Comparison of approaches for assessing detection and quantitation limits in bioanalytical methods using HPLC for sotalol in plasma

Abstract

The limit of detection (LOD) and limit of quantification (LOQ) stand elements in the validation of analytical and bioanalytical methods as emphasized in numerous guidelines. Despite this, the absence of a universal protocol for establishing these limits has led to varied approaches among researchers and analysts in literature. In this work we present the latest graphical strategy of validation known as the uncertainty profile to assess the LOQ and LOD. Then, we conducted a comparative study of this approach with those based on accuracy profile and parameters of the calibration curve. We realize the uncertainty profile from the uncertainty parameter calculated from the tolerance interval. Furthermore, we provide a succinct overview of alternative methods for computing these limits. This includes the classical strategy based on statistical concepts and graphical one using the method of accuracy profile. In pursuit of this objective, these strategies are implemented in the same experimental results of an HPLC method dedicated for the determination of sotalol in plasma using atenolol as internal standard. The classical strategy based on statistical concepts provides underestimated values of LOD and LOQ. In the other side, the two graphical tools give a relevant and realistic assessment, and the values LOD and LOQ found by uncertainty and accuracy profiles are in the same order of magnitude, especially the method of uncertainty profile. It provides precise estimate of the measurement uncertainty. The graphical strategies of validation, uncertainty profile and accuracy profile, based on tolerance interval are a reliable alternative to the classic strategy, based on classical concepts, for assessment of LOD and LOQ.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-83474-5.

Introduction

The validation is the whole of necessary operations to prove that the analytical procedure is sufficiently accurate and reliable to have confidence in the results and this provided for a specific purpose^1,2. Generally, all analytical and bioanalytical procedures are characterized by several technical operating parameters such as: precision, bias, accuracy, linearity, and percent recovery³. Among these parameters, two are crucial: the lowest detectable concentration called the limit of detection (LOD) and the lowest quantifiable concentration called the limit of quantification (LOQ)^4–6. Indeed, it is of great significance for researchers and analysts to determine these limits so that would allow them to be aware of the concentration that could identify, with certainty, the presence of the sought analyte (LOD) and the threshold concentration beyond which the bioanalytical procedure can guarantee the reliability of its results (LOQ)⁷.

Regarding the nomenclatures, several designations have been suggested for these limits such as the limit of determination, limit of reporting, limit of application⁸.

LOD and LOQ parameters are related but have distinct definitions and should not be confused. According to the guide Q2R1 of International Conference on harmonization (ICH) (ICH Q2R2 2022)¹, the detection limit of an individual bioanalytical method corresponds to the lowest amount of the substance analyzed detectable by the method, without necessarily providing the exact value. The quantification limit of an individual analysis method corresponds to the lowest amount of the substance analyzed that the method allows the determination with an acceptable precision and accuracy degree. There has often been a lack of agreement within the chemical and biological analysis field as to the terminology best suited to describe these parameters. Likewise, there have been various methods for estimating them. The various possible approaches to estimate these limits seem to be far from equivalence, regarding the values and their degree of reliability⁹.

In that respect, given the number of proposed methods for determining these limits, a question that arises” what is the best way to assess the LOD and the LOQ?”

In this context, we propose to use an innovative approach called “uncertainty profile”, presented in the works of Saffaj et al., 2011 and 2013 for a good estimation of these two critical parameters of the bioanalytical validation [6]– [8]. Our proposal allows at the same time examination of the validity of bioanalytical procedures as well as estimation of the measurement uncertainty.

As well, we briefly present other methods for computing these limits, based on statistical and graphical concepts as suggested by the guidelines: ICH, IUPAC and SFSTP¹. In addition, we conducted a comparative study of these approaches since nothing guarantees their equivalence.

Experimental

Methodologies

Theory of uncertainty profile

SAFFAJ & IHSSANE introduced the uncertainty profile as an original validation approach based on the tolerance interval and measurement uncertainty¹⁰. Uncertainty profile is a decision-making graphical tool aiming to help the analyst in deciding whether an analytical procedure is valid. It is based on the combination in the same graphic of the uncertainty interval and the acceptability limit. One way to build the uncertainty profile is to compute the β-content tolerance interval (β-TI) introduced by Mee¹¹, and to compare it to acceptance limits. Indeed, this β-TI is an interval that one can claim to contain a specified proportion β of the population with a specified degree of confidence γ. A method is claimed to be valid when uncertainty limits assessed from tolerance intervals are fully included within the acceptability limits. A validity domain is, then, defined between the limit of quantitation and the upper tested concentration.

