D Mourard

D Bonneau

J.-M Clausse

F Hénault

A Marcotto

A Blazit

S Bosio

Y Bresson

T Ten Brummelaar

P Kervella

S Lagarde

H A Mcalister

A Mérand

G Merlin

N Nardetto

R Petrov

A Roussel

K Rousselet-Perraut

P Stee

J Sturmann

L Sturmann

I Tallon-Bosc

VEGA: a visible spectrograph and polarimeter for CHARA

Keywords: Optical interferometry, visible wavelengths, spectroscopy, polarimetry

INTRODUCTION

This report describes a project for a visible focal instrument for the CHARA Array , named VEGA for Visible spEctroGraph and polArimeter. The goal is to give more complete access to the visible wavelengths, and add spectral and polarimetric capabilities. It is a new scientific niche for the CHARA Array 1 , and will provide access to observational phase space available no where else. By offering access to the visible band and a high spectral resolution, along with good sensitivity, VEGA is very complementary to existing or planned near-infrared instruments in many ways. First, both infrared and visible measurements are mandatory for correct and complete radiative transfer modeling. Second, the spatial resolution will reach 0.3 mas at 0.6 µm on the longest baseline of 331m. Third, many astrophysical signals investigated by interferometry have stronger signatures at visible wavelengths than in the near infra-red. All this will allow us to investigate new physical processes involved in stars, and open the field of astrophysical domains already covered by the Array. Finally, high spectral resolution is quite easy to achieve in the visible domain with VEGA, and this will be mandatory for progress in stellar activity studies.

Our previous experience has lead to the design of an instrument focusing on one mode of operation: high spectral resolution with a dispersed fringe mode called X-λ. This instrument will offer the combination of up to 4 telescopes, giving simultaneous access to 6 squared visibilities, 6 differential phases and 3 closure phases per spectral channel. The spectral resolutions offered will be 1500, 5000 and 30000 and the spectral range is from 0.45 to 0.87 µm (and if possible from 0.4 to 1.0 µm). The spectrograph is equipped with polarimetric capabilities allowing us to simultaneously record interferograms in two polarizations, either linear or circular. Finally, the field of view extension in the slit direction is 4''. This new beam combiner is based on the visible spectrograph of the GI2T interferometer, which is now decommissioned, and offers the unique opportunity of moving the existing spectrograph to the CHARA Array for use in initial experiments. An important point is that the spectrograph already exists, is equipped with two new generation photon counting detectors and we have already observe with it on the GI2T. Furthermore, the data reduction process is quite mature. So all the major difficulties of the project are clearly already solved and we can now focus our attention to the implementation of VEGA at the CHARA Array and on future improvements.

In summary, the VEGA project already has a spectrograph along with calibration sources, two new generation photon counting detectors, a global control system and a data reduction pipeline. In order to deploy this instrument at the CHARA Array we will have to modify the magnification of the cameras in the spectrograph as well as the injection of the calibration sources. We will also have to develop the interface optical system, adapt the control system to fit into the CHARA framework, upgrade the software control for the 3/4 telescopes mode, and expand the data reduction pipeline. Our current plan is to install VEGA at Mt Wilson in mid 2007 and to begin operation during summer 2007. The science case of this project is described in Stee et al., at this meeting.

MAIN CHARACTERISTICS OF THE VEGA PROJECT

Operating mode

The experience we gained while operating the GI2T interferometer has lead us to focus our attention on the most consolidated mode of the REGAIN spectrograph and especially the one which requires the least development for implementation at the CHARA Array. We have a great deal of experience with the X-λ mode or dispersed fringe mode and so we have decided to put the highest priority on this operating mode at the CHARA Array in order to optimize the scientific returns. Future implementation of the COURTES mode (or Multiple Band Passes) will be possible but until Adaptive Optics are installed on the CHARA Array telescopes; the best use of the spectroscopic capabilities will be with X-λ mode. For the present configuration of the CHARA Array, we propose a visible instrument based on the multispeckle approach. Our experience, in both speckle interferometry and long baseline interferometry on the GI2T, coupled with theoretical predictions based on the typical seeing conditions at Mt Wilson, leads to a maximum spectral band of the order of 50 nm, during median to best or excellent seeing.

Calibration of multi mode visibility has already been demonstrated with the GI2T 2 at the level of σ( V²)/V²=1.3%. We observed the well know A0V star Vega (α Lyrae) on different nights in June 2001, at different hour angles, different baselines and at different wavelengths. These measurements are therefore done during different atmospheric conditions. We do not use any reference star and just measured the raw visibilities. The data processing was done by using the following steps: 1) correct for the geometrical distortion of the photon counting detectors, 2) select correctly superimposed images by rejecting the sequences where tracking errors are out of bounds, 3) integrate the autocorrelation of short exposures images, in a spectral bandwidth of 9 nm centered around 645 nm, 4) calculate the raw spectral density, 5) Remove the so-called "photon counting hole" and 6) subtract the photon noise bias.

