Accepted Manuscript Univariate analytical calibration methods and procedures. A review Paweł Kościelniak, Marcin Wieczorek PII:
S0003-2670(16)31108-4
DOI:
10.1016/j.aca.2016.09.024
Reference:
ACA 234807
To appear in:
Analytica Chimica Acta
Received Date: 4 April 2016 Revised Date:
9 September 2016
Accepted Date: 11 September 2016
Please cite this article as: P. Kościelniak, M. Wieczorek, Univariate analytical calibration methods and procedures. A review, Analytica Chimica Acta (2016), doi: 10.1016/j.aca.2016.09.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Univariate analytical calibration methods and procedures. A review.
Paweł Kościelniak, Marcin Wieczorek Department of Analytical Chemistry, Faculty of Chemistry, Jagiellonian University,
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30-060 Krakow, Poland
Abstract
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An original focus on univariate calibration as an experimental process of quantitative analysis is presented. A novel classification system is introduced against the background of the present
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situation concerning nomenclature of calibration methods. Namely, it has been revealed that four methods well-known in analytical chemistry: the conventional method, the internal standard method, the indirect method and the dilution method, can be split into those carried out in both the interpolative and the extrapolative mode. It is then shown that the basic
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procedures of all these methods can be modified including different approaches, such as matrix-matched technique, spiking the sample with a reactant, bracketing calibration, and others. For the first time (as compared to monographies dealing with univariate calibration) it
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is reviewed how the methods are mixed and integrated with one another thereby creating new calibration strategies of extended capabilities in terms of enhanced resistance to the
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interference and non-linear effects – as the main sources of systematic calibration errors. As additional novelty, rationally possible combinations of the calibration methods – not met hitherto in the literature – have been predicted. Finally, some general rules relating to calibration are formulated and the main calibration problems that still need to be solved are displayed. Keywords: Univariate calibration; Calibration methods; Calibration procedures; Classification of calibration methods
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ACCEPTED MANUSCRIPT Contents 1. Introduction 2. Calibration methods 3. Modified calibration procedures 4. Combination of calibration methods
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4.1. Mixed methods 4.2. Integrated methods 5. Other methods and procedures
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6. Conclusions
1. Introduction
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In spite of broad improvements in methods and instrumentation, analytical calibration still remains a key element in analytical chemistry. Calibration is a crucial step in almost every chemical procedure leading to evaluation of the concentration of a determined substance (analyte) in a sample. This is so because the instruments used for chemical analysis
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provide, in principle, information in the form of an intensity of the analytical signal that only corresponds to the analyte concentration but does not directly determine this concentration. The main task of calibration is thus to transform the signal intensity into the analyte
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concentration as accurately and precisely as possible. Although billions of chemical analyses are performed every day all over the world and
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the great majority of them lead to the determination of analytes on the basis of univariate calibration, this kind of analytical calibration is not very popular among analysts as an individual, methodologically and practically interesting subject. In particular, the most recent literature comprehensively dealing with calibration problems is related to multivariate analysis [1-4] with special attention paid to elaboration of the calibration data with the use of chemometric methods. The last article relating broadly to the methodological aspects of univariate calibration appeared in 2001 [5] and it covered the subject in the context of
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ACCEPTED MANUSCRIPT chemical standards rather than generally. In some other papers the fundamental issues of univariate calibration were reviewed either from metrological [6] and statistical [7-9] points of view or in relation to individual analytical methods, including mainly chromatography [10,11] and ICP-MS [12]. Specific calibration approaches that are applied when a sample is
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analyzed by flow techniques [13-15] or is prepared by solid phase micro-extraction [16] have also been summarized recently.
The above situation motivated us to write the present paper, in which we have
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attempted to discuss univariate calibration methodologies and strategies in analytical chemistry more generally and comprehensively. The main goal of this paper is to present the
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advantages and potential applicability of a wide range of various calibration methods, including those which have been forgotten to some extent or exploited very rarely in practice. It was decided to reveal the possibilities of how they can be performed in two essentially different modes (interpolative and extrapolative) and in accordance with different modified
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procedures, as well as how they can be mixed and integrated with each other. The calibration methods and procedures are primarily discussed from the perspective of their capability of overcoming the interference effects as the principal source of systematic errors that can occur
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during the calibration process.
The article is essentially focused on purely practical and chemical calibration aspects,
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whilst bypassing mathematical and metrological approaches to the subject. Our own experiences in calibration – in comparison with other works – are widely presented and some of our own concepts and ideas are suggested and put forward for general review. The paper is organized as follows: the principles of analytical calibration are first desrcibed, along with a discussion on calibration classification. Then, examples of various calibration procedures and methods exploited in both batch and flow analysis are then
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ACCEPTED MANUSCRIPT presented and discussed. The final section is dedicated to more specific calibration approaches, which are however widely applied in particular analytical areas.
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2. Calibration methods According to the last recommendation of the Joint Committee for Guides in Metrology calibration is defined as “operation that, under specified conditions, in a first step, establishes a relation between the quantity values with measurement uncertainties provided by
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measurement standards and corresponding indications with associated measurement
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uncertainties and, in a second step, uses this information to establish a relation for obtaining a measurement result from an indication” [17]. This general definition can be adapted to analytical practice and purposes through the following expression: “calibration is a process encompassing modeling of the theoretical relationship. i.e. calibration dependence, between the intensity of the analytical signal and the concentration of a substance (usually an analyte)
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in the form of an experimental dependence, i.e. calibration function (plotted as calibration graph), formulated with the use of chemical standards and transformation of the intensity of
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the signal measured for a sample assayed into the analyte concentration in the sample [18]. The purely analytical character of above definition can be understood by drawing an
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analogy with the relationship between calibration dependence and calibration function and the real (true) and found (experimental) concentration of the analyte in a sample: the calibration dependence, like the real analyte concentration, can only be experimentally evaluated but never ideally accurately determined. Another relationship is as follows: the better (more accurate) the modeling of the calibration dependence by means of a calibration function, the greater the accuracy of the analyte concentration in a sample. From a practical point of view, the calibration process consists of four stages: the preparative stage (i.e., the preparation of sample and standards), the measurement stage (i.e.,
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ACCEPTED MANUSCRIPT reading the signal intensities for sample and standards), the modeling stage (formulation of the calibration function), and the transformation stage (calculation of the analytical result) [15]. As the great majority of analyses are performed with the use of liquid samples and standards (even if solid or gaseous samples are originally to be analyzed) as well as the
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samples are sometimes prepared together with and not separately from the standards, the term “calibration solutions” instead of “sample and standards” will be used throughout the paper. In the analytical literature, two further useful terms relating to calibration are also
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often encountered: calibration procedure and calibration method. Unfortunately, they are not strictly defined and distinguished in any relevant official documents or unofficial articles. In
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our approach [19], the calibration procedure is a detailed mode of performing all the abovementioned four stages of the calibration process. If the calibration procedures are followed according to specified rules – which determine a more general way of proceeding, leading to the attainment of (besides the main calibration objective) certain additional analytical goals –
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they constitute a calibration method.
Whichever of the calibration method is used, different errors can be committed in the calibration stage. Certainly, the trueness of standards is the ultimate limit to producing ever
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improved calibrations. However, the most characteristic “calibration” source of systematic errors is the interference effect, i.e., a change in the intensity of the analytical signal due to
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substances (usually sample components) other than the analyte. The interference effect of additive (unspecific) and multiplicative (specific) nature is revealed by constant and proportional dislocation of the calibration dependence in relation to the calibration function formulated on the basis of just an analyte. An effect of more complex nature is able to make this dependence non-linear in the entire range of the analyte concentrations or even linear and non-linear in certain intervals of this range. In many cases, a mixture of all kinds of interferences is also encountered.
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ACCEPTED MANUSCRIPT Another disadvantageous situation, which can be named the non-linear effect, is when the sample matrix is free of interferents but the calibration dependence is non-linear for some other reasons (e.g. instrumental). If the dependence is modeled by a linear calibration plot function (which is commonly done), erroneous analytical results can be expected again.
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In analytical chemistry, several calibration strategies of varying popularity are known that can be considered to be ‘calibration methods’ in the light of the definition mentioned above. The crucial questions are: how are they linked to both calibration effects (i.e.,
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interference and non-linear effects) and can these effects be overcome if they have not been recognized and eliminated in the initial steps of an analytical procedure?
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The calibration method that is definitely most often used is the calibration curve method (CCM). The basic procedure is simple and relatively fast. It involves the modeling of a calibration function using several standards containing the analyte (usually alone) at various concentrations – encompassing the analyte concentration in the assayed sample. The sample is
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subjected to measurements separately from the standards; the signal intensity obtained is related to the calibration function and the analyte concentration in the sample is calculated in an interpolative way (see Fig. 1A). In this form, the method evidently leads to an inaccurate
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analytical result if the interference effect occurs (see Fig.1SA). On the other hand, a nonlinear calibration dependence can be modeled exactly if the number of standards is great
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enough, and even if it is not, the error cannot be expected to be great. Another quite popular calibration method introduced into analytical practice in 1935
[20] is the standard addition method (SAM). The basic SAM procedure consists in the addition of a standard solution of increasing analyte concentrations to the sample by such means that the concentrations of analyte and of other components of the sample are kept constant. A calibration function is created for the analyte added to the sample, and the analyte concentration in the sample is calculated by extrapolation of this function to the zero signal
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ACCEPTED MANUSCRIPT (see Fig. 2A). As in all calibration solutions, the increasing concentrations of analyte are accompanied by the same amount of interferents, the method gives a chance to eliminate (compensate for) interference effects, but only ones of a multiplicative nature. On the other hand, the systematic error caused by the non-linear effect may be quite great because of the
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extrapolation process (see Fig. 1SB).
A method especially often exploited in separation and ICP techniques is the internal standard method (ISM). In this case, the standard solutions and the sample are spiked with the
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same and known amounts of a substance (internal standard), which is absent from the (unspiked) sample. It is chosen in such a way to be similar to the analyte in terms of
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physicochemical properties and to provide a signal that is readily distinguishable by the instrument from the signal produced by the analyte. The calibration function is created on the basis of the analyte-to-internal signal intensity ratios measured for the calibration solutions and the signal intensity ratio measured for the sample leads to interpolative calculation of the
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analyte concentration in the sample (see Fig. 1B). An additional objective of the method is to determine the analyte with increased precision if the size and direction of random fluctuations of the intensities of analytical signals measured for both components are similar. In general,
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the interference effect is not expected to be compensated for, apart from such exceptional cases where the interferents change both signal intensities (from the analyte and the internal
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standard) in the same direction and to the same extent. As in the above version, the method has an interpolative character, and its resistance to the non-linear effect is similar to that of CCM.
The indirect method (IM), is usually applied when the analytical instrument used is not able to produce the signals (or produces signals of too weak intensity) directly for the sample component required to be determined. Although the method is mentioned very rarely in the contemporary literature, it is exploited quite often in both chemical analysis (e.g. for the
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ACCEPTED MANUSCRIPT determination of anions by FAAS) and biochemical analysis (e.g. using antigen-labelled competitive ELISA technique [21]). In accordance with the basic IM procedure, a constant, known and excessive amount of substance that reacts with the analyte is added to the sample and to standard solutions of increasing analyte concentrations, and the signal intensities are
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measured for either the reaction product or the reagent remaining after the reaction. An increasing or decreasing, (respectively) calibration function is then formulated and the analytical result is obtained in an interpolative way (see Fig. 1C). In principle, the method is
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linked to both calibration effects in a similar way to CCM. However, if it is known or suspected that the analyte is strongly affected by interferents, then the reactant can be chosen
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in such a way as to avoid effects caused by the same or other interferents.
In the dilution method (DM), brought to analytical chemistry in 1959 [22], a single standard solution containing the given analyte is used. Both the standard and the sample are progressively diluted and at each dilution stage, both solutions are measured. Each pair of
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analytical signals obtained allows the apparent concentration of the analyte in a sample of a given dilution to be estimated with the help of a two-point calibration function (see Fig. 1D). On the basis of the obtained results, the apparent concentration corresponding to a sample of
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infinite dilution is and it is considered as the final analytical result. The dilution method can be successfully applied when the non-linear effect and an interference effect of a different
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nature (except an additive one) occur, as both effects usually tend to be diminished in the course of sample dilution and disappear in the infinitely diluted sample. Where calibration methods are classified (e.g., in textbooks), they are commonly
divided into absolute methods and others. Among the latter methods, three are usually mentioned, namely CCM, SAM and ISM, under different names [21,23-25]. In recent years, a tendency has been noticeable to differentiate them according to whether the standard is prepared separately from the sample or added to the sample, and the terms: external and
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ACCEPTED MANUSCRIPT internal methods are respectively recommended [24-27]. SAM and ISM are sometimes included in the group of internal methods [8], and a negative consequence of this is that SAM can be considered to be a version of ISM [28]. The classification suggested by the authors of this paper is related to the
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aforementioned definition of chemical calibration [19]. We propose to divide calibration methods depending on how the signal measured for a sample is transformed into the analyte concentration in the sample. It seems to be a more fundamental criterion than one based on
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laboratorial and measurement aspects (as in the hitherto prevailing approach).
A fundamental reflection is that, in principle, at the preparative stage, SAM is similar
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to CCM except that the standards are prepared together with and not separately from the sample. As a consequence, the calibration dependence can be partly, but not entirely, modeled and the analytical result is calculated in an extrapolative and not interpolative way. It has been revealed that the remaining methods, i.e. ISM, IM and DM, can be also carried out at the
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preparative stage not only in the mode described above (i.e. separately from the sample), but also by spiking the sample with standards. The consequences of the changes made are the same as in SAM, including that the analytical results are in all cases calculated in an
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extrapolative and not interpolative way.
Thus, the calibration methods mentioned above can be split into those carried out in
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interpolative and extrapolative modes. As CCM and SAM can be considered the most conventional ones, it is proposed to name them the interpolative conventional method (I-CM) and the extrapolative conventional method (E-CM), respectively. For the remaining methods, we recommend that their traditional names should be kept, but qualified with the adjectives ‘interpolative’ or ‘extrapolative’. All the new names with their abbreviations are listed in Table 1 (the same abbreviations will be used in further sections of this paper).
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ACCEPTED MANUSCRIPT Concerning literature examples of the extrapolative calibration approaches, the E-ISM method was used (probably first time) by Matisova et al. [29] as a modified standard addition method in gas chromatography and then it was exploited in, e.g., photon activation analysis [30] and ICP [31]. In a textbook relating to liquid chromatography it is mentioned as the mean
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with addition of determined substance [27]. E-IM was developed by Kościelniak et. al [32] on analytical examples using flame AAS and UV/VIS spectrophotometry. E-DM was introduced by Pszonicki et al. [33] as the standard addition and successive dilution method, carried out in
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accordance with batch procedure and then adapted to flow injection analysis as the gradient ratio standard addition method [34].
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On the basis of theoretical and experimental studies, the following general conclusion can be drawn: the essential difference between interpolative and extrapolative calibration methods is that the latter can compensate for interference effects of a multiplicative character, whilst the remaining features (advantages and disadvantages) of all individual interpolative
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methods are preserved in their extrapolative versions.
Figs. 1 and 2 show how the interpolative and extrapolative methods are interpreted in comparison with each other, i.e. how the calibration functions are formulated and how the
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analytical results are calculated. The most controversial and very curious interpretation of the measurement results pertains to the E-IM method where the calibration function has a
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negative slope (see Fig. 2C). It was revealed [32] that in such a case the analytical result is obtained by extrapolation of the function to the signal intensity corresponding to the standard solution containing the reactant alone, and hence such a solution has to be prepared in addition to the other solutions required by the method. We have suggested also how the calibration procedures developed in flow analysis can be specifically arranged [13,14]. In this case, it was decided to define the type of calibration not only on the basis of how the signal intensity is transformed into the analytical result, but
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ACCEPTED MANUSCRIPT also how the calibration is modeled in the form of calibration functions. The flow approaches were categorized as interpolative [13] and extrapolative [14], and most of them were further divided according to the number of calibration functions (graphs) that needed to be formulated: one-graph, several-graph and multi-graph procedures. In addition, all methods
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were also distinguished with respect to either direct or indirect transformation of measurement data.
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3. Modified calibration procedures
Calibration methods can be performed according to different procedures that have
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been modified in relation to the basic ones (described above). The purpose is to improve the calibration method in terms of either the analytical performance (e.g., to avoid calibration errors) or the efficiency (e.g., to make the analytical process faster). From among many procedures reported in the literature, the most versatile examples are briefly described below,
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i.e. those that can be performed with the use of various analytical techniques (including those exploited in biochemical analysis), methods and detectors. Besides, even if they are originally presented as used in analysis of liquid samples, most of them can be adapted to analysis of
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samples in solid and gaseous states.
One of the procedures relating exclusively to the preparation of calibration solutions is
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the matrix-matched technique. Being highly versatile in principle, in the literature it has been adapted to both the I-CM and I-ISM methods, and recently has especially often been used in chromatography [35,36] and ICP-MS [37,38]. It is also routinely exploited in biochemical analysis using such immunoassay techniques as RIA and ELISA [21]. The procedure It is usually performed in such a way that the standards of different analyte concentrations are mixed with the analyte-free matrix of the sample analyzed. In principle the result is similar to that when E-CM is used, i.e., the calibration dependence is modeled accurately by the
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ACCEPTED MANUSCRIPT calibration function and the multiplicative interference effect is compensated for. As the analytical result is calculated in an interpolative way, it is possible for the analyte to be determined even more accurately than by using E-CM. What is more, if the matrix contains a component causing an interference effect of an additive nature, the effect can be estimated
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through the measurement of the matrix alone (unspiked by the analyte).