Tolerance interval computation: To build the uncertainty profile, we must firstly estimate the β-content tolerance interval from the following equation:

1

where:

2

and are the estimates of the reproducibility variance, the between conditions variance and the within conditions variance (repeatability).

Mee uses the Satterthwaite approximation to compute k_tol (11). The tolerance factor is given by:

3

With

4

And

5

Where

• F is the mean square ratio MSb/MSe and Fη is the 100η percentile of an F distribution with ν1 = a(n-1) andν2= (a-1). However, based on numerical results, the recommended values of η are 0.85, 0.905 and 0.975, corresponding to γ = 0.90, 0.95 and 0.99, respectively.

• denotes the β quantile of a noncentral chi-square distribution with degrees of freedom f and non-centrality parameter h.

• a is the number of series.

• n is the number of independent replicates per series.

And

6

With and and and represent the two variance components in the model.

Measurement uncertainty assessment: Once the tolerance intervals are calculated, we deduct the measurement uncertainty through the following formula:

7

where

• U is upper β-content tolerance interval.

• L is lower β-content tolerance interval.

• t(ν) is the (1 + γ)/2quantile of Student t distribution with ν degrees of freedom. For balanced data, ν can be estimated by the Satterthwaite formula (11).

Construction of the uncertainty profile: After calculating the uncertainty through the Eq. (7), we have used the following formula to build the uncertainty profile.

8

where:

• k is a Coverage factor. The choice of the factor k is based on the level of confidence desired. For an approximate level of confidence of 95%, k = 2.

• is the estimate of the mean results.

• λ is the acceptance limits.

  The validation strategy based on the uncertainty profile can be achieved through the following steps:

• Choice of the appropriate acceptance limits taking into account the intended use of the method;

• Generate all possible calibration models, using the calibration data.

• Calculation of the inverse predicted concentrations of all validation standards according to the selected calibration model;

• Compute the two-sided β-content γ-confidence tolerance intervals for each level, according to one of three approaches proposed below;

• Determination of the uncertainty for each level using the Eq. (7);

• Construct the uncertainty profile according to the Eq. (8) and make 2D-graphical representation results for the acceptability and uncertainty limits;

• Compare the interval of uncertainty (L, U) to the acceptance limits (-λ, λ);

• If (L, U) falls totally within (-λ, λ), the method is accepted; otherwise, the method is not valid.

Assessment of Quantitation and detection limits from the uncertainty profile: From the uncertainty profile, experimental uncertainty intervals as a function of the concentration are obtained for the accuracy, which have to be compared with acceptability limits fixed by the analyst. The intersection at low concentrations of acceptability limits and uncertainty intervals defines the lowest value of the validity domain for which the analytical method can be applied, and corresponds to a limit of quantitation.

The LOQ can be obtained accurately by calculating the intersection point coordinate of the upper (or lower) uncertainty line and the row that represent high (or low) acceptability. For this purpose, the linear algebra was applied to calculate the intersection point coordinate between these two lines.

Indeed, the calculation of the coordinates of the point of intersection of two lines is an algebra problem¹².

To calculate the LOQ between two levels, the following notations are used:

• ♣ Z the absolute values read from the y-axis value;

• ♣ X reference values provided by the x-axis;

• ♣X_A and X_B the abscissa of the two levels A and B;

• ♣And: intercepts of the upper (or lower) tolerance interval limit.

• ♣And: intercepts on the upper (or lower) acceptability limit.

• ♣The abscissa of the point of intersection with LOQ equal XLOQ.

These two lines can be represented by a system of two equations, the first concern the tolerance interval limit (t₀ the intercept and slope t₁ parameters) and the second represent the acceptability limit (parameters a0 and a1):

9

10

With a₀ must be equal to because the limit of acceptability always passes through zero.

