Validation of liquid chromatography mass spectrometry (LC-MS) methods

9.3. Estimation of LoD

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Different approaches to estimate

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https://www.youtube.com/watch?v=xk2Ou3jaovg

There are a multitude of different approaches that can be used to estimate LoD, and no clear consensus exists on which approach is the best in different situations. The approach that we recommend in this course is discussed below in the video “Important aspects of estimating LoD and , ” and in the tutorial review [ and ]. A general overview of approaches from most prominent guidelines to estimate LoD from most prominent guidelines can be found in Table 1 (NB! For more specific overview of procedures, see the specific guideline!). These approaches can result in widely varying LoD estimates. Different guidelines often suggest different approaches and it is up to the analyst to choose which approach to use. If a specific approach is not demanded by the guideline, this choice must be made based on the necessities and properties of the analytical method.

An Excel sheet with example calculations of LoD with approaches in Table 1 can be found at the end of this chapter.

Table 1. Different approaches for determining LoD, CC_α and CC_β.

Group	Reference	What is obtained?	Equation
1	[, , , ]	LoD (considers and negative results – the probability of false positive and negative values depends on the choice of t)	(Eq 1) is mean value of blank samples or 0; t is Student’s Coefficient; S(y) is standard deviation of blank or fortified samples. Equation gives LoD in intensity scale.
Description: Concentration of fortified samples in LoD range (e.g. lowest level where S/N > 3) or at maximum residue limit (); t is taken 3 or 4.65; 6 to 10 repeated measurements for blank and fortified samples; all signal intensities and standard deviations have to be over 0; Assumptions, simplifications: ; normal distribution of the replicates; variability of the slope and intercept are not taken into account; of the calibration data; t value is rounded and does not take into account the degrees of freedom; only for single sample measurement results. Notes: Care must be taken when integrating blank samples; Erroneous calibration function can lead to negative LoD results; Note that is not necessary (i.e.is equal as 0) if subtraction with intercept (or with) is done to all measured results before calculations.
2	[]	LoD essentially equivalent to CC_α (considers only false positive results)	(Eq 2) S(x) is the standard deviation or pooled standard deviation of the analyte concentrations from the replicate measurements.
Description: A detailed procedure is given to choose fortified sample concentration (incl. estimating an approximate LoD first, measuring only 2 of the needed repeated samples before measuring the rest of the 7 samples); t is taken depending on the degrees of freedom; Recommended analyte concentration range in fortified samples is 1-5 times LoD. Assumptions, simplifications: Normal distribution of replicates; variability of the slope and intercept are not taken into account; linearity of the calibration data; is somewhat considered by careful choice of fortification concentration; only for single sample measurement results. Notes: LoD as equivalent to CC_α ( results are not accounted for); the background (mean of blank values or the intercept value) is subtracted from all other results. It is then suggested to iteratively check the LoD by estimating it again.
3	[]	LoD (considers false positive and negative results – the probability of false positive and negative values depends on choice of t)	(Eq 3) (Eq 4) where is the average intercept; n is the number of repeated measurements of the sample; S(y) is the standard deviation of the blank or fortified samples; n_b is the number of repeated measurements of blank samples. Equations give LoD in intensity scale.
Description: Second equation is used if LoD is estimated from single day measurement results and blank values are used for correction; t is taken as 3. Assumptions, simplifications: Homoscedasticity; normal distribution of the replicates;linearity of the calibration data; variability of the slope and intercept are not taken into account. t value is rounded and does not take into account the degrees of freedom. Allows taking into account the averaging of sample measurement results. Notes: Using (not standard deviation) to estimate LoD is suggested. Monitoring of and regular recalculation of LoD values is suggested if LoD is used for making decisions.
4	[]	LoD (considers false positive and negative results)	(Eq 5) b is the slope of the calibration function, S_d can be chosen as a standard deviation of the blank samples, residuals (S_y.x) or intercept. Instructions to calculate a standard deviation of the residuals and an intercept in Excel can be found in the video “Calculating LoD” below.
Description: Regression line must be in the range of LoD. Calibration function is used to estimate the slope and the standard deviation of the residuals and the intercept. Number of repeated measurements is not specified. Assumptions, simplifications: Homoscedasticity; normal distribution of the replicates; linearity of the calibration data; variability of the slope and intercept are not taken into account. If repeated results at each calibration level are averaged and standard deviation of the residuals is used for estimate LoD then the number of repeated measurements must be the same as repeated measurements for each calibration level. Notes: The standard deviation of the intercept underestimates the variance of the results at 0 concentration and should not be used. Due to the conservative LoD estimates, simple calculation procedure and reasonable workload (S_d is taken from values), this is the suggested approach if a rigorous LoD estimate is not needed [, ].
