Second-order calibration

Calibration is a method in analytical chemistry in which an instrumental response is related to a concentration of a chemical in a sample. Once this relationship is established the instrument can be used to measure the concentration of the analyte calibrated for in a future sample [Booksh Kowalski 1994]. 

A further distinction in second-order instrumentation is based on the mathematical properties of the matrix resulting from a measurement. If the instrument is such that every analyte ideally contributes with a rank one to the instrumental (matrix) response, then this is called a rank-one second-order instrument. The term bilinear is sometimes used in the literature to describe this property. This terminology is not strictly correct, because a bilinear form of a matrix X is X = AB holds for any matrix and does not necessarily mean that the rank of X is one (see also Elaboration 10.1). Not all second-order instruments are of the rank-one type, e.g. tandem mass spectrometry is not a rank-one second-order instrument [Smilde etal. 1994], 

With data coming from hyphenated instruments, low-rank bilinearity of a single-sample measurement can often be assumed. A hypothesized model of the data from one sample 

When the underlying model is low-rank bilinear, the data matrices produced are often close to this bilinear model, except for noise and nonlinearities. In such a case, a low-rank bilinear model can be fitted, for example, by curve resolution methods. When certain added conditions such as selective channels, nonnegative parameters etc. are fulfilled, unique models can sometimes be obtained [Gemperline 1999, Manne 1995], 

With a stack of low-rank bilinear data matrices such as the one above, a three-way array can be built. The underlying model of such an array is not necessarily low-rank trilinear if for example there are retention time shifts from sample to sample. However, the data can be still be fitted by a low-rank trilinear model. If the deviations from the ideal trilinearity (low-rank - one component per analyte) are small, directly interpretable components can be found. If not, components are still found in the same sense as in PCA, where the components together describe the data, although no single component can necessarily be related to a single analyte. 

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Table 4 Comparison of the observed signal intensity with calculated response based on the best fit of a linear or a second-order calibration line...

ISO 8466 describes how to perform calibration. Part 1 is covering the linear regression and part 2 the second order calibration strategy. 

Part 2 Calibration strategy for non-linear second order calibration functions... 

If the relationship between the signal and the concentration is not linear, we may apply a second order calibration function. For details about this slightly more difficult calculation see ISO 8466-2. 

If possible, the linearcalibrationfunctjonshouldbe used, onlyinspedal circumstances second order calibration should be used... 

ISO 8466-2 2001 - Calibration and evaluation of analytical methods and estimation of performance characteristics - Part 2 Calibration strategy for non-linear second-order calibration functions... 

Smilde, A.K., Tauler, R., Henshaw, J.M., Burgess, L.W., and Kowalski, B.R., Multicomponent determination of chlorinated hydrocarbons using a reaction based sensor, 3 medium-rank second-order calibration with restricted Tucker models, Anal. Chem., 66, 3345-3351, 1994. 

There is some confusion in the second-order calibration literature [Sanchez Kowalski 1988] where the term bilinearity is reserved for equations of the form of Equation (2.14) with a one component PCA model for X. However, bilinearity is more general and holds for an arbitrary number of components [Kruskal 1984],... 

At first sight the treatment and concepts of rank might seem very theoretical. However, the concept of rank plays an important role in second-order calibration and curve resolution. Three-way rank may also have repercussions for degrees of freedom and hence for significance testing. 

As an example of the relationship between chemical sources of variation and three-way rank consider second-order calibration. In that type of calibration, instruments are used that give a matrix response for measuring a single sample. The data can, for example, come from fluorescence (emission-excitation) spectroscopy or liquid chromatography-ultraviolet spectroscopy. A standard X (J x K) in which certain analytes are present in known concentrations is used to quantify for those analytes in a mixture X2(/ x K), in which unknown interferents might be present. This results in a three-way array X where Xi and X2 are the two individual slices. Second-order calibration usually comes down to building a PARAFAC model for that X. 

Suppose that in second-order calibration a standard Xi contains one analyte (hence, there is one chemical source of variation) and this standard is measured in such a way that the pseudo-rank of Xi equals one. The mixture X2, measured under the same experimental circumstances, contains the analyte and one unknown interferent. If the instrumental profiles (e.g. spectra and chromatograms) of the analyte and interferent are different, then the three-way array X having Xi and X2 as its two individual slices has two chemical sources of variation. This equals the number of PARAFAC components needed to model the systematic part of the data, which is the three-way rank of X, the systematic part of X. 

Constrained or restricted Tucker models have found use in analytical chemistry, specifically, in second-order calibration [Kiers Smilde 1998, Smilde el al. 1994a, Smilde el al. 1994b, Tauler et al. 1994] as well as in batch process modelling [Gurden el al. 2001, Gurden etal. 2002],... 

One of the most attractive features of the PARAFAC model is its uniqueness property. Many of the applications of PARAFAC models are based on this property, such as second-order calibration and resolution of spectral data [Leurgans Ross 1992, Sanchez Kowalski 1988], It is, therefore, worthwhile to treat this topic in detail. 

In some cases, the Kruskal condition is not met and unique resolution of the data cannot be obtained. However, it may still be possible to obtain partial uniqueness in the sense that some parameters are uniquely determined. One often occurring example is in second-order calibration (Chapter 10) if the pure standard contains only the analyte of interest and the mixture contains the analyte and two or more interferents. 

The model is not fully identified. Unidentified models appear when the k-rank conditions (p.lll) are not satisfied. Consider for example a second-order calibration problem in which two samples are modeled using PARAFAC. One sample may be an unknown sample with, say, R underlying components and the other sample a pure standard with only one underlying component. It follows that the true sample-mode loading ideally holding the estimated relative concentrations has the form... 