To obtain the slope t₁:

11

For the intercept t₀:

12

By analogy:

13

And:

14

The equation system resolution leads to:

15

Other methods of estimating LOD and LOQ

There are numerous possible conceptual methods concerning LOD and LOQ, each providing a somewhat different definition. Depending on the definition chosen, the values of LOD and LOQ can vary greatly which make it difficult for comparative purposes. In this section, the two methods based on International Conference of Harmonization (ICH) and Société Francaise des Sciences et Techniques Pharmaceutiques (SFSTP) guidelines are discussed.

ICH’s method: Since the majority of analytical methods using instrumental techniques, the International Conference on Harmonization (ICH) Q2R2 guideline recommends estimating LOD and LOQ based on the Standard Deviation of the instrument Response and the Slope¹³.

The LOD and LOQ can be expressed as:

16

17

where the standard deviation of the instrument response and b is is the slope of the calibration curve.

The standard deviation of the response can be estimated by the standard deviation of either y-residuals, or y-intercepts, of regression of calibration curve.

SFSTP’s method: Several years earlier a SFSTP commission has developed a statistical and graphical decision-making tool called accuracy profile used to assess the validity of analytical methods. This accuracy profile uses the statistical methodology of expectation tolerance intervals. The SFSTP approach estimates the total error of analytical methods using method validation data. Therefore, this profile assesses the validity of the method by comparing the total error to a predefined acceptance limit stating the maximum error suitable for the method under study. The complete theory and calculation of accuracy profile are described in^14–17, and a fully developed application is presented in¹⁴ .

The uncertainty profile of Saffaj et al. and the accuracy profile of the SFSTP commission are both fulfilling the same final purpose.

Regarding the estimation of the limit of quantification, it is possible to derive an algebraic formula for calculating this important parameter using the total error approach. Indeed, the same procedure for determining the limit of detection and quantification from the uncertainty profile, presented in Sect. 2.1.4 can be used to calculate these limits from the accuracy profile.

Application

The evaluation of different approaches described above for determining the limit of detection and limit of quantification is shown in an example of a validation of bioanalytical method.

The example, which has been the object of this illustration, concerns the validation of a method for the determination of sotalol in plasma by HPLC using atenolol as an internal standard. The data of this method is taken from SFSTP guide¹⁶.

Since the validity of this method was demonstrated by the uncertainty profile and the accuracy profile in previous studies^10,18–20, we observed that some response functions are better than others i.e., calibration models most appropriate to quantify, with uncertainties (or errors) not exceeding the limits of acceptability (± 20%) are: the linear regression model, the linear regression after square root transformation and the weighted regression model with a weight equal to 1/X.

The validity of this method was tested by both approaches: the profile of accuracy proposed by the guide SFSTP¹⁴ and the profile of uncertainty proposed by Saffaj and al¹⁰.

Results and discussion

Construction of accuracy profile

Before building each profile we must choose the best regression model i.e. that produces goshawks of the responses regeneration of validation data and minimum observed error^14–16,21. Usually, eight main models could be tested. In the present study, three models have been chosen which are: linear regression model, linear regression model after processing square root, weighted regression model with a weight equal to1/X, and accuracy profiles build for each of them as shown in Fig. 1, the limits of acceptability were set at ± 20% (Table 1).

Fig. 1.

Accuracy profile to 95% of the analytical procedure using: (A) linear regression model, (B) linear regression model after processing square root, (C) weighted regression model with a weight equal to1/X as functions of responses. The dashed lines represent the acceptance limits (± 20%); the dotting lines the 95% tolerance interval connected and the black dashed line represents the estimated relative bias line.

Table 1.

Estimation of low and high relative tolerances limits for each response based on β-expectation tolerance interval calculated using Mee’s approach for the determination method of sotalol in plasma.