5	[, ]	LoD (considers false positive and negative results)	Cut-off approach: number of repeated measurements (usually 10) are made at different concentrations near LoD. The lowest concentration at which all the samples are „detected“ is used as the LoD. The detection threshold can be established for example based on the S/N, visual evaluation or automatic integration for chromatographic methods.
Assumptions, simplifications: Uses robust statistics. This approach does not assume normal distribution. Visual evaluation of the presence of a peak depends on the analyst. Notes: This approach is very work intensive; If repeated estimations of the LoD need to be made, this approach is not recommended for LC-MS/MS methods; It has also been suggested to plot the portion of the positive responses against concentration to find the lowest concentration at which necessary number of samples give the decision „detected“; Each sample should be independent of the others.
6	[, ]	CCα and CCβ	CC_α: 1. Calculated as (Eq 6) LCL is the lowest concentration at which the measuring system has been calibrated (in case MRL has been set the concentration must be at MRL), t is Student’s Coefficient and its value is based on the specific validation experiment, and u is combined standard at LCL (or MRL). NB! If Gaussian distribution (i.e. n = ∞; one-sided) is taken as basis then t can be taken as 2.33 (or 1.64 if the substance has a set MRL). 2. Blank matrices are analyzed to estimate the noise in the analyte time window. The level at which S/N > 3 can be calculated and this level is stated as CC_α. It is stated in 2021/808 that this approach can only be used until 1 January 2026 – if a method is validated after this date, then using this approach is not acceptable. CC_β: 1. Calculated as (Eq 7) In this equation u is combined standard measurement uncertainty at or above CC_α. NB! In this equation if Gaussian distribution (i.e. n = ∞; one-sided) is taken as basis then t can be taken as 1.64 whether the substance has a set MRL or not. 2. Lowest concentration level is found where ≤ 5% of samples are compliant. This concentration is taken as CC_β. For this, at least 20 samples at each concentration are necessary. If MRL has not been set for the analyte, the samples are considered compliant if the analysis result is below CC_α. If MRL has been set, the sample is considered compliant if the result is below MRL. Equations give LoD in intensity scale.
Description: Some simple approaches suggested to estimate CC_α and CC_β; Similarly CC_α and CC_β estimation approaches are suggested in case a MRL is set; After estimating the intensity value corresponding to CC_α and CC_β, the calibration function should be used to convert them to the concentration scale; Approach 2 for estimating the CC_α and approach 3 for estimating CC_β demand at least 20 replicates (at each level for CC_β). Assumptions, simplifications: Linearity of the calibration data; variability of the slope and intercept are not taken into account. Possible heteroscedasticity is considered to some extent: CC_α and CC_β are not found using the same variance. In these approaches the α value is 1 % if MRL has not been set for the analyte and 5 % if MRL has been set for the analyte, and the β value is 5 %. The coefficients in equations do not take into account the degrees of freedom. Notes: CC_α and CC_β are found for minimum required performance level or MRL. Identification requirements have to be followed (only after identification of the analyte can the sample be used for CC_α and CC_β evaluation).
7	[]	CC_α and CC_β	(Eq 8) (Eq 9) (Eq 10) is the estimated slope, is the estimated residual standard deviation, t_0.95 is the 95% one-sided quantile of t-distribution (where ν = I*J – 2), δ is non-centrality parameter of the non-central t-distribution (similar to t_0.95), K is the number of repeated preparations of the (unknown) sample, I is the number of calibration levels, J is the number of separate sample preparations at each concentration level, is the mean value of the concentration levels, x_i is the concentration of ith calibration level. These equations are for homoscedastic data, for calculations in case of heteroscedastic data see [].
Description: Given equations are for homoscedastic data; iterative approach to estimate CC_α and CC_β, suggested for heteroscedastic data, is also given in the guideline; Requirements of the approaches: K should equal J I should be at least 3 (5 is recommended) J should be at least 2 Number of measurements per sample (L) should be at least 2 and identical for all samples. The blank measurements are required to also be included in the calibration points. Assumptions, simplifications: Normal distribution of the replicates; linearity of the calibration data; It is suggested to estimate whether the data are heteroscedastic based on prior knowledge and visual evaluation of the data; In heteroscedastic approach standard deviation of results is assumed to increase linearly with concentration Notes: In this guideline the concentration scale is called the net state variable and the intensity scale is called the response variable. Notice that 2 measurements are recommended for each preparation and the mean of these measurements is then used in the following calculations.

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Calculating CC_α and CC_β

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https://www.youtube.com/watch?v=BQ_dOEMDoDs

Estimating CC_α and CC_β

The approaches that are usually suggested to estimate CC_α and CC_β are more complex than the approaches suggested for LoD. This is so because their definition is statistically more rigorous (demanding a known probability level of false positive and negative results) but the results are also more reliable. Some approaches to estimate CC_α and CC_β suggested in the guidelines and articles can be found in Table 1.

The CC_α and CC_β calculations [] take into account the standard deviation of the used linear regression line parameters (slope and intercept). This variance is propagated into the concentration values that are calculated by using these parameters. As CC_α and CC_β are used in the concentration scale (similarly to LoD) the variance of the slope and intercept must be taken into account when estimating them.

