand is regressed, for each X-ray peak underanalysis, against the lead concentration ofthe calibration standards to produce a cali-bration line.

The in vivo signal from a subject is mea-sured for each lead X-ray to be analyzed andis compared to the established calibrationline to obtain one or more estimates of thesubject's bone-lead level. The individual X-ray estimates are then combined, usually inan inverse-variance weighted manner, toproduce the result.

The remainder of this paper addressesmethods for the mathematical treatment of

the measurement uncertainty; corrects typo-graphical errors in the published method ofGordon et al. (3); presents a crude estimateof the measurement error that can beacquired without computational peak-fittingprograms; and addresses the measurementerror attributable to covariance.

Materials and MethodsGordon et al. (3) published a study of thereproducibility of109Cd-based X-ray fluores-cence (XRF) measurements of bone lead.Their paper contained an "Appendix"wherein they gave a near-complete descrip-tion of the mathematical method by whichthey calculated the variance of an in vivobone-lead measurement. In brief, they mademultiple measurements of a series of plaster-of-paris phantoms doped with a range oflead concentrations. The spectrum of scat-tered radiation showed characteristic peaksfrom the emission of lead K X-rays that var-ied in size depending, in part, on the leadconcentration of the phantom. Gordon et al.used four of the lead K X-rays for analysis:those with Siegbahn (International Union ofPure and Applied Chemistry notation inparentheses) Kul (K-L3), Ku2 (K-L2), Kil(K-M3), and K,3 (K-M2). For clarity andease of comparison, I will use the notation ofGordon et al.: xi denotes the amplitude ofeach X-ray peak, coh denotes the coherentpeak amplitude, and RJ denotes the ratio ofthe two peak amplitudes. Peak amplitudesand SDs are extracted from the spectra byapplying a nonlinear least-squares technique.A calibration line is constructed for the ratioof the X-ray-to-coherent peak amplitudesagainst lead concentration. The ratio is usedbecause it is independent, to a good approxi-mation, of two important factors that affectin vivo and phantom measurements; namely,source-to-skin distance and overlying tissuethickness. Each calibration line is calculated

using least-squares regression. I performweighted least-squares regression and I suspectthat Gordon et al. did also, although they didnot state what method they used. However,the method of least-squares regression is irrel-evant to the arguments of this paper.

Regression gives estimates of the calibra-tion line slope (mi), the slope's variance(a2A), the intercept (C), the intercept's vari-ance (aci), and the covariance between theslope and intercept (Gcimi). The X-ray-to-coherent ratios from an in vivo measurementcan be converted, using the calibration lines,into estimates of the in vivo lead concen-tration (Pbi). A matrix correction termaccounts for the difference between phan-tom (plaster-of-paris) and human (bone)matrices, and the estimates of the in vivobone-lead concentration are combined intoan inverse-variance weighted mean to give asingle estimate (Pb,,). The inverse-varianceweighted estimate has a variance that isdenoted cypb P'

For each of the lead X-rays in use

R. - CiPbi = 1A46 ' , [Gordon 1]mi

where 1.46 is "the ratio of coherent scatteringcross-sections of bone mineral to hydratedplaster of paris at 88 keV and 160` (3) and

coh' [Gordon 2]

The variance of the ratio, a 2 , is given by

2 xi ~~~~~~~~~~~+aco xi,{(x,) coh I}Lcoh

[Gordon 3]

An expression for "a crude underestimate ofthe measurement variance a (3), which

Address correspondence to A.C. Todd, MountSinai School of Medicine, Department ofCommunity and Preventive Medicine, 1 GustaveL. Levy Place, New York, NY 10029 USA.Telephone: (212) 241-1668. Fax: (212) 996-0407.E-mail: todd@mssm.edu

I thank S. Carroll for manuscript preparation,J.H. Godbold for statistical guidance, and D.R.Chettle for helpful discussions on the Gordon et al."Appendix."This project was supported by grants ES05697

and ES06616 from the National Institute ofEnvironmental Health Sciences.Received 26 July 1999; accepted 21 October

1999.

Environmental Health Perspectives -VOLUME 1081 NUMBER 5 1 May 2000 383

Commentaries * Todd

ignores the calibration line uncertainties, isgiven by

2

2pb = (1.46)2Umi [Gordon 4]

Gordon et al. then gave two expressions forthe inverse-variance weighted mean (Pb )and the variance thereof (aP2b.) derived fromthe individual lead X-ray peak estimates:

Pb2

Pb, = , [Gordon 5]2

iCPbi

which has a

obtained from an inverse-variance weightedmean of the estimates from each of the leadX-rays considered. The final term inside thebracket of Gordon Equation 8,

