Ninon Burgos
email: ninon.burgos@cnrs.fr

Sotirios A Tsaftaris

David Svoboda

Future trends in medical image synthesis

of approaches able to exploit unpaired data, but also of weakly-supervised or unsupervised methods, comes as a natural solution. But such approaches face difficulties due to the residual mismatches or shifts that often exist between domains. For example, generative adversarial networks with cycle-consistency loss allow non-unique unpaired image translation as a one-to-many mapping exists between the source and target domains in the forward cycle, and a one-to-many mapping exists between the target and source domains in the backward cycle. Data-related difficulties are not the only obstacles in medical image synthesis, computational complexity is also of concern.

The attentive reader has surely observed between the lines that computational complexity is indeed an Achilles heel in the world of simulations. The problem of complexity can also be viewed as a scale bar where each tick mark uniquely expresses how difficult it is to design and develop a new simulation engine. On the very left tip of the bar are the complex simulation engines that try to properly model all the physical phenomena, which makes the engines rather complex. Typically such a kind of simulation can require days or weeks. These methods include for example Monte Carlo based approaches. In the middle of the scale are image-based parametric models. These models are significantly simpler because their objective is to offer a good visual imagination of the studied objects whereas the detailed description is intentionally omitted to reduce the amount of parameters. These simulation engines are nevertheless still quite complicated and require a good expert knowledge during the initial design. Finally, deep learning based simulation methods, that occupy the right most part of the bar, do not require as much expert knowledge as the learned model tends to replace it. This however comes at the cost of limited controllability. Regardless of the choice of the simulation or synthesis method, we must be aware of high demands for hardware performance. The simulations stand and fall together with the availability of high performance computing. However, the computational complexity of Monte Carlo and living system simulations or of deep learning should become less of an issue with the increased availability of highly-parallel architectures (GPU).

While generative models continue to improve, naturally the utility of synthetic data depends on how they are intended to be used. For example, if synthetic will be used to augment real data, one expects that the synthetic do not merely replicate what they have seen in the training set but are able to synthesise new and unseen samples. This ability of generative models to generalise remains a topic of active investigation. In fact, there is broad consensus now that generative models tend to faithfully be able to reproduce regions of the underlying distribution that contain the majority of the mass, whereas they find difficulty to learn how to generate rare data (closer to the distributional boundaries and tails). Of course the ability to generate only the mass and thus replicate what is frequently encountered in the training data may Biomedical Image Synthesis and Simulation, Chapter 27 have suitable applications as well. For example, in settings where data cannot be distributed due to privacy concerns, synthetic proxies can (and have been shown to) be suitable stand-ins.

This highlights another limitation preventing the wider use of synthetic images: the lack of extensive validation and comparison of the methods proposed. Synthetic data must be validated qualitatively and quantitatively as well as for their usability for particular medical or biomedical tasks. The first two are gradually becoming a standard. Usability, however, appears very rarely and has to be tailored to the intended use for the synthetic data. Usability remains hard to demonstrate. One can show that with data similar decisions can be made or show that by adding synthetic data models can be trained better, but perhaps this is not enough to convince clinical users downstream.

Readers will have noticed that, for many applications, deep learning has supplanted the approaches in place. However, classical methods must not be forgotten. Their combination with deep learning might improve the overall performance. Hybrid solutions combining deep learning and systematic modeling, or deep learning and sparse representation would exploit the advantages from each approach. If we have physical models that we know how to mathematically describe, spending data and deep model parameters to capture these degrees of freedom is not efficient use of resources. Thus, one can imagine the marriage between physical systems capturing degrees of freedom we know with learned (from data) models capturing degrees of freedom that describe physically/mathematically we either do not know how to describe or are intractable. Future methodological developments could also focus on systematic modeling to eliminate magic constants and ad-hoc parameters. The model should be well mathematically and biologically based. The still wider availability of performant computing resources will enable the larger development of synthesis of longitudinal/temporal data. Finally, future simulation frameworks need to be flexible and continually develop as current systems are too task-or dataspecific. Harmonization may help reach this goal.

Overall, despite the limitations we identified and the trends which we observe, we are optimistic that synthetic and simulated data will continue to play a considerable role in medical image analysis and beyond.

Biomedical Image Synthesis and Simulation, Chapter 27

Acknowledgments

N. Burgos received funding from the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" programme reference ANR-10-IAIHU-0006 (Agence Nationale de la Recherche-10-IA Institut Hospitalo-Universitaire-6).


T H A Nguyen

T Laurent

C Thomas-Agnan

A Ruiz-Gazen

Analyzing the impacts of socio-economic factors on French departmental elections with CODA methods

Keywords:

The proportions of votes by party on a given subdivision of a territory form a vector called composition (mathematically, a vector belonging to a simplex). It is interesting to model these proportions and study the impact of the characteristics of the territorial units on the outcome of the elections. In the political economy literature, such regression models are generally restricted to the case of two political parties. In the statistical literature, there are regression models adapted to share vectors including CODA models (for COmpositional Data Analysis), but also Dirichlet models, Student models and others. Our goal is to use CODA regression models to generalize political economy models to more than two parties. The models are fitted on French electoral data of the 2015 departmental elections.

Introduction

Recently, models for elections focus on analyzing impacts of socio-economic factors for two-party systems using classical regression models [START_REF] Lewis | Estimating Regression Models in which the Dependent Variable is based on Estimates[END_REF]. In this paper, we propose a statistical model for studying the multiparty system using compositional data analysis (CODA) with departmental level data. The dependent variable will be the vector of votes shares for the French departmental election in 2015. The explanatory variables include some compositional and continuous socio-economic variables.

Among papers concentrating on the relationship between socio-economic variables and election results, [START_REF] Beauguitte | Analyser les comportements électéraux à l'échelle du bureau de vote[END_REF] [START_REF] Beauguitte | Analyser les comportements électéraux à l'échelle du bureau de vote[END_REF] study a linear regression at three levels of aggregation (polling stations, cities and electoral districts) and show that the socio-economic variables are significant. [START_REF] Kavanagh | A Geographically Weighted Regression Analysis of Election Specific Turnout in the Republic of Ireland[END_REF] [3] use geographically weighted regression, which produces parameter estimates for each data point, i.e. for each electoral division. On the other hand in the statistical literature, people have developped CODA regression models where the dependent and independent variables may be compositional variables (see [START_REF] Mert | Compositional data analysis in epidemiology[END_REF] [START_REF] Mert | Compositional data analysis in epidemiology[END_REF] for a review). [START_REF] Morais | Using compositional and Dirichlet models for market share regression[END_REF] [START_REF] Morais | Using compositional and Dirichlet models for market share regression[END_REF] study the impact of media investments on brand's market shares with a CODA regression model. Trinh and Morais (2017) [START_REF] Trinh | Impact of socioeconomic factors on nutritional diet in Vietnam from 2004 to 2014: New insights using compositional data analysis[END_REF] use a CODA regression model to highlight the nutrition transition and to explain it according to household characteristics. [START_REF] Honaker | A fast, Easy and Efficient Estimator for Multiparty Electoral Data[END_REF] [START_REF] Honaker | A fast, Easy and Efficient Estimator for Multiparty Electoral Data[END_REF], Katz and King (1990) [START_REF] Katz | A statistical Model for Multiparty Electoral Data[END_REF] use a statistical model for multiparty electoral data assuming that the territorial units yield independent observations. In Section 2, we present the data set. Subsection 3.1 (resp: 3.2) recalls the principles of compositional data analysis (resp: of compositional regression models). In subsection 3.3, we implement the CODA model on the election data set and present several plots to explore the impact of explanatory variables of a classical type illustrated by the case of unemployment rate as well as variables of a compositional type illustrated by the diploma variable.