The final visibility and its formal error are calculated as described in Berio et al. 3,[START_REF] Bério | Spectrally resolved Michelson interferometry : II Noise analysis[END_REF] . We only include the measurements where no problems are identified during the acquisition (e.g. bad positioning of the output pupils, bad fringe tracking, and bad star tracking...). No corrections are applied to the raw visibilities. Final adjustment of the visibility points is done using a model of a uniform disk of angular diameter Φ UD multiplied by the squared instrument visibility V² inst . Both V² inst and Φ are simultaneously adjusted to fit the data. We finally get the following results: V² inst = 0.367 ± 0.005 and Φ = 3.07 ± 0.06

The angular diameter of Vega as determined here is in very good agreement with a previously published measurement (3.08±0.07) made with the Narrabri Intensity Interferometer. The most interesting result is the very good stability of the transfer function of the GI2T/REGAIN interferometer as shown on Fig. 1. This will allow visibility measurements with a final accuracy of the order of 1%. and demonstrates the feasibility of accurate visibility measurement in multimode interferometry.

The dispersed fringe concept has already been demonstrated in a large number of papers as well as the potential and accuracy of the differential phase measurements (Vakili et al. [START_REF] Vakili | Subtles structures in the wind of P Cyg[END_REF][START_REF] Vakili | Evidence for one-armed oscillations in the equatorial disk of ζ Tau from GI2T spectrally resolved interferometry[END_REF] and Bério et al. [START_REF] Berio | Interferometric insight of gamma Cas long-term variability[END_REF] ). 1° of phase precision is easily obtainable, and with a better centroiding system, such as the one we intend to develop for VEGA, we estimate that 10 -3 rd should be possible.

We propose the combination of 4 telescopes simultaneously at the CHARA Array by using the spectral+spatial encoding of the information, as explained below:

Pupil plane Dispersed fringes in the image plane (X-λ) Spatio-Spectral Density plane The operation of REGAIN has so far been in the 2 telescopes mode, but the spectrograph has been designed to accept 3 pupils located at positions 1, 2 and 3 in figure 2. When REGAIN was first being developed, we did not foresee 4 telescope operation. Still, our currently installed internal source allows us to simulate 4 telescope operation and we plan to make intensive tests in the coming months in order to correctly establish the limits of this mode. Thanks to the dispersed fringe mode, we are able to accommodate 4 telescopes without changing the highest spatial frequency with respect to the non redundant 3T mode. In the spectral density plane, conjugate to the X-λ plane, the fringe frequencies could be separated in the wave direction by simply adding a fixed delay on some of the pupils.

For a given spatial frequency, for example the one corresponding to the three baselines 1-2, 2-4 and 4-3, the spectral separation of the fringe signals in the spectral density plane is related to the optical path delay and to the spectral bandwidth. This means that if one wants to measure the fringe visibilities in one narrow spectral channel, the three signals will be blurred and there is no way to separate them. However, by use of the cross spectral analysis, it is possible to preserve the spectral resolution of the instrument with this compact and redundant configuration. The principle of this technique is to integrate cross-correlations of short exposures between one wide spectral band and one narrow spectral channel. The fringes of baseline 1-2 in the wide spectral channel are not correlated with the fringes of baselines 2-4 and 4-3 in the narrow spectral channel but only with the 1-2 fringes. This is due to the presence of a random atmospheric phase on each baseline. Thus, as with the 2T experiment where the fringes of one speckle are only correlated with the fringes of the same speckle, in this special 4T configuration the fringes in one speckle of one baseline can be only correlated with the fringes of the same speckle and the same baseline. This permits us to separate the redundant baselines and to preserve the spectral resolution.

Spectral coverage and resolution. Sampling considerations.

The spectrograph is designed to cover the visible band from 0.45 to 0.87 µm. It is equipped with two simultaneous cameras, each one equipped with a new generation photon counting detector (intensified CCD). The blue camera covers the domain [0.45,0.75] µm and the red one the domain [0.58,0.87] µm. The spectral distance between the two cameras depends on the selected grating: The spectrograph is equipped with three different gratings allowing the resolutions set out in Table 1.

Table 1: Spectral resolution and spectral band for the three gratings of VEGA. Spectral separation between red and blue cameras.

Grating X-λ mode

Spectral distance between red and blue cameras R1: 1800tr/mm R=35000 ∆λ=6.7nm

18 nm R2: 300tr/mm R=5000 ∆λ=40nm 140 nm R3: 100tr/mm R=1700 ∆λ=120nm Not usable simultaneously

For installation of VEGA at the CHARA Array, we plan to adapt the magnification of the cameras for the best use of the new photon counting detectors. For a correct sampling on the detector of the narrowest fringes (in the case of the 3/4Telescopes operation), we have to make a compromise between sampling of the fringes, the extension of the field of view in the slit direction, and the number of simultaneous spectral channels. Indeed, the detector characteristics (size of the entrance plane φ=16mm, FWHM of the photon events at the output of the intensified stages=50µm, internal magnification 1/2.5, pixel size on the CCD=10µm), gives us a number of spatial resolution elements in the spectrograph image plane which is a disk of 640 in diameter. Based on these data we have optimized the size of the slit, the magnification of the spectrograph and the extension of the field of view and obtain the following parameters:

Polarization capabilities

VEGA integrates a polarizing device able to simultaneously view both polarizations, either linear or circular [START_REF] Rousselet-Perraut | Polarization effects in stellar interferometry[END_REF] on the detectors.

