Multiway partial least-squares coupled to residual trilinearization: A genuine multidimensional tool for the study of third-order data. Simultaneous analysis of procaine and its metabolite p-aminobenzoic acid in equine serum

Article abstract of DOI:10.1021/ac070596+

A new third-order multivariate calibration approach, based on the combination of multiway-partial least-squares with a separate procedure called residual trilinearization (N-PLS/RTL), is presented and applied to multicomponent analysis using third-order data. The proposed chemometric algorithm is able to predict analyte concentrations in the presence of unexpected sample components, which require strict adherence to the second-order advantage. Results for the determination of procaine and its metabolite p-aminobenzoic acid in equine serum are discussed, based on kinetic fluorescence excitation-emission four-way measurements and application of the newly developed multiway methodology. Since the analytes are also the reagent and product of the hydrolysis reaction followed by fast-scanning fluorescence spectroscopy, the classical approach based on parallel factor analysis is challenged by strong linear dependencies and multilinearity losses. In comparison, N-PLS/RTL appears an appealing genuine multiway alternative that avoids the latter complications, yielding analytical results that are statistically comparable to those rendered by related unfolded algorithms, which are also able to process four-way data. Prediction was made on validation samples with a qualitative composition similar to the calibration set and also on test samples containing unexpected equine serum components.

Full text of DOI:10.1021/ac070596+

Anal. Chem. 2007, 79, 6949-6958
Multiway Partial Least-Squares Coupled to
Residual Trilinearization: A Genuine
Multidimensional Tool for the Study of Third-Order
Data. Simultaneous Analysis of Procaine and Its
Metabolite p-Aminobenzoic Acid in Equine Serum
Patricia C. Damiani,† Isabel Dura´ n-Mera´s,‡ Alejandro Garcı´a-Reiriz,† Ana Jime´ nez-Giro´n,‡
Arsenio Mun˜oz de la Pen˜a,‡ and Alejandro C. Olivieri*,†
Departamento de Qu´ımica Anal´ıtica, Facultad de Ciencias, Universidad de Extremadura (06071) Badajoz, Spain, and
Departamento de Qu´ımica Anal´ıtica, Facultad de Ciencias Bioqu´ımicas y Farmace´uticas, Universidad Nacional de Rosario,
and Instituto de Qu´ımica Rosario (IQUIR), Consejo Nacional de Investigaciones Cient´ıficas y Te´cnicas (CONICET),
Suipacha 531, Rosario (S2002LRK), Argentina
is by resorting to higher-order data coupled to the second-order
advantage.2 Examples on the use of second-order spectrofluoro-
metric data to analyze complex samples can be found in recent
reports.3,4 However, very few works have been published based
on third-order data. Specifically, they describe the processing of
time-modulated luminescence excitation-emission matrices,
either by following the kinetics of a reaction using spectrofluoro-
metry5-11 or by the time-resolved decay of phosphorescence
signals.12 A recent publication deals with two-dimensional liquid
chromatography coupled to diode array detection for metabolom-
ic studies.13
The classical multiway multivariate algorithm parallel factor
analysis (PARAFAC)14 is capable of processing second- and higher-
order data and obtaining the second-order advantage. The success
of PARAFAC in this regard is directly related to the trilinearity in
the processed data: fluorescence excitation-emission matrices
are not merely a collection of two-way data sets, but there is
actually an internal relationship between each of the matrices, and
as a consequence, the decomposition of the data array built with
A new third-order multivariate calibration approach,
based on the combination of multiway-partial least-
squares with a separate procedure called residual trilin-
earization (N-PLS/RTL), is presented and applied to
multicomponent analysis using third-order data. The
proposed chemometric algorithm is able to predict analyte
concentrations in the presence of unexpected sample
components, which require strict adherence to the second-
order advantage. Results for the determination of procaine
and its metabolite p-aminobenzoic acid in equine serum
are discussed, based on kinetic fluorescence excitation-
emission four-way measurements and application of the
newly developed multiway methodology. Since the ana-
lytes are also the reagent and product of the hydrolysis
reaction followed by fast-scanning fluorescence spectros-
copy, the classical approach based on parallel factor
analysis is challenged by strong linear dependencies and
multilinearity losses. In comparison, N-PLS/RTL appears
an appealing genuine multiway alternative that avoids the
latter complications, yielding analytical results that are
statistically comparable to those rendered by related
unfolded algorithms, which are also able to process four-
way data. Prediction was made on validation samples with
a qualitative composition similar to the calibration set and
also on test samples containing unexpected equine serum
components.
(2) Booksh, K. S.; Kowalski, B. R. Anal. Chem. 1994, 66, 782A-791A.
(3) Smilde, A.; Bro, R.; Geladi, P. Multi-way analysis. Applications in the chemical
sciences; Wiley: Chichester, 2004.
(4) Bro, R. Crit. Rev. Anal. Chem. 2006, 36, 279-293.
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Acta 2003, 475, 137-150.
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412, 195-202.
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130, 85-93.
Owing to its intrinsic sensitivity, easiness of use, and availability
of instruments, fluorescence spectroscopy provides a valuable tool
to analytical chemists.1 When a given sample carries unexpected
fluorescent constituents that have not been modeled during the
calibration phase, a convenient way of quantifying a given analyte
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Espinosa Mansilla, A. Anal. Chem. 2004, 76, 5657-5666.
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Chem. 2006, 385, 1289-1297.
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C. Talanta 2007, 71, 806-815.
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Chem. 2006, 78, 5559-5569.
* To whom correspondence should be addressed. E-mail: aolivier@fbioyf.
unr.edu.ar.
† Universidad Nacional de Rosario and Instituto de Qu´ımica Rosario.
‡ Universidad de Extremadura.
(1) Ichinose, N.; Schwedt, G.; Schnepel, F. M.; Adachi, K. Fluorometric analysis
in biomedical chemistry; John Wiley & Sons: New York, 1987.
