Application of parallel factor analysis to total synchronous fluorescence spectrum of dilute multifluorophoric solutions: Addressing the issue of lack of trilinearity in total synchronous fluorescence data set

Abstract

In recent years, total synchronous fluorescence (TSF) spectroscopy has become popular for the analysis of multifluorophoric systems. Application of PARAFAC, a popular deconvolution tool, requires trilinear structure in the three-way data array. The present work shows that TSF based three-way array data set of dimension sample × wavelength × Δλ does not have trilinear structure and hence it should not be subjected to PARAFAC analysis. This work also proposes that a TSF data set can be converted to an excitation–emission matrix fluorescence (EEMF) like data set which has trilinear structure, so that PARAFAC analysis can be performed on it. This also enables the retrieval of PARAFAC-separated component TSF spectra.

Introduction

Parallel factor (PARAFAC) analysis is a mathematical deconvolution technique which is used for the analysis of multi-way array and in particular three-way array data sets [1], [2], [3], [4], [5], [6]. The technique is known to have second order advantages i.e. it can quantify a component even in the presence of unknown interferences [7], [8], [9], [10], [11], [12]. It is considered as a generalization of principal component analysis (PCA) [13] to deal with three or higher order arrays [4], [5], [14]. Decomposition of three-way array X of dimension I × J × K by PARAFAC is shown graphically in Fig. 1.

Mathematically decomposition by PARAFAC can be written as

$$x_{ijk}=\sum _{f=1}^F{a}_{if}{b}_{jf}{c}_{kf}+e_{ijk}$$

where F is the number of components and $x_{ijk}$ is an element of X. $a_{if}$, $b_{jf}$, and $c_{kf}$ are the elements of loading matrices A, B, and C, respectively. e_ijk is the element of residual three-way data set E. Dimensions of A, B, C and E are I × F, J × F, K × F, and I × J × K respectively. Unlike other decomposition methods such as PCA [5], [13], Tucker3 [5], [15], [16], multivariate curve resolution alternating least square (MCR-ALS) analysis [17], [18], [19], PARAFAC gives unique solutions i.e. there is no rotation ambiguity [1], [2], [3], [4], [5], [6]. One of the necessary requirements to perform the PARAFAC analysis is that three-way array data X should be trilinear in nature [1], [2], [3], [4], [5], [6]. A three-way array data is called trilinear if it satisfies two conditions (i) each mode has equal number of components and (ii) every component has a profile along each mode and shape of which does not change with the variation in other two modes.

PARAFAC has been used successfully for the analysis of data generated from different techniques such as nuclear magnetic resonance (NMR) [20], [21], [22], high pressure liquid chromatography (HPLC) [23], [24], [25] or gas chromatography (GC) [26], [27], [28], [29]. It has also been used for the sensor array processing [30].

PARAFAC has been applied extensively for the analysis of excitation–emission matrix fluorescence (EEMF) [31] based three-way array of dimension sample × emission × excitation. It is mainly because of the trilinear structure of EEMF data set which makes it compatible with PARAFAC [4], [5], [32], [33], [34], [35]. Trilinear structure to the EEMF data set comes from the fact that every fluorophore (component) has a concentration, excitation, and emission profiles and shape of profile along a particular mode does not change with the changes in other two modes. Combination of PARAFAC and EEMF has been used successfully for the analysis of polycyclic aromatic hydrocarbons (PAHs) [9], [36], [37] pesticides [9] and other dissolved organic matters in water samples [38], [39], [40]. One major drawback with EEMF data set is the Rayleigh scattering which does not contain any fluorescence information appears as a prominent ridge [32], [41], [42], [43]. In dilute solutions, scattering signals are much more intense than the fluorescence signals. In addition, scattering lines do not have the trilinear structure and therefore they cannot be modelled by the PARAFAC [32], [41], [44], [45]. Thus, it is necessary that the Rayleigh scattering signals should be removed from EEMF data set before subjecting it to the PARAFAC or any other multivariate analysis methods. Several mathematical techniques are reported in literature to handle the Rayleigh scattering in EEMF [32], [33], [34], [44], [45], [46].