The principle of SPIN (Spectro-Polarimetric INterferometry) is based on a Wollaston prism located (optionally) just before the grating. The prism is followed by a λ/4 plate in order to equalise the transmission of the grating after the polarizer. The Wollaston introduces an angular separation of the two polarizations, so that both dispersed images arrive on the same detector. For this SPIN operation, the slit is automatically reduced by a factor 2 in its long direction in order to avoid polarization overlaps. In front of the Wollaston, another λ/4 plate could be inserted in order to go from linear to circular polarization. The first tests of this design were made on the GI2T in 2004 on the stars β UMa, α Cep, and α Lyrae. Results are in press 9 (see fig. 3).

L

Fig. 3: left: prototype of the SPIN device. center: long exposure showing both polarizations on the same detector right: spectral density of the two parts of the images, showing the presence of fringes simultaneously on both polarizations.

Imaging capabilities

In the X-λ mode, up to 4 telescopes are recombined. With 3 closure phases, 50% of the phase information could be recovered. We do not aspire to complex imaging capabilities but simple images could be reconstructed in this configuration. If one also considers the large number of spectral channels simultaneously recorded by VEGA and the use of earth rotation to complete the (u,v) place coverage, imaging could be possible. Our groups are collaborating on the 'image reconstruction' software development (mainly the Lyon group) and we plan to integrate the specific configuration of VEGA into this general purpose user software.

Limiting magnitude and signal to noise considerations

In multimode operation, we usually consider two domains for the signal to noise ratio, depending on the number of photons per speckle per single exposure. If this number is larger than 1, then the signal to noise ratio increases as the square root of the number of photons, whereas if it is smaller than 1, the signal to noise ratio increases as the number of photons [START_REF] Roddier | Interferometric imaging in optical astronomy[END_REF] . In consequence, we first defined the limiting magnitude as the magnitude giving 1 photon per speckle per single exposure. The signal to noise ratio is then equal to one and should be multiplied by the square root of the number of speckles and by the square root of the number of images. We usually consider in speckle techniques that the limiting magnitude is defined as this limit. It is clear however that the signal to noise ratio could reach high values at this limit, since one can integrate a large number of single exposures containing a large number of speckles.

In the differential regime, it has been shown by Petrov 11 , that the signal to noise ratio is the geometrical mean of the signal to noise ratios in the reference channel and in the science channel. Then a good signal to noise ratio could also be achieved in a science channel where the number of photons per speckle per frame is much less than 1, thanks to the fact that we can have more than 1 photon per speckle per frame in the reference channels. These considerations will be used for the following calculation. We also consider the following hypothesis for the calculations:

1. For a V=0 star, the number of photons received is equal to N 0 =1000 ph/s/cm 2 /A 2. Transmission in the visible Q Total =0.15% assuming a. Q CHARA =0.03 b. Q Instrument =0.15 (13 mirrors @ 0.98 + 1 grating 0.6 + the slit 0.3) c. Q Detector =0.3 3. Exposure time t 0 =20ms, integration time=1800s 4. r 0 estimations (for 650 nm) at Mt Wilson: a. Median conditions, r 0 =8.0*(650/500) 6/5 =11.0 cm (seeing=1.25'') b. Excellent conditions, r 0 =15.0*(650/500) 6/5 =20.6 cm (seeing 0. The calculations are made for a seeing of 1.25" (median conditions) and 30mn of integration time.

As already described above, the principle of multimode operation relies on a coherence (group delay) tracker and not on a fringe (phase) tracker. Therefore the previous calculations do not assume an external fringe tracker system but only a coherence tracking system able to stabilize the optical path difference to less than 2 microns over the duration of the integration of short exposures. A good model of baselines and an optical path difference measurement at a period of a few seconds are enough for correct operation of VEGA. Real time processing of the data is used for optical delay stabilisation, a technique that has already been demonstrated on the GI2T with the RAFT experience [START_REF] Koechlin | Dispersed fringe tracking with the multi-r0 apertures of the Grand Interféromètre à 2 Télescopes[END_REF] .

FUNCTIONAL ANALYSIS

Functional Analysis and product breakdown

We can break down the VEGA instrument into two sub-systems, the Interface Optics (Acronym: IOP) and the Spectrograph (Acronym: SPE)

The main functions of the Interface Optics sub-system are to: inject the beams coming from CHARA to VEGA adjust the pupil location (see Annex A) reduce the size of the beams (from 3/4 inches to 5 mm) configure the beams at the entrance of the spectrograph (vertical separation of 10mm) provide artificial beams to calibrate and align the instrument adjust the optical path difference (the zero OPD plan will be the same as the current CHARA visible combiner) The high spectral dispersion of the VEGA instrument means that it is not necessary to correct for the chromatic image shift caused by atmospheric refraction. In addition, the fact that the main part of the optical path between the telescopes and the instrument is inside vacuum tubes allows us to avoid the use of a longitudinal OPD compensator (see Annex B). The main functions of the Spectrograph sub-system are to combine the beams, disperse the fringe signal, optimize the fringe sampling and to select the polarization The following diagram shows the different modules which make up the sub-systems:

DETECTOR

The following table summarizes the different functions of each module:

Implantation

We have studied two possible implantations for the VEGA spectrograph at the CHARA Array. The first option was to remove the current CHARA visible table and to replace it by a larger one able to host the VEGA interface optic. Unfortunately, this solution has a large impact on the current configuration of the CHARA Array and does not minimize the number of reflections for VEGA. We present here the preferred solution for the location of VEGA.

Interfaces with CHARA

The existing elements of the CHARA interferometer will provide some functions necessary for the VEGA instrument. These functions are the following:

• Telescope control.