A special role in the matrix-matched procedure play the certified reference materials (CRM’s) [39]. If CRM is suited to the sample analyzed with regard to its composition it can
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serve itself as the standard for one-standard I-CM calibration (see below) performed in accordance with the matrix-matched procedure. If, in addition, CRM is free of analyte, it can
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be added to the set of standards considered as the matrix of the sample analyzed. It should be, however, pointed that the preparation of a set of standards in a way of dilution of a single CRM is not consistent with the principle of the matrix-matched procedure and, consequently, can not lead to elimination of interferences. As the most CRM’s are available in the solid
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state, they are especially useful for matching the matrix of solid samples (in particular, using laser-ablation ICP-MS [12,38]).
Another procedure undertaken quite often in analytical practice at the preparative
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calibration stage is spiking the sample with a special reactant. The reactant is referred to as ‘special’ because it is chosen in a way that selectively eliminates a given interference effect
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(usually multiplicative). As a rule, the standards are also spiked with the same reactant in order to compensate for its possible influence on the analytical signal. Such a procedure is usually adapted to calibration by I-CM, but it evidently has wider applicability. In particular, if E-DM is used, it is suggested that the calibration solutions should be diluted with a reactant chosen as a spectral buffer, and its concentration should be kept constant in order to support the dilution process in the elimination of interferences [33]. It was also revealed that the same aim can be achieved even more effectively if both calibration solutions are injected in a flow
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ACCEPTED MANUSCRIPT system into a carrier stream containing spectral buffer, as then a decreasing amount of interferents is able to meet a progressively increasing amount of buffer [40]. If the calibration function is modeled by a non-linear dependence in a region corresponding to low analyte concentrations, a simple modification of I-CM, which can be
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termed the signal increment procedure, can be applied [41]. It involves the preparation of a set of standards in the usual manner, but the sample is spiked with a known amount of the analyte in order to take the total analyte concentration into the linear region of the calibration
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dependence. The analyte concentration in the sample is calculated in an interpolative way, with the initial analyte concentration taken into account. Recently, the above approach has
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also been applied to E-CM [42] (see Fig. 2S), what is even more reasonable, as the nonlinearity of the low-concentration calibration dependence can lead to much more inaccurate results in E-CM than in I-CM .
The calibration methods (except I-DM and E-DM) are usually performed with the use
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of several standards, enabling creation of multipoint calibration functions. However, the multi-standard procedure is quite often limited to preparation of a single standard solution allowing one-point calibration (which should rather be named ‘one-standard calibration’) to
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be performed [43]. If the procedure is adapted to I-CM, a calibration function is formulated on the basis of the zero-signal (or of the signal intensity measured for a blank solution) and the
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signal intensity corresponding to the standard [43,44]. In the case of E-CM, the signal intensities measured for the sample with and without the standard added are taken for linear extrapolative calibration [45]. It is evident that when using such a procedure one needs to be sure that the calibration dependence in the concentration range considered is linear. A single standard solution can be exploited more efficiently than in the one-point calibration procedure when the signal intensity produced by the standard is controllably changed by a change of selected instrumental parameters. In such a case, the one-point
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ACCEPTED MANUSCRIPT several-graph procedure can be carried out, i.e. two-point calibration functions are formulated corresponding individually to each parameter value. The signal intensities measured in the same conditions are then referred to appropriate calibration functions, allowing a set of apparent analyte concentrations and their mean value (as the final result) to be calculated. The
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calibration data are interpreted similarly to I-DM (see Fig. 1D), but the difference is that the sample is not diluted and the apparent concentrations are not considered to be systematically, but randomly changed. Such calibration is often encountered in flow injection analysis, as the
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flow systems can treat calibration solutions in a special manner (e.g., inject them from different loops [46] or transport them after injection for different lengths of time [47]) in order
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to differentiate the signal intensities significantly and repeatedly. The one-point several-graph procedure was also adapted in flow injection analysis to calibration by E-CM [48]. A common drawback of both versions (interpolative and extrapolative) of this modification is a risk of leading to inaccurate results if the calibration dependence is non-linear.
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I-CM is occasionally performed in accordance with bracketing calibration [49,50], which is in fact a version of the two-point calibration procedure. In this case, two standards are selected: with the analyte at lower and higher concentrations than the analyte
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concentration in the sample. If the concentrations chosen are close enough to each other, the method becomes resistant to the non-linear effect, i.e. the segment of the calibration function
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can be considered linear even if the calibration dependence is really non-linear. Thus, such a procedure helps in the improvement of analytical results in terms of their accuracy. In contrast to the above approaches, one that can be termed the consecutive one-point
calibration is focused exclusively on the modeling and transformation stages of the calibration procedure [51]. It was originally applied to multipoint E-CM, and consists in extrapolative calculation of the apparent concentrations of the analyte in the sample basing on measurements performed for the sample and for each consecutive solution of the sample with
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ACCEPTED MANUSCRIPT standard addition (see Fig. 3A). The apparent concentrations are then expressed as a function of the added analyte concentrations, and the final result is calculated by extrapolation of this function to zero added concentration. As seen in Fig. 3B, such an approach allows, in particular, for determination of the analyte with improved accuracy when the calibration
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dependence modeled in accordance with the basic E-CM procedure is non-linear.
If E-CM is performed in accordance to the basic procedure, i.e. keeping the concentration of all the sample components constant in the series of calibration solutions, the
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multiplicative interferences can be eliminated but the price to be paid is the need to calculate the analytical result extrapolatively. The extrapolation process causes worsening of the results
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(in comparison with those obtained by I-CM in the same experimental conditions) in terms of precision and uncertainty [52,53]. Besides, serious systematic errors can be expected when the calibration function is nonlinear [54] or when the calibration dependence in the extrapolation region is rmodeled by a linear calibration graph function [55]. This is the reason for attempts
extrapolation process.
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to modify the basic E-CM procedure in the preparative stage in order to avoid the
One such approach is the interpolative standard addition method (ISAM), carried out
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using flow injection techniques [56,57]. It involves successive injection (addition) of a set of standard solutions to the sample flowing continuously to the detector. As a result, the positive
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and negative peaks are recorded depending on whether the analyte concentration in a standard is greater or less than the signal intensity measured for the sample. If the differences between the intensities of transient signals and steady-state signal are expressed as a function of the analyte concentration in the standards, the intercept of the calibration function with the concentration axis interpolatively indicates the analyte concentration in the sample (see Fig. 4). Although the procedure allows the extrapolation process to be avoided, the method is not able to compensate for multiplicative interferences [58]. The point is that the sample before
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ACCEPTED MANUSCRIPT and after standard addition is of different dilution, which contravenes the fundamental prerequisite of E-CM. The only advantage of ISAM over I-CM is the possibility of overcoming effects caused by interferents present in the sample in great excess relative to the analyte.
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A modification of the ISAM procedure was proposed in the form of the sample-tostandard additions method [59]. In this approach, a sample and then a blank solution containing all the components of the sample matrix are added to the standard solution. In such
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an approach, being in fact a combination of one-point E-CM with the matrix matching
matrix free of analyte is available.
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procedure, the analytical result can be calculated interpolatively, provided that a sample
Another attempt of the same kind was the standard addition and indicative dilution method (SAIDM) [60] (also reported later as the method of serial dilutions [61]). It consists of the addition of a single standard solution to the sample and successive dilution of this solution
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until the signal intensity measured is equal to or less of the signal intensity produced by the undiluted sample. If the calibration function is formulated as the relationship between the signal intensities measured and the dilution factor, the analytical results can be obtained in an
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interpolative way from equation shown in Fig. 5. It was revealed that SAIDM is able to provide more accurate results than E-CM in some cases when the interferences are not
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multiplicative but of a more complex nature, as well as when the calibration function formulated in accordance with the basic E-CM procedure is nonlinear [60]. A modified extrapolative E-CM procedure of an original interpretation of
measurement data was developed as is the sequential injection standard addition procedure [62]. It involves the creation of a calibration function based on the signal intensities measured for an undiluted sample and for a sample diluted with two standard solutions of different analyte concentrations. As the calibration solutions contain interferents at different
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ACCEPTED MANUSCRIPT concentrations, the analytical result calculated extrapolatively from this function is considered to be the apparent concentration only, rather than the true concentration. In order to obtain an accurate result, it is suggested to apply a mathematical approach: in a few calculation cycles, the analyte concentrations added to the sample are successively corrected on the basis of the
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apparent concentration value obtained one step earlier and the iteration procedure is ended when the difference between two succeeding apparent concentrations is less than a chosen value.
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A procedure that was developed and implemented a long time ago but only studied more deeply quite recently is sequential standard addition calibration (SSAC) [63-66]. It
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consists in successive addition of a single standard solution to a sample and measurement of the signal intensity after each addition. As a consequence, the sample is more and more diluted with the standard, and the total analyte concentration gradually increases or decreases, depending on the standard:sample analyte concentration ratio. The analytical result is
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calculated in the extrapolative way with some mathematical correction compensating the dilution process. The SSAC procedure is recommended to be used especially when the analyte concentration in the sample is relatively high (in particular, too high for the E-CM
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procedure with respect to the limitations of the instrument response). However, SSAC seems to be more similar to ISAM than to E-CM in terms of capability of compensating for
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interferences.
In Table 2, all the above procedures have been characterized in terms of their rational
adaptability to different calibration methods and capability of overcoming calibration errors. As suggested, each procedure can be adapted to a few methods at least, even if originally (in the literature) it was applied with just one of them. When most of these procedures are adapted to interpolative methods, they are able to make them resistant (to a greater or lesser degree) to interference effects. Besides, if some of them are the basis of extrapolative
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ACCEPTED MANUSCRIPT methods, they are able to make the methods more resistant to the non-linear effect. Thus the procedures provide an opportunity to improve both kinds of calibration methods with regard to these effects (which, if unaddressed, potentially create the most serious systematic errors in
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both cases).
4. Combination of analytical methods
Another possible way to achieve some analytical benefits through the use of
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appropriate calibration is to combine calibration methods. In the literature, quite a lot of such approaches can be found. Usually, mixing the methods is recommended, i.e. performing them
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separately (but, as a rule, in the same experimental conditions) in accordance with their individual procedures. There are also some attempts to integrate the methods into a single procedure, thereby in fact creating a new calibration strategy. Both kinds of approaches are
4.1. Mixed methods
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presented below.
The simplest example of mixed methods is to use both I-CM and E-CM according to
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their basic procedures for the determination of an analyte in the same sample. Assuming that the possible multiplicative interference effect is compensated for by E-CM, but not by I-CM,
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then the effect can be detected and estimated by comparison of either the analytical results obtained by both methods [46] or just the slopes of the calibration functions formulated in both cases [67,68]. Thus such an approach can be named a test for the interference effect and recommended as an initial experiment performed especially when no information about the sample composition (and, consequently, about interferents) is provided. In the further course of analysis, the sample should be analyzed in a way that is dependent on the results of this test (i.e. using one of either the interpolative or extrapolative calibration methods). One should
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ACCEPTED MANUSCRIPT remember, however, that the test only allows interferences of a multiplicative nature to be detected and avoided. Even more advantages as described above can be achieved by a different interpretation of the results obtained by I-CM and E-CM if applying the approach originally named a new
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kind of standard addition method [69,70]. It involves the preparation of a set of calibration solutions according to the basic E-CM procedure and the determination of the analyte in all these solutions in accordance with the I-CM procedure (using a separately formulated
SC
calibration function, see Fig. 6A). The results obtained in the interpolative way (i.e. sums of the analyte concentrations in the sample and in the standard additions) are then presented
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against the analyte concentrations added to the sample (as shown in Fig. 6B). Interestingly, on the basis of a single calibration function such as this, one is able not only to evaluate the analytical result in an extrapolative way but also to conclude about the occurrence of the multiplicative interference effect in the assayed sample: namely, if the function is inclined at
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an angle of 45°, the effect is not indicated; otherwise, it is suspected to occur. It was also revealed that such a calibration strategy leads to analytical results of better accuracy than ECM itself when the calibration dependence is non-linear [5]. Other advantages listed in the
EP
same paper, i.e., the simplicity and rapidity of the calibration procedure, are rather doubtful. When an analyte is required to be determined in a series of samples of similar
AC C
composition and the interference effect is suspected to occur – but at approximately the same level in all samples – I-CM and E-CM can be combined in another way, which can be named simply the mixed method [71]. At first, E-CM is applied to the determination of the analyte in a single selected sample and then the created calibration function (including its extrapolative part) is used for interpolative determination of the analyte in the remaining samples (see Fig. 3S). By doing so, the multiplicative interferences have a chance to be compensated for in all samples, hence avoiding the need to do so by the use of E-CM in relation to each of them
19
ACCEPTED MANUSCRIPT individually. One should note, however, that the selected sample has to be analyzed very carefully as the result of the analyte determination in this sample affects the remaining results in terms of accuracy. The mixture of I-CM and E-CM is also represented by the recovery method [17,72]. It
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is very often used in analytical chemistry when the analyte is suspected to have been lost during the preparative stages of the analytical procedure (e.g. digestion, extraction, preconcentration, separation). In the most common version, it consists in addition of the standard
SC
solution to one of two portions of a sample and in subjection of both portions (in equal volumes) to the required preparative operations and measurements. In such a way, a two-point
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calibration function is formulated according to E-CM principles. Then the analyte concentrations in both portions are found in the interpolative way using a calibration function prepared separately according to I-CM principles. The (percentage) ratio of the difference between the concentrations to the concentration added to the sample is a measure of the
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analyte recovery from the sample. This value is also an indication of both the efficiency of the preparative technique used and the accuracy of the analytical result obtained. The important thing to remember is, however, that the results obtained by the recovery method can be
EP
interpreted correctly only when the effects caused by all preparative operations and, in addition, by the interferents possibly present in the sample have a multiplicative nature
AC C
[73,74].
In all the above approaches, calibration by E-CM (by itself or together with I-CM)
only gives the possibility of overcoming the multiplicative interference effect. As it turns out, when E-CM is applied twice (i.e. ‘mixed’ with itself) in different controllable conditions, it offers a quite new valuable opportunity to additionally eliminate additive interferences caused by unknown interferents present in a sample. This method, absolutely unique in analytical chemistry, is named the H-point standard addition method (HSAM) [75]. In its original
20
ACCEPTED MANUSCRIPT version,
adapted
for
elimination
of
spectral
additive
interferences
in
UV/VIS
spectrophotometry, it involves the preparation of two calibration functions for an analyte determined in the sample in accordance with the basic E-CM procedure. The functions are formulated at two different wavelengths chosen in such a way as to be sure that the signal
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intensities measured for the analyte are as different as possible, but the signal intensities produced by the interferent are equal. Under these conditions, both function have a chance to cross at a point (H-point) indicating both the analyte concentration in the sample and the
SC
signal intensity corresponding to the additive effect (see Fig. 4S). If the latter signal is produced by a known interferent, this component can be determined by, e.g., E-CM using a
of two-component analysis [76].
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calibration function formulated separately. This is why HSAM is usually used for realization
Due to its great analytical significance, HSAM has been intensively developed in order to avoid the above restrictive conditions [77-79] and to adapt it to other instrumental
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methods than UV/VIS spectrophotometry [80-83]. In particular, in the kinetic version, the reaction between the analyte and a reactant added in excessive amount to the calibration solutions (sample alone and with standards added) is involved, and the signal intensity for the
EP
reaction product is measured at two well-defined times after the reaction initiation [83,84]. In this case, two calibration function are then formulated at a single wavelength, but at two
AC C
reaction times, on the assumption that the additive interference effect remains the same under these two conditions. Interestingly, such an calibration approach can be considered as, in fact, being a mixture of two extrapolative indirect methods (E-IM+E-IM). The aforementioned examples of mixed calibration methods are presented in Table 3. When considering them with regard to calibration effects (omitting their other features), they are mostly resistant to multiplicative interference effects alone, but predominantly not resistant to the non-linear effect. This is so because one of the methods that forms part of the
21
ACCEPTED MANUSCRIPT original mix is always E-CM. However, as revealed, they can also be mixed in other configurations of individual interpolative and extrapolative methods, thereby extending their capabilities towards the required analytical and practical objectives.
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4.2. Integrated methods
The primary examples of integrated calibration methods are the extrapolative methods, E-ISM, E-IM and E-DM, which are proposed to be introduced to our extended classification
SC
(see Section 3.2.). The basic procedure of each of them has been combined with the E-CM basic procedure – hence three new methods have been created that are carried out in
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accordance with specific individual procedures at all four calibration stages: preparative, measurement, modeling and transformation stages. As a consequence, each of the new methods is typified by its original features (i.e., also characteristic for I-ISM, I-IM and I-DM, respectively) combined with the features of E-CM. For instance, when using E-ISM there is a
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possibility of determining the analyte with increased precision and of compensating for multiplicative interferences.
An attempt to combine I-ISM with E-CM in another way has been made in an
EP
approach called standard dilution analysis (SDA) [85,86]. It uses two initially prepared solutions: a sample solution and a solution containing the same volume of the sample spiked
AC C
with a known amount of both analyte and internal standard. A set of calibration solutions is prepared by gradual dilution of the second solution with the sample solution. By doing so the concentration of the analyte present in the sample remains constant, but the concentrations of both the internal standard and the analyte added successively decrease to the same extent. A calibration function is formulated as the relationship between of the analyte-to-internal standard signal intensities ratio versus the inverse of the internal standard concentration and the analytical result is calculated from equation presented in Fig. 7.
22
ACCEPTED MANUSCRIPT As seen, the difference between SDA and extrapolative E-CM is that the internal standard is measured at decreasing and not constant concentrations. In spite of this, SDA is able to offer the same main advantages, i.e., correction of both random signal fluctuations and systematic interference effects. It provides results of similar accuracy to E-CM and of
easily automated by using, e.g., a flow technique [87].
