A strategy to obtain accurate analytical solutions in second-order

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A strategy to obtain accurate analytical solutions in secondorder multivariate calibration with curve resolution methods Mahdiyeh Ghaffari, Alejandro Cesar Olivieri, and Hamid Abdollahi Anal. Chem., Just Accepted Manuscript • Publication Date (Web): 24 Jul 2018 Downloaded from http://pubs.acs.org on July 25, 2018

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Analytical Chemistry

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A strategy to obtain accurate analytical solutions in second-order

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multivariate calibration with curve resolution methods

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Mahdiyeh Ghaffari1, Alejandro C. Olivieri2, Hamid Abdollahi1*

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Department of Chemistry, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran

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2

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Universidad Nacional de Rosario, Instituto de Química de Rosario (IQUIR-CONICET),

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Suipacha 531, Rosario S2002LRK, Argentina.

Departamento de Química Analítica, Facultad de Ciencias Bioquímicas y Farmacéuticas,

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1 ACS Paragon Plus Environment

Analytical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Abstract

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A novel procedure is described for processing the second-order data matrices with

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multivariate curve resolution-alternating least-squares; while the data set is non-trilinear and

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severe profile overlapping occurs in the instrumental data modes. The area of feasible solutions

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can be reduced to a unique solution by including/considering the area correlation constraint,

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besides the traditional constraints (i.e. non-negativity, unimodality, species correspondence, etc.).

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The latter is not only implemented for the unknown samples, but also for all calibration samples,

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regardless of their interferent content. Area of correlation constraint was specially designed to

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remove rotational ambiguity in the chemical data sets when information about calibration

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samples is at hand. In this contribution a comprehensive strategy is developed to uniquely

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unravel non-trilinear data sets or data sets with severely overlapped profiles in the instrumental

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data modes. The approach is illustrated with simulated and experimental data sets. Borgen plots

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are employed to adequately visualize the extent of rotational ambiguity under non-negativity

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constraint.

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Key words:

Multivariate curve resolution - Alternating least-squares; Area of feasible

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solutions; Area correlation constraint; Unique MCR solutions; second order

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advantage; non-trilinear data sets;


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1. Introduction

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Multivariate curve resolution is a promising tool which has the ability of solving the

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mixture analysis problem, providing chemically meaningful pure component contributions from

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experimental data sets. Multivariate curve resolution-alternating least-squares (MCR-ALS) is an

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iterative algorithm that solves the bilinear model based on Bouguer-Beer-Lambert’s law. The

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output consists of pure concentration and spectral profiles resulting from suitable alternating

7

least-squares optimization subjected to different constraints. The latter is imposed based on the

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chemical knowledge of the studied systems1-4.

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When data sets are analyzed by MCR-ALS, the result might be challenging due to lack of

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unique solutions, which is an intrinsic characteristic of bilinear matrix decompositions if

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incomplete information is available about the system5. Rotational ambiguities are an important

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source of uncertainty in MCR results, and probably the most difficult problem to solve. Non-

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unique MCR solutions derived from rotational ambiguity are less reliable and difficult to

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evaluate. The effect of rotational ambiguity on the accuracy of quantitative results obtained from

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soft-modeling methods has been investigated in detail, even in the presence of constraints. The

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resulting uncertainty may be dramatically large in certain cases, e.g., when extensive or complete

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profile overlapping occurs in one of the data modes6.

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Although non-uniqueness is omnipresent in MCR methods, it can be alleviated and in

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some cases completely avoided by means of a judicious use of the data structure and well-suited

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constraints. For example, imposing different constraints such as non-negativity, unimodality,

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selectivity and hard modeling may dramatically decrease the extent of rotational ambiguity by

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adding more information to the MCR analysis of the system under study7-12. Area correlation is a

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recently developed additional constraint which includes pseudo-univariate local regressions

24

using the area of resolved profiles against reference concentration values during the ALS

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optimization13.

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In the usual extended MCR-ALS analysis of multiple data sets with an appropriate design

27

of the simultaneously analyzed experiments, including the use of proper information as

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constraints, the solutions can be significantly improved through/by reducing or even eliminating

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rotational ambiguities and achieving uniqueness.

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It is important to notice that a carefully design of the augmented data set, should include,

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if possible, samples containing single pure analytes or pure interferents. This might allow the 3 ACS Paragon Plus Environment


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decomposition to reach unique solutions using the species correspondence constraint, also called

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sample selectivity constraint, which forces resolved concentration profiles to have zero area for

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the known absent species in certain samples as a consequence of the well-known Manne

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theorems14-16.

