Selecting the Correct Weighting Factors for Linear and Quadratic

Aug 26, 2014 - weighting factors for linear and quadratic calibration curves with least-squares regression algorithm in bioanalytical LC-MS/MS assays...
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Selecting the Correct Weighting Factors for Linear and Quadratic Calibration Curves with Least-Squares Regression Algorithm in Bioanalytical LC-MS/MS Assays and Impacts of Using Incorrect Weighting Factors on Curve Stability, Data Quality, and Assay Performance Huidong Gu,* Guowen Liu, Jian Wang, Anne-Françoise Aubry, and Mark E. Arnold Bioanalytical Sciences, Research & Development, Bristol-Myers Squibb Company, Route 206 & Province Line Road, Princeton, New Jersey 08543, United States S Supporting Information *

ABSTRACT: A simple procedure for selecting the correct weighting factors for linear and quadratic calibration curves with least-squares regression algorithm in bioanalytical LC-MS/MS assays is reported. The correct weighting factor is determined by the relationship between the standard deviation of instrument responses (σ) and the concentrations (x). The weighting factor of 1, 1/x, or 1/x2 should be selected if, over the entire concentration range, σ is a constant, σ2 is proportional to x, or σ is proportional to x, respectively. For the first time, we demonstrated with detailed scientific reasoning, solid historical data, and convincing justification that 1/x2 should always be used as the weighting factor for all bioanalytical LC-MS/MS assays. The impacts of using incorrect weighting factors on curve stability, data quality, and assay performance were thoroughly investigated. It was found that the most stable curve could be obtained when the correct weighting factor was used, whereas other curves using incorrect weighting factors were unstable. It was also found that there was a very insignificant impact on the concentrations reported with calibration curves using incorrect weighting factors as the concentrations were always reported with the passing curves which actually overlapped with or were very close to the curves using the correct weighting factor. However, the use of incorrect weighting factors did impact the assay performance significantly. Finally, the difference between the weighting factors of 1/x2 and 1/y2 was discussed. All of the findings can be generalized and applied into other quantitative analysis techniques using calibration curves with weighted least-squares regression algorithm.

T

nonweighted or weighted least-squares regression algorithm. Although the least-squares regression and curve weighting are well-established statistics techniques, the “Test and Fit” strategy is still widely used for selection of calibration curves and their weighting factors in the bioanalytical LC-MS/MS community because of its simplicity and lack of statistics data for the regression model and weighting selection. With the “Test and Fit” strategy, an incorrect weighting factor can be easily selected because this strategy is based on the analyst’s subjective interpretation of a limited set of test results. Besides the “Test and Fit” strategy, several methods for selecting regression model have been reported. Kiser and Dolan proposed a systemic way to test and identify the best fit curve.8 By applying regression with different weighting factor (1/x0, 1/x0.5, 1/x, 1/x2, and 1/x3) on a set of STD data, the suitable weighting

he importance of establishing an accurate calibration curve in quantitative analysis has been addressed by many research groups.1−5 A well-established and interpreted calibration curve is essential for the assay performance and data quality,1 especially for bioanalytical liquid chromatography-tandem mass spectrometry (LC-MS/MS) assays used in drug development, where the FDA guidance6 and EMA guideline7 on “Bioanalytical Method Validation” must be strictly followed. The FDA guidance states very clearly regarding the establishment of a calibration curve as the following: “The simplest model that adequately describes the concentration-response relationship should be used and selection of weighting and use of a complex regression equation should be justified”. Furthermore, selection of a regression model correctly during the method development and validation is critical for the smooth assay transfer from method validation to sample analysis, from analyst to analyst, and from laboratory to laboratory. Two most commonly used regression models for LC-MS/ MS calibration curves are linear and quadratic regressions using © 2014 American Chemical Society

Received: February 17, 2014 Accepted: August 26, 2014 Published: August 26, 2014 8959