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superior accuracy to I-CM and I-ISM [85]. Besides, the calibration procedure is simple and
A simple way to integrate I-DM with E-DM has been presented in the form of the
SC
generalized dilution method (GDM) developed in our laboratory [88]. This approach involves the initial preparation of three solutions: the sample solution, the standard solution and a
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mixture of both solutions (keeping the analyte concentrations in both solutions the same). All of them are then gradually diluted to the same extent, and at each dilution step the apparent concentrations are calculated in both an interpolative and extrapolative way. The results are presented as two plots of apparent concentrations versus dilution factor, and the analyte
dilution (see Fig. 5S).
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concentration in the sample is calculated as the apparent concentration value at infinite
It was revealed that the method can serve not only for elimination of both the
EP
interference and non-linear effects (see Table 4), but also as an effective tool allowing the interference effect to be detected and examined. The shapes and the mutual position of both
AC C
plots are a source of information as to whether the effects occur and, if so, whether they only have a multiplicative or a more complex nature. Besides, as two estimations of the analytical result are obtained in different and independent ways, they can be verified against each other in terms of accuracy. In particular, if both of them are equal, the probability that they are close to the true value is quite great. The advantages of GDM motivated us to continue the integration process of calibration methods in a similar direction and, as a result, two novel approaches to calibration
23
ACCEPTED MANUSCRIPT were proposed: first, the integrated calibration method (ICM) [89,90] and then, the generalized calibration strategy (GCS) [91]. In the most advanced of the developed ICM versions, I-CM and E-CM are combined with each other. At the preparative stage, six calibration solutions are prepared in such a way that the sample and the standard solutions,
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both in two different volumes, are complementarily diluted with each other and with the blank solution [14]. The measurement data then allow four one-point calibration functions to be prepared and four estimations of the analytical result to be calculated (see Fig. 8). As two of
SC
them are obtained in an interpolative way, and two in an extrapolative way, as well as being
for ascertaining their accuracy.
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obtained in pairs for samples of different volumes, their mutual comparison is a strong basis
When developing ICM, it was shown that the measurement data obtained for six calibration solutions are a great enough information source to estimate the analytical result through the additional pair of results obtained by ‘semi-extrapolative’ way [92]. Several flow
TE D
systems operating in accordance with various flow techniques were recommended to implement ICM automatically and effectively [93,94]. What is more, a versatile flow injection manifold was designed enabling integration (through dedicated preparation of the calibration
EP
solutions) of different calibration methods: not only I-CM with E-CM, but also I-ISM with EISM, and I-IM with E-IM, according to the ICM principles [95].
AC C
GCS, similarly to GCM, involves the integration of two dilution methods: I-DM with
E-DM. However, in this case, the procedure is based on the integration process applied to ICM – namely, at the initial stage, six solutions of the same composition as in ICM are prepared and then all of them are gradually diluted [7]. Hence, at each dilution step, four (or even six) apparent concentrations are calculated from such a family of one-point calibration functions as presented in Fig. 8. Due to the dilution process, all apparent concentrations usually lead all together to the true analyte concentration as the sample is more and more
24
ACCEPTED MANUSCRIPT diluted (or if they do not, the dilution process can be supported with special reagents that eliminate the interference effect) and, according to the principles of the dilution methods, indicate the final analytical result for infinite dilution of the sample. In order to ensure both good repeatability of the results throughout the dilution process and satisfactory speed of the
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analytical procedure it is recommended to perform GCM with the use of a dedicated flow system [96,97].
It was shown that GCM calibration can be successfully performed on the basis of
SC
calibration solutions of compositions that are characteristic for indirect methods (I-IM, E-IM) [91]. As a consequence, a ‘three-dimensional’ integration was created by combining indirect
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and dilution methods with interpolative and extrapolative modes (I-IM + E-IM / I-DM + EDM). There is nothing stopping us then from integrating the internal standard method with the dilution method in the same way – in accordance with GCM principles (I-ISM + E-ISM / IDM + E-DM), and, as seen in Table 4, the same is supposed for GDM.
TE D
Thus, GCM can serve – similarly to but even better than GDM – in the diagnosis of an analytical system with respect to interferences. As shown in Table 4, it is able to eliminate not only multiplicative interferences (due to the extrapolation calibration mode) but also more
EP
complex interferences (due to the dilution process), and thereby improve analytical results in terms of accuracy. What is more, it can be additionally integrated with other calibration
AC C
methods. If such advantages of GCM as the ability to work in the non-linear range of the calibration dependence and the possibility of controlling the accuracy of the analytical results during the calibration are taken into account, the method can indeed be considered as being ‘generalized’.
5. Other methods and procedures
25
ACCEPTED MANUSCRIPT The procedures and methods discussed above evidently do not constitute the full package of calibration approaches proposed in the literature. In particular, in this paper, we have omitted those procedures and methods that were recommended many years ago and are not very useful nowadays or even cannot be used with modern instrumentation. Besides,
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many calibration procedures deal with specific conditions (analytical systems, measurement instruments, etc.) and their more comprehensive adaptation is simply not realizable.
An example of a procedure of the latter kind is calibration by internal normalization,
SC
which is also called ‘response-normalized calibration’ (see Section 3.2.) [5]. Such an approach (being in fact a one-point multicomponent procedure) is used when several analytes
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can be determined during a single analytical run (e.g., using separation techniques or ICP). If analytes are different with different sensitivities, the ratios of the signal intensities measured for the analytes and internal standard (named the normalization factors) are then determined and they are the basis for the calculation of the concentrations of each analyte individually in
TE D
the sample. If the analytes reveal the same sensitivity, one normalization factor is sufficient for determination of all analytes. In both versions, the concentration of each analyte is usually expressed as a percentage of the sum of the concentrations of all analytes determined.
EP
As said before, E-ISM offers a chance to compensate not only for random signal fluctuations but also for interference effects, provided that the analyte and internal standard
AC C
are very similar to each other in terms of their physicochemical properties. If the analysis is performed with the use of mass spectrometry, the best choice in this regard is to use an isotopically labeled standard [98] in the role of the internal standard. The drawback is that such kinds of standards are not available in relation to some analytes, or if they are, are very expensive. A further example of a specific calibration approach is the isotope dilution method (99,100). In the simplest version, it consists in the addition of a known amount of
26
ACCEPTED MANUSCRIPT isotopically-enriched substance to the analyzed sample. The analyte concentration in the sample is calculated from the isotope abundance ratios measured for the sample, standard and spiked sample by mass spectrometry. Although, the method is very useful and possible to be exploiting in different versions, its application is limited to the instrumental systems allowing
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different isotopes to be measured and determined (e.g. the mass spectrometry).
A calibration procedure that is also applied very rarely (e.g., when using electroanalytical methods) is the subtraction method, which is presented erroneously as a
SC
variation of the standard addition method [101], but methodologically is more similar to the titration procedure. In this approach, the sample is spiked with a known amount of species
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reacting with the analyte so that the concentration of the remaining analyte in the sample is less than the original concentration. The analytical result can be found on the basis of the signal intensity measured before and after spiking as well as on the basis of the reaction stoichiometry.
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Special attention needs to be paid to titration. In our opinion –contrary to the mainstream – it is a method that belongs fully to the family of chemical calibration methods. It is characterized by a typical calibration procedure encompassing, e.g., preparation of a
EP
standard solution (a titrant), creation of a calibration function plot (titration curve) and evaluation of the analyte concentration in the sample assayed. However, the titration method
AC C
is slightly outside the classifications presented in Section 3.2., especially concerning the way of calculating the analytical result. Namely, the equivalence volume corresponding to the analytical result is not estimated via the signal intensity obtained for the sample (as it is in interpolative methods) but is usually indicated by a characteristic point (e.g., inflection point) of the calibration function. For this reason, titration should rather be referred to as an indicative (and not interpolative or extrapolative) calibration method [18,19].
27
ACCEPTED MANUSCRIPT Another question that is still open is how to classify titration carried out in flow injection analysis by the gradient technique [102]. In this approach, a sample is injected into a stream of titrant flowing continuously and due to the subsequent reaction, a peak cut-off value is produced, the width of which represents the intensity of the analytical signal. As such, a
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signal value cannot be directly attributed to the equivalence volume, and the analyte concentration in the sample is obtained with the use of additional standard solutions in accordance with the I-CM interpolative procedure. It has been revealed that the gradient
SC
titration process is in fact equivalent to the I-IM process [103]. If so, the whole gradient
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titration procedure could be considered as a mixture of two methods: I-CM and I-DM.
6. Conclusions
It has been revealed in Tables 1-4, the calibration approaches, when appropriately selected and applied, can above all considerably contribute to the elimination of systematic
TE D
errors that can occur throughout the calibration procedure, i.e., errors caused by both the interference and non-linear effect. As a consequence, analytical results can be obtained with improved accuracy. In addition, some of the calibration approaches can help to improve
EP
analytical methods in areas such as: precision (I-ISM, E-ISM, interpolative versions of the standard addition method), general analytical applicability (I-IM, E-IM) and efficiency (I-CM,
AC C
one-point calibration procedures, mixed method) or even give a chance to control (to some extent) the accuracy of analytical results throughout the calibration process (ICM, GCS). As shown, the preparative stage of a calibration method determines the subsequent
stages of the calibration procedure, including the way of calculating the analytical result and, consequently, makes the method resistant (or not resistant) to different interference effects. The rules in this domain are as follow:
28
ACCEPTED MANUSCRIPT • if the standard solutions and the sample are prepared separately from each other, the analytical result is calculated in an interpolative way and the interference effect (irrespective of its character) cannot be eliminated (unless additional efforts are undertaken), • if standard solutions are merged with the sample in such a manner that the sample
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components are kept at the same concentration, the analytical result is calculated in an extrapolative way and the multiplicative interference effect can be eliminated,
• if standard solutions are merged with the sample in a different manner from above, the
SC
analytical result can be calculated in the interpolative way, but the interference effect can only be eliminated in special cases (in particular, when it is either constant independent of the
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interferent concentration in a sample or proportional to the analyte-to-interferent concentration ratio [60]),
• if the sample is prepared separately from or together with the standard solutions, and the calibration solutions are progressively diluted, the analytical results calculated at each dilution
TE D
step (interpolatively or extrapolatively) have a chance to be more and more accurate as interference effects of both a multiplicative and complex nature can be decreased; however, one should take into account that the effect in a diluted sample can also be constant or even
EP
increased in some cases [40]).
As seen in Table 2, the actual and potential adaptability of different calibration
AC C
procedures to calibration methods is really quite extensive. Similarly, the implementation of mixing and integrating of methods is also quite great (see Tables 3 and 4). By skillfully combining procedures and methods, there is a chance of creating a novel calibration tools with new analytical features appropriable in both chemical and biochemical analysis.. Certainly, such attempts lead to progress in analytical chemistry, especially when they are directed towards problems which have not been fully solved yet.
29
ACCEPTED MANUSCRIPT One of the problems strongly connected with analytical calibration (but mentioned, recognized and studied very rarely) arises when the analyte is present in the standard solutions in a different chemical form than in the sample, and the signal intensities produced by equal concentrations of both forms are different from each other. Such a phenomenon, termed the
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species effect, can evidently lead to serious systematic errors of analytical results independently of which calibration method is used. The development of a procedure making the calibration process resistant enough to the species effect is certainly one of the greatest
SC
challenge that analysts face.
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Acknowledgments
The study was supported by Polish National Science Centre, Project 2013/11/B/ST4/00864.
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[48] P. Kościelniak, J. Janiszewska, Z. Fang, Flow-injection calibration procedures with the use of fully rotary valve, Chem. Anal. 41 (1996) 85-93. [49] J.H. Moffett, Bracketing Standards Calibration and PROMT Measurement Mode for the Analysis of Bronze Using SpectrAA Instruments, The Application Note, Agilent Technologies, 2010. [50] L.H.J. Lajunen, P. Perämäki, Spectrochemical Analysis by Atomic Absorption and Emission, Royal Society of Chemistry, Cambridge, 2004.
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[51] I.G. Zenkevich, T.E. Morozova, Areas of application and characteristics of quantitative chromatographic analysis by the consecutive standard addition method, J. Anal. Chem., 69 (2014) 327-335. [52] M.J. Gardner, A.M. Gunn, Optimizing precision in standard additions determinations, Fresenius Z. Anal. Chem. 325 (1986) 263-266.
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[53] J.M. Andrade, J. Terán-Baamonde, R.M. Soto-Ferreiro, A. Carlosena, Interpolation in the standard additions method, Anal. Chim. Acta 780 (2013) 13-19.
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[54] P. Kościelniak, Non-linear calibration by the standard addition method, Chemometr. Intell. Lab. 47 (1999) 275-287. [55] J.W. Hosking, K.R. Oliver, B.T. Sturman, Errors in the atomic absorption determination of calcium by the standard addition method, Anal. Chem. 51 (1979) 307-310. [56] Tyson F.J., Low cost continuous flow analysis, Anal. Proc. 18 (1981) 542-545. [57] F. Mas, A. Cladera, J. M. Estela, V. Cerdá, New approach to sequential injection analysis: using the sample as carrier, Analyst 123 (1998) 1541-1546. [58] P. Kościelniak, A critical look at the interpolative standard addition method, Analusis 24 (1996) 24-27. [59] Y. Israel, R. M. Barnes, Flow injection sample-to standard addition method. Spectrophotometric determination of hydrochloric acid and orthophosphate, Analyst 114 (1989) 843-848.
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ACCEPTED MANUSCRIPT [60] P. Kościelniak, J. Kozak, M. Wieczorek, Calibration by the standard addition and indicative dilution method in flame atomic absorption spectrometry, J. Anal. Atom. Spectrom. 26 (2011) 1387-1392. [61] W. Hyk, Z. Stojek, Quantifying uncertainty of determination by standard additions and serial dilutions methods taking into account standard uncertainties in both axes, Anal. Chem. 85 (2013) 5933-5939.
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[62] O. Elsholz, C. Frank, B. Stachel, H. Reincke, R. Ebinghaus, Sequential injection standard addition for on-line measurement of mercury in the river Elbe, Anal. Chim. Acta 438 (2001) 251-258. [63] R.J.C. Brown, M.R. Roberts, M.J.T. Milton, Systematic error arising from ‘sequential’ standard addition calibrations: quantification and correction, Anal. Chim. Acta, 587 (2007) 158-163.
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[64] R.J.C. Brown, C.L. Mustoe, Demonstration of a standard dilution technique for standard addition calibration, Talanta 122 (2014) 97-100.
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[65] R.J.C. Brown, Comparison of the extrapolation precision of sequential and conventional standard addition calibrations, Measurement 44 (2011) 1487-1490. [66] R.J.C. Brown, Systematic error arising from ‘sequential’ standard addition calibrations. 2: Determination of analyte mass fraction in blank solutions, Anal. Chim. Acta 648 (2009) 153-156. [67] M. Thompson, S.R. Ellison, R. Wood, Harmonized guidelines for single laboratory validation of methods of analysis, Pure Appl. Chem. 74 (2002) 835–855.
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[68] R.C. Castells, M.A. Castillo, Systematic errors: detection and correction by means of standard calibration, Youden calibration and standard additions method in conjunction with a method response model, Anal. Chim. Acta 423 (2000) 179-185. [69] T. Li, L.-J. Yu, W. Li, M.-T. Li, Analysis of zinc in Vitix negundo foliage by AAS and application of a new kind of standard addition method, Microchim. Acta 150 (2005) 153-157.
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[70] T. Li, L.-J. Yu, M.-T. Li, W. Li, A New Approach to the standard addition method for the analysis of F, Al and K content in green tea, Microchim Acta 153 (2006) 109-114.
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[71] S. Walas, A. Tobiasz, M. Gawin, B. Trzewik, M. Strojny, H. Mrowiec, Application of a metal ion-imprinted polymer based on salen-Cu complex to flow injection preconcentration and FAAS determination of copper, Talanta 76 (2008) 96-101. [72] M. Thompson, S.L.R. Ellison, A. Fajgelj, P. Willetts, R. Wood, Harmonised guidelines for the use of recovery information in analytical measurement, Pure Appl. Chem. 71 (1999) 337-348. [73] P. Kościelniak, On analytical usefulness of the recovery method, Anal. Lett. 37 (2004) 2097-2112. [74] M. Stafiński, M. Wieczorek, P. Janicki, P. Kościelniak, Theoretical and experimental examination of recovery in the context of trueness, Talanta 96 (2012) 39-43. [75] F. Bosch-Reig, P. Campina, P. Campins-Falco, H-point standard additions method. Part 1. Fundamentals and application to analytical spectroscopy, Analyst 113 (1988) 10111016.
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ACCEPTED MANUSCRIPT [76] T.N. Al-Sabha, A.A. Bunaciu, H.Y. Aboul-Enein, H-point standard addition method (HPSAM) in simultaneous spectrophotometric determination of binary mixtures: An overview, Appl. Spectr. Rev. 46 (2011) 607-623.
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[77] J. Verdú-Andrés, F. Bosch-Reig, P. Campíns-Falcó, Analyte estimation using the generalized H-point standard additions method and a new methodology for locating linear spectral intervals for unknown interferents, J. Chemometr. 12 (1998) 27-40. [78] K. Asadpour-Zeynali, M. Bastami, Net analyte signal standard addition method (NASSAM) as a novel spectrofluorimetric and spectrophotometric technique for simultaneous determination, application to assay of melatonin and pyridoxine, Spectrochim. Acta A 75 (2010) 589-597. [79] M. Hasani, M. Mohammadi, M. Shariati-Rad , H. Abdollahi, H-point curve isolation method for determination of catechol in complex unknown mixtures, Spectrochim. Acta A 96 (2012) 563-568.