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Finally, the trilinearity constraint can guarantee accurate unique profiles for constituents

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with trilinear structure17-21. To fulfill the trilinearity condition, the augmented data matrix should

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contain some similar factors in its sub-matrices, which should not be affected by experimental

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conditions; so that they share a common shape. Only their areas (and vertical heights) should

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proportionately change according to the constituent concentration. It is a very strong constraint

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that enforces the decompositions to give unique solutions under mild conditions. However,

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trilinearity is seldom achieved in chromatographic experiments due to run-to-run differences in

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peak shapes and positions of pure concentration profiles, which can commonly be found in

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practice. As a consequence, non-trilinear data sets might always be subjected to rotational

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ambiguity to some extent generating uncertainty in analyte prediction. The problem is especially

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severe in cases which selectivity is compromised in the direction of the augmented data, due to

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extensive or total profile overlapping among analyte and interferents.

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In the present report, the possibility of achieving uniqueness in non-trilinear data sets has

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studied by applying area correlation constraint and including proper calibration samples in the

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augmented data matrix, even in cases with total profile overlapping in the augmented direction.

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The proper calibration samples should contain analyte(s) at known nominal concentrations and

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all interferents that might be expected in the following unknown samples. It has already been

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shown that in the absence of these samples, MCR is unable to generate uniqueness, especially

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under severe or total profile overlapping in the concentration profiles. To indicate how area

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correlation constraint can lead to uniqueness of MCR results in the presence of proper calibration

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samples, the extent of rotational ambiguity was evaluated by analysis of different simulated and

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experimental data sets. The latter involves the quantification of ciprofloxacin in a second-order

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luminescence excitation-time decay data of human serum.

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2. Theoretical background

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2.1. Multivariate curve resolution-alternating least-squares (MCR-ALS) 4 ACS Paragon Plus Environment

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In the extended version of MCR-ALS, data matrices can be augmented row-wise,

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column-wise or row- and column-wise to form a multiset structure by appending individual

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matrices in a proper direction and respectively having/keeping the same number of rows,

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columns or both. Individual data matrices should share information with the remaining ones. A

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column-wise augmented matrix Daug can be written as [D1; D2; D3; ...; DK], where the semicolon

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‘;’ MATLAB notation is used to indicate that the different data matrices are appended column-

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wise and k indicates the total number of samples.

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From the mathematical point of view, Daug can be described by a bilinear MCR model, a factor decomposition which can be written as:

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Daug = Caug SaugT + Eaug = [D1; D2; D3; ...; DK] = [C1; C2; C3; ...; CK] SaugT + [E1; E2; E3; ...; EK]

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Where Caug (augmented concentration profiles) and Saug (spectra) are the factor matrices obtained

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by the bilinear decomposition of the experimental data matrix Daug and Eaug collects the model

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errors. This decomposition is performed for a number of components which are contributing to

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the observed data variance in Daug. Alternating least-squares (ALS) optimization is a possible

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way to solve equation (1). The main advantage of this algorithm is the amount of information

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that can be included in the optimization process and the ability for working with either single

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data matrices or multiset data structures. Drawbacks include permutation, intensity and rotational

18

ambiguities. While the first two can be easily resolved, rotational ambiguity is the most

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dangerous one, because it may eventually lead to inaccurate analytical results. Including

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chemical information in the decomposition process, via the above mentioned constraints, can

21

significantly reduce or even eliminate the extent of rotational ambiguity.

(1)

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Usually MCR-ALS optimization starts with the initial estimates of either concentration or

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spectral profiles. Consequently, different methods can be used to find suitable initial estimates to

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start the MCR-ALS. The so-called purest variables method was used to calculate the initial

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spectral estimates.

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2.2. Borgen plots

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Borgen plots22,23 can be considered as a microscope to observe the details of abstract

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space geometry in a three-component system. The subspace of the pure components in a three-

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component system is three dimensional. However, the geometry of the bilinear model in this

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system can be shown in a 2D plane using proper normalization. In a three-component system, the 5 ACS Paragon Plus Environment


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shape of the area of feasible solutions which fulfill the non-negativity constraint and construct

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the whole data matrix is a triangle enclosing the inner polygon. Every vertex of triangles belongs

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to one pure component and locates somewhere between the inner and outer polygons or on the

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edges of the outer polygon. In the reduced abstract space, the projected rows or columns of the

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data span the special parts of space around the origin. This section is enclosed by a convex

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polygon named inner. In addition, the outer polygon is the boundary between positive and

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negative parts of the abstract space.

8

Figure 1 shows an arbitrary Borgen plot of a typical data in U-space for better understanding.

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After proper normalization of the columns of the three-component data matrix, the original three-

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dimensional problem is reduced and visualized in a two-dimensional abstract space; this is the

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reason why only two principal components are required for plotting.

