Hamza Hafidi Alaoui
email: hamzaalaouihafidi@gmail.com

Samuel Rodriguez
email: samuel.rodriguez@u-bordeaux.fr

Marc Deschamps
email: marc.deschamps@u-bordeaux.fr

Detection of defects in a 2D fluid-solid periodic cluster

Keywords: Nondestructive Testing, Ultrasonic Waves, Topological Imaging, Heterogeneous Medium, Multiple scattering, Periodic Cluster. Bullet Points

The present work deals with locating a defect buried in a medium composed of a fluid matrix and small solid inhomogeneities. Classical imaging methods are based on delay and sum principle and would implicitly assume that the undamaged medium is homogeneous. The topological imaging framework however allows to take into account the heterogeneous nature of the undamaged medium and potentially to take advantage of it. In this work, it is applied to a demanding test case with different assumptions on the knowledge of the medium properties using a specifically-designed fluid-solid compatible imaging function. It leads to the definition of three imaging processes whose results are compared using respectively synthetic and experimental data. The results show the relevance of using the inhomogeneities' locations information, but not necessarily at all steps of the imaging process, leading to the definition of so called hybrid topological imaging method.

Introduction

The detection, localization, and monitoring of the evolution of defects in diffusive media have received a lot of attention because of the underneath theoretical interest and the large number of industrial applications. Wave propagation in such media is complex and the challenge is that the signature of the defect is hidden in the acoustic field reflected by the structure itself independently of the defect. In the present work, only active ultrasonic inspection is considered. The principle of active inspection is summarized in the following steps: one or several source(s) emit(s) a wave into the medium; it is scattered by the inhomogeneities and the resulting wave is measured by the same or other sensor(s). The measurement contains single scattering and multiple scattering contributions. Several methods have been implemented to process the information contained in the backscattered waves in order to characterize or image the medium. In a globally homogeneous one-polarisation non-dispersive isotropic infinite or semi-infinite medium, the presence of a single or a few inhomogeneities modify the wave field in a manner that multiple scattering contribution is small in comparison to single scattering. In this case, we consider that the incident wave undergoes only one scattering event before propagating to the sensor(s). This is the contribution used in conventional imaging techniques, that rely on the direct relation between the arrival time t of the echo and the distance d between the sensors and the scatterers, t = 2d/c (c is the sound velocity).

These analytical models of propagation allow the definition of efficient imaging methods based on the delay and sum principle such as Synthetic Aperture Focusing Technique [START_REF] Busse | Three-Dimensional Imaging Using a Frequency-Domain Synthetic Aperture Focusing Technique[END_REF] and Total Focusing method [START_REF] Holmes | Post-processing of the full matrix of ultrasonic transmit â receive array data for non-destructive evaluation[END_REF]. In more complex media, these methods require some improvements. Dispersion can be compensated for [START_REF] Sicard | A SAFT algorithm for lamb wave imaging of isotropic plate-like structures[END_REF] whereas anisotropy requires more complex processing [START_REF] Spies | Synthetic aperture focusing for defect reconstruction in anisotropic media[END_REF]. In the presence of several simultaneous wave polarizations, an experimental 2D sampling of the medium allows discriminating them [START_REF] Leleux | Long range detection of defects in composite plates using lamb waves generated and detected by ultrasonic phased array probes[END_REF]. This step is required as each polarization corresponds to a different wave velocity.

In reverberating or diffusive media, the conventional imaging methods based on the delay and sum principle fail to detect or to locate the defects because the single scattering assumption is not valid as multiple scattering contribution is dominant. In random heterogeneous media, like concrete for instance, the waves are scattered multiple times by heterogeneities. The challenge in inspecting such media is to deal with all the effects of the different objects in the wave propagation model. The issue of characterization and imaging of random or even heterogeneous media has received special attention in the three past decades. The methods developed for this purpose can be classified into two distinct categories: imaging methods without the prior knowledge of the local properties of the heterogeneous medium, and others that use a partial or complete scattering model obtained from the local properties of the medium. The methods presented in the present work belong to the second category. Short reviews of both categories are respectively given in the next subsections.

Imaging methods for unknown heterogeneous media

The diffusion model is one of the most used models to describe the wave propagation phenomena in unknown heterogeneous media because the energy of the waves is transported in a way which is similar to the diffusion of heat. G. Bal and O. Pinaud [START_REF] Bal | Time-reversal-based detection in random media[END_REF] use the macroscopic model of the diffusion equation to detect an inclusion buried in highly heterogeneous random media. The imaging of the medium is based on the analysis of the fluctuations of the diffusion coefficient (estimated from energy measurements) caused by the buried inclusions. The method is illustrated by numerical simulations in two-dimensional space. Most of the results presented in this article have been generalized to other macroscopic models such as the radiative transfer equation [START_REF] Bal | Experimental validation of a transport-based imaging method in highly scattering environments[END_REF]. However, this approach needs heavy numerical computations. Moreover, the final resolution of the image is limited by the transport mean free path of the scattering medium instead of half the wavelength. A.

Aubry and A. Derode present another method to solve the same problem. Using a transducer array, the impulse responses between all array elements are measured and form a matrix. A filtration operation is then applied to this matrix in order to separate the single scattering contribution of the inclusion from the multiple scattering contribution. The detection of the target is achieved applying the DORT method [START_REF] Prada | Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media[END_REF] (french acronym for the decomposition of the time reversal operator) to the filtered matrix. Finally, the random matrix theory is used to give a detection criterion. This method is successfully applied to detect and image defects in highly scattering experimental media [START_REF] Aubry | Detection and imaging in a random medium: A matrix method to overcome multiple scattering and aberration[END_REF], and weakly scattering media [START_REF] Aubry | Multiple scattering of ultrasound in weakly inhomogeneous media: Application to human soft tissues[END_REF]. It is shown that the quality of detection is significantly better than what is obtained with classical imaging methods, but this technique is limited to the detection of a strong scatterer and is not sensitive to weak perturbations. In many practical applications such as monitoring volcanoes and bioacoustical oceanography, the medium changes over time. M. Cowman et al. [START_REF] Cowan | Diffusing acoustic wave spectroscopy[END_REF] adapted the diffusing wave spectroscopy technique [START_REF] Pine | Diffusing wave spectroscopy[END_REF] to acoustic waves in order to relate the temporal field fluctuations with the dynamics of the multiple scattering medium. J. D. Rosny and P. Roux [START_REF] De Rosny | Multiple scattering in a reflecting cavity: Application to fish counting in a tank[END_REF] used the multiple scattering theory to measure the density and the scattering strength of moving scatterers inside a highly reverberating cavity. They apply this theory to fish counting in a tank, but it can also be used to estimate its volume, the sound velocity, the absorption cross-sections, and other properties. A few months later, a parallel is established between these results and the diffusing acoustic wave spectroscopy [START_REF] Demer | Absolute measurements of total target strength from reverberation in a cavity[END_REF][START_REF] De Rosny | Field fluctuation spectroscopy in a reverberant cavity with moving scatterers[END_REF]. Excellent agreement is found between experiment and theory for steels balls moving in a water tank. Failing to accurately model the wave propagation in the random heterogeneous medium, some methods use it as an interferometer.

The idea is to compare the response of the inspected medium before and after a change has been introduced. Thereby, the major part of the incoherent field is eliminated. Nevertheless, the residual coherent field can lead to detect, locate, or characterize a perturbation [START_REF] Feng | Acoustical nondestructive evaluation of heterogeneous materials in the multiple scattering regime[END_REF]. This technique can be useful in warning mode, or in the detection of temporal changes in the medium. Coda wave interferometry is an example of this type of method. It was used to estimate the variations in the positions of small changes in granite from multiply scattered waves by cross-correlating perturbed and unperturbed field [START_REF] Snieder | Coda wave interferometry for estimating nonlinear behavior in seismic velocity[END_REF]. Other studies use this method to detect larger changes [START_REF] Lobkis | Coda-wave interferometry in finite solids: Recovery of p-to-s conversion rates in an elas-todynamic billiard[END_REF] or to determine the relative location of earthquakes [START_REF] Snieder | Constraining the source separation with coda wave interferometry: Theory and application to earthquake doublets in the hayward fault, california[END_REF]. Despite their noisy and chaotic appearance, coda waves are highly repeatable. If a perturbation occurs, even if very small, an observable change will be produced in the coda waves. Thereby, the coda wave interferometry is able to determine the average motion of scatterers or change in the propagation velocity from the temporal change of multiply scattered waves. However, it cannot determine the exact location or the spatial extent of the change in the medium [START_REF] Pacheco | Time-lapse travel time change of multiply scattered acoustic waves[END_REF]. Inspired by coda wave interferometry C. Pacheco, and R. Snieder [START_REF] Pacheco | Time-lapse travel time change of multiply scattered acoustic waves[END_REF] use it in conjunction with a diffusive model to obtain an expression that relates the mean travel time change of multiply scattered waves with the localized perturbation in the propagation velocity of the medium. E. Larose et al. [START_REF] Larose | Locating a small change in a multiple scattering environment[END_REF] use the coherent field part to formulate an inverse problem which is solved by a maximum likelihood approach in order to image and locate a weak perturbation in concrete. S. T. Rakotonarivo et al. [START_REF] Rakotonarivo | Localization of a small change in a multiple scattering environment without modeling of the actual medium[END_REF] backpropagate the coherent field in a homogeneous reference medium in order to focus it on the position of the defect. In the latter application, the medium background is known. Only the heterogeneities nature and localization are unknown. These two studies are supported by conclusive experiments. However, the target must be approximately the same size as the heterogeneities of the medium in order to be correctly imaged.

