Anal. Chem. 2004, 76, 1295-1301
Precision, Limit of Detection and Range of Quantitation in Competitive ELISA Yuzuru Hayashi,* Rieko Matsuda, and Tamio Maitani
National Institute of Health Sciences, 1-18-1 Kami-Yoga, Setagaya, Tokyo 158-8501, Japan Kazuhiro Imai
Kyoritsu College of Pharmacy, 1-5-30 Shibakoen, Minato, Tokyo 105-8512, Japan Waka Nishimura, Katsutoshi Ito, and Masako Maeda
School of Pharmaceutical Sciences, Showa University, 1-5-8 Hatanodai, Shinagawa, Tokyo 142-8555, Japan
This paper develops a mathematical model for describing the within-plate variation as the RSD (relative standard deviation) of absorbance measurements in a wide concentration range in competitive ELISA and proposes a method for determining the limit of detection (LOD) and range of quantitation (ROQ). The ELISA for 17r-hydroxyprogesterone is taken as an example. The theoretical RSD description involves analyte concentration as an independent variable and error sources as parameters which concern the pipetting and absorbance measurement. Our model can dispense with repeated experiments of real samples, but the error parameters should be determined experimentally. The theory is in good agreement with the experiments. The most influential error sources at low and high sample concentrations are shown to be the pipetting of a viscous solution of antiserum and the absorbance inherent to the wells of a plate, respectively. The LOD and ROQ are defined as the concentration with 30% RSD and the region with 5 µg/L). Maeda and her colleagues observed the similar within-assay precision (∼1-6% RSD) in bioluminescent enzyme immunoassays.4,5,7 The theoretical estimation of RSD (s) is based on eq 8, and not a least-squares fit to the experimental results (b). The parameters of eq 8 (variables other than X) can all be obtained with experiments, and no arbitrary constants are included (also see Table 1). The values used are: G ) 0.1 µg/L; FG ) FX ) 0.9% RSD (from repeated pipetting of the labeled antigen, n ) 24); FB ) 1.86% RSD (from repeated pipetting of the antiserum solution, n ) 24); FS ) 0.61 ) (2/3) × 0.92% RSD (from repeated pipetting of the labeled antigen, n ) 24); σW ) 0.002 absorbance (from the between-well SD of averaged measurements of empty wells, n ) 30); σN ) 0. For simplicity, the variability in the absorbance measurements of a fixed well is neglected by averaging the measurements (n ) 30). The edge effect was observed to be so small (∼1% RSD, data not shown) that it is not involved in eq 8. Substituting the above values for eq 8, we can obtain the actual form of the uncertainty equation,
FT2 )
X2 × 2 × (0.9)2 + (X + 0.1)2 (1.86)2 + (0.61)2 +
× 100) (0.002 Y
2
(12)
where the RSD is expressed as percentage (%). The measurement, Y, is replaced by the calibration curve,
Y)
C0 - C3 + C3 X C1 1+ C2
()
(13)
(also see Figure 2C and 2C′). The coefficients, C0, C1, C2, and C3, 1298
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are given by the least-squares fitting (see the legend of Figure 2). The theoretical lines in Figure 2A and 2A′ (s) are drawn on the basis of eqs 12 and 13. Figure 2B and 2B′ demonstrates the contribution of the individual errors to the total error of the competitive ELISA. At low concentrations, the major error source is the pipetting of the solution of rabbit anti-17-OHP antiserum (b). This solution is viscous, and its pipetting error is more critical than that for the solutions of analyte, labeled antigen, and chromogen substrate. A well of the microplate actually has an absorbance of its own, though very low, and the absorbance values of the empty wells are different from well to well in the detection system used. The absorbance inherent to the wells in a plate (2) is relatively important at higher concentrations than 5 µg/L. The other factors, pipetting of analyte and labeled antigen (O) and that of substrate (4), are of no significance over the entire region of concentration. The precision profiles of Figure 2A and 2A′ as well as the error contributions of Figure 2B and 2B′ result from the same ELISA experiments, except for the incubation times, 3 and 20 h. However, the incubation times are not included in the uncertainty equation (eqs 8 and 12), and all the values of the parameters, G, FG, FX, FB, FS, and σW, are common to both the long and short incubations. The uncertainty equation distinguishes the different profiles through the calibration curves (C and C′). At concentrations higher than 5 µg/L, the measurement RSD is much larger for the 3-h incubation (A′) than that for 20 h (A). In the 3-h incubation, the absorbance of analyte is low at the high concentrations (C′), and the relative importance of the absorbance of the wells in a plate is more enhanced (2 of B′). At concentrations around 0.01 µg/L, the analyte absorbance for the long incubation is ∼1.5 times that for the short incubation. Irrespective of this difference, the measurement RSD is indistinguishable for both the assays (A and A′). This is because the major error source at the low concentrations is not the absorbance measurement, but the pipetting error of the viscous antiserum solution. From the above discussion, we can obtain the approximate equation of uncertainty for the competitive ELISA,
FT2 ) FB2 +
( ) σW Y
2
) (1.86)2 +
(
0.002 × 100 Y
)
2
(14)
This equation comprises the major error sources only, that is, pipetting of the viscous solution and absorbance of the wells. The uncertainty equation neglects the blank measurements. Therefore, the RSD of measurements, FT, should be calculated from blank-subtracted measurements. This rule holds true for the calibration curve (eq 13). Figure 2D and 2D′ shows the precision profiles for concentration estimates in the ELISA systems with 20- (A) and 3-h (B) incubations. The theoretical RSD (s) comes from the experiments which are independent of the results (b). The precision profiles are open u-shaped and quite similar to ones shown in the literature.12 The theory and practice are in good agreement. The actual RSD values at the edges of the concentration range (b) in Figure 2D and 2D′ seem to be apart from the theoretical (32) Hayashi, Y.; Matsuda, R. Anal. Sci. 1994, 10, 881-88.
