Anal. Chem. 1988, 60, 1281-1285
cz = geometrical parameter
D = diffusion coefficient i = current
K = Michaelis constant
4 = axial length
R, R’ = geometrical parameters r = radial distance t- = $me V, V = dimensionless reaction rates V,, = maximum velocity of reaction z = axial distance Greek Letters a = partition coefficient 6 = mass-transfer boundary layer thickness
e = aspect ratio 4 = Thiele modulus K = dimensionless Michaelis constant X, X‘ = ratio of diffusion coefficients v = stoichiometry coefficient p = dimensionless radial distance u = dimensionless relative catalytic activity 7 = dimensionless time { = dimensionless axial distance
Subscripts B = bulk solution e = electrode f = final value value g = glucose gmo = glucose-modulated oxygen-dependent
1281
H = value in the outer hydrophobic membrane I = value in the inner hydrophobic membrane i = initial value; inner o = oxygen; outer r = radial t = total z = axial Registry No. Glucose, 50-99-7.
LITERATURE CITED (1) Gough, D. A.; Leypoidt, J. K.; Armour, J. C. Diabetes Care 1982, 5 , 190-198. (2) Leypoidt, J. K.; Gough. D. A. Anal. Chem. 1884, 56, 2896-2904. (3) Gough, D. A.; Lucisano, J. Y.; Tse, P. H. S. Anal. Chem. 1985, 57, 235 1-2357. (4) Conway, P. J. Ph.D. Dissertation, University of California, San Diego, 1986. Tse, P. H. S.; Gough. D. A. Blotechnol. Bioeng. 1987, 29, 705-713. Gough, D. A.; Armour. J. C.; Lucisano, J. Y.; McKean, B. D. Trans. Am. Soc. Artlf. Intern. Organs 1888, 32, 148-150. Tse, P. H. S.; Gough, D. A. Anal. Chem. 1887, 59, 2339-2344. Aris, R. The Mathematical Theory of Reactlon and Dlffuslon In Permeable Catalysts; Clarendon: Oxford, England, 1975. Lucisano. J. Y. Ph.D. Dissertation. University of California, San Diego, 1987. (10) Peaceman, D. W.; Rachford. H. H. J . SOC.Ind. Appl. Math. 1855, 3 , 28-4 1. (11) Weibei, M. K.; Brlght. H. J. J . Biol. Chem. 1971. 246, 2734-2744. (12) Diffusion in Polymefs; Crank, J., Park, G. S., Eds.; Academic: London, 1968.
RECEIVED for review October 23, 1987. Accepted March 1, 1988. This work was supported by grants from the National Institutes of Health and the American Diabetes Association.
Bias-Free Adjustment of Analytical Methods to Laboratory Samples in Routine Analytical Procedures Ricard Ferrfis* and Francesc Torrades Escola d’Enginyers, Colom 11, 08222 Terrassa, Spain
The Youden oneample regresslon and the standard addition method, SAM, are Jolntly applied to ascertain whether less cumbersome experhmis can be used in routine work, when analyzing glven kinds of laboratory samples. The suggested steps are illustrated by means of barlum sulfate gravimetry (the analytical method) to determine sulfate Ion in calcium carbonate of reagent quality (through the analyticai procedure). I t Is shown that, in this particular case, the latter can be reduced to a minimum, slnce the blank correction Is zero and the analytical signal vs analyte concentratlon slope agrees wlth that derived from the stolchiometric model. Two SAM shortcomlngs are dlscussed and a way to dlagnose one of them Is given, while for the other a one-point calibration using a matrix reference material Is suggested as a posslble breakthrough. Also, the sultabillty of substituting the solvent (slmpllfied matrix) for real matrlces In callbration work Is consldered.