In second-order calibration, the concentration of the analyte in the standard is known. This is held in one element of the diagonal of Di whereas the remaining diagonal elements are zero, when the standard only contains the analyte of interest. The matrix D2 then contains the concentrations of all the chemical analytes in the unknown mixture. From the known Di and the estimated D21Di, the value in Di corresponding to the analyte can be obtained. 

Another development, also with a history in chemistry, is second-order calibration, where rank annihilation was developed for analyzing data from typically hyphenated instruments. This includes excitation-emission fluorescence spectra of different samples, liquid chromatography with ultraviolet (UV) detection for different samples and gas chromatography with mass spectrometric detection for different samples, giving an array. An illustration is given in Figure 10.2. 

Figure 10.2. Example of typical structure of data for second-order calibration. LC, liquid chromatography GC, gas chromatography UV, ultraviolet MS, mass spectrometry.

For a second-order instrument there is an even stronger property with respect to interferents. Even in the case of a new interferent in a future sample, accurate analyte concentration estimates are possible. This is called the second-order advantage and is a very powerful property indeed. The first publications on second-order calibration were from Ho and co-workers [Ho et al. 1978, Ho et al. 1980, Ho et al. 1981], who used rank annihilation to quantify the analyte. Unfortunately, in order to obtain the second-order advantage, the second-order instrument has to operate in a very reproducible way, which is not always the case. Hence, up to now, it is not widespread in analytical practice. 

To set the stage for the examples, the most simple rank-one second-order problem is described in mathematical terms and its relation to three-way analysis is established. The standard sample (i.e. the sample that contains a known amount of analyte) is denoted Xi(/ x K) and the mixture (i.e. the unknown sample containing an unknown amount of analyte and possibly some interferents) is called X2 (J x K). It is assumed that the first instrumental direction (e.g. the liquid chromatograph) is sampled at J points, and the second instrumental direction (e.g. the spectrometer) at K points. Then the rank-one second-order calibration problem for the case of one analyte and one interferent can be presented as... 

Depending on the situation, there are several unknowns in Equation (10.4). In its most general case the unknowns are a, a2, bi, b2, c2ji and c2j2. The standard Xi has pseudorank one (see Chapter 2, Section 6), and, hence, can be decomposed in its contributions ai and bi. Note that there is still intensity ambiguity because aibl equals aiQ Q 1b y, for any nonzero a. The quantification of the analyte requires the estimation c2,i and sometimes it is also convenient to obtain estimates of ai, a2, bi and b2 for diagnostics purposes. This is the curve resolution aspect of second-order calibration. 

There are more complicated situations possible, e.g. (i) a standard containing several analytes, (ii) mixtures containing more than one interferent, (iii) multiple standards, (iv) multiple mixtures or combinations of those four. The properties of these second-order calibration problems can be deduced by writing them in a PARAFAC structure (as in Equation (10.4)) and examining their properties. Two such examples are given in Appendix 10.B... 

The second-order calibration example shown next is from the field of environmental analytical chemistry. A sensor was constructed to measure heavy metal ions in tap and lake water [Lin et al. 1994], The two heavy metal ions Pb2+ and Cd2+ are of special interest (the analytes) and there may be interferents from other metals, such as Co2+, Mn2+, Ni2+ and Zn2+. The principle of the sensor is described in detail in the original publication but repeated here briefly for illustration. The metal ions diffuse through a membrane and enter the sensor chamber upon which they form a colored complex with the metal indicator (4-(2-pyridylazo) resorcinol PAR) present in that chamber. Hence, the two modes (instrumental directions) of the sensor are the temporal mode related to the diffusion through the membrane, and the spectroscopic mode (visible spectroscopy from 380 to 700 nm). Selectivity in the temporal mode is obtained by differences in diffusion behavior of the metal ions (see Figure 10.22) and in the spectroscopic mode by spectral differences of the complexes formed. In the spectroscopic mode, second-derivative spectra are taken to enhance the selectivity (see Figure 10.23). The spectra were measured every 30 s with a resolution of 1 nm from 420 to 630 nm for a period of 37 min. This results in a data matrix of size 74 (times) x 210 (wavelengths) for each sample. 

Second-order calibration is a set of methods developed to deal with second-order instrumental data instruments that generate a matrix of measurements for a single sample. Second-order calibration problems can be written in the form of a PARAFAC model, which facilitates understanding the properties of the problem and of the solution. 

Second-order calibration depends critically on the underlying assumptions of linearity. One of the problems, e.g., in using a hyphenated technique such as chromatography-spectroscopy, is the poor reproducibility of the retention time axis. This destroys the trilinearity of the calibration model. For certain situations, this can be counteracted by using different types of three-way models (e.g. PARAFAC2 [Bro el al. 1999]), but in most of the cases extensive (manual) preprocessing is needed to linearize the data. This renders second-order calibration a less practical technique for such data. 

Xie et al. [1996] compared the merits of second-order calibration for quantifying binary mixtures of p-7 o, and m-aminobenzoic acids and orciprenaline reacting with diazotized sulfanilamide in a kinetic UV-vis study. The samples were measured at 31 times and at different numbers of wavelengths. 

In the following, several approaches to constructing a regression model will be tested on some simple example data sets. In these examples, traditional second-order calibration is not possible, as there is no causal or direct relationship between specific measured variables and the responses. The purpose of this example is mainly to illustrate the relative merits of different calibration approaches on different types of data, whereas the model building (residual and influence analysis) itself is not treated in detail. 

Appendix 10.B Other Types of Second-Order Calibration Problems... 

This appendix contains examples of other types of second-order calibration, as mentioned in the main text. 

Appendix 10.C The Multiple Standards Calibration Model of the Second-Order Calibration Example... 

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