Nathalie Coquillé

Valentin Dupraz

Dominique Ménard

Larissa Haugarreau

Rossana Sussarellu

Sabine Stachowski-Haberkorn

Toxicity Of Two

IRGAROL Antifouling Biocides

AND DIURON, ON TWO MARINE PHYTOPLANKTON SPECIES COMMONLY USED IN AQUACULTURE

Since the ban on tributyltin (TBT) from antifouling paints due to serious environmental concerns [1], alternative compounds have been used to keep protecting ship hulls from the unwanted attachment and growth of fouling organisms (e.g. microorganisms, algae and animals such as mussels and barnacles).

Irgarol 1051 and diuron are usually used as booster biocides in copper-based antifouling paints, in replacement of TBT [2]. Because they target directly photosystem II, these two biocides might lead to severe damages on nontarget photosynthetic organisms in case of environment contamination. This study aimed : to assess the sensitivity of two different phyla towards these biocides, and to investigate the toxicity of irgarol towards a diuron-resistant mutant strain.

Since the ban on tributyltin (TBT) from antifouling paints due to serious environmental concerns [1], alternative compounds have been used to keep protecting ship hulls from the unwanted attachment and growth of fouling organisms (e.g. microorganisms, algae and animals such as mussels and barnacles).

Irgarol 1051 and diuron are usually used as booster biocides in copper-based antifouling paints, in replacement of TBT [2]. Because they target directly photosystem II, these two biocides might lead to severe damages on nontarget photosynthetic organisms in case of environment contamination. This study aimed : to assess the sensitivity of two different phyla towards these biocides, and to investigate the toxicity of irgarol towards a diuron-resistant mutant strain.

Introduction & Objectives Introduction & Objectives

The effects of these two compounds have been studied in two marine phytoplankton species: the chlorophyte Tetraselmis suecica (two strains: "wild" strain (WT) and diuron-resistant mutant (MT) [3]) and the diatom Chaetoceros calcitrans (a single "wild" strain: WC).

Six-day exposure experiments were run: each strain was exposed to each biocide alone (irgarol: 0.05, 0.1 and 0.5 µg.L -1 ; diuron: 0.5, 1 and 5 µg.L -1 ) and to mixtures of irgarol and diuron (µg.L -1 : MixA: diuron 5 + irgarol 0.5; MixB: diuron 5 + irgarol 0.1; MixC: diuron 1 + irgarol 0.5; MixD: diuron 1 + irgarol 0.1).

Cultures were carried out in triplicate (exposed conditions) or in quadruplicate (control condition). The effects of herbicides were assessed on the 6 th day on: the doubling time (T D , hours); the effective quantum yield (Y eff ) (PAM-fuorometry); the intracellular relative ROS content (H 2 DCFDA) and the intracellular relative lipid content (BODIPY), both by flow cytometry. At the same time, a sequencing of the psbA gene was performed in both T.suecica strains with the aim to identify the possible mutation responsible for diuron resistance in MT strain.

The effects of these two compounds have been studied in two marine phytoplankton species: the chlorophyte Tetraselmis suecica (two strains: "wild" strain (WT) and diuron-resistant mutant (MT) [3]) and the diatom Chaetoceros calcitrans (a single "wild" strain: WC).

Six-day exposure experiments were run: each strain was exposed to each biocide alone (irgarol: 0.05, 0.1 and 0.5 µg.L -1 ; diuron: 0.5, 1 and 5 µg.L -1 ) and to mixtures of irgarol and diuron (µg.L -1 : MixA: diuron 5 + irgarol 0.5; MixB: diuron 5 + irgarol 0.1; MixC: diuron 1 + irgarol 0.5; MixD: diuron 1 + irgarol 0.1).

Cultures were carried out in triplicate (exposed conditions) or in quadruplicate (control condition). The effects of herbicides were assessed on the 6 th day on: the doubling time (T D , hours); the effective quantum yield (Y eff ) (PAM-fuorometry); the intracellular relative ROS content (H 2 DCFDA) and the intracellular relative lipid content (BODIPY), both by flow cytometry. At the same time, a sequencing of the psbA gene was performed in both T.suecica strains with the aim to identify the possible mutation responsible for diuron resistance in MT strain.

Materials & Methods Materials & Methods

This study demonstrates the high toxicity of these two biocides, in particular irgarol 1051, for marine phytoplankton. It shows that species sensitivity depends on the molecule tested (even with a similar mode of action) but also on the strain, that raises the question of intra-specific genetic diversity and adaptation mechanisms in micro-algae exposed to chemical contamination.

In addition, the results of the experiment with the mixtures of diuron and irgarol provided information for a better understanding of combined effects that occur in the environment (synergism). Finally, this study highlights the need to monitor the fate of these two compounds in the environment, especially for Irgarol since it is still authorized in many European countries.

This study demonstrates the high toxicity of these two biocides, in particular irgarol 1051, for marine phytoplankton. It shows that species sensitivity depends on the molecule tested (even with a similar mode of action) but also on the strain, that raises the question of intra-specific genetic diversity and adaptation mechanisms in micro-algae exposed to chemical contamination.

In addition, the results of the experiment with the mixtures of diuron and irgarol provided information for a better understanding of combined effects that occur in the environment (synergism). Finally, this study highlights the need to monitor the fate of these two compounds in the environment, especially for Irgarol since it is still authorized in many European countries.

Conclusions Conclusions

PsbA gene mutation: resistance to diuron and comparison with the wild strain

The results of the sequencing showed a mutation from G to A in the quinone binding domain at the position 661 of the cds sequence (data not shown). This mutation corresponded to a change in the codons from GTA to ATA, leading to a switch in the amino acid sequence from Val to Ile at the position 221 of the protein.

When exposed to diuron, the wild strain was significantly affected by the herbicide, with strong effects on each parameter at diuron 5 µg.L -1 (Fig1): increase of T D (+125%) and ROS related fluorescence (+43%), decrease of Y eff (-25%) and lipid related fluorescence (-38%).

Irgarol induced the same kind of significant effects, especially at 0.5 µg.L -1 (Fig1): increase of T D (+19%) and ROS related fluorescence (+45%), decrease of Y eff (-2%) and lipid related fluorescence (-21%).

Among the four mixtures, Mix A, B and C showed significant effects, with the strongest effects induced by MixA (Fig1): increase of T D (+356%) and ROS related fluorescence (+95%), decrease of Y eff (-29%) and lipid related fluorescence (-37%). The toxicity of the four mixtures was gradual considering the whole parameters: MixA > MixB > MixC > MixD.

PsbA gene mutation: resistance to diuron and comparison with the wild strain

The results of the sequencing showed a mutation from G to A in the quinone binding domain at the position 661 of the cds sequence (data not shown). This mutation corresponded to a change in the codons from GTA to ATA, leading to a switch in the amino acid sequence from Val to Ile at the position 221 of the protein.