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scheme could be easily identified by choosing the one generating the smallest sum of the relative errors. The whole process can be programmed in a spreadsheet and takes only seconds for the selection. However, this simply represents an automated version of the “Test and Fit” approach. Almeida et al.1 suggested that a weighting should only be used if homoscedasticity was not met for the analytical data. Recently, Steliopoulos et al.9 used several different functions to model the dispersion of measurements with the concentration and then to estimate the weights needed for the regression. By using Monte Carlo calculations, Tellinghuisen investigated the effect of improper weighting on linear calibration models.2 It was found that for analytical data with heteroscedastic proportional error, neglecting the weightings could result in as high as 1 order of magnitude of precision loss in the low concentration region. Another significant finding from this study was that the fitness of calibration curve should not be used as the criteria in selecting regression model and weighting factor because of the “self-fulfilling” nature of the weighted least-squares regression algorithm. Many research works recommended that 1/x2 weighting should be used for bioanalytical LC-MS/MS assays; however, the recommendations were largely based on the “Test and Fit” strategy.10−12 More recently, Hoffman introduced a quantitative procedure for selecting weighting factors in bioanalytical LC-MS/MS assays without using “Test and Fit”.13 It was mentioned that this procedure had been adopted by some bioanalytical laboratories in selecting weighting factors for many years. In this paper, we examined the justifications for using a weighting factor based on the experimental data and proposed a simple procedure for selecting the correct weighting factors for linear and quadratic calibration curves with least-squares regression algorithm in bioanalytical LC-MS/MS assays. Using this simple procedure on historical data, we demonstrated that 1/x2 should be the only right weighting factor for LC-MS/MS assays. The impacts of using incorrect weighting factors on the curve stability, data quality, and assay performance were investigated. The difference between the weighting of 1/x2 and 1/y2 was also discussed. Several critical findings from this investigation can be generalized and applied in other quantitative analysis techniques using calibration curves with least-squares regression algorithm.

Figure 1. (a) Instrument responses of 16 STD curves in eight analytical runs for BMS-708163 in human plasma LC-MS/MS assay validation.16 (b) Standard deviation (σ) and variance (σ2) for instrument responses of 16 STD curves in eight analytical runs for BMS-708163 in human plasma LC-MS/MS assay validation (STD curve range: 0.1−100 ng/mL).16

deviations (σ, blue line) and variance (σ2, red line) for instrument responses at each concentration level are shown in Figure 1b. It was found that, from 0.1 to 100 ng/mL, σ increased in an approximately concentration-proportional manner, and σ2 increased in a much more than concentration proportional manner. To address the heteroscedatic errors, a modified least-squares algorithm, weighted least-squares, is warranted. The idea is to assign an appropriate weight at each concentration level which reflects the different measurement uncertainties at different concentration levels.17 When the measurements were uncorrelated and had different uncertainties, Aitken demonstrated that the best unbiased estimates could be obtained if each weight used was equal to the reciprocal of the variance of the measurement.18,19 This is easy to understand because the importance of each squared residual at all concentration levels are normalized/equalized with the reciprocal of the variance at the corresponding concentration level.18,20,21 (Please see Supporting Information A for more detailed discussion on linear and quadratic regression with least-squares algorithm.) Weight Estimation for LC-MS/MS Bioanalytical Assay Calibration Curves Using Linear or Quadratic Regression with Weighted Least-Squares Algorithm. For the estimation of weights for calibration curves in LC-MS/MS bioanalysis, it is impractical to directly use 1/σi2 at each concentration level for each individual run because the data set is normally too small to estimate the σi, and the weights could vary from run to run. The most common practice is to find an

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RESULTS AND DISCUSSION The assumptions for the nonweighted linear and quadratic regressions in bioanalytical LC-MS/MS assays are that the errors from the dependent variable (y, instrument response) are independent from each other (random), uncorrelated with the independent variable (x, concentration), and have equal variance (σ2) or standard deviation (σ) across the different concentration levels (homoscedasticity14). However, a quick review of any set of real life standard (STD) or quality control (QC) data from assay validations or sample analysis will reveal that the instrument response errors at different concentration levels are actually correlated with the independent variable (concentration) and have a larger variance (σ2) or standard deviation (σ) at higher concentrations as long as the sample size at each concentration level is large enough to provide a rough estimate of variance. This is a clear indication that the instrument responses have heteroscedatic error.15 Figure 1a shows the instrument responses of 16 calibration curves in eight analytical runs for a test compound (BMS-708163) in human plasma LC-MS/MS assay validation.16 The standard 8960

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Table 1. Theoretical Data for x and σ To Justify the Selection of 1, 1/x, or 1/x2 as the Weighting Factor weighting factor 1

