Responsivity is a conversion qualification of a measurement device given by the functional dependence between the input and output quantities.
A concentrationresponsedependent calibration curve represents the most simple experiment for the measurement of responsivity in mass spectrometry.
The cyanobacterial hepatotoxin microcystinLR content in complex biological matrices of food additives was chosen as a model example of a typical problem.
The calibration curves for pure microcystin and its mixtures with extracts of green alga and fish meat were reconstructed from the series of measurement. A novel
approach for the quantitative estimation of ion competition in ESI is proposed in this paper. We define the correlated responsivity offset in the intensity values using
the approximation of minimal correlation given by the matrix to the target mass values of the analyte. The estimation of the matrix influence enables the
approximation of the position of
Mass spectrometry connected with highperformance liquid chromatography (HPLCMS) is a widely used analytical tool for the analysis of complex biological samples and the detection of different kinds of organic compounds. Recently, the potential of HPLCMS for metabolomic studies has been highlighted due to its capability of routinely handling large sequences of samples. This instrument provides excellent reproducibility and usefulness for qualitative analysis. However, some questions have been raised about the quantitative abilities of HPLCMS analysis. Several studies have discussed the fact that ion competition among different analytes exists when they are simultaneously ionized [
The effectiveness of an analysis depends on two key features of the measurement: (i) experimental performance by the operator and (ii) performance of the instrument. The first feature comprises the precision of the operator during sample preparation as well as during the measurement itself. The second feature may be characterized by the proper mathematical description of the measurement device attributes according to the theory of measurement (general descriptions of basic attributes of every measurement device), which is done to encapsulate the analysis into the appropriate mathematical space. The layout of the possible domain values ensures that the interpretation of the measured datasets also fulfills the mathematical presumptions of the measurement process. Unfortunately, this point of view is not often practically supported.
The basic attribute of the measurement is responsivity. As was already pinpointed in the literature [
The retention time is determined by the separation process on the chromatographic column. The responsivity of the rt quantity is therefore based on the sampling frequency, gradient time, peak capacity, column temperature, and flow rate. The mass to charge ratio depends on the MS detector accuracy and precision and, therefore, on the resolution (or distinguishability). The intensity values for every individual measurement run are generated for all possible pairings of rt
One of the most used representations of measurement capabilities is the limit of detection, which is related to the responsivity, as will be shown later. The limit of detection is usually expressed as the lowest concentration or amount of the analyte that can be clearly detected with a stated degree of reliability from the background or blank sample. However, the evaluation of LOD in concentrations is just a recommendation; generally the formula is valid even for intensity units, the interpretation of LOD is then slightly different as will be shown, but for our purpose it remains consistent. The blank sample is a sample that does not contain analyte but has a matrix that is identical to that of the average analyzed sample. Therefore, the limit of detection (LOD) should vary in different matrices. To verify this, two different matrices of food additives, that is, filamentous green alga
MCYSTLR is heptapeptide with a molecular weight of 994.5 Da that is produced by different cyanobacterial taxa, for example,
The aim of the present study is to characterize and describe the responsivity and minimal ion suppression in mass spectrometer with linear ion trap. We focus on three connected subtopics: (i) the determination of the responsivity function for intensity values in HPLCMS, (ii) the discussion of the LOD and its statistical interpretation, and (iii) an estimation of the ion competition in different biological matrices using the knowledge of responsivity.
The calibration curve for pure MCYSTLR was constructed from the analysis of 10 MCYSTLR concentration measurements. The pure MCYSTLR standard (Sigma no. 33893) was diluted in methanol to obtain the required concentrations. Two food additives were mixed with known concentrations of MCYSTLR: (i) filamentous green algae
The extract composition was analyzed using an HP 1100 Agilent liquid chromatogaphy with an HP 100 XD SLIon trap. The extract was separated on a reversed phase column (Zorbax XBD C8, 4.6 × 150 mm, 5
Obtained data were processed using both manual (supervised) and automatic (unsupervised) tools to compare the two approaches. The analysis of the pure MCYSTLR calibration and mixed samples was carried out to evaluate the influence of food additives extracts on the quantitative responsivity of the mass spectrometer. Supervised parametrized analysis was carried out using the Bruker Daltonik software DataAnalysis 3.3 for data obtained using the LC/MSD trap, which is the standard tool used for the Agilent device. Raw TICs were preprocessed for peak integration in Data Analysis using supervised parameters. The parameters used consist of a Gaussian smoothing of width equal to 4 points in 2 cycles. The outputs of the supervised analysis were exported as Cmpd Mass Spec List Report—MS (P) Layout and contained information on the retention time, maximal intensity, and area after smoothing. The unsupervised nonparametric analysis was carried out using EMP (Expertomica Metabolite Profiling) [
