A key assumption in the calibration process is that the signal-to-concentration relationship is fully conserved between the calibration material matrix and the clinical sample matrix. A long-standing recommendation is that calibrator standards should ideally be prepared in matrix-matched materials to avoid bias resulting from matrix differences between patient samples and calibrators. However, the effectiveness of a matrix-matched calibration approach is related to the commutability of the calibration matrix and how representative it is of clinical patient samples [4]. If there are significant matrix differences between calibrators and clinical patient samples, the signal-to-concentration relationship may not be conserved, leading to biased measurements.

For the measurement of exogenous analytes, blank matrices (i.e., matrices devoid of target analytes) from commercial sources or inhouse preparations are readily available. The measurement of endogenous analytes poses a greater challenge, as a “proxy” blank matrix is required for the preparation of calibrators. These matrices (commercial or inhouse-prepared) are often generated through the removal of analytes by dialysis, stripping the native matrix with activated charcoal, or using synthetic matrix materials [5]. Subjecting the matrix to more rounds of stripping (e.g., triple-stripped serum) may reduce the quantity of endogenous analytes in the matrix. However, these additional processes may cause the blank matrix to deviate from the native human matrix and become less representative of the clinical patient samples.

Other variables to consider for calibrator matrix preparation are when the calibration matrix cannot be depleted (e.g., in the case of amino acids), when there is qualitative or quantitative difference in binding proteins of stripped matrix, compared to the ones in native human matrix [6], and in the matrix preparation of unstable molecules. These cases may require spiking of additional components into the blank matrix, such as bovine serum albumin—for nonspecific binding, and antioxidants or preservatives for stabilizing unstable analytes. A synthetic matrix or solvent-based calibrator may be considered when endogenous analytes cannot be effectively removed [5].

It may be desirable to verify the commutability of the calibrator matrix during method development, which can be performed following the CLSI EP07 guideline [7]. Evaluation of differences between the calibrator matrix and native patient matrix, such as spike and recovery experiments, can be conducted to determine the presence of matrix effects [5, 7, 8].

A matrix effect may enhance or suppress ionization of the analyte or internal standard (IS), leading to over- or underestimation of the analyte concentration, respectively. The extent of matrix effect interference can be variable and unpredictable; it may be dependent on interactions between the target and co-eluting molecules or may have a nonspecific effect on instrument responses. The same analyte can give different responses in different matrices, and the same matrix can affect various analytes differently [4, 9]. Matrix effects can be assessed by examining the recovery of a spiked analyte in the matrix under investigation or by observing signal enhancement or suppression by co-infusing a blank matrix with a pure labeled/unlabeled standard [10, 11]. These effects can be reduced using selective sample extraction steps, diverting flow to waste to reduce ion source contamination, or adopting matrix-matched calibration strategies; however, these practices may not completely negate all matrix effects [4]. The matrix effect may be better mitigated by chromatographically optimizing the resolution between regions of suppression/enhancement and the analyte of interest during method development to separate the analyte from the bulk waste of unretained species.

Use of a stable isotope-labeled (SIL)-IS helps to minimize the issues caused by matrix ion suppression or enhancement during quantitation [9]. An SIL-IS allows for accurate quantitation without the need for matrix-matched calibrators by compensating for matrix effects and any potential losses in recovery during cleanup or extraction processes [9, 12]. Although the absolute response can vary, the response ratio (or relative response) of the analyte to SIL-IS remains the same [13]. As described above, other matrices should be evaluated to determine the effects of matrix differences.

Ideally, the SIL-IS should exactly mimic the target analyte(s) to correct matrix-related responses. An SIL-IS must behave in the same way as the target analyte in both the sample extraction and ionization processes. For this compensation to be effective, it is necessary for the IS to resemble the analyte in terms of physical and chemical properties. The coincidental retention time of two structurally unrelated molecules is insufficient to ensure ideal behavior as an IS. The overall variability of experimental results increases as the structures of the target analyte and IS become more divergent [14]. An IS must be structurally similar and chromatographically co-eluted with the target analyte to effectively compensate for non-proportional ionization due to the matrix effect [9, 15].