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10.1021/ac070596+ CCC: $37.00 © 2007 American Chemical Society
Published on Web 08/10/2007
Analytical Chemistry, Vol. 79, No. 18, September 15, 2007 6949

the response matrices measured for a number of samples is often
unique. If a unique decomposition is achieved, the second-order
advantage can be obtained, allowing a robust estimation of analyte
concentrations in mixtures, even in the presence of unknown
interferences.
zoic acid,26 and therefore, the simultaneous determination of both
compounds is important in monitoring fair horse sports. Several
analytical methods for the determination of either procaine or
p-aminobenzoic acid in a variety of samples are known;27-35
however, only few methods were developed for the simultaneous
determination of both compounds, mainly based on high-
performance liquid chromatographic techniques.36-38
In this report, we follow the hydrolysis of procaine to give
strongly fluorescent p-aminobenzoic acid in aqueous alkaline
media,39 by recording fluorescence excitation-emission matrices
as a function of reaction time, using a fast scanning detector, and
processing the obtained four-way information with the classical
PARAFAC model, as well as with the presently introduced
multiway latent structured method N-PLS/RTL and some of its
unfolded predecessors. We also compare the advantages and
disadvantages of the above-mentioned chemometric techniques
as regards the resolution of the presently studied multicomponent
mixtures.
Recently, alternatives based on data unfolding, such as trilinear
least-squares (TLLS)15 and unfolded-partial least-squares (U-PLS),16
have been introduced, which deal with this information by
concatenating (or unfolding) the original data into unidimensional
arrays (vectors). In the presence of unexpected compounds of
unknown origin, both of these methods should be complemented
with an additional procedure to achieve the second-order advan-
tage. A technique that is useful in this regard, called residual
trilinearization (RTL), has been developed by our group as an
extension of residual bilinearization (RBL),17,18 which was origi-
nally proposed for second-order data.15 Latent structured methods
such as U-PLS/RBL are particularly interesting, especially because
recent applications in the second-order domain have highlighted
its abilities in modeling nontrilinear data sets, when compared to
the more classical PARAFAC and bilinear least-squares/RBL
(BLLS/RBL) methods.19-21
However, it appears that a genuine multiway latent variable
method capable of achieving the second-order advantage is still
needed. We have recently combined multiway PLS (N-PLS)22 with
RBL for three-way data, producing a technique that is still under
testing in our laboratories, using both simulated and experimental
second-order data sets. In the present report, where four-way
kinetic fluorescence excitation-emission data are analyzed, we
introduce the required combination of N-PLS with RTL for third-
order data sets, extending the approach one further dimension.
The aim is the simultaneous quantitation of the anaesthetic
procaine and its metabolite p-aminobenzoic acid in equine serum
samples, where the background interference makes the use of
the second-order advantage mandatory.
THEORY
General Considerations. Before discussing specific aspects
of the employed algorithms, it is useful to consider the different
manners in which they process four-way data arrays. For a given
test or problem sample, PARAFAC builds a joined model, which
includes the third-order signals for all the calibration samples and
the analyzed test sample data. This must be done for each of the
test samples. In the first step of PARAFAC processing, the four-
way signal array is decomposed, while the calibration concentra-
tions are employed, in a subsequent step, to estimate a given
component concentration in the test sample. This means that a
new PARAFAC model is calculated for each new test or problem
sample to be predicted.
On the other hand, in the N-PLS/RTL, U-PLS/RTL, and TLLS/
RTL methodologies, the first step consists of establishing a
relationship between the measured calibration signals and the
known calibration concentrations, without considering the test
sample data. Once this is done, a postcalibration RTL procedure
applied to the test sample signals is introduced, allowing one to
model the presence of unexpected components and to accurately
estimate the analyte concentration.
Procaine (PRO, 2-diethylaminoethyl p-aminobenzoate hydro-
chloride) is used as a local anaesthetic. In human plasma, it is
almost completely hydrolyzed by the enzyme pseudocholinest-
erase to p-aminobenzoic acid (PAB) and diethylaminoethanol.23
The latter compound has both circulatory and neurological effects,
producing blood vessel constriction and euphoria.24 Procaine is
usually used in horses as an anaesthetic and a stimulating drug,
and also accompanying penicillin to treat infections, with plasma
In this way, PARAFAC resolves each component information
separately, and this is the reason why this model inherently
(26) Committee for Veterinary Medicinal Products, Procaine. Summary Report
EMEA/MRL/217/97-FINAL of the Veterinary Medicines Evaluation Unit,
European Agency for the Evaluation of Medicine Products, January 1998.
(27) Storms, M. L.; Stewart, J. T. J. Pharm. Biomed. Anal. 2002, 30, 49-58.
(28) Berzas Nevado, J. J.; Murillo Pulgar´ın, J. A.; Reillo Escudero, O. I. Appl.
Spectrosc. 2000, 54, 1678-1683.
concentrations on the order of 10 mg mL .
-1 25 Horse plasma does
not promote the complete hydrolysis of procaine into p-aminoben-
(15) Arancibia, J. A.; Olivieri, A. C.; Bohoyo Gil, D.; Mun˜oz de la Pen˜a, A.; Dura´n-
Mera´s, I.; Espinosa Mansilla, A. Chemom. Intell. Lab. Syst. 2006, 80, 77-
86.
(29) Einosuke, T.; Yuji, N.; Xuan, Z. S.; Shogo, M.; Yukio, K. Jpn. J. Forensic
Toxicol. 1995, 13, 11-16.
(16) Wold, S.; Geladi, P.; Esbensen, K.; O¨ hman, J. J. Chemom. 1987, 1, 41-56.
(17) O¨ hman, J.; Geladi, P.; Wold, S. J. Chemometrics 1990, 4, 79-90.
(18) Olivieri, A. C. J. Chemom. 2005, 19, 253-265.
(30) Carretero, A. S.; Cruces-Blanco, C.; Peinado, S. F.; Bergmi, R. E. I.; Gutie´rrez,
A. F. J. Pharm. Biomed. Anal. 1999, 21, 969-974.
(31) Badea, I.; Moja, D.; Vladescu, L. Anal. Biochem. 2002, 374, 51-53.
(32) Vanquerp, V.; Rodriguez, C.; Coiffard, C.; Coiffard, L. J. M.; De Roeck-
Holtzhauer, Y. J. Chromatogr., A 1999, 832, 273-277.
(19) Bohoyo Gil, D.; Mun˜oz de la Pen˜a, A.; Arancibia, J. A.; Escandar, G. M.;
Olivieri, A. C. Anal. Chem. 2006, 78, 8051-8058.