Imaging methods for known heterogeneous media

In the situations where all geometrical and physical properties of the inspected medium are known, several methods allows performing NDT. Time-reversal exploits the reciprocity of wave equations to refocus an incident wavefield, measured by an array of transmit-receive transducers, to the location of the original source, regardless of the complexity of the propagation medium. Therefore, NDT is still possible, even with very dense cluttering as long as the full Green function is known [START_REF] Derode | Robust acoustic time reversal with high-order multiple scattering[END_REF]. The use of time reversal in imaging of cluttered media is also made in [START_REF] Dubois | Localization and characterization of two-dimensional targets buried in a cluttered environment[END_REF][START_REF] Haider | Time-reversal simulations for detection in randomly layered media[END_REF] and in [START_REF] Liu | Electromagnetic target detection in uncertain media: Time-reversal and minimumvariance algorithms[END_REF] for electromagnetic waves applications. The sensitivity of the refocusing properties of the timereversed waves to the Green function accuracy is studied in [START_REF] Vigoa | Robustness of time reversal for waves in timedependent random media[END_REF][START_REF] Bal | Time reversal in changing environments[END_REF]. Usually, sufficiently accurate knowledge of the media is out of reach in many applications. Borcea et al. [START_REF] Borcea | Imaging and time reversal in random media[END_REF] use the principle of time-reversal in order to estimate the location of small scatterers in a randomly heterogeneous medium. The resulting imaging methods are based on the computation of an objective functional whose maxima are an estimation of the location of the scattering objects. This method is efficient if the number of inclusions is less than the number of transducers composing the array. The more the inclusions are separated, the more accurate the localization is. This approach leads to imaging methods such as MUSIC [START_REF] Devaney | Time reversal imaging of obscured targets from multistatic data[END_REF] and SAT [START_REF] Borcea | Imaging and time reversal in random media[END_REF]. Topological optimization was first developed to solve static structural inverse problems. It was then transposed to acoustics [START_REF] Samet | The Topological Asymptotic for the Helmholtz Equation[END_REF] and elastodynamics [START_REF] Bonnet | Sounding of finite solid bodies by way of topological derivative[END_REF]. The parameter to be optimized is the location of an object in the structure. This object can be penetrable or a Dirichlet or a Neumann border inside the medium. Similarly to the Full WaveForm Inversion [START_REF] Virieux | An overview of full-waveform inversion in exploration geophysics[END_REF] where the variables to be optimized are continuous parameters of the medium (wave speed, density...), addressing the inverse problem implies the definition of a cost function and the study of its sensitivity to the optimized parameters. In topological optimization, this sensitivity is also referred as the topological gradient. It is a space dependent variable that takes high magnitude negative values where the insertion of the object will tend to strongly decrease the cost function. When looking for small Dirichlet or Neumann objects in a medium, simple analytical formulations are obtained as functions of two wavefields that are to be computed in the defect-free reference medium [START_REF] Guzina | Topological derivative for the inverse scattering of elastic waves[END_REF]. Similar formulas are also derived in Full Wave inversion when optimizing the inverse of the squared wave velocity [START_REF] Goncharsky | Inverse problems of 3D ultrasonic tomography with complete and incomplete range data[END_REF]. Using these observations, the topological gradient (or one of its derivative) can be used as the imaging function [START_REF] Dominguez | Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection[END_REF] and is obtained with two numerical computations performed in the defect-free medium. This corresponds to the so called topological imaging. Topological Imaging (TI) is thus a non-iterative process that is able to take into account the complexity of a medium through those two numerical simulations whose boundary conditions depend on the experimental excitation and measurements respectively. It has been applied to strongly dispersive waves [START_REF] Rodriguez | Guided wave topological imaging of isotropic plates[END_REF] as well as to reverberating [START_REF] Rodriguez | One channel defect imaging in a reverberating medium[END_REF] and bounded media [START_REF] Tokmashev | Experimental validation of the topological sensitivity approach to elastic-wave imaging[END_REF][START_REF] Lubeigt | Topological imaging in bounded elastic media[END_REF]. It is shown with numerical experiments that defects are localized in anisotropic homogeneous [START_REF] Rodriguez | Topological Imaging of Defects in Anisotropic Plates[END_REF] and anisotropic heterogeneous media [START_REF] Metwally | Weld inspection by focused adjoint method[END_REF]. Bellis and Bonnet also showed that cracks and their orientation could be retrieved in shells using the topological optimization approach [START_REF] Bellis | Qualitative identification of cracks using 3D transient elastodynamic topological derivative: formulation and FE implementation[END_REF].

Outline

The inspection of heterogeneous media covers a wide application field. When the effect of the heterogeneity on the acoustic field is small in comparison to the effect of the defect or easy to distinguish from, classical delay and sum methods allow appropriate imaging. Still, there are cases where neglecting the heterogeneity lead to strong artifacts or prevent from defect detection or medium characterization. For instance, weld inspection [START_REF] Metwally | Weld inspection by focused adjoint method[END_REF] or multi-material stratified media require the heterogeneity to be taken into account for accurate defect location. In vivo bone characterization [START_REF] Renaud | In vivo ultrasound imaging of the bone cortex[END_REF] is another example where taking the heterogeneity into account allows previously unreleased medium characterization. In the present paper, a more academic example is chosen in order to assess the approach validity with a demanding test case, to compare the effect of choosing different reference media and to propose an imaging function compatible with fluid-solid media. The authors believe this could also be a first step toward phononic crystal nondestructive evaluation.

The work focuses on Nondestructive Testing (NDT) applications through active ultrasonic imaging processes. Our aim is to apply the TI method to a known heterogeneous fluid-solid medium whose heterogeneities are arranged in an even manner. The defect to be localized is not an object but the absence of one of the heterogeneities. The paper is divided into four section. Section 2 explains the principle of TI and the way it is here applied to a known heterogeneous medium. The specimen under study and the numerical approach to model the acoustic wave propagation are described in detail. Depending on how accurately the medium is known, three different imaging methods are proposed. In section 3, the experimental apparatus and the imaging results obtained with synthetic and with experimental data respectively are presented and analyzed. Finally, applications and limitations of the techniques are discussed in section 4.

Applying topological imaging to a known heterogeneous medium

The topological imaging framework

The topological imaging process consists first in an active acoustic investigation of the medium, then in computing two specific wavefields and finally in computing the imaging from both wavefields. The wavefields are computed in a reference medium that includes all the knowledge of the medium, except the possible defects. Some known acoustic sources generate waves propagating and interacting with the medium and some known sensors measure the acoustic response. This corresponds to the step described in Fig. 1a. Then, this response is simulated in the reference medium, i.e. in the absence of defect (Fig. 1b). Finally, the residue is obtained as the difference between the two (Fig. 1c). This corresponds to the acoustic signature of the defect.

The two required numerical wavefields correspond to the solutions of the direct and adjoint prob- Second, the whole space-and time-or frequency-dependent wavefield is stored to be used in the image computation. The adjoint problem consists in simulating the backpropagation of the residue from the measuring sensors in the same reference medium as the one used in the direct problem.

The image is then obtained by computing a zero-lag convolution of direct and adjoint wavefields (Fig. 2).

It has to be emphasized that residue computation may be difficult when the wave response of the reference medium is complex. In the topological optimization approach, the residue is defined as the difference between a measured signal and a numerical simulation. This implies a quantitative measurement which is not easy when using for instance piezoelectric sensors. An alternative when available is to make a measurement before damaging the medium. It corresponds to the defect-free medium response and is used in residue computation to replace simulation [START_REF] Rodriguez | One channel defect imaging in a reverberating medium[END_REF].

The specimen under study

The specimen under study is a fluid-solid periodic cluster. This medium is made up of 275 steel rods immersed in water. The steel rods are arranged in an even manner forming a rectangle of 11 rows and 25 columns. The inter-row space and the inter-column space are 9.6 and 11.9 mm respectively. As the diameter of each rod is one millimeter, the surface concentration of the solid inclusions in the water is C 0.7 %. The elastic properties of steel are given by:

Definition of the reference medium and the three corresponding imaging methods

Defining the reference medium is an important step of the method as it embeds all the knowledge of the medium that we want to be taken into account in the final image. In the present application, the natural choice is to simulate the steel bar cluster immersed in water assuming the locations of the bars and of the transducers are known. Thus, the reference medium is made of steel bars immersed in water and the transducers are facing the cluster according to the experiments. It correspond to what is here defined as heterogeneous TI.

Nevertheless, the case where the steel bars are not taken into account is also studied. In this second case, the reference medium is simply a semi-infinite water medium. This medium is of interest as it corresponds to the one implicitly considered when applying classical delay and sum methods. Delay and sum and topological methods have some physics in common. It is shown in the Appendix that assuming a homogeneous infinite and single-polarization reference medium, applying topological imaging allows retrieving the classical delay and sum method imaging function.