The outputs of both software, that is, Bruker Daltonics DataAnalysis, and EMP were used for comparison of the selected protonated MCYSTLR molecular ion and doubly charged sodium adduct of the molecular ion peaks. The responsivity (dependency of the detector response on known concentrations) for pure and mixed MCYSTLR samples was fitted. The fitting process was carried out for maximal intensity and for the area values of the selected peaks using Matlab cftool. An evaluation of the measurement attributes (responsivity, LOD, CRO, correlation coefficients) was also performed in Matlab.
Generally, the responsivity is defined as a derivative of the transfer function:
The most puzzling issue is the task of the fitting function type specification [
The limit of detection is often incorrectly [
The detection decision at the LOD leads to a risk of false detects. The LOD is constructed as the level of false nondetects with some probability. This definition offers the possibility to detect an analyte below the LOD because the proper values of risk and probability are sample dependent. Hypothesis testing involves the distribution of results under the null hypothesis only. The probability of false nondetect increases with decreasing analyte concentration. However, the risk of false positive remains small as long as the result exceeds some critical level. The IUPAC definition of LOD is based on the homoscedastic assumption that the uncertainty does not depend on the actual analyte level. This assumption is usually violated [
In this paper, we present a definition of the correlated responsivity offset (CRO) of detection, which is derived from the common interpretation of the LOD. The connection of HPLC with MS is advantageous in comparison to other chromatographic methods, as MS adds an extra dimension to the measured dataset. Therefore, the technical operating parameters of mean values and standard deviations should be computed independently for each measured
In statistics, covariance provides a measure of the correlation between the changes of two variables. The covariance of two random variables is evaluated as the difference of the mean value of the multiplicity of the variables minus the multiplicity of the mean values of the variables. Variance of one random value is a special case of covariance and is used when the two variables are identical. The standard deviation is the square root of the variance.
The evaluation of many variables produces a covariant matrix in which the elements represent the covariance between two given variables. The most familiar measure of dependence between two variables is the correlation coefficient. The correlation coefficient is computed from standardized random variables, that is, the covariance of multiple variables’ standard deviations. The square of the correlation coefficient times 100 is called the strength of relation, in percentage.
The total covariance of many variables is simply approximated as the maximal covariance of a given variable with all other variables. This approximated total covariance is always a bit lower than the true total covariance. Therefore, the maximal covariance of two variables is the minimal covariance of all variables. Correlated standard deviation should be also approximated as the square root of the approximated total covariance value.
We therefore define the correlated responsivity offset (CRO) as the mean blank intensity value of a target
We measure two blanks of the matrices,
From each blank, we compute several statistical attributes of the target mass
The mean values (2) and correlated standard deviations (2) are used for the evaluation of correlated responsivity offsets, CRO1 and CRO2 (defined in previous section).
The measured mass
We also measure the calibration curve of mass
The hypothetical linear responsivity function is
We want to know how the calibration curve of target mass
The correlation coefficient is described as a slope between correlated variables. We can assume some “ideal curve”
The same assumption as in previous step (6) is made for matrix
From the hypothetical responsivity (5) and the ideal curve (6), both in matrix
Accordingly, we put the estimated ideal curve (8) into the equation for an ideal curve (7) in matrix
There remains the question of the proper offset in matrix
The final equation for the estimation of the calibration curve in matrix
This equation evaluates the minimal competition for target mass
For the simple estimation of calibration curve
The chromatographic peak for MCYSTLR was observed at a retention time of approximately 17.1 min under the gradient conditions described previously (Figure
Total ion chromatogram (TIC) of pure MCYSTLR (10
Calibration curves were reconstructed using linear and exponential functions for pure MCYSTLR in MeOH (10 concentrations in duplicates), MCYSTLR in
Evaluation of root mean square errors (RMSEs) for linear and exponential fittings. Fittings were done for concentrationresponse dependences for areas and maximal intensities of the MCYSTLR molecular ion of
Expertomica  Manual  