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decomposition (SVD) can be employed to define the basis row (V) and column vectors (U) of a

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data matrix:

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D = U S + = X + = U +

15

where X and Y are coordinates of the rows and columns of D in its row and column space,

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and is the matrix of residuals or non-modeled parts of D. A common procedure for

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normalization in a Borgen plot of U-space is to force the elements of the first eigenvector to be

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one.

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The whole abstract space is the area of feasible solutions (AFS) for all components in the

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absence of constraints. However, considering the non-negative property of a data set, the abstract

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space is delimited by the outer polygon indicated in magenta in Figure 1. The coordinates of the

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abstract representation of the data columns are presented as black circles. Three sections shown

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in blue, green and red are the three feasible regions which belong to the three components. In

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addition, the inner polygon is shown in yellow in Figure 1.

Singular value

(2)

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Figure 1. Visualization of the geometry of abstract space with feasible regions.

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3. Using/applying/imposing area correlation constraint in the presence of proper

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calibration samples

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In the extended MCR-ALS analysis of second-order data, relative quantitative

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estimations of a constituent in the different simultaneously analyzed samples (different data

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matrices), can be easily derived from the relative areas of the concentration profiles of this

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resolved component. A calibration model can be built in this case if the concentration of this

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constituent (analyte) is known in some samples. It can usually be done for the pure calibration

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samples, but here we have proposed to extend the concept to proper calibration samples by

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expecting interferents in future/following samples.

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To apply the area correlation constraint, the integrated concentration sub-profiles for the

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analyte are regressed to known analyte concentrations at each iteration step of the ALS process,

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by means of the following linear model:

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aknown,k = b0 + b1 ck

(3)

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where aknown,k represents the area of the analyte concentration profile in the kth calibration sample,

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ck is the corresponding nominal analyte concentration and b0 and b1 are the intercept and slope of

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the regression line, respectively. The linear model parameters b0 and b1 are estimated for k



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known calibration samples and employed to rescale the elements of the analyte profiles in all the

2

calibration samples, i.e., the vectors , are given by:

3

,, = ,

(4)

,

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The vector ,, is the rescaled analyte concentration profile in the kth calibration

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sample. For the unknown sample, an expression analogous to equation (4) applies: the analyte

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concentration profile in the unknown sample is also scaled:

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,!"# =

$

(5)

$

8

Where aunk is the area of the resolved analyte concentration profile in the unknown sample and

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the analyte concentration in the unknown mixture cunk is estimated as:

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cunk = (aunk – b0) / b1

(6)

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The above procedure is implemented in every ALS step, therefore at convergence to the

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optimal fit, the estimation of the concentration and spectral profiles will also be optimal as well

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as the analyte concentration estimates. Proper calibration samples, which have similar

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compositions as unknown mixtures, lead to unique solutions in trilinear and non-trilinear data

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sets. The number of these samples is an important factor in application of this constraint; it

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should be one unit less than the chemical rank of the unknown sample. As an example, for a

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three-component unknown mixture, two proper calibration samples with similar qualitative

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compositions to the unknown mixture are necessary. It is clear from the literature that analyzing

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full rank second-order chemical data sets with trilinear structure by second-order calibration

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methods can generate uniqueness. In these cases, it is not necessary to employ the proposed

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procedure. On the other hand, if the data are definitely non-trilinear or its trilinearity is unsure,

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the present approach would be beneficial for generating a unique solution.

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4. Data simulations

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In this section, three different examples are described to demonstrate the power of

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extended MCR to resolve diverse chemical problems through applying area correlation

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constraint. The examples involve the simultaneous analysis of: (1) several kinetic-spectroscopic

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data sets involving three parallel first-order kinetic processes with equal rate constant for all

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reactions, (2) different non-trilinear chromatographic runs of a complex mixture and (3) several

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kinetic-spectroscopic data sets without calibration samples containing interferents. 8 ACS Paragon Plus Environment

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In the first two cases, four data matrices were simulated. The first data matrix consists of

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only pure analyte data, whereas the remaining three ones contain the analyte and all interferents.

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The analyte concentration was not only known in the first data matrix, but also in the second and

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third, which were used as proper calibration samples. The composition of the last sample is

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qualitatively similar to the proper calibration samples; however the analyte concentration is

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regarded as unknown. The last simulation consists of only pure analyte as standard data sets and

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a mixture of three components as unknown sample.

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4.1. Dataset 1: kinetic-spectrophotometric data with identical concentration profiles

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The first example is generated from artificially constructed UV-Vis spectra and kinetic

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profiles. The simulated kinetic models are three simple parallel first-order processes with equal

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rate constants monitored by spectrophotometric measurements (the products are supposed to be

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spectrally inactive). First row of figure 2 shows pure concentration profiles, pure spectral profiles

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and initial concentration of three components from left to right, respectively. Analyte specified

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by blue and interferents by green and red. In addition, the Borgen plot in the U-space of this case

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is shown in Figure 3a. The coordinates of the abstract representation of the data columns are

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presented as black circles. Three sections shown in blue, green and red are the three feasible

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regions which belong to the three components (blue is used for the analyte). In addition, the inner

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and outer polygons are shown in yellow and magenta, respectively.