This case corresponds to what is here defined as homogeneous TI.

It has to be reminded that the reference medium is used for both direct and adjoint field and that direct field is used for both residue and image computation. That's why an alternative method is also studied. It consists in taking a reference medium for residue computation and another one for direct and adjoint fields that are necessary in image computation. In the present case, the hybrid method consists in measuring the experimental residue by substracting the measure obtained in the experimental defect free medium with that obtained in the real medium. Thus, the heterogeneity of the medium is fully taken into account in the residue computation. Hence, the reference medium is the heterogeneous medium. Then the experimental information (transducer excitation and residue) is used to compute direct and adjoint fields in a homogeneous reference medium. It has to be noted that this method is similar to that described by Rakotonarivo et al. [START_REF] Rakotonarivo | Localization of a small change in a multiple scattering environment without modeling of the actual medium[END_REF]. To sum up, three methods are here tested: the heterogeneous, the homogeneous and the hybrid TI methods (see Table 1).

Reference Medium Caption for residue for direct & adjoint fields

Heterogeneous TI Heterogenous Heterogeneous

Residue Dir./Adj.

Homogeneous TI Homogeneous Homogeneous

Residue Dir./Adj.

Hybrid TI Heterogenous Homogeneous

Residue Dir./Adj.

Table 1: The imaging methods compared in this study. This table specifies the reference media used to obtain the images for each method.

Numerical modelling

The experiments are performed in three-dimensional space at a central frequency of 500 kHz.

The steel rods are all oriented along the x 3 axis and so is the length of the transducers. All have large dimensions in comparison to the wavelength in water. Thus the propagation is assumed twodimensional. The propagation space is noted (x 1 , x 2 ) whereas the third dimension is noted x 3 (Fig. 3). The plane strain assumption is applied along x 3 . Numerical simulations are performed using frequency-domain finite elements in the 200 to 800 kHz bandwidth with the COMSOL Multiphysics software 2 . Plane elasticity equations model the wave propagation in the fluid-solid periodic cluster.

The scalar wave equation is used in water and the vectorial equations are applied in the steel rod sections. The continuity of normal displacements and normal stresses at the fluid/solid interfaces is managed by the Multiphysics module of COMSOL software. The excitation source corresponding to each transducer is modelled with normal velocity conditions applied at the boundary of the medium. The semi-infinite nature of the medium is modeled by absorbing regions similar to those defined in [START_REF] Ke | 3D finite element simulations of an air-coupled ultrasonic NDT system[END_REF]. The absorbing region is defined as a region where the imaginary part of wave velocity is progressively increasing in order to generate progressively increasing absorption.

Simultaneously, the density is modified so that the acoustic impedance is constant to achieve minimal reflection. This method is especially suited to solid anisotropic media where PMLs are harder to apply. Still it is also efficient in fluids. In this application, the thickness of the absorbing region is three times the largest wavelength which is about 7.4 mm. The same mesh is used for all computation frequencies. It is clear to the authors that this is not optimized but it eases the processing and avoids re-meshing at each frequency. The mesh is composed of quadratic triangular elements carrying second-order polynomial interpolation functions. The minimal size of an element is taken as a fourth of the shortest wavelength. This would correspond to the eight of the shortest wavelength using linear interpolation functions. The shortest wavelength is obtained at 800 kHz.

Thus, at the central frequency of 500 kHz, the mesh elements are smaller than a sixth of the wavelength. This compromise was found to ensure low numerical dispersion.

Figure 3: Numerical model of the wave propagation in the investigation environment. The ultrasonic probe is represented by the bold broken line at the bottom of the rectangle. On the one hand, the light gray rectangle represents water. On the other hand, the dark gray rectangle represents the artificial absorbent layer. The black discs represent the steel rods composing the periodic cluster.

The target is buried deep in the periodic cluster, located at the intersection of 13th column and 10th row.

The imaging function

Following the topological imaging approach, the imaging function is defined as the norm product between direct and adjoint wavefields integrated over the frequency domain. It can also be obtained in the time-domain by taking the spatial envelope of the zero-lag convolution between both fields [START_REF] Rodriguez | Guided wave topological imaging of isotropic plates[END_REF]. The specificity of present application is that the medium is made of fluid and solid. On the one hand, the numerical degree of freedom is the hydrostatic pressure P f . On the other hand, it is the displacement vector u s . The common physical variable chosen to compute the image is the isostatic pressure noted P . It is expressed as:

P =    P f (x) if x ∈ fluid -1 2 Tr (σ s (x)) if x ∈ solid ( 1
)
where Tr is the trace function and σ s (x) is the 2D stress tensor. Thus, the image is simply given by:

I(x) = R + P d (x, ω)P a (x, ω)dω (2)
where P d (x, ω) and P a (x, ω) are the isostatic pressure fields in direct and adjoint wavefields respectively. This definition is used for both pure fluid and fluid-solid reference media.

Theoretical and experimental results

This section is dedicated to the images of the test-specimen obtained with the three different methods of Table 1. In what follows, the target is not an object but the absence of the rod located in the 13th column and the 10th row of the cluster (Fig. 3). First, the experimental setup is described, then the results obtained with synthetic and experimental data are presented for each imaging method listed in Table 1.

The experimental apparatus

The experimental apparatus is presented Fig. 4. The cubic structure presented in the front view (Fig. 4a) is composed of two horizontal thin steel plates. The plates are identical and perforated according to a known periodic geometric pattern in order to hold the steel bars vertically. Each of them is clamped between two thick PVC plates making the setup stiff and stable. To inspect the medium in a pulse-echo configuration, an Imasonic ultrasonic transducer array is placed in front of the medium of investigation, at a distance a=37 mm. In addition, the probe is centered relative to the periodic cluster of steel rods and is parallel to its first row similarly to Fig. 3. The ultrasonic array is composed of 128 programmable transmitters/receivers with a 500 ± 10 KHz central frequency, the acoustic adaptation layer mechanical impedance is 3.2 MRayl. Each array element is 2 mm wide and the array pitch p is 2.5 mm, which makes a total active length of 319.5 mm. The transducer array is connected to a multi-channel Lecoeur Electronique measurement system. This device is used at a 10 MHz sampling frequency. Note that the experiments take place in a water tank as shown in the perspective view (Fig. 4b). (3) The perforated steel plates. ( 4) The PVC plates. ( 5) The water tank.

For each measure, the medium is insonified once by firing simultaneously all transducers with the same three-period windowed sinusoidal signal in order to generate a quasi-plane wave. This choice is not a restriction, any known excitation signal can be used. The residue that compares the response of the unperturbed medium with the response of the perturbed medium requires two acquisitions.

The first one is the response of the medium with all the rods of the periodic cluster. The second one is the response of the medium after removing the rod. A typical experimental waveform is plotted to the electrical excitation signal. In the simulations the piezoelectric behaviour of the transducers is here not simulated. Thus, the normal velocity signal applied to the boundary of the medium (Fig. 3) should be proportional to the generated pressure signal. In order to identify this excitation signal, a calibration of the physical experience is performed so that experimental and numerical simulation are in accordance. The excitation signal obtained is presented Fig. 6.

Results

The imaging methods presented in Table 1 are tested using successively synthetic and experimental data. The difference lies in the way the measured signals are obtained, either with simulations or with experiments respectively. The experimental data are obtained using the ultrasonic acquisitions of the real medium, in the absence and in the presence of the defect. This data is then used to calculate the residue and to compute the adjoint problem associated with each method. The synthetic data are obtained by computing the propagation of acoustic waves in the virtual defective medium and in the virtual defect-free medium. Then, the residue is used to compute the adjoint problem according to the method studied. The direct and the adjoint fields are computed in the frequency domain according to the numerical model introduced in paragraph 2.4. The results obtained with the three methods described in Table 1 are successively presented.

Heterogeneous TI results

In heterogeneous TI, the heterogeneous reference medium is used to calculate the residue, the direct, and the adjoint fields. Therefore, the residue contains the ballistic wave corresponding to the defect (here, the absence of a rod) and the corresponding multiple interactions with the other inclusions of the periodic cluster. On the one hand, the direct problem corresponds to the simulation of the physical experiment in the defect-free medium (the one containing all the rods).

On the other hand, the adjoint problem corresponds to the backpropagation of the residue from the measurement surface in the defect-free medium. The imaging function is normalized. Thus it is defined by I(x)/max(I(x)) where I(x) is given by Eq. ( 2). The images obtained with synthetic and experimental data are presented Fig. 7(a) and (b) respectively. The defect appears as a single point.

Artefacts are small so that no wrong defect detection is likely to occur. In the zoom dial, the white circle indicates the true edges of the missing rod (the defect) in the medium. The spot maxima are located at the lower edge of the defect. This can be explained by the single insonification coming from the bottom that implies that the lower edge of the defect is more insonificated than the rest of the defect. In both cases, the defect is accurately located and no artefacts could indicate a defect where there is none.

The -6dB width of the spots is about 1.7 and 1.8 mm for the synthetic and the experimental data cases respectively. These values correspond to 0.57λ 0 ans 0.6λ 0 where λ 0 is the wavelength in water at the central frequency 500 kHz. Such a high resolution is achieved with a single insonification.