Calibration  Salmon  Stigeoclonium  Calibration  Salmon  Stigeoclonium  
Linear RMSE  
Area NA adduct  0.008977  0.128800  0.025770  0.012140  0.058260  0.047040 
Area molecular ion  0.028660  0.082940  0.041840  0.027280  0.147200  0.040940 
Max intensity NA adduct  0.018420  0.142500  0.126200  0.022360  0.034610  0.021190 
Max intensity molecular ion  0.009455  0.038150  0.069690  0.033760  0.015690  0.015250 
Exp RMSE  
Area NA adduct  0.008033  0.080920  0.022590  0.007450  0.034090  0.036680 
Area molecular ion  0.001540  0.048970  0.020280  0.008426  0.097570  0.037800 
Max intensity NA adduct  0.017030  0.091840  0.129800  0.010750  0.023110  0.021510 
Max intensity molecular ion  0.009579  0.034330  0.075650  0.010090  0.004033  0.014200 
Slopes of the linear fittings and exponents for the exponential fittings. Fittings were carried out for concentrationresponse dependences for the areas and maximal intensities of the MCYSTLR molecular ion of
Expertomica  Manual  

Calibration  Salmon  Stigeoclonium  Calibration  Salmon  Stigeoclonium  
Linear slope  
Area NA adduct  0.10060  4.98100  0.97450  0.10080  3.49100  0.90970 
Area molecular ion  0.09800  4.76000  1.03900  0.09802  3.20800  0.98200 
Max intensity NA adduct  0.10010  4.37200  0.86500  0.10160  3.99900  0.92120 
Max intensity molecular ion  0.10080  3.62500  0.93860  0.09746  3.47800  0.99960 
Exp exponent  
Area NA adduct  −0.01077  1.00100  −0.31260  −0.02181  1.00100  −0.69510 
Area molecular ion  0.06452  0.99960  −0.67920  0.05913  1.00200  −0.66400 
Max intensity NA adduct  −0.02063  1.00400  1.26000  −0.04363  1.00100  0.09998 
Max intensity molecular ion  −0.00655  0.99900  0.36720  0.07418  0.99950  0.15920 
Linear and exponential fittings of concentrationresponse dependence for the peak area of the pure MCYSTLR molecular ion of
In order to examine the responsivity of the three different sample types, we compared the parameters of the fitting functions for all of them. The concentrationresponse curve parameters exhibited remarkable differences when reconstructed by both linear and exponential functions for pure MCYSTLR and MCYSTLR in
All measurements were analysed using two methods: (i) manual and (ii) nonparametric analysis. Nonparametric Expertomica metabolomic profiling enabled automatic noise subtraction as well as automatic peak decomposition. Expertomica was able to retain all important ions for microcystinLR in the
Example of lines in the Expertomica metabolite profiling software PRT report for detected peaks of the MCYSTLR molecular ion of
Mass  Probability  Relative content  RT  Begin time  End time  Max intensity  Area  