• Tilt/tilt measurement and correction

• Optical path length equalization

• Optical path difference and delay line control

• Differential field rotation compensation

• Telescope selection (thanks to the Beam Sampler)

• Beam selection (thanks to the shutters located on the CHARA visible combiner)

• Instrument/Interferometer co-alignment (thanks to the different artificial sources, integration tools and optical targets located in the beam combination laboratory)

VISIBLE

Software aspects

All the software aspects of VEGA are based on PC with WindowsXP and Visual C++. We dedicate one PC (single or bi-processor) for each main subsystem (motors, cameras, fringe tracking and main control) so to ensure real time processing.

Each subsystem controller has the same structure (see software structure diagram below):

A server which controls the device(s) according to the reception of high-level commands (change state command or information request, etc).

A client which can be launched locally on the same computer (for debugging or local use) or launched on another computer connected through a standard network remote operation. The main control of sequencing is based on an observing script file composed of elementary commands organized in blocks. These commands are sent one by one to the specified devices. The script file can be easily read and edited in a standard text editor but a dedicated observing preparation tool allows coherent editing of this file. The central control is also responsible for collecting various information representative of the observing conditions and logs them in a file.

The PC and digital camera(s) structure is based on a multicast dialog which allows us to have the same frame on 2 or more computer at the same time. We can than clearly separate real time functions (intensified camera security, data acquisition, fringe tracking, science Quick Look). For this purpose we need one, or better yet 2, gigabit dedicated fiber links between the technical lab and the control room. A one gigabit link is quiet enough for the dataflow of one camera (1000 pixels *1000 pixels * 8bits * 150 frames per second).

Fig. 1 :

 1 Fig. 1: Multi mode measurement of instrumental visibility of the GI2T/REGAIN interferometer.

Fig. 2 :

 2 Fig. 2: schematic representation of the dispersed fringes mode of the VEGA project in the 4 telescope configuration.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Limiting magnitude for 30mn of integration, a signal to noise ratio of 10 on the differential visibility measurement with ∆λ 1 = 50 nm (resp. 40, 6.7) and ∆λ 2 = 0.4 nm (resp. 0.13, 0.02) for the Low LR (resp. medium MR, high HR) spectral resolution and a central wavelength equal to 650nm. The vertical lines correspond to the median and excellent seeing conditions.

Fig. 6 :

 6 Fig. 6: Sub system decomposition of VEGA

Fig. 7 :

 7 Fig. 7: VEGA implantation in the CHARA focal laboratory.

Fig. 8

 8 Fig. 8 VEGA software structure inside the CHARA organisation

Table 2 :

 2 Parameters of the red and blue cameras of the spectrograph.

	Parameters	Red camera	Blue camera
	λ min λ max λ ref	0.58µm 0.87µm 0.7µm	0.45µm 0.75µm 0.57µm
	Slit width	61µm	50µm
	Maximum field of view (center of detector)	5.4''	4.2''
	Number of spectral channels	173	156
	Internal magnification of the spectrograph	1.4	1.8
	(between the slit and the image plane)

Table 3 :

 3 Functional analysis of the different subsystems of VEGA.

	SUB-SYSTEM : INTERFACE OPTICS (IOP)
	MODULE NAME	ACR.	COMPOSITION	M #	FONCTIONS
	Picking Optics	PGO	Set of 4 periscopes	*	Injects the beam coming from CHARA into
					VEGA
					Avoids beam vignetting due to the mounts of the
	Beam Compression	BCP	Parabolic and spherical		optical elements of the CHARA visible combiner Reduces the beam size (0.75 inches → 5 mm)
			mirrors		Adjusts the pupil location
					Adjusts the ZOPD
	Beam Configuration	BCF	Set of 4 flat mirrors		Configures the beam (with a beam separation of
					10 mm)
					Puts the beam in a vertical configuration
	Calibration and	CAU	Artificial sources	*	Produces light for internal alignment and for the
	Alignment Unit		(spectral and halogen		co-alignment between the instrument and
			lamps, laser)		CHARA
			Collimator		Produces light for the calibration (flat-field,
			Mask		distortion grid, spectral calibration, …)
	CHARA/Sources	CSS	Beam splitter	*	Feeds the spectrograph with the internal sources
	Selector		Flat mirror		or with the beams coming from CHARA
	SUB-SYSTEM : SPECTROGRAPH (SPE)
	MODULE NAME	ACR.	COMPOSITION	M #	FONCTIONS
	Focusing Optics	FOC	Parabolic mirror		Produces an image at the spectrograph slits level
	Slit Module	SLM	Slits	*	Selects a part of the image
	Anamorphosis Optics	ANO	2 cylindrical mirrors		Optimizes the sampling (ratio 12 between the
					spectral and the spatial directions)
	Collimating Optics	COL	Parabolic mirror		Collimates the beams
	Polarization Module	POL	Wallaston prism λ/4 blades	*	Allows to separate the 2 polarization directions
	Dispersive Optics	DIS	3 gratings	*	Disperses the light
					Selects between 3 different spectral resolutions
					(1700, 5000, 35000)
	Camera Optics	DFC	Parabolic and flat mirrors		Produces a dispersed fringes image at the detector
					level
	Neutral Densities	NDN	Neutral densities	*	Attenuates the flux
	Detector	DET	Detector	*	Detects the dispersed image

#: if *, motorized or remotely controlled module

The data reduction principles have been published in two main papers 3,4 . A complete description of the current data reduction pipeline (the GI2T one) is available. Adaptations are foreseen for a larger use of this facility and for more automatic processing. Of course adaptations will be made for the 3T and 4T configuration.