This can be explained by the fact that topological imaging takes the complex nature of the medium into account. It thus takes advantage of the enhanced refocusing properties of the time reversal operation in such media when computing the adjoint field.

The spot of the experimental data based image is a little off the defect and a secondary spot is present. They can be explained by a small transducer array location estimation in the experiments. Experimental data. The real defect is delimited by the white hollow circle.

Homogeneous TI results

In homogeneous TI, the heterogeneity of the medium is taken into account neither in the residue computation, nor in the direct and adjoint fields computations. The direct problem corresponds to the propagation of a quasi-plane wave propagating in a homogeneous semi-infinite medium.

The adjoint problem corresponds to the backpropagation of the pressure field measured in the experiments into the same homogeneous semi-infinite reference medium. The imaging function is given by I(x)/max(I(x)). The images obtained with synthetic and experimental data are presented Fig. 8(a) and (b) respectively. As the rods are not in the reference medium, they all corresponds to perturbations and should thus appear as defects in the image. In practice, resolution and contrast decrease with the investigation depth so that the rods of first six rows can be identified but deeper, the image is poorly defined. In the image obtained with synthetic data (Fig. 8(a)) the last three rows are too blurry to make any sure conclusion about the presence or the absence of the rods. In the image obtained with experimental data (Fig. 8(b)), the missing rod can be better identified but wrong missing rod detection occur elsewhere. It is shown in appendix 4 that in this case, topological imaging is similar to classical delay and sum methods. As a conclusion, when not taking the heterogeneity of the medium in the reference model, some defect may thus be missed or detected even when not existing.

It should be noticed that the image obtained with the synthetic data (Fig. 8(a)) looks nosier than that obtained with experimental data (Fig. 8(b)). This can be explained by a discrepancy between synthetic and experimental apparent attenuation when measuring the acoustic field reflected by the rods. As a matter of fact, experimental signals are attenuated in a shorter time than synthetic signals. That means that synthetic data contain more multiple interactions between the rods themselves and between the rods and the transducer boundary than the experimental data. Thus, when considering a reference medium that is too different from the one investigated, the artifacts are stronger. This explains the nosier look of the image when using the homogeneous reference medium.

The discrepancy between synthetic and experimental apparent attenuation can be explained by several phenomena. First, the transducer array is neither generating a perfect plane wave nor a perfectly collimated wave field and the induced diffraction in the x 3 axis is not taken into account in the 2D-model used to generate the synthetic data. A second possibility is that despite our efforts, the rods may not be perfectly normal to the incident wave field. This could imply some more diffraction along the x 3 axis. Experimental data. The real defect is delimited by the white hollow circle.

Hybrid TI results

The implementation of this hybrid TI method requires two distinct reference media. The first reference medium used to calculate the residue is heterogeneous. The second reference medium in which the direct and adjoint fields are calculated is homogeneous. The spot in the images indicate accurately the location of the defect and no misinterpretation is possible like in homogeneous TI.

When processing the synthetic data, the defect and the corresponding spot are superimposed in the image. When processing the experimental data, the spot is shifted 1.1 mm to the right and 0.8 mm to the front. We interpret this error as an array location estimation error. We consider it as small in comparison to the wavelength in water at the central frequency λ 0 = 3 mm and to the active size of the array (320mm). The -6dB width of the spot is respectively 2.5 mm and

Discussions and conclusions

The same heterogeneous medium is studied following three different methods, all deriving from the topological imaging approach. The defect to be located is here the absence of one of the elements constituting the periodic cluster. The topological imaging framework allows the heterogeneous complex nature of the defect-free medium to be taken into account in the imaging process. This corresponds to the so called heterogeneous topological imaging. It allows an accurate location of the defect both when applying it to synthetic or to experimental data. When ignoring the heteroeneous nature of the medium, the reference medium used to build the image is homogeneous. Thus, all cluster elements should appear as defects in the image. In practise, the first rows are accurately described but the resolution and contrast decrease with the investigation depth so that the missing element is hardly located and wrong defect detection may occur. It is demonstrated that the so called homogeneous topological imaging is in fact similar to classical delay and sum methods. In the hybrid topological imaging method, the heterogeneous nature of the defect-free medium is only required for the residue measurement. It allows accurate location of the defect at the cost of a degraded resolution in comparison with the heterogeneous TI. It has to be noted that hybrid TI is in fact similar to the method presented in [START_REF] Rakotonarivo | Localization of a small change in a multiple scattering environment without modeling of the actual medium[END_REF]. The advantage is that the wavefields required to obtain the image are computed in a homogeneous medium which allows fast computation methods. Still, in both, hybrid and heterogeneous TI, experimental residue is in practise required which implies that the experimental acoustic response of the defect-free medium has to be known.

The medium investigated here present a small inclusion concentration. This work is a first step towards more concentrated media investigation in order to prepare phononic crystal nondestructive testing.

us apply the topological imaging method to this medium. The transducer network consists of N transducers located at t i on the x 1 axis, where 1 < i < N (Fig. 10). The emitted and received signals are noted e(t) and s i (t) respectively. In the topological framework, the residue is defined as the difference between responses obtained in the defect free medium and that in the presence of possible defects. In a homogeneous infinite medium, no signal is measured in the absence of defect as waves propagate to infinity. Thus, in that case, residue signals are simply the measured signal.

In that context the infinite medium hypothesis simply implies that the boundaries of the domain are sufficiently far away regarding the acquisition duration. Let h(t e , x, t) be the impulse response between a source located at t e and any given point x. Depending on the type of insonification the distance d f traveled by the wave to reach point x is given by:

1. d f = xt e if a single transducer is used to insonify the medium.

2. d f = n.x if a plane wave is used (given by the direction n). The direction of the wave front is typically obtained applying appropriate delay laws to e(t). The imaging function of topologcial imaging is given by the spatial envelope of the function I(x) written as:

I(x) = ∞ -∞ u(x, t)v(x, -t)dt, (3)
where u and v are the solutions of the direct and the adjoint problems respectively. According to the type of insonification, the direct field takes one of the following two forms:

The adjoint field is given by:

v(x, t) = Nr i=1 [s * i * h (t i , x, •)] (t), (6)
where s * i (t) = s i (-t) is the time reversed signal. In the hypothesis of ray propagation, the impulse response is approximated by: h(t e , x, t) = δ t -x-te c regardless of the size or shape of the source. Therefore, the fields u and v are respectively expressed as follows:

u(x, t) = e * δ • - d f c (t) = e t - d f c , (7)
v(x, t) = Nr i=1

s * i * δ • - x -t i c (t). (8)
The time reversed adjoint field is thus given by: v(x, -t) = Nr i=1

s i t + x -t i c . (9)
Inserting Eq. ( 7) and Eq. ( 9) in Eq. ( 3) leads to:

I(x) = u(x,
s i τ + d f + x -t i c dτ, (10) 22
where τ = t -d f c . Assuming that the emission signal is a Dirac impulse e(t) = δ(t), the function I is given by:

I(x) = Nr i=1 s i d f + x -t i c , (11)
As a matter of fact, Env(I(x)) corresponds to the definition of the imaging function of the delay and sum method [START_REF] Holmes | Post-processing of the full matrix of ultrasonic transmit â receive array data for non-destructive evaluation[END_REF][START_REF] Ihn | Pitch-catch Active Sensing Methods in Structural Health Monitoring for Aircraft Structures[END_REF]. This result can easily be generalized to any excitation with several emitters located at x j and emitting with relative delays noted β j . In this case, the imaging function is given by:

I(x) = i,j s i x -t j + x -t i c + β j . (12)
Unlike conventional imaging methods, IT involves separate processing of transmitted and measured signals. Thus, the explicit computations of the distances between the emitters, the potential defect and the receiver are not carried out. They are implicitly taken into account in the two computation of direct and adjoint fields.

To sum up, assuming ray-tracing wave propagation, Dirac-like emission signals, the delay and sum imaging functions are retrieved from the topological imaging framework. Let us recall that delay and sum imaging function validity is limited to medium that are homogeneous and that can be considered infinite or semi-infinite (it is to say whose boundaries are far enough regarding the signal acquisition duration). Thus, topological imaging framework offers more general imaging heuristics as it can also be applied to bounded and heterogeneous media.

Figure 1 :

 1 Figure 1: The active ultrasonic inspection for residue computation. (a) Experiments: acquisition of the response of the perturbed medium S exp (t). (b) Simulation: computation of the response of the reference medium at the transducer location U sim (t). (c) Residue: difference between experimental and simulated signals r(t).

Figure 2 :

 2 Figure 2: Formation of the image with topological imaging method. (a) Computation of the direct field. (b) Computation of the adjoint field. (c) Image computation.

E = 210.10 9

 9 Pa for Young's modulus, ρ steel = 7900 kg/m 3 for density and ν = 0.285 for the Poisson coefficient. That corresponds to a longitudinal wave velocity c L = 5902 m/s and a transversal wave velocity c T = 3236 m/s. During the physical experiment, the water temperature was 19.8 • C, which implies that the velocity of the longitudinal waves is c water = 1477 m/s and that the density is ρ water = 998 kg/m 3 .