Figure 2. Precision plot for measurements (A, A′, D, and D′), contribution of errors (B and B′) and calibration curve (C and C′) in 20-h incubation (A, B, C, and D) and 3 h incubation (A′, B′, C′, and D′). The precision profiles of A and A′ are different from those of D and D′ in the scale of the Y axis. A and A′: b, the experimental RSD of absorbance measurements (n ) 8); s, the theoretical RSD from eq 12. B and B′: Y axis is the relative contribution of error: (the individual term of the right side of eq 12)/(the left side of eq 12); O, first term, [X2/(X + G)2][FG2 + FX2]; b, second term, FB2; 4, third term, FS2; 2, forth term, (σW/f(X))2. C and C′: b, averaged measurements (n ) 8); s, the fitted curve (eq 13). C: C0 ) 1.25; C1 ) 8.81 × 10-1; C2 ) 1.11 × 10-1; C3 ) 3.61 × 10-2. C′: C0 ) 7.97 × 10-1; C1 ) 1.57; C2 ) 2.02 × 10-1; C3 ) 8.0 × 10-3. D and D′: b, experimental RSD of concentration estimates (n ) 8); s, theoretical RSD of concentration estimates.
estimation (s). However, the profiles are drawn on the semilogarithmic scale, and the agreement is better than it looks at low concentrations, but not at high concentrations. The causes of the high RSD values at the edges of the calibration curves can be identified by the numerical examination of the stepwise conversion from the measurement SD to the RSD of concentration estimates. At high concentrations around 10 µg/ L, the major error source is the extremely gentle slope, f ′, of the calibration curve. On the other hand, the small values of
concentration lead to the high RSD values at low concentrations. The actual slope, dY/dX, of the calibration curve on the natural scale, which is used for the conversion, is steeper in the low concentration region than it looks on the semilogarithmic scale of Figure 2C and 2C′. Figure 3A and 3A′ shows the method for determining the LOD and ROQ. The precision profiles (s) are the same as those of Figure 2D and 2D′, respectively. As described in the theoretical section, the LOD and ROQ are characterized by the RSD values Analytical Chemistry, Vol. 76, No. 5, March 1, 2004
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Figure 3. Detection limit and quantitation range (A and A′) and the corresponding B/B0 values (B and B′) for 20- (A and B) and 3-h incubation (A′ and B′) times. A and A′: arrow D, the detection limit; arrow Q, the lower and upper limits of quantitation range. B and B′: the X positions of the vertical arrows are the same as those in A and A′, respectively.
of 30 and 10% and are spotted in the precision profiles for the long (A) and short (A′) incubation times. The LOD for the long incubation is lower () 0.007 µg/L) than for the short incubation () 0.028 µg/L). The ROQ for the former is from 0.029 to 7.0 µg/L, wider than the ROQ for the latter from 0.061 to 2.6 µg/L. The LOD for the short incubation () 0.028 µg/L) is almost equal to the lower limit of ROQ for the long incubation () 0.029 µg/L). Obviously, the competitive ELISA system with the long incubation time is superior in precision to that with the short incubation time. Figure 3B and 3B′ shows the relationship between our approach and the well-known plot, B/B0 () (Y - C3)/(C0 - C3)). The LOD concentrations determined in Figure 3A and 3A′ both correspond to ∼0.95 in the B/B0 plot, irrespective of the different calibration curves. The plot, B/B0, is often used for the limit of detection, and the above result demonstrates the validity of the plot, though not rejecting the coincidence. The lower limit of ROQ gives the different B/B0 values () 0.76 and 0.87), but the upper limit of ROQ leads to almost the same value () 0.3). CONCLUSION This paper presents a mathematical model for estimating the RSD of absorbance measurements in the competitive ELISA for 17R-hydroxyprogesterone. The exact equation (eq 8) and simplified equation (eq 14) of uncertainty are proposed. If the RSD of pipetted volumes of the viscous solution of antiserum and SD of the absorbances of the wells in a plate are known, we can predict the precision from eq 14 over a wide concentration range. The repetition using real samples is not necessary in this approach. Although the uncertainty structure of ELISA as shown in Figure 2Band 2B′ is different from assay to assay, the principle of the uncertainty estimation (eq 8) will be common to most, if 1300
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not all, competitive ELISAs. Some error sources (e.g., scattering of incubation time, etc.) are neglected in the uncertainty equation. However, if some error components are found to be serious in competitive immunoassays, our approach can integrate them to improve the reliability of the uncertainty estimation. This paper also presents a proposal for the limit of detection and range of quantitation in competitive ELISA. Our proposal retains the probabilistic aspect of the original definitions. That is, the limit of detection is characterized with 30% RSD, and the range of quantitation is