Data

Vote share data of the 2015 French departmental election for 95 departments in France are collected from the Cartelec website 1 and corresponding socio-economic data (for 2014) have been downloaded from the INSEE website 2 . Table 1 summarizes our data set. Employment has five categories: AZ (agriculture, fisheries), BE (manufacturing industry, mining industry and others), FZ (construction), GU (business, transport and services) and OQ (public administration, teaching, human health). Diploma has three levels: <BAC for people with at most some secondary education, BAC for people with at least some secondary education and at most a high school diploma, and SUP for people with a university diploma. The Age variable has three levels: Age 1840 for people from 18 to 40 years old, Age 4064 for people from 40 to 64 years old, and Age 65 for elderly. For the vote share variable, the Cartelec website provides a very detailed information. The list of political parties which present candidates at that election is higher than 15. However, at the end of the election, it is common to present the results by grouping the political parties into three main components : Left, Right and Extreme-Right. 4From the CODA point of view, when compositional data have three components, they can be represented in a ternary diagram. For instance, the vote shares of the 95 departments for the Left and Right wings and the Extreme Right party are the blue points in Figure 1. The red triangle corresponding to the Aube department on Figure 1 shows that its vote shares of the Left wing, the Right wing and the Extreme Right party are respectively 17.4%, 54.6%, and 28% . Figure 2 illustrates the positions of the French departments on the ternary diagram whose components correspond to the three levels of the diploma variable, and the red triangle figures the geometric mean (adapted mean for compositional data) of all departments. In order to analyze the impacts of the socio-economic factors on the election results, a CODA regression model is proposed where the dependent variable is a compositional variable (vote shares) and the independent variables are compositional or classical variables or a mixture of both. This model is based on the log-ratio transformation approach.

Principles of compositional data analysis

A composition x is a vector of D parts of some whole which carries relative information. A D-composition x lies in the so-called simplex space S D defined by: S D = {x = (x 1 , ..., x D ) : x j > 0, j = 1, ..., D;

D j=1 x j = 1}
The simplex S D can be equipped with the Aitchison inner product ( [START_REF] Aitchison | A General Class of Distributions on the Simplex[END_REF] and [START_REF] Pawlowsky-Glahn | Modeling and Analysis of Compositional Data[END_REF]) in order to define distances. Classical regression models cannot be used directly in the simplex because the constraints that the components are positive and sum up to 1 are not compatible with their usual distributional assumptions. To overcome this difficulty, one way out is to use a log-ratio transformation from the simplex space S D to the Euclidean space R D-1 . The classical transformations are alr (additive log-ratio transformation), clr (centered log-ratio transformation), and ilr (isometric log-ratio transformation). The coordinates in the clr transformed vector are linearly dependent, and the coordinates in the alr transformed vector are not compatible with the geometry (distance between the components in the simplex space is different from distance between the coordinates in the Euclidean space). For these reasons people generally use one of the ilr transformation for compositional regression models. An isometric log-ratio transformation (ilr) is defined by: ilr

(x) = V T D ln(x)
where the logarithm of x is understood componentwise, V T D is a transposed contrast matrix [START_REF] Pawlowsky-Glahn | Modeling and Analysis of Compositional Data[END_REF] associated to a given orthonormal basis (e

1 , • • • , e D-1 ) of S D by V D = clr(e 1 , • • • , e D-1 ).
Note that such a contrast matrix V D of size D × (D -1) satisfies the following property:

1. V D V T D = I D -1 D 1 D×D where I D is the D × D identity matrix, 1 D×D is a D × D matrix of ones. 2. V T D V D = I D-1
where I D-1 is the identity matrix with dimension (D -1).

3. V T D j D = 0 D-1 where j D is a D × 1 column vectors of ones. The following D × (D -1) matrix V D defined by [START_REF] Egozcue | Isometric Logratio Transformations for Compositional Data Analysis[END_REF] [START_REF] Egozcue | Isometric Logratio Transformations for Compositional Data Analysis[END_REF] is an example of contrast matrix for D = 3

V 3 = 2/ √ 6 0 -1/ √ 6 1/ √ 2 -1/ √ 6 -1/ √ 2
This particular matrix defines the following ilr coordinates

ilr 1 (x) = 1 √ 6 (2 log x 1 -ln x 2 -log x 3 ) = 2 √ 6 log x 1 √ x 2 x 3 ilr 2 (x) = 1 √ 2 (log x 2 -log x 3 ) = 1 √ 2 log x 2 x 3
The first ilr coordinate contains information about the relative importance of the first component x 1 with respect to the geometric mean of the second and the third components g = √ x 2 x 3 . The second ilr coordinate contains information about the relative importance of the second component x 2 with respect to the third component x 3 . In our case, the first ilr coordinate opposes the Left wing to the group of the Right wing and the Extreme Right party and the second opposes the Right wing to the Extreme Right party. The inverse ilr transformation is given by:

x = ilr -1 (x * ) = C(exp(V D x * )) for x * ∈ R D-1
where the exponential of vector x is understood componentwise and

C(x) =   x 1 / D j=1 x j , • • • , x D / D j=1 x j   is the closure operation.
The vector space structure of the simplex S D is defined by the perturbation and powering operations:

x ⊕ y = C(x 1 y 1 , . . . , x D y D ), x, y ∈ S D λ x = C(x λ 1 , . . . , x λ D ), λ is a scalar, x ∈ S D .
The compositional inner product (C-inner product) of x and y in S D is defined by

x, y c = 1 D D-1 i=1 D j=i+1 log x i x j • log y i y j = D i=1 log x i g(x)
• log y i g(y)

where g(x) = D √ x 1 x 2 ...x D is the geometric mean of the components. The compositional distance (C-distance) between x and y in S D is derived from the inner product

d c (x, y) =   1 D D-1 i=1 D j=i+1 log x i x j -log y i y j 2   1/2 = D i=1 log x i g(x)
-log y g(y)

2 1/2
The expected value E ⊕ Y of a simplex-valued random composition Y ∈ S D (Pawlowsky [START_REF] Pawlowsky-Glahn | Modeling and Analysis of Compositional Data[END_REF]) is defined by

argmin z∈S D E(d 2 c (Y, z))
and it is equal to

E ⊕ Y = C(exp(E log Y)) = clr -1 (E clr(Y)) = ilr -1 (E ilr(Y)) = ilr -1 (EY * )
where Y * = ilr(Y).

Compositional regression models

The notations used in this paper are summarized in Table 2 Variable Notation Coordinates

Dependent Y i = (Y 1 , . . . , Y L ) ilr(Y i ) = Y * i Compositional explanatory X (q) i = (X (q) i1 , . . . , X (q)
iD q ) ilr(X

(q) ip ) = X (q) * ip Classical explanatory Z ik General notations L Number of components of the dependent variable i = 1, . . . , n Index of observations (n = 95 ) q = 1, . . . , Q Index of compositional explanatory variables (Q = 3) p = 1, . . . , D q
Index of the coordinates for the compositional explanatory variables k = 1, . . . , K Index of classical explanatory variables (K = 5)

Table 2: Notations

We now describe the CODA regression model. Y i ∈ S L denotes the compositional response value of the ith observation, and X (q) i ∈ S D q , q = 1, . . . , Q, denotes the value of the qth compositional covariate for the ith observation, where Y ∈ S L and X (q) ∈ S D q , q = 1, . . . , Q, belong to the simplex spaces S L and S D q . Z ik , k = 1, . . . , K, denotes the kth classical covariate of the ith observation. Let be the compositional matrix product, which corresponds to the matrix product in the coordinate space through the ilr transformation