Hiske Klaassen

Sophie Tissot

Jordan Meliani

Justine Boutry

Anna Miltiadous

Peter A Biro

David J Mitchell

Beata Ujvari

Aaron Schultz

Frédéric Thomas

Antoine M Dujon
email: antoine.dujon@yahoo.fr

Behavioural ecology meets oncology: quantifying the recovery of animal behaviour to a transient exposure to a cancer risk factor

Keywords: Biological applications Subject Areas: behaviour, environmental science, health and disease and epidemiology neoplasm, movement ecology, freshwater ecology, heavy metals, pollution

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Human activities are increasingly degrading the environment, often creating evolutionary mismatches between the risk of developing malignancies and the effectiveness of natural cancer defences [START_REF] Ujvari | Ecology and evolution of cancer[END_REF]. As a result, many wildlife species are now developing cancers [START_REF] Giraudeau | Human activities might influence oncogenic processes in wild animal populations[END_REF][START_REF] Pesavento | Cancer in wildlife: patterns of emergence[END_REF][START_REF] Dujon | Cancer risk landscapes: a framework to study cancer in ecosystems[END_REF][START_REF] Sepp | Urban environment and cancer in wildlife: available evidence and future research avenues[END_REF][START_REF] Dujon | Sea turtles in the cancer risk landscape: a global meta-analysis of fibropapillomatosis prevalence and associated risk factors[END_REF]. Because of the exacerbated need to activate anti-cancer defences [START_REF] Boutry | Ecological and evolutionary consequences of anticancer adaptations[END_REF], and/or because of the adverse effects of the disease itself [START_REF] Thomas | The importance of cancer cells for animal evolutionary ecology[END_REF], many phenotypic traits are substantially altered in these individuals, which may even ultimately influence ecosystem functioning [START_REF] Vittecoq | Cancer: a missing link in ecosystem functioning?[END_REF][START_REF] Dujon | On the need for integrating cancer into the One Health perspective[END_REF]. For example, oncogenic processes can alter life-history traits in their host at both early and late stages of tumour development [START_REF] Arnal | Cancer brings forward oviposition in the fly Drosophila melanogaster[END_REF][START_REF] Boutry | Tumors alter life history traits in the freshwater cnidarian[END_REF][START_REF] Jones | Life-history change in diseaseravaged Tasmanian devil populations[END_REF][START_REF] Dujon | The effect of placentation type, litter size, lactation and gestation length on cancer risk in mammals[END_REF], increase the risk of predation [START_REF] Boutry | Tumors (re)shape biotic interactions within ecosystems: experimental evidence from the freshwater cnidarian Hydra[END_REF][START_REF] Duneau | Gut cancer increases the risk of Drosophila being preyed upon by hunting spiders[END_REF] and modify their commensal, parasitic and microbiome communities [START_REF] Boutry | Tumors (re)shape biotic interactions within ecosystems: experimental evidence from the freshwater cnidarian Hydra[END_REF][START_REF] Work | Bacteraemia in free-ranging Hawaiian green turtles Chelonia mydas with fibropapillomatosis[END_REF][START_REF] Work | Retrospective pathology survey of green turtles Chelonia mydas with fibropapillomatosis in the Hawaiian Islands, 1993-2003[END_REF][START_REF] Boutry | Spontaneously occurring tumors in different wild-derived strains of hydra[END_REF]. By causing massive mortalities in populations of key ecosystem species, cancers can have major cascading effects on ecosystem biodiversity [START_REF] Hamede | The ecology and evolution of wildlife cancers: applications for management and conservation[END_REF]. In addition, tumour progression can cause major modifications in the behaviour of the host [START_REF] Vittecoq | Animal behaviour and cancer[END_REF]. Examples of this phenomenon include: tumourous Drosophila selectively choosing their social environment to mediate tumour progression [START_REF] Dawson | Social environment mediates cancer progression in Drosophila[END_REF]; Tasmanian devils affected with devil facial tumour diseases (two transmissible cancers) reducing their social interactions with other conspecifics as tumour size increases [START_REF] Hamilton | Infectious disease and sickness behaviour: tumour progression affects interaction patterns and social network structure in wild Tasmanian devils[END_REF]; and green sea turtles affected by fibropapillomatosis spending extended periods of time resting on the seabed compared to healthy individuals [START_REF] Brill | Daily movements, habitat use, and submergence intervals of normal and tumorbearing juvenile green turtles (Chelonia mydas L.) within a foraging area in the Hawaiian Islands[END_REF].