Model	Concentration level (ng mL− 1)	Bias (%) ( β = 80%)	Relative		Relative
			Tolerance limits (%)		Tolerance limits (%)
			β = 80%		β = 90%
			Lower	Upper	Lower	Upper
Linear regression after square root transformation	25.3533	-0.52	-12.52	12.41	-15.505	15.402
	48.2417	-1.38	-13.27	10.52	-16.391	13.635
	437.8235	-1.52	-11.2	8.165	-13.959	10.927
	838.6479	0.31	-10.13	10.75	-13.479	14.105
Linear regression	25.3533	-0.0602	-13.64	13.49	-1.453	32.829
	48.2417	-2.67	-12.47	1.405	-9.500	13.743
	437.8235	-9.19	-12.15	7.953	-9.232	4.262
	838.6479	11.78	-10.33	13.14	-5.278	6.244
Weighted (1/X) linear regression	25.3533	2.14	-3.534	20.40	-6.872	15.527
	48.2417	-0.675	-12.72	9.916	-12.672	5.861
	437.8235	-10.24	-12.08	7.404	-8.988	4.649
	838.6479	7.326	-9.692	11.44	-4.642	7.004

The accuracy profile of the method is obtained by connecting the lower bounds of the tolerance range between them and the other upper bounds between them as in Fig. 1:

Uncertainty profile building

As decision rule, we selected the 4-6-20 rule (c.o.d. β = 66.7% and γ = 90%)^10,17,20 (Tables 2 and 3).

Table 2.

Estimation of relative expanded uncertainties and uncertainty limits for each response based on the β-content tolerance interval determined by Mee’s approach for the method of determination ofof sotalol in plasma.

Model	Concentration level (ng mL− 1)	Relative expanded uncertainty (%)	Relative uncertainty		Relative expanded uncertainty (%)	Relative uncertainty
			Limits (%)			Limits (%)
			β = 80%			β = 73%
			Lower	Upper		Lower	Upper
Linear regression	25.3533	18.17	−18.24	18.09	21.071	-20.731	20.572
	48.2417	6.39	−11.92	0.86	15.132	-12.816	1.751
	437.8235	12.73	−14.83	10.63	6.470	-16.577	12.378
	838.6479	12.99	−11.59	14.39	3.578	-13.364	16.172
Linear regression after square root transformation	25.3533	16.88	−16.94	16.83	20.146	-19.243	19.140
	48.2417	15.97	−17.34	14.59	18.658	-19.256	16.770
	437.8235	12.33	−13.84	10.81	14.081	-15.531	12.499
	838.6479	11.98	−11.67	12.29	13.219	-13.302	13.928
Weighted (1/X) linear regression	25.3533	16.29	-7.8646	24.73	7.167	-10.104	26.972
	48.2417	15.07	-16.47	13.67	6.567	-18.541	15.742
	437.8235	12.43	-14.77	10.1	7.019	-16.476	11.799
	838.6479	12.19	-11.32	13.06	3.616	-12.985	14.732

Table 3.

The LOD and LOQ assessment based on the statistical approach for the three chosen models.

Model	SD-Intercept	SD-Residue	LOD		LOQ
Linear regression	0.01024	0.04538	13.68	60.63	41.46	183.73
Square root transformation	0.00733	0.02281	0.47	1.47	1.43	4.45
weighted (1/X) linear regression	0.00355	0.0021	4.71	2.78	14.26	8.43

Like accuracy profile, the uncertainty profile is also obtained by joining the extremes of the 95% tolerance interval, i.e. the interval that will contain 95%of the future individual results.

According to the previous conditions and as can be seen from Figs. 1 and 2 the linear regression after root square transformation provides the best uncertainty and accuracy profiles for the procedure. Contrariwise, for the other models, especially the weighted linear regression, they are not adopted for a raison that they are not better contributing to the crucial aim of the experiences.

Fig. 2.

Uncertainty profiles to 95% of the analytical procedure using: (A) linear regression model, (B) linear regression model after processing square root, (C) weighted regression model with a weight equal to1/X as functions of responses. The red dashed lines represent the acceptance limits (± 20%); the dotting lines the 95% tolerance interval connected.

Determination of LOD and LOQ by statistical approach

To determine the “3.3 ” LOD or the “10 ” LOQ, two steps are necessary. The first step is to obtain the minimum response or the minimum signal, and the second is to find the corresponding concentration at the signal from the calibration curve, it is often assumed that the calibration curve is linear to LOD and LOQ and that all points are aligned to the calibration curve^8,12,22. These assumptions are fair for two points virtually and are never valid because the regression leads to uncertainties on the slope and on the intercept. The determination of these parameters can be done based on the calibration curve and by studying samples with a low level of analyte that we can detect or quantify, y-intercept standard deviation of the calibration curve or residual standard deviation of the calibration curve (ICH Q2R2 2022).