B x = C   D j=1 x b 1j j , • • • , D j=1 x b Lj j   T
where x ∈ S D and B = ((b ij )), i = 1, . . . , L, j = 1, . . . , D, is a parameter matrix such that the column vectors belong to S D , j T L B = 0 D , Bj D = 0 L , where j L is a L × 1 column vector of ones, and j T L is the transposed of j L . Let us first introduce the CODA regression model in the ilr coordinate space as follows:

ilr(Y i ) = b 0 * + Q q=1 ilr(X (q) i )B * q + K k=1 Z ik c * k + ilr( i ) (1)
where ilr(Y i ), ilr(X

(q) i ) are the ilr coordinates of Y i , X (q) i
(q = 1, . . . , Q) respectively; b 0 * , B * q , c * k are the parameters in the coordinate space, and ilr( i ) are the residuals. The distributional assumption is that ilr( ) follows the multivariate normal distribution with zero mean and covariance matrix Σ Σ Σ. This regression model can be written in the simplex as

Y i = b 0 Q q=1 B (q) X (q) i K k=1 Z k c k ⊕ i , i = 1, . . . , n (2)
where b 0 , B (1) , . . . , B (Q) , c 1 , . . . , c K are the parameters satisfying b 0 ∈ S L , B (q) ∈ S D q , q = 1, . . . , Q, c k ∈ S L , k = 1, . . . , K, j T L B (q) = 0 D q , B (q) j D q = 0 L , The distributional assumption is that i ∈ S L follows the normal distribution on the simplex (see [START_REF] Aitchison | A General Class of Distributions on the Simplex[END_REF]).

It is classical to estimate model (2) using OLS thus assuming the independence between the ilr coordinates. Chen et al (2016) [START_REF] Chen | Multiple linear regression with compositional response and covariates[END_REF] give different formulas relating the parameters in the simplex to the parameters in the coordinate space. Following Chen et al.(2016) [START_REF] Chen | Multiple linear regression with compositional response and covariates[END_REF] (Property 2.1 and Property 2.3(3)), we extend this result to the case of an additional non-compositional covariate: we calculate the sum of the squared norm of the residuals and derive the estimators from the normal equations. We obtain that: Theorem 1 In model ( 1)- [START_REF] Beauguitte | Analyser les comportements électéraux à l'échelle du bureau de vote[END_REF], the relationship between the parameters in the simplex and their counterpart in coordinate space is given by

     b 0 = exp(b 0 * T V L ) = ilr -1 (b 0 * ) B q = V T D q B * q V L c k = exp(c * k V L ) = ilr -1 (c * k ) (3)
where V L and V D q , q = 1, . . . , Q are contrast matrices associated to the selected ilr transformations.

Impact of compositional and classical explanatory variables

Because the interpretation of the parameters of these models is not so straightforward [START_REF] Morais | Interpretation of explanatory variables impacts in compositional regression models[END_REF], we rather concentrate on illustrating graphically the relationship between the predicted vote shares and the explanatory variables. The prediction of the dependent variable for the above models are given by ( 4):

Ŷi = b0 Q q=1 Bq X (q) i K k=1 Z ik ĉk i = 1, ..., n (4)
where b0 , Bq and ĉk are the estimated parameters. We can rewrite (4) as

Ŷi = C b0 . Q q=1 X (q) i Bq . K k=1 ĉZ ik k i = 1, ..., n (5)
In order to illustrate these formulas, we will focus on graphing the predicted values of the dependent variable as a function of one specific variable of interest: two cases must be considered depending on whether the specific variable is classical or compositional. In both cases, we will create a grid of potential values of the specific explanatory and fix the other explanatory variables at the values they take for one selected point of the dataset (we repeat for several selected points). For the sake of simplicity let us take L = 3.

For the case when the specific variable is a classical covariate Z ik , from (5) there exists â0 ∈ S L (this term contains the impacts of all other explanatory but is constant when Z ik alone varies) such that

Ŷi = â0 Z ik ĉ = C â01 ĉZ ik 1 , • • • , â0L ĉZ ik L With T = b01 ĉZ ik 1 + • • • + b0L ĉZ ik L , we get Ŷi1 = â01 ĉZ ik 1 T ; Ŷi2 = â02 ĉZ ik 2 T ; • • • ; ŶiL = â0L ĉZ ik L T .
Now for the case when the specific variable is a compositional variable X (q)

i , let us take for the sake of simplicity D q = 3. As before, from (5), there exists â0 ∈ S L (this term contains the impacts of all other explanatory but is constant when

X (q) i alone varies) such that Ŷi = â0 X (q) i Bq = C(â 01 X (q) i1 b(q) 11 X (q) i2 b(q) 12 X (q) i3 b(q) 13 , â02 X (q) i1 b(q) 21 X (q) i2 b(q) 22 X (q) i3 b(q) 23 , â03 X (q) i1 b(q) 31 X (q) i2 b(q) 32 X (q) i3 b(q) 33 )
We now fit a CODA regression model describing the impacts of socio-economic factors on vote shares in the 2015 French departmental election. After including all explanatory variables from our data set in the regression model, and eliminating one by one the variables which are not significant, we obtain the results in Table 3. This model shows that the age of people, the proportion of people who own assets, the proportion of foreigners do not have any impact on the vote shares. However, the levels of education, the working areas, the unemployment rate and the proportion of people who pay income tax really affect the result of the French departmental election in 2015. In order to illustrate the impact of unemployment on the predicted shares, we choose three departments Ariège, Cantal and Bas-Rhin with different characteristics: Ariège has the maximum Left wing share, Cantal the maximum Right wing share and Bas Rhin has the minimum Left wing share. We fix the values of the covariates at the values of each of the three departments and create a grid of fictive values of unemployment rates. Figure 3 shows the predictions of vote shares in these departments Ariège, Cantal and Bas-Rhin as a function of unemployment rate (its minimum and maximum in the data base are figured by the dotted vertical lines). We first of all see the non linear nature of the relationship, and the fact that they differ from one department to the other. Note that the predicted shares using this model satisfy the constraint of unit sum and it clearly shows on the graph. In all cases, when the unemployment rate increases up to a given threshold of around 15%, the Left wing and the Extreme Right party gain votes at the expense of the Right wing. However, if unemployment keeps increasing beyond 15%, the Left wing starts loosing votes while the Right wing keeps decreasing and the Extreme Right keeps increasing. Overall, we can say that as unemployment rate varies, the Left wing proportion is more stable than the other two parties and that the other two parties affect each other like interconnecting pipes. Even though the three departments curves have the same general shape, we note differences: the maximum of the Left wing share is the highest in Ariège and lowest in Bas-Rhin; it is striking that the point at which the Left wing share and the Right wing share are equal is obtained at approximately the same value of unemployment rate in the three department but corresponds to different values of the common Left wing-Right wing share; this value is lower than the maximum Left wing share in Ariège whereas it is slightly higher in Bas-Rhin. A major difference between the three departments is revealed when one looks at the highest of the three predictions: in Ariège, all realistic scenarios (between two vertical lines) result in a victory of the Left wing, in Cantal, all three parties may win depending on the value of unemployment and finally in Bas-Rhin there is no scenario leading to a victory of the Left wing . To represent this differently, we plot on Figure 4 a ternary diagram showing the curve of predicted shares as a function of unemployment rates together with a small square figuring the observed position of the given department in the triangle and a small diamond the corresponding prediction on the curve. The curve, a line in the simplex, is colored according to the value of unemployment rate. The Cantal department is better predicted than the Ariège and Bas-Rhin departments. We also note that the maximum predicted proportion for the Left wing is lower in the Bas-Rhin than in the other two departments. Finally, the triangle is divided in three parts with respect to the highest shares to highlight the winning party as in Figure 3. Left Right XR q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Cantal Left Right XR q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Bas-Rhin Left Right XR q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 4: The vote share prediction ternary diagrams for fixed covariates given by three departments: Ariège, Cantal, Bas-Rhin respectively as a function of the unemployment rate. The green squares show the observed vote share of these departments and the green triangles on the red curve the corresponding predictions.