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4.2. Dataset 2: three-component mixture in a chromatographic system

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As a second example, four chromatographic-spectrophotometric data sets were simulated.

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In this case, the analyte elution profile has similar widths with different retention times in the

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various chromatographic runs, causing the system to be non-trilinear. Second row of figure 2

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shows pure concentration profiles, pure spectral profiles and initial concentration of three

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components from left to right, respectively. Analyte specified by blue and interferents by green

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and red. In addition, the Borgen plot in the U-space of this case is shown in Figure 3b. The

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coordinates of the abstract representation of the data columns are presented as black circles.

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Three sections shown in blue, green and red are the three feasible regions which belong to the



1

three components (blue has used for the analyte). In addition, the inner and outer polygons are

2

shown in yellow and magenta, respectively.

3 4

4.3. Dataset 3: three-component mixture and kinetic-spectrophotometric data (No proper

5

calibration samples)

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The third example is generated from artificially constructed UV-Vis spectra and kinetic

7

profiles. The simulated kinetic models are three simple parallel first-order processes monitored

8

by spectrophotometric measurements (the products are supposed to be spectrally inactive). The

9

third row of Figure 2 shows the pure concentration profiles, the pure spectral profiles and the

10

initial concentration of the three components from left to right, respectively. The analyte is

11

specified by blue and the interferents by green and red. In addition, the Borgen plot in the U-

12

space for this case is shown in Figure 3c. The coordinates of the abstract representation of the

13

data columns are presented by black circles. Three sections shown in blue, green and red are the

14

three feasible regions which belong to the three components (blue is used for the analyte). In

15

addition, the inner and outer polygons are shown in yellow and magenta, respectively.

16 17 18 19 20 21 22 23 24 25 26 27


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1 2

Figure 2. A summary of three different simulated cases. Each row represents a designed case

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(the first, second and third row correspond to three simulated cases, simultaneously). Each panel

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represents real profiles. From left to right: pure concentration profiles, pure spectral profiles and

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initial concentration of three components. The analyte is specified by blue and the interferents by

6

green and red.

7 8 9 10 11 12 13



a

b

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c

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Figure 3. a, b and c are the Borgen plots in U-space of three different simulated cases,

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respectively. The AFS of the analyte is specified in blue and the interferents in by green and red.

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5. Experimental data

7

This data set consists of terbium (III) -sensitized luminescence excitation-time decay

8

matrices, measured for human serum samples spiked with the fluoroquinolone antibiotic

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ciprofloxacin, in the presence of the potential interferent salicylate. The analyte concentration

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ranges were all within the therapeutic range, i.e., 0-6 mg L–1 in serum, with final concentrations

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in the measuring cell in the order of 0.2 mg L–1. For additional experimental details, see ref24.

12 13 14 15

5.1. Software Data analysis was run under MATLAB 8.3 and to construct the Borgen plots, the generalized Borgen plots module of the FACPACK software was used25.

16 17 18

6. Visualization

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Analysis and visualization of multivariate data sets are important aspects of

20

chemometrics. Thus, in this section we discuss the visualization process in the simulated data

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sets, to reach a better understanding of the effect of area correlation constraint in the presence

22

and absence of proper calibration samples. In order to visualize this concept, it is necessary to

23

prove that in the presence of proper calibration samples, a single point exists in the U-space of


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the data which fulfills the area correlation constraint. Consequently, in the absence of the latter it

2

is not possible to reach uniqueness.

3

Each point in the U-space of the three simulated cases in the Borgen plots has two

4

coordinates, α and β. Based on the coordinates of points and column eigenvectors, it is possible

5

to generate the profile corresponding to the point. The values of first coordinates are ones

6

because of normalization.

7

p = [1 α β] %

8

p is the translated profile belonging to each α and β. Applying the area correlation constraint on

9

'. Based on the explained procedure in part 3 the area correlation constraint must be p, generates &

10

applied not only on pure and mixture calibration samples also the unknown samples will be

11

scaled. Pure analyte standards and mixture standards can enter into the internal calibration model

12

as long as no sample matrix effect exists. Otherwise, only mixture standards act to define the

13

calibration model.

14

'. If the residual shows a Consequently, it is possible to calculate the residual between p and &

15

clear minimum, p fulfills the area correlation constraint.

16

'; ssq= ∑ *,+ r=p– &

17

Figure 4 displays the value of log(ssq) vs. α and β in the column space of the three simulated

18

cases from left to right.

(7)

(8)

19

20

Figure 4. Values of log(ssq) vs. α and β in the column space of simulated cases 1, 2 and 3,

21

respectively.