Although several options for SIL-IS are available, ¹³C- or ¹⁵N-labeled compounds are preferred over deuterium (²H)-labeled compounds because they demonstrate better labeling stability and often have greater purities. They also display identical or almost identical chromatographic and ionization behaviors to those of unlabeled target analytes [16]. SIL-IS is expected to reduce the effects of matrix ion suppression or enhancement, provided they co-elute from the column. For an SIL-IS labeled with deuterium, the position of deuterium isotopes may lead to hydrogen-deuterium exchange, causing IS response variability, leading to the loss of deuterium isotopes in some cases. An SIL-IS heavily labeled with deuterium isotopes may not co-elute with the target analyte because of slight differences in physiochemical properties, resulting in partial ionization differences in the matrix [17, 18]. Importantly, a certificate of analysis for an SIL-IS usually indicates purity as a function of liquid chromatography-UV analysis, which is not capable of indicating isotopic purity.

Signal drifts can be observed during an analytical run, which may be caused by fluctuations in liquid chromatography hardware performance, variations in the electrospray process, changes in ion transfer caused by fouled or moved optics, and changes in detector sensitivity [9, 19]. Among these variations, electrospray ionization at the ion source is considered the major cause of instrumental response fluctuations. An IS is commonly used in quantitative analyses to compensate for these drifts using the analyte-to-IS ratio [19, 20].

A minimum of two data points is needed to draw a straight-line graph and for linear regression modeling (first-degree polynomial) according to Equation 1:

where y is the instrument response (or normalized ratio), x is the concentration of the analyte, a is the analytical sensitivity of the instrument (i.e., the signal per unit change in the concentration of an analyte [21], or slope), and b is the instrument response when no analyte is present (or intercept).

Increasing the polynomial order of the equation to a second-degree polynomial would require a minimum of three data points to plot the curve, and a further increase to a third-degree polynomial requires four data points. In general, a higher number of data points is required for non-linear regression. Non-linearity in calibration curves is commonly observed in LC-MS. Common causes of the observed non-linearity are matrix effects, saturation during ionization, dimer or multimer formation, isotopic effects, and detector saturation [19, 22]. If known causes of non-linearity can be mitigated during method development, they should be implemented to achieve a linear calibrator response. However, if non-linearity remains at the end of method development, non-linear curve fitting may be considered, as elaborated below.

The aim of regression modeling calibration data is to minimize the error in the estimation of parameters describing the relationship between the standard concentration and observed signal response. Increasing the number of points in the calibration curve reduces the uncertainty in the estimated parameters. A sufficient number of calibration standards is needed to define the response profile in relation to the concentration range of the standards. There is no agreement on calibration strategies from regulatory bodies and organizations regarding the number and concentrations of calibrators required, which are often arbitrarily selected. One common recommendation by commercial guidelines and regulatory authorities is that a calibration curve should include a minimum of six non-zero samples covering the intended calibration range, a zero sample (matrix with only the IS), and a blank sample (matrix without any analyte or IS) [2, 23]. The non-zero calibrators should also encompass the lower limit of quantitation (LLOQ) and upper limit of quantitation (ULOQ) [2, 23]. Zero and blank samples (also known as blank and double blank, respectively) should not be included in the calibration curve regression [5]. They are to be utilized to ensure that the veracity of the system does not unduly bias the intercept.

Having a greater number of calibration standards leads to a smaller error estimate for the calculated concentration and, consequently, better precision and a narrower confidence interval at the determined concentrations. Although more calibration points can improve the accuracy and precision of the model, this must be balanced against operational and financial costs such as greater laboratory effort in the preparation and analysis of additional calibration standards or replicates [24].

Similar accuracy and precision can be achieved by using as few as one calibration standard (with the regression being forced through the origin) or two calibration standards (without the regression being forced through the origin) to construct the calibration curve as against using eight or ten standards [25-27]. Under the two-calibration standard approach, previously established linear multipoint calibration data were retrospectively linearly regressed using two critical concentrations, the LLOQ and ULOQ, with no additional batch failure or QC rejections observed. More specifically, the differences in mean QC concentrations were within −0.8% to 2.5%, and the differences in %CV were within −0.7% to 0.9% for the 12 measurement procedures studied [27]. These studies involved exogenous analyte applications, which may be associated with better opportunities for matrix matching between the calibrators and samples. It is important to determine the susceptibility of a low-number calibration strategy to calibration drift or shift errors, and its ruggedness over time.

An increased number of points in a calibration curve is only an indication of the back-calculated values of the standard curve points and is not directly correlated with the inaccuracy or imprecision of points that were not used to generate the model (e.g., QCs or samples). Additional calibration points/replicates may improve the precision of the regression model, but this does not directly correlate with improved performance of the assay. This may be especially true for assays with modified matrices (e.g., endogenous analytes using a stripped matrix).

