Chemometrics and Intelligent Laboratory Systems 126 (2013) 91–99 Contents lists available at SciVerse ScienceDirect Chemometrics and Intelligent Laboratory Systems journal homepage: www.elsevier.com/locate/chemolab How to correct inner ﬁlter effects altering 3D ﬂuorescence spectra by using a mirrored cell X. Luciani ⁎, R. Redon, S. Mounier Université de Toulon, PROTEE, EA 3819, 83957 La Garde, France a r t i c l e i n f o Article history: Received 7 February 2013 Received in revised form 17 April 2013 Accepted 22 April 2013 Available online 2 May 2013 Keywords: EEM Inner ﬁlter effects PARAFAC Fluorescence Mirrored cell Multi-way a b s t r a c t In this paper we present a new correction method of inner ﬁlter effects that occurs when measuring ﬂuorescence Excitation–Emission Matrices (EEM) of concentrated solutions. While traditional method requires absorption measurement or sample dilution(s), the Mirrored Cell Approach (MCA) only requires two different EEM of the considered sample: a ﬁrst one using a traditional cell and a second one using a mirrored cell. The mathematical relationship between both models is originally exploited to obtain a simple numerical correction. Method is validated using a set of known mixtures. In addition we show that advanced multilinear analysis can be efﬁciently applied on to the corrected EEM. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Standard ﬂuorometers allow to measure the ﬂuorescence intensity emitted by a solution for a given couple of excitation and emission wavelengths. A ﬂuorescence Excitation Emission Matrix (EEM) (also called 3D spectrum) is then obtained by scanning both wavelength domains [1] and allows to characterize the solution. EEMs are now widely used in various scientiﬁc domains such as medicine [2], analytical chemistry [3] or environmental sciences, in particular for Dissolved Organic Matter (DOM) tracing and characterization purpose [4,5]. Measured signal relies on the spectroscopic properties of each ﬂuorescent component of the solution (ﬂuorophore) and their concentration in the solution. EEM analysis consists of deducing these underlying parameters from the measured spectrum and thus characterizing mixture components and possibly estimating their relative contribution to the sample signal. A signiﬁcant panel of chemometric tools has been adapted or speciﬁcally developed to solve this inverse problem, from basic peak picking [6] to most advanced approaches based on multidimensional algebra [7]. All of these methods resort to a model of the measured ﬂuorescence signal. Classical multilinear model assumes that the EEM of a single ﬂuorophore, has a magnitude proportional to the ﬂuorophore concentration in the solution and that its pattern is given by the outer product between the excitation spectrum and the emission ⁎ Corresponding author at: Laboratoire PROTEE, Batiment R, Université du Sud ToulonVar, BP 20132, 83957 La Garde Cedex, France. Tel.: +33 4 94 14 23 18. E-mail addresses: [email protected] (X. Luciani), [email protected] (R. Redon), [email protected] (S. Mounier). 0169-7439/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chemolab.2013.04.014 spectrum of the ﬂuorophore. Hence according to this model, EEM of a mixture of ﬂuorophores is described by a linear combination of the individual EEM of each ﬂuorophore. This model is very useful in practice, notably when considering several mixtures (samples) of the same ﬂuorophores. Indeed, the corresponding excitation–emission matrices set deﬁnes a three-way array and the linear model of ﬂuorescence is nothing else than the Canonical Polyadic decomposition (CP) of this array [8], also known in the community as CANonical DECOMPosition (CANDECOMP) [9] or PARAllel FACtor analysis (PARAFAC) [10,7]. We keep here the acronym CP that honors the original work and also stands for CANDECOMP/PARAFAC. This multilinear decomposition usually admits a unique solution [11] and many algorithms allow to perform the decomposition [12–14]. Thereby it has rapidly become one of the most popular EEM analysis method. However it is well known that the pertinence of the linear model decreases with the solution absorbancy [1] meaning that the gradual absorption by the solution of both exciting and ﬂuorescent lights cannot be neglected. These effects are known as inner ﬁlter effects (IFE) and affect both EEMs magnitude and pattern. They are observed and studied for a long time now [15,16] and it had been shown that IFE are still perceptible at low absorption and occur in practical applications such as DOM analysis [17,18]. The result is that traditional linear excitation–emission matrices analysis methods cannot be directly applied on the raw measured EEMs. Two main methods are currently used to restore EEM multilinear properties (we then speak of corrected EEM). The ﬁrst one is to strongly dilute the solution so that absorption of the diluted solution is lower than a certain threshold [19]. This threshold is hard to deﬁne precisely so that it is usually preferable to realize dilution series and compare the corresponding 92 X. Luciani et al. / Chemometrics and Intelligent Laboratory Systems 126 (2013) 91–99 measured EEM to effectively unsure the linearity. Another drawback of this method is that contamination or physico-chemical changes can occur during the dilution, thus modifying the ﬂuorescence properties of the sample. The most common alternative [20,21], namely the Absorption Correction Approach (ACA) resorts to mathematical model of inner ﬁlter effects derived from the Beer–Lambert law [22–24]: −ðAðλex ÞþAðλem ÞÞ 2 F ðλex ; λem Þ ¼ Lðλex ; λem Þ10 ; ð1Þ where F is the measured EEM, L the corrected one and A is the solution absorption. Rigorous physical justiﬁcation of this mathematical model can be found in [25] and [18]. ACA uses the measured absorption spectrum