1/x

1/x2

concn x (arbitrary units)

1

2.5

5

10

50

100

250

350

500

RSD%

σ/x0 σ/x0.5 σ/x1 σ/x0 σ/x0.5 σ/x1 σ/x0 σ/x0.5 σ/x1

0.5000 0.0050 0.5000 0.0354 0.0354 0.0354 0.0050 0.0500 0.0050

0.5000 0.0079 0.2000 0.0559 0.0354 0.0224 0.0125 0.0316 0.0050

0.5000 0.0112 0.1000 0.0791 0.0354 0.0158 0.0250 0.0224 0.0050

0.5000 0.0158 0.0500 0.1118 0.0354 0.0112 0.0500 0.0158 0.0050

0.5000 0.0354 0.0100 0.2500 0.0354 0.0050 0.2500 0.0071 0.0050

0.5000 0.0500 0.0050 0.3536 0.0354 0.0035 0.5000 0.0050 0.0050

0.5000 0.0791 0.0020 0.5590 0.0354 0.0022 1.2500 0.0032 0.0050

0.5000 0.0935 0.0014 0.6614 0.0354 0.0019 1.7500 0.0027 0.0050

0.5000 0.1118 0.0010 0.7906 0.0354 0.0016 2.5000 0.0022 0.0050

0 88.8 171.1 88.8 0 106.0 130.1 106.0 0

empirical function between weights (1/σ2) and x or y to approximate the variance with x or y.22,23 Fortunately, the empirical function is likely predictable for data arising from these assays and normally can be expressed in a simple power relationship as σ = cyn (or σ = cxn, where c and n are constants).24,25 Therefore, the weighting factor of 1/y2n (or 1/x2n) can be used as the weighting factor. As 1, 1/x, and 1/x2 are the most commonly used empirical functions for weighting factor in LC-MS/MS assays, they were used in the initial investigation, and the use of 1/y and 1/y2 were discussed in the last part of this paper. Selection of Weighting Factor Using Plots of σ ∼ x and σ2 ∼ xA Qualitative Approach. The theoretical relationships between σ and x, and between σ2 and x to justify the selection of 1, 1/x, and 1/x2 weighting factors are shown in Supporting Information B, Figure SI 1a−1c, respectively, and the corresponding data are listed in Table 1.18,19 As shown in Figure SI 1a−1c, the weighting factor of 1 should be used if a linear relationship between σ and x0 is observed (σ and σ2 are constants). The weighting factor of 1/x should be used if a linear relationship between σ and x0.5 (or between σ2 and x1) and a rightward quadratic relationship between σ and x1 are observed. Similarly, the weighting factor of 1/x2 should be used if a linear relationship between σ and x1 and a concave upward quadratic relationship between σ2 and x are observed. In regulated bioanalytical LC-MS/MS analysis supporting nonclinical and clinical studies, a method has to be validated before it can be used in sample analysis. A full method validation consists of at least three accuracy and precision runs, which can provide sufficient instrument response data from STD or QC samples for the calculation of σ. Figure 1b and Figure 2a−c16,26,27 show the relationships between σ and x (blue line), and between σ2 and x (red line) from STD curve instrument responses from four assay validations. A quick eyeball of these plots suggested that 1/x2 should be used as the weighting factor for 3 out of 4 assays listed, as very good straight line relationships between σ and x and concave upward quadratic relationships between σ2 and x were found over their corresponding STD ranges. For the last assay, in most of the STD curve range (0.1 to 80 ng/mL), the concave upward quadratic curvature between σ2 and x was not obvious, and it looked like that the relationships between σ and x, and between σ2 and x followed the same linear trend. Actually, this was due to σ being very close to 1 at 80 ng/mL so that the difference between σ and σ2 over the concentration range of 0.1 to 80 ng/ mL was small and hard to discern graphically. However, over the entire STD curve range, better linearity was found for the relationship between σ and x. Therefore, 1/x2 should also be used as the weighting factor for that assay.