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[80] P. Campíns-Falcó, F. Bosch-Reig, R. Herráez-Hernández, A. Sevillano-Cabeza, Determination of theophylline and paraxanthine in urine samples by liquid chromatography using the H-point standard additions method, Anal. Chim. Acta 268 (1992) 73-80. [81] P. Campíns-Falcó, J. Verdú-Andrés, F. Bosch-Reig, Development of the H-point standard additions method for the use of spectrofluorimetry and synchronous spectrofluorimetry, Analyst 119 (1994) 2123-2127. [82] E. Shams, H. Abdollahi, M. Yekehtaz, R. Hajian, H-point standard addition method in the analysis by differential pulse anodic stripping voltammetry: Simultaneous determination of lead and tin, Talanta 63 (2004) 359-364.
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[83] M.A. Karimi, M. Mazloum Ardakani, H. Abdollahi, F. Banifatemeh, Application of Hpoint standard addition method and partial least squares to the simultaneous kineticpotentiometric determination of hydrazine and phenylhydrazine, Anal. Sci., 24 (2008) 261-266.
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[84] F. Bosch-Reig, P. Campins-Falco, A. Sevillano-Cabeza, B. Harraez-Hernandez, C. Molins-Lequa, Development of the H-point standard-additions method for ultravioletvisible spectroscopic kinetic analysis of two-component systems, Anal. Chem. 63 (1991) 2424-2429.
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[85] W.B. Jones, G.L. Donati, C.P. Calloway, Jr., B.T. Jones, Standard dilution analysis, Anal. Chem. 87 (2015) 2321-2327. [86] F.M. Fortunato, M.A. Bechlin, J.A. Gomes Neto, G.L. Donati, B.T. Jones, Internal standard addition calibration: Determination of calcium and magnesium by atomic absorption spectrometry, Microchem. J. 122 (2015) 63-69. [87] F.M. Fortunato, M.A. Bechlin, J.A. Gomes Neto, A. Virgilio, G.L. Donati, B.T. Jones, Standard dilution analysis in flow system: Sodium determination by flame atomic emission spectrometry, Microchem. J. 124 (2016) 662-667. [88] P. Kościelniak, M. Kozak, A. Karocki, Detection and examination of interferences basing on the dilution process, Chem. Anal. 41 (1996) 363-376. [89] P. Kościelniak, G. Akcin, M. Herman, J. Kozak, M. Wieczorek, A new approach to the integrated calibration in flow injection analysis, Ann. Chim. 903 (2003) 1045-1052.
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ACCEPTED MANUSCRIPT [90] P. Kościelniak, J. Kozak, M. Herman, M. Wieczorek, A. Fudalik, Complementary dilution method – a new version of calibration by the integrated strategy, Anal. Lett. 37 (2004) 1233-1253. [91] P. Kościelniak, M. Wieczorek, J. Kozak, M. Herman, Generalized calibration strategy in analytical chemistry, Anal. Lett. 44 (2011) 411-430.
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[92] J. Kozak, M. Wójtowicz, A. Wróbel, P. Kościelniak, Novel approach to calibration by the complementary dilution method with the use of a monosegmented sequential injection system, Talanta 77 (2008) 587-592. [93] P. Kościelniak, M. Wieczorek, J. Kozak, J. Kozioł, The integrated calibration method – comparison of various flow approaches, Anal. Lett. 44 (2011) 398-410.
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[94] M. Wieczorek, P. Kościelniak, P. Świt, K. Marszałek, Novel multicommuted flow manifold dedicated to the integrated calibration method, Anal. Methods 6 (2014) 92769282. [95] P. Kościelniak, M. Wieczorek, J. Kozak, M. Herman, Versatile flow injection manifold for analytical calibration, Anal. Chim. Acta 600 (2007) 6-13.
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[96] M. Wieczorek, P. Kościelniak, P. Świt, J. Paluch, J. Kozak, Solenoid micropump-based flow system for generalized calibration strategy, Talanta 133 (2015) 21-26. [97] K. Medinskaia, S. Garmonov, J. Kozak, M. Wieczorek, V. Andruch, P. Kościelniak, A. Bulatov, Stepwise injection determination of isoniazid in human urine samples coupled with generalized calibration method, Microchem. J. 123 (2015) 111-117. [98] Stokvis E, Rosing H, Beijnen JH., Stable isotopically labeled internal standards in quantitative bioanalysis using liquid chromatography/mass spectrometry: necessity or not?, Rapid Commun. Mass Sp. 19 (2005) 401-407.
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[99] L. Wu, M.R. Mashego, J.C. van Dam, A.M. Proell, J.L. Vinke, C. Ras, W.A. van Winden, W.M. van Gulik, J.J. Heijnen, Quantitative analysis of the microbial metabolome by isotope dilution mass spectrometry using uniformly 13C-labeled cell extracts as internal standards, Anal. Biochem. 336 (2005) 164-171.
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[100] D.L. Achat, M.R. Bakker, L. Augusto, E. Saur, L. Dousseron, C. Morel, Evaluation of the phosphorus status of P-deficient podzols in temperate pine stands: combining isotopic dilution and extraction methods, Biogeochemistry 92 (2009) 183-200.
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[101] A.D. McNaught and A. Wilkinson, Compendium of chemical terminology IUPAC, second ed., Blackwell Science, Oxford 1997. [102] J. Růžička, E.H. Hansen, H. Mosbaek, Flow injection analysis : Part IX. A new approach to continuous flow titrations, Anal. Chim. Acta 92 (1977) 235-249. [103] M. Wieczorek, J. Kozak, P. Kościelniak, P. Knihnicki, E. Pieprzyca, Critical approach to flow injection gradient titration as a calibration method, Talanta 96 (2012) 34-38.
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Applicability in the presence of calibration effects Proposed name
AE
calibration curve method
CCM
interpolative conventional method
I-CM
internal standard method
SAM
interpolative internal standard method
I-ISM
n
indirect method
IM
interpolative indirect method
I-IM
dilution method
DM
interpolative dilution method
standard addition method
SAM
-
ME
CE
SC
Calibration method Old name
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Table 1. Nomenclature and characterization of the calibration methods in accordance with the proposed classification (AE, ME, CE - additive, multiplicative and complex interference effect, respectively, NE - nonlinear effect; symbols y, p, n denote applicability, possible applicability and no applicability, respectively)
Specific ability
NE
n
p
to perform analyses in relatively simple and fast way
p
n
p
to compensate random signal fluctuations, to give a chance to compensate the multiplicative interference effect
n
n
n
p
to extend the analytical applicability
I-DM
n
p
y
y
to eliminate interference effects (multiplicative and complex ones) and non-linear effect
extrapolative conventional method
E-CM
n
y
n
n
to compensate random signal fluctuations, to compensate multiplicative interference effect
-
extrapolative internal standard method
E-ISM
n
y
n
n
-
-
extrapolative indirect method
E-IM
n
y
n
n
-
-
extrapolative dilution method
E-DM
n
y
y
y
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to offer correspondingly the same advantages as interpolative methods and to compensate multiplicative interference effect
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Table 2. Original or commonly met ( y) and possible (p) rational adaptation of the calibration procedures to interpolative and extrapolative methods as well as their potential capability ( y) of overcoming non-linear effect (NE) and interference effects of multiplicative (ME), complex (CE) and additive (AE) characters Adaptability to calibration methods Calibration procedure
Interpolative methods I-IDM
Extrapolative methods
I-DM E-CCM E-ISM E-IDM E-DM
AE
ME
CE
NE
y
y
y
y
y
y
y
y
y
p
spiking the sample with special reactant
y
y
p
signal increment procedure
y
p
p
y
p
p
y
one-point calibration
y
y
p
y
p
p
y
one-point several-graph procedure
y
p
p
y
p
p
bracketing calibration
y
p
p
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matrix-matched technique
p
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consecutive two-point calibration
p
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I-CCM I-ISM
Capability of overcoming the calibration effects
y
y
y y
y
p
p
y
y
p
p
y
sample-to-standard additions method
y
p
p
y
y
p
p
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interpolative standard addition method
standard addition and indicative dilution method
y
y y
y
sequential injection standard addition procedure
y
p
p
y
sequential standard addition calibration
y
p
p
y
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y
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Table 3. Characterization of mixed calibration methods (AE, ME, CE – additive, multiplicative and complex interference effect, respectively, NE – nonlinear effect; symbols y and n denote applicability and no applicability, respectively)
Mixture of calibration methods Name
Applicability in the presence of calibration effects
rationally possible mixture
AE
ME
CE
NE
test for interference effect
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
n
new kind of standard addition method
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
y
mixed method
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
n
recovery method
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
n
H-point standard addition method (original version)
E-CM+E-CM
E-ISM+E-ISM, E-IM+E-IM
y
y
n
n
H-point standard addition method (kinetic version)
E-IM + E-IM
E-IM + E-IM / E-ISM+ E-ISM
y
y
n
n
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original mixture
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Integration of calibration methods Name
E-CM + I-ISM
generalized dilution method
I-DM + E-DM
integrated calibration method
I-CM + E-CM
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I-DM + E-DM I-IM + E-IM / I-DM + E-DM
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generalized calibration strategy
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Applicability in the presence of calibration effects AE
ME
CE
NE
E-IM + I-ISM
n
y
n
n
I-ISM + E-ISM / I-DM + E-DM I-IM + E-IM / I-DM + E-DM
n
y
y
y
I-ISM + E-ISM, I-IM + E-IM
n
y
n
n
I-ISM + E-ISM / I-DM + E-DM
n
y
y
y
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standard dilution analysis
rationally possible integration
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original integration
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Table 4. Characterization of integrated calibration methods (AE, ME, CE – additive, multiplicative and complex interference effect, respectively, NE – nonlinear effect; symbols y and n denote applicability and no applicability, respectively)
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Figure captions:
Fig. 1. Principles of interpolative calibration methods: I-CM (A), I-ISM (B), I-IM (C), I-DM (D); Yai, Ybi, Ysi – signal intensities measured for analyte, reactant and internal standard in the standards, respectively; ci – analyte concentrations in the standards (i = 1,…N); Yax, Ybx, Ysx – signal intensities measured for sample; cx – found analyte concentration; , – experimental
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points obtained for standards and sample, respectively Fig. 2. Principles of extrapolative calibration methods: E-CM (A), E-ISM (B), E-IM (C), E-DM (D); ∆ci – analyte concentrations added to sample (i = 1,…N); (remaining symbols as in Fig. 1) Fig. 3. Consecutive one-point calibration: apparent concentrations cx1 – cx3 obtained in extrapolative
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way in accordance to one-point E-CM procedure (A) are presented as a function of analyte concentrations ∆c1 – ∆c3 added to a sample (B); cx – found analyte concentration
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Fig. 4. Interpolative standard addition method: differences between signal intensity Yx measured for a sample and intensities Y0 – Y3 measured for standards added to the sample (A) are presented as a function of added analyte concentrations (B); cx – found analyte concentration Fig. 5. Standard addition and indicative dilution method: calibration graph obtained for a sample spiked with analyte of concentration c1 and diluted to such extent (of dilution factor kx) until concentration
cx – found analyte
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signal intensity Yx measured for undiluted sample is reached;
Fig. 6. New kind of standard addition method: apparent analyte concentrations determined in the interpolative way in a sample, cx1, and in the sample spiked with analyte, ct1 – ct3, (A) and then
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presented as a function of added analyte concentrations ∆c1 – ∆c3 (B); cx – found analyte concentration
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Fig. 7. Standard dilution analysis: calibration graph presented as a function of the analyte-to-internal standard signal intensities ratio, Y/Ys, and the inverse of internal standard concentration, 1/cs;
c1, cs1 – concentrations of analyte and internal standard, respectively, in the standard solutions, cx – found analyte concentration
Fig. 8. Integrated calibration method: calibration graphs based on the signal intensities, Y1 - Y6, corresponding to six calibration solutions prepared from a single standard solution of concentration c1 (for details see text) and allowing the analyte concentration in sample to be calculated in interpolative (cx1, cx2) and extrapolative (cx3, cx4) ways
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A
Ya
B
Ya/Ys
YaN
(Ya/Ys)N
(Ya/Ys)3
Ya2
(Ya/Ys)2
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~
~ Ya3
(Ya/Ys)x
Yax Ya1
0
c1
cx
c2
c3
~
c
0
cN
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Yb
(Ya/Ys)0
c1
cx
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(Ya/Ys)1
Ya
Yb0
c2
c3
~
c
cN
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Ya1
Yb1
Ya2
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Ybx Yb2
Ya3
Yb3
c1
cx
c2
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0
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~ YbN
c3
~
Yax1 YaN Yax2 Yax3 YaxN
c
c
Y0 0
cN
cx
c1
Fig. 1. Principles of interpolative calibration methods: I-CM (A), I-ISM (B), I-IM (C), I-DM (D); Yai, Ybi, Ysi – signal intensities measured for analyte, reactant and internal standard in the standards, respectively; ci – analyte concentrations in the standards (i = 1,…N); Yax, Ybx, Ysx – signal intensities measured for sample; cx – found analyte concentration; , – experimental points obtained for standards and sample, respectively
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A
Ya
(Ya/Ys)N
YaN
(Ya2/Ys)2
Ya1
(Ya/Ys)1
Yax
(Ya/Ys)x
∆c1
∆c2
∆c -cx
∆cN
C
Yb
(Ya/Ys)0
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0
~
0
∆c1
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Ya0
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~
~ Ya2
-cx
B
Ya/Ys
Ya
Yb0
∆c2
~
∆c
∆cN
D
Ya1
Ya2
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Ybx Yb1 Yb2
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~ YbN
0
∆c1
∆c2
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-cx
~
Ya3 YaN Yax1 Yax2 Yax3 YaxN
∆c
∆c
Y0 -cx
∆cN
0
∆c1
Fig. 2. Principles of extrapolative calibration methods: E-CM (A), E-ISM (B), E-IM (C), E-DM (D); ∆ci – analyte concentrations added to sample (i = 1,…N); (remaining symbols as in Fig. 1)
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A
Y3
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Y2
0 Y1
cx
cx1 cx2 cx3
∆c1
∆c2
∆c3
∆c
∆c 0
∆c1
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cx2 cx1
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Yx
cx3
B
c
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Y
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∆c3
Fig. 3. Consecutive one-point calibration: apparent concentrations cx1 – cx3 obtained in extrapolative
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way in accordance to one-point E-CM procedure (A) are presented as a function of analyte concentrations ∆c1 – ∆c3 added to a sample (B); cx – found analyte concentration
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A
B
∆Y
Y3 Y2
∆Y
Yx
∆c
cx
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0
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Y
Y0
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t
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Y1
Fig. 4. Interpolative standard addition method: differences between signal intensity Yx measured for a sample and intensities Y0 – Y3 measured for standards added to the sample (A) are presented
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as a function of added analyte concentrations (B); cx – found analyte concentration
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Y
kx ⋅ c1 1− kx
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cx =
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Yx
k
kx
0
1.0
c
cx
cx+c1
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Fig. 5. Standard addition and indicative dilution method: calibration graph obtained for a sample spiked with analyte of concentration c1 and diluted to such extent (of dilution factor kx) until signal intensity Yx measured for undiluted sample is reached;
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concentration
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cx – found analyte
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c ct3
Y2
ct2
Y1
ct1
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Y3
B
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A
Y
cx1
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Yx
∆c
c
0
cx1
ct2
ct1
ct3
cx
0 c
∆c1
∆c2
∆c3
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Fig. 6. New kind of standard addition method: apparent analyte concentrations determined in the interpolative way in a sample, cx1, and in the sample spiked with analyte, ct1 – ct3, (A) and then presented as a function of added analyte concentrations ∆c1 – ∆c3 (B); cx – found analyte
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tg α c1 • intercept cs1
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intercept
α
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cx =
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Y/Ys
1/cs
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Fig. 7. Standard dilution analysis: calibration graph presented as a function of the analyte-to-internal standard signal intensities ratio, Y/Ys, and the inverse of internal standard concentration, 1/cs; c1, cs1 – concentrations of analyte and internal standard, respectively, in the standard
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solutions, cx – found analyte concentration
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Y Y2 Y5
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Y1
Y6
SC
Y4
Y0 cx3, cx4
M AN U
Y3
cx1, cx2
c c1
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Fig. 8. Integrated calibration method: calibration graphs based on the signal intensities, Y1 - Y6, corresponding to six calibration solutions prepared from a single standard solution of concentration c1 (for details see text) and allowing the analyte concentration in sample to be
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calculated in interpolative (cx1, cx2) and extrapolative (cx3, cx4) ways
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ACCEPTED MANUSCRIPT Highlights
• Fundamental terms related to univariate calibration are defined. • Classification of calibration methods is presented.
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• Calibration methods and procedures are reviewed.
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• Real and potential possibility of combining analytical methods is provided.
ACCEPTED MANUSCRIPT Marcin Wieczorek was born in 1978 in Kielce, Poland. He received M.Sc. and Ph.D. degree in analytical chemistry at the Faculty of Chemistry Jagiellonian University in Kraków in 2002 and 2007, respectively. In 2008, he was honoured with the award of the Committee of Analytical Chemistry of the Polish Academy of Sciences for his PhD thesis. Since 2007, he
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has been working in the Department of Analytical Chemistry at the Faculty of Chemistry Jagiellonian University. His main scientific interests focus on development of new analytical calibration strategies and on new methods in chemical analysis realized with the use of flow
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techniques.