Let us now turn attention to the case of a compositional explanatory variable impact. Figure 5 presents the vote share predictions according to the Diploma variable in the same three departments (Ariège, Cantal and Bas-Rhin). The principle is the same: all explanatory variables are fixed to the value of the given department except Diploma. We create of grid in the Diploma triangle and compute the predicted shares at each point of this grid. However since it is impossible to plot a function from the simplex to the simplex, we choose to summarize the predicted shares by the winning party (corresponding to the highest share) and color the triangle in the Diploma space according to the winning party color. The observed shares are also figured by black points in this ternary diagram thus showing the realistic values. This figure shows that there is a large proportion of fictive situations (in terms of diploma proportions) where the Left party would win.

Ariège

Right Left XR <BAC BAC SUP q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 5: Predictions of vote shares according to Diploma for fixed covariates given by the departments Arige (left plot), Cantal (middle plot) and Bas-Rhin (right plot)

Conclusion

The above analysis demonstrates that the CODA regression models can be useful in the context of political economy. We analyze how the predicted values in these models vary with the predictors and propose new graphical tools to explore the impact of some socio-economic variables on election results. Our future perspectives are to introduce the geographical dimension in the model and to use the Student distribution (Katz and King, 1999 [7]) instead of the normal distribution. At last, we plan to compute the elasticities as in [START_REF] Morais | Interpretation of explanatory variables impacts in compositional regression models[END_REF] to characterize the impacts of the covariates in a more quantitative way.

Figure 1 :

 1 Figure 1: Vote shares in the 95 departments (blue points) with the Aube department as the red triangle

Figure 2 :

 2 Figure 2: Components of Diploma in the 95 departments (blue points) and their geometric mean (red triangle)

Figure 3 :

 3 Figure 3: The vote share prediction curves in three departments: Ariège, Cantal, Bas-Rhin respectively (the grey dotted line show the minimum and the maximum observed unemployment rates)

Table 1 :

 1 Data description

	Variable name Description	Averages 3
	Vote share	Left(L), Right(R), Extreme Right(XR)	0.37, 0.388, 0.242
	Age	Age 1840, Age 4064, Age 65.	0.313, 0.432, 0.255
	Diploma	<BAC, BAC, SUP.	0.591, 0.16, 0.239
	Employment	AZ, BE, FZ, GU, OQ	0.031, 0.099, 0.049, 0.439, 0.382
	unemp	the unemployment rate	0.117
	employ evol	Mean annual growth rate of employment (2009-2014) -0.145
	owner	The proportion of people who own assets	0.616
	income tax	The proportion of people who pay income tax	0.552
	foreign	The proportion of foreigners	0.050

Table 3 :

 3 Regression with compositional and classical variables

for more details, see https://fr.wikipedia.org/wiki/Elections départementales francaises de 2015

Acknowledgments. We thank professor Le Breton for introducing us to this topic of political science and for nice discussions.


Keywords: Chlordecone, ID-LC-MS/MS, bovine tissues, correlation factors

Occurrence and concentrations of chlordecone in fat, muscle and liver in 200 bovines.

Chlordecone concentrations are correlated in fat, muscle and liver.

Introduction

Chlordecone (CLD) is an organochlorine pesticide that was widely used in the French West Indies (Martinique and Guadeloupe) to protect banana plantations from black weevil (cosmopolites sordidus) between 1972 and 1993. CLD is a bi-cyclic chlorinated molecule. As its octanol/water coefficient exceeds 3, CLD is classified as a lipophilic pollutant. CLD also exhibits a ketone function (C=O) and the presence of a non-saturated carbon-carbon bond that can lead to keto-enol equilibrium.

CLD was classified in 2009 by the Stockholm Convention as a persistent pollutant. It is known to be carcinogenic, mutagenic and reprotoxic, and is suspected of being an endocrine disrupter [START_REF] Multigner | Chlordecone exposure and risk of prostate cancer[END_REF]. In a cohort of a study carried out in Guadeloupe (Timoun cohort), an effect of CLD exposure was found on the duration of pregnancy [START_REF] Hervé | Prenatal exposure to chlordecone, gestational weight gain, and birth weight in a Guadeloupean birth cohort[END_REF], and on the thyroid hormone system [START_REF] Cordier | Perinatal exposure to chlordecone, thyroid hormone status and neurodevelopment in infants: The Timoun cohort study in Guadeloupe (French West Indies)[END_REF], as well as impairment of fine motor function in boys [START_REF] Boucher | Exposure to an organochlorine pesticide (chlordecone) and development of 18-month-old infants[END_REF][START_REF] Dallaire | Cognitive, visual, and motor development of 7-month-old Guadeloupean infants exposed to chlordecone[END_REF], and visual contrast sensitivity [START_REF] Saint-Amour | Visual contrast sensitivity in school-age Guadeloupean children exposed to chlordecone[END_REF]. For male adults, an increase in the risk of prostate cancer has also been documented [START_REF] Multigner | Chlordecone exposure and risk of prostate cancer[END_REF].

Due to the former use of this pesticide, certain Martinican and Guadeloupean environmental compartments are widely contaminated with CLD. Soil contamination, linked to former use of CLD, depends on soil composition [START_REF] Cabidoche | Long-term pollution by chlordecone of tropical volcanic soils in the French West Indies: A simple leaching model accounts for current residue[END_REF][START_REF] Woignier | Soil microstructure and organic matter: Keys for chlordecone sequestration[END_REF]. Studies on the fate of CLD have demonstrated low and poor degradation in soils with decontamination projected to take decades to centuries [START_REF] Cabidoche | Long-term pollution by chlordecone of tropical volcanic soils in the French West Indies: A simple leaching model accounts for current residue[END_REF]. More recent studies have also indicated that transformation of CLD occurs in soils through highly complex pathways [START_REF] Benoit | Categorizing chlordecone potential degradation products to explore their environmental fate[END_REF][START_REF] Chevallier | Natural Chlordecone Degradation Revealed by Numerous Transformation Products Characterized in Key French West Indies Environmental Compartments[END_REF][START_REF] Clostre | Comparative fate of an organochlorine, chlordecone, and a related compound, chlordecone-5b-hydro, in soils and plants[END_REF]. Food crops (including vegetables) through root uptake [START_REF] Clostre | Comparative fate of an organochlorine, chlordecone, and a related compound, chlordecone-5b-hydro, in soils and plants[END_REF][START_REF] Florence | Organochlorine (chlordecone) uptake by root vegetables[END_REF][START_REF] Létondor | Uptake and distribution of chlordecone in radish: Different contamination routes in edible roots[END_REF], and ruminants, through involuntary soil ingestion, can also be contaminated by CLD [START_REF] Clostre | Effect of home food processing on chlordecone (organochlorine) content in vegetables[END_REF][START_REF] Collas | Cattle exposure to chlordecone through soil intake. The case-study of tropical grazing practices in the French West Indies[END_REF][START_REF] Fournier | Toxicokinetics of chlordecone in goats: Implications for risk management in French West Indies[END_REF].