Selection of Weighting Factor Using Linearity IndicatorsA Quantitative Approach. In some cases, such as the last assay discussed above, the selection of a relationship with better linearity may not be obvious using the plots. An alternative, less empirical approach using three linearity indicators is proposed for the selection of weighting factor. The linearity indicators were defined as the relative standard deviation (RSD%) of σ/x0, RSD% of σ/x0.5, and RSD % of σ/x1, respectively. It is easy to see that, to justify the use of 1, 1/x or 1/x2 as the weighting factor, the RSD% of σ/x0, σ/x0.5, or σ/x1 should be 0% as an exact linear relationship exists between σ and x0, σ and x0.5, or σ and x, respectively, as shown in Table 1. Therefore, in evaluating a real STD or QC data set, the best linear relationship can be assumed to be the one that gives the smallest linearity indicator. The linearity indicators were calculated for all of the four assays discussed above. The results (Table 2) confirmed the selection of 1/x2 as the weighting factor for all of the four assays. More than 20 LC-MS/MS assays developed in our laboratories over a period of 6 years have been surveyed using both the qualitative and quantitative approaches. The assays covered small molecule and protein analysis with different types of sample preparation and extraction techniques, using different MS platforms for detection. The survey result showed that 1/x2 was the only correct weighting factor for all of the assays without any exception. This result is in agreement with most bioanalysts’ “experience” that 1/x2 is the most commonly used weighting, which have probably been built up using the “Test and Fit” technique based on the overall accuracy of the STD or/and QC data. The selection of 1/x2 for all of the assays was also confirmed by weighted residual plots as the weighted residuals are randomly distributed over the entire concentration range.23 Another study we conducted on uncertainty from LC-MS/MS measurement by calculating the propagation of uncertainties from preparation of STD samples to MS detection showed that a linear relationship actually existed between σ and x, and therefore, the weighting factor 1/x2 should be used for all of LC-MS/MS assays (to be published elsewhere).28 Any issues for bioanalytical assays (e.g., nonspecific adsorption, cross contamination, systematic bias, failed QC due to preparation/storage/matrix difference between STD and QC samples, etc.) may obscure this fact. However, people should not manipulate the weighting factor to deal with these issues for the sake of passing an analytical run. All these issues should be resolved during assay development. Impacts of Using Incorrect Weighting Factors on Curve Stability, Data Quality, and Assay Performance. In most cases, 1/x weighting and in some rare cases, even no weighting, can also generate very good STD curves and QC data, although the correct weighting factor should be 1/x2. 8961

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Figure 2. (a) Standard deviation (σ) and variance (σ2) for instrument responses of 10 STD curves in five analytical runs for BMS-512148 in dog plasma LC-MS/MS assay validation (STD curve range: 50−2000 ng/mL). (b) Standard deviation (σ) and variance (σ2) for instrument responses of 14 STD curves in seven analytical runs for a PEGylated adnectin protein in monkey plasma LC-MS/MS assay validation (STD curve range: 50−25 000 ng/mL).27 (c) Standard deviation (σ) and variance (σ2) for instrument responses of eight STD curves in four analytical runs for BMS-708163 in human CSF LC-MS/MS assay validation (STD curve range: 0.1−100 ng/mL).16

factors of 1, 1/x, and 1/x2 for each run using the original data in Table 3a. Table 3c shows the regression results using the original data with a couple of assumed instrument response biases to test the curve stability. The quantitative comparisons

Here we show an example with STD data (Table 3a) from BMS-708163 in mouse plasma validation with three accuracy and precision runs, and the correct weighting factor is 1/x2. Table 3b shows the linear regression results with weighting 8962