When exposed to diuron, the wild strain was significantly affected by the herbicide, with strong effects on each parameter at diuron 5 µg.L -1 (Fig1): increase of T D (+125%) and ROS related fluorescence (+43%), decrease of Y eff (-25%) and lipid related fluorescence (-38%).

Irgarol induced the same kind of significant effects, especially at 0.5 µg.L -1 (Fig1): increase of T D (+19%) and ROS related fluorescence (+45%), decrease of Y eff (-2%) and lipid related fluorescence (-21%).

Among the four mixtures, Mix A, B and C showed significant effects, with the strongest effects induced by MixA (Fig1): increase of T D (+356%) and ROS related fluorescence (+95%), decrease of Y eff (-29%) and lipid related fluorescence (-37%). The toxicity of the four mixtures was gradual considering the whole parameters: MixA > MixB > MixC > MixD.

Results & Discussion Results & Discussion

Herbicide toxicity towards the two wild strains

Among the three concentrations of diuron and irgarol tested, only the highest (5 and 0.5 µg.L -1 , respectively) exhibited significant effects on the diatom (Fig3). Diuron 5 µg.L -1 exposure induced a 21% increase in the doubling time when the increase was 52% after irgarol 0.5 µg.L -1 exposure. Y eff was reduced by 8% and 18% when exposed to diuron 5 µg.L -1 and irgarol 0,5 µg.L -1 respectively. Mix A, B and C showed significant effects on all parameters, with the strongest effects induced by the MixA: increase in doubling time (+87%) and decrease in Y eff (-26%), FL1 ROS (-61%) and FL1 Lipids (-62%). Toxicity of other mixtures were lower and gradual, the most toxics being the ones with the higher irgarol concentration: MixA > MixC > MixB > MixD.

Unlike Tetraselmis sp., the diatom was more sensitive to irgarol on all parameters, especially for T D and Y eff . Effects on FL1 Lipids were two-fold higher for the diatom. Effects on FL1 ROS on WC were quite unexpected. When an increase in FL1 ROS was expected as compared to the results obtained with Tetraselmis sp., the values decreased. Herbicides might stimulate the ROS scavenging system in this diatom, which would explain the results.

Herbicide toxicity towards the two wild strains

Among the three concentrations of diuron and irgarol tested, only the highest (5 and 0.5 µg.L -1 , respectively) exhibited significant effects on the diatom (Fig3). Diuron 5 µg.L -1 exposure induced a 21% increase in the doubling time when the increase was 52% after irgarol 0.5 µg.L -1 exposure. Y eff was reduced by 8% and 18% when exposed to diuron 5 µg.L -1 and irgarol 0,5 µg.L -1 respectively. Mix A, B and C showed significant effects on all parameters, with the strongest effects induced by the MixA: increase in doubling time (+87%) and decrease in Y eff (-26%), FL1 ROS (-61%) and FL1 Lipids (-62%). Toxicity of other mixtures were lower and gradual, the most toxics being the ones with the higher irgarol concentration: MixA > MixC > MixB > MixD.

Unlike Tetraselmis sp., the diatom was more sensitive to irgarol on all parameters, especially for T D and Y eff . Effects on FL1 Lipids were two-fold higher for the diatom. Effects on FL1 ROS on WC were quite unexpected. When an increase in FL1 ROS was expected as compared to the results obtained with Tetraselmis sp., the values decreased. Herbicides might stimulate the ROS scavenging system in this diatom, which would explain the results. MT strain showed no significant effect after exposure to diuron even at the highest concentration (except a slight decrease of Y eff , -3%) (Fig2) [3]. On the contrary, irgarol significantly impacted the whole parameters tested, with quite similar effects than WT for T D and the fluorescence related to lipids levels. However, increase of ROS related fluorescence was lower (+19%) than for WT, and decrease of Y eff was higher (+8%).

When exposed to the mixtures A and C, MT exhibited significant effects on the whole parameters, probably due to the presence of irgarol at the highest concentration. Mixtures B and D induced no significant effects except for Yeff with a slight decrease (-4% and -2% respectively).

[ 1 ]

 1 Oehlmann J, Fioronib P, Strobenb E, Markert B. 1996. Tributyltin (TBT) effects on Ocinebrina aciculata (Gastropoda : Muricidae) : imposex development, sterilization, sex change and population decline. The Sci. of the Tot. Env. 188, 205-223. [2] Manzo S, Buono S, Cremisini C. 2006. Toxic Effects of Irgarol and Diuron on Sea Urchin Paracentrotus lividus Early Development, Fertilization, and Offspring Quality. Arch. Env. Contam. Tox. 51, 61-68. [3] Stachowski-Haberkorn S, Jérôme M, Rouxel J, Khelifi C, Rincé M, Burgeot T. 2013. Multigenerational exposure of the microalga Tetraselmis suecica to diuron leads to spontaneous long-term strain adaptation. Aqu. Tox. 140-141, 380-388.

Figure 1 -

 1 Figure 1 -Variation (%, as compared with Control) of doubling time (T D ), photosynthetic efficiency (Y eff ), relative ROS content (FL1 ROS ) and relative lipid content (FL1 Lipids ) of Tetraselmis suecica wild strain after exposure to diuron and irgarol, alone and in mixtures. Only significant effects are shown (ANOVA, p<0.05).

Figure 1 -

 1 Figure 1 -Variation (%, as compared with Control) of doubling time (T D ), photosynthetic efficiency (Y eff ), relative ROS content (FL1 ROS ) and relative lipid content (FL1 Lipids ) of Tetraselmis suecica wild strain after exposure to diuron and irgarol, alone and in mixtures. Only significant effects are shown (ANOVA, p<0.05).

Figure 2 -

 2 Figure 2 -Variation (%, as compared with Control) of doubling time (T D ), photosynthetic efficiency (Y eff ), relative ROS content (FL1 ROS ) and relative lipid content (FL1 Lipids ) of Tetraselmis suecica mutant strain after exposure to diuron and irgarol, alone and in mixtures. Only significant effects are shown (ANOVA, p<0.05).

Figure 2 -

 2 Figure 2 -Variation (%, as compared with Control) of doubling time (T D ), photosynthetic efficiency (Y eff ), relative ROS content (FL1 ROS ) and relative lipid content (FL1 Lipids ) of Tetraselmis suecica mutant strain after exposure to diuron and irgarol, alone and in mixtures. Only significant effects are shown (ANOVA, p<0.05).

Figure 3 -

 3 Figure 3 -Variation (%, as compared with Control) of doubling time (T D ), photosynthetic efficiency (Y eff ), relative ROS content (FL1 ROS ) and relative lipid content (FL1 Lipids ) of Chaetoceros calcitrans wild strain after exposure to diuron and irgarol, alone and in mixtures. Only significant effects are shown (ANOVA, p<0.05).