Peak  509.2  0.005  0.05847  17.1129  16.9146  17.5930  1693675 

Peak  861.0  0.952  0.00296  17.0323  17.0179  17.1732  260411 

Peak  995.7  0.952  0.00062  17.0323  17.0323  17.0395  256928 

Total ion chromatogram (TIC) of MCYSTLR (0.01
The PRT reports show relevant information about MCYSTLR in a reasonable way and is simple to be used in postprocessing. Manual data analysis (by Brucker Daltonic DataAnalysis) requires supervised parameterization for data smoothing to reliably integrate the peaks that were manually selected (the
The LOD was computed as the mean blank value of the target mass plus 3 times the blank standard deviation of the target mass. The computed parameters are shown in Table
Computed statistical parameters of target mass values (995 for [M + H]^{+}, 509 for [M + H + Na]^{2+}, and 861 for [M + H − 135]^{+}) in three different blanks (MeOH,
Blank  MeOH  Stig  Hymc  

Ion  995  509  861  995  509  861  995  509  861 
LOD 









Covariance 









Correlation  0.1953  0.5972  0.2759  0.3723  0.6806  0.4428  0.5574  0.6037  0.5133 
Strength of relation 









Correlated standard deviation 









Mean 









CRO 









Computed LODs were compared with the measured calibration curves for all three matrices (Figure
Calibration curves for MCYSTLR in three different matrices (MeOH,
Therefore, univariate LOD represents the basic offset as the blank mean value plus 3 times the univariate (noncorrelated) sensitivity. Therefore, this LOD is the level of false nondetects and should also be interpreted as the univariate offset. The LOD for false detects is better to evaluate as Boqué’s multivariate detection limit (MDL). The correlated offset, as a step between the univariate and multivariate approach, is introduced as the correlated responsivity offset, CRO. The CRO value represents the minimal estimation of the matrix contribution to the analyte signal offset via the correlated standard deviation (~correlated sensitivity). Thus, CRO is a useful quantity for ion competition estimation.
Reference responsivity (in MeOH) was fitted using Matlab cftool. The RMSE of the responsivity function linear fitting was very small; however, exponential fitting produced a better fit (Table
The estimation of ion competition in food additive matrices (Stig and Hymc) was computed via (
the reference correlation coefficient,
the correlation coefficient,
the reference correlated responsivity offset, CRO1, of the reference matrix (MeOH),
the correlated responsivity offset, CRO2 or CRO3, of the food additive matrix blank (Stig or Hymc),
and the parameters of linear fitting (linear slope
Results of the estimation are shown in Figures
Calibration curves of the protonated MCYSTLR molecular ion in MeOH and estimated calibration curves in food additives matrices (salmon hydrolyzate and
Calibration curves of the doubly charged sodium adduct of the MCYSTLR molecular ion in MeOH and the estimated calibration curves in food additive matrices (salmon hydrolyzate and
Calibration curves of the MCYSTLR molecular ion with cleavage of the Adda moiety in MeOH and the estimated calibration curves in food additives matrices (hydrolyzate and
Manual fragmentation, the mother ion 509.2 [M + H + Na]^{2+} provided MCYSTLR molecular ion 995.3 and dehydrated ion 977.5 in the MS2 spectrum.
The estimated calibration curves for the food additives represent the most probable position of the measured calibration curves according to the matrices offsets and correlations. The exact intensity values are, of course, sample dependent and based on all influences of the matrix on the analyte. Information on the total influence is not present in the blank measurement; therefore, the complete information cannot be estimated before the measurement of the analyte in the matrix is done, at which point the estimation is no longer necessary. However, partial information, that is, correlated responsivity offsets (CROs) and minimal correlations (
It is quite obvious that some compounds will coelute with the analyte and interfere with ionization. In other words, it is very predictable that matrices may influence and compete with analyte ionization by ESI. However, up until now, an evaluation of the influence of matrices was not possible. In this paper, we propose a method for estimating the competition of analyte hepatotoxin MCYSTLR ions in the measurement of calibration curves in food additives by HPLCMS. The influence of the matrix comprises two major parts.
The chemical noise contributes as the offset to the intensity value of target mass values.
The matrix composition affects analyte ionization by correlation of the theoretical offset and even more by the correlation of the responsivity slope in the measurement of the calibration curve.