22 23



1

The two left panels of Figure 4 demonstrate that applying the area correlation constraint in the

2

presence of two proper calibration samples in a three-component system will generate a unique

3

solution, as it is clear in cases 1 and 2. In addition, in case 3, which does not contain proper

4

calibration samples, it is not possible to get a unique solution, because all profiles in the space

5

fulfill the constraint as well.

6 7 8

7. Results and discussion

9

In order to investigate the effect of area correlation constraint on the accuracy of MCR-

10

ALS analyte quantification results in the presence and absence of proper calibration samples, the

11

simulated and experimental examples have been studied and the results are discussed below.

12 13 14

7.1. Dataset 1: kinetic-spectrophotometry

15

In this case, the four simulated data matrices were augmented column-wise, i.e. in the

16

time direction. As explained above, the first matrix corresponds to a standard analyte solution

17

with known concentration and the two next ones are corresponding to proper calibration samples,

18

also with known analyte concentrations, but in the presence of interferents. The last sample

19

contains interferents and analyte at an unknown concentration with the aim of its quantification

20

in this mixture.

21

First, the area of feasible solutions for the analyte was computed in the concentration

22

space of the augmented multivariate data matrix, by only applying the non-negativity constraint.

23

Since every point in this AFS can be transformed to a concentration profile, the feasible band for

24

the analyte profile is shown in Figure 5a. In the next step all of the analyte’s concentration

25

profiles which were calculated under non-negativity constraint were normalized to the maximum

26

value of the standard part as illustrated in 5b. It is clear that the result is non-unique, except for

27

the pure analyte sample (the first one). To reach uniqueness, additional information should be

28

provided during the ALS procedure. Thus, in the next step, the area correlation constraint was

29

applied to all concentration profiles from the area of feasible solutions of the analyte under non-

30

negativity. In all the profiles a single one occurs which fulfills the area correlation constraint, as

31

seen in the visualization section. The corresponding concentration profile is shown in Figure 5c, 14 ACS Paragon Plus Environment

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1

illustrating the unique resolution of the analyte profile in this case. The extent of rotational

2

ambiguity on the accuracy of quantitative results obtained from soft-modeling methods has been

3

investigated in details in the presence of non-negativity and species correspondence. The

4

resulting uncertainty was dramatically large in the case with complete profile overlapping in one

5

of the data modes26. However, the proposed procedure can guarantee the accuracy of results for

6

this case in the presence of proper calibration samples and area correlation constraint as shown in

7

simulation.

8 9 10 11

a

b

c

12 13

Figure 5. a, b and c are the concentration profiles of the analyte in dataset 1, under non-

14

negativity constraint, under non-negativity and normalize to the maximum value of standard part

15

and under non-negativity and area correlation constraint, respectively.

16 17 18

7.2. Dataset 2: non-trilinear chromatography with spectral detection

19

Non-trilinearity is an intrinsic property of chromatographic-spectral data sets because

20

total reproducibility in the concentration profile of compounds in different experimental runs can

21

seldom be achieved. A data structure of this type was employed to show the ability of area

22

correlation constraint in dealing with non-trilinear data sets. The above described four simulated

23

data matrices were augmented column-wise, i.e. in the direction of the elution time. The first

24

sample is taken as a standard analyte solution with known concentration and the next two are 15 ACS Paragon Plus Environment


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proper calibration samples with similar qualitative composition as unknown mixtures and with

2

known analyte concentration. The last sample is a truly unknown mixture with the aim of analyte

3

quantitation.

4

Similar to the previous case, the area of feasible solutions for the analyte was computed

5

in the concentration space of the augmented multivariate data matrix, by only applying the non-

6

negativity constraint. Since every point in this AFS can be transformed to a concentration profile,

7

the feasible band for the analyte profile is shown in Figure 6a. In the next step, all of the

8

concentration profiles of analyte which were calculated under non-negativity constraint were

9

normalized to the maximum value of the standard part as illustrated in 6b. It is clear that the

10

result is non- unique except for the pure analyte sample (the first one).In the next step, the area

11

correlation constraint was applied to all concentration profiles from the area of feasible solutions

12

of the analyte under non-negativity. In all profiles a single one occurs which fulfills the area

13

correlation constraint. The corresponding concentration profile is shown in Figure 6c, illustrating

14

the unique resolution of the analyte profile in this case. Again, the range of feasible solution of

15

analyte in concentration mode is reduced to a single point, owing to useful information added to

16

the analysis.

17 18

a

b

c

19 20


21

negativity constraint, under non-negativity and normalize to the maximum value of standard part

22

and under non-negativity and area correlation constraint, respectively.