(20) Culzoni, M. J.; Goicoechea, H. C.; Pagani, A. P.; Cabezo´n, M. A.; Olivieri,
A. C. Analyst 2006, 131, 718-732.
(21) Garc´ıa-Reiriz, A.; Damiani, P. C.; Olivieri, A. C. Talanta 2007, 71, 806-815.
(22) Bro, R. J. Chemom. 1996, 10, 47-61.
(33) Kastel, R.; Rosival, I.; Blahovec, J. Biomed. Chromatogr. 1994, 8, 294-296.
(34) Stokes, D. L.; Vo-Dinh, T. Sens. Actuators, B 2000, 69, 28-36.
(35) Chen, L. C.; Hu, M. L. J. Food Drug Anal. 1996, 4, 75-87.
(36) Yang, H.; Thyrion, F. C. J. Liq. Chromatogr. Rel. Technol. 1998, 21, 1347-
1357.
(23) Reynolds, J. E. F., Ed. W. Martindale, Extra Pharmacopoeia, Pharmaceutical
Press: London, 1996; Vol. XXXI, p 76.
(24) Turner, P.; Volans, G.; Wiserman, H. Drugs Handbook; McMillan: London,
1993; p 76.
(37) Rop, P. P.; Grimaldi, F.; Bresson, M.; Fornaris, M.; Viala, A. J. Liq.
Chromatogr. 1993, 16, 2797-2811.
(38) Dhananjeyan, M. R.; Bykowski, C.; Trendel, J. A.; Sarver, J. G.; Andob, H.;
(25) Olse´n, L.; Ingvast-Larsson, C.; Brostro¨m, H.; Larsson, P.; Tj¨alve, H. J. Vet.
Pharmacol. Therap. 2007, 30, 201-207.
Erhardt, P. W. J. Chromatogr., B 2007, 847, 224-230.
(39) Kondritzer, A. A.; Zvirblis, P. J. Am. Pharm. Assoc. 1957, 46, 531-535.
6950 Analytical Chemistry, Vol. 79, No. 18, September 15, 2007

presents the second-order advantage (the analyte signal is
separated from that of the potential interferences). On the other
hand, the remaining methods intend to find the relationship
among the measured calibration responses and particular con-
centration information (used to build the model). This is why the
potential interferents should be present in the calibration samples;
otherwise a separate procedure, such as residual trilinearization,
should be combined with the calibration model to obtain the
second-order advantage.
PARAFAC Model. After measuring third-order data for a set
of samples, each of them as a J × K × L array (J, K, and L are the
number of data points in each of the three dimensions), the I
training arrays Xi,cal are joined with the unknown sample array
Xu into a four-way data array X, whose dimensions are [(I + 1) ×
J × K × L]. Provided X follows a quadrilinear PARAFAC model,
it can be written in terms of four vectors for each responsive
component, designated as an, bn, cn, and dn, and collecting the
relative concentrations [(I + 1) × 1] for component n and the
profiles in the three modes (J × 1), (K × 1), and (L × 1),
respectively. The specific expression for a given element of X is40
The number of responsive components (N) can be estimated
by several methods. A useful technique is CORCONDIA, a
diagnostic tool considering the PARAFAC internal parameter
known as core consistency.43,44 Another useful technique is the
consideration of the PARAFAC residual error, i.e., the standard
deviation of the elements of the array E in eq 1.14 Usually this
parameter decreases with increasing N, until it stabilizes at a value
compatible with the instrumental noise (the latter can be assessed
by blank replicate measurements). A reasonable choice for N is
thus the smallest number of components for which the residual
error is not statistically different than the instrumental noise.
Identification of the chemical constituents under investigation
is done with the aid of the three estimated mode profiles, emission
spectrum, excitation spectrum, and kinetic profile, and comparing
them with those for a standard solution of the analyte of interest.
This is required since the components obtained by decomposition
of X are sorted according to their contribution to the overall
spectral variance, and this order is not necessarily maintained
when the unknown sample is changed.
Absolute analyte concentrations are obtained after calibration,
because the four-way array decomposition only provides relative
values (A). Calibration is done by means of the set of standards
with known analyte concentrations (contained in an I × 1 vector
y) and regression of the first I elements of column an against y:
N
Xijkl
)
ainbjnckndln + Eijkl
(1)
∑
n)1
k ) y+ × [a1n|...|aIn]
(2)
where N is the total number of responsive components, ain is the
relative concentration of component n in the ith. sample, and bjn,
ckn, and dln are the fluorescence intensities at the emission
wavelength j, excitation wavelength k, and time l, respectively.
The values of Eijkl are the elements of the array E, which is a
residual error term of the same dimensions as X. The column
vectors an, bn, cn, and dn are collected into the corresponding
loading matrices A, B, C, and D (bn, cn, and dn are usually
normalized to unit length).
where “+” implies taking the pseudoinverse. Conversion of relative
to absolute concentration of n in the unknown proceeds by division
of the last element of column an [a(I+1)n] by the slope of the
calibration graph k:
yu ) a(I+1)n/k
(3)
The model described by eq 1 defines a decomposition of X,
which provides access to emission (B) and excitation spectral
profiles (C), time profiles (D), and relative concentrations (A) of
individual components in the (I + 1) mixtures, whether they are
chemically known or not. This constitutes the basis of the second-
order advantage. The decomposition is usually accomplished
through an alternating least-squares minimization scheme.14,41
Issues relevant to the application of the PARAFAC model for
the calibration of four-way data are as follows: (1) initializing the
algorithm, (2) establishing the number of responsive components,
(3) identifying specific components from the information provided
by the model, and (4) calibrating the model in order to obtain
absolute concentrations for a particular component in an unknown
sample.
The above procedure is repeated for each new test sample
analyzed.
N-PLS Model. In the N-PLS method applied to third-order
data, concentration information is employed in the calibration step,
without including data for the unknown sample. The I calibration
data arrays, together with the vector of calibration concentrations
y (size I × 1) are employed to obtain sets of loadings Wj, Wk, and
Wl (of sizes J × A, K × A, and L × A, where A is the number of
latent factors), as well as regression coefficients b (size A × 1).43
The parameter A can be selected by techniques such as leave-
one-out cross-validation.45 If no unexpected components occurred
in the test sample, b could be employed to estimate the analyte
concentration according to
Initializing PARAFAC for the study of four-way arrays can be
done using singular value decomposition vectors, spectral and
kinetic data that are known in advance for pure components, or
by the loadings giving the best fit after small PARAFAC runs
involving both singular value decomposition vectors and several
sets of orthogonal random loadings. These options are imple-
mented in the PARAFAC package.42
yu ) tuT b
(4)
where tu is the test sample score vector, obtained by appropriate
projection of the test data onto the calibration loading matrices.