Figure 3 -

 3 Figure 3 -Variation (%, as compared with Control) of doubling time (T D ), photosynthetic efficiency (Y eff ), relative ROS content (FL1 ROS ) and relative lipid content (FL1 Lipids ) of Chaetoceros calcitrans wild strain after exposure to diuron and irgarol, alone and in mixtures. Only significant effects are shown (ANOVA, p<0.05).


Keywords: Échantillonnage préférentiel, mélanges de lois, méthodes de Monte-Carlo adaptatives, marches aléatoires

Les algorithmes Population Monte Carlo (PMC) sont des schémas d'échantillonnage préférentiel adaptatifs qui ont été introduits afin d'ajuster la distribution d'importance à la loi cible. Dans cet article, nous comparons les performances du schéma PMC classique à celles d'une version modifiée qui diminue la variabilité des poids d'importance en utilisant une technique de double Rao-Blackwellisation. Il est montré que cette modification permet d'améliorer significativement les capacités d'exploration de l'algorithme dans le cas de lois cibles multi-modales.

Introduction

When [START_REF] Cappé | Population Monte Carlo[END_REF] introduced Population Monte Carlo (or PMC) as a repeated Sampling Importance Resampling procedure, the purpose was to overcome the shortcomings of Importance Sampling (abbreviated to IS in the following) procedures, while preserving the advantages of IS compared to alternatives such as Markov Chain Monte Carlo methods, namely the possibility of developing parallel implementations, which becomes more and more important with the generalisation of multiple core machines and computer clusters, and of allowing for easy assessment of the Monte Carlo error and, correlatively, for the development of on-line calibration mechanisms.

Indeed, adaptive Monte Carlo naturally answers the difficulties of finding a "good" importance function in IS by gradually improving this importance function against a given target density, based on some form of Monte Carlo approximation. Previous simulations are used (see, e.g., Douc et al., 2007a,b) to modify proposal distributions represented as mixtures of fixed proposals in order to increase the weights of the most appropriate components. When the proposal is inspired from random walk structures as in Metropolis-Hastings algorithms and when the update of those weights is too crude, as in [START_REF] Cappé | Population Monte Carlo[END_REF], the improvement is so shortsighted that multiple iterations do not increase the efficiency of the method, unless a Rao-Blackwell step is added to replace the actual proposal with the mixture proposal in the importance weight (see Douc et al., 2007a). More recently, [START_REF] Cappé | Adaptive importance sampling in general mixture classes[END_REF] have thus replaced random walk proposals based on earlier samples with a new PMC scheme with parameterised proposals whose parameters are estimated from earlier samples, leading to a monotonic decrease in the entropic distance to the target distribution. There is indeed a myopic feature in random walk proposals, namely that, once a (first) sample has exhibited some high density regions for the target distribution, the algorithm is reluctant to allow for a wider exploration of the support of the target and it may thus miss important density regions, missing the energy to reach forward to these other regions of interest.

In this paper, we nonetheless focus on random walk proposals because those kernels are more open to complex settings than on independent proposals-the later indeed require some preliminary knowledge about the target or else an major upgrade in computing power to face a much larger number of components in the mixture. More precisely, we present an experimental assessment of the use of a so-called double Rao-Blackwellisation scheme towards a better exploration of the modes of the target distribution. The second Rao-Blackwellisation step used in this double Rao-Blackwellisation scheme is essentially an integration over the particles of the previous sample used in the random walk proposal. While this leads to an additional computing cost in the derivation of the importance weights, double Rao-Blackwellisation undoubtedly brings a significant improvement in the stability of those importance weights and therefore justifies (in principle) the use of this additional step. Nevertheless, since an analytic proof of this improvement brought by double Rao-Blackwellisation is too delicate to contemplate, we use an intensive Monte Carlo experiment to establish the clear improvement in the case of a posterior distribution associated with a Gaussian location mixture and a sample with outliers-in order to increase the number of modes.

The paper is organised as follows: In Section 2, we recall the basics of our population Monte Carlo algorithm and we introduce the double Rao-Blackwellisation modification. In Section 3, we motivate the choice of Gaussian mean mixtures as a worthy Monte Carlo experiment to test the mode finding abilities of both algorithm. Section 4 describes the implementation of the Monte Carlo experiment and describes the results. Section 5 contains a short discussion.

Population Monte Carlo

Given a target distribution with density π that is known up to a normalising constant, the grand scheme of PMC is the same as with other Monte Carlo-including MCMC-methods, namely to produce a sample that is distributed from π without resorting to direct simulation from π. Let us recall that, once a sample (X 1 , . . . , X N ) is produced by sampling importance resampling (see, e.g., Robert andCasella, 2004, Marin andRobert, 2007), i.e., by first simulating X i ∼ f (x), second producing importance weights ω i ∝ π(X i )/f (X i ), and third resampling the X i 's by multinomial/bootstrap sampling with weights ω i , this SIR sample provides an approximation to the target distribution π and can be used as a stepping stone towards a better approximation to π.

PMC basics

In the PMC algorithm of [START_REF] Cappé | Population Monte Carlo[END_REF], if (X 1 , . . . , X N ) is a sample approximately distributed from π, it is modified stochastically using an arbitrary Markov transition kernel q(x, x ′ ) so as to produce a new sample

(X ′ 1 , . . . , X ′ N )
as X ′ i ∼ q(X i , x). Conducting a resampling step based on the IS weights ω i = π(X ′ i )/q(X i , X ′ i ), we then produce a new sample ( X1 , . . . , XN )

that equally constitutes an approximation to the target distribution π. It is however necessary to stress that, as established in [START_REF] Douc | Minimum variance importance sampling via population Monte Carlo[END_REF], repeating this scheme in an iterative manner is only of interest if samples that have been previously simulated are used to update (or adapt) the kernel q(x, x ′ ): a kernel q that is fixed over iterations does not modify the statistical properties of the samples. The choice made in [START_REF] Douc | Minimum variance importance sampling via population Monte Carlo[END_REF] of an adaptive proposal kernel q represented as a mixture of fixed transition kernels q d ,

q α (x, x ′ ) = D d=1 α d q d (x, x ′ ) , D d=1 α d = 1 , (1)
can improve the efficiency of the kernel in terms of deviance (or relative entropy) from the target density. To achieve such an improvement, the weights α 1 , . . . , α D must be tuned adaptively at each iteration of the PMC algorithm.