2CUcoh x2coh X

2 2mi wi

should be

22 2C c.h x(i

tcoh) coh)m2 2

giving a corrected version of GordonEquation 8 of

variance of 2

a-2 ( < [Gordon 6] yb.= 1.462 Co )4 +

DiscussionCrud estimates of measurement error. Thecrude estimate of measurement error givenby Gordon et al. (Equation Gordon 4) canbe simplified into a form that can beobtained at the time of measurement("online") and with slightly less computa-tional effort than the estimate of Gordon etal. A cruder estimate of measurement errorcan be obtained by using the fact that thevariance of the X-ray peak amplitude (orarea) dominates the variance of the ratio ofthe X-ray to coherent peak amplitudes (orareas). The fractional error in the ratio istherefore approximately equal to the frac-tional error in the X-ray peak:

a22 x,

coh2

whereupon

An expression that combines the vari-ance of the XRF response and the varianceof the calibration line is then given:

UPb, = 1.462 ( )Ii +UT i( c i

2m{coh ci) coh)4Ix+b3 + 24

[Gordon 7]

This formula for O2?3b is incorrect in thefourth term inside the square bracket; thisterm should be

JCO \2( )2

mi2

giving a corrected equation of

= 1462[Ci + a ijx ,y

+ + ( coh )(coh)

[Gordon 7 (corrected)]

The typographical error in Equation 7 ofGordon et al. is propagated through theirEquations 8 and 9. Gordon Equation 8 gavethe variance of the lead concentration

2a Xi -c ch 11Xi'coh i) coh )(coh)+ 32 222

[Gordon 8 (corrected)]

Similarly, the final term of GordonEquation 9, 2

C Ucoh x2coh2 4

mi Pbi

should be( F 2 2

coh )coh)2 4

mi YUPbi

resulting in the corrected version of GordonEquation 9:

2()_ 14y+ 2cjaPb~~~ 24x 442X -c

2a2( X 1-c Ucoh' ('X N

2 2my Coh-'J) cohJ(coh)+ M3 24 4

m,UP4b mi0Pb J

[Gordon 9 (corrected)]

Gordon et al. (3) then pointed out that"each of the terms (xi/coh) has a mutualdependence on coh, the coherent scatteramplitude," and that this dependence can beaccounted for by adding a further term toEquation Gordon 9 (corrected):

2 xSPb.m= 1.46 mIoh .i Ijco ojmi Pbi aPbj

2 (1.46) a,Pb'. m2coh2

Several spectroscopy package regions ofinterest give a2 allowing an online estimateofOpbito be obtained (assuming the calibra-tion line slope is already known). An onlineestimate of the measurement error has beenuseful when physicians require rapid assess-ment of a patient, and it may prove useful ifa target measurement error is needed for allsubjects. The expression for variance thataccounts for only the X-ray amplitude isindistinguishable from the expression thataccounts for the variances in both the coher-ent and the X-ray amplitudes. Table 1 illus-trates this using the data of Gordon et al.

The error arising from covariance.Equation Gordon 7 warrants further exami-nation because it contains two assumptionsthat are not explicitly stated by Gordon et al.Equation Gordon 7 is derived from a gener-alized treatment of error propagation thatcan be represented in matrix form (4):

Vy=CaY= y XY)dxl dX2 dX3

( var xi coV(xl,x2)

X cov(xI,x2) varx2

COV(x1,x3) COV(X2 Xx3)

dydx1

X O2dX2dydx3

cov(x1,x3)

cov(x2,x3)

varx3

where V is the variance in y = f(xl, x2, x3),ycoy is the covariance, and var is the variance.

VOLUME 1081 NUMBER 5 May 2000 * Environmental Health Perspectives384

Commentaries * Calculating bone-lead measurement variance

Upon multiplication:

Vy+=(St Ivar c(i

+ 2 dy dy

)C°V(Xi,xj)

When I revert from generalized notationto the notation of Gordon et al. and writeout the summation terms, the result is

a 2 2 22 UR. + i +(R-CR j-Ci 2Pbi + m2+ 2 Mi

+( Ri-RCi CJ2

Using

K2 Uch+ ~ XO

( + coh2 ) Xohi= coh22 cimioh

and

Xi

we obtain

(a2i 22 2 2 2

<P=(~oh2 + oh4 ) m2 m

+ It]o,2 <2- R2C

2( Xii +Xi 2( X-C

2coh2C ch2 hIIcoh h23 aRim +

x3

This equation has all of the terms inGordon's (corrected) Equation 7 but alsocontains two additional terms that involvethe covariance between the response Ri andthe calibration line slope (G7imi) and inter-cept (ajRici). The unstated assumptions ofGordon et al. are that both of these covari-ances are zero:

2 2 =2 2

URimi aR2C

These assumptions are valid and are statedhere only for completeness.