Regarding food safety, in 2007, the French Food Safety Agency re-examined toxicological reference values based on previous studies in rats dating back to 2003. No changes were made: the chronic health-based guidance value was established at 0.5 µg/kg body weight/day, and the acute value at 10 µg/kg body weight/day. The impact of food consumption, including local food supply habits, was also examined based on consumption data (AFSSA, 2007): it was determined that setting maximum concentration limits in foodstuffs would be of limited effectiveness considering subsistence production [START_REF] Dubuisson | Impact of subsistence production on the management options to reduce the food exposure of the Martinican population to Chlordecone[END_REF]. Moreover, it was concluded that the maximum residue levels Concerning livestock, the target tissue for official controls is perirenal fat, since it is the tissue in which non-polar organochlorine pesticides are supposed to be mostly distributed. However, some studies have shown that CLD has atypical behaviour regarding its tissue distribution, and found that CLD is not only distributed in fat but also in muscle in certain species, such as poultry (Jondreville et al., 2014a), ducks (Jondreville et al., 2014b), growing male Alpine kids [START_REF] Lastel | Comparison of chlordecone and NDL-PCB decontamination dynamics in growing male kids after cessation of oral exposure: Is there a potential to decrease the body levels of these pollutants by dietary supplementation of activated carbon or paraffin oil?[END_REF]), ewes Saint-Hilaire et al., 2018) and pigs [START_REF] Fourcot | Modeling chlordecone toxicokinetics data in growing pigs using a nonlinear mixed-effects approach[END_REF]. This raises the question of distribution in other species such as bovines, which are largely consumed and are one of the most targeted species for official food controls.

CLD concentrations can be determined either by gas or liquid chromatography; methods are mostly adapted from Bordet et al. [START_REF] Bordet | In-house validation for analytical methods and quality control for risk evaluation of chlordecone in food[END_REF]. For gas chromatography, detection can be performed with an electron capture detector (GC/ECD) or mass spectrometry, with low or high resolution [START_REF] Clostre | Effect of home food processing on chlordecone (organochlorine) content in vegetables[END_REF][START_REF] Florence | Organochlorine (chlordecone) uptake by root vegetables[END_REF][START_REF] Fournier | Toxicokinetics of chlordecone in goats: Implications for risk management in French West Indies[END_REF][START_REF] Lastel | Comparison of chlordecone and NDL-PCB decontamination dynamics in growing male kids after cessation of oral exposure: Is there a potential to decrease the body levels of these pollutants by dietary supplementation of activated carbon or paraffin oil?[END_REF][START_REF] Méndez-Fernandez | From banana fields to the deep blue: Assessment of chlordecone contamination of oceanic cetaceans in the eastern Caribbean[END_REF]. This second type of detection is recommended in the case of positive results (i.e. > the limit of quantification -LOQ) for confirmation purposes.

Although gas chromatography is still used, most of the current methods, including the official method for controls in France, are based on isotopic dilution followed by reversed-phase liquid chromatography analysis and tandem mass spectrometry detection with negative electrospray ionization [START_REF] Clostre | Comparative fate of an organochlorine, chlordecone, and a related compound, chlordecone-5b-hydro, in soils and plants[END_REF][START_REF] Lastel | Comparison of chlordecone and NDL-PCB decontamination dynamics in growing male kids after cessation of oral exposure: Is there a potential to decrease the body levels of these pollutants by dietary supplementation of activated carbon or paraffin oil?[END_REF][START_REF] Saint-Hilaire | Development and validation of an HPLC-MS/MS method with QuEChERS extraction using isotopic dilution to simultaneously analyze chlordecone and chlordecol in animal livers[END_REF]. Isotopic dilution is recommended to ensure accurate identification and quantification. Recent publications also report the use of high-resolution mass spectrometry with LC separation for the identification of CLD transformation products in the environment through biodegradation or food processing [START_REF] Martin | Effect of home cooking processes on chlordecone content in beef and investigation of its by-products and metabolites by HPLC-HRMS/MS[END_REF][START_REF] Ollivier | Transport of chlordecone and two of its derivatives through a saturated nitisol column (Martinique, France)[END_REF].

The present study aims at determining CLD levels in fat, muscle and liver tissues of 200 endemically contaminated bovines originating primarily from the most contaminated areas of Martinique and Guadeloupe. The analytical method used was the official method for monitoring controls in France, and is based on isotopic dilution liquid chromatography hyphenated with mass spectrometry (ID-LC-MS/MS). These data were then used to assess the correlations between CLD concentrations in fat, muscle and liver.

Material and methods

Chemicals and reagents

All solutions were prepared with analytical reagent grade chemicals and ultrapure water (18.2 Milli-Q system (Merck Millipore, Saint-Quentin-en-Yvelines, France). All solvents (acetone, acetonitrile, dichloromethane, hexane, and methanol) were HPLC grade and were purchased from Fisher Scientific (Illkirch, France). Sodium hydroxide and formic acid (98%) were also purchased from Fisher Scientific (Illkirch, France). CLD (98%) and CLD 13 C10 (98%) standards were purchased from Cluzeau Info Labo (Sainte-Foy-La-Grande, France). Working solutions were prepared by dilution of the commercial standards in methanol before use, to prepare calibration standards and fortify the samples for quality controls and method validation. All solutions were stored at 5 ± 3°C.

Equipment

LC-MS/MS measurements were performed using a 1200 series HPLC binary pump system (Agilent Technologies, Courtaboeuf, France) coupled with an API 5500 QTRAP hybrid tandem mass spectrometer (AB Sciex, Les Ulis, France). An Aqua C18 column (150 mm x 2.0 mm i.d., 3 µm particle size) equipped with an Aqua C18 guard column (4.0 mm x 3.0 mm i.d., 3 µm particle size) was used for chromatographic separation. An Eppendorf centrifuge 5810 (Sigma Aldrich, Saint-Quentin-Fallavier, France), a refrigerated Hettich 32 R centrifuge (Sigma Aldrich, Saint-Quentin-Fallavier, France), and a Genie 2 vortex (Scientific Industries, Bohemia, NY, United States) were used for the extraction. A Grindomix GM 200 (Retsch Industries, Haan, Germany) and a Polytron PT 3100 (Fisher Scientific, Illkirch, France) were used for sample preparation. An IKA Ultraturrax T25 digital Disperser (Retsch Industries) was used for both sample preparation and extraction. An XS 204 balance (Mettler-Toledo, Viroflay, France) was used to weigh the samples. A water bath heater from Fisher (Fisher Scientific, Illkirch, France) was used for fat melting. A Reacti-Therm Pierce heating module (Thermofisher, Les Ulis France) was used for solvent evaporation.

Sampling

From January to October 2016, an oriented exploratory survey plan was implemented by the French authorities in charge of food control (Directorate for food, agriculture and forestry of Guadeloupe and Martinique) toward the most probable contaminated bovines in the French West Indies. In all, 200 bovines were selected, mostly in Martinique (120 bovines). In Guadeloupe, open-air farms located in the most contaminated areas were chosen, whereas in Martinique, animals were selected as a priority when the livestock was found to have positive results in previous official control plans. Livestock number, age, and sex of the animals were reported, as well as slaughterhouse locations (18 locations for Martinique and a single location for Guadeloupe). For each bovine, the three tissues perirenal fat, skirt for muscle, and liver were sampled and clearly identified for traceability.

For quality controls and method validation, liver and muscle samples were purchased from a local supermarket (Choisy-le-Roi, France) and perirenal fat was collected from a local butcher (Saint-Maur-des-Fossés, France). These matrices were determined to be CLD-free.