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Paweł Kościelniak was born in Krakow, Poland, in 1952. He received his Ph.D. in 1981 from the Faculty of Chemistry, Jagiellonian University, Krakow. He is full professor (since 2000) and head of the Department of Analytical Chemistry at the Jagiellonian University (since 1997). He is a member of the Committee of Analytical Chemistry of the Polish Academy of Sciences. He published more than 200 scientific articles, reviews and book chapters. The main areas of his scientific interest are flow analysis, forensic chemistry and environmental analysis with special attention paid to analytical calibration and interference effects. He supervised twenty Ph.D.theses.
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S0003-2670(16)31108-4
DOI:
10.1016/j.aca.2016.09.024
Reference:
ACA 234807
To appear in:
Analytica Chimica Acta
Received Date: 4 April 2016 Revised Date:
9 September 2016
Accepted Date: 11 September 2016
Please cite this article as: P. Kościelniak, M. Wieczorek, Univariate analytical calibration methods and procedures. A review, Analytica Chimica Acta (2016), doi: 10.1016/j.aca.2016.09.024. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Univariate analytical calibration methods and procedures. A review.
Paweł Kościelniak, Marcin Wieczorek Department of Analytical Chemistry, Faculty of Chemistry, Jagiellonian University,
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30-060 Krakow, Poland
Abstract
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An original focus on univariate calibration as an experimental process of quantitative analysis is presented. A novel classification system is introduced against the background of the present
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situation concerning nomenclature of calibration methods. Namely, it has been revealed that four methods well-known in analytical chemistry: the conventional method, the internal standard method, the indirect method and the dilution method, can be split into those carried out in both the interpolative and the extrapolative mode. It is then shown that the basic
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procedures of all these methods can be modified including different approaches, such as matrix-matched technique, spiking the sample with a reactant, bracketing calibration, and others. For the first time (as compared to monographies dealing with univariate calibration) it
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is reviewed how the methods are mixed and integrated with one another thereby creating new calibration strategies of extended capabilities in terms of enhanced resistance to the
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interference and non-linear effects – as the main sources of systematic calibration errors. As additional novelty, rationally possible combinations of the calibration methods – not met hitherto in the literature – have been predicted. Finally, some general rules relating to calibration are formulated and the main calibration problems that still need to be solved are displayed. Keywords: Univariate calibration; Calibration methods; Calibration procedures; Classification of calibration methods
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ACCEPTED MANUSCRIPT Contents 1. Introduction 2. Calibration methods 3. Modified calibration procedures 4. Combination of calibration methods
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4.1. Mixed methods 4.2. Integrated methods 5. Other methods and procedures
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6. Conclusions
1. Introduction
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In spite of broad improvements in methods and instrumentation, analytical calibration still remains a key element in analytical chemistry. Calibration is a crucial step in almost every chemical procedure leading to evaluation of the concentration of a determined substance (analyte) in a sample. This is so because the instruments used for chemical analysis
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provide, in principle, information in the form of an intensity of the analytical signal that only corresponds to the analyte concentration but does not directly determine this concentration. The main task of calibration is thus to transform the signal intensity into the analyte
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concentration as accurately and precisely as possible. Although billions of chemical analyses are performed every day all over the world and
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the great majority of them lead to the determination of analytes on the basis of univariate calibration, this kind of analytical calibration is not very popular among analysts as an individual, methodologically and practically interesting subject. In particular, the most recent literature comprehensively dealing with calibration problems is related to multivariate analysis [1-4] with special attention paid to elaboration of the calibration data with the use of chemometric methods. The last article relating broadly to the methodological aspects of univariate calibration appeared in 2001 [5] and it covered the subject in the context of
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ACCEPTED MANUSCRIPT chemical standards rather than generally. In some other papers the fundamental issues of univariate calibration were reviewed either from metrological [6] and statistical [7-9] points of view or in relation to individual analytical methods, including mainly chromatography [10,11] and ICP-MS [12]. Specific calibration approaches that are applied when a sample is
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analyzed by flow techniques [13-15] or is prepared by solid phase micro-extraction [16] have also been summarized recently.
The above situation motivated us to write the present paper, in which we have
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attempted to discuss univariate calibration methodologies and strategies in analytical chemistry more generally and comprehensively. The main goal of this paper is to present the
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advantages and potential applicability of a wide range of various calibration methods, including those which have been forgotten to some extent or exploited very rarely in practice. It was decided to reveal the possibilities of how they can be performed in two essentially different modes (interpolative and extrapolative) and in accordance with different modified
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procedures, as well as how they can be mixed and integrated with each other. The calibration methods and procedures are primarily discussed from the perspective of their capability of overcoming the interference effects as the principal source of systematic errors that can occur
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during the calibration process.
The article is essentially focused on purely practical and chemical calibration aspects,
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whilst bypassing mathematical and metrological approaches to the subject. Our own experiences in calibration – in comparison with other works – are widely presented and some of our own concepts and ideas are suggested and put forward for general review. The paper is organized as follows: the principles of analytical calibration are first desrcibed, along with a discussion on calibration classification. Then, examples of various calibration procedures and methods exploited in both batch and flow analysis are then
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ACCEPTED MANUSCRIPT presented and discussed. The final section is dedicated to more specific calibration approaches, which are however widely applied in particular analytical areas.
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2. Calibration methods According to the last recommendation of the Joint Committee for Guides in Metrology calibration is defined as “operation that, under specified conditions, in a first step, establishes a relation between the quantity values with measurement uncertainties provided by
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measurement standards and corresponding indications with associated measurement
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uncertainties and, in a second step, uses this information to establish a relation for obtaining a measurement result from an indication” [17]. This general definition can be adapted to analytical practice and purposes through the following expression: “calibration is a process encompassing modeling of the theoretical relationship. i.e. calibration dependence, between the intensity of the analytical signal and the concentration of a substance (usually an analyte)
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in the form of an experimental dependence, i.e. calibration function (plotted as calibration graph), formulated with the use of chemical standards and transformation of the intensity of
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the signal measured for a sample assayed into the analyte concentration in the sample [18]. The purely analytical character of above definition can be understood by drawing an
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analogy with the relationship between calibration dependence and calibration function and the real (true) and found (experimental) concentration of the analyte in a sample: the calibration dependence, like the real analyte concentration, can only be experimentally evaluated but never ideally accurately determined. Another relationship is as follows: the better (more accurate) the modeling of the calibration dependence by means of a calibration function, the greater the accuracy of the analyte concentration in a sample. From a practical point of view, the calibration process consists of four stages: the preparative stage (i.e., the preparation of sample and standards), the measurement stage (i.e.,
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ACCEPTED MANUSCRIPT reading the signal intensities for sample and standards), the modeling stage (formulation of the calibration function), and the transformation stage (calculation of the analytical result) [15]. As the great majority of analyses are performed with the use of liquid samples and standards (even if solid or gaseous samples are originally to be analyzed) as well as the
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samples are sometimes prepared together with and not separately from the standards, the term “calibration solutions” instead of “sample and standards” will be used throughout the paper. In the analytical literature, two further useful terms relating to calibration are also
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often encountered: calibration procedure and calibration method. Unfortunately, they are not strictly defined and distinguished in any relevant official documents or unofficial articles. In
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our approach [19], the calibration procedure is a detailed mode of performing all the abovementioned four stages of the calibration process. If the calibration procedures are followed according to specified rules – which determine a more general way of proceeding, leading to the attainment of (besides the main calibration objective) certain additional analytical goals –
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they constitute a calibration method.
Whichever of the calibration method is used, different errors can be committed in the calibration stage. Certainly, the trueness of standards is the ultimate limit to producing ever
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improved calibrations. However, the most characteristic “calibration” source of systematic errors is the interference effect, i.e., a change in the intensity of the analytical signal due to
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substances (usually sample components) other than the analyte. The interference effect of additive (unspecific) and multiplicative (specific) nature is revealed by constant and proportional dislocation of the calibration dependence in relation to the calibration function formulated on the basis of just an analyte. An effect of more complex nature is able to make this dependence non-linear in the entire range of the analyte concentrations or even linear and non-linear in certain intervals of this range. In many cases, a mixture of all kinds of interferences is also encountered.
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ACCEPTED MANUSCRIPT Another disadvantageous situation, which can be named the non-linear effect, is when the sample matrix is free of interferents but the calibration dependence is non-linear for some other reasons (e.g. instrumental). If the dependence is modeled by a linear calibration plot function (which is commonly done), erroneous analytical results can be expected again.
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In analytical chemistry, several calibration strategies of varying popularity are known that can be considered to be ‘calibration methods’ in the light of the definition mentioned above. The crucial questions are: how are they linked to both calibration effects (i.e.,
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interference and non-linear effects) and can these effects be overcome if they have not been recognized and eliminated in the initial steps of an analytical procedure?
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The calibration method that is definitely most often used is the calibration curve method (CCM). The basic procedure is simple and relatively fast. It involves the modeling of a calibration function using several standards containing the analyte (usually alone) at various concentrations – encompassing the analyte concentration in the assayed sample. The sample is
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subjected to measurements separately from the standards; the signal intensity obtained is related to the calibration function and the analyte concentration in the sample is calculated in an interpolative way (see Fig. 1A). In this form, the method evidently leads to an inaccurate
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analytical result if the interference effect occurs (see Fig.1SA). On the other hand, a nonlinear calibration dependence can be modeled exactly if the number of standards is great
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enough, and even if it is not, the error cannot be expected to be great. Another quite popular calibration method introduced into analytical practice in 1935
[20] is the standard addition method (SAM). The basic SAM procedure consists in the addition of a standard solution of increasing analyte concentrations to the sample by such means that the concentrations of analyte and of other components of the sample are kept constant. A calibration function is created for the analyte added to the sample, and the analyte concentration in the sample is calculated by extrapolation of this function to the zero signal
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ACCEPTED MANUSCRIPT (see Fig. 2A). As in all calibration solutions, the increasing concentrations of analyte are accompanied by the same amount of interferents, the method gives a chance to eliminate (compensate for) interference effects, but only ones of a multiplicative nature. On the other hand, the systematic error caused by the non-linear effect may be quite great because of the
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extrapolation process (see Fig. 1SB).
A method especially often exploited in separation and ICP techniques is the internal standard method (ISM). In this case, the standard solutions and the sample are spiked with the
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same and known amounts of a substance (internal standard), which is absent from the (unspiked) sample. It is chosen in such a way to be similar to the analyte in terms of
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physicochemical properties and to provide a signal that is readily distinguishable by the instrument from the signal produced by the analyte. The calibration function is created on the basis of the analyte-to-internal signal intensity ratios measured for the calibration solutions and the signal intensity ratio measured for the sample leads to interpolative calculation of the
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analyte concentration in the sample (see Fig. 1B). An additional objective of the method is to determine the analyte with increased precision if the size and direction of random fluctuations of the intensities of analytical signals measured for both components are similar. In general,
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the interference effect is not expected to be compensated for, apart from such exceptional cases where the interferents change both signal intensities (from the analyte and the internal
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standard) in the same direction and to the same extent. As in the above version, the method has an interpolative character, and its resistance to the non-linear effect is similar to that of CCM.
The indirect method (IM), is usually applied when the analytical instrument used is not able to produce the signals (or produces signals of too weak intensity) directly for the sample component required to be determined. Although the method is mentioned very rarely in the contemporary literature, it is exploited quite often in both chemical analysis (e.g. for the
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ACCEPTED MANUSCRIPT determination of anions by FAAS) and biochemical analysis (e.g. using antigen-labelled competitive ELISA technique [21]). In accordance with the basic IM procedure, a constant, known and excessive amount of substance that reacts with the analyte is added to the sample and to standard solutions of increasing analyte concentrations, and the signal intensities are
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measured for either the reaction product or the reagent remaining after the reaction. An increasing or decreasing, (respectively) calibration function is then formulated and the analytical result is obtained in an interpolative way (see Fig. 1C). In principle, the method is
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linked to both calibration effects in a similar way to CCM. However, if it is known or suspected that the analyte is strongly affected by interferents, then the reactant can be chosen
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in such a way as to avoid effects caused by the same or other interferents.
In the dilution method (DM), brought to analytical chemistry in 1959 [22], a single standard solution containing the given analyte is used. Both the standard and the sample are progressively diluted and at each dilution stage, both solutions are measured. Each pair of
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analytical signals obtained allows the apparent concentration of the analyte in a sample of a given dilution to be estimated with the help of a two-point calibration function (see Fig. 1D). On the basis of the obtained results, the apparent concentration corresponding to a sample of
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infinite dilution is and it is considered as the final analytical result. The dilution method can be successfully applied when the non-linear effect and an interference effect of a different
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nature (except an additive one) occur, as both effects usually tend to be diminished in the course of sample dilution and disappear in the infinitely diluted sample. Where calibration methods are classified (e.g., in textbooks), they are commonly
divided into absolute methods and others. Among the latter methods, three are usually mentioned, namely CCM, SAM and ISM, under different names [21,23-25]. In recent years, a tendency has been noticeable to differentiate them according to whether the standard is prepared separately from the sample or added to the sample, and the terms: external and
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ACCEPTED MANUSCRIPT internal methods are respectively recommended [24-27]. SAM and ISM are sometimes included in the group of internal methods [8], and a negative consequence of this is that SAM can be considered to be a version of ISM [28]. The classification suggested by the authors of this paper is related to the
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aforementioned definition of chemical calibration [19]. We propose to divide calibration methods depending on how the signal measured for a sample is transformed into the analyte concentration in the sample. It seems to be a more fundamental criterion than one based on
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laboratorial and measurement aspects (as in the hitherto prevailing approach).
A fundamental reflection is that, in principle, at the preparative stage, SAM is similar
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to CCM except that the standards are prepared together with and not separately from the sample. As a consequence, the calibration dependence can be partly, but not entirely, modeled and the analytical result is calculated in an extrapolative and not interpolative way. It has been revealed that the remaining methods, i.e. ISM, IM and DM, can be also carried out at the
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preparative stage not only in the mode described above (i.e. separately from the sample), but also by spiking the sample with standards. The consequences of the changes made are the same as in SAM, including that the analytical results are in all cases calculated in an
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extrapolative and not interpolative way.
Thus, the calibration methods mentioned above can be split into those carried out in
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interpolative and extrapolative modes. As CCM and SAM can be considered the most conventional ones, it is proposed to name them the interpolative conventional method (I-CM) and the extrapolative conventional method (E-CM), respectively. For the remaining methods, we recommend that their traditional names should be kept, but qualified with the adjectives ‘interpolative’ or ‘extrapolative’. All the new names with their abbreviations are listed in Table 1 (the same abbreviations will be used in further sections of this paper).
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ACCEPTED MANUSCRIPT Concerning literature examples of the extrapolative calibration approaches, the E-ISM method was used (probably first time) by Matisova et al. [29] as a modified standard addition method in gas chromatography and then it was exploited in, e.g., photon activation analysis [30] and ICP [31]. In a textbook relating to liquid chromatography it is mentioned as the mean
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with addition of determined substance [27]. E-IM was developed by Kościelniak et. al [32] on analytical examples using flame AAS and UV/VIS spectrophotometry. E-DM was introduced by Pszonicki et al. [33] as the standard addition and successive dilution method, carried out in
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accordance with batch procedure and then adapted to flow injection analysis as the gradient ratio standard addition method [34].
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On the basis of theoretical and experimental studies, the following general conclusion can be drawn: the essential difference between interpolative and extrapolative calibration methods is that the latter can compensate for interference effects of a multiplicative character, whilst the remaining features (advantages and disadvantages) of all individual interpolative
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methods are preserved in their extrapolative versions.
Figs. 1 and 2 show how the interpolative and extrapolative methods are interpreted in comparison with each other, i.e. how the calibration functions are formulated and how the
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analytical results are calculated. The most controversial and very curious interpretation of the measurement results pertains to the E-IM method where the calibration function has a
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negative slope (see Fig. 2C). It was revealed [32] that in such a case the analytical result is obtained by extrapolation of the function to the signal intensity corresponding to the standard solution containing the reactant alone, and hence such a solution has to be prepared in addition to the other solutions required by the method. We have suggested also how the calibration procedures developed in flow analysis can be specifically arranged [13,14]. In this case, it was decided to define the type of calibration not only on the basis of how the signal intensity is transformed into the analytical result, but
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ACCEPTED MANUSCRIPT also how the calibration is modeled in the form of calibration functions. The flow approaches were categorized as interpolative [13] and extrapolative [14], and most of them were further divided according to the number of calibration functions (graphs) that needed to be formulated: one-graph, several-graph and multi-graph procedures. In addition, all methods
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were also distinguished with respect to either direct or indirect transformation of measurement data.
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3. Modified calibration procedures
Calibration methods can be performed according to different procedures that have
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been modified in relation to the basic ones (described above). The purpose is to improve the calibration method in terms of either the analytical performance (e.g., to avoid calibration errors) or the efficiency (e.g., to make the analytical process faster). From among many procedures reported in the literature, the most versatile examples are briefly described below,
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i.e. those that can be performed with the use of various analytical techniques (including those exploited in biochemical analysis), methods and detectors. Besides, even if they are originally presented as used in analysis of liquid samples, most of them can be adapted to analysis of
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samples in solid and gaseous states.
One of the procedures relating exclusively to the preparation of calibration solutions is
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the matrix-matched technique. Being highly versatile in principle, in the literature it has been adapted to both the I-CM and I-ISM methods, and recently has especially often been used in chromatography [35,36] and ICP-MS [37,38]. It is also routinely exploited in biochemical analysis using such immunoassay techniques as RIA and ELISA [21]. The procedure It is usually performed in such a way that the standards of different analyte concentrations are mixed with the analyte-free matrix of the sample analyzed. In principle the result is similar to that when E-CM is used, i.e., the calibration dependence is modeled accurately by the
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ACCEPTED MANUSCRIPT calibration function and the multiplicative interference effect is compensated for. As the analytical result is calculated in an interpolative way, it is possible for the analyte to be determined even more accurately than by using E-CM. What is more, if the matrix contains a component causing an interference effect of an additive nature, the effect can be estimated
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through the measurement of the matrix alone (unspiked by the analyte).