as an estimation of A and then deduces L from F using Eq. (1). Similar methods were proposed in [25–27]. However ACA requires another experimental device. In addition the measured absorption spectrum is sometimes a poor estimate of A because in case of highly absorbing solutions it is obtained from the ratio of two weak signals. In order to avoid these limitations a mathematical correction method has been proposed in [28]. More recently we have introduced the Controlled Dilution Approach (CDA) [18]. CDA does not require absorption measurement but uses a second EEM of the sample to correct the ﬁrst one. This second EEM is measured from a controlled (and reduced) dilution of the considered sample. The dilution is not intended to suppress IFE by itself so that the dilution factor can be chosen arbitrarily small in order to avoid the drawback of the dilution approach. The key point of CDA is that this second EEM can be modelized by a second equation involving F and L. Then it is shown that the linear term L can be directly deduced from the combination of both equations. Hence CDA still involves additional experimental modiﬁcations of the sample. Therefore we introduce in this paper a Mirrored Cell Approach (MCA) which is based on a similar idea but suppresses the dilution step. More precisely, instead of measuring the second EEM from a dilute sample, this one is obtained from the same sample but put into a mirrored cell. MCA principle and implementation are described in the next section. We ﬁrst derive some new mathematical models of IFE that take into account main multiple reﬂections into the mirrored cell. Then we describe MCA as an EEM correction method. In the third section we show how to set MCA parameters and correct practical issues. Third and fourth sections are dedicated to the experimental validation. MCA ability for IFE correction is studied onto a set of various mixtures of three ﬂuorophores, along with its effectiveness as a CP pretreatment of non-linear sets of EEMs. We then conclude about the reliability of this new approach. 2. Theory 2.1. Physical models of inner ﬁlter effects We derive here from physical considerations a new mathematical model of inner ﬁlter effects. The proposed model takes into account multiple reﬂections into the cell. Although several simpliﬁcations (described below) are still assumed, it makes this model suitable for mirrored cell. We extend here the reasoning used in [18]. We consider a mixture of N ﬂuorophores. For each ﬂuorophore n(n = 1 ⋯ N), cn, εn(λex), εn(λem), Φn, γn(λem) denotes the concentration in the solution, the molar extinction coefﬁcient at the excitation wavelength λex, the molar extinction coefﬁcient at the emission wavelength λem, the quantum yield and the emission probability at wavelength λem respectively. Absorption coefﬁcient is deﬁned by αn(λ) = cnεn(λ). In the following model, we also take into consideration possible presence of chromophores in the solution, i.e.: species that are absorbing light but not ﬂuorescing. Hence we denote α0 the total absorption coefﬁcient of chromophores and thus the total absorption coefﬁcient α of the solution is given by: α = ∑nN = 0 αn. Fig. 1 recalls basically the experimental device of right angle standard ﬂuorometers, with multiple reﬂection on the facets. 0 0 Fig. 1. Scheme of the mirrored cell and considered optical paths, view from above. In order to simplify model equations, one classically considers perfectly collimated light beams in both excitation and emission. Although, this is usually not the case in practice (notably with our system) this simpliﬁcation is mathematically justiﬁed by small angle approximation. The excitation light I0(λex) is absorbed through the sample cell (length l) by the solution (primary inner ﬁlter effect). This induces the ﬂuorescent light which is also partly absorbed by the solution (secondary inner ﬁlter effect). A fraction Fmir(λem) of this ﬂuorescence signal is then collected perpendicularly to the exciting beam. We make the following main approximations. First of all, fraction of the exciting light which do not reach the “inﬂuence zone” Z in Fig. 1 is neglected as well as the ﬂuorescence light issued from the region outside Z. In other words two main optical paths are considered which represent the excitation beam and emission beam in Fig. 1 scheme. Furthermore, diffusion and re-emission effects are neglected. A model of inner ﬁlter effects for mirrored cells has been proposed in [29] that take into account reﬂections due to the mirrored facet of the cell. In the present study we assume that the reﬂection coefﬁcients of both the mirrored and non-mirrored facets are non-null. These are denoted Rmir and R respectively. Hence we consider the multiple reﬂexions depicted in Fig. 1 scheme. However variations of Rmir and R according to the wavelength are neglected. Finally, wavelength dependence of I0(λex) is corrected by the apparatus so that I0(λex) = I0. The cell is then divided into horizontal and vertical elementary strips of respective dimensions dy × l and l × dx. At the entry of the cell (x = 0) each horizontal strip is supposed to receive an equal elementary fraction of the exciting light from the rectilinear exciting beam: dyI Δ . Beer–Lambert law then quantiﬁes the intensity transmitted to a position x in the cell, where a fraction αndx of this intensity is absorbed by the ﬂuorophore n. Hence, taking into account the primary inner ﬁlter effect (α(λex)) and the multiple reﬂections that depend on R and Rmir values, the total intensity absorbed by ﬂuorophore n in an elementary horizontal strip of the “inﬂuence zone” (An) is given by: 0 y An ðλex Þ ¼ ∞ X l þ Δx dyI0 p p −α ðλex Þð2plþxÞ α n ðλex Þ∫l−Δx R Rmir e þ Δy p¼0 ! ∞ X p−1 p −α ðλex Þð2pl−xÞ R Rmir e dx: 2 2 p¼1 ð2Þ

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