Synchronous fluorescence (SF) spectra are usually acquired by simultaneously varying the excitation and emission monchromator with a constant wavelength offset (Δλ = λ_em–λ_ex) [47], [48], [49], [50]. Fluorescence intensity of a fluorophore in SF mode is given by the Eq. (2),

$$I_{\text{s}}({\lambda }_{\text{ex}},{\lambda }_{\text{em}})=Kcd{E}_{\text{X}}({\lambda }_{\text{ex}})E_{\text{M}}({\lambda }_{\text{ex}}+\Delta \lambda )$$

where I_s is the SF intensity, λ_ex is excitation wavelength, λ_em is the emission wavelength, K is constant, c is concentration of the fluorophore, d is path length, E_X and E_M are excitation and emission profile, respectively [48], [50]. SF spectra acquired with different Δλ’s are used to create total synchronous fluorescence (TSF) spectroscopy data set [49], [50]. Similar to EEMF technique, TSF spectroscopy has also been used successfully for the analysis of food samples [51], essential oils [52], hydrocarbon based fuel [49], [50], [53], [54], [55], diagnosis of cancer [56], [57], quantification of PAHs in water [42] and PAT application in bioprocess [58]. In TSF, first and second order Rayleigh scattering which is problem in EEMF data is easily avoided by acquiring the SF data set with suitable wavelength offsets and hence no mathematical processing is required to eliminate it [42], [43], [55]. TSF data sets collected for number of samples can be used to create three-way array data of dimension sample × excitation × Δλ. At low concentrations inner filter effects are easily avoided and spectral responses of the fluorophores are additive (i.e. fluorescence intensity is directly proportional to the concentration). As a result, TSF response of a dilute mixture of fluorophores will be equal to the sum of the TSF response of individual fluorophores. Thus, in principal TSF data can be deconvoluted with a suitable multivariate technique to get the pure TSF profile of each component (fluorophore). In a recent work [43], TSF data was unfolded along the sample mode to obtain the unfolded-TSF data which has bilinear structure and it was subjected to MCR-ALS analysis to obtain the SF profile at various Δλ. Obtained SF profiles were further folded back to create the pure component TSF profile. However, it is known that MCR-ALS solutions are not unique because of rotation, permutation and intensity ambiguities [17], [18], [19]. Therefore, deconvolution of a data set by a method such as PARAFAC which gives unique solution is always preferred.

In contrast to EEMF, TSF based three-way arrays are not expected to be trilinear in nature. It is because shape of the SF profile of a fluorophore changes with the change in Δλ. In a true trilinear data set, the shape of the spectral profile along a given mode should not change with the variations along other two modes. Therefore, in principal TSF data should not be subjected to the PARAFAC analysis to retrieve the TSF spectra of individual components from the TSF data of mixtures of fluorophores. In other words, TSF profiles generated for various components by the PARAFAC analysis would not match with their actual TSF profiles. However, so far there are no reports in literature where the trilinearity of the TSF based three-way arrays of dimension sample × excitation × offset (Δλ), has been tested.

EEMF is a collection of emission and excitation spectra at various excitation and emission wavelengths, respectively. Fluorescence intensity of a fluorophore in EEMF is given by Eq. (3)

$$I({\lambda }_{\text{ex}},{\lambda }_{\text{em}})=Kcd{E}_{\text{X}}({\lambda }_{\text{ex}})E_{\text{M}}({\lambda }_{\text{em}})$$

The close correspondence between the intensity expressions of SF and EEMF spectroscopy and the fact that Δλ = λ_em–λ_ex, enables the representation of the TSF data in EEMF layout. Application of PARAFAC on EEMF based three-way array gives two spectral loading vectors and a concentration loading vector for each of the fluorophore. The two spectral loading vectors of a fluorophore can further be multiplied together to generate its pure EEMF [40]. Thus, application of the PARAFAC on the TSF data in EEMF layout would also give the two spectral loading vectors and one concentration loading vector for each component of the mixture. These two spectral loading can be multiplied together to create EEMF layout which contains the SF spectra at various offsets of a fluorophore. From there, data at various Δλ’s can be extracted which can further be used to create its TSF fingerprint. These steps are summarized in Fig. 2.

The motivation behind deconvoluting the TSF data of a mixture of fluorophores and getting the pure TSF based fingerprint of a fluorophore using PARAFAC is due to the three reasons (i) in TSF spectroscopy Rayleigh scattering is easily avoided which is an advantage with TSF data set [42], [43], [55], (ii) TSF based fingerprints are more unique than the EEMF based fingerprints [43], and (iii) PARAFAC gives unique solution [1], [2], [3], [4], [5], [6].

From the above discussions, it follows that it is possible to perform PARAFAC on TSF data provided the data set is properly arranged. The main objectives of the present work are to show that (i) TSF based three-way arrays do not have trilinear structure and hence they should not be subjected to PARAFAC and (ii) trilinear decomposition of TSF data set is possible by representing it in a suitable layout such as EEMF and subsequently subjecting it to PARAFAC analysis. For this purpose, TSF data was acquired for the dilute aqueous mixtures of chrysene, pyrene, and benzo[a]pyrene as the test set. Selected fluorophores have substantial spectral overlap with each other. The match of the deconvoluted TSF fingerprints of chrysene, pyrene, and benzo[a]pyrene with their actual TSF fingerprints would verify the feasibility of the scheme.