Another property that must be considered is homo- and heteroscedasticity. Homoscedasticity means that the variance of signal is constant in case the concentration changes and heteroscedasticity therefore means that the variance changes with the concentration (see example in Figure 1). Analytical methods are often heteroscedastic – as the concentration increases, the standard deviation of the measurements also increases. If it is shown that the collected calibration data collected is heteroscedastic then, weighted linear regression (WLS) should be used to take the variance of the slope and intercept more accurately into account. A simplified approach that usually works sufficiently well is presented below.

Figure1_section_9_3

Figure 1. Data in Plot A are homoscedastic and data in Plot B are heteroscedastic. In plot A as the concentration increases the variability of results in intensity scale does not increase. However, in Plot B the variability of intensity values increases as the concentration increases.

With WLS the propagated errors of the slope and intercept to the concentration value significantly decrease at lower concentration levels. Therefore, the CC_α and CC_β values are also significantly influenced. Using WLS can be complex and a possibility to avoid this is to select a narrow concentration range at lower concentrations from the calibration data that can be shown to be reasonably homoscedastic. These data can then be used to estimate the slope and the intercept with ordinary linear regression (OLS) which assumes that the data are homoscedastic. As a result calculating the CC_α and CC_β estimates also becomes simpler.

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Important aspects of estimating LoD and CCα, CCβ

http://www.uttv.ee/naita?id=23350

https://www.youtube.com/watch?v=9GFMa0AYkdA

It should be considered how important the LoD value for a given analytical method is. Based on this knowledge it can be chosen whether a simple approach to estimate a LoD is enough or a more complex approach that makes less assumptions (e.g. about homoscedasticity) and therefore gives more accurate results should be used. The important assumptions made by different approaches are summarized in Table 1. Further details about how to evaluate whether these assumptions can be made is discussed in the following references [, ]. If the analyte concentration will never come close to a LoD value then LoD does not have to be estimated at all. However, often LoD is still estimated in these cases just for proving that the samples are significantly above the LoD of the method. For example, when measuring calcium in milk by complexometric titration, we do not have to worry that in some samples the concentration of calcium might be so low that it would be below a LoD for a reasonable titration procedure. However, if the LoD estimate is an important parameter used to interpret the results of an analysis, more complex and accurate approaches must be used to estimate LoD. For example, when analyzing blood samples of athletes for doping, the method must interpret the results correctly even if only very small amounts of the analyte is detected. Therefore, CC_α and CC_β values estimated with complex approaches that make less assumptions (e.g. ISO []) must be used.

In some cases the analytical method can have properties that do not allow the use of some LoD estimation approaches. For example, it can be difficult to estimate the standard deviation of the blank for LC-MS/MS methods as the noise can be zero due to the signal processing. As the blank values all give intensity of 0, the LoD value cannot be calculated from them but the standard deviation at 0 can be still estimated by other approaches: from the standard deviation of the intercept value or from the standard deviation of the residuals. A more thorough discussion about the problems of processing chromatograms of samples at low concentrations can be found in the following references [, ]. In conclusion, in general the analyst must understand which approaches cannot be used for a given analytical method.

It should always be kept in mind that LoD is an estimated value and never represents the true LoD as it is calculated from the parameters that deviate randomly from their true value between measurements. Moreover, the true value around which the results deviates can change randomly between days. For example, the slope of the LC-MS/MS changes significantly between days – this means that the true intensity value given by a concentration changes between days. For this reason, the within-day standard deviation is lower than the standard deviation of results collected on different days (see section 4.1). Therefore, LoD also changes between days. To take this fact into account, the LoD should be estimated over a long period of time (e.g. a month) and the median LoD value can then be used []. If it can be seen that the LoD estimate changes significantly between days (meaning the variation of LoD value within a day is significantly smaller than between days) and the estimate is important for the correct interpretation of the results on that day, then the LoD should be estimated on that day and that value should be used for the interpretation. However, it can also be noted here that if the LoD is used only for simple characterization of the method and not used further (see above), then the LoD does not have to be estimated on multiple days. It must also be noted that the previous discussion also applies for CC_α and CC_β.

As the different approaches can give differently biased values, it should be always stated which approach is used to evaluate the LoD. If different approaches are used (to characterize the lab or the new method), then the comparison should be made with caution.

A different concept for estimating LoD is by using the signal-to-noise ratio (S/N). This approach is mostly used in chromatographic methods. Modern chromatography programs determine this value automatically. The signal value for this is found from the height of the peak and noise values are found from either the standard deviation of the noise or from so called peak-to-peak value (meaning the difference between the highest and lowest points in the noise). From this it can be seen that S/N can be found for only one measurement of a sample. A single measurement however does not take into account the variability between measurements and therefore the LoD should not be evaluated from this result. A scheme has been suggested by Eurachem where 10 samples are measured on different concentration levels and the lowest concentration where all 10 are detected is taken as the LoD. Here the decision that an analyte has been detected can be made from the fact that the S/N is equal to or over 3. However, this means that many measurements have to be made to estimate the LoD and due to the S/N being conceptually different from other approaches, it will be difficult to compare the LoD estimates found with other approaches.