Sample preparation and extraction

The analytical procedure employed in this study was adapted from [START_REF] Bordet | In-house validation for analytical methods and quality control for risk evaluation of chlordecone in food[END_REF], with minor changes regarding sample preparation and major changes regarding separation and detection: GC-ECD was switched to LC-MS/MS². This is the official method to be implemented in France for monitoring and control plans for food of animal origin by approved laboratories.

Preparation and extraction procedure for fat

Frozen fat samples were cut into pieces, blended with a Grindomix apparatus, heated to a maximum of 60 °C and filtered through a strainer to clean samples of residues. A total of 0.5 g was precisely weighed in a test tube. Then, 75 µL of internal standard CLD-13 C10 at a concentration of 0.5 ng.µL -1 was added to melted fat. After 1 min vortex-stirring, the internal standard was kept in contact for 1 h with the matrix. CLD was extracted from the melted fat sample 3 times with 3 mL of a mix of acetonitrile/dichloromethane 75/25 (v/v). For each extraction, the test tube was briefly shaken with a vortex homogenizer and then centrifuged for 20 min at 1,200 g at a temperature of -20 °C. The supernatants were collected into another test tube and solvent was evaporated to dryness at a temperature not exceeding 40 °C under a gentle stream of nitrogen. The dry extract was dissolved in 15 mL of a mixture of hexane/acetone 85/15 (v/v) and then transferred to a 50 mL centrifuge tube. 5 mL of 0.5 M aqueous sodium hydroxide solution was added, and the tube was vortexed for 15 s. The lower aqueous phase was collected in another centrifuge tube. This extraction step was repeated twice, and the three aqueous phases were combined in the same tube, and then washed with 5 mL of hexane by gentle manual shaking. The hexane phase was discarded after a 3 min centrifugation step at 750 g. Then, 5 mL of aqueous sulfuric acid solution at 60% were added to the aqueous extract, and the tube was shaken for 15 s with a vortex homogenizer and gently left to cool at room temperature. CLD was extracted 3 times with 5 mL of a mixture of hexane/acetone 85/15 (v/v) by shaking for 15 s with a vortex homogenizer, then centrifuged for 3 min at 750 g, and the upper organic phases were collected in a 15 mL centrifuge tube and washed with 2 mL of ultrapure water. After final centrifugation for 3 min at 750 g, the organic phase was evaporated to dryness at a temperature not exceeding 40 °C under a gentle stream of nitrogen. The dry extract was dissolved in 1 mL of methanol and transferred to a vial for chromatographic analysis.

Preparation and extraction procedure for muscle and liver

Due to the lower fat content of liver and muscle compared to perirenal fat, a different sample preparation method was used. Samples were cut into pieces and homogenized first with a Grindomix apparatus, then a Polytron and a T25 Ultra-Turrax apparatus. Then, 2 g of homogenized sample were precisely weighed in a 50 mL centrifuge tube before the addition of 50 µL of internal standard CLD-13 C10 at a concentration of 0.6 ng.µL -1 . After 1 min vortexstirring, the internal standard was kept in contact for 1 h with the matrix. CLD was extracted by 10 mL of a mixture of hexane/acetone 85/15 (v/v) by using a T25 Ultra-Turrax for 1 min at 10,000 rpm. The organic phase was collected in another 50 mL centrifuge tube. The extracts then followed the same basic and acidic transformations as for the fat extracts.

LC-MS/MS analysis

A total amount of 5 µL of the individual final extract was injected into the chromatographic system at room temperature (Phenomenex, Le Pecq, France). Ultrapure water containing 0.1% (v/v) formic acid (mobile phase A) and methanol containing 0.1% (v/v) formic acid (mobile phase B) were used for the gradient elution at a constant flow rate (200 µL min -1 ). The gradient started with 40% of phase B and increased to 100% within 6 min. Phase B was kept constant for the next 8 min and was then set back to 60% in 2 min before a 5 min equilibrium time, before the next injection.

Electrospray ionization was run in negative mode. The spray voltage was set at -4,500 V. The curtain gas was set at 25 psi of nitrogen. Spray gas and auxiliary gas were set at 40 and 60 psi of air, set at 600 °C. The instrument was operated in multiple reaction-monitoring mode (MRM). CLD and its isotopic standard were detected under the deprotonated hydrated form [START_REF] Bichon | Ultratrace quantification method for chlordecone in human fluids and tissues[END_REF]. The quantifying and qualifying transitions for CLD were 506.7 > 426.7 and 506.7 > 428.7, respectively. For these two transitions, the decluttering potential was -160 V, entrance potential -10 V, collision cell exit potential -15 V, and collision energy -25 eV. The transition for CLD-13 C was 516.7 > 435.7 with the same parameters, except the collision energy of -28 eV. The transition for CLD-13 C was 516.7 > 435.7 with the same parameters, except the collision cell exit potential -10 V, and the collision energy -30 eV. The dwell time for each transition was set at 100 ms. Analyst 1.51 Software (AB Sciex) was used for system control and data acquisition and processing.

2.6.

Method validation and quality assurance/quality control

The method was validated as per standard NF V03-110: 2010 (AFNOR, 2010), based on tolerance intervals and overall determination of the performance of the method through the accuracy profile [START_REF] Mermet | Potential of accuracy profile for method validation in inductively coupled plasma spectrochemistry[END_REF].

The accuracy profile provides a graphical representation of the risk of error for each -expectation limits lie also evaluates the performance of the calibration model, the accuracy, -120% in accordance with European Union guidelines (SANTE/11945/ 2015(SANTE/11945/ , 2015) ) set to 80%, meaning that the risk of results lying outside the limits is below 20% on average.

Initially, the method was validated and implemented to cover two commodity groups mentioned in the SANTE imal ori

For the second commodity group, as fishes were targeted for official controls, it was decided to select barracuda, a fish consumed in the French West Indies. Since (i) sample preparation is identical for fish and meat samples, and (ii) the analysis is based on isotopic dilution, the method was considered to be validated for the whole commodity group. This was confirmed by the conformity of routine quality control, implemented for the present study, including recoveries on meat as detailed below. Later, the method was extended to liver tissues. Therefore, three separate profiles were built: one for the , and one covering fat regarding its specific preparation. Five series spiked at 4 levels (3 levels for liver) from 3 to 500 µg.kg -1 fw were analyzed, each on different days in duplicate. The performances of the method are presented in Table 1.

An overall limit of quantification (LOQ) (3 µg.kg -1 fw) was experimentally tested and the range of application was tested up to 500 µg.kg The limit of detection (LOD) was set according to the calibration model used for quantification with standard solutions and to the enrichment factor between the matrix and the extracts: it was set at 2 µg.kg -1 fw for fat, and 0.5 for µg.kg -1 fw for muscle. For liver, the LOD was set at one third of the LOQ, i.e. 1 µg.kg -1 fw.

The repeatability coefficients of variation for all matrices were within the range 1.2 8.3%, all concentration levels considered. The intermediate precision coefficients of variation ranged from 2.5 to 9.9%. Both criteria met the requirements of the SANTE Guidelines (SANTE/11945/ 2015, 2015), which recommend a maximum value of 20%.

The expanded uncertainty of the method was estimated taking into account the standard deviation of the tolerance interval and an estimation of the bias. It was set at 20 %, corresponding to the maximum estimated uncertainty for each matrix and tested concentrations.

For routine analysis, quality controls were included in each sample batch, with criteria recommended by SANTE guidelines (SANTE/11945/ 2015, 2015). Reagent blanks were injected after each sample to check the absence of cross-contamination, with a tolerance corresponding to 30% of the LOQ. Bracketing calibration (6 levels) was performed, and linearity was controlled by residual deviation below 20% from the calibration curve. In every sample batch, recovery at a level corresponding to 1.5 x LOQ was checked to be in the 70 120% range. Retention times (± 0.1 min between CLD and CLD 13 C retention times) and ion ratio deviations (± 30%) were systematically examined to ensure correct analyte identification in case of a result above the LOQ. Results were validated only if all criteria were within the acceptable limits.

