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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