When unexpected constituents occur in the unknown sample, the
(43) Bro, R. Multi-way analysis in the food industry. Doctoral Thesis, University
of Amsterdam, The Netherlands, 1998.
(44) Bro, R.; Kiers, H. A. L. J. Chemom. 2003, 17, 274-286.
(45) Haaland, D. M.; Thomas, E. V. Anal. Chem. 1988, 60, 1193-1202.
(40) Leurgans, S.; Ross, R. T. Stat. Sci. 1992, 7, 289-319.
(41) Paatero, P. Chemom. Intell. Lab. Syst. 1997, 38, 223-242.
(42) http://www.models.kvl.dk/source/.
Analytical Chemistry, Vol. 79, No. 18, September 15, 2007 6951

latter scores are unsuitable for analyte prediction through eq 4.
In this case, it is useful to consider the residuals of the N-PLS
modeling of the test sample signal (sp; see eq 5 below) before
prediction is made. These residuals will be abnormally large in
comparison with the typical instrumental noise level:
that model by N-PLS before the RTL procedure, while su (eq 7)
arises from the difference after the RTL modeling of the interferent
effects. Hence, it is the latter one that should be comparable to
the instrumental noise level if RTL is successful.
Finally, we note that N-PLS/RTL can easily been extended to
even higher data orders, giving rise to N-PLS coupled to residual
multilinearization, or N-PLS/RML, which will certainly be more
and more applied in the near future, as data of higher complexity
are produced by modern instruments.
sp ) || ep ||/(JKL-A)1/2 ) || vec(Xu) - vec(Xˆ u) ||/
(JKL-A)1/2 (5)
Software. All routines employed to carry out the calculations
described in this paper were written in MATLAB.46 Those for
applying PARAFAC and N-PLS (without the second-order advan-
tage provided by RTL) are available in the Internet thanks to Bro.42
N-PLS/RTL, U-PLS/RTL, and TLLS/RTL are available from the
authors on request, including a useful graphical user interface
for data input and parameter setting, of the type already described
for first-order multivariate calibration.47
where Xˆ u is the sample three-way data array (Xu) reconstructed
by the N-PLS model and || ‚ || indicates the Euclidean norm.
This situation can be handled by a separate procedure called
residual trilinearization, based on the Tucker3 model of the
unexpected effects.8 RTL aims at minimizing the residuals
computed while fitting the sample data to the sum of the relevant
contributions:
Xu ) reshape{tu[(Wj| X |Wk)| X |Wl]} +
Tucker3(Xˆ u - Xu) + Eu (6)
EXPERIMENTAL SECTION
Apparatus. Fluorescence spectral measurements were per-
formed on a fast-scanning Varian Cary Eclipse fluorescence
spectrophotometer, equipped with two Czerny-Turner mono-
chromators and a xenon flash lamp, and connected to a PC
microcomputer via an IEEE 488 (GPIB) serial interface. Excita-
tion-emission-time data arrays were recorded in a 10 mm quartz
cell, in the following ranges: excitation, 240-300 nm each 6 nm;
emission, 310-390 nm each 4 nm; and time, 0-18 min each 2
min. Thus, a third-order data array for a given sample was of size
11 × 21 × 10, consisting of a total of 2310 data points. The rapid-
scanning instrument allows the acquisition of a complete excita-
tion-emission matrix in 0.2 min at a wavelength scanning speed
of 24 000 nm/min, considerably smaller than the time between
successive measurements. This is important, as it should be
noticed that, if reaction significantly evolves during spectral
acquisition, the sample data would not be strictly quadrilinear.
The cell was thermostated at 40 °C.
Reagents. All experiments were performed with analytical
reagent grade chemicals: p-aminobenzoic acid and trichloroacetic
acid from Sigma-Aldrich (Steinheim, Germany); and sodium
hydroxyde from Merck (Darmstadt, Germany). Procaine was
obtained from Laboratorio Scott Cassara (Rosario, Argentina), as
an aqueous solution containing 452 mg mL-1, as determined by
a titrimetric method described in the Pharmacopoeia.48 Equine
sera were obtained from the laboratory of the Rosario Racetrack.
Calibration, Validation, and Spiked Serum Samples. A
calibration set was constructed composed of 11 samples, using a
central composite design with the central point replicated three
times. The concentrations were in the ranges 0-900 ng mL-1 for
procaine and 0-600 ng mL-1 for p-aminobenzoic acid. The analyte
concentrations were selected by considering the linear fluores-
cence concentration range, the drug therapeutic levels, and the
molecular weight relation, since procaine is converted into
p-aminobenzoic acid by alkaline hydrolysis. For preparing a given
calibration sample, the analytes were mixed in the measuring cell,
where “reshape” indicates transforming a JKL × 1 vector into a J
× K × L three-way array, and | X | indicates the Kathri-Rao
operator.43 During this RTL procedure, the weight loadings Wj,
Wk, and Wl are kept constant at the calibration values, and tu is
varied until the final RTL residual error su is minimized using a
Gauss-Newton procedure, with su given by
su ) || Eu ||/(JKL)1/2
(7)
where Eu is from eq 6. Once this is done, the analyte concentra-
tions are provided by eq 4, by introducing the final tu vector found
by the RTL procedure.
To analyze the presently discussed data, the Tucker3 model
in eq 6 is constructed by restricting the loadings to be orthogonal
and with no special constraints on the core elements. For a single
unexpected component, this analysis is straightforward and
provides the corresponding interferent profiles in the three
dimensions. For additional unexpected constituents, however, the
retrieved profiles no longer resemble true spectra (or time
profiles). Moreover, in this latter case, several different Tucker3
models could in principle be constructed, because the number of
loadings may be different in each dimension. We notice that the
aim which guides the RTL procedure is the minimization of the
residual error term su of eq 7 to a level compatible with the degree
of noise present in the measured signals. Therefore, if two
unexpected components are considered, for example, one should
explore the possible Tucker3 models having one or two loadings
in each dimension and select the simplest model giving a residual
value of su that is not statistically different from the minimum one.