Weight updating

If α t = (α t 1 , . . . , α t D ) denote the mixture weights at the t-th iteration of the algorithm (where t = 1, . . . , T ), the update of the α t 's of [START_REF] Douc | Minimum variance importance sampling via population Monte Carlo[END_REF] takes advantage of the latent variable structure that underlines the mixture representation, resulting in an integrated EM (Expectation-Maximisation) scheme. In the current setting, the latent variable Z is the standard component indicator in the mixture (see, e.g., [START_REF] Marin | Bayesian modelling and inference on mixtures of distributions[END_REF], with values in {1, . . . , D} such that the joint density f of x ′ and z given x satisfies f (z) = α z and f (x ′ |z, x) = q z (x, x ′ ) .

The updating mechanism for the α d 's then corresponds to setting the new parameter α t+1 equal to

arg max α E X,X ′ π×π E Z α t {log(α Z q Z (X, X ′ ))|X, X ′ } = arg max α E X,X ′ π×π E Z α t {log(α Z )|X, X ′ } ,
where the inner expectation is computed under the conditional distribution of Z for the current value of the parameter, α t , i.e.

f (z|x, x ′ , α t ) = α t z q z (x, x ′ ) D d=1 α t d q d (x, x ′ ) , with solution α t+1 d = E X,X ′ π×π f (d|X, X ′ , α t ) .
(2)

Single Rao-Blackwellisation updates

Given that (2) is not available in closed form, the adaptivity of the proposed procedure is achieved by approximating this actualising expression based on the sample that has been previously simulated. The implementation of the standard PMC algorithm relies on an arbitrary initial proposal µ 0 that produces a pre-initial sample (X 1,-1 , . . . , X N,-1 ) , with importance weights π(X i,-1 )/µ 0 (X i,-1 ). This initial sample allows for the derivation of a sample ( X1,-1 , . . . , XN,-1 ) approximately distributed from π, using importance sampling based on those weights. The algorithm then picks arbitrary starting weights α 0 d in the N -dimensional simplex to produce a (truly) initial sample (X 1,0 , . . . , X N,0 ) from the mixture

X i,0 ∼ D d=1 α 0 d q d ( Xi,-1 , x) .
In the detail of its implementation, the production of this initial sample is naturally associated with latent variables (Z i,0 ) 1≤i≤N that indicate from which component d of the mixture the corresponding X i,0 has been generated (i = 1, . . . , n). From this stage, [START_REF] Douc | Minimum variance importance sampling via population Monte Carlo[END_REF] proceed recursively. Starting at time t from a sample (X 1,t , . . . , X N,t ) , associated with (Z i,t ) 1≤i≤N and with the current value of the weights α t,N , the normalised importance weights of the sample point X i,t is provided by

ωi,t = π(X i,t ) D d=1 α t,N d q d ( Xi,t-1 , X i,t ) N j=1 π(X j,t ) D d=1 α t,N d q d ( Xj,t-1 , X j,t )
.

(3)

To approximate the update step (2) in practice, an initial possibility is

α t+1,N d = N i=1 ωi,t ✶{Z i,t = d} ,
where the sum does not need renormalisation because the ωi,t sum up to 1 (over i).

This choice is however likely to be highly variable when N is small and/or the number of components D becomes larger. To robustify this update, [START_REF] Cappé | Adaptive importance sampling in general mixture classes[END_REF] introduce a Rao-Blackwellisation step (see, e.g., Robert and Casella, 2004, Section 4.2) that consists in replacing the Bernoulli variable ✶{Z i,t = d} with its conditional expectation given X i,t and Xi,t-1 , that is,

α t+1,N d = N i=1 ωi,t f d X i,t , Xi,t-1 , α t,N .
This replacement does not involve a significant increase in the computational cost of the algorithm, while both approximations are converging to the solution of (2) as N grows to infinity.

Double Rao-Blackwellisation

While [START_REF] Cappé | Adaptive importance sampling in general mixture classes[END_REF] showed through two experimental examples that the above Rao-Blackwellisation step does provide a significant improvement in the stability of the PMC weights (and, thus, in fine, a reduction in the number of iterations), there remains an extra-source of variation in the importance weights (3), namely the dependence of ωi,t on the previous sample point (or particle) Xi,t-1 . Even though this dependence illustrates the fact that X i,t is indeed generated from the mixture

D d=1 α t,N d q d ( Xi,t-1 , x) ,
and is thus providing a correct IS weight, it also shows a conditioning on the result of a (random) multinomial sampling in the previous iteration that led to the selection of Xi,t-1 with probability ωi,t-1 . In other words, by de-conditioning, it is also the case that the sample point X i,t has been generated from the (integrated) distribution

N j=1 ωj,t-1 D d=1 α t,N d q d ( Xj,t-1 , x) , (4)
when averaging over all multinomial outputs. This double averaging over both the components of the mixture and the initial sample points is the reason why we call this representation double Rao-Blackwellisation.

The appeal of using ( 4) is that not only does the averaging over all possible sample points provide a most likely stabilisation of the weights, but it also eliminates a strange feature of the original approach, namely that two identical values x i,t and x j,t = x i,t could have different importance weights simply because their conditioning sample values Xi,t-1 and Xj,t-1 were different.

Naturally, the replacement of the importance sampling distribution in (3) by ( 4) has a cost of O(N 3 ) compared with the original O(N 2 ), but this is often negligible when compared with the cost of computing π(X i,t ). (We also stress that some time-saving steps could be taken in order to avoid computing all the q d ( Xj,t-1 , X i,t ) by considering first the distance between X j,t-1 and X i,t and keeping only close neighbours within the sum although the increase in computing time in our case did not justify the filtering.) In the following experiment, the additional cost resulting from the double Rao-Blackwellisation does not induce a considerable upsurge in the computing time, even though it is not negligible. We indeed found an increase in the order of three to five times the original computing time for the same number N of sampling values. This obviously fails to account for the faster stabilisation of the IS approximation resulting from using double Rao-Blackwellisation. Note also that double Rao-Blackwellisation does not remove the need to sample the Xj,t-1 's: indeed, when simulating each new X i,t from (4), we need to first select which Xj,t-1 is going to be conditioned upon, then second determine which component is to be used.

The Gaussian mean mixture benchmark

As benchmark for our Monte Carlo experiment, we consider the case of a one-dimensional Gaussian mean mixture distribution, namely

x 1 , . . . , x n ∼ pN (µ 1 , σ 2 1 ) + (1 -p)N (µ 2 , σ 2 1 ) , (5)
with p, σ 2 1 and σ 2 2 known and θ = (µ 1 , µ 2 ) being the parameter of the model, as in [START_REF] Marin | Bayesian modelling and inference on mixtures of distributions[END_REF]. Using a flat prior on θ within a square region, we are thus interested in simulating from the posterior distribution associated with a given sample (x 1 , . . . , x n ). The appeal of this example is that it is sufficiently simple to allow for an explicit characterisation of the attractive points for the adaptive procedure, being of dimension two, while still illustrating the variety of situations found in more realistic applications. In particular as already explained in [START_REF] Marin | Bayesian modelling and inference on mixtures of distributions[END_REF], the model contains at least one attractive point that does not correspond to the global minimum of the entropy criterion as well as some regions of attraction that can eventually lead to a failure of the algorithm when the data (x 1 , . . . , x n ) is not simulated from the above mixture but by clumps that induce more modes in the posterior. Figures 1 and2 illustrate quite well the diversity of posterior features when changing the repartition of the sample (x 1 , . . . , x n ). We stress the fact that those samples, called artificial samples, are not resulting from simulations from (5) but from simulations from a normal mixture with five equal and well-separated components centred in µ 1 = 0 and ±µ 2 , ±2µ 2 , and with variances 0.1. This choice was made in order to increase the potential number of modes on the posterior surface when modelling the data with (5).