Tan, et al. [26] demonstrated that the impact of using different high-concentration levels as calibration points was almost identical. Hence, the ULOQ calibrator can be represented by adjacent high-concentration standards close to the ULOQ, and extrapolation at the higher concentration end is generally linear and acceptable [26]. The addition of a concentration point at the geometric mean of the LLOQ and ULOQ can establish quadratic regression models. Placing a calibration point at the geometric mean also provides better overall accuracy than using other adjacent points, although extrapolation beyond the highest concentration is not advised [26]. These results should be interpreted within the specific simulation parameters. For example, the degree of quadraticity of the regression curve may vary from day to day depending on the condition of the source/ion optics/detector. The application of such a strategy in other LC-MS/MS techniques warrants examination.

It may be difficult to determine the true linearity of a calibration curve using only two points. Musuku, et al. [27] demonstrated alternative approaches to verify the linearity of the curve, such as performing a regression using the two calibrators together with QC samples; however, this may not always be sufficient to conclude linearity. It is advisable to validate the method using more calibrator concentrations first to map out the detector response and investigate polynomial regression models, while conducting experiments to critically stress the linearity assumption before subsequently removing some calibration levels to maintain the performance specifications [2, 26]. It may be necessary to remap the detector response when a significant analytical shift or drift is observed.

The above studies show that optimal accuracy for regression statistics can be obtained using fewer calibration concentrations with a higher number of replicates at each concentration, instead of the commonly adopted practice of using more calibration concentrations with fewer replicates [26]. Additional replicates help to reduce the imprecision of the estimated parameters [24]. Having fewer calibration standards also reduces the risk of error from calibration standard preparation. The number of replicate calibrator injections is also an arbitrarily set criterion. Duplicate injection of calibrators is recommended because it improves the precision of the observed result without requiring additional calibration standard preparations [23, 24, 26].

The number of calibrator points and replicates analyzed, as well as their concentration levels, should be dependent on the characteristics of the particular bioanalytical measurement procedure, such as the presence of linearity or non-linearity and the precision of the analysis [26].

The order of calibrator assessment does not significantly influence the performance and can be conducted in either ascending or descending order [23]. Nonetheless, the addition of a blank sample after injection of the highest-concentration calibrator can assist in the detection of carryover [23].

Having the calibrations bracket an analytical run, with one set at the beginning and one at the end of a sequence, can help reduce the effect of analytical drift throughout the run. Combining both sets of data points to construct a calibration curve helps to compensate for any variations in instrument sensitivity (slope changes) during the run. If only one replicate injection of the calibration is adopted, interspersing the calibrators in the run can achieve the same compensation [24].

Another related issue is how calibration levels should be distributed across the measurement range. The concentration levels can be evenly (equidistant) or unevenly distributed. Unequal spacing of calibration points with clustering at lower concentrations improves the accuracy and stability of the calibration curve at the lower end, which subsequently improves the precision of measurements near the limit of detection and LLOQ. Having a single very high calibrator concentration (i.e., spaced far away from other points on the x-axis) may lead to a high degree of leverage, with a small error having a disproportionately large influence on the regression model estimates [2].

Although not thoroughly examined in the literature covered for this review, different distributions of calibrators can influence stability of the regression model when an incorrect weighting factor is used. With an appropriate weighting factor, the curve was stable regardless of the calibrator distribution [28]. Weighting the response also reduces leverage effects [2]. Further research is required in this area of calibration data heteroscedasticity, precision profiles, and appropriate regression model weighting.

It is generally considered suboptimal to prepare calibrators by serial dilution from a single stock because this carries a higher risk of error if the primary stock is prepared incorrectly.

As mentioned above, it is common to observe non-linear calibration data using LC-MS methods. Non-linear behavior may not be evident from visual inspection of the calibration data but may display significant bias when fitted with an inappropriate curve. When non-linearity is determined in calibration data, there are two options to overcome this limitation. First, a quadratic or higher-order calibration regression model can be used. Second, the calibration data can be divided into two separate ranges. A linear regression model may be used to fit the lower calibration ranges. This lowers the calibration range to narrower linear ranges, reducing the dynamic range of the measurement procedure [14, 15, 30].

Although r and R² are commonly used indicators to assess the goodness of fit for calibration models, they are not appropriate for assessing linearity [31]. Both r and R² are statistical measures; r is an indicator of the degree of correlation between two variables (signal and concentration), and R² is an indicator of the proportion of variability in the response explained by the regression. The correlation and response variability are only loosely related to linearity, and using these two coefficients to determine linearity may be misleading. These coefficients used in isolation are not adequate to assess linearity because values close to unity (e.g., R² >0.999) can be obtained even when the data show signs of curvature [3, 32]. Linearity should instead be assessed using appropriate statistical methods (e.g., ANOVA) and/or other mathematical measures (e.g., residual plot), which will be further discussed in Section 5 below.