Figure 4 :

 4 Figure 4: The experimental setup and the specimen under study before immersion. (a) Front view. (b) Perspective view . (1) The ultrasonic transducers array. (2) The periodic cluster of steel rods.

Fig. 5 .Figure 5 :

 55 Fig.5. The backscattered wave is complex, as it combines the simple scattering contribution and the multiple scattering contribution. First 11 wave packets corresponding to the rows of the periodic cluster can be distinguished inside rectangle 2. The signal part in rectangle 3 is the superimposition of late arrival partial waves. The later the wave arrives, the more multiple scattering it underwent and the more it explored the medium. Rectangle 1 corresponds to the saturation of the receiving channel during emission as it is also used to generate the acoustic wave. In practice, the electrical excitation signal is distorted at the output of the transducers as the piezoelectric transducers have their own transfer function. Consequently, the acoustic wave generated does not correspond exactly

Figure 6 :

 6 Figure 6: The experimental excitation signal

Figure 7 :

 7 Figure 7:

Figure 8 :

 8 Figure 8:

3. 5

 5 mm for the synthetic data and the experimental data image. They corresponds to 0.83λ 0 and to 1.16λ 0 respectively which are very reasonable values for a one-insonification image. The spots are clearly wider than those in the heterogeneous TI images. Taking into account the complexity of the medium in the Heterogeneous TI gives better focusing in the backpropagation and thus a better resolution in the image. This observation corroborates the results demonstrated in[START_REF] Derode | Robust acoustic time reversal with high-order multiple scattering[END_REF].

Figure 9 :

 9 Figure 9:

Figure 10 :

 10 Figure 10: Schematic representation of medium and excitation types. The red line represents the wave front generated by all the transducers and given by the direction n. The blue line represents the excitation by a single transducer located at t e .

  1. A single emitter located at t e :u(x, t) = [e * h (t e , x, •)] (t),(4) 2.A plane wave front propagating in direction n:

u(x, t) = e * δ • -n.x c (t).

http://www.comsol.com

Acknowledgments

This work was funded by the french Ministry of Higher Education, Research and Innovation.

Appendix: Equivalency between Delay and Sum methods and homogeneous Topological imaging in infinite media

In this appendix, it is demonstrated that the delay and sum imaging function can be retrieved from the application of topological imaging method to an homogeneous infinite medium. For pedagogical purposes, the calculation is performed in the time domain. It could also be done in the frequency domain.

Let us consider a homogeneous infinite fluid medium characterized by the wave velocity c and let


Jean-François Lemaître

Jeffrey Lemaître

Benjamin Carbillet

Rupert Rey

Hannah Palme

Rachael V Froy

Sarah L Wilbourn

Louise Underwood

Jean-Michel Cheynel

A J Mark Gaillard

Hewison

Jean-François Lemaître

Jeffrey Carbillet

Benjamin Rey

Rupert Palme

Hannah Froy

Rachael V Wilbourn

Sarah L Underwood

Louise Cheynel

Jean-Michel Gaillard

; A J Mark Hewison

Hélène Verheyden

François Débias

Jeanne Duhayer

Corinne Régis

Sylvia Pardonnet

Maryline Pellerin

Daniel H Nussey

Emmanuelle Gilot-Fromont

Short-term telomere dynamics is associated with glucocorticoid levels in wild populations of roe deer

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Telomeres are non-coding and repetitive DNA sequences located at the extremity of eukaryotic linear chromosomes. The shortening of telomere sequences observed in somatic cells over the lifetime of most species is expected to be associated with the aetiology of age-related diseases and an increased risk of mortality (see [START_REF] Blackburn | Human telomere biology: a contributory and interactive factor in aging, disease risks, and protection[END_REF][START_REF] Wilbourn | The relationship between telomere length and mortality risk in non-model vertebrate systems: a meta-analysis[END_REF] for evidence in humans and birds). Therefore, identifying ecological factors modulating telomere dynamics has become an important challenge [START_REF] Blackburn | Human telomere biology: a contributory and interactive factor in aging, disease risks, and protection[END_REF]Monaghan et al., 2018). In vertebrates, including humans [START_REF] Epel | Accelerated telomere shortening in response to life stress[END_REF], it has been repeatedly suggested that environmental stressors (e.g. exposure to predators, food shortage, psychosocial stress) should accelerate telomere attrition rate [START_REF] Epel | Accelerated telomere shortening in response to life stress[END_REF][START_REF] Haussmann | Telomeres: Linking stress and survival, ecology and evolution[END_REF][START_REF] Monaghan | Organismal stress, telomeres and life histories[END_REF]. In agreement with this prediction, a recent meta-analysis based on 109 studies investigating the association between environmental stressors and telomere length across vertebrates revealed that individuals facing stressful conditions have, on average, shorter telomeres (Chatelain et al. 2020). This relationship was particularly pronounced in birds and mammals, and was consistent irrespective of the type of stressor considered (Chatelain et al. 2020). From a mechanistic point of view, environmental stressors have been suggested to modulate telomere dynamics through stimulation of the hypothalamic-pituitary-adrenal axis, triggering the release of glucocorticoids into the bloodstream by the adrenal gland cortex (reviewed in [START_REF] Haussmann | Telomeres: Linking stress and survival, ecology and evolution[END_REF]. Chronically elevated glucocorticoids are thought to accelerate telomere shortening through diverse physiological pathways [START_REF] Haussmann | Telomeres: Linking stress and survival, ecology and evolution[END_REF], especially through increased oxidative stress [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF][START_REF] Gil | Harsh conditions during early development influence telomere length in an altricial passerine: Links with oxidative stress and corticosteroids[END_REF][START_REF] Reichert | Does oxidative stress shorten telomeres in vivo? A review[END_REF].

In the wild, most of our knowledge about the influence of glucocorticoids on telomere length or dynamics comes from avian studies [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF]. These studies have revealed that, in most cases, high levels of plasma corticosterone are associated with short telomeres. This is well illustrated in black-legged kittiwakes (Rissa tridactyla) where individuals carrying a corticosterone implant during the reproductive season experienced more marked telomere shortening than control individuals [START_REF] Schultner | Migration and stress during reproduction govern telomere dynamics in a seabird[END_REF] and in blackbrowed albatross (Thalassarche melanophrys) where a negative association was found between baseline corticosterone levels and telomere length in both sexes [START_REF] Angelier | Is telomere length a molecular marker of individual quality? Insights from a long-lived bird[END_REF].

Interestingly, some studies have reported no association between glucocorticoid level and telomere length, or even a positive relationship in some cases (reviewed in [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF], and empirical studies have suggested that the direction of the association might be influenced by both environmental conditions and sex [START_REF] Bauch | Telomere length reflects reproductive effort indicated by corticosterone levels in a long-lived seabird[END_REF][START_REF] Jiang | Basal cortisol, cortisol reactivity, and telomere length: A systematic review and meta-analysis[END_REF][START_REF] Young | Telomere length and environmental conditions predict stress levels but not parental investment in a long-lived seabird[END_REF]. Our knowledge is much more limited in mammals, although a few experiments on laboratory rodents suggest that stressful environments can induce shorter telomeres [START_REF] Ilmonen | Telomere attrition due to infection[END_REF]. Overall, our current understanding of the link between physiological markers of stress and telomere dynamics remains limited, especially in wild mammals. In this study, we tested whether individuals with high glucocorticoid levels had shorter telomeres and a faster year-to-year telomere attrition rate in both sexes of two free-ranging populations of roe deer (Capreolus capreolus) living in environments that differ markedly in terms of habitat quality.

We predicted that the relationship between glucocorticoid level and telomere loss should be steeper in the roe deer population facing marked resource limitation (Chizé) than in the population living in a more productive environment (Trois-Fontaines). We also predicted that this relationship should be steeper in males, as they might experience a higher baseline stress level than females [START_REF] Carbillet | The neutrophil to lymphocyte ratio indexes individual variation in the behavioural stress response of wild roe deer across fluctuating environmental conditions[END_REF]. A stronger impact of glucocorticoids on telomere shortening in males compared to females could explain, at least to some extent, why male roe deer have a shorter lifespan compared to females (e.g. [START_REF] Garratt | High juvenile mortality is associated with sexspecific adult survival and lifespan in wild roe deer[END_REF].