Monte Carlo experiment

Given the target distribution defined by ( 5), we want to study the improvement brought by the double Rao-Blackwellisation (4) in terms of mode degeneracy, as well as to ascertain the additional cost of using double Rao-Blackwellisation. We therefore study the performance of both single and double Rao-Blackwellisation PMC over a whole range of samples, with various values of n and of the parameters p, µ 2 and σ 2 (since we can always use µ 1 = 0 and σ 1 = 1 without loss of generality).

Automatic mode finding and classification

A first difficulty, when building this experiment, is to determine the number of modes on the posterior surface for the data at hand. We achieve this goal by first discretising the parameter space in (µ 1 , µ 2 ), and then recovering the modal points by a brute-force search for maxima and saddle-points over the grid and then identifying their basins of attraction. (The algorithm is based on multidirectional calls to the function turnpoints() of the pastecs package.) This is obviously prone to overlook some small local modes but it also allows for a quick identification of the modal regions and of their exploration by the PMC algorithm. Table 1 illustrates the results of this procedure for a range of values of (n, p, µ 2 , σ 2 ). When conditioning upon (µ 2 , σ 2 ) the number of modes is increasing in µ 2 (since the simulated samples include five clusters that are better separated) and slightly decreasing in σ 2 (for apparently the same reason). A similar table varying upon the pair (n, p) does not show much variation in the number of modes (which does make sense, since only the relative magnitudes of the modes are changing). The code for both implementing both versions of PMC and for evaluating their mode-finding abilities was written in R (R Development Core Team, 2006) and is available as http:\\www.ceremade.dauphine.fr\~xian\2RB.R. Further documentation is available as http:\\www.ceremade.dauphine.fr\~xian\2RB.R.pdf.

Output

The experiment was run on 4860 samples on seven different machines for a total of 238140 simulations, using the same machine for both single and double Rao-Blackwellised versions at a given value of the parameters in order to keep the CPU comparison sensible. As seen from Table 2, the CPU time required to implement the double Rao-Blackwellised scheme is five to three time higher than the original PMC scheme, the additional time decreasing with n. (This CPU time does not include the mode finding and storing steps that are required for the comparison. It only corresponds to the execution of a regular PMC run with 10 iterations.) Note also that the increase in computing time is certainly less than linear. Both single and double Rao-Blackwellisation PMC samplers were used on the same samples, with different values of p, σ 2 and µ 2 .

Turning now to the central tables, Tables 3456, we can see that the influence of σ 2 on the detection of the modes is relatively limited, in opposition to the influence of µ 2 . As µ 2 increases (recall that µ 1 = 0), a larger fraction of modes gets undetected. Both after 5 and 10 iterations of PMC, the performances of the double Rao-Blackwellised scheme are superior to those of the single Rao-Blackwellisation in terms of mode detection, by a factor of about 5%. Running the PMC algorithm longer clearly has a downward impact on the number of detected modes, even though this impact is quite limited. It however points out the major issue that importance sampling algorithms like PMC are mostly unable to recover lost modes. Another interesting feature is that single Rao-Blackwellisation suffers more from the loss of modes between 5 and 10 iterations, compared with double Rao-Blackwellisation. As expected, the later is more robust because of the average over all past values in the PMC sample. Figure 3 presents the evolution of the capture rate for the single and the double Rao-Blackwellisations as a function of µ 2 after 5 and 10 iterations, obtained by averaging Tables 3456over σ 2 .

If we consider instead the output of this simulation experiment decomposed by the values of n and p, in Tables 7-10, a main feature is the quick deterioration in the exploration of the modes as n increases. This is obviously to be expected given that the more observations, the steeper the slopes of the posterior surfaces. Thus, for small values of n, both types of Rao-Blackwellised PMC algorithms achieve high coverage rates, but larger values of n lead to poorer achievements. Once again, the robustness against the number of PMC iterations is superior in the case of the double Rao-Blackwellisation. Figure 4 presents the evolution of the capture rate for the single and the double Rao-Blackwellisations as a function of the sample size n after 5 and 10 iterations, obtained by averaging Tables 7-10 over p.

Conclusions

The double Rao-Blackwellised algorithm provides a more robust framework for adapting general importance sampling densities represented as mixtures in the sense that the influence of the starting points diminishes when compared with the original algorithm. It is thus unnecessary to rely on a large value of the number T of iterations of the PMC algorithm: with large enough sample sizes N at each iteration-especially on the initial iteration that requires many points to counter-weight a potentially poor initial proposal-, it is quite uncommon to fail to spot a stabilisation of both the estimates and of the entropy criterion within a few iterations.

Figure 1 :Figure 2 :

 12 Figure 1: Series of posterior distributions associated with (5) represented as level sets for various artificial samples (x 1 , . . . , x n ) and different values of n, µ 1 , µ 2 , σ 2 1 and σ 2 2 .

Figure 3 :

 3 Figure 3: Evolution of the capture rate for the single and the double Rao-Blackwellised versions as a function of µ 2 after 5 and 10 iterations, obtained by averaging Tables 3-6 over σ 2 .

Figure 4 :

 4 Figure 4: Evolution of the capture rate for the single and the double Rao-Blackwellisations as a function of the sample size n after 5 and 10 iterations, obtained by averaging Tables 7-10 over p.

Figure 5 :

 5 Figure 5: Comparison between PMC samples using a single Rao-Blackwellisation (left) and a double Rao-Blackwellisation (right).

Figure 5

 5 Figure 5 illustrates the superior performances of the double Rao-Blackwellisation strategy, compared to the single one in terms of modes detection. Those three graphs plot side by side for a given sample the repartition of the PMC samples in the modal domains. The colour codes are associated with the five possible variances used in the 5 kernel proposal. In those three experiments, the persistence of the samples in all modes, as well as the concentration in the regions of importance, is noticeable. Both rows of Figure 5 interestingly contain a ridge structure in one of the modal basins.