For more unexpected components, a similar procedure is recom-
mended. The final Tucker3 model selected to model the unex-
pected effects is the simplest one, which provides a value of su,
which is not statistically different from the noise level.
(46) MATLAB 6.0, The MathWorks Inc., Natick, MA 2000.
(47) Olivieri, A. C.; Goicoechea, H. C.; In˜o´n, F. A. Chemom. Intell. Lab. Syst. 2004,
73, 189-197.
(48) British Pharmacopoeia; Her Majesty’s Stationary Office: London, 2003, pp
7, 94, and 258.
We note that two different residual parameters appear in the
above discussion, which should not be confused: sp (eq 5)
corresponds to the difference between the test sample signal and
6952 Analytical Chemistry, Vol. 79, No. 18, September 15, 2007

Table 1. Validation Results Using the Different
Algorithmsa
predicted/ng mL-1
nominal/ng mL-1
PARAFAC
N-PLS
TLLS
U-PLS
Procaine
218
607
587
251
657
390
700
668
263
221
602
583
284
683
414
722
672
273
223
603
595
275
684
401
724
674
259
221
607
598
275
687
402
728
678
258
225
603
595
275
681
401
722
672
260
RMSE/ng mL-1
REP%
18
4.0
15
3.3
17
3.8
14
3.1
p-Aminobenzoic Acid
456
307
484
233
290
167
383
394
170
473
299
475
243
287
170
367
392
185
469
307
489
233
300
165
385
410
173
475
304
487
237
295
166
380
406
177
469
307
489
233
300
165
386
410
173
RMSE/ng mL-1
REP%
11
3.7
8
2.7
8
2.7
8
2.7
a RMSE = root mean square error; REP% = relative error of
prediction.
by taking appropriate volumes of PRO and PAB solutions and
diluting with 0.1 M NaOH to the desired concentration. The
excitation-emission-time fluorescence arrays of these solutions
were then recorded, and the data were subjected to four-way
analysis.
A validation set was also prepared, composed of nine samples,
in the same form as those for calibration, but using a random
design, i.e., selecting the target concentrations of both analytes
at random from each calibration range (see Table 1 for details of
the composition of these samples).
Spiked equine serum samples were prepared as follows.
Typically, 0.50 mL of a horse serum was spiked with procaine
and p-aminobenzoic acid, treated with 1.50 mL of 2% (v/v)
trichloroacetic acid and centrifuged at 3000 rpm during 5 min for
deproteinization. Finally, 0.40 mL of the supernatant was diluted
to a final volume of 2.00 mL with 0.1 M NaOH, and thus, the
overall degree of dilution of the equine serum was 1:20. Several
deproteinization agents were tried, but tricholoroacetic acid
seemed to provide better results in terms of absence of inner filter
effects and level of analyte recovery from protein binding. As will
be clear below, a first set of equine sera containing five duplicate
concentration levels of procaine was employed to estimate the
degree of recovery for procaine after deproteinization. This set
(T1) did not contain p-aminobenzoic acid. The second set of sera,
(set T2) spiked with both analytes, was prepared for specific
prediction purposes, and contained five duplicate samples, with
final concentrations of procaine and p-aminobenzoic acid selected
at random from the corresponding calibration ranges. In both of
these sets, each sample contained a different equine serum, with
different concentrations of the analyte or analytes of interest.
Figure 1. (A) Contour plot of the excitation-emission matrix of a
solution of PRO (900 ng mL-1) at pH 5.0 to avoid hydrolysis. The
intensities have been multiplied by a factor of 10. The color bar on
the right shows the fluorescence intensities. (B) Contour plot of the
excitation-emission matrix for a 500 ng mL-1 solution of PAB at pH
13. (C) Contour plot of the excitation-emission matrix for a typical
equine serum, after deproteinization and 1:20 dilution at pH 13. Notice
the higher fluorescence intensities of p-aminobenzoic acid and serum
in comparison with procaine.
RESULTS AND DISCUSSION
Kinetic and Spectral Characteristics. Procaine is weakly
fluorescent (compare the excitation-emission matrices contours
of the analytes procaine and p-aminobenzoic acid in Figure 1A
and B), but its hydrolysis in basic medium generates p-aminoben-
zoic acid, which is a strongly fluorescent compound (Figure 2).
In the working wavelength regions, the signal of the latter is
also overlapped with that for a typical horse serum sample
(Figure 1C), making necessary the use of advanced multiway
modeling techniques for quantifying the analytes in the presence
of the serum background.
Analytical Chemistry, Vol. 79, No. 18, September 15, 2007 6953

Figure 2. Contour plots of the excitation-emission matrices as a function of time (as indicated) for a solution of procaine (700 ng mL-1). The
color bar on the right shows the fluorescence intensities.
The experimental parameters for the hydrolysis reaction were
studied in order to obtain appropriate kinetics and fluorescence
properties of the product originated. Increasing the pH and the
temperature led to increasing reaction rates and to decreasing
fluorescence intensity for p-aminobenzoic acid. Several pH values
were tested in the range between 9 and 13, and temperatures in
the range from 13 to 46 °C. This study suggested that pH 13 and
a temperature of 40 °C were a sensible choice, mainly because
the reaction rate was compatible with the time required to record
a single excitation-emission matrix. At this latter temperature,
the fluorescence intensity of p-aminobenzoic acid was readily
measured.
Validation Data. The first step in processing the data sets
with the different third-order algorithms is the assessment of the
correct number of sample constituents. With PARAFAC, the
standard procedure involves the so-called core consistency
diagnostic test (CORCONDIA),44 which involves observing the
changes in the core consistency parameter as the number of trial
fluorescent components is increased. The number of components
is taken as the largest number for which the latter parameter is
larger than ∼50. However, in the presently studied system, a high
correlation is observed between the concentration of procaine and
that for the p-aminobenzoic acid which forms during hydrolysis
of the former. In this case, CORCONDIA analysis could not be
applied. One alternative is to judge the correct number of
components mainly on the basis of the knowledge of the chemical
composition of samples and kinetic behavior of analytes.49,50
Another technique selects N by considering the residuals of the
PARAFAC least-squares fit for models including an increasing trial
number of components.14 The appropriate value of N corresponds
to the residuals stabilizing at a value compatible with the
instrumental noise. This latter value, estimated through several
blank replicates, was on the order of 5.0 arbitrary fluorescence
units.