The distinction between analytical method and analytical procedure (1) might be summarized by saying that, in developing the latter, the former is applied within an experiment designed to give bias-free results with a particular kind of matrix. The analytical procedure includes also the sample 0003-2700/88/0360-1281$01.50/0
treatment, which changes the test portion, TP, into the treated sample, TS (2). It is possible to follow a generalized approach while going from an analytical method to the corresponding quickest analytical procedure still compatible with the attainment of bias-free results. Bias in chemical analysis is usually resolved (3) into two components: (a) a constant one that does not depend upon the analyte concentration in the laboratory sample, LS (2), and (b) a proportional component that shows a first-order dependence on such a concentration. Higher power terms (quadratic, etc.) can be ignored, provided that the dynamic range is so chosen that the relationship between analytical signal, XI, and analyte concentration, C, in the LS, is always first order, i.e., XI= a bC. The advantages of linear relationships within the calibration work have been stressed (4, 5). The blank measurement through the Youden one-sample regression gives the constant component of bias, the total Youden blank TYB (6-8). When extrapolation to TP equal to zero is done, the analyte amount from the LS is also zero, so the proportional component vanishes from the intercept on the ordinate, leaving alone the constant component of bias. Once the blank is known, it is subtracted from the raw analytical signal, X I , thus resulting a net analytical signal, X,
+
0 1988 American Chemical Society
1282
ANALYTICAL CHEMISTRY, VOL. 60, NO. 13, JULY 1, 1988
which still retains the proportional component of bias. A bias-free analytical result can be obtained with the standard addition method, SAM. Here the blank-corrected analytical signal is plotted either vs the added standard amount (3)or the added standard concentration in the TP (9, IO), and the intercept on the abscissa is obtained from extrapolation. Provided that the relationship between X and C is X = bC, b constant, we have a t the abscissa intercept X = 0 and, also, C = 0. Therefore, the numerical value given by the intercept on the abscissa is free from both the constant and the proportional bias components. Although there are many other sources of bias besides the matrix effects, these are particularly insidious and they will be specially considered here. Keeping ourselves within effects of an order not higher than one, we have (a) zero-order (constant) matrix effect, (b) first-order matrix effect, independent from the analyte amount within the dynamic range, and (c) matrix effect first order with respect to the analyte amount, but independent from the matrix/analyte ratio. Class a is eliminated when applying the TYB and will be considered no longer. Class b is a direct matrix effect; it appears, e.g., as an increase in the mass of the precipitate collected in the filter, due to silica and other insoluble matter, when determining sulfate ion by barium sulfate gravimetry. Another example is chromatographic peak overlapping (11). Effects of this kind are proportional to the matrix amount and they are not corrected by the TYB, thus invalidating the SAM extrapolation. Finally, class c, which may be designed as interactive matrix effect, invalidates the SAM and, accordingly, interferes with the analytical result, if the independence from the matrix/analyte ratio is not fulfilled, as pointed out by Tyson (12). We shall deal with a generalized approach for going from analytical method to analytical procedure. Misleading abstraction will be avoided, as far as possible, by applying the suggested approach to the sulfate ion quantitation in a calcium carbonate of reagent quality, with a claimed limit for sulfate ion of 0.02%. EXPERIMENTAL SECTI 0N Test Portion Curve. Samples containing 3, 6, 11, or 16 g, weighed to 0.001 g, of precipitated calcium carbonate, AR, Merck Art.2066, are dissolved in a light excess of 32% hydrochloricacid, AR, Merck Art. 319. The solution is evaporated to dryness, the residue taken with distilled deionized water, acidified with 12 mmol hydrochloric acid, and diluted to 200 mL. The resulting solution is brought to boiling and precipitated dropwise with 10 mL of hot, 10% (m/v), barium chloride dihydrate solution. The precipitated barium sulfate is gathered in a glass filter crucible (no. 4 pore size),filtered by suction, washed, dried at 105 "C, and weighed. Further details for the analytical method can be found elsewhere (13). Standard Addition Curve. Either 3 or 6 g of calcium carbonate is dissolved as indicated above and, before dilution to 200 mL, the solution is spiked with given volumes of 0.2082 mM sodium sulfate. After dilution to 200 mL the spiked solutions are subjected to the analytical method. Simplified Matrix Standard Curve. Given volumes of 0.2082 mM sodium sulfate are acidified with 12 mmol hydrochloric acid and diluted to 200 mL. The resulting solutions are subjected to the analytical method. Reagent grade chemicalsand distilled deionized water are used throughout. RESULTS AND DISCUSSION The pertinent data for test portion (Youden's one-sample regression), standard addition, and simplified matrix standard curves are in Table I. In all, there are 12 treatments, i.e., 12 sets of measurements, any two of which differing from each other in the calcium carbonate level, in the added standard sulfate ion level, or in both. Three replicate measurements are made at each treatment. The 12 treatments in Table I
Table I. Data for Test Portion, Standard Addition. and Simplified Matrix Standard Curves run no.