23


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7.3. Dataset 3: three-component mixture and kinetic-spectrophotometric data (No proper

2

calibration)

3

In this case, the four simulated kinetic-spectrophotometric data matrices were augmented

4

column-wise, i.e. in the time direction. As explained above, the first three matrices correspond to

5

a standard analyte solution with known concentration and in the absence of interferents. The last

6

sample contains interferents, and the analyte at an unknown concentration, with the aim of its

7

quantification in this mixture. Similar to other cases, in the first step the analyte area of feasible

8

solutions was computed in the concentration space of the augmented multivariate data matrix, by

9

only applying the non-negativity constraint. Since every point in this AFS of analyte is

10

transformed to a concentration profile, the feasible band for the analyte profile is shown in Figure

11

7a. In the next step, all of the analyte’s concentration profiles which were calculated under non-

12

negativity constraint were normalized to the maximum value of the first standard part as

13

indicated in 7b. It is clear that the result is non-unique, except for the pure analyte samples. It is

14

clear that the result is not unique, except for the pure analyte sample (the first one). It is clear that

15

the result is not unique, except for the pure analyte samples.

16

In the next step, the area of correlation constraint was applied to all the solutions in the AFS of

17

analyte under non-negativity. All profiles in the AFS fulfill the constraint as well, so the extent

18

of rotational ambiguity did not change (7c). a

b

c

19 20


21

negativity constraint, under non-negativity and normalize to the maximum value of the first

22

standard part and under non-negativity and area correlation constraint, respectively.

23 24

7.4. Effect of noise in the performance of the proposed procedure 17 ACS Paragon Plus Environment


1

Analysis of a simulated noisy data set helps to provide an insight about the problem in

2

real experimental systems. Data set 1 is used to show the efficiency of the proposed method in

3

non-ideal situations. Hence a homoscedastic noise equivalent to 0.7% of the maximum values of

4

the signals was added to data set 1. Figure 8 displays of the values of log(ssq) vs. α and β in the

5

column space of data set 1 in the presence of noise. It is clear that a single point exists in the U-

6

space of the data which fulfills the area correlation constraint in the presence of proper

7

calibration samples.

8 9 10

Figure 8. Values of log(ssq) vs. α and β in the column space of the non-ideal data set 1.

11 12

7.5. Experimental data set

13

In order to discuss the ability of area correlation constraint in the presence of noise, a real

14

example is discussed. In this case, twelve kinetic-fluorescence data matrices were augmented

15

column-wise in the direction of decay time. Two pure standards are available, three proper

16

calibration samples which contain interferent and the remaining seven ones were considered as

17

unknown mixtures. Similar to the simulated cases, the analyte area of feasible solutions was first

18

computed in the score space of the augmented data matrix, only under non-negativity. All points

19

in the AFS were transformed to analyte concentration profiles as shown in figure 9a. More

20

information is available about the studied system, i.e. known concentration of analyte in the first

21

five samples. Thus, in the next step, besides non-negativity, the area correlation constraint was

22

applied as additional information to obtain uniqueness. Only one profile fulfill the area


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correlation constraint as well which is shown in figure 9b, that illustrates the achieved

2

uniqueness.

3 4 5

a

b

6 7

Figure 9. The concentration profiles of the analyte in the experimental data, under non-

8

negativity constraint (a), and under non-negativity and area correlation constraint (b).

9 10

The specific quantitation results are displayed in Table 1 in both conditions for

11

application of constraint. In the presence of only non-negativity, a range of analyte

12

concentrations can be predicted. Table 1 provides the upper and lower analyte predictions in this

13

particular case. The smaller concentration value corresponds to the lower analyte profile of its

14

AFS, while the larger corresponds to the upper AFS profile. Large relative uncertainties are

15

derived from these lower and upper analyte predictions, which from Table 1 its range can be

16

estimated from –40% to +80%. However, in the case with both non-negativity and area

17

correlation constraints, the predicted values are unique (Table 1). Moreover, figure 10 shows the

18

satisfactory regression plot of predicted ciprofloxacin concentration versus nominal values in the

19

unknown samples under both constraints, leading to a reasonable average error of 0.03 mg L–1

20

(ca. 10% of relative error with respect to the mean calibration concentration).



1 2

Figure 10. The regression plot of predicted ciprofloxacin concentration vs. nominal values in the

3

unknown samples under non-negativity and area correlation constraint.