Satisfactory PARAFAC fit was achieved by initializing the
algorithm with loadings in the spectral and time dimensions
carrying spectral and kinetic information recorded for pure
p-aminobenzoic acid. However, this procedure lacks the generality
(49) Espinosa-Mansilla, A.; Mun˜oz de la Pen˜a, A.; Goicoechea, H. C.; Olivieri, A.
C. Appl. Spectrosc. 2004, 58, 83-90.
(50) Marsili, N. R.; Lista, A.; Fernandez Band, B. S.; Goicoechea, H. C.; Olivieri,
A. C. J. Agric. Food. Chem. 2004, 52, 2479-2484.
6954 Analytical Chemistry, Vol. 79, No. 18, September 15, 2007

required for checking different trial values of N. A good alternative
was to run several PARAFAC models started with singular value
decomposition vectors and random orthogonal loadings using a
few iterations and selecting the best-fitting model. During PARAFAC
least-squares minimization, all four modes were constrained to
be non-negative, a common practice when facing linear depen-
dency problems.49,50 When applying this procedure to a typical
validation sample, the standard deviation of the PARAFAC residu-
als were, in arbitrary fluorescence units, 15.2, 5.5, and 5.0 for N
) 1, 2, and 3, respectively. Similar results were obtained for all
validation samples, suggesting that two components is a sensible
choice in all cases.
The need of two PARAFAC components can be explained on
the following basis. Assuming that only p-aminobenzoic acid is
fluorescent, and that hydrolysis of procaine originates p-aminoben-
zoic acid, the excitation-emission matrix for a given sample will
vary as a function of time as
Xi(t) ) SPAB,i cPAB + SPAB,i cPRO f (t)
(8)
where Xi(t) is the excitation-emission matrix data matrix at time
t for sample i, SPAB,i is the excitation-emission data matrix for
pure p-aminobenzoic acid at unit concentration, cPAB and cPRO are
the nominal concentrations of p-aminobenzoic acid and procaine
in this particular sample, and f (t) is a function of time describing
the kinetics of the conversion of procaine into p-aminobenzoic acid.
Since SPAB,i is the same in both terms of the right-hand side of
eq 8, we may write
Xi(t) ) SPAB,i [cPAB + cPRO f (t)]
(9)
If the factor [cPAB + cPRO f (t)] were independent of the nominal
concentrations cPAB and cPRO, due to a closure relationship present
in the system, then one may consider that a single PARAFAC
component is enough to describe the time evolution of the data.
However, in our case, procaine and p-aminobenzoic acid are
independent analytes, and hence, we have employed designed
calibration data, where cPAB and cPRO are different for each sample.
Consequently, there is no single function of time that can describe
this evolution, and hence two PARAFAC components are required.
This implies the need of a B loading matrix having two columns,
both equal to the (normalized) emission spectrum of p-aminoben-
zoic acid. Likewise, the C matrix will contain two columns with
the excitation p-aminobenzoic acid spectra. The loading matrix
D, in turn, would be given by
Figure 3. Spectral profiles recovered by PARAFAC after processing
a typical validation sample, containing procaine (218 ng mL-1) and
p-aminobenzoic acid (456 ng mL-1): solid line, first PARAFAC
component; dashed line, second PARAFAC component. (A) Spectral
emission profiles. (B) Spectral excitation profiles. (C) Time profiles.
All profiles are normalized to unit length.
produced by the small fluorescence signal of procaine
(Figure 1), which is not only very low but does also decrease
significantly with time.
Analysis of a typical validation sample rendered the profiles
in the three dimensions which are shown in Figure 3. The
emission and excitation profiles for both calibrated components
were almost identical and match the expected spectral properties
of p-aminobenzoic acid, but the time profiles differed. The
retrieved kinetic profiles are composed of the following: (1) a
time-decreasing profile for one of the spectrally active components
and (2) a time-increasing profile for the other component. These
profiles are not the expected ones on the basis of the monitored
phenomenon, i.e., a constant signal due to the p-aminobenzoic
acid originally present in the solution, and a growing signal due
to procaine being hydrolyzed to p-aminobenzoic acid. These
unexpected profiles suggest the presence of linear combinations
in the time mode, which are expressed into corresponding
correlations in the PARAFAC A or score matrix (the decomposi-
D ) [N1 cPAB 1 | N2 cPRO f (t)]
(10)
where 1 is a K × 1 vector having all elements equal to 1, t is a
vector having the K time values, f (t) is the function of time of
eq 9, and N1 and N2 are normalization constants. However, a D
matrix composed of linear combinations of the two columns of
eq 10 will also describe the time variation of the instrumental data.
This means that the PARAFAC least-squares fit may give linear
combinations of the expected ideal kinetic profiles, which may
depend on the following: (1) the starting values for the least-
squares PARAFAC minimization and (2) possible perturbations
Analytical Chemistry, Vol. 79, No. 18, September 15, 2007 6955

unexpected interferents. In this sense, the application of N-PLS
to the presently studied samples was considerably simpler than
PARAFAC, requiring no special initialization conditions or restric-
tions.
The N-PLS results are collected in Table 1 and are similar to
those obtained with its unfolded companions U-PLS and TLLS,
but better than those from PARAFAC as regards the content of
procaine. The better predictive ability of N-PLS can be related to
its more flexible internal structure in comparison with PARAFAC,
which makes the former methodology better fitted for processing
four-way information, which is not strictly quadrilinear. It is
interesting to note that the presently applied method is based on
a kinetic phenomenon, and therefore, the precise timing of the
instrumental measurements may be crucial for the success of a
given algorithm. The following causes may lead to multilinearity
losses: (1) the time needed to record the complete excitation-
emission matrix of a sample in comparison with the reaction
progress and (2) the fact that the analytes are the reagent and
product of the hydrolysis reaction. These two sources of multi-
linearity losses may seriously affect the less robust algorithm
PARAFAC as compared with a latent structured methodology such
as N-PLS/RTL.
Figure 4. PARAFAC A scores as a function of nominal analyte
concentrations in the calibration set of samples, after processing a
typical validation sample. (A) First column of matrix A as a function
of p-aminobenzoic acid calibration concentrations. (B) Second column
of A as a function of p-aminobenzoic acid . (C) First column of A as
a function of procaine. (D) Second column of A as a function of
procaine .