Table 1 :

 1 Average number (and standard deviation) of identified modes on a collection of 1470 samples.

	(σ2, µ2)	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
	1.0	2.053	2.184	2.450	2.863	3.265	3.524	3.641	3.707	3.742
		(0.477) (0.486) (1.273) (1.085) (0.895) (0.754) (0.706) (0.668) (0.627)
	1.5	2.037	2.124	2.135	2.165	2.294	2.494	2.731	2.958	3.182
		(0.707) (0.362) (0.404) (0.565) (0.580) (0.668) (0.696) (0.729) (0.706)
	2.0	2.026	2.067	2.121	2.059	2.083	2.132	2.259	2.546	2.807
		(0.862) (0.337) (0.359) (0.342) (0.350) (0.369) (0.505) (0.648) (0.636)
	2.5	1.976	2.024	2.107	2.067	2.073	2.184	2.472	2.757	2.893
		(1.027) (0.354) (0.344) (0.302) (0.313) (0.434) (0.570) (0.533) (0.500)
	3.0	1.993	1.982	2.073	2.203	2.137	2.361	2.743	2.933	3.040
		(1.147) (0.359) (0.329) (0.599) (0.469) (0.580) (0.572) (0.518) (0.486)
	3.5	1.979	1.964	2.076	2.186	2.367	2.662	2.973	3.181	3.354
		(1.315) (0.432) (0.415) (0.519) (0.938) (0.844) (0.712) (0.678) (0.719)
	4.0	1.961	1.939	2.078	2.209	2.412	2.871	3.214	3.466	3.752
		(1.469) (0.507) (0.490) (0.500) (0.851) (0.922) (0.807) (0.806) (0.860)
	4.5	1.858	1.893	2.046	2.220	2.524	3.101	3.418	3.777	4.073
		(1.471) (0.504) (0.445) (0.540) (0.832) (1.034) (0.898) (0.910) (0.901)
	5.0	1.698	1.886	2.063	2.273	2.678	3.187	3.673	4.046	4.295
		(1.247) (0.605) (0.566) (0.612) (1.128) (1.013) (1.033) (0.970) (0.867)

Table 2 :

 2 Overall mean CPU times and their standard error (in seconds) vs. sample size. Each value is obtained by averaging over 4860 generated samples, for a range of parameters (p = .1, . . . , .8, µ 2 = 1, . . . , 5, σ 2 = 1, . . . , 5).

	n	t 1RB	t 2RB
	20	3.60	15.59
		(0.52) (1.12)
	30	3.65	15.51
		(0.52) (1.13)
	40	3.68	15.59
		(0.52) (1.13)
	50	3.72	15.61
		(0.52) (1.13)
	100	3.89	15.74
		(0.53) (1.16)
	500	5.13	17.19
		(0.62) (1.26)
	1000	6.85	18.92
		(0.74) (1.42)

Table 3 :

 3 Average detection rate after 5 iterations of the single Rao-Blackwellised PMC algorithm as a function of σ 2 and µ 2 . The number of samples per entry is 1470, obtained for 7 values of n and 6 values of p.

	(σ2, µ2)	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
	1.0	0.794	0.801	0.725	0.618	0.528	0.474	0.463	0.438	0.440
		(0.258) (0.247) (0.266) (0.285) (0.265) (0.241) (0.238) (0.232) (0.229)
	1.5	0.747	0.811	0.753	0.700	0.627	0.558	0.487	0.439	0.386
		(0.275) (0.240) (0.252) (0.257) (0.256) (0.248) (0.223) (0.216) (0.174)
	2.0	0.679	0.832	0.763	0.708	0.669	0.613	0.566	0.500	0.443
		(0.279) (0.241) (0.255) (0.253) (0.248) (0.232) (0.212) (0.206) (0.191)
	2.5	0.679	0.832	0.756	0.697	0.648	0.586	0.514	0.447	0.416
		(0.291) (0.240) (0.257) (0.251) (0.238) (0.220) (0.212) (0.185) (0.160)
	3.0	0.697	0.826	0.758	0.672	0.625	0.555	0.451	0.411	0.386
		(0.304) (0.242) (0.256) (0.263) (0.237) (0.226) (0.179) (0.159) (0.137)
	3.5	0.712	0.807	0.745	0.657	0.591	0.488	0.424	0.376	0.355
		(0.312) (0.251) (0.260) (0.257) (0.244) (0.212) (0.178) (0.143) (0.131)
	4.0	0.728	0.788	0.740	0.646	0.571	0.457	0.386	0.350	0.317
		(0.312) (0.258) (0.260) (0.257) (0.238) (0.204) (0.153) (0.138) (0.123)
	4.5	0.758	0.795	0.738	0.627	0.540	0.431	0.363	0.318	0.298
		(0.307) (0.255) (0.262) (0.249) (0.229) (0.199) (0.146) (0.125) (0.121)
	5.0	0.782	0.783	0.709	0.617	0.532	0.413	0.341	0.305	0.273
		(0.292) (0.259) (0.264) (0.253) (0.241) (0.182) (0.141) (0.124) (0.105)

Table 4 :

 4 Same legend as Table3for the double Rao-Blackwellised PMC algorithm.

	(σ2, µ2)	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
	1.0	0.707	0.641	0.570	0.482	0.413	0.376	0.365	0.356	0.355
		(0.259) (0.244) (0.226) (0.220) (0.199) (0.187) (0.187) (0.181) (0.185)
	1.5	0.656	0.599	0.537	0.516	0.478	0.443	0.403	0.371	0.340
		(0.262) (0.222) (0.170) (0.146) (0.129) (0.129) (0.125) (0.119) (0.102)
	2.0	0.638	0.622	0.533	0.518	0.512	0.491	0.474	0.427	0.384
		(0.269) (0.231) (0.157) (0.119) (0.115) (0.094) (0.108) (0.115) (0.107)
	2.5	0.656	0.641	0.531	0.510	0.507	0.483	0.436	0.386	0.366
		(0.285) (0.237) (0.158) (0.111) (0.100) (0.102) (0.117) (0.095) (0.092)
	3.0	0.673	0.660	0.542	0.497	0.499	0.456	0.388	0.359	0.346
		(0.302) (0.243) (0.168) (0.130) (0.112) (0.114) (0.098) (0.084) (0.080)
	3.5	0.700	0.660	0.554	0.499	0.472	0.416	0.362	0.335	0.315
		(0.311) (0.245) (0.182) (0.127) (0.122) (0.122) (0.096) (0.085) (0.071)
	4.0	0.723	0.670	0.560	0.498	0.462	0.390	0.337	0.308	0.284
		(0.312) (0.248) (0.186) (0.139) (0.126) (0.121) (0.095) (0.075) (0.073)
	4.5	0.755	0.700	0.572	0.496	0.445	0.365	0.319	0.285	0.267
		(0.308) (0.255) (0.201) (0.142) (0.129) (0.126) (0.090) (0.076) (0.079)
	5.0	0.780	0.700	0.578	0.501	0.438	0.357	0.303	0.270	0.249
		(0.293) (0.258) (0.208) (0.159) (0.149) (0.120) (0.101) (0.078) (0.068)

Table 5 :

 5 Average detection rate after 10 iterations of the single Rao-Blackwellised PMC algorithm as a function of σ 2 and µ 2 .