A special role in the matrix-matched procedure play the certified reference materials (CRM’s) [39]. If CRM is suited to the sample analyzed with regard to its composition it can
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serve itself as the standard for one-standard I-CM calibration (see below) performed in accordance with the matrix-matched procedure. If, in addition, CRM is free of analyte, it can
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be added to the set of standards considered as the matrix of the sample analyzed. It should be, however, pointed that the preparation of a set of standards in a way of dilution of a single CRM is not consistent with the principle of the matrix-matched procedure and, consequently, can not lead to elimination of interferences. As the most CRM’s are available in the solid
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state, they are especially useful for matching the matrix of solid samples (in particular, using laser-ablation ICP-MS [12,38]).
Another procedure undertaken quite often in analytical practice at the preparative
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calibration stage is spiking the sample with a special reactant. The reactant is referred to as ‘special’ because it is chosen in a way that selectively eliminates a given interference effect
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(usually multiplicative). As a rule, the standards are also spiked with the same reactant in order to compensate for its possible influence on the analytical signal. Such a procedure is usually adapted to calibration by I-CM, but it evidently has wider applicability. In particular, if E-DM is used, it is suggested that the calibration solutions should be diluted with a reactant chosen as a spectral buffer, and its concentration should be kept constant in order to support the dilution process in the elimination of interferences [33]. It was also revealed that the same aim can be achieved even more effectively if both calibration solutions are injected in a flow
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ACCEPTED MANUSCRIPT system into a carrier stream containing spectral buffer, as then a decreasing amount of interferents is able to meet a progressively increasing amount of buffer [40]. If the calibration function is modeled by a non-linear dependence in a region corresponding to low analyte concentrations, a simple modification of I-CM, which can be
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termed the signal increment procedure, can be applied [41]. It involves the preparation of a set of standards in the usual manner, but the sample is spiked with a known amount of the analyte in order to take the total analyte concentration into the linear region of the calibration
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dependence. The analyte concentration in the sample is calculated in an interpolative way, with the initial analyte concentration taken into account. Recently, the above approach has
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also been applied to E-CM [42] (see Fig. 2S), what is even more reasonable, as the nonlinearity of the low-concentration calibration dependence can lead to much more inaccurate results in E-CM than in I-CM .
The calibration methods (except I-DM and E-DM) are usually performed with the use
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of several standards, enabling creation of multipoint calibration functions. However, the multi-standard procedure is quite often limited to preparation of a single standard solution allowing one-point calibration (which should rather be named ‘one-standard calibration’) to
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be performed [43]. If the procedure is adapted to I-CM, a calibration function is formulated on the basis of the zero-signal (or of the signal intensity measured for a blank solution) and the
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signal intensity corresponding to the standard [43,44]. In the case of E-CM, the signal intensities measured for the sample with and without the standard added are taken for linear extrapolative calibration [45]. It is evident that when using such a procedure one needs to be sure that the calibration dependence in the concentration range considered is linear. A single standard solution can be exploited more efficiently than in the one-point calibration procedure when the signal intensity produced by the standard is controllably changed by a change of selected instrumental parameters. In such a case, the one-point
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ACCEPTED MANUSCRIPT several-graph procedure can be carried out, i.e. two-point calibration functions are formulated corresponding individually to each parameter value. The signal intensities measured in the same conditions are then referred to appropriate calibration functions, allowing a set of apparent analyte concentrations and their mean value (as the final result) to be calculated. The
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calibration data are interpreted similarly to I-DM (see Fig. 1D), but the difference is that the sample is not diluted and the apparent concentrations are not considered to be systematically, but randomly changed. Such calibration is often encountered in flow injection analysis, as the
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flow systems can treat calibration solutions in a special manner (e.g., inject them from different loops [46] or transport them after injection for different lengths of time [47]) in order
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to differentiate the signal intensities significantly and repeatedly. The one-point several-graph procedure was also adapted in flow injection analysis to calibration by E-CM [48]. A common drawback of both versions (interpolative and extrapolative) of this modification is a risk of leading to inaccurate results if the calibration dependence is non-linear.
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I-CM is occasionally performed in accordance with bracketing calibration [49,50], which is in fact a version of the two-point calibration procedure. In this case, two standards are selected: with the analyte at lower and higher concentrations than the analyte
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concentration in the sample. If the concentrations chosen are close enough to each other, the method becomes resistant to the non-linear effect, i.e. the segment of the calibration function
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can be considered linear even if the calibration dependence is really non-linear. Thus, such a procedure helps in the improvement of analytical results in terms of their accuracy. In contrast to the above approaches, one that can be termed the consecutive one-point
calibration is focused exclusively on the modeling and transformation stages of the calibration procedure [51]. It was originally applied to multipoint E-CM, and consists in extrapolative calculation of the apparent concentrations of the analyte in the sample basing on measurements performed for the sample and for each consecutive solution of the sample with
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ACCEPTED MANUSCRIPT standard addition (see Fig. 3A). The apparent concentrations are then expressed as a function of the added analyte concentrations, and the final result is calculated by extrapolation of this function to zero added concentration. As seen in Fig. 3B, such an approach allows, in particular, for determination of the analyte with improved accuracy when the calibration
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dependence modeled in accordance with the basic E-CM procedure is non-linear.
If E-CM is performed in accordance to the basic procedure, i.e. keeping the concentration of all the sample components constant in the series of calibration solutions, the
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multiplicative interferences can be eliminated but the price to be paid is the need to calculate the analytical result extrapolatively. The extrapolation process causes worsening of the results
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(in comparison with those obtained by I-CM in the same experimental conditions) in terms of precision and uncertainty [52,53]. Besides, serious systematic errors can be expected when the calibration function is nonlinear [54] or when the calibration dependence in the extrapolation region is rmodeled by a linear calibration graph function [55]. This is the reason for attempts
extrapolation process.
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to modify the basic E-CM procedure in the preparative stage in order to avoid the
One such approach is the interpolative standard addition method (ISAM), carried out
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using flow injection techniques [56,57]. It involves successive injection (addition) of a set of standard solutions to the sample flowing continuously to the detector. As a result, the positive
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and negative peaks are recorded depending on whether the analyte concentration in a standard is greater or less than the signal intensity measured for the sample. If the differences between the intensities of transient signals and steady-state signal are expressed as a function of the analyte concentration in the standards, the intercept of the calibration function with the concentration axis interpolatively indicates the analyte concentration in the sample (see Fig. 4). Although the procedure allows the extrapolation process to be avoided, the method is not able to compensate for multiplicative interferences [58]. The point is that the sample before
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ACCEPTED MANUSCRIPT and after standard addition is of different dilution, which contravenes the fundamental prerequisite of E-CM. The only advantage of ISAM over I-CM is the possibility of overcoming effects caused by interferents present in the sample in great excess relative to the analyte.
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A modification of the ISAM procedure was proposed in the form of the sample-tostandard additions method [59]. In this approach, a sample and then a blank solution containing all the components of the sample matrix are added to the standard solution. In such
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an approach, being in fact a combination of one-point E-CM with the matrix matching
matrix free of analyte is available.
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procedure, the analytical result can be calculated interpolatively, provided that a sample
Another attempt of the same kind was the standard addition and indicative dilution method (SAIDM) [60] (also reported later as the method of serial dilutions [61]). It consists of the addition of a single standard solution to the sample and successive dilution of this solution
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until the signal intensity measured is equal to or less of the signal intensity produced by the undiluted sample. If the calibration function is formulated as the relationship between the signal intensities measured and the dilution factor, the analytical results can be obtained in an
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interpolative way from equation shown in Fig. 5. It was revealed that SAIDM is able to provide more accurate results than E-CM in some cases when the interferences are not
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multiplicative but of a more complex nature, as well as when the calibration function formulated in accordance with the basic E-CM procedure is nonlinear [60]. A modified extrapolative E-CM procedure of an original interpretation of
measurement data was developed as is the sequential injection standard addition procedure [62]. It involves the creation of a calibration function based on the signal intensities measured for an undiluted sample and for a sample diluted with two standard solutions of different analyte concentrations. As the calibration solutions contain interferents at different
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ACCEPTED MANUSCRIPT concentrations, the analytical result calculated extrapolatively from this function is considered to be the apparent concentration only, rather than the true concentration. In order to obtain an accurate result, it is suggested to apply a mathematical approach: in a few calculation cycles, the analyte concentrations added to the sample are successively corrected on the basis of the
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apparent concentration value obtained one step earlier and the iteration procedure is ended when the difference between two succeeding apparent concentrations is less than a chosen value.
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A procedure that was developed and implemented a long time ago but only studied more deeply quite recently is sequential standard addition calibration (SSAC) [63-66]. It
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consists in successive addition of a single standard solution to a sample and measurement of the signal intensity after each addition. As a consequence, the sample is more and more diluted with the standard, and the total analyte concentration gradually increases or decreases, depending on the standard:sample analyte concentration ratio. The analytical result is
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calculated in the extrapolative way with some mathematical correction compensating the dilution process. The SSAC procedure is recommended to be used especially when the analyte concentration in the sample is relatively high (in particular, too high for the E-CM
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procedure with respect to the limitations of the instrument response). However, SSAC seems to be more similar to ISAM than to E-CM in terms of capability of compensating for
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interferences.
In Table 2, all the above procedures have been characterized in terms of their rational
adaptability to different calibration methods and capability of overcoming calibration errors. As suggested, each procedure can be adapted to a few methods at least, even if originally (in the literature) it was applied with just one of them. When most of these procedures are adapted to interpolative methods, they are able to make them resistant (to a greater or lesser degree) to interference effects. Besides, if some of them are the basis of extrapolative
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ACCEPTED MANUSCRIPT methods, they are able to make the methods more resistant to the non-linear effect. Thus the procedures provide an opportunity to improve both kinds of calibration methods with regard to these effects (which, if unaddressed, potentially create the most serious systematic errors in
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both cases).
4. Combination of analytical methods
Another possible way to achieve some analytical benefits through the use of
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appropriate calibration is to combine calibration methods. In the literature, quite a lot of such approaches can be found. Usually, mixing the methods is recommended, i.e. performing them
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separately (but, as a rule, in the same experimental conditions) in accordance with their individual procedures. There are also some attempts to integrate the methods into a single procedure, thereby in fact creating a new calibration strategy. Both kinds of approaches are
4.1. Mixed methods
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presented below.
The simplest example of mixed methods is to use both I-CM and E-CM according to
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their basic procedures for the determination of an analyte in the same sample. Assuming that the possible multiplicative interference effect is compensated for by E-CM, but not by I-CM,
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then the effect can be detected and estimated by comparison of either the analytical results obtained by both methods [46] or just the slopes of the calibration functions formulated in both cases [67,68]. Thus such an approach can be named a test for the interference effect and recommended as an initial experiment performed especially when no information about the sample composition (and, consequently, about interferents) is provided. In the further course of analysis, the sample should be analyzed in a way that is dependent on the results of this test (i.e. using one of either the interpolative or extrapolative calibration methods). One should
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ACCEPTED MANUSCRIPT remember, however, that the test only allows interferences of a multiplicative nature to be detected and avoided. Even more advantages as described above can be achieved by a different interpretation of the results obtained by I-CM and E-CM if applying the approach originally named a new
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kind of standard addition method [69,70]. It involves the preparation of a set of calibration solutions according to the basic E-CM procedure and the determination of the analyte in all these solutions in accordance with the I-CM procedure (using a separately formulated
SC
calibration function, see Fig. 6A). The results obtained in the interpolative way (i.e. sums of the analyte concentrations in the sample and in the standard additions) are then presented
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against the analyte concentrations added to the sample (as shown in Fig. 6B). Interestingly, on the basis of a single calibration function such as this, one is able not only to evaluate the analytical result in an extrapolative way but also to conclude about the occurrence of the multiplicative interference effect in the assayed sample: namely, if the function is inclined at
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an angle of 45°, the effect is not indicated; otherwise, it is suspected to occur. It was also revealed that such a calibration strategy leads to analytical results of better accuracy than ECM itself when the calibration dependence is non-linear [5]. Other advantages listed in the
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same paper, i.e., the simplicity and rapidity of the calibration procedure, are rather doubtful. When an analyte is required to be determined in a series of samples of similar
AC C
composition and the interference effect is suspected to occur – but at approximately the same level in all samples – I-CM and E-CM can be combined in another way, which can be named simply the mixed method [71]. At first, E-CM is applied to the determination of the analyte in a single selected sample and then the created calibration function (including its extrapolative part) is used for interpolative determination of the analyte in the remaining samples (see Fig. 3S). By doing so, the multiplicative interferences have a chance to be compensated for in all samples, hence avoiding the need to do so by the use of E-CM in relation to each of them
19
ACCEPTED MANUSCRIPT individually. One should note, however, that the selected sample has to be analyzed very carefully as the result of the analyte determination in this sample affects the remaining results in terms of accuracy. The mixture of I-CM and E-CM is also represented by the recovery method [17,72]. It
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is very often used in analytical chemistry when the analyte is suspected to have been lost during the preparative stages of the analytical procedure (e.g. digestion, extraction, preconcentration, separation). In the most common version, it consists in addition of the standard
SC
solution to one of two portions of a sample and in subjection of both portions (in equal volumes) to the required preparative operations and measurements. In such a way, a two-point
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calibration function is formulated according to E-CM principles. Then the analyte concentrations in both portions are found in the interpolative way using a calibration function prepared separately according to I-CM principles. The (percentage) ratio of the difference between the concentrations to the concentration added to the sample is a measure of the
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analyte recovery from the sample. This value is also an indication of both the efficiency of the preparative technique used and the accuracy of the analytical result obtained. The important thing to remember is, however, that the results obtained by the recovery method can be
EP
interpreted correctly only when the effects caused by all preparative operations and, in addition, by the interferents possibly present in the sample have a multiplicative nature
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[73,74].
In all the above approaches, calibration by E-CM (by itself or together with I-CM)
only gives the possibility of overcoming the multiplicative interference effect. As it turns out, when E-CM is applied twice (i.e. ‘mixed’ with itself) in different controllable conditions, it offers a quite new valuable opportunity to additionally eliminate additive interferences caused by unknown interferents present in a sample. This method, absolutely unique in analytical chemistry, is named the H-point standard addition method (HSAM) [75]. In its original
20
ACCEPTED MANUSCRIPT version,
adapted
for
elimination
of
spectral
additive
interferences
in
UV/VIS
spectrophotometry, it involves the preparation of two calibration functions for an analyte determined in the sample in accordance with the basic E-CM procedure. The functions are formulated at two different wavelengths chosen in such a way as to be sure that the signal
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intensities measured for the analyte are as different as possible, but the signal intensities produced by the interferent are equal. Under these conditions, both function have a chance to cross at a point (H-point) indicating both the analyte concentration in the sample and the
SC
signal intensity corresponding to the additive effect (see Fig. 4S). If the latter signal is produced by a known interferent, this component can be determined by, e.g., E-CM using a
of two-component analysis [76].
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calibration function formulated separately. This is why HSAM is usually used for realization
Due to its great analytical significance, HSAM has been intensively developed in order to avoid the above restrictive conditions [77-79] and to adapt it to other instrumental
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methods than UV/VIS spectrophotometry [80-83]. In particular, in the kinetic version, the reaction between the analyte and a reactant added in excessive amount to the calibration solutions (sample alone and with standards added) is involved, and the signal intensity for the
EP
reaction product is measured at two well-defined times after the reaction initiation [83,84]. In this case, two calibration function are then formulated at a single wavelength, but at two
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reaction times, on the assumption that the additive interference effect remains the same under these two conditions. Interestingly, such an calibration approach can be considered as, in fact, being a mixture of two extrapolative indirect methods (E-IM+E-IM). The aforementioned examples of mixed calibration methods are presented in Table 3. When considering them with regard to calibration effects (omitting their other features), they are mostly resistant to multiplicative interference effects alone, but predominantly not resistant to the non-linear effect. This is so because one of the methods that forms part of the
21
ACCEPTED MANUSCRIPT original mix is always E-CM. However, as revealed, they can also be mixed in other configurations of individual interpolative and extrapolative methods, thereby extending their capabilities towards the required analytical and practical objectives.
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4.2. Integrated methods
The primary examples of integrated calibration methods are the extrapolative methods, E-ISM, E-IM and E-DM, which are proposed to be introduced to our extended classification
SC
(see Section 3.2.). The basic procedure of each of them has been combined with the E-CM basic procedure – hence three new methods have been created that are carried out in
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accordance with specific individual procedures at all four calibration stages: preparative, measurement, modeling and transformation stages. As a consequence, each of the new methods is typified by its original features (i.e., also characteristic for I-ISM, I-IM and I-DM, respectively) combined with the features of E-CM. For instance, when using E-ISM there is a
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possibility of determining the analyte with increased precision and of compensating for multiplicative interferences.
An attempt to combine I-ISM with E-CM in another way has been made in an
EP
approach called standard dilution analysis (SDA) [85,86]. It uses two initially prepared solutions: a sample solution and a solution containing the same volume of the sample spiked
AC C
with a known amount of both analyte and internal standard. A set of calibration solutions is prepared by gradual dilution of the second solution with the sample solution. By doing so the concentration of the analyte present in the sample remains constant, but the concentrations of both the internal standard and the analyte added successively decrease to the same extent. A calibration function is formulated as the relationship between of the analyte-to-internal standard signal intensities ratio versus the inverse of the internal standard concentration and the analytical result is calculated from equation presented in Fig. 7.