Spiked Serum Samples. In the case of the set of spiked
serum samples, application of PARAFAC followed guidelines
similar to those commented on above in connection with the study
of the validation samples, in the sense of the need of setting
initialization conditions (the best-fitting model of several models
run with a few iterations proved to be a useful procedure) and
also of the requirement of non-negative constraints during the
least-squares fit. The assessment of the number of components
could not be done using core consistency diagnostics, and was
made by increasing a trial number of components, and inspecting
the PARAFAC least-squares residues until they matched the
instrumental noise level. Using this approach, a typical spiked
serum samples rendered the following values for the standard
deviation of PARAFAC residuals: 18.5, 14.5, 5.2, and 4.9, in
arbitrary fluorescence units, for N ) 1, 2, 3, and 4, respectively.
Similar results were obtained for all spiked samples. These values,
together with the instrumental noise level of ∼5.0 arbitrary
fluorescence units, suggest that three responsive components are
a sensible selection, with the additional component with respect
to the validation samples being ascribed to the serum background.
It is noteworthy that PARAFAC decomposition retrieved correctly
the spectral (both excitation and emission) and time profiles for
serum, which are shown in Figure 5. Indeed, Panels A and B in
Figure 5 favorably compare the recovered spectral profiles with
those recorded for a typical unspiked blank serum sample. Figure
5C, in turn, compares the time evolution with that expected, i.e.,
a constant value at all times (normalized to unit length). Though
the agreement is reasonable, it is however noticed that the
PARAFAC time profile suffers an initial decrease until it stabilizes
after the first minutes. We attribute this observation, which was
repeated consistently for all samples, to temperature instabilities
upon placement of the measuring cell in the spectrofluorometer,
rather than to any reaction of serum under the applied conditions.
Since the fluorescence intensity usually decreases with increasing
temperature, the initial decrease in Figure 5C for the serum
background is ascribed to changes upon equilibrating the tem-
tion of the four-way data renders the profiles in the three
dimensions, which are usually collected into the B, C, and D
matrices). These correlations translate into pseudounivariate
calibration graphs that do not present the familiar properties with
respect to nominal concentrations of either analyte. Figure 4
clearly shows that none of the A scores are linearly related to
procaine, although one of them provides a distorted central
composite design (typical of the presence of correlations between
analytes), whereas the correlation with p-aminobenzoic acid
calibration concentrations, although seemingly linear, indicates
the presence of a significant intercept.
One alternative to cope with the above problem is to correlate
all columns of A with concentrations for each analyte using
multilinear regression (MLR) analysis instead of classical
pseudounivariate methodology. This approach has been previously
applied in the modeling of a kinetics using four-way fluorescence
data.5 When PARAFAC results were processed in this latter way,
analyte concentrations could be reasonably predicted in the
validation set of samples. The statistical summary of this analysis
is presented in Table 1. As can be seen, good figures of merit are
obtained, in terms of root-mean-square error (RMSE) and relative
error of prediction (REP%), computed with respect to the mean
calibration concentration of each analyte. It should be noticed that
these results came at the expense of a somewhat complicated
program operation, in terms of initialization, fitting restrictions
and postdecomposition calibration procedures.
On the other hand, when using N-PLS (and also U-PLS), a
valid procedure for establishing the number of latent variables A
is leave-one-out cross-validation, according to the Haaland and
Thomas criterion, as described in the literature.45 Within the TLLS
methodology, the number of components is established as the
number of calibrated analytes (two in the present case). None of
these approaches required the use of the separate RTL procedure
for analyzing the validation samples, since they contain no
6956 Analytical Chemistry, Vol. 79, No. 18, September 15, 2007

Table 2. Spiked Sera Results Using the Different
Algorithms on Set T2a
predictedb/ng mL-1
nominal/ng mL-1
PARAFAC
N-PLS
TLLS
U-PLS
Procaine
0
57
0
38
-38
614
834
580
-35
-88
529
759
494
-20
20
580
1003
700
0
540
747
567
651
919
612
RMSE/ng mL-1
REP%
97
22
55
12
54
12
109
24
p-Aminobenzoic Acid
300
0
320
70
297
54
320
70
297
55
110
250
400
170
310
445
147
317
416
150
325
445
149
310
405
RMSE/ng mL-1
REP%
54
18
42
14
54
18
40
13
a RMSE = root mean square error; REP% = relative error of
prediction. Predictions are averages of duplicate samples. b Predicted
values were corrected using the correction factor for deproteinization
(see text).
acid showed a good correlation of predicted versus nominal values,
but in the case of procaine, the linear correlation exhibited a slope
that was smaller than the ideal value of 1. In particular, linear
regression of predicted versus nominal procaine concentrations
in the set T1 of serum samples (see Experimental Section)
rendered the following results: slope ) 0.47(2) (standard deviation
in the last significant figure in parenthesis) with an r2 ) 0.9819.
The value of the slope was subsequently employed to correct for
the deproteinization loss of procaine when predicting its concen-
tration in the independent set T2. Interestingly, the degree of
binding or procaine to serum proteins has been estimated to be
∼45%,51 and thus, we attribute the lower value of the slope in the
case of procaine to this binding phenomenon.
To demonstrate how consistent the results are on analyte
recovery after serum deproteinization, a set of sera spiked with
both analytes (T2) was studied, and the predicted procaine
concentrations by PARAFAC were adjusted according to the slope
value. The results on the set T2 are shown in Table 2. It is apparent
that the RMSE and REP% values are reasonable in view of the
complexity of the presently studied biological samples. It may be
noticed that even when the test sample composition is more
complex than the validation samples, a successful PARAFAC
model could be built, based on the one employed to model the
calibration data, expanded by adding an interfering component.
With a similar starting strategy, non-negativity restrictions and
MLR calibration, PARAFAC achieves prediction results that are
comparable to those for the validation samples.