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consequence was that the STD sample with −18.4% bias became an outlier (−18.8% deviation) and had to be excluded from the STD curve, and this was exactly what should happen. However, when the weighting factor of 1/x was used, the same instrument bias caused −15.0% (at the STD curve low end) and 4.0% (at the STD curve high end) changes in the predicted concentrations. Therefore, the low STD sample with −23.2% deviation had to be excluded and the regression had to be performed again. As a result, an obvious instrument response bias at the STD high end failed a total of six STD samples at the STD low end one by one, and eventually failed the run. More substantial curve change can be observed with only −1.5% bias from one original instrument response for a curve without weighting when 1/x2 weighting is actually needed, as shown in Tables 3b−3d, run 3. It should be pointed out that different spacing of the STD calibrators can influence the curve stability differently only when a wrong weighting factor is used. When the correct weighting factor of 1/x2 is used, the curve is always stable regardless the calibrator spacing. (Please see Supporting Information C for the detailed discussion.) Data Quality. With the three established curves (using the weighting factors of 1, 1/x, and 1/x2) for each individual run, the QC concentrations were calculated, and QC data for run 3 are shown in Supporting Information D, Table SI 2a−2c. The results showed that “good” STD curves generated “good” QC data and “bad” STD curves generated “bad” QC data no matter which weighting factor was used for the STD curves. As discussed above, any STD curves generated using incorrect weighting factors are not stable. Good accuracy could sometimes also be achieved in cases when those curves were “luckily” overlapped with or very close to the corresponding STD curves generated with the correct weighting factors (Table 3d). Therefore, a STD curve with good accuracy or even both the STD curve and QC data with good accuracy does not necessarily mean that a correct weighting factor has been used, and this should not be used as the criteria in selecting of weighting factor. Fortunately, even with the incorrect weighting factors, there was no or very insignificant impact on the concentration data reported because the concentrations were always reported with passing curves which actually overlapped with or were very close to the curves using the correct weighting factor, as shown in Table 3d. One thing worth mentioning is that, in some cases, “bad” QC data could also be generated with a “good” STD curve using a correct weighting factor. This is almost always caused by some differences between QC and STD samples, such as the differences between storage conditions, preparation procedures, and matrix for STD and QC samples, and therefore, QC performance should not be used in the selection of weighting factor. Assay Performance. However, the use of incorrect weighting factors did impact the assay performance. A survey

Table 2. Results of Linearity Indicator (RSD% of σ, σ/x0.5, and σ/x) and Selection of Weighting Factor assay Figure 1b Figure 2a Figure 2b Figure 2c (STD curve range: 0.1 to 100 ng/mL) Figure 2c (STD curve range: 0.1 to 80 ng/mL)

RSD% of σ/x0

RSD% of σ/x0.5

RSD% of σ/x1

weighting factor selected

141.2 104.2 146.6 144.7

108.4 60.2 104.3 104.9

19.1 7.5 26.1 26.3

1/x2 1/x2 1/x2 1/x2

147.5

98.8

28.0

1/x2

of curve stability for run 1 and 3 are shown in Table 3d. Regulatory guidelines (e.g., FDA and EMA bioanalytical guidance) were followed to discuss the data quality, assay performance, and set the assay acceptance criteria. For nonLLOQ STD and QC samples, ≤ ± 15.0% was used. For LLOQ STD and QC samples, ≤ ± 20.0% was used.6,7 Curve Stability. Curve stability was defined as the resistance of a STD curve against significant errors from one or more STD samples. As the correct weighting factor for this assay was 1/x2, the influences of the errors at different concentration levels on the calculation of the best fitting curve were not taken into account at all when no weighting was applied. Therefore, the STD curves were extremely unstable, and a small bias at the high end of the STD curves could change the STD curves significantly, especially at the low end of the STD curves, as shown in Tables 3b, 3c, and 3d (run 3 with no weighting). By using 1/x as the weighting factor for the assay, the influences of the errors at different concentration levels on the calculation of the best fitting curve were taken into account to some degree, but it was still not enough to equalize the importance of the errors at different concentration levels. As a result, the instability of the curve could only be observed clearly when there were significant biases at the high end of the STD curve, as shown in Table 3b, 3c (run 1 and 2 with 1/x weighting) and 3d (run 1 with 1/x weighting). When the correct weighting factor 1/x2 was used, the most stable curves were generated. Any big biases from one or a couple of STD samples at any concentration levels affected only those STD samples with big biases, and had no or very little impact on the curves, as shown in Table 3b and 3c (run 1 and 2 with 1/x2 weighting) and 3d (run 1 with 1/x2 weighting). This can be easily understood as a correct STD curve is “voted” equally by all of the STD samples in a run (16 STD samples in our examples) when a correct weighting factor is used. Therefore, it should not be sensitive (or it should be stable) to any biases from only one or a couple of STD samples. For example, as shown in Tables 3b−3d, when the correct weighting factor of 1/x2 was used for run 1, only −0.8% (at the STD curve low end) and 1.7% (at the STD curve high end) predicted concentration changes were observed with a −18.4% bias from one original instrument response. The only

Table 3a. STD Instrument Responses for BMS-708163 in Mouse Plasma LC-MS/MS Assay Validation concn (ng/mL)