22
ACCEPTED MANUSCRIPT As seen, the difference between SDA and extrapolative E-CM is that the internal standard is measured at decreasing and not constant concentrations. In spite of this, SDA is able to offer the same main advantages, i.e., correction of both random signal fluctuations and systematic interference effects. It provides results of similar accuracy to E-CM and of
easily automated by using, e.g., a flow technique [87].
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superior accuracy to I-CM and I-ISM [85]. Besides, the calibration procedure is simple and
A simple way to integrate I-DM with E-DM has been presented in the form of the
SC
generalized dilution method (GDM) developed in our laboratory [88]. This approach involves the initial preparation of three solutions: the sample solution, the standard solution and a
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mixture of both solutions (keeping the analyte concentrations in both solutions the same). All of them are then gradually diluted to the same extent, and at each dilution step the apparent concentrations are calculated in both an interpolative and extrapolative way. The results are presented as two plots of apparent concentrations versus dilution factor, and the analyte
dilution (see Fig. 5S).
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concentration in the sample is calculated as the apparent concentration value at infinite
It was revealed that the method can serve not only for elimination of both the
EP
interference and non-linear effects (see Table 4), but also as an effective tool allowing the interference effect to be detected and examined. The shapes and the mutual position of both
AC C
plots are a source of information as to whether the effects occur and, if so, whether they only have a multiplicative or a more complex nature. Besides, as two estimations of the analytical result are obtained in different and independent ways, they can be verified against each other in terms of accuracy. In particular, if both of them are equal, the probability that they are close to the true value is quite great. The advantages of GDM motivated us to continue the integration process of calibration methods in a similar direction and, as a result, two novel approaches to calibration
23
ACCEPTED MANUSCRIPT were proposed: first, the integrated calibration method (ICM) [89,90] and then, the generalized calibration strategy (GCS) [91]. In the most advanced of the developed ICM versions, I-CM and E-CM are combined with each other. At the preparative stage, six calibration solutions are prepared in such a way that the sample and the standard solutions,
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both in two different volumes, are complementarily diluted with each other and with the blank solution [14]. The measurement data then allow four one-point calibration functions to be prepared and four estimations of the analytical result to be calculated (see Fig. 8). As two of
SC
them are obtained in an interpolative way, and two in an extrapolative way, as well as being
for ascertaining their accuracy.
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obtained in pairs for samples of different volumes, their mutual comparison is a strong basis
When developing ICM, it was shown that the measurement data obtained for six calibration solutions are a great enough information source to estimate the analytical result through the additional pair of results obtained by ‘semi-extrapolative’ way [92]. Several flow
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systems operating in accordance with various flow techniques were recommended to implement ICM automatically and effectively [93,94]. What is more, a versatile flow injection manifold was designed enabling integration (through dedicated preparation of the calibration
EP
solutions) of different calibration methods: not only I-CM with E-CM, but also I-ISM with EISM, and I-IM with E-IM, according to the ICM principles [95].
AC C
GCS, similarly to GCM, involves the integration of two dilution methods: I-DM with
E-DM. However, in this case, the procedure is based on the integration process applied to ICM – namely, at the initial stage, six solutions of the same composition as in ICM are prepared and then all of them are gradually diluted [7]. Hence, at each dilution step, four (or even six) apparent concentrations are calculated from such a family of one-point calibration functions as presented in Fig. 8. Due to the dilution process, all apparent concentrations usually lead all together to the true analyte concentration as the sample is more and more
24
ACCEPTED MANUSCRIPT diluted (or if they do not, the dilution process can be supported with special reagents that eliminate the interference effect) and, according to the principles of the dilution methods, indicate the final analytical result for infinite dilution of the sample. In order to ensure both good repeatability of the results throughout the dilution process and satisfactory speed of the
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analytical procedure it is recommended to perform GCM with the use of a dedicated flow system [96,97].
It was shown that GCM calibration can be successfully performed on the basis of
SC
calibration solutions of compositions that are characteristic for indirect methods (I-IM, E-IM) [91]. As a consequence, a ‘three-dimensional’ integration was created by combining indirect
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and dilution methods with interpolative and extrapolative modes (I-IM + E-IM / I-DM + EDM). There is nothing stopping us then from integrating the internal standard method with the dilution method in the same way – in accordance with GCM principles (I-ISM + E-ISM / IDM + E-DM), and, as seen in Table 4, the same is supposed for GDM.
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Thus, GCM can serve – similarly to but even better than GDM – in the diagnosis of an analytical system with respect to interferences. As shown in Table 4, it is able to eliminate not only multiplicative interferences (due to the extrapolation calibration mode) but also more
EP
complex interferences (due to the dilution process), and thereby improve analytical results in terms of accuracy. What is more, it can be additionally integrated with other calibration
AC C
methods. If such advantages of GCM as the ability to work in the non-linear range of the calibration dependence and the possibility of controlling the accuracy of the analytical results during the calibration are taken into account, the method can indeed be considered as being ‘generalized’.
5. Other methods and procedures
25
ACCEPTED MANUSCRIPT The procedures and methods discussed above evidently do not constitute the full package of calibration approaches proposed in the literature. In particular, in this paper, we have omitted those procedures and methods that were recommended many years ago and are not very useful nowadays or even cannot be used with modern instrumentation. Besides,
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many calibration procedures deal with specific conditions (analytical systems, measurement instruments, etc.) and their more comprehensive adaptation is simply not realizable.
An example of a procedure of the latter kind is calibration by internal normalization,
SC
which is also called ‘response-normalized calibration’ (see Section 3.2.) [5]. Such an approach (being in fact a one-point multicomponent procedure) is used when several analytes
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can be determined during a single analytical run (e.g., using separation techniques or ICP). If analytes are different with different sensitivities, the ratios of the signal intensities measured for the analytes and internal standard (named the normalization factors) are then determined and they are the basis for the calculation of the concentrations of each analyte individually in
TE D
the sample. If the analytes reveal the same sensitivity, one normalization factor is sufficient for determination of all analytes. In both versions, the concentration of each analyte is usually expressed as a percentage of the sum of the concentrations of all analytes determined.
EP
As said before, E-ISM offers a chance to compensate not only for random signal fluctuations but also for interference effects, provided that the analyte and internal standard
AC C
are very similar to each other in terms of their physicochemical properties. If the analysis is performed with the use of mass spectrometry, the best choice in this regard is to use an isotopically labeled standard [98] in the role of the internal standard. The drawback is that such kinds of standards are not available in relation to some analytes, or if they are, are very expensive. A further example of a specific calibration approach is the isotope dilution method (99,100). In the simplest version, it consists in the addition of a known amount of
26
ACCEPTED MANUSCRIPT isotopically-enriched substance to the analyzed sample. The analyte concentration in the sample is calculated from the isotope abundance ratios measured for the sample, standard and spiked sample by mass spectrometry. Although, the method is very useful and possible to be exploiting in different versions, its application is limited to the instrumental systems allowing
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different isotopes to be measured and determined (e.g. the mass spectrometry).
A calibration procedure that is also applied very rarely (e.g., when using electroanalytical methods) is the subtraction method, which is presented erroneously as a
SC
variation of the standard addition method [101], but methodologically is more similar to the titration procedure. In this approach, the sample is spiked with a known amount of species
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reacting with the analyte so that the concentration of the remaining analyte in the sample is less than the original concentration. The analytical result can be found on the basis of the signal intensity measured before and after spiking as well as on the basis of the reaction stoichiometry.
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Special attention needs to be paid to titration. In our opinion –contrary to the mainstream – it is a method that belongs fully to the family of chemical calibration methods. It is characterized by a typical calibration procedure encompassing, e.g., preparation of a
EP
standard solution (a titrant), creation of a calibration function plot (titration curve) and evaluation of the analyte concentration in the sample assayed. However, the titration method
AC C
is slightly outside the classifications presented in Section 3.2., especially concerning the way of calculating the analytical result. Namely, the equivalence volume corresponding to the analytical result is not estimated via the signal intensity obtained for the sample (as it is in interpolative methods) but is usually indicated by a characteristic point (e.g., inflection point) of the calibration function. For this reason, titration should rather be referred to as an indicative (and not interpolative or extrapolative) calibration method [18,19].
27
ACCEPTED MANUSCRIPT Another question that is still open is how to classify titration carried out in flow injection analysis by the gradient technique [102]. In this approach, a sample is injected into a stream of titrant flowing continuously and due to the subsequent reaction, a peak cut-off value is produced, the width of which represents the intensity of the analytical signal. As such, a
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signal value cannot be directly attributed to the equivalence volume, and the analyte concentration in the sample is obtained with the use of additional standard solutions in accordance with the I-CM interpolative procedure. It has been revealed that the gradient
SC
titration process is in fact equivalent to the I-IM process [103]. If so, the whole gradient
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titration procedure could be considered as a mixture of two methods: I-CM and I-DM.
6. Conclusions
It has been revealed in Tables 1-4, the calibration approaches, when appropriately selected and applied, can above all considerably contribute to the elimination of systematic
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errors that can occur throughout the calibration procedure, i.e., errors caused by both the interference and non-linear effect. As a consequence, analytical results can be obtained with improved accuracy. In addition, some of the calibration approaches can help to improve
EP
analytical methods in areas such as: precision (I-ISM, E-ISM, interpolative versions of the standard addition method), general analytical applicability (I-IM, E-IM) and efficiency (I-CM,
AC C
one-point calibration procedures, mixed method) or even give a chance to control (to some extent) the accuracy of analytical results throughout the calibration process (ICM, GCS). As shown, the preparative stage of a calibration method determines the subsequent
stages of the calibration procedure, including the way of calculating the analytical result and, consequently, makes the method resistant (or not resistant) to different interference effects. The rules in this domain are as follow:
28
ACCEPTED MANUSCRIPT • if the standard solutions and the sample are prepared separately from each other, the analytical result is calculated in an interpolative way and the interference effect (irrespective of its character) cannot be eliminated (unless additional efforts are undertaken), • if standard solutions are merged with the sample in such a manner that the sample
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components are kept at the same concentration, the analytical result is calculated in an extrapolative way and the multiplicative interference effect can be eliminated,
• if standard solutions are merged with the sample in a different manner from above, the
SC
analytical result can be calculated in the interpolative way, but the interference effect can only be eliminated in special cases (in particular, when it is either constant independent of the
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interferent concentration in a sample or proportional to the analyte-to-interferent concentration ratio [60]),
• if the sample is prepared separately from or together with the standard solutions, and the calibration solutions are progressively diluted, the analytical results calculated at each dilution
TE D
step (interpolatively or extrapolatively) have a chance to be more and more accurate as interference effects of both a multiplicative and complex nature can be decreased; however, one should take into account that the effect in a diluted sample can also be constant or even
EP
increased in some cases [40]).
As seen in Table 2, the actual and potential adaptability of different calibration
AC C
procedures to calibration methods is really quite extensive. Similarly, the implementation of mixing and integrating of methods is also quite great (see Tables 3 and 4). By skillfully combining procedures and methods, there is a chance of creating a novel calibration tools with new analytical features appropriable in both chemical and biochemical analysis.. Certainly, such attempts lead to progress in analytical chemistry, especially when they are directed towards problems which have not been fully solved yet.
29
ACCEPTED MANUSCRIPT One of the problems strongly connected with analytical calibration (but mentioned, recognized and studied very rarely) arises when the analyte is present in the standard solutions in a different chemical form than in the sample, and the signal intensities produced by equal concentrations of both forms are different from each other. Such a phenomenon, termed the
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species effect, can evidently lead to serious systematic errors of analytical results independently of which calibration method is used. The development of a procedure making the calibration process resistant enough to the species effect is certainly one of the greatest
SC
challenge that analysts face.
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Acknowledgments
The study was supported by Polish National Science Centre, Project 2013/11/B/ST4/00864.
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[51] I.G. Zenkevich, T.E. Morozova, Areas of application and characteristics of quantitative chromatographic analysis by the consecutive standard addition method, J. Anal. Chem., 69 (2014) 327-335. [52] M.J. Gardner, A.M. Gunn, Optimizing precision in standard additions determinations, Fresenius Z. Anal. Chem. 325 (1986) 263-266.
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[53] J.M. Andrade, J. Terán-Baamonde, R.M. Soto-Ferreiro, A. Carlosena, Interpolation in the standard additions method, Anal. Chim. Acta 780 (2013) 13-19.
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[54] P. Kościelniak, Non-linear calibration by the standard addition method, Chemometr. Intell. Lab. 47 (1999) 275-287. [55] J.W. Hosking, K.R. Oliver, B.T. Sturman, Errors in the atomic absorption determination of calcium by the standard addition method, Anal. Chem. 51 (1979) 307-310. [56] Tyson F.J., Low cost continuous flow analysis, Anal. Proc. 18 (1981) 542-545. [57] F. Mas, A. Cladera, J. M. Estela, V. Cerdá, New approach to sequential injection analysis: using the sample as carrier, Analyst 123 (1998) 1541-1546. [58] P. Kościelniak, A critical look at the interpolative standard addition method, Analusis 24 (1996) 24-27. [59] Y. Israel, R. M. Barnes, Flow injection sample-to standard addition method. Spectrophotometric determination of hydrochloric acid and orthophosphate, Analyst 114 (1989) 843-848.
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ACCEPTED MANUSCRIPT [60] P. Kościelniak, J. Kozak, M. Wieczorek, Calibration by the standard addition and indicative dilution method in flame atomic absorption spectrometry, J. Anal. Atom. Spectrom. 26 (2011) 1387-1392. [61] W. Hyk, Z. Stojek, Quantifying uncertainty of determination by standard additions and serial dilutions methods taking into account standard uncertainties in both axes, Anal. Chem. 85 (2013) 5933-5939.
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[62] O. Elsholz, C. Frank, B. Stachel, H. Reincke, R. Ebinghaus, Sequential injection standard addition for on-line measurement of mercury in the river Elbe, Anal. Chim. Acta 438 (2001) 251-258. [63] R.J.C. Brown, M.R. Roberts, M.J.T. Milton, Systematic error arising from ‘sequential’ standard addition calibrations: quantification and correction, Anal. Chim. Acta, 587 (2007) 158-163.
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[64] R.J.C. Brown, C.L. Mustoe, Demonstration of a standard dilution technique for standard addition calibration, Talanta 122 (2014) 97-100.
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[65] R.J.C. Brown, Comparison of the extrapolation precision of sequential and conventional standard addition calibrations, Measurement 44 (2011) 1487-1490. [66] R.J.C. Brown, Systematic error arising from ‘sequential’ standard addition calibrations. 2: Determination of analyte mass fraction in blank solutions, Anal. Chim. Acta 648 (2009) 153-156. [67] M. Thompson, S.R. Ellison, R. Wood, Harmonized guidelines for single laboratory validation of methods of analysis, Pure Appl. Chem. 74 (2002) 835–855.
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[68] R.C. Castells, M.A. Castillo, Systematic errors: detection and correction by means of standard calibration, Youden calibration and standard additions method in conjunction with a method response model, Anal. Chim. Acta 423 (2000) 179-185. [69] T. Li, L.-J. Yu, W. Li, M.-T. Li, Analysis of zinc in Vitix negundo foliage by AAS and application of a new kind of standard addition method, Microchim. Acta 150 (2005) 153-157.
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[70] T. Li, L.-J. Yu, M.-T. Li, W. Li, A New Approach to the standard addition method for the analysis of F, Al and K content in green tea, Microchim Acta 153 (2006) 109-114.
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[71] S. Walas, A. Tobiasz, M. Gawin, B. Trzewik, M. Strojny, H. Mrowiec, Application of a metal ion-imprinted polymer based on salen-Cu complex to flow injection preconcentration and FAAS determination of copper, Talanta 76 (2008) 96-101. [72] M. Thompson, S.L.R. Ellison, A. Fajgelj, P. Willetts, R. Wood, Harmonised guidelines for the use of recovery information in analytical measurement, Pure Appl. Chem. 71 (1999) 337-348. [73] P. Kościelniak, On analytical usefulness of the recovery method, Anal. Lett. 37 (2004) 2097-2112. [74] M. Stafiński, M. Wieczorek, P. Janicki, P. Kościelniak, Theoretical and experimental examination of recovery in the context of trueness, Talanta 96 (2012) 39-43. [75] F. Bosch-Reig, P. Campina, P. Campins-Falco, H-point standard additions method. Part 1. Fundamentals and application to analytical spectroscopy, Analyst 113 (1988) 10111016.
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ACCEPTED MANUSCRIPT [76] T.N. Al-Sabha, A.A. Bunaciu, H.Y. Aboul-Enein, H-point standard addition method (HPSAM) in simultaneous spectrophotometric determination of binary mixtures: An overview, Appl. Spectr. Rev. 46 (2011) 607-623.
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[77] J. Verdú-Andrés, F. Bosch-Reig, P. Campíns-Falcó, Analyte estimation using the generalized H-point standard additions method and a new methodology for locating linear spectral intervals for unknown interferents, J. Chemometr. 12 (1998) 27-40. [78] K. Asadpour-Zeynali, M. Bastami, Net analyte signal standard addition method (NASSAM) as a novel spectrofluorimetric and spectrophotometric technique for simultaneous determination, application to assay of melatonin and pyridoxine, Spectrochim. Acta A 75 (2010) 589-597. [79] M. Hasani, M. Mohammadi, M. Shariati-Rad , H. Abdollahi, H-point curve isolation method for determination of catechol in complex unknown mixtures, Spectrochim. Acta A 96 (2012) 563-568.