Figure 5. (A) Emission profiles recovered for the component not
present in the calibration set after processing a typical spiked equine
serum sample. In the case of PARAFAC (solid line), it is one of the
three components used to model the data, which is only present in
the spiked test serum sample. In the case of N-PLS/RTL (dashed
line), it is the corresponding Tucker3 loading in the emission
dimension, as obtained through the RTL procedure. The dashed-
dotted line shows the spectrum for a typical unspiked serum sample
The emission spectrum of the serum has been registered at a
constant excitation wavelength of 280 nm. (B) Spectral excitation
profiles, similar to (A). The excitation spectrum of the serum has been
registered at a constant emission wavelength of 360 nm. (C) Time
profiles, similar to (A). In the time profiles, solid and dashed lines are
almost coincident. The dashed-dotted line is a constant at all times.
All profiles have been normalized to unit length.
perature to 40 °C, which was selected for obtaining a sensible
reaction rate for procaine hydrolysis.
As was the case with the validation data, prediction of analyte
concentrations in spiked sera required MLR calibrated with the
relevant A-scores (i.e., the columns corresponding to the analytes)
against nominal calibration concentrations, because of obtaining
linear combination of analyte scores. Specific prediction results
showed that there was a consistent degree of loss of procaine
during the deproteinization procedure. A series of spiked sera with
several different concentrations of procaine and p-aminobenzoic
When applying N-PLS to these spiked sera, it was necessary
to assess the number of unexpected components (Nunx) to be
employed in the RTL procedure. This can be done by analyzing
the residuals su as a function of a trial number of unexpected
components, as has already been described. A typical result for
one of the analyzed samples was as follows: su ) 45 and 6.2 for
(51) Tobin, T.; Blake, J. W. Br. J. Sport Med. 1976, 10, 109-116.
Analytical Chemistry, Vol. 79, No. 18, September 15, 2007 6957

Nunx ) 0 and 1, respectively. For Nunx ) 2, all possibilities having
a different number of loadings in each dimension were tried in
order to construct the Tucker3 model for the potential interferents,
giving su ) 4.8 [2 2 2], 6.1 [1 1 2], 6.1 [1 2 1], 6.2 [2 1 1], 4.8 [2
2 1], 6.0 [2 1 2], and 6.1 [1 2 2] (the values are in arbitrary
fluorescence units, with the number of loadings used in the
emission, excitation, and time dimension, respectively, between
brackets). Since blank replication allowed us to estimate an
instrumental noise level of ∼5.0 arbitrary fluorescence units, for
this particular sample Nunx ) 1 is a reasonable choice. This is
because the value of (su)2, measuring the residual variance, is
statistically indistinguishable from the squared noise level (25
arbitrary fluorescence units2) using an appropriate Fischer F test
for the comparison of variances. Similar results were obtained for
the remaining samples. The choice of Nunx ) 1 can also be
confirmed to be reasonable because, using this value, the
predicted analyte concentrations did not significantly change with
respect to the use of higher number of unexpected components.
Application of the N-PLS/RTL methodology to this set led to
the retrieval of unexpected serum spectral and time profiles, which
can be compared with those expected for background serum.
Panels A-C in Figure 5 show the profiles obtained by the RTL
procedure, i.e., the three Tucker3 loadings, which are employed
to model the effect of the background serum. Figure 5 shows that
the ability of RTL in recovering profiles is of a quality comparable
to that of PARAFAC, implying that the former approach is valid
for achieving the second-order advantage. Moreover, the subtle
effects discussed above concerning the slight decrease of fluo-
rescence intensity for serum in the first reaction minutes are also
observable in the RTL time profile of Figure 5C.
results: U-PLS/RTL, slope ) 0.45(2) (r2 ) 0.9890); and TLLS/
RTL, slope ) 0.51(2) (r2 ) 0.9874). Prediction on set T2, in turn,
provided comparable results to N-PLS/RTL, except in the case
of U-PLS/RTL for procaine (Table 2), which are comparable to
PARAFAC.
CONCLUSIONS
The use of kinetics in combination with excitation-emission
fluorescence matrices measured with a fast-scanning fluorometric
detector allows the recording of four-way data sets. Several
unfolded algorithms are available for processing four-way informa-
tion, although they do not take full advantage of the higher-order
data structure. In comparison, multiway partial least-squares is a
genuine higher-order latent variable method, which, once com-
bined with the successful residual trilinearization technique,
achieves the second-order advantage needed to successfully
quantify analytes in complex biological samples. The latter
combination of techniques, which is proposed in this report, i.e.,
N-PLS/RTL, is currently the only competitor available to parallel
factor analysis in the realm of truly higher-order analysis achieving
the important second-order advantage. It certainly deserves a place
among the few available multiway chemometric methods for
advanced modeling studies and is demonstrated to perform better
than PARAFAC in the presently studied samples. This is because
of the more flexible internal structure of N-PLS/RTL in compari-
son with PARAFAC, which makes the former a more appropriate
approach for processing four-way data, which are not strictly
quadrilinear.
ACKNOWLEDGMENT
Concerning the specific prediction results, better figures of
merit were obtained with N-PLS/RTL in comparison with those
furnished by PARAFAC. First, the set T1 of serum samples spiked
only with procaine was studied, in order to assess the level of
recovery after deproteinization. This study yielded, for N-PLS/
RTL, slope ) 0.45(2) (r2 ) 0.9930), i.e., similar to PARAFAC (see
above). Subsequent prediction on the second independent set T2
of spiked equine sera gave the results shown in Table 2. As can
be seen, the results are better than those for PARAFAC, again
pointing to an improved predicting ability of the latent structured
methodology N-PLS/RTL.
Universidad Nacional de Rosario, CONICET (Consejo Nacional
de Investigaciones Cient´ıficas y Te´cnicas, Project No. PIP 5303),
ANPCyT (Agencia Nacional de Promocio´n Cient´ıfica y Tecno-
lo´gica, Project No. PICT-25825), Ministerio de Educacio´n y Ciencia
of Spain (Project CTQ2005-02389), and AECI (Programa Inter-
campus Iberoame´rica/PCI Project A/6576/06) are gratefully
acknowledged for financial support. A.J.-G. thanks Ministerio de
Eudacio´n y Ciencia of Spain and A.G.-R. thanks CONICET for
their fellowships.
Received for review March 26, 2007. Accepted June 15,
2007.
Application of the unfolded methodologies TLLS/RTL and
U-PLS/RTL gave results that are similar to those obtained with
N-PLS/RTL. Recovery of procaine on the set T1 gave the following
AC070596+
6958 Analytical Chemistry, Vol. 79, No. 18, September 15, 2007