TP/g of CaC03
added std, SO-: massfmg concnf %" X,lmgb
variance est/wg2
3.000
0.71 0.61 0.78
7.3
4 5 6
6.000
1.40 1.35 1.24
6.7
I
11.000
2.39 2.53 2.31
12.4
10 11 12
16.000
3.44 3.60 3.39
12.0
13 14 15
3.000
1.96 1.91 1.85
3.0
1 2 3
8 9
16 17 18 19 20 21
6.000
22 23 24 25 26 21
0.000
0.500
0.0166
1.000
0.0333
3.04 3.20 3.34
22.5
0.500
0.0083
2.60 2.53 2.71
8.2
1.000
0.0166
3.88 3.72 3.88
8.5
0.02 0.01 -0.03
0.7
0.000
28 29 30
0.600
1.50 1.72 1.47
18.6
31 32 33
1.100
2.75 2.69 2.68
1.4
34 35 36
1.600
3.68 3.60 3.71
3.2
Expressed as parts of added sulfate ion per 100 parts of calcium carbonate. bAnalyticalsignal, mg of barium sulfate collected in the glass filter crucible. are needed to obtain four straight lines, by means of leastsquares fitting, according to the following design: test portion curve, runs 1-12; standard addition curve with TP equal to 3 g of calcium carbonate, runs 1-3 and 13-18; standard addition curve with TP equal to 6 g of calcium carbonate, runs 4-6 and 19-24; simplified matrix standard curve, runs 25-36. Outliers among Triplicates. The Grubbs' test (14) is applied with 5% significance level, critical value 1.155. No outlier is found. Cochran's Test for Variances. Cochran's maximum variance test (15) is applied at significance level 1% , critical value 0.8643 for four variances, and 0.9423 for three variances, before proceeding to the straight line fitting. The test statistic in the least favorable case for homoscedasticity is 18.6/(0.7 + 18.6 1.4 + 3.2) = 0.778 < 0.8643
+
Therefore, we go on by the ordinary least-squares fitting. Test for Lack-of-Fit to the Straight Line. As pointed out in the introduction, straight-line relationships are needed if we want to keep the model with just the constant and the proportional components of bias. Accordingly, a nonsignificant lack-of-fit of experimental points to the least-squares straight line is required. Least-squares fitting and lack-of-fit tests are jointly performed by way of an ANOVA table (16). As an
ANALYTICAL CHEMISTRY, VOL. 60, NO. 13, JULY 1, 1988
Table 11. ANOVA (Showing Lack-of-Fit) ( 1 6 ) for a Simplified Matrix Standard Curve source
SS
df
MS
F ratio
regression 1 22.2600 residual 10 0.1548 0.01548 = s2 lack of fit pure error total
2
8
0.1068 0.0534 0.0480 0.0060
8.90
> F:
(1%) = 8.65
11 22.4149
example, Table I1 shows the only case where a significant lack of fit is established. Besides the lack-of-fit statistic, LF, Table I11 shows a number of other statistics which will be needed later on. Constant Component of Bias. The test portion curve does not differ significantly from a straight line, as shown by the LF statistic, Table 111. The intercept on the ordinate and its 95% confidence interval are calculated in the usual way (I6),eq 1. The result is a = 0.05 f 0.12 mg of BaS04.