The two most commonly used regression models for constructing LC-MS/MS calibration curves are linear and quadratic regression equations, which use either weighted or non-weighted fitting techniques [1]. To determine whether the calibration model should be weighted, the calibration data should be examined for evidence of heteroscedasticity.

Heteroscedasticity in regression analysis refers to the error term or residuals, which are unequal across the values of the dependent variable (calibration standards in this case). Calibration data may be homoscedastic, where the variance of each concentration level is constant and independent of the concentration range, or heteroscedastic, where the variance increases as a proportion of the concentration range. When calibration data are heteroscedastic, a scatter plot of these variances often shows a funnel shape, in which the variance widens or narrows in response to the standard concentration [32]. A standard non-weighted (ordinary least-squares) calibration regression model assumes that the measurement error is homoscedastic (i.e., exhibits constant variance). If calibration data are heteroscedastic, it is more appropriate to use weighted least-squares regression [1].

When the calibration data are heteroscedastic, but no weighting is applied during regression modeling, the influence of errors at different concentration levels on the estimation of the parameters is ignored, which reduces the stability of the calibration models. Model stability is defined as the resistance of a calibration to significant errors from calibration samples [28]. Using a non-weighted regression means that a small bias at higher calibration concentrations could change the curve significantly and cause large deviations at low concentrations. This is especially detrimental for heteroscedastic data because the variance at each level could increase with increasing concentrations.

To account for the heteroscedasticity of the data, weighted regression analysis has been used to maintain constant variance through the measured concentration range. Weighted regression models minimize the influence of higher concentrations by balancing the regression line to distribute the variance uniformly throughout the calibration range [31]. Appropriate weighting factors can be calculated using the inverse of variance (1/σ²). However, this practice requires several determinations for each calibration point [31]. Instead, weighting methods commonly involve adjusting the data using a factor related to an inverse function of the concentration of the standards. Weights of 1/x² have been recommended as LC-MS/MS bioanalytical methods [26, 28, 33]. Simply following historical practices may not be appropriate, as many of these recommendations are based on the “test-and-fit” strategy, which involves fitting the calibration data points with different models and weighting factors, starting with the most simplistic unweighted linear regression and progressively fitting more complex models and subsequently assessing which model provides the best fit. This is often largely based on the analyst’s subjective interpretation that the data points are close to the trend line and that the regression has an R² value close to unity.

Good recovery of the calibration standard concentrations and/or QC performance does not necessarily mean that a correct weighting factor has been applied. Acceptable recovery may be achieved when an inappropriately weighted calibration model coincidentally overlaps or is close to the underlying true relationship. Gu, et al. [28] demonstrated that, in some cases, no weighting or 1/x weighting could generate good calibration curves and QC performance, although the weighting factor determined from the collected data should have been 1/x². Hence, recovery and assay performance data should not be used as criteria for the selection of weighting factors. The choice of weighting depends on the relationship of the variance for the data with other variables, and correct application of weighting factors generally results in better longitudinal assay performance and stability, as explained further below in Section 5.

Other weighting factors that are less commonly used in clinical LC-MS/MS applications are 1/y and 1/y². In most cases, the effects of using either 1/x or 1/y or 1/x² and 1/y² are similar for bioanalytical LC-MS/MS measurement procedures, as linear or quadratic models with very mild curvatures are commonly encountered. Instrument response errors (or variances) are directly related to the y-axis instead of the x-axis on the calibration curve plot. When the curves are linear or close to linear, the empirical functions between 1/σ² and y can be translated into the same empirical functions between 1/σ² and x [28].

When the quadratic curve has strong curvature, or for the four-parameter logistic and five-parameter logistic 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. In such cases, 1/y or 1/y² weighting factors are preferable [28, 34].

Heteroscedasticity in calibration data should not be overlooked. Heteroscedasticity can lead to a significant loss of precision, particularly at low concentrations, in the calibration model. This is crucial in clinical mass spectrometry as it affects the limits of detection and quantitation of the measurement procedure.

Non-weighted regression modeling of calibration data that are heteroscedastic leads to an increase in imprecision, particularly at lower concentrations, resulting in falsely higher limits of detection and quantitation and incorrect performance characteristics of the measurement procedure. Alternatively, application of weighting to homoscedastic calibration data could lead to a falsely lowered calculation of detection and quantitation limits [14, 30-32].