Gordon et al. did not derive the expres-sion that accounts for the covariance intro-duced by the mutual dependence of theratios of X-ray-to-coherent amplitudes (R)on the same coherent peak amplitude. Itmay, however, be derived from the productof crude estimates of Gpb from two X-rays iand j aPbi and oPbj

c2i=1.462 im

whereupon

Ifwe then use

2 ('2 X2 22 + cobhcoh2 coh4

we obtain

1.464 2 X 22= 2 2 2+m, mm coh coh

2( 2 2

coh2 coh4

[Todd 1]

It may be that Gordon et al. then ignoredall of the terms except the last one in the

bracket (the only one that contains the prod-uct of the X-ray peak amplitudes) to obtain:

4 X(X2242 2 1.46 (XiXjIC0hm2mi coh'

The square root of the expression givesthe term given by Gordon et al. if the weight-ing factors (w?, w) are added and the X-rayproduct is evaluated for all pairs ofX-rays

2o x X~Xz 1.462 co,h m,mij

coh4#jM.M.W.Wj

In the notation of Gordon et al., wi =2pbi and w == ba making Gordon's actual

expression

142O~ X 222 coh i j

chiiij(7Pbi ( Pbj

If the assumption about how this termwas derived is correct, there is a potentialproblem because the final term of EquationTodd 1 is not the largest term. Using thedata of Table A2 in Gordon et al. (3), Table2 of this paper shows that the first term,(2.2./coh4), contributes > 95% of thevalue of the whole, whereas the final termused by Gordon et al. contributes very littleto 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-to-coherent peak ratio for two human subjects measured by Gordon et al.

Subject, peak Coherent al a2 131 P3Subject B (male)Amplitudea 2,523 421.5 313.9 81.29 34.61Amplitude ±a 15.16 24.81 38.52 7.399 7.532Error in peak amplitude (%) 0.601 5.886 12.271 9.102 21.762Error in peak/coherent (%) - 5.917 12.286 9.122 21.771(Error in peak amplitude)/ - 99.483 99.880 99.783 99.962(Error in peak/coherent) (%)

Subject C (female)Amplitudea 3,436 31.74 85.46 9.106 6.438Amplitude ±a 17.69 29.96 50.07 8.102 8.442

Error in peak amplitude (%) 0.515 94.392 58.589 88.974 131.128Error in peak/coherent(%) - 94.393 58.591 88.976 131.129(Error in peak amplitude)/ - 99.999 99.996 99.998 99.999(Error in peak/coherent) (%)

*Data from Gordon et al. 13), Table A2.

Table 2. The contribution to an expression for the covariance from two terms, aa2.(/coh4 andxx?a4cohcoh8, expressed as a percentage of Equation Todd 1 and derived from the data of Gordon et al. (3).

Human subject B' Human subject c'Covariance oQ 11 133 o2 131 13cs2. 2./coh4alc 98.7 98.5 98.9 98.4 98.1 98.5a2 - 99.3 99.7 - 99.2 99.611 - - 99.5 - 99.3 -

X..Goh/coh8375x1-

a1 3.715 x 10-5 3.708 x 10-5 3.721 x 10-5 3.703 x 10-5 3.692 x 10-5 3.707 x 10-5a2 - 8.598 x 10-6 8.629 x 10-6 - 8.586 x 10-6 8.621 x 10-6131 - - 1.565 x 10-5 - 1.563 x105 -

'Percentage contribution to the total covariance expression.

Environmental Health Perspectives * VOLUME 1081 NUMBER 51 May 2000 385

Commentaries * Todd

Irrespective of the assumption abouthow Gordon et al. derived the covarianceterm, the quantitative values for the increasein error arising from the covariance termreported by Gordon et al. cannot be repro-duced, probably because of rounding errors.Table A2 ofGordon et al. gives the measure-ment uncertainty resulting from accountingfor the covariance as 0.009 (3.417 - 3.408)jig Pb/g bone mineral for human subject B.For Gordon et al. human subject C, themeasurement uncertainty due to the covari-ance term is 0.00006 (100.0-99.998% of2.972). My calculation of the contributionof the covariance term directly from the peakamplitudes given by Gordon et al., but pre-sumably rounded, is 0.021 and 0.002 pgPb/g bone mineral for subjects B and C,

respectively. For Gordon et al.'s high-leadsubject (B), my calculation of the errorincrease resulting from adding the covari-ance term is greater than that calculated byGordon et al. by a factor of 2.3. For the low-lead subject (C), my calculations yield anincrease in error 35 times of that of Gordonet al. For both subjects, the covariance cor-rection is still small. If the differences areindeed a result of rounding errors, I questionthe use of making a correction to a bone-lead measurement error that can vary somuch solely as a result of rounding.

ConclusionsI have corrected typographical errors in thepublished method of Gordon et al. (3) and Iprovided a crude estimate of measurement

error that may be of some use. I proposethat the correction for the mutual depen-dence of the X-rays on the coherent peaknot be used because of the small size of thecovariance correction and the variability dueto rounding.

REFERENCES AND NOTES

1. Todd AC, Chettle DR. In vivo X-ray fluorescence of leadin bone: review and current issues. Environ HealthPerspect 102:172-177 (1994).

2. Hu H, Rabinowitz M, Smith D. Bone lead as a biologicalmarker in epidemiologic studies of chronic toxicity: con-ceptual paradigms. Environ Health Perspect 106:1-8(1998).

3. Gordon CL, Webber CE, Chettle DR. The reproducibility ofl09Cd-based X-ray fluorescence measurements of bonelead. Environ Health Perspect 102:690-694 (1994).

4. Morrison DF. Multivariate Statistical Methods. 2nd ed.NewYork:McGraw-Hill, 1976.

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