There are diversified possible conceptual methods regarding LOD and LOQ, each providing somewhat different results. Depending on the chosen approach, the values of LOD and LOQ can vary greatly which make it difficult for comparative purposes.

The parameters of regression were calculated based on the analysis of four replicates at four different concentration levels. The obtained data were used to compute the two coefficients of the calibration curve and also to perform a lack-of-fit test, which was used to verify that the selected calibration domain was truly linear.

In this approach the standard deviation of the blank is estimated by extrapolation of the standard deviation of the measurement of the analyte concentration zero, this is done by estimate of the standard deviation of the intercept of the regression line.

Evaluation of the LOD and LOQ based on they-intercept standard deviation

In the majority of cases, we use Eq. 16 and Eq. 17 for the LOD and LOQ determination. In this subsection we present an example of the LOD and LOQ calculation using Eq. 16 and Eq. 17, to do this we took the dispersion parameters of the linear regression after square root transformation model. The obtained results for the two other models will be displayed below in the Table 3. The calibration curve has been obtained from the measures in each concentration of the analyte.

The instrumental sensitivity of the signal (Ratio) is represented by the slope of the calibration curve, the slope is equal to 0.0513 and intercept is 0.0733 where [ng/ml] is a unit of concentration.

Standard deviation obtained from the intercept of the linear regression in the figure above is 0.01431. As already cited, this standard deviation represents that of blank, then by applying the Eq. 16 and Eq. 17, the calculation of LOD and LOQ gives us:

Therefore the concentrations resulting in the level of detection and quantification are respectively 0.47 and 1.43 [ng.mL-1].

Evaluation of the LOD and LOQ based on deviation of residue of the regression

In this method the variability of the blank is represented by the linear regression residue standard deviation, this residue reflects the contribution of the variability of the blank in the determination of an analyte for each concentration response⁴, the value of the standard deviation obtained is equal to 0.0166, the sensitivity is equal to 0.0513, the Eq. 16 and Eq. 17 we give:

Concentrations at levels of detection and quantification are 1.47 and 4.45 [ng.mL^− 1].

Graphical assessment

It is known that the limit of quantification depends on the chosen values limits of acceptability and the proportion.represents the intersection of the tolerance limit curve with that of acceptability. From data presented in Table 1 and after the calculation from the limits of tolerance interval we obtained values of LOD and LOQ from uncertainty and accuracy profiles.

The results shown in Tables 4 and 5 that the lowest detected value of sotalol is 0.47 (ng / mL) calculated based on the intercept standard deviation of the linear regression model after square root transformation. The pessimistic founded value is 13.68 (ng / mL) calculated from the standard deviation of the intercept of the simple linear regression model, the weighted model provides a value of 4.71 (ng / mL).

Table 4.

The LOD and LOQ assessment based on the uncertainty profile for the three models chosen. Two different values of β have been chosen.

Future measurements	β = 66.7%		β = 73%
Model	LOD	LOQ	LOD	LOQ
Linear regression	7.39	22.39	8.49	25.72
Square root transformation	3.31	10.02	7.14	21.63
Weighted (1/X) Linear regression	10.5	31.81	11.86	35.94

Table 5.

LOD and LOQ assessment based on the accuracy profile for three chosen models. Two different values of β have been chosen.

Future measurements	β = 80%		β = 90%
Model	LOD	LOQ	LOD	LOQ
Linear regression	2.34	7.08	7.73	23.43
Square root transformation	2.24	6.79	---	---
Weighted (1/X) Linear regression	6.66	20.19	9.88	29.95

The lowest limit of quantification value of sotalol is 1.43 (ng / mL) which is calculated based on the intercept standard deviation of the linear regression model after square root transformation. The highest founded value is 14.26 (ng / mL) calculated from the intercept standard deviation of the weighted regression model.