0.2

0.4

0.8

4

20

100

160

200

run 1

0.045576 0.048468 0.052042 0.056547 0.051634 0.054632

0.090377 0.097604 0.096329 0.104977 0.095067 0.098076

0.179375 0.181369 0.187270 0.197225 0.190177 0.190476

0.886359 0.905368 0.864844 0.920856 0.891820 0.926106

4.312755 4.477241 4.360585 4.642914 4.376717 4.703780

20.806649 21.436947 21.658589 23.277822 22.056011 22.764870

32.974830 34.527598 33.931792 36.537529 33.947702 36.046021

41.307287 42.261225 41.885419 46.105747 43.279825 46.925214

run 2 run 3

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Table 3b. Linear Regression Results for STD Curves in BMS-708163 in Mouse Plasma LC-MS/MS Assay Validation (Original Data) run 1 (Dev%a)

run 2 (Dev%a)

weighting factor

a

run 3 (Dev%a)

weighting factor

weighting factor

concn (ng/mL)

1

1/x

1/x2

1

1/x

1/x2

1

1/x

1/x2

0.2 0.2 0.4 0.4 0.8 0.8 4 4 20 20 100 100 160 160 200 200

−162.8 −155.9 −78.0 −69.3 −35.9 −34.7 −2.8 −0.5 1.2 5.1 −1.0 2.0 −1.8 2.8 −1.6 0.7

−14.3 −7.4 −3.9 4.6 0.9 2.0 4.1 6.4 2.2 6.1 −1.2 1.8 −2.1 2.5 −1.9 0.3

−5.6 1.1 −0.9 7.5 1.1 2.3 2.1 4.3 −0.2 3.6 −3.6 −0.7 −4.5 0.0 −4.3 −2.1

−125.5 −115.3 −62.5 −52.7 −29.6 −24.0 −9.0 −2.7 −2.5 3.9 −2.0 5.4 −3.9 3.5 −5.1 4.5

−9.1 1.1 −4.5 5.3 −0.8 4.8 −3.6 2.8 −1.7 4.7 −2.1 5.2 −4.1 3.2 −5.3 4.2

−6.2 3.9 −3.5 6.2 −0.7 4.8 −4.2 2.1 −2.5 3.9 −2.9 4.3 −4.9 2.4 −6.1 3.4

2.2 8.9 −0.2 3.1 3.2 3.4 −0.7 3.1 −2.0 5.3 −1.1 2.1 −4.9 1.0 −3.0 5.2

−5.0 1.8 −3.8 −0.4 1.4 1.6 −1.0 2.8 −2.1 5.3 −1.1 2.1 −4.9 1.0 −3.0 5.2

−3.1 3.6 −3.1 0.3 1.5 1.6 −1.5 2.4 −2.6 4.7 −1.7 1.5 −5.4 0.5 −3.5 4.6

Results expressed in percentage deviation from nominal concentrations.

Table 3c. Curve stability test using STD curves in BMS-708163 in mouse plasma LC-MS/MS assay validation

concn (ng/mL) 0.2 0.2 0.4 0.4 0.8 0.8 4 4 20 20 100 100 160 160 200 200 curve stability

run 1 (Dev%a)

run 2 (Dev%a)

run 3 (Dev%a)

weighting factor

weighting factor

weighting factor

1

1/x

1/x2

1

1/x

1/x2

1

1/x

1/x2

−720.9 −713.6 −353.8 −344.7 −170.7 −169.4 −24.8 −22.4 1.6 5.8 3.7 6.9 3.2 8.1 3.6 −13.6 (34.5) extremely unstable

−23.2 −16.0 −6.3 2.7 1.8 3.0 7.7 10.0 6.2 10.2 2.7 5.8 1.8 6.5 2.0 −14.8 (34.5) unstable

−6.3 0.5 −0.4 8.1 2.2 3.4 3.7 5.9 1.4 5.3 −2.0 0.9 −2.9 1.6 −2.7 −18.8 (34.5) stable

−288.6 −277.8 −141.1 −130.7 −65.8 −59.9 −11.7 −5.0 1.7 8.5 −6.8 (19.5) 11.3 −10.2 (30.0) 9.5 −11.3 (37.0) 10.6 extremely unstable