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[80] P. Campíns-Falcó, F. Bosch-Reig, R. Herráez-Hernández, A. Sevillano-Cabeza, Determination of theophylline and paraxanthine in urine samples by liquid chromatography using the H-point standard additions method, Anal. Chim. Acta 268 (1992) 73-80. [81] P. Campíns-Falcó, J. Verdú-Andrés, F. Bosch-Reig, Development of the H-point standard additions method for the use of spectrofluorimetry and synchronous spectrofluorimetry, Analyst 119 (1994) 2123-2127. [82] E. Shams, H. Abdollahi, M. Yekehtaz, R. Hajian, H-point standard addition method in the analysis by differential pulse anodic stripping voltammetry: Simultaneous determination of lead and tin, Talanta 63 (2004) 359-364.
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[83] M.A. Karimi, M. Mazloum Ardakani, H. Abdollahi, F. Banifatemeh, Application of Hpoint standard addition method and partial least squares to the simultaneous kineticpotentiometric determination of hydrazine and phenylhydrazine, Anal. Sci., 24 (2008) 261-266.
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[84] F. Bosch-Reig, P. Campins-Falco, A. Sevillano-Cabeza, B. Harraez-Hernandez, C. Molins-Lequa, Development of the H-point standard-additions method for ultravioletvisible spectroscopic kinetic analysis of two-component systems, Anal. Chem. 63 (1991) 2424-2429.
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[85] W.B. Jones, G.L. Donati, C.P. Calloway, Jr., B.T. Jones, Standard dilution analysis, Anal. Chem. 87 (2015) 2321-2327. [86] F.M. Fortunato, M.A. Bechlin, J.A. Gomes Neto, G.L. Donati, B.T. Jones, Internal standard addition calibration: Determination of calcium and magnesium by atomic absorption spectrometry, Microchem. J. 122 (2015) 63-69. [87] F.M. Fortunato, M.A. Bechlin, J.A. Gomes Neto, A. Virgilio, G.L. Donati, B.T. Jones, Standard dilution analysis in flow system: Sodium determination by flame atomic emission spectrometry, Microchem. J. 124 (2016) 662-667. [88] P. Kościelniak, M. Kozak, A. Karocki, Detection and examination of interferences basing on the dilution process, Chem. Anal. 41 (1996) 363-376. [89] P. Kościelniak, G. Akcin, M. Herman, J. Kozak, M. Wieczorek, A new approach to the integrated calibration in flow injection analysis, Ann. Chim. 903 (2003) 1045-1052.
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ACCEPTED MANUSCRIPT [90] P. Kościelniak, J. Kozak, M. Herman, M. Wieczorek, A. Fudalik, Complementary dilution method – a new version of calibration by the integrated strategy, Anal. Lett. 37 (2004) 1233-1253. [91] P. Kościelniak, M. Wieczorek, J. Kozak, M. Herman, Generalized calibration strategy in analytical chemistry, Anal. Lett. 44 (2011) 411-430.
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[92] J. Kozak, M. Wójtowicz, A. Wróbel, P. Kościelniak, Novel approach to calibration by the complementary dilution method with the use of a monosegmented sequential injection system, Talanta 77 (2008) 587-592. [93] P. Kościelniak, M. Wieczorek, J. Kozak, J. Kozioł, The integrated calibration method – comparison of various flow approaches, Anal. Lett. 44 (2011) 398-410.
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[94] M. Wieczorek, P. Kościelniak, P. Świt, K. Marszałek, Novel multicommuted flow manifold dedicated to the integrated calibration method, Anal. Methods 6 (2014) 92769282. [95] P. Kościelniak, M. Wieczorek, J. Kozak, M. Herman, Versatile flow injection manifold for analytical calibration, Anal. Chim. Acta 600 (2007) 6-13.
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[96] M. Wieczorek, P. Kościelniak, P. Świt, J. Paluch, J. Kozak, Solenoid micropump-based flow system for generalized calibration strategy, Talanta 133 (2015) 21-26. [97] K. Medinskaia, S. Garmonov, J. Kozak, M. Wieczorek, V. Andruch, P. Kościelniak, A. Bulatov, Stepwise injection determination of isoniazid in human urine samples coupled with generalized calibration method, Microchem. J. 123 (2015) 111-117. [98] Stokvis E, Rosing H, Beijnen JH., Stable isotopically labeled internal standards in quantitative bioanalysis using liquid chromatography/mass spectrometry: necessity or not?, Rapid Commun. Mass Sp. 19 (2005) 401-407.
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[99] L. Wu, M.R. Mashego, J.C. van Dam, A.M. Proell, J.L. Vinke, C. Ras, W.A. van Winden, W.M. van Gulik, J.J. Heijnen, Quantitative analysis of the microbial metabolome by isotope dilution mass spectrometry using uniformly 13C-labeled cell extracts as internal standards, Anal. Biochem. 336 (2005) 164-171.
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[100] D.L. Achat, M.R. Bakker, L. Augusto, E. Saur, L. Dousseron, C. Morel, Evaluation of the phosphorus status of P-deficient podzols in temperate pine stands: combining isotopic dilution and extraction methods, Biogeochemistry 92 (2009) 183-200.
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[101] A.D. McNaught and A. Wilkinson, Compendium of chemical terminology IUPAC, second ed., Blackwell Science, Oxford 1997. [102] J. Růžička, E.H. Hansen, H. Mosbaek, Flow injection analysis : Part IX. A new approach to continuous flow titrations, Anal. Chim. Acta 92 (1977) 235-249. [103] M. Wieczorek, J. Kozak, P. Kościelniak, P. Knihnicki, E. Pieprzyca, Critical approach to flow injection gradient titration as a calibration method, Talanta 96 (2012) 34-38.
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Applicability in the presence of calibration effects Proposed name
AE
calibration curve method
CCM
interpolative conventional method
I-CM
internal standard method
SAM
interpolative internal standard method
I-ISM
n
indirect method
IM
interpolative indirect method
I-IM
dilution method
DM
interpolative dilution method
standard addition method
SAM
-
ME
CE
SC
Calibration method Old name
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Table 1. Nomenclature and characterization of the calibration methods in accordance with the proposed classification (AE, ME, CE - additive, multiplicative and complex interference effect, respectively, NE - nonlinear effect; symbols y, p, n denote applicability, possible applicability and no applicability, respectively)
Specific ability
NE
n
p
to perform analyses in relatively simple and fast way
p
n
p
to compensate random signal fluctuations, to give a chance to compensate the multiplicative interference effect
n
n
n
p
to extend the analytical applicability
I-DM
n
p
y
y
to eliminate interference effects (multiplicative and complex ones) and non-linear effect
extrapolative conventional method
E-CM
n
y
n
n
to compensate random signal fluctuations, to compensate multiplicative interference effect
-
extrapolative internal standard method
E-ISM
n
y
n
n
-
-
extrapolative indirect method
E-IM
n
y
n
n
-
-
extrapolative dilution method
E-DM
n
y
y
y
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to offer correspondingly the same advantages as interpolative methods and to compensate multiplicative interference effect
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Table 2. Original or commonly met ( y) and possible (p) rational adaptation of the calibration procedures to interpolative and extrapolative methods as well as their potential capability ( y) of overcoming non-linear effect (NE) and interference effects of multiplicative (ME), complex (CE) and additive (AE) characters Adaptability to calibration methods Calibration procedure
Interpolative methods I-IDM
Extrapolative methods
I-DM E-CCM E-ISM E-IDM E-DM
AE
ME
CE
NE
y
y
y
y
y
y
y
y
y
p
spiking the sample with special reactant
y
y
p
signal increment procedure
y
p
p
y
p
p
y
one-point calibration
y
y
p
y
p
p
y
one-point several-graph procedure
y
p
p
y
p
p
bracketing calibration
y
p
p
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matrix-matched technique
p
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consecutive two-point calibration
p
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I-CCM I-ISM
Capability of overcoming the calibration effects
y
y
y y
y
p
p
y
y
p
p
y
sample-to-standard additions method
y
p
p
y
y
p
p
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interpolative standard addition method
standard addition and indicative dilution method
y
y y
y
sequential injection standard addition procedure
y
p
p
y
sequential standard addition calibration
y
p
p
y
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y
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Table 3. Characterization of mixed calibration methods (AE, ME, CE – additive, multiplicative and complex interference effect, respectively, NE – nonlinear effect; symbols y and n denote applicability and no applicability, respectively)
Mixture of calibration methods Name
Applicability in the presence of calibration effects
rationally possible mixture
AE
ME
CE
NE
test for interference effect
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
n
new kind of standard addition method
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
y
mixed method
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
n
recovery method
I-CM+E-CM
I-ISM+E-ISM, I-IM+E-IM
n
y
n
n
H-point standard addition method (original version)
E-CM+E-CM
E-ISM+E-ISM, E-IM+E-IM
y
y
n
n
H-point standard addition method (kinetic version)
E-IM + E-IM
E-IM + E-IM / E-ISM+ E-ISM
y
y
n
n
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original mixture
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Integration of calibration methods Name
E-CM + I-ISM
generalized dilution method
I-DM + E-DM
integrated calibration method
I-CM + E-CM
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I-DM + E-DM I-IM + E-IM / I-DM + E-DM
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generalized calibration strategy
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Applicability in the presence of calibration effects AE
ME
CE
NE
E-IM + I-ISM
n
y
n
n
I-ISM + E-ISM / I-DM + E-DM I-IM + E-IM / I-DM + E-DM
n
y
y
y
I-ISM + E-ISM, I-IM + E-IM
n
y
n
n
I-ISM + E-ISM / I-DM + E-DM
n
y
y
y
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standard dilution analysis
rationally possible integration
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original integration
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Table 4. Characterization of integrated calibration methods (AE, ME, CE – additive, multiplicative and complex interference effect, respectively, NE – nonlinear effect; symbols y and n denote applicability and no applicability, respectively)
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Figure captions:
Fig. 1. Principles of interpolative calibration methods: I-CM (A), I-ISM (B), I-IM (C), I-DM (D); Yai, Ybi, Ysi – signal intensities measured for analyte, reactant and internal standard in the standards, respectively; ci – analyte concentrations in the standards (i = 1,…N); Yax, Ybx, Ysx – signal intensities measured for sample; cx – found analyte concentration; , – experimental
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points obtained for standards and sample, respectively Fig. 2. Principles of extrapolative calibration methods: E-CM (A), E-ISM (B), E-IM (C), E-DM (D); ∆ci – analyte concentrations added to sample (i = 1,…N); (remaining symbols as in Fig. 1) Fig. 3. Consecutive one-point calibration: apparent concentrations cx1 – cx3 obtained in extrapolative
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way in accordance to one-point E-CM procedure (A) are presented as a function of analyte concentrations ∆c1 – ∆c3 added to a sample (B); cx – found analyte concentration
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Fig. 4. Interpolative standard addition method: differences between signal intensity Yx measured for a sample and intensities Y0 – Y3 measured for standards added to the sample (A) are presented as a function of added analyte concentrations (B); cx – found analyte concentration Fig. 5. Standard addition and indicative dilution method: calibration graph obtained for a sample spiked with analyte of concentration c1 and diluted to such extent (of dilution factor kx) until concentration
cx – found analyte
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signal intensity Yx measured for undiluted sample is reached;
Fig. 6. New kind of standard addition method: apparent analyte concentrations determined in the interpolative way in a sample, cx1, and in the sample spiked with analyte, ct1 – ct3, (A) and then
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presented as a function of added analyte concentrations ∆c1 – ∆c3 (B); cx – found analyte concentration
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Fig. 7. Standard dilution analysis: calibration graph presented as a function of the analyte-to-internal standard signal intensities ratio, Y/Ys, and the inverse of internal standard concentration, 1/cs;
c1, cs1 – concentrations of analyte and internal standard, respectively, in the standard solutions, cx – found analyte concentration
Fig. 8. Integrated calibration method: calibration graphs based on the signal intensities, Y1 - Y6, corresponding to six calibration solutions prepared from a single standard solution of concentration c1 (for details see text) and allowing the analyte concentration in sample to be calculated in interpolative (cx1, cx2) and extrapolative (cx3, cx4) ways
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A
Ya
B
Ya/Ys
YaN
(Ya/Ys)N
(Ya/Ys)3
Ya2
(Ya/Ys)2
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~
~ Ya3
(Ya/Ys)x
Yax Ya1
0
c1
cx
c2
c3
~
c
0
cN
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Yb
(Ya/Ys)0
c1
cx
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(Ya/Ys)1
Ya
Yb0
c2
c3
~
c
cN
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Ya1
Yb1
Ya2
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Ybx Yb2
Ya3
Yb3
c1
cx
c2
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0
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~ YbN
c3
~
Yax1 YaN Yax2 Yax3 YaxN
c
c
Y0 0
cN
cx
c1
Fig. 1. Principles of interpolative calibration methods: I-CM (A), I-ISM (B), I-IM (C), I-DM (D); Yai, Ybi, Ysi – signal intensities measured for analyte, reactant and internal standard in the standards, respectively; ci – analyte concentrations in the standards (i = 1,…N); Yax, Ybx, Ysx – signal intensities measured for sample; cx – found analyte concentration; , – experimental points obtained for standards and sample, respectively
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A
Ya
(Ya/Ys)N
YaN
(Ya2/Ys)2
Ya1
(Ya/Ys)1
Yax
(Ya/Ys)x
∆c1
∆c2
∆c -cx
∆cN
C
Yb
(Ya/Ys)0
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0
~
0
∆c1
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Ya0
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~
~ Ya2
-cx
B
Ya/Ys
Ya
Yb0
∆c2
~
∆c
∆cN
D
Ya1
Ya2
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Ybx Yb1 Yb2
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~ YbN
0
∆c1
∆c2
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-cx
~
Ya3 YaN Yax1 Yax2 Yax3 YaxN
∆c
∆c
Y0 -cx
∆cN
0
∆c1
Fig. 2. Principles of extrapolative calibration methods: E-CM (A), E-ISM (B), E-IM (C), E-DM (D); ∆ci – analyte concentrations added to sample (i = 1,…N); (remaining symbols as in Fig. 1)
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A
Y3
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Y2
0 Y1
cx
cx1 cx2 cx3
∆c1
∆c2
∆c3
∆c
∆c 0
∆c1
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Yx
cx3
B
c
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Y
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∆c3
Fig. 3. Consecutive one-point calibration: apparent concentrations cx1 – cx3 obtained in extrapolative
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way in accordance to one-point E-CM procedure (A) are presented as a function of analyte concentrations ∆c1 – ∆c3 added to a sample (B); cx – found analyte concentration
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A
B
∆Y
Y3 Y2
∆Y
Yx
∆c
cx
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0
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Y
Y0
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t
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Y1
Fig. 4. Interpolative standard addition method: differences between signal intensity Yx measured for a sample and intensities Y0 – Y3 measured for standards added to the sample (A) are presented
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as a function of added analyte concentrations (B); cx – found analyte concentration
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Y
kx ⋅ c1 1− kx
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cx =
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Yx
k
kx
0
1.0
c
cx
cx+c1
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Fig. 5. Standard addition and indicative dilution method: calibration graph obtained for a sample spiked with analyte of concentration c1 and diluted to such extent (of dilution factor kx) until signal intensity Yx measured for undiluted sample is reached;
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concentration
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cx – found analyte
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c ct3
Y2
ct2
Y1
ct1
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Y3
B
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A
Y
cx1
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Yx
∆c
c
0
cx1
ct2
ct1
ct3
cx
0 c
∆c1
∆c2
∆c3
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Fig. 6. New kind of standard addition method: apparent analyte concentrations determined in the interpolative way in a sample, cx1, and in the sample spiked with analyte, ct1 – ct3, (A) and then presented as a function of added analyte concentrations ∆c1 – ∆c3 (B); cx – found analyte
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tg α c1 • intercept cs1
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intercept
α
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cx =
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Y/Ys
1/cs
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Fig. 7. Standard dilution analysis: calibration graph presented as a function of the analyte-to-internal standard signal intensities ratio, Y/Ys, and the inverse of internal standard concentration, 1/cs; c1, cs1 – concentrations of analyte and internal standard, respectively, in the standard
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solutions, cx – found analyte concentration
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Y Y2 Y5
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Y1
Y6
SC
Y4
Y0 cx3, cx4
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Y3
cx1, cx2
c c1
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Fig. 8. Integrated calibration method: calibration graphs based on the signal intensities, Y1 - Y6, corresponding to six calibration solutions prepared from a single standard solution of concentration c1 (for details see text) and allowing the analyte concentration in sample to be
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calculated in interpolative (cx1, cx2) and extrapolative (cx3, cx4) ways
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ACCEPTED MANUSCRIPT Highlights
• Fundamental terms related to univariate calibration are defined. • Classification of calibration methods is presented.
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• Calibration methods and procedures are reviewed.
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• Real and potential possibility of combining analytical methods is provided.
ACCEPTED MANUSCRIPT Marcin Wieczorek was born in 1978 in Kielce, Poland. He received M.Sc. and Ph.D. degree in analytical chemistry at the Faculty of Chemistry Jagiellonian University in Kraków in 2002 and 2007, respectively. In 2008, he was honoured with the award of the Committee of Analytical Chemistry of the Polish Academy of Sciences for his PhD thesis. Since 2007, he
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has been working in the Department of Analytical Chemistry at the Faculty of Chemistry Jagiellonian University. His main scientific interests focus on development of new analytical calibration strategies and on new methods in chemical analysis realized with the use of flow
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techniques.
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Paweł Kościelniak was born in Krakow, Poland, in 1952. He received his Ph.D. in 1981 from the Faculty of Chemistry, Jagiellonian University, Krakow. He is full professor (since 2000) and head of the Department of Analytical Chemistry at the Jagiellonian University (since 1997). He is a member of the Committee of Analytical Chemistry of the Polish Academy of Sciences. He published more than 200 scientific articles, reviews and book chapters. The main areas of his scientific interest are flow analysis, forensic chemistry and environmental analysis with special attention paid to analytical calibration and interference effects. He supervised twenty Ph.D.theses.
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