Therefore, the constant component of bias can be considered here as negligible, a t least for analytical signals between 0.6 and 3.6 mg of BaS04, and the blank measurement may be disregarded in routine work. The chemical blank, as defined in ref 7, is also measured because of its usefulness as a trouble-shooting diagnostic device (8). From runs 25-27, Table I, we have 0 f 0.07 (df = 2, a = 0.05). Analytical Result from the Standard Addition Method. Plotting X vs. m,the standard amount or mass added to the TP, the analyte amount, po, in TP is obtained as the intercept on the abscissa by extrapolating to X = 0 (3). The intercept is calculated according to eq 2, which is derived from eq 2.11 in ref 17, for situations where homoscedasticity can be assumed, as in the present case.
Xb,
bm2 - (t2s2/S,)
\ E'
. . .
n
From the SAM data we obtain, after extrapolation to X = = (0.280 f 0.068) mg of SO -: in TP = 3 g, and = (0.542 f 0.073) mg of S042-in TP = 6 g. From these amounts and
0,
the corresponding TP's, we arrive at the sulfate ion percentages in the analyzed calcium carbonate, 0.0093 f 0.0023 and 0.0090 f 0.0012. According to a t test there is no significant difference between these results. As a pooled percentage (18), we can take 0.0092 f 0.0018% (w/w) sulfate ion. Reliability of the SAM Analytical Results. Direct matrix interference cannot be detected from within the analytical procedure. This kind of interference requires for its detection the application of an analytical procedure based on another analytical method. On the other hand, in the interactive situation the SAM analytical results are bias-free, only if the proportional interactive effect does not change with the matrix/analyte ratio (12,19,20). This independence is evident from the observed linearity in the SAM plot (20), provided that the spike range is sufficiently large. An alternative way of showing the noneffect of the matrix/analyte ratio is to try another SAM plot at a different TP level. This is why we have done two series of SAM measurements, one at TP = 3 and the other at TP = 6 g of calcium carbonate. Since the SAM slope, b, in Table 111, is the same in both cases, we have sufficient grounds to accept the analytical result 0.0092% (w/w) sulfate ion in calicum carbonate, as free from interactive bias, because Tyson's requirement (12) is satisfied. The (Analytical Resu1t:Net Analytical Signal) Scale. It i obtained from the slope of the S A M plot, as a factor (l/b). The slope confidence interval (16)is (3) Thus we get 2.49 f 0.19 mg of BaS04 (mg of S042-)-' for TP = 3 g of calcium carbonate, and 2.49 f 0.16 for 6 g of calcium carbonate, df = 7, a = 0.05. These values agree with that expected from the stoichiometric model BaSO, ts/S,'J2
-
(137.34 + 96.063)/96.063 = 2.430 mg of BaS04 (mg of SO:-)-', so we may take the stoichiometricgravimetric factor (2.430)-' = 0.4116 mg of SO4 (mg of BaS04)-l as the (net analytical signal to the analytical result) scale or, in short, the scale (21). Nevertheless, agreement with the predictions of a stoichiometric model is irrelevant, as far as the bias-free nature of the SAM analytical result goes. Focused SAM Plot. Thus far, we have dealt only with the constant slope SAM plot (3), which gives the amount of analyte in the TP, as the intercept on the abscissa. There is also the focused SAM plot, Figure 3B in ref 8 with the proper axis labeling (IO). The focused SAM plot (9) gives directly
Table 111. Numerical Values of Some Statistics Used in This Paper curve standard addition to TP/g of CaC03 statistic
simplified matrix standard 12 8.90b 1.983 0.825
90 2.24c 2.644
1283
test portion
3
6
12
9
9