−18.9 −8.1 −6.6 3.7 0.9 6.8 0.9 7.6 3.5 10.3 −7.1 (19.5) 10.9 −10.7 (30.0) 8.8 −11.8 (37.0) 9.9 unstable

−7.5 2.9 −2.6 7.4 1.2 6.9 −1.6 4.9 0.4 6.9 −10.0 (19.5) 7.4 −13.5 (30.0) 5.4 −14.6 (37.0) 6.4 stable

−43.5 −36.8 −22.8 −19.5 −7.9 −7.7 −2.5 1.3 −2.0 5.4 −0.7 2.5 −4.5 1.4 −2.6 4.0 (46.2) extremely unstable

−5.7 1.1 −4.0 −0.6 1.5 1.7 −0.8 3.1 −1.7 5.6 −0.8 2.4 −4.5 1.4 −2.6 3.9 (46.2) unstableb

−3.1 3.6 −3.1 0.3 1.6 1.7 −1.3 2.5 −2.4 4.9 −1.5 1.7 −5.2 0.6 −3.4 3.2 (46.2) stable

a

Results expressed in percentage deviation from nominal concentrations. bAlthough the curve with 1/x weighting is unstable, the instability is almost invisible with only one small instrument response bias (∼1.5%) from one STD sample. In this case, the small instrument response bias was used to test the curve stability for the curve without weighting. The curve instability for curves with 1/x weighting can only be detected with one or a couple of big instrument response biases, such as the examples shown in the table for run 1 and 2. Note: instrument response biases listed in parentheses, please see Table 3a for the original instrument responses.

was conducted on more than 200 successful LC-MS/MS sample analysis runs, and the correct weighting factor is 1/x2 for all of the runs. By using the acceptance criteria for regulated GLP bioanalytical assays,6 about 80% and 5−10% of the runs would have been failed runs if the wrong weighting factors of 1 and 1/x, respectively, were used. This will eventually result in the significant cost increase for bioanalytical LC-MS/MS assay validation, assay transfer, and sample analysis. The curve stability and assay performance drop quickly while the weighting factor used moving away from the correct weighting

factor. This is probably the reason that, although most likely, the correct weighting factor should be 1/x2, 1/x is still being used for quite a few assays as it is hardly to be noticed with “Test and Fit” (more than 90% of curves still pass), while the weighting factor of 1 is rarely used as the 80% of curve failure rate is easily noticeable. Using Curve Stability Test in Selecting of Correct Weighting Factor. The stability of a linear STD curve to any random variations from only one or a couple of STD samples is a quick and reliable indicator to check if a correct weighting 8964

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Table 3d. Quantative comparison of curve stability using STD curves in BMS-708163 in mouse plasma LC-MS/MS assay validation original passing curve with 1/x2 weighting (y = 0.215830x + 0.004819)

run 1

% difference for the predicted concentration at STD low end original passing curve, 1/x weighting (y = 0.210535x + 0.009489) passing curve with one assumed instrument response biasc, 1/x2 weighting (y = 0.212305x + 0.005788) failed curve with one assumed instrument response biasc, 1/x weighting (y = 0.202458x + 0.014473)

% difference for the predicted concentration at STD high end

−9.2%a −0.8%a

2.5%a 1.7%a

−22.0%a (−15.0%)b

6.6%a (4.0%)b original passing curve with 1/x2 weighting (y = 0.224190x+0.008166)

run 3

original passing curve with no weighting (y = 0.223013x+0.006071) passing curve with one assumed instrument response biasc, 1/x2 weighting (y = 0.223861x+0.008257) failed curve with one assumed instrument response biasc, no weighting, (y = 0.221928x + 0.026574)

% difference for the predicted concentration at STD low end

% difference for the predicted concentration at STD high end

5.4%a −0.1%a −41.8%a (−43.8%)b

0.5%a 0.1%a 1.0%a (0.5%)b

a

% differences for the predicted concentration comparing to the original passing curve with the weighting factor of 1/x2. b% differences for the predicted concentration comparing to the original curve with the same weighting factor of 1 or 1/x. cPlease see Table 3a and 3c for the assumed instrument response bias.