Material and Methods

Study population and sample collection

We studied two populations of roe deer living in enclosed forests. Trois-Fontaines forest (TF -1,360 ha), located in north-eastern France (48°43'N, 4°55'E), has rich soils and provides high quality habitat for roe deer. In contrast, Chizé forest (CH -2,614 ha), located in western France (46°05'N, 0°25'W), has a low productivity due to poor soils and frequent summer droughts, providing a less suitable habitat for roe deer [START_REF] Pettorelli | Using a proxy of plant productivity (NDVI) to find key periods for animal performance: the case of roe deer[END_REF]. For the last 40 years, 10-12 days of capture have been organized each winter as part of a long-term Capture-Mark-Recapture program [START_REF] Gaillard | Roe deer survival patterns: a comparative analysis of contrasting populations[END_REF]. During winter, the diet composition is very similar at both study sites (mostly brambles, Rubus sp. and ivy, Hedera helix, see [START_REF] Tixier | Are European roe deer browsers? A review of variations in the composition of their diets[END_REF] and should not generate differences in faecal composition between populations which otherwise may cause a bias in FGM measurements (see below). In two consecutive years (2016 and 2017), blood samples were collected between January and February, minimizing potential confounding variation in stress levels due to individual reproductive state [START_REF] Cheynel | Immunosenescence patterns differ between populations but not between sexes in a long-lived mammal[END_REF]. Indeed, at this time of the year, nearly all females are in the very early phase of gestation, while males have not yet established their mating territories [START_REF] Andersen | The European roe deer: the biology of success[END_REF]. Upon capture, individuals are sexed, weighed (± 50g) and a basic clinical examination is performed. We collected blood samples (up to 1mL/kg) from the jugular vein. We sampled only individuals of known age, which have been captured during the first year of life. Within 30 min of sampling, whole blood was spun at 2000 g for 10 min and the plasma layer drawn off and replaced with the same quantity of 0.9% w/v NaCl solution and spun again. The intermediate buffy coat layer, comprising mainly leukocytes, was collected in a 1.5-mL Eppendorf tube and immediately frozen at -80 °C in a portable freezer (Telstar SF 8025) until further use.

Telomere assays

Relative telomere length (hereafter RTL) was measured by quantitative PCR as previously described for these populations [START_REF] Wilbourn | Age-dependent associations between telomere length and environmental conditions in roe deer[END_REF]. Genomic DNA was extracted from white blood cells using the Macherey-Nagel NucleoSpin® Blood QuickPure kit. DNA yield and purity was quantified using a Nanodrop ND-1000 spectrophotometer (Thermo Scientific, Wilmington DE, USA) and DNA integrity was assessed by running 200 ng total DNA on a 0.5% agarose gel and DNA bands scored on a scale of 1-5 by visual examination. Samples passed QC with a DNA yield of ≥20 ng/µl, an acceptable purity absorption range of 1.7 -2.0 for the 260/280 nm ratio and > 1.8 for the 260/230 nm ratio, and a DNA integrity score of either 1 or 2 [START_REF] Seeker | Method specific calibration corrects for DNA extraction method effects on relative telomere length measurements by quantitative PCR[END_REF]. We measured relative leukocyte telomere length (RTL) using a real-time quantitative PCR method (qPCR; Cawthon 2002) which has previously been optimized and validated in sheep and cattle [START_REF] Seeker | Method specific calibration corrects for DNA extraction method effects on relative telomere length measurements by quantitative PCR[END_REF] and previously used in roe deer [START_REF] Wilbourn | Age-dependent associations between telomere length and environmental conditions in roe deer[END_REF]. This method measures the total amount of telomeric sequence present in a DNA sample relative to the amount of a non-variable copy number reference gene (beta-2-microglobulin (B2M)). B2M was previously identified as an appropriate reference gene for use in qPCR studies of telomere length in sheep [START_REF] Fairlie | Lifelong leukocyte telomere dynamics and survival in a free-living mammal[END_REF] and cattle [START_REF] Seeker | Bovine telomere dynamics and the association between telomere length and productive lifespan[END_REF][START_REF] Seeker | Method specific calibration corrects for DNA extraction method effects on relative telomere length measurements by quantitative PCR[END_REF]. The selection was based on comparison of panels of candidate genes supplied as part of the GeNorm kit by Primerdesign (12 candidate reference genes for sheep, 6 for cattle).

B2M showed a consistent amplification profile, clean melting curve and stable qPCR results in preliminary analyses with roe deer samples, and was therefore deemed to be a suitable reference gene for our study [START_REF] Wilbourn | Age-dependent associations between telomere length and environmental conditions in roe deer[END_REF]. For telomere reactions, we used the following HPLC purified primers, Tel 1b (5'-CGG TTT GTT TGG GTT TGG GTT TGG GTT TGG GTT TGG GTT-3') and Tel 2b (5'-GGC TTG CCT TAC CCT TAC CCT TAC CCT TAC CCT TAC CCT-3') (from [START_REF] Epel | Accelerated telomere shortening in response to life stress[END_REF]). For B2M reactions, primers were supplied by Primer Design (Catalogue number: HK-SY-Sh-900, Southampton, UK).

Using an automated liquid handling robot (Freedom Evo-2 150; Tecan), we were able to load both the DNA samples and qPCR master mix in 384 well plates, allowing us to run both telomeric and B2M reactions in separate wells, but on a single plate. A separate master mix for each primer set was prepared containing 5µl LightCycler 480SYBR Green I Master Mix (Cat # 04887352001, Roche, West Sussex, UK), 0.5µl B2M (300nm) primer or 0.6µl each telomeric primer (900nm), and 1 ng of sample DNA per individual PCR reaction. DNA was amplified in 10µl reactions. Each plate included a non-treated control (water; NTC) for each amplicon, a calibrator sample (1ng) on each row to account for plate to plate variation and robot pipetting error, as well as a five step 1:4 serial dilution starting at 20ng/µl to inspect visually the qPCR curves. The calibrator sample was DNA that had been extracted from a large quantity of blood obtained from a single wild roe deer. In this case, the calibrator was extracted using the Qiagen DNeasy Blood and Tissue kit (Cat# 69581. Manchester. UK), pooled and quality controlled in the same way as our DNA samples of interest. All samples, calibrators and NTC's were run in triplicate and all qPCR were performed using a Roche LC480 instrument using the following reaction protocol: 10min at 95°C (enzyme activation), followed by 50 cycles of 15s at 95°C (denaturation) and 30s at 58°C (primer annealing), then 30s at 72°C (signal acquisition).

Melting curve protocol was 1 min at 95°C, followed by 30s at 58°C, then 0.11°C/s to 95°C followed by 10s at 40°C. We used the LinRegPCR software package (version 2016.0) [START_REF] Ruijter | Amplification efficiency: linking baseline and bias in the analysis of quantitative PCR data[END_REF] to correct for baseline fluorescence, to set a window of linearity for each amplicon group and to calculate well-specific reaction efficiencies and Cq values [START_REF] Fairlie | Lifelong leukocyte telomere dynamics and survival in a free-living mammal[END_REF]. A constant fluorescence threshold was set within the window of linearity for each amplicon group, calculated using the average Cq across all three plates. The threshold values used were 0.304 and 0.394, and the average efficiency across all plates was 1.89 for both the B2M and telomere amplicon groups. Samples were excluded from further analysis if the coefficient of variation (CV) across triplicate Cq values for either amplicon was > 5%, or if at least one of their triplicate reactions had an efficiency that was 5% higher or lower than the mean efficiency across all wells on that plate for the respective amplicon. Overall, thirteen samples failed quality control at either the DNA extraction or qPCR stage and were excluded from the study. We calculated relative telomere length (RTL) for each sample following [START_REF] Pfaffl | A new mathematical model for relative quantification in real-time RT-PCR[END_REF] as follows:

RTL = (ETEL(CqTEL[Calibrator] -CqTEL[Sample]))/ (EB2M (CqB2M[Calibrator] - CqB2M[Sample]))
where ETEL and EB2M are the mean reaction efficiencies for the respective amplicon group across all samples on a given plate; CqTEL[Calibrator] and CqB2M[Calibrator] are the average Cqs for the relevant amplicon across all calibrator samples on the plate; and CqTEL [Sample] and CqB2M[Sample] are the average of the triplicate Cqs for the sample for each amplicon.

Data on RTL change between years were available for 43 individuals (Chizé: 12 females, 6 males; Trois Fontaines: 13 females, 12 males, see Table 1 for sample sizes).

Glucocorticoid assays

Glucocorticoid levels were assessed by measuring fecal glucocorticoid metabolites (hereafter FGMs). In roe deer, FGM levels represent an integrated measure of adrenocortical activity about half a day prior to capture [START_REF] Dehnhard | Noninvasive monitoring of adrenocortical activity in roe deer (Capreolus capreolus) by measurement of fecal cortisol metabolites[END_REF][START_REF] Palme | Non-invasive measurement of glucocorticoids: advances and problems[END_REF]. Feces were collected rectally and frozen immediately at -80°C until assayed (except in 2016 at TF when samples were stored at -20°C). FGMs were extracted following a methanol-based procedure and assayed using a group-specific 11-oxoetiocholanolone enzyme immunoassay (EIA) as previously described and validated for roe deer [START_REF] Möstl | Measurement of cortisol metabolites in faeces of ruminants[END_REF][START_REF] Zbyryt | Do wild ungulates experience higher stress with humans than with large carnivores?[END_REF]. To determine the amount of glucocorticoid metabolites in feces of roe deer from the 2016 and 2017 field seasons, we sampled 500 ± 5 mg of each homogenized fecal sample that we vortexed for 30 min with 5 ml of 80% methanol before being centrifuged (15 min at 2500 g) [START_REF] Palme | Steroid extraction: get the best out of faecal samples[END_REF]. We determined the amount of FGMs (with a 3α, 11-oxo structure) in an aliquot of the supernatant (after a further 1+9 dilution with assay buffer). Measurements were carried out in duplicate (intra-and inter-assay coefficients of variation were <10% and <15%, respectively) and results expressed as nanograms per gram of wet feces. To ensure that there were no differences in the moisture content of feces between populations that could bias our FGM measurements, we measured the moisture content of 60 randomly selected fecal samples (30 per population). In each sample, the moisture content was calculated by subtracting dry mass (after total dehydration in an oven at 60°C for 120 hours) from wet mass. The weighing was carried out to the nearest 0.1 mg on an electronic scale and the result was expressed as % of wet mass. The results did not reveal any difference in feces moisture content between sites (mean moisture content ± SD: 43 ± 7% vs. 44 ± 11% at Chizé and Trois Fontaines, respectively).