	(σ2, µ2)	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
	1.0	0.786	0.785	0.712	0.610	0.521	0.467	0.453	0.431	0.432
		(0.259) (0.251) (0.268) (0.284) (0.262) (0.240) (0.236) (0.228) (0.227)
	1.5	0.745	0.795	0.733	0.679	0.612	0.543	0.479	0.431	0.379
		(0.275) (0.246) (0.253) (0.255) (0.249) (0.240) (0.216) (0.208) (0.167)
	2.0	0.677	0.813	0.736	0.684	0.647	0.600	0.555	0.494	0.437
		(0.280) (0.246) (0.256) (0.247) (0.242) (0.223) (0.207) (0.200) (0.188)
	2.5	0.675	0.809	0.726	0.665	0.623	0.567	0.500	0.435	0.409
		(0.291) (0.247) (0.256) (0.244) (0.229) (0.208) (0.199) (0.169) (0.152)
	3.0	0.694	0.804	0.723	0.643	0.606	0.537	0.443	0.404	0.380
		(0.304) (0.251) (0.257) (0.257) (0.227) (0.211) (0.167) (0.154) (0.134)
	3.5	0.711	0.783	0.706	0.624	0.568	0.477	0.413	0.372	0.350
		(0.311) (0.257) (0.256) (0.244) (0.233) (0.202) (0.169) (0.135) (0.124)
	4.0	0.727	0.764	0.704	0.609	0.542	0.446	0.377	0.346	0.312
		(0.313) (0.259) (0.255) (0.241) (0.218) (0.193) (0.145) (0.131) (0.116)
	4.5	0.757	0.766	0.693	0.596	0.518	0.416	0.355	0.313	0.293
		(0.308) (0.259) (0.258) (0.237) (0.215) (0.185) (0.139) (0.117) (0.116)
	5.0	0.782	0.764	0.665	0.589	0.512	0.404	0.334	0.300	0.268
		(0.293) (0.260) (0.255) (0.238) (0.228) (0.174) (0.134) (0.120) (0.098)

Table 6 :

 6 Same legend as Table5for the double Rao-Blackwellised PMC algorithm.

Table 7 :

 7 Average detection rate after 5 PMC iterations of the single Rao-Blackwellised algorithm as a function of n and p. The number of samples is 2835, obtained for 7 values of µ 1 and 5 values of σ 2 .

	(p, n)	20	30	40	50	100	500	1000
	0.10	0.652	0.619	0.601	0.577	0.542	0.463	0.445
		(0.291) (0.286) (0.292) (0.283) (0.283) (0.249) (0.232)
	0.20	0.644	0.629	0.613	0.604	0.566	0.482	0.453
		(0.286) (0.284) (0.283) (0.283) (0.279) (0.250) (0.229)
	0.30	0.653	0.645	0.630	0.628	0.588	0.500	0.467
		(0.288) (0.288) (0.285) (0.284) (0.280) (0.256) (0.228)
	0.40	0.664	0.662	0.654	0.643	0.598	0.513	0.472
		(0.291) (0.289) (0.290) (0.281) (0.283) (0.260) (0.227)
	0.50	0.668	0.651	0.648	0.639	0.614	0.524	0.482
		(0.298) (0.290) (0.290) (0.284) (0.275) (0.241) (0.211)
	0.60	0.665	0.651	0.627	0.623	0.582	0.482	0.450
		(0.291) (0.288) (0.285) (0.276) (0.272) (0.222) (0.189)

Table 8 :

 8 Same legend as Table7in the double Rao-Blackwellised case.

	(p, n)	20	30	40	50	100	500	1000
	0.10	0.577	0.541	0.519	0.495	0.457	0.408	0.398
		(0.281) (0.266) (0.259) (0.243) (0.223) (0.175) (0.161)
	0.20	0.545	0.514	0.492	0.485	0.447	0.413	0.402
		(0.265) (0.242) (0.222) (0.218) (0.183) (0.164) (0.152)
	0.30	0.541	0.510	0.489	0.480	0.456	0.424	0.414
		(0.257) (0.233) (0.214) (0.201) (0.180) (0.159) (0.146)
	0.40	0.536	0.516	0.496	0.487	0.458	0.431	0.418
		(0.252) (0.234) (0.219) (0.207) (0.178) (0.162) (0.143)
	0.50	0.564	0.542	0.528	0.517	0.497	0.459	0.443
		(0.269) (0.246) (0.232) (0.219) (0.197) (0.166) (0.156)
	0.60	0.544	0.519	0.499	0.487	0.463	0.426	0.420
		(0.258) (0.238) (0.224) (0.206) (0.180) (0.146) (0.134)

Table 9 :

 9 Average detection rate after 10 PMC iterations for the single Rao-Blackwellised algorithm as a function of n and p.

	(p, n)	20	30	40	50	100	500	1000
	0.10	0.639	0.608	0.590	0.564	0.529	0.458	0.441
		(0.291) (0.285) (0.288) (0.277) (0.275) (0.243) (0.227)
	0.20	0.622	0.612	0.594	0.585	0.549	0.475	0.448
		(0.284) (0.281) (0.278) (0.276) (0.267) (0.246) (0.220)
	0.30	0.629	0.623	0.608	0.604	0.571	0.493	0.461
		(0.284) (0.283) (0.279) (0.277) (0.274) (0.250) (0.223)
	0.40	0.641	0.636	0.628	0.620	0.582	0.506	0.468
		(0.287) (0.285) (0.286) (0.276) (0.274) (0.254) (0.224)
	0.50	0.642	0.630	0.627	0.619	0.600	0.518	0.480
		(0.293) (0.284) (0.283) (0.277) (0.269) (0.237) (0.208)
	0.60	0.641	0.628	0.608	0.604	0.569	0.475	0.447
		(0.287) (0.281) (0.278) (0.271) (0.264) (0.215) (0.184)

Table 10 :

 10 Same legend as Table9for the double Rao-Blackwellised PMC algorithm.

RR n➦ 0123456789

INRIA


Philippe Bados

F Combaluzie

Marina Coquery

Cecile Miege

F Combaluzier

C Miège

Method development and validation for the analysis of 20 hormones (including estrogens, androgens and progestagens compounds) in various aqueous matrices

 prove that linear model for calibration is the proper one (F test),  verify LOQ on accuracy criteria by spiking various aqueous matrices,  study accuracy and repeatability of the entire method at a lower concentration level,  determine uncertainties at each concentration level and in various types of water (ISO 11352:2012 [4] ),  control matrix effects in various types of water and show specificity of this method (T test).