Some of the analyses (73 out of 200 whole triplets) were subcontracted to the laboratory Inovalys, Le Mans, France. This laboratory, belonging to the network approved for French official controls, applied the same official method described above after in-house validation and demonstration of equivalent performance. In both laboratories, analyses were performed under the international standard ISO/IEC 17025:2005 (ISO/IEC 17025, 2005) by qualified operators.

Statistical analysis

The correlation study of the CLD concentrations in the three tissues was performed in multiple steps.

Shapiro

Wilk test was used to determine the normality of the concentration distributions for each of the three studied tissues. muscle

Results and discussion

Occurrence of CLD content in bovine perirenal fat, muscle, and liver

The main statistical descriptors are presented in Table 2, and Figure 1 illustrates the distribution of the determined concentrations. The whole dataset is available in the Supplementary Information. The most contaminated triplets exhibit atypically high values compared to the whole dataset: CLD concentrations in liver and muscle (812 µg.kg -1 fw and 152 µg.kg -1 fw) are almost two fold higher than for the penultimate triplet (420.6 µg.kg -1 fw and 67.6 µg.kg -1 fw). This triplet was excluded from the dataset for description and statistical analysis. Irrespective of the type of matrix, CLD was detected in 68% of samples (404 samples above LOD) and quantified in 59% of samples (332 samples above LOQ). Considering the three tissues, CLD was mostly quantified in liver (65%, 130 out of 199 samples), followed by fat (53%, 106 out of 199 samples), and muscle (48%, 96 out of 199 samples). Regarding contamination levels, liver covers a broader range of concentrations (LOQ up to 420.6 µg.kg -1 fw) than fat (LOQ up to 124.6 µg.kg -1 fw), and muscle (LOQ up to 67.6 µg.kg -1 fw), as mentioned in Table 2.

For muscle and liver, a previous study regarding food exposure of Martinicans and Guadeloupeans reported that CLD was detected in 19% of beef meat samples (n=38) [START_REF] Dubuisson | Impact of subsistence production on the management options to reduce the food exposure of the Martinican population to Chlordecone[END_REF]. The samples were collected during a sampling plan that was not only focussed on contaminated areas (without precision) and was organised between 2002 and 2004.

The type of tissue collected was not detailed. The observed concentration range started at 0.83 µg.kg -1 fw for the average lower-bound residue level, and at 1.16 µg.kg -1 fw for the average upper-bound, and reached up to 2.88 µg.kg -1 fw for the highest values. The higher detection occurrence (59% for muscle and 85% for liver) and contamination ranges in the present study are difficult to compare since they might be linked to a bias in sample collection, since it is oriented toward most-probable contaminated bovines in the present study.

Considering muscle tissue, the most consumed part, the results can be compared with those reported in the Kannari study (ANSES, 2017), where meat samples were collected between 2009 and 2015 from different supply chains, including slaughterhouses, local markets, subsistence production, and medium and large wholesalers. The Kannari study differentiates contaminated and non-contaminated areas of Martinique and Guadeloupe. In slaughterhouses, 64% (n=293) of the samples were above the LOQ of 1 10 µg.kg -1 fw in contaminated areas.

These figures are consistent with those found in the present study oriented toward contaminated areas: 48%, 96 samples out of 199 above the LOQ for muscle, and 65%, 130 samples out of 199 for liver. The median level calculated in the Kannari study (11 µg.kg -1 fw in contaminated areas) was higher than the median level of 4.1 µg.kg -1 fw found in the present study. Moreover, considering the lower and upper bound hypotheses, the same trend is observed (mean concentrations of 5.8 and 6.2 µg.kg -1 fw versus 0.6 and 4 µg.kg -1 fw in the Kannari study). The differences regarding these figures may be due to the type of animals sampled (not detailed in the Kannari study) and to the sampling strategy that included imported commodities for the purpose of the Kannari plan, whereas only local production was considered in the present work.

Regarding other species, our results in endemically contaminated bovines are similar to those reported in endemically contaminated pigs originating from Guadeloupe (ANSES 2019). In pig skirt sample (n=77), CLD concentrations ranged from 10 to 56 µg.kg -1 fw with a median at 23 µg.kg -1 fw. In three other muscles (longissimus dorsal muscle, semi-membranous muscle and flank) collected on another set of 15 pigs, the concentration range was also similar (11 34 µg.kg -1 fw and median within 18 27 µg.kg -1 fw). These figures document the potential endemic background contamination range of non-artificially fed bovines and pigs bred in Martinique and Guadeloupe. Another study measured CLD concentrations in fat, legs and liver of freerange ducks bred in endemically contaminated areas of Martinique (Jondreville et al., 2014b).

Again, concentration ranges are consistent with those reported in the present study: 47 1,215 µg.kg -1 fw for liver, 9 278 µg.kg -1 fw for fat, and 6 145 µg.kg -1 fw for legs (without skin). In duck tissues, livers were also more highly contaminated than fat and muscle.

The distribution of CLD in fat, muscle and liver has also been reported to document the contamination and/or decontamination kinetics for male Alpine kids [START_REF] Lastel | Comparison of chlordecone and NDL-PCB decontamination dynamics in growing male kids after cessation of oral exposure: Is there a potential to decrease the body levels of these pollutants by dietary supplementation of activated carbon or paraffin oil?[END_REF], Muscovy ducks (Jondreville et al., 2014b) and pigs (ANSES 2019). It was also documented for poultry, where liver and fat were investigated (Jondreville et al., 2014a). The same structure of contamination was observed: liver followed by fat and muscle. Since the above-mentioned studies concerned artificially contaminated animals, the comparison of both CLD levels and occurrence is not relevant.

The contamination of all studied tissues, both occurrence and concentration levels, documents the atypical behaviour of CLD in different species, including bovines. It is distributed in different tissues and does not accumulate mainly in fat. This raises the question of a potential correlation factor between CLD concentrations in tissues.

Statistical correlation between tissues

For the purposes of the correlation study, only quantified results in the three tissues were taken into account, thus limiting the dataset to 96 animals. As mentioned above, the most contaminated triplet (concentrations in fat/muscle/liver: 172/152/812 µg.kg -1 fw) was excluded from data processing.

As mentioned in Table 3, Pearson correlation coefficients were above 0.884, indicating a strong correlation. Moreover, a significance test was performed: values were below 0.01, which supports the strong correlation.

A linear relationship of the observed CLD concentrations was found for muscle/fat, muscle/liver, and liver/fat. The intercept was forced to zero to avoid negative values at low concentrations. The slope of the curves represents the CLD correlation factors between the tissues (Figure 2). The CLD concentration ratios were estimated: 0.54 for muscle/fat, 3.75 for liver/fat, and 0.14 for muscle/liver. The coefficients of correlation (R²) of the model were 0.973, 0.901, and 0.906, respectively. To go further, prediction intervals were also determined, based on the dispersion of the measurements (ANSES 2018.). Depending on a two standard deviations confidence level, it was possible to determine a prediction range of CLD concentration in fat (tissue considered for controls) from a target concentration in skirt or liver (as consumed parts).