Overall, FGM concentrations were similar in Chizé and Trois Fontaines (Chizé: 973.98ng/g, 95% CI [716.55;1231.42]; Trois Fontaines: 800.59ng/g, 95% CI [649.60;951.47]).

One male from the TF population had an extremely low non-physiological (and unexplained) FGM level in 2017 (8 ng/g, see Fig. S1) and was removed from the analyses. Individual concentrations of FGMs in 2016 and 2017 were correlated (r = 0.42, p = 0.02, n = 30, Fig. 1).

However, this relationship was strongly influenced by a single male in Chizé, which had the highest level of FGMs in the dataset in both 2016 and 2017. Once this individual was removed, individual concentrations of FGMs in 2016 and 2017 were no longer correlated (r = 0.18, p = 0.34, n = 29), but the repeatability of FGM concentrations (i.e. ratio of among-individual variance to total variance, [START_REF] Nakagawa | Repeatability for Gaussian and non-Gaussian data: a practical guide for biologists[END_REF]) between 2016 and 2017 was 0.35, which is within the range generally observed for glucocorticoids in vertebrates [START_REF] Taff | The repeatability of glucocorticoids: A review and meta-analysis[END_REF] and more specifically in roe deer [START_REF] Carbillet | Under cover of the night: contextdependency of anthropogenic disturbance on stress levels of wild roe deer Capreolus capreolus[END_REF].

Statistical analyses

We performed a set of analyses seeking to test whether either raw RTL values (in 2016 and 2017) or the change in RTL between years (RTLc, computed as the difference in RTL values between 2016 and 2017 for a given individual) were related to glucocorticoid levels. We first analyzed the relationship between FGMs on RTL within each year. As both habitat and sex have been suggested to mediate the relationship between FGMs and telomere length in wild animals [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF], we included possible additive or interactive effects of sex and population. Finally, we included age to control for possible age-dependence of telomere length [START_REF] Wilbourn | Age-dependent associations between telomere length and environmental conditions in roe deer[END_REF], and body mass (log-transformed) to control for possible growthmediated covariation between body mass and telomere length [START_REF] Monaghan | Somatic growth and telomere dynamics in vertebrates: relationships, mechanisms and consequences[END_REF] as covariates (see Tables S1,S2 Appendix A for the full set of models).

To test for a link between FGMs and RTLc, we analyzed the relationship between RTLc and FGM levels (log-transformed) measured in 2016 and 2017 (e.g. Table S4). In addition to the effects of FGMs measured in 2016 or 2017, we tested the influence of the mean FGM level (log-transformed) as a proxy of the mean baseline stress level experienced during two consecutive winters on RTLc. We followed the same model selection procedure as the one described above using RTL as the response variable. We thus included log-transformed body mass (using body mass measured in 2016, 2017 or mean body mass for models with FGMs in 2016, FGM in 2017 and mean FGMs, respectively) and age as potential covariates. We also included possible additive or interactive effects of sex and population on RTLc (see Table S3 for a full list of models).

We repeated the analyses excluding individuals sampled in their first year of life (i.e. at about eight months of age, Tables S2 andS4), because telomere attrition may be faster during early life compared to adulthood [START_REF] Fairlie | Lifelong leukocyte telomere dynamics and survival in a free-living mammal[END_REF], and including the birth cohort (2003 to 2015) as a random effect (Table S5) to control for non-independence among animals born in the same year using linear mixed effects models (using the R-package lme4). In all cases, results were qualitatively unchanged (Tables S2, S4 andS5). Model selection was based on the Akaike Information Criterion (AIC) and we retained the model with the lowest AIC, except when the difference in AIC (ΔAIC) between two competing models was less than 2, in which case we retained the simplest model (Burnham and Anderson 2002).

Results

Within years, we detected no association between FGMs and RTL (Table S1, b±SE = -0.04 ± 0.07, n= 38 in 2016; b = -0.11 ± 0.08 in 2017, n=32; Fig. 2). The repeatability of RTL values was 0.24, while we observed a decrease in telomere length between 2016 and 2017 in 51.2% (22 out of 43) of individuals (Fig. 3). Our model selection procedure revealed that high average FGM levels in 2016 and 2017 were associated with a stronger decrease in RTL between these two consecutive years (Table S3, b= -0.23 ± 0.11, n= 30, Fig. 4a). Likewise, FGM levels of roe deer captured in 2017 were negatively associated with RTLc (b = -0.20 ± 0.09, n= 32, Fig. 4b).

Population, sex and body mass had no detectable influence on RTLc. The level of FGMs measured in 2016 was not related to RTLc (the constant model was selected, Table S3, Fig 4c).

Finally, our results were not impacted by individual differences in the number of days elapsed between the 2016 and 2017 capture sessions (Table 2).

Discussion

Previous reports in birds have revealed that high levels of glucocorticoids are associated with shorter telomeres in nestlings (e.g. [START_REF] Pegan | Brief increases in corticosterone affect morphology, stress responses, and telomere length but not postfledging movements in a wild songbird[END_REF][START_REF] Powolny | Corticosterone mediates telomere length in raptor chicks exposed to chemical mixture[END_REF] and among adults (see [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF] for a review; but see Cerchiara et al., 2017 for a counter-example in Magellanic penguins, Spheniscus magellanicus). In our study, we did not observe any withinyear association between relative telomere length and FGM level in roe deer. While substantial inter-individual differences in telomere dynamics occur in wild vertebrates [START_REF] Fairlie | Lifelong leukocyte telomere dynamics and survival in a free-living mammal[END_REF][START_REF] Spurgin | Spatio-temporal variation in lifelong telomere dynamics in a long-term ecological study[END_REF], the large variation in telomere length observed among juveniles [START_REF] Wilbourn | Age-dependent associations between telomere length and environmental conditions in roe deer[END_REF] might explain these results if, to some extent, telomere length in adulthood can be predicted by telomere length during early-life. However, we found that high average FGM levels in two consecutive winters predicted the rate of telomere attrition in both roe deer populations. Here, it is noteworthy that, between 2016 and 2017, telomere length increased with age in 48.8% of individuals, in line with a few recent reports of within-individual elongation of telomere length in wild populations of vertebrates (e.g. Soay sheep, Ovis aries, [START_REF] Fairlie | Lifelong leukocyte telomere dynamics and survival in a free-living mammal[END_REF]edible dormouse, Glis glis, Hoelzl et al., 2016; Seychelles warbler, Acrocephalus sechellensis [START_REF] Spurgin | Spatio-temporal variation in lifelong telomere dynamics in a long-term ecological study[END_REF]. While measurement error could contribute to such an apparent elongation of telomere length [START_REF] Steenstrup | The telomere lengthening conundrum-artifact or biology?[END_REF], we cannot totally ruled out that this pattern might have a biological foundation [START_REF] Bateson | The telomere lengthening conundrum-it could be biology[END_REF]. For instance, this pattern could arise from year-to-year changes in the leukocyte formula of individuals [START_REF] Spurgin | Spatio-temporal variation in lifelong telomere dynamics in a long-term ecological study[END_REF], which has already been observed in roe deer [START_REF] Cheynel | Immunosenescence patterns differ between populations but not between sexes in a long-lived mammal[END_REF]. In addition, the elongation of telomeres with increasing age could result from a possible expression of telomerase [START_REF] Blackburn | Human telomere biology: a contributory and interactive factor in aging, disease risks, and protection[END_REF]. Currently, our knowledge on telomerase activity in cervids is both limited and unclear (no expression in the fibroblasts of two Muntjac species, Muntiacus muntjak and Muntiacus reevesi, see [START_REF] Gomes | Comparative biology of mammalian telomeres: hypotheses on ancestral states and the roles of telomeres in longevity determination[END_REF]; expression in the antlers of sika deer, Cervus nippon, see [START_REF] Sun | Detection of telomerase activity and expression of TERT gene in antler of sika deer[END_REF] so that complementary analyses are thus required to disentangle these different scenarios that challenge the common view of an inevitable telomere loss over the life course. In roe deer, repeatability in telomere length was relatively high (i.e. 0.24) compared to previous published values (e.g. 0.068 in Seychelles warbler, Acrocephalus sechellensis, but see [START_REF] Bichet | Telomere length is repeatable, shortens with age and reproductive success, and predicts remaining lifespan in a long-lived seabird[END_REF], which might be due to the fact that our measurements were performed in two consecutive years, so that a lower rate of change is expected compared to measurements taken across a longer time span.