4


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1


Table 1. The quantification result of experimental data set in two different conditions. Predictions under constraint(s) Sample

Nominal

Only non-negativity Upper

Lower

Non-negativity and area correlation

1

0.22

0.27

0.12

0.24

2

0.24

0.35

0.11

0.3

3

0.19

0.26

0.4

0.22

4

0.14

0.21

0.03

0.16

5

0.13

0.19

0.08

0.16

6

0.17

0.25

0.10

0.22

7

0.18

0.21

0.02

0.21

8

0.08

0.13

0.05

0.10

9

0.08

0.14

0.04

0.09

10

0.08

0.14

0.03

0.09

2 3 4

8. Conclusions and outlook

5

Noise free simulated data sets were analyzed by MCR-ALS optimization under proper

6

constraints during the least-squares optimization to retrieve the final profiles, namely non-

7

negativity in all profiles in both data modes and area correlation constraint for the analyte

8

concentration profile. In the two first simulated data sets, the resolved analyte profiles were

9

virtually the same as the simulated ones in the simulated data sets. Area correlation constraint

10

and non-negativity for both concentration and spectral profiles were imposed as suitable

11

constraints whish lead to the acceptable analytical results.

12

The uniqueness property, which ensures the accuracy of qualitative and quantitative

13

application of soft modeling results, has immense consequences in second-order analytical

14

calibration. Uniqueness provides the possibility of quantitating analytes in a sample containing

15

unexpected constituents. The main conclusion of the present study is that the area correlation

16

constraint in the presence of proper calibration samples which have qualitatively similar

17

composition as unknown data sets can guarantee accurate solutions in multivariate curve

18

resolution methods even in data sets with non-trilinear structure. The composition of proper 21 ACS Paragon Plus Environment


1

calibration samples is a key factor in using this constraint. The proper calibration data sets should

2

contain all interferents which are expected in future/later unknown samples. The number of

3

calibration samples which should contain all interferents must be one unit less than the chemical

4

rank of the unknown sample. The presently described procedure is a new possibility for general

5

cases of second-order multivariate calibration data in the presence of unknown interferents or in

6

more difficult cases.

7 8

Acknowledgements

9

A.C.O. thank Universidad Nacional de Rosario (Project No. 19B/487), CONICET (Consejo

10

Nacional de Investigaciones Científicas y Técnicas, Project No. PIP 0163) and ANPCyT

11

(Agencia Nacional de Promoción Científica y Tecnológica, Project No. PICT-2016-1122) for

12

financial support.

13


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References (1) Tauler, R.; Kowalski, B.; Fleming, S.: Multivariate curve resolution applied to spectral data from multiple runs of an industrial process. Analytical chemistry 1993, 65, 2040-2047. (2) Tauler, R.; Barceló, D.: Multivariate curve resolution applied to liquid chromatography—diode array. TrAC: Trends in Analytical Chemistry 2013, 12, 319. (3) Tauler, R.: Multivariate curve resolution applied to second order data. Chemometrics and intelligent laboratory systems 1995, 30, 133-146. (4) Tauler, R.; Smilde, A.; Kowalski, B.: Selectivity, local rank, three‐way data analysis and ambiguity in multivariate curve resolution. Journal of Chemometrics 1995, 9, 31-58. (5) Golshan, A.; Abdollahi, H.; Beyramysoltan, S.; Maeder, M.; Neymeyr, K.; Rajkó, R.; Sawall, M.; Tauler, R.: A review of recent methods for the determination of ranges of feasible solutions resulting from soft modelling analyses of multivariate data. Analytica chimica acta 2016, 911, 1-13. (6) Ahmadi, G.; Abdollahi, H.: A systematic study on the accuracy of chemical quantitative analysis using soft modeling methods. Chemometrics and Intelligent Laboratory Systems 2013, 120, 59-70. (7) Ahmadi, G.; Tauler, R.; Abdollahi, H.: Multivariate calibration of first-order data with the correlation constrained MCR-ALS method. Chemometrics and Intelligent Laboratory Systems 2015, 142, 143-150. (8) Rajkó, R.; Abdollahi, H.; Beyramysoltan, S.; Omidikia, N.: Definition and detection of databased uniqueness in evaluating bilinear (two-way) chemical measurements. Analytica chimica acta 2015, 855, 21-33. (9) Beyramysoltan, S.; Abdollahi, H.; Rajkó, R.: Newer developments on self-modeling curve resolution implementing equality and unimodality constraints. Analytica chimica acta 2014, 827, 1-14. (10) Beyramysoltan, S.; Rajkó, R.; Abdollahi, H.: Investigation of the equality constraint effect on the reduction of the rotational ambiguity in three-component system using a novel grid search method. Analytica chimica acta 2013, 791, 25-35. (11) Olivieri, A. C.; Tauler, R.: The effect of data matrix augmentation and constraints in extended multivariate curve resolution–alternating least squares. Journal of Chemometrics 2017, 31, e2875. (12) Omidikia, N.; Abdollahi, H.; Kompany-Zareh, M.; Rajkó, R.: Analytical solution and meaning of feasible regions in two-component three-way arrays. Analytica chimica acta 2016, 939, 42-53. (13) de Oliveira Neves, A. C.; Tauler, R.; de Lima, K. M. G.: Area correlation constraint for the MCR− ALS quantification of cholesterol using EEM fluorescence data: A new approach. Analytica chimica acta 2016, 937, 21-28. (14) Manne, R.: On the resolution problem in hyphenated chromatography. Chemometrics and Intelligent Laboratory Systems 1995, 27, 89-94. (15) Alinaghi, M.; Rajkó, R.; Abdollahi, H.: A systematic study on the effects of multi-set data analysis on the range of feasible solutions. Chemometrics and Intelligent Laboratory Systems 2016, 153, 22-32. (16) Tavakkoli, E.; Rajkó, R.; Abdollahi, H.: Duality based direct resolution of unique profiles using zero concentration region information. Talanta 2018, 184, 557-564. (17) Olivieri, A. C.; Arancibia, J. A.; Muñoz de la Peña, A.; Duran-Meras, I.; Espinosa Mansilla, A.: Second-order advantage achieved with four-way fluorescence excitation− emission− kinetic data processed by parallel factor analysis and trilinear least-squares. Determination of methotrexate and leucovorin in human urine. Analytical chemistry 2004, 76, 5657-5666. (18) Cattell, R. B.: “Parallel proportional profiles” and other principles for determining the choice of factors by rotation. Psychometrika 1944, 9, 267-283. (19) Alier, M.; Felipe, M.; Hernández, I.; Tauler, R.: Trilinearity and component interaction constraints in the multivariate curve resolution investigation of NO and O3 pollution in Barcelona. Analytical and bioanalytical chemistry 2011, 399, 2015-2029. (20) Sanchez, E.; Kowalski, B. R.: Generalized rank annihilation factor analysis. Analytical Chemistry 1986, 58, 496-499.