The main advantage of our approach is the evaluation of the minimal correlation given by the matrix to the target mass values. Therefore, we can estimate both the responsivity offset and the correlation of that offset and the calibration (responsivity) slope. This correlation information is directly evaluated from the blanks of the matrices. In combination with the known measurement of the calibration curve in a known matrix, it can be used for the estimation of the position of the calibration curve in other matrices (food additives) that are known only from their blanks.
There are several disadvantages that require deeper induction. First of all, our estimated correlation does not represent all of the correlations in the matrices. Correlation is computed as the maximal correlation of the target mass and all of the other masses. From the statistics, it is known that this computed correlation is just the minimal total correlation. The value of the total correlation should be computed via the recurrence equation as the multiplication of primitive and partial correlation coefficients, if those are known. Unfortunately, this is still just the correlation to the matrix noise at the target mass and not to the analyte ion. Correlation of all analyte ions cannot be performed until the measurement is done. However, once the measurement is done, no estimation of the correlation is required.
The situation of the unknown precision of the total correlation to the analyte ions means that the estimated responsivity slope and real measured responsivity slope will differ. The slope of the responsivity should be slightly higher or lower, which leads to the important point: the estimated calibration curve cannot be extrapolated. The estimated values are valid only in the short interval of linearization. The additional error contribution to extrapolation is the nonlinearity of the responsivity function.
The correlated responsivity offset, CRO, is computed from the blank noise mean and the correlated sensitivity (correlated standard deviation) of that noise. The CRO is useful for low concentrations above the critical limit of detection (MDL). However, it is expected that the “strength” of the analyte amount will influence the offset during ionization, especially for very high concentrations. The magnitude of the effect of this influence remains unknown.
Therefore, the exact values of correlation and offset are sample dependent and should be performed only via experiments. On the other hand, the estimation of the minimal influence is hidden in the blank measurements of the matrices. Once again, it is the matrix noise influence and not the total correlation of the analyte. Even so, it is the best approximation of the responsivity and, therefore, of the ion competition and the calibration curve, which should be easily revealed with available knowledge. Therefore, the matrix blank represents the minimal required set of information. Estimation of ion competition via the correlated responsivity offset offers a simple approach for the evaluation of the probable position of the calibration curve in a given matrix. This method is derived directly from the basic properties of the theory of measurement.
The change in responsivity of pure MCYSTLR and mixtures of MCYSTLR in complex biological samples indicates the influence of coeluting compounds. The phenomenon of ion competition in MS (ESI) has been discussed previously in the literature. In our study, the type of responsivity function (calibration curve) was tested and the exponential function was fitted to the measured calibration curves. For small intervals of three consecutive concentrations, it is sufficient to use approximation via a linear function.
We confirmed that the standard limit of detection (LOD) approach typically leads to the neglect of data points that are well within the range of the response curve. With the knowledge of the blanks’ mean value and correlated standard deviation (sensitivity), we proposed a method for evaluating the correlated responsivity offset (CRO) for individual target masses of the analyte ions in any given matrices. This value should be used for the estimation of quantitative ion competition among different analytes when they are ionized at the same retention time.
The evaluation is valid only for congruent measurement conditions, including the device settings, mobile phase composition, and gradient changes. Agreement between the theoretical and experimental values is sufficient. The proposed algorithm of the correlated responsivity estimation is computationally easy and is promising for wider usage in LCMS. However, further investigation and verification of additional multivariate responsivity properties remain our focus.
This work was supported and cofinanced by the South Bohemian Research Center of Aquaculture and Biodiversity of Hydrocenoses (CENAKVA CZ 1.05/2.1.00/01.0024), by the Center for Algal Biotechnology TřeboňALGATECH (CZ. 1.05/21.00/03.0110), by the South Bohemia University Grant GA JU 152/2010/Z, and by the Postdok JU (CZ. 1.07/2.3.00/30.0006) in part.