Progesterone-D9 Dienestrol-D6

CONCLUSION & PERSPECTIVES

These preliminary results are acquired in the context of a conscientious and complete method validation process. More experiments have to be driven according to experimental design described in our reference standard (NF T90-210 [3] ) in order to:

INTRODUCTION

During the last twenty years, consumption of hormones compounds (natural hormones or synthetic analogues), mainly for human medicine, has considerably increased. Both androgens, progestogens and estrogens compounds are usually not entirely metabolized and reach aquatic environment mainly via effluents of wastewater treatment plants (WWTP). All of them can be considered as endocrine disrupting compounds because of undesirable effects in biota. Hence, a highly sensitive and selective analytical method is needed to identify and quantify these emerging contaminants in various environmental compartments, especially in surface waters and wastewaters.

Method synoptic

Initially based on previous estrogenic analytical method [1] METHOD PERFORMANCES

Estimated

Experimental design approach

We built a full factorial experimental design. 3 parameters ranging between 2 or 3 levels lead to 18 experiments that were driven on spiked samples of Evian ® mineral. Results presented are absolute recoveries (external calibration). Data treatments have been done with Statgraphics® software (Screening experimental design, Pareto diagrams, ACPs). Best desirability (0,87) corresponding to optimum extraction parameters was attempted with sample SPE percolation flow rate of 15 mL/min, on 500 mg HLB SPE phase and with ethyl acetate/MeOH 75/25 V/V as elution solvent.

UHPLC/MS-MS parameters

PURIFICATION OPTMISATION

Recoveries and repeatability increase with a second MeOH elution. Following results are absolute recoveries (external calibration) for n=3 experiments.

st RESULTS

The whole analytical method has been applied on real samples. Following results show concentration levels for compounds that occur in 3 different matrices.

  C. Miege, P. Bados, C. Brosse, M. Coquery. Method validation for the analysis of estrogens (including conjugated compounds) in aqueous matrices, Trends in Analytical Chemistry, Vol. 28, No. 2, 2009 [2] 2002/657/EC: Commission Decision of 08/12/2002 implementing Council Directive 96/23/EC concerning the performance of analytical methods and the interpretation of results [3] AFNOR: NF T90-210 (2009) -Qualité de l'eau : Protocole d'évaluation initiale des performances d'une méthode dans un laboratoire [4] ISO 11352:2012 -Water quality -Estimation of measurement uncertainty based on validation and quality control data

  Contact : philippe.bados@irstea.fr Sample filtration: Calcinated glass fiber filter, GF/F 0,7 µm Sample volume: 250 mL (river waters and WWTP effluents) SPE automated extraction: Oasis ® HLB 6mL SPE manual purification: Florisil ® 6mL, 1g UHPLC/MS-MS (Waters BEH C18, Shimadzu Nexera X2, AB SCIEX API4000) ESI (+): 13 hormones MilliQ® Water+0,1% FA/ACN+0,1% FA ESI (-): 7 hormones MilliQ® Water/ACN

  1st elution step w/ acetone/heptane 75/25 V/V Florisil 1g, 6mL -1st and 2d elution steps w/ 5 mL MeOH

  analogue labelled with stable(s) isotope(s) (D and/or 13 C) as internal standard (IS)

Compound Correlation coefficient r² Concentration range for river waters and WWTP effluent (ng/L) Concentration range for WWTP influent (ng/L)

  S/N = signal to noise (peak to peak determination method) **CF = concentration factor (sample volume is 250 mL for river water and for WWTP effluent and 100 mL fot WWTP influent)

														Linearity range in waters
	Compound	S/N* in calibration standard at 0,1 µg/L or 0,25 µg/L	Calculated instrumental LOQ (µg/L)	Estimated LOQ river water and WWTP effluent (ng/L) (CF**= 1000)	Estimated LOQ in WWTP influent (ng/L) (CF**= 400)
	CORT-OH	62		0,02		0,02			0,04					CORT-OH		0,9995		0,02-100	0,04-100
	CORT	91		0,01		0,01			0,03					CORT		0,9971		0,01-100	0,03-100
	DEXA	106		0,01		0,01			0,02					DEXA		0,9997		0,01-100	0,02-100
	EPI-T	45		0,02		0,02			0,10					EPI-T		0,9992		0,02-100	0,10-100
	NORE	40		0,03		0,03			0,10					NORE		0,9985		0,03-100	0,10-100
	TESTO ANDRO DROSPI LEVO MEDROX ANDROSTER MEG_AC PROG E3	46 50 35 19 97 10 167 61 31		0,02 0,02 0,03 0,14 0,01 0,10 0,01 0,02 0,10		0,02 0,02 0,03 0,14 0,01 0,10 0,01 0,02 0,10			0,05 0,05 0,10 0,30 0,03 0,20 0,01 0,04 0,20					TESTO ANDRO DROSPI LEVO MEDROX ANDROSTER MEG_AC PROG		0,9989 0,9991 0,9994 0,9980 0,9987 0,9899 0,9997 0,9987		0,02-100 0,02-100 0,03-100 0,14-100 0,01-100 0,10-100 0,01-100 0,02-100	0,05-100 0,05-100 0,10-100 0,30-100 0,03-100 0,20-100 0,01-100 0,04-100
	β-E2	13		0,10		0,10			0,20					E3		0,9989		0,10-100	0,20-100
	α-E2	13		0,10		0,10			0,20					β-E2		0,9995		0,10-100	0,20-100
	EE2	11		0,20		0,20			0,60					α-E2		0,9995		0,10-100	0,20-100
	E1	11		0,10		0,10			0,24					EE2		0,9989		0,20-100	0,60-100
	DES	29		0,03		0,03			0,10					E1		0,9996		0,10-100	0,24-100
	DIEN	52		0,02		0,02			0,05					DES		0,9995		0,03-100	0,10-100
															DIEN		0,9960		0,02-100	0,05-100
	Accuracy & repeatability
	For all experiments in intermediate fidelity conditions (2 operators and one duplicate, n=4. Relative recoveries (internal calibration) ranging
	from 88 to 130% and relative standard deviations smaller than 11% show good accuracy and repeatability. Native concentrations for E3
	and ANDROSTER in WWTP influent are out of calibration range and are too high comparatively to spiking concentration level (80 ng/L) 
	no data for these compounds in this matrices ().
	100
	Relative (%) recoveries	50																	WWTP Influent WWTP Effluent
																			River water
																			*these concentrations are out of calibration range. We have to dilute
																			and reanalyze them to get more accurate values.
		0																	ND = Not detected
		CORT-OH	CORT	DEXA	EPI-T	NORE	TESTO	ANDRO	DROSPI	LEVO	MEDROX ANDROSTER MEG-AC	PROG	E3	bE2	aE2	EE2	E1	DES	DIEN

*