Our findings indicate that the relationship between glucocorticoid levels and telomere dynamics is not sex-specific in roe deer. This contrasts with the relationship between baseline corticosterone levels and telomere length (measured in two consecutive years) reported in males only in a study in common terns (Sterna hirundo) [START_REF] Bauch | Telomere length reflects reproductive effort indicated by corticosterone levels in a long-lived seabird[END_REF]. In terns, corticosterone levels reflect male, but not female, reproductive expenditure [START_REF] Bauch | Telomere length reflects reproductive effort indicated by corticosterone levels in a long-lived seabird[END_REF] and males allocate three times more effort to chick feeding than females [START_REF] Wiggins | Parental care of the Common Tern Sterna hirundo[END_REF], which may explain these sex-specific responses of telomere attrition. In roe deer, males grow antlers each year and actively defend mating territories, while female reproductive allocation is high, because they generally give birth to twins that have fast post-natal growth rates [START_REF] Andersen | The European roe deer: the biology of success[END_REF]. Given that the overall reproductive effort is high in both males and females and that this species is weakly dimorphic in size, this is consistent with the lack of sex differences in telomere dynamics that we report. From this, it appears particularly important to accurately assess how reproductive expenditure differs between males and females and to consider how the stress response following increased reproductive effort might differ between the sexes when investigating the possible sex-specific effects of stress on telomere dynamics.

In recent years, a few studies in birds have investigated whether the relationship between telomere length and corticosterone level can be modulated by environmental conditions. A comparison of nestlings from two populations of Thorn-tailed Rayadito (Aphrastura spinicauda) located at different latitudes and thus exposed to different climatic conditions revealed that individuals living at high latitude have higher levels of baseline corticosterone and somewhat shorter telomeres than individuals living at low latitude [START_REF] Quirici | The relationship of telomere length to baseline corticosterone levels in nestlings of an altricial passerine bird in natural populations[END_REF]. However, the negative association found between telomere length and baseline corticosterone level was broadly similar across the two populations [START_REF] Quirici | The relationship of telomere length to baseline corticosterone levels in nestlings of an altricial passerine bird in natural populations[END_REF]. On the contrary, a comparison of three colonies of thick-billed murre (Uria lomvia) revealed a much more contrasted picture [START_REF] Young | Telomere length and environmental conditions predict stress levels but not parental investment in a long-lived seabird[END_REF]. While no relationship between telomere length and baseline corticosterone levels was found in individuals living in habitat of medium quality, the relationship was negative in the high quality habitat but, surprisingly, positive in the low quality habitat, emphasizing the complex role played by the environment in mediating these relationships [START_REF] Young | Telomere length and environmental conditions predict stress levels but not parental investment in a long-lived seabird[END_REF]. One possible approach to decipher these relationships is to focus on telomere loss rather than telomere length to control for inter-individual differences in telomere length [START_REF] Quirici | The relationship of telomere length to baseline corticosterone levels in nestlings of an altricial passerine bird in natural populations[END_REF]. In the present study, we found that the slope of the relationship between telomere length and glucocorticoid level did not differ between our two populations of roe deer living in contrasting environments. Since glucocorticoid levels also did not differ between the two populations, we can hypothesize that an adaptive down-regulation of the stress response in Chizé, where the roe deer experience the harshest environmental conditions (see [START_REF] Saout | Levels of fecal glucocorticoid metabolites do not reflect environmental contrasts across islands in black-tailed deer (Odocoileus hemionus sitkensis) populations[END_REF] for similar observations in black-tailed deer, Odocoileus hemionus sitkensis) might lead to a similar pattern of telomere loss as a function of physiological stress in both habitats. Note, however, that we cannot rule out that the absence of sex-or population-specific variation in our study might be due to low statistical power.

In vertebrates, the release of glucocorticoids is the last stage in a cascade of physiological responses initiated by environmental stressors [START_REF] Haussmann | Telomeres: Linking stress and survival, ecology and evolution[END_REF]. While these pathways are well described [START_REF] Boonstra | Reality as the leading cause of stress: rethinking the impact of chronic stress in nature[END_REF], the physiological mechanisms linking high levels of glucocorticoids to a faster pace of telomere attrition -as we report here -are yet to be identified in wild vertebrate populations. Studies performed to date have focused on the alteration of the redox homeostasis [START_REF] Angelier | Do glucocorticoids mediate the link between environmental conditions and telomere dynamics in wild vertebrates? A review[END_REF][START_REF] Haussmann | Embryonic exposure to corticosterone modifies the juvenile stress response, oxidative stress and telomere length[END_REF] and on the modulation of the mitochondrial metabolism [START_REF] Casagrande | Increased glucocorticoid concentrations in early life cause mitochondrial inefficiency and short telomeres[END_REF][START_REF] Casagrande | Telomere attrition: metabolic regulation and signalling function?[END_REF] by glucocorticoids. In addition, in vitro experiments have shown that high levels of glucocorticoids reduce the activity of telomerase, the main enzyme responsible for preserving chromosome length through telomeric DNA synthesis, in human leukocytes [START_REF] Choi | Reduced telomerase activity in human T lymphocytes exposed to cortisol[END_REF]. However, modulation of telomerase activity by stress is a complex phenomenon that depends not only on glucocorticoid level, but also on the magnitude of stress reactivity and recovery [START_REF] Beery | Chronic stress elevates telomerase activity in rats[END_REF][START_REF] Epel | Dynamics of telomerase activity in response to acute psychological stress[END_REF]. Moreover, the relative influence of this third pathway remains unknown in large herbivores, as our knowledge on telomerase activity is particularly limited within this taxa [START_REF] Gomes | Comparative biology of mammalian telomeres: hypotheses on ancestral states and the roles of telomeres in longevity determination[END_REF]. Future work should seek to integrate multiple measures of oxidative damage and antioxidant defenses with measures of mitochondrial and telomerase activity in leukocytes to decipher the physiological connections linking environmental stressors and telomere dynamics.

Overall, our study highlights that the level of stress (as measured by FGM levels) influences telomere dynamics, even over a short time scale, which could potentially generate substantial individual differences in telomere length in late adulthood. As short telomeres are associated with poor health [START_REF] Beirne | Age-related declines and diseaseassociated variation in immune cell telomere length in a wild mammal[END_REF], and thereby greater mortality risk in populations of vertebrates in the wild [START_REF] Bichet | Telomere length is repeatable, shortens with age and reproductive success, and predicts remaining lifespan in a long-lived seabird[END_REF][START_REF] Wilbourn | The relationship between telomere length and mortality risk in non-model vertebrate systems: a meta-analysis[END_REF], we suggest that telomere dynamics might mediate the long-term survival cost of repeated exposure to environmental stressors.

Declaration of Competing Interest

No conflict of interest. first draft of the paper and then received input from all other co-authors. All authors approved the final version of the manuscript and agree to be held accountable for the content therein.

CAPTION FOR FIGURES:

Fig. 1: Relationship between the concentration of fecal glucocorticoid metabolites (FGMs) measured in 2017 (ng/g, log-transformed) and the FGM concentration measured in 2016 (ng/g, log-transformed). Individual FGM concentrations in 2016 and 2017 were correlated (r = 0.42, p = 0.02, n = 30), with a repeatability of 0.22. This relationship was strongly influenced by a single male in Chizé with the highest FGMs in both 2016 and 2017. After removing this individual the relationship was no longer statistically significant: r = 0.18, p = 0.34, n = 29, repeatability of 0.32). Males in Chizé are in dark blue, females in Chizé are in light blue, males in Trois-Fontaines are in dark red and females in Trois-Fontaines are in light red.

Fig. 2 :

 2 Fig.1

Acknowledgements

We thank all the OFB staff, in particular Claude Warnant, Gilles Capron, Florian Chaudat and the field volunteers for organizing the roe deer captures. We thank Edith Klobetz-Rassam for FGM analysis. Lastly, we are grateful to two anonymous reviewers for helpful comments.

Funding J.F.L., J.M.G., E.G.F., B.R. and L.C. were supported by a grant from the Agence Nationale de la Recherche (ANR-15-CE32-0002-01). This work was performed within the framework of the LABEX ECOFECT (ANR-11-LABX-0048), within the program 'Investissements d'Avenir' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). D.H.N. and H.F. were supported by a BBSRC grant no. (BB/L020769/1).

Authors' contributions

JFL & EGF conceived and designed the study. JFL, BR, LC, JMG, AJMH, HV, FD, JD, SP, MP, EGF performed fieldwork. BR, CR and LC extracted DNA. RP, JC, BR ran FGM assays. HF, RW, SLU and DN ran telomere assays. JFL performed the statistical analysis, wrote the

Ethics

The protocol of capture and blood sampling under the authority of the Office Français de la Biodiversité (OFB) was approved by the Director of Food, Agriculture and Forest (Prefectoral order 2009-14 from Paris). The land manager of both sites, the Office National des Forêts (ONF), permitted the study of the populations (Partnership Convention ONCFS-ONF dated 2005-12-23). All experiments were performed in accordance with guidelines and regulations of the Ethical Committee of Lyon 1 University (project DR2014-09, June 5, 2014).

Data accessibility

Data available have been uploaded on dryad (https://datadryad.org/stash/share/YuyEniPCdMFLAZzmIja29re6JmRktx09ITzi15UMoL8) and will be made publicly available upon acceptance.