considering the x weighting is widely adopted for bioanalytical LC-MS/MS assays and is built in most of the commercial software, our recommendation is that 1/x2 should be used for all bioanalytical LC-MS/MS assays. For other nonlinear curves using least-squares regression algorithm, such as 4PL and 5PL curves in LBA assays, 1/y2 should be used. One exception is when quadratic curves or other nonlinear curves are severely bended, and the linear relationship between σ and y does not hold anymore, the correct weighting factor should be selected using the proposed qualitative or quantitative approach. (Please see Supporting Information E for more detailed discussion.)

factor is used even with just one STD curve, where the calculation of σ is impossible. Among all of the STD curves generated from a set of STD data within a single run using different weighting factors, such as 1, 1/x, 1/x2, 1/x3, and so forth, the curve with the best stability can be easily identified with a single instrument response bias (>20% bias away from the original curve) at the STD high end for over weighting at the high end, such as use of 1, 1/x for the curve with the correct weighting factor of 1/x2, as shown in Table 3c, or at the STD low end for over weighting at the low end, such as use of 1/x3 for the curve with the correct weighting factor of 1/x2 (data not shown). As there is one more degree of freedom in determining quadratic curves, the curve stability test is less sensitive for quadratic curves. Therefore, the test should be applied on the linear curves only. Weighting Factor Using 1/x2 or 1/y2 and Boundary of the Application. As we have discussed, the weighting factor was used to normalize the importance of the instrument response errors in the estimation of the best-fit curve. The instrument response errors are related to y directly, instead of x. Therefore, an empirical function between 1/σ2 and y can be established using either the plot or the linearity indicator approach. In most cases, the effects of using the weighting factor of 1/x2 or 1/y2 (1/x or 1/y) are the same for bioanalytical LC-MS/MS assays because only linear or quadratic curves with very mild curvature (close to linear) are used. Therefore, the empirical functions between 1/σ2 and y can be exactly (for linear curves) or approximately (for quadratic curves with very mild curvature) translated to the same empirical functions between 1/σ2 and x. However, for quadratic curves with strong curvature and for 4-parameter logistic (4-PL) and 5-PL curves commonly used in ligand binding assays, the empirical function between 1/σ2 and y cannot be translated to the same empirical function between 1/σ2 and x. As a result, finding an empirical function between σ and x becomes very difficult for nonlinear STD curves while a linear relationship always exists between σ and y and the weighting factor of 1/y2 can be used directly. Therefore, the weighting factor of 1/y2 is commonly used for 4PL and 5PL curves in ligand binding assays. As the 1/x2 and 1/y2 weightings are equivalent for bioanalytical LC-MS/MS assays, and also

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CONCLUSIONS

The procedure for the selection of the right weighting factor for linear or quadratic calibration curves with least-squares regression algorithm in bioanalytical LC-MS/MS assay was presented and demonstrated. The weighting factor of 1, 1/x or 1/x2 should be selected if the standard deviation of the instrument response (σ) is a constant, σ2 is proportional to x, or σ is proportional to x, respectively. Further investigations showed that 1/x2 should be used for all LC-MS/MS bioanalytical assays as a linear relationship existed between σ and x. The most stable curve could be obtained when the correct weighting factor was used, whereas curves using incorrect weighting factors were unstable. It was also found in this work that there was no or very insignificant impact on the concentrations reported with STD curves using incorrect weighting factors as concentrations were only reported with passing curves which actually overlapped with or were very close to the curve using the correct weighting factor. However, the run success rates dropped by about 80% and 5% to 10% for assays using the incorrect weighting factors of 1 and 1/x, respectively, when 1/x2 should be used. Finally, 1/x2 weighting was recommended for all bioanalytical LC-MS/MS assays, and 1/y2 weighting should be used for other types of assays, especially for assays using nonlinear calibration curves. All of the findings in this work can be generalized and applied in other quantitative analysis techniques using calibration curves with weighted least-squares regression algorithm. 8965

dx.doi.org/10.1021/ac5018265 | Anal. Chem. 2014, 86, 8959−8966

Analytical Chemistry

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Article

(27) Dawes, M. L.; Gu, H.; Wang, J.; Schuster, A. E.; Haulenbeek, J. J. Chromatogr. B 2013, 934, 1−7. (28) Gu, H. et al. Bristol-Myers Squibb, Princeton, New Jersey. Unpublished work, 2013.

ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: 609-252-6171. Tel.: 609252-7636. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Ms. Carol Gleason and Dr. Qin Ji for their helpful comments and discussions. REFERENCES

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