The studies of the effect of tumoural processes on the host's behaviour are currently mainly limited to late stages of tumour progression, because late-stage tumours are easier to observe and we generally lack the tools to detect early stage tumours in wildlife [START_REF] Dujon | Machine learning is a powerful tool to study the effect of cancer on species and ecosystems[END_REF]. However, the exposure of species to cancer risk factors, and/or the very early stages of tumourigenesis are also predicted to have strong effects on the host behaviour [START_REF] Dujon | Cancer risk landscapes: a framework to study cancer in ecosystems[END_REF][START_REF] Thomas | Cancer is not (only) a senescence problem[END_REF][START_REF] Tissot | The impact of food availability on tumorigenesis is evolutionarily conserved[END_REF]. For example, damage to the DNA must be repaired to avoid the accumulation of deleterious mutations leading to the emergence of tumourous cells (see for instance [START_REF] Fortunato | Upregulation of DNA repair genes and cell extrusion underpin the remarkable radiation resistance of Trichoplax adhaerens[END_REF], an energetically costly process for the host [START_REF] Hanahan | Hallmarks of cancer: the next generation[END_REF]). These costs of being exposed to sublethal levels of cancer risk factors can be further increased when species are in an unfavourable ecological setting (for example under increased risk of predation [START_REF] Alton | The energetic cost of exposure to UV radiation for tadpoles is greater when they live with predators[END_REF], or intraspecific competition [START_REF] Almeida | Oxidative stress biomarkers and aggressive behavior in fish exposed to aquatic cadmium contamination[END_REF]).

The effects of tumoural processes on host behaviour are currently primarily quantified at the population level. This may be achieved by estimating the mean value of a given trait, for example distance travelled, number of contacts or distance between conspecifics [START_REF] Dawson | Social environment mediates cancer progression in Drosophila[END_REF][START_REF] Hamilton | Infectious disease and sickness behaviour: tumour progression affects interaction patterns and social network structure in wild Tasmanian devils[END_REF], the duration of time spent resting [START_REF] Brill | Daily movements, habitat use, and submergence intervals of normal and tumorbearing juvenile green turtles (Chelonia mydas L.) within a foraging area in the Hawaiian Islands[END_REF] or the mean number of prey consumed [START_REF] Boutry | Tumors (re)shape biotic interactions within ecosystems: experimental evidence from the freshwater cnidarian Hydra[END_REF]. A focus on the mean, while ignoring individual variation, necessarily limits our inferences and ultimate understanding because individuals are known to differ in susceptibility to cancer; some individuals have greater ability to fight cancer and this ability is often correlated with behaviour [START_REF] Vittecoq | Animal behaviour and cancer[END_REF] (figure 1). For instance, recent conceptual advances and literature reviews indicate that individuals with a high energetic capacity are better able to fight cancer and that high capacity encourages higher activity, and vice versa, in humans and other animals [START_REF] Biro | Can energetic capacity help explain why physical activity reduces cancer risk?[END_REF][START_REF] Biro | A novel perspective suggesting high sustained energy expenditure may be net protective against cancer[END_REF]. In addition, and unlike most other animal attributes, behaviour is a highly labile trait with low repeatability [START_REF] Bell | The repeatability of behaviour: a meta-analysis[END_REF], meaning that repeated measures of behaviour should be employed to estimate an individuals' phenotype, otherwise biased estimates of population means may be obtained [START_REF] Wolak | Guidelines for estimating repeatability[END_REF].

Individual differences in behaviour can be assessed using repeated measures of behaviour, and may be evident as individual differences in activity, individual differences in how activity changes through time, and even differences in how unpredictable an individual is-termed residual intraindividual variation (rIIV). An individual's rIIV may be viewed as a form of unexplained behavioural plasticity or represent within-individual endogenous factors such as energetic constraints, variation in epigenetics and hormonal cycles among others [START_REF] Mitchell | Understanding the unexplained: the magnitude and correlates of individual differences in residual variance[END_REF][START_REF] Cleasby | Quantifying the predictability of behaviour: statistical approaches for the study of betweenindividual variation in the within-individual variance[END_REF][START_REF] Laskowski | The emergence and development of behavioral individuality in clonal fish[END_REF]. Given that more than 60% of overall behavioural variation is unexplained residual variation [START_REF] Bell | The repeatability of behaviour: a meta-analysis[END_REF], individual differences in rIIV might be an important but mostly overlooked individual attribute to consider when studying behaviour [START_REF] Mitchell | Understanding the unexplained: the magnitude and correlates of individual differences in residual variance[END_REF]. Given the individual differences described above, and if they are linked to underlying physiology, we might expect to see a different expression of trait variance under stressful conditions as some individuals may be more affected than others (figure 1).

Here, we used wild planaria (Girardia tigrina), a common freshwater invertebrate, as a study species to study how exposure to cancer risk factors may impact on animal behaviour, and individual behavioural variation at various hierarchical levels. The planaria model has several advantages to study the effects of exposures to sublethal doses of cancer risk factors. Early stages of carcinogenesis can be induced by exposing them to heavy metals or hydrocarbons [START_REF] Roten | A carcinogenic trigger to study the function of tumor suppressor genes in Schmidtea mediterranea[END_REF][START_REF] Voura | Planarians as models of cadmiuminduced neoplasia provide measurable benchmarks for mechanistic studies[END_REF][START_REF] Dujon | A review of the methods used to induce cancer in invertebrates to study its effects on the evolution of species and ecosystem functioning[END_REF]. Their DNA reparation pathways and tumour suppressor genes are activated by an exposure to carcinogens which increase the costs associated with these functions for the animals [START_REF] Roten | A carcinogenic trigger to study the function of tumor suppressor genes in Schmidtea mediterranea[END_REF][START_REF] Wang | Cadmium sulfate induces apoptosis in planarians[END_REF][START_REF] Barghouth | DNA damage and tissue repair: what we can learn from planaria[END_REF]. They are also ecologically relevant, as G. tigrina has a widespread distribution, including in Australia, and is an invasive species which feeds on native invertebrates [START_REF] Grant | Biodiversity of Australian freshwater planarians (Platyhelminthes: Tricladida: Paludicola): new species and localities, and a review of paludicolan distribution in Australia[END_REF]. They are also easy to maintain and experiment with in the laboratory [START_REF] Sousa | Maintenance of Schmidtea mediterranea in the laboratory[END_REF].