The value targeted in skirt was set at 20 µg.kg -1 fw on the basis of national recommendations. For the two standard deviation interval, the CLD concentration in fat ranged from 27 to 47 µg.kg -1 fw: a CLD concentration of 27 µg.kg -1 fw in fat was associated at 97.5%

to a CLD concentration below the target value of 20 µg.kg -1 fw in skirt. For the three standard deviation interval, the range was 32 to 42 µg.kg -1 fw: a CLD concentration of 32 µg.kg -1 fw in fat was associated at 84% to a concentration of CLD below the target value of 20 µg.kg -1 fw in skirt. For liver, it was not possible to determine a range of concentrations of CLD in fat corresponding to 20 µg.kg -1 fw in liver for any of the two confidence levels (95 nor 68%). It was only possible to conclude that a concentration of 20 µg.kg -1 fw in liver, corresponded to a value of 5 µg.kg -1 fw in fat in average. These prediction interval were used to define a CLD mitigation concentration of 27 µg.kg -1 fw in fat for official control plans (Ministerial decree of Mai the 23 rd , 2019), corresponding to a MRL of 20 µg.kg -1 fw in muscle and liver.

As no data on CLD distribution in fat/muscle/liver are reported in the literature for endemically contaminated bovines, these results were compared to those obtained for other species, i.e. The CLD tissue distribution in artificially fed growing kids was also assessed [START_REF] Lastel | Comparison of chlordecone and NDL-PCB decontamination dynamics in growing male kids after cessation of oral exposure: Is there a potential to decrease the body levels of these pollutants by dietary supplementation of activated carbon or paraffin oil?[END_REF]. The observed ratios (liver/fat: 31 during the contamination period, 28 during the decontamination period muscle (skirt) /fat: 0.5 during the contamination period and 0.6 during the decontamination period ) are very different from those estimated in the present bovine study:

the liver/fat ratio is 55 times higher and the skirt muscle/fat ratio is 7 times lower in goats. These differences could be linked to the general tissue composition of the different studied species, including the differences in lipid classes, the overall fat content of the whole animal, as mentioned by Lastel et.al., and the metabolic pathways of CLD, as mentioned above.

Conclusion

An in-house validated ID-LC-MS/MS method was used to analyze CLD in 200 samples of bovine fat, liver and muscle (skirt) originating from contaminated areas of Martinique and Guadeloupe. CLD was mostly quantified in liver (65%), followed by fat (53%), and muscle (48%). These results confirmed that CLD can be distributed in the studied bovine tissues, as previously reported for goats, poultry and ducks. Strong CLD concentration correlations were found between these three tissues, with concentration ratios estimated at 0.54 for muscle/fat, 3.75 for liver/fat, and 0.14 for muscle/liver. These correlations were used to define a CLD mitigation concentration of 27 µg.kg -1 fw in fat for official control plans (Ministerial decree of Mai the 23 rd , 2019), corresponding to an MRL of 20 µg.kg -1 fw in muscle and offal. Moreover, those data might be the support for MRL revision at European level.

It would be interesting to include other types of muscle tissue in a similar study, to take into account their different composition, including fat content. This could reinforce the correlation factors to further support consumer protection through dietary exposure.

  (MRLs) established by the European Regulation on MRLs of pesticides in and on food and feed of plant and animal origin (Commission Regulation (EU) No 212/2013, 212AD ; Commission Regulation (EC) No 839/2008) , provided sufficient protection of the population regarding CLD exposure via food of animal origin (ANSES, 2017).

Figure 1 :

 1 Figure 1: CLD concentrations (µg.kg -1, fat

Figure 2 :

 2 Figure 2: Linear model and confidence intervals on means and observances for muscle/fat and

Figure 1 :Figure 2 :

 12 Figure 1: CLD concentrations (µg.kg -1 fw (a) muscle (n=96) and fat (n=106), and (b) in liver (n=130) for the studied bovines. Boxes represent the concentrations between percentiles 25 and 75. The line dividing the box represents the median concentration. The whiskers below and above the box represent percentiles 10 and 90.

  -1 fw, with success. Compliance of the test concentrations with performance requirements laid down in the SANTE document(SANTE/11945/ 2015(SANTE/11945/ , 2015) ) was checked, and the accuracy profiles were validated. The LOQ is consistent with previous studies[START_REF] Lastel | Comparison of chlordecone and NDL-PCB decontamination dynamics in growing male kids after cessation of oral exposure: Is there a potential to decrease the body levels of these pollutants by dietary supplementation of activated carbon or paraffin oil?[END_REF][START_REF] Saint-Hilaire | Development and validation of an HPLC-MS/MS method with QuEChERS extraction using isotopic dilution to simultaneously analyze chlordecone and chlordecol in animal livers[END_REF] and the European regulation on MRLs of pesticides (Commission Regulation (EU) No 212/2013, 212AD ; Commission Regulation (EC) No 839/2008), which sets the MRL at 100 µg.kg -1 fw for fat, muscle and liver.

  Large White and Creole pigs) contaminated with intravenous administration of CLD (n=7 for each breed). To allow comparison with our data, we calculated a muscle/fat ratio from the published figures in Fourcot et al.: total amount of CLD estimated in muscle or fat (Muscle: 8.9 mg for Creole pigs and 12.5 mg for Large White pigs-Fat: 13.2 mg for Creole pigs and 10.0 for Large White pigs) is divided by muscle or fat percentage of the animals respectively (Muscle: 43.3 % for Creole pigs and 54.8% for Large White pigs. Fat: 34.7% for Creole pigs and 22.4% for Large White pigs). Muscle/fat ratios are similar between bovine (0.54), Creole pigs (0.54) and Large White pigs (0.51).

	pigs (
	Muscovy ducks: 3.9 for liver/abdominal fat, 0.6 for muscle with skin/fat, and 0.5 for muscle
	without skin/fat. The muscle/fat ratio was also modelled with an intercept not forced to zero for
	endemically contaminated pigs from Guadeloupe (n=77 animals), where skirt muscle was
	investigated, and another set of pigs (n=15 animals) where skirt muscle but also three other
	types of muscles were considered (longissimus dorsi muscle, semi-membranous muscle, and
	flank) (ANSES 2019). A muscle/fat ratio of 0.64 was determined, considering the two groups
	of animals, and thus including muscle type variability. This ratio is similar to that modelled in
	the present study, even though animals and muscle types are different.

ducks and pigs. The observed ratios of CLD concentrations in liver/fat and muscle/liver for bovines match those reported by Jondreville et al.

(Jondreville et al., 2014b)

on fresh matter for

Table 1 :

 1 Analytical performances of the method

Table 2 :

 2 Main descriptors of the dataset (n=199 animals)

Table 3 :

 3 Pearson correlation coefficients

					Concentration (µg.kg -1 fw)
		Number of	Number
			of		Lower bound hypothesis	Upper bound hypothesis
		detections a	quantifications b	Maximum
					Mean	Median	Mean	Median
	Fat	117	106	124.6	11.4	4.2	12.2	4.2
	Muscle	117	96	67.6	5.8	0.5	6.2	3.0
	Liver	170	130	420.6	40.0	19.0	40.5	19.0
	a:>LOD
	b: >LOQ

Table 3 :

 3 Pearson correlation coefficients

		Fat	Liver
	Fat	-	0.884
	Muscle	0.974	0.871

Acknowledgments

The authors wish to acknowledge the French Directorate General for Food of the French Ministry for Food, Agriculture and Fisheries for their financial support and for organizing the oriented exploratory survey plan. The authors also acknowledge the ANSES Risk Assessment Department-Methodology and Survey Study Unit for its support for the statistical analysis. The authors are also grateful to Inovalys (Le Mans, France) for its participation in the analyses

List of figures and tables

Funding sources

This work is funded by the French Directorate for food of the French Agriculture ministry (DGAL)

Conflicts of interest

The authors declare no conflict of interest.