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(21) Ghaffari, M.; Abdollahi, H.: Duality based interpretation of uniqueness in the trilinear decompositions. Chemometrics and Intelligent Laboratory Systems 2018, 177, 17-25. (22) Borgen, O. S.; Kowalski, B. R.: An extension of the multivariate component-resolution method to three components. Analytica Chimica Acta 1985, 174, 1-26. (23) Rajkó, R.; István, K.: Analytical solution for determining feasible regions of self‐modeling curve resolution (SMCR) method based on computational geometry. Journal of Chemometrics: A Journal of the Chemometrics Society 2005, 19, 448-463. (24) Lozano, V. A.; Ibañez, G. A.; Olivieri, A. C.: Second-order analyte quantitation under identical profiles in one data dimension. A dependency-adapted partial least-squares/residual bilinearization method. Analytical chemistry 2010, 82, 4510-4519. (25) Sawall, M.; Jürß, A.; Neymeyr, K.: FACPACK. (26) Lozano, V. A.; Tauler, R.; Ibañez, G. A.; Olivieri, A. C.: Standard addition analysis of fluoroquinolones in human serum in the presence of the interferent salicylate using lanthanide-sensitized excitation-time decay luminescence data and multivariate curve resolution. Talanta 2009, 77, 1715-1723.

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Caption of figures:

18

Figure 1. Visualization of the geometry of abstract space with feasible regions.

19

Figure 2. A summary of three different simulated cases. Each row represents a designed case

20

(the first, second and third row correspond to three simulated cases, simultaneously). Each panel

21

represents real profiles. From left to right: pure concentration profiles, pure spectral profiles and

22

initial concentration of three components. The analyte is specified by blue and the interferents by

23

green and red.

24

Figure 3. The Borgen plots in U-space of three different simulated cases. The AFS of the analyte

25

is specified in blue and the interferents in by green and red.

26

Figure 4. Values of log(ssq) vs. α and β in the column space of simulated cases 1, 2 and 3,

27

respectively.

28


29

negativity constraint, under non-negativity and normalize to the maximum value of the standard

30

part and under non-negativity and area correlation constraint, respectively.

31


32

negativity constraint, under non-negativity and normalize to the maximum value of the standard

33

part and under non-negativity and area correlation constraint, respectively.

34


35

negativity constraint, under non-negativity and normalize to the maximum value of the first

36

standard part and under non-negativity and area correlation constraint, respectively. 24 ACS Paragon Plus Environment

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Figure 8. Values of log(ssq) vs. α and β in the column space of the non-ideal data set 1.

2

Figure 9. The concentration profiles of the analyte in the experimental data, under non-

3

negativity constraint (A), and under non-negativity and area correlation constraint (B).

4

Figure 10. The regression plot of predicted ciprofloxacin concentration vs. nominal values in the

5

unknown samples under non-negativity and area correlation constraint.

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31



1

Figure 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


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Figure 4

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Figure 6 a

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Figure 8

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Figure 9 a

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Figure 10

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A strategy to obtain accurate analytical solutions in second-order

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