To simulate an exposure to a transient cancer risk factor, for the duration of one week we exposed planaria to cadmium, a heavy metal known to induce carcinogenesis in planaria [START_REF] Roten | A carcinogenic trigger to study the function of tumor suppressor genes in Schmidtea mediterranea[END_REF][START_REF] Voura | Planarians as models of cadmiuminduced neoplasia provide measurable benchmarks for mechanistic studies[END_REF]. Activity rates were monitored during this interval, and during a recovery interval of equal duration, to quantify the effect of both periods on the behaviour of the animals. This protocol aimed to mimic the early stage of tumourigenesis, which may not lead to malignant cancer in a wildlife species, but is likely to activate anti-toxicant and anti-cancer defences [START_REF] Fortunato | Upregulation of DNA repair genes and cell extrusion underpin the remarkable radiation resistance of Trichoplax adhaerens[END_REF]. We then combined tools from behavioural ecology, statistical modelling and cancer ecology to explore the data both in terms of quantifying the effect of cadmium exposure and recovery on the population means of measured traits, but also on individual variation in means and individual variation in their rIIV. With this paper we aim to demonstrate the usefulness of this multidisciplinary approach to obtain valuable insights and a more complete picture on the role that tumoural processes play on the ecology of species [START_REF] Dujon | Identifying key questions in the ecology and evolution of cancer[END_REF]. Such insights are important considering how human activities are increasing the exposure of species to a range of cancer risk factors, likely contributing to the degradation of ecosystem functioning and altering the evolutionary course of species.

Material and methods (a) Planaria trapping

Wild freshwater planaria (G. tigrina) were collected from Saint Augustine's lagoon (Highton, Victoria, Australia, 38°11 0 30 00 S, 144°18 0 48 00 N) in September 2022 using custom made planarian traps baited with beef liver and left in the water overnight. Saint Augustine's lagoon is a roughly triangular, 150 m long, and 140 m wide stormwater fed pond that receives urban runoff from the adjacent areas, with Australian eelgrass (Vallisneria australis) as the primary aquatic vegetation. It is regularly stocked with fish for recreational fishing. After capture, the planaria were transported to the laboratory and transferred into spring water (Coles Natural Water Spring) in 1 l food grade polypropylene plastic containers, at a density of approximatively 1 individual per 10 ml of water and left in a dark cupboard in a laboratory at 20°C without feeding for one week to standardize their metabolism. The planaria were left unfed in the subsequent experiments.

(b) Cadmium exposure experiments

Cadmium is a heavy metal and non-essential element which is carcinogenic in humans and wildlife [START_REF] Satarug | A global perspective on cadmium pollution and toxicity in non-occupationally exposed population[END_REF], including planaria, activating anti-cancer defences (e.g. tumour suppressor genes, DNA repair mechanisms) and causing tumours over long periods of time [START_REF] Roten | A carcinogenic trigger to study the function of tumor suppressor genes in Schmidtea mediterranea[END_REF][START_REF] Voura | Planarians as models of cadmiuminduced neoplasia provide measurable benchmarks for mechanistic studies[END_REF][START_REF] Hall | Neoplastic transformation in the planarian: I. Cocarcinogenesis and histopathology[END_REF]. Cadmium contaminates aquatic ecosystems during the mining and processing of zinc or iron ore, the use of phosphate fertilizers in agriculture, waste incineration and royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 291: 20232666 disposal, and via natural emissions such as volcanic eruptions [START_REF] Satarug | A global perspective on cadmium pollution and toxicity in non-occupationally exposed population[END_REF][START_REF] Hutton | Sources of cadmium in the environment[END_REF], and as a result bioaccumulates in the trophic chain [START_REF] Warren | Contamination of sediments by lead, zinc and cadmium: a review[END_REF][START_REF] Dietz | Comparison of contaminants from different trophic levels and ecosystems[END_REF]. Cadmium increases the risk of developing cancer by inducing oxidative stress and impairing major DNA repair pathways, thus increasing the probability of mutations in the genome [START_REF] Valko | Free radicals, metals and antioxidants in oxidative stress-induced cancer[END_REF][START_REF] Hartwig | Cadmium and cancer[END_REF]. In relatively undisturbed Australian aquatic ecosystems, the concentrations of cadmium in water are low, in the order of 0.1 ppb [START_REF] Mackey | Trace elements and organic matter in a pristine environment: Bathurst Harbour[END_REF].