We notice that we are interested only by the values obtained (LOD and LOQ) was lower field validation. The intercept standard deviation and residue standard deviation of the simple linear regression model are higher compared to other standard deviations from other selected models; this latter is due to the heterogeneity of variances. This large difference between leads to the values ​​of LOD and LOQ not being on the same order of magnitude for the three selected models. Then a test of homogeneity of variances is required.

According to the results of the example, the statistical approach provided incomparable estimates of LOD for each model because the founded values are sometimes underestimated or overestimated. The same situation occurs for LOQ. Indeed, the determination of these limits based on the statistical approach is strongly influenced by the choice of the statistical model and data variability. So, for biological matrices the statistical concepts are not the best methods to be adopted for bioanalytical methods. Otherwise, theoretically the statistical approach gives relevant results in case that the analytical response is linear down the extreme lower dosage, which does not appear for bioanalytical methods. If small quantities of analyte produce a relatively large signal response, perhaps the statistical method is more feasible.

Long and Winefordner have explained, in the statistical approach, that the founded values of the slope of the assay’s calibration curve, the intercept on the concentration values, the signal response²³, and the imprecision associated with these values, are all important factors. As for the weighted model 1/X, the procedure is not validated on all levels of concentration; however, it is validated for the two chosen models, it can be concluded, that the model type clearly influences on the estimation of the LOD and LOQ values. The values of LOD and LOQ obtained by a regression method can vary depending on the number of concentration levels, range of concentration used, number of measurements, influenced by and data heteroscedasticity.

To highlight the influence of the chosen on the estimating of the limits (LOD) and (LOQ), we do the calculation with different values of . All results are displayed in Tables 4 and 5.

Based on the obtained results, the uncertainty and accuracy profiles shown in Tables 4 and 5 are comparable, particularly for the square root transformation model and the weighted model. However, slight differences are observed between these values, which can be attributed to the calculation methods for tolerance limits. These differences arise because the profiles are based on two distinct tolerance intervals: the β-content γ-confidence tolerance interval and the β-expectation tolerance interval.

uncertainty and accuracy profiles rely on tolerance intervals, it is important to distinguish their conceptual and practical differences. As described in our study, the accuracy profile is based on the β-expectation tolerance interval, while the uncertainty profile is constructed using the β-content γ-confidence tolerance interval. The distinction between these two types of intervals has been thoroughly outlined by Mee¹¹.

The β-expectation tolerance interval aims to predict the range in which future individual measurements are expected to fall, considering both random errors and systematic biases, with a specified confidence level. In contrast, the β-content γ-confidence tolerance interval estimates the range that contains a certain proportion (β) of the population, with a specified confidence level (γ). The latter provides a more probabilistic framework for describing the distribution of results, emphasizing the proportion of the population within the specified range.

In practice, these differences affect how the profiles are interpreted. The accuracy profile focuses on the overall performance of the analytical method in meeting predefined acceptance limits across the calibration range. It evaluates whether the total error (bias and imprecision combined) remains within acceptable bounds. On the other hand, the uncertainty profile emphasizes the precision of individual measurements, providing a detailed representation of measurement uncertainty as a function of analyte concentration.

These conceptual differences are reflected in their application. The accuracy profile is particularly useful for determining whether a method consistently meets quality requirements, while the uncertainty profile is better suited for detailed error analysis and decision-making at specific concentration levels.

A method for determination of the LOD and LOQ based on visual assessment proposes a way to estimate a part, often important, of the total uncertainty. It is given from validation data, but it can’t replace an exhaustive procedure for estimating uncertainty.

For the uncertainty profile, we do the computation of the uncertainty limits with two values of the β-content, the first is 66.7% and the second is 73%. Whereas for the accuracy profile we chose two values for the β-expectation.

The results show that the chosen proportion of  has a remarkable effect on the estimate of the tolerance limits for the two strategies: accuracy and uncertainty profiles. Then, the LOQ values will be effectively influenced byvalues, this influence appeared by the difference of the results obtained for the values in Tables 4 and 5. Moreover, the choice of the proportion should be done with caution because it has a direct effect on the validity of the analytical procedure.

It is also important to note that β-values have different interpretations depending on whether they are applied in the context of β-expectation tolerance intervals or β-content γ-confidence tolerance intervals. As outlined in Mee¹¹, the β-expectation tolerance interval focuses on the expected distribution of individual results within acceptable limits, providing a broader assessment of overall method performance. Conversely, the β-content γ-confidence tolerance interval ensures that a defined proportion of the population falls within specified bounds with a given confidence level, which is particularly relevant for decision-making in quality assurance contexts.

Our study aligns with these principles, drawing on the approach described by Saffaj and Ihssane¹⁰, where the choice of β-values reflects the intended application and the variability inherent in the method. In the revised manuscript, we will clarify the rationale for the selected β-values, emphasizing their critical role in defining the acceptability of the method and their adaptability to different application domains.

On the other hand, the estimated LOQ is practically the x-coordinate of the intersection of the limit of tolerance and the acceptability limit . This clearly demonstrates that this latter has a direct effect on this critical parameter.

The meaningful question has been how to best assess the LOD and LOQ. In our situation, the two graphical tools give a relevant and realistic assessment, and since the values found by uncertainty and accuracy profiles are in the same order of magnitude, we recommend the graphical tool to assess these two crucial parameters, especially the method of uncertainty profile which is better estimate the measurement uncertainty. Otherwise, note that the accuracy and uncertainty profiles, constructed respectively from the chosen values of β-expectation tolerance interval and β-content tolerance interval and thus the acceptability criterion λ set by the customer or a regulatory requirement, are directly influenced by the chosen values of these parameters as shown in Tables 4 and 5, which directly influences the values of the limit of quantification and detection.

Conclusion

In this study we have examined two different approaches for the estimation of the limits of detection (LOD) and quantification (LOQ) of the HPLC method of determination of sotalol in plasma using atenolol as internal standard.

The results show that the graphical strategies of validation, uncertainty profile and accuracy profile, based on tolerance interval are a reliable alternative to the classic strategy, based on classical concepts, for assessment of LOD and LOQ. Additionally, LOD and LOQ values produced by uncertainty profile and accuracy profile are not significantly varied.

The ICH method for calculating LOD and LOQ typically relies on the standard deviation of the blank. However, in practice, this is often replaced with the standard deviation of the residuals from the linear model or the standard deviation of the intercept (ordinate at the origin) of the regression model. These statistical models, which have already been validated for analytical methods, are then used to estimate the LOD and LOQ. However, relying on these standard deviations can result in overestimated or underestimated limits of quantification and detection, particularly in the absence of homoscedasticity, as variability across the calibration range is not uniform.

Comparing all LODs and LOQs values produced by both statistical and graphical approaches shows that the LODs and LOQs obtained from graphical approach are more pessimistic, but they are selected as the most realistic estimates for the method above.

Electronic Supplementary Material

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Author contributions

Author contributionsLamia Zaari Lambarki : Performing calculus, writing and interpreting resultsFayssal Jhilal: Supervising, checking, performing calculus, writing and interpreting resultsLamia Slimani: Writing, referencesRidouan El Hajji: Checking results and interpretationsFadil Bakkali: Reviewing the whole articles and improving interpretationsSamy Iskandar: Data visualizing and collectingMariam El Jemli: Performing calculus, writing and interpreting resultsWafaa EL Ghali: Performing calculus and elaborating figures and tablesBouchaib Ihssane: Supervising and writing Taoufiq Saffaj: Supervising the work, creating the idea, reviewing the article.

Funding

There is no funding for this research work.

Data availability

The used data in this articles are provided in this one intitled : Validation des procédures analytiques quantitatives : Harmonisation des démarchesPartie II - StatistiquesCommission SFSTP, P. Hubert, J.J. Nguyen-Huu, B. Boulanger, E. Chapuzet, N. Cohen, P.A. Compagnon, W. Dewé, M. Feinberg, M. Laurentie, N. Mercier, G. Muzard, L. Valat.STP Pharma Pratiques, - volume 16 - N° 1 - janvier/février 2006The full text of the paper entitled : Validation des procédures analytiques quantitatives: Harmonisation des démarches Partie II-Statistiques is available at the following website permalink : http://hdl.handle.net/2268/22157.

Declarations

Competing interests

The authors declare no competing interests.