We first determined the lethal concentration 50 (LC50, measured at 96 h) of wild G. tigrina planaria exposed to cadmium. We exposed planaria to increasing nominal concentrations of 0, 0.15, 0.3, 0.6, 1.2, 2.4, 4.8 and 9.6 ppm of Cd 2+ prepared from a 1000 ppm Cd 2+ stock solution obtained by dissolving CdSO 4 salts in spring water (Sigma Aldrich, catalogue no. 481882). We trialled 24 planaria (in six-well plates with one individual per well, Thermo Fisher Scientific, catalogue no. 140675) per treatment for a total of 168 animals and recorded their mortality over a period of 96 h. A LC50 of 2.08 ppm of Cd 2+ (95% CI: 1.83-2.34 ppm) was estimated after analysing the data using a logistic regression (electronic supplementary material, S1). Based on this value we decided to use a nominal concentration of 0.5 ppm of Cd 2+ (close to 25% of the LC50) to perform the behavioural experiments. This concentration was chosen in order to challenge the anticancer defences of planaria without causing mortality. The concentrations of cadmium in the water and sediments of Saint Augustine's Lagoon are not known. In urban environments, Once the animal is moved into a clean environment the mean distance increases over time as it recovers. The variance of the residuals is the same for both phases of the experiment, thus the predictability is the same. (b) As Individual 2 is damaged by cadmium, the mean distance travelled during the first half of the experiment decreases over time. Once the animal is moved into a clean environment the mean distance increases over time as it recovers. The variance of the residuals also increases during the second phase of the experiment. Thus, the predictability decreases during the second phase. (c) The exposure of Individual 3 to cadmium has no effect on the mean distance travelled during each phase of the experiment; however, the variance of the residuals increases during the second phase of the experiment. The predictability decreases during the second phase. (d ) The three individuals are pooled into a statistical population which may be obtained at the end of an experiment. All three individuals have different responses to exposure to cadmium with variations in their mean (solid grey lines) but also in their residual variance. This generates a relatively complex dataset in which both the information about the mean distance travelled by an animal and the residual intraindividual variability (rIIV) must be quantified to capture changes in residual variance to obtain an accurate picture.

Allinson et al. [START_REF] Allinson | Pesticide and trace metal occurrence and aquatic benchmark exceedances in surface waters and sediments of urban wetlands and retention ponds in Melbourne, Australia[END_REF] found that surface sediments of 24 wetlands and retention ponds in Melbourne, located in the same state as our study site, and with relatively similar urbanism, have a median concentration of 2 ppm of cadmium (range: 2-3 ppm). Similarly, Sharley et al. [START_REF] Sharley | Linking urban land use to pollutants in constructed wetlands: implications for stormwater and urban planning[END_REF] measured a mean concentration of 1.2 ppm of cadmium (range: less than 1-11 ppm) in 111 urban wetlands in and around Melbourne. Thus, we considered a nominal concentration of 0.5 ppm of Cd 2+ as appropriate to simulate a sublethal exposure to a cancer risk factor.

To investigate the effects of cadmium exposure on behaviour, 60 planaria were transferred into ten six-well plates, with one individual per well, filled with either 10 ml of 0.5 ppm Cd 2+ spring water solution (treatment) or 10 ml of spring water only (control) and then placed in the dark in a 20°C incubator (Infors HT Ecotron) for 7 days. On day eight of the experiment, the cadmium solution was replaced in both the control and the treatment group with 10 ml of clean spring water after gently rinsing each well with water after which the plate was replaced in the incubator for an additional 7 days. Thus, in this publication we refer to the first 7 days of the experiment as the exposure phase, and the subsequent 7 days as the recovery phase. The experiment length was selected to simulate a short-term exposure to cadmium but also as a trade-off between the logistics involved in the experiment and the number of measurements required to quantify rIIV (here, seven repeated measurements per animal per phase, with an initial 60 individuals per treatment group, see [START_REF] Cleasby | Quantifying the predictability of behaviour: statistical approaches for the study of betweenindividual variation in the within-individual variance[END_REF]).

A behavioural assay was performed on every individual for each day of the 14-day long experiment. The whole experiment was repeated three times with 40 animals (20 in the control group and 20 in the cadmium exposed group) for a total of 120 individuals.

(c) Behavioural assay and movement recording

Once per day, each six-well plate was removed from the incubator and placed on a filtered light pad (Born LED Light Pad, model BOLEDLPA4, 279 lumens, white light) housed within a dark box that blocked out external light sources. When exposed to light, G. tigrina exhibit a photonegative response activity which stimulates their movement [START_REF] Dasheiff | Photonegative response in brown planaria (Dugesia tigrina) following regeneration[END_REF][START_REF] Prados | Short-and long-term habituation of photonegative and exploratory responses in the flatworm planaria (Dugesia)[END_REF]. Planaria activity was filmed using a tripod-mounted Canon camera for 11 min before being replaced in the incubator.

Individual planaria trajectories were reconstructed from video recordings by applying a background subtraction method provided by the OpenCV python library (that we optimized for planaria activity), calculating the average coordinates of the pixels identified as being part of an animal (BackgroundSubtrac-torCNT method [START_REF] Bradski | The OpenCV library[END_REF]). The trajectories were then standardized by interpolating them to one location per second and the activity of each planaria was calculated as the total distance (in mm) travelled by an individual during a trial [START_REF] Réale | Integrating animal temperament within ecology and evolution[END_REF]. The video footage was also used to measure planaria body length (mm). The first minute of the trajectory was discarded to allow the algorithm to accurately detect and track individual planaria (as it uses optical flow between different frames of the video to identify individuals) and the next 10 min were used to determine activity (a duration used in multiple studies, see for example [START_REF] Jordan | Conditioned place avoidance in the planaria Schmidtea mediterranea: a pre-clinical invertebrate model of anxiety-related disorders[END_REF][START_REF] Martinez | Behavioral responses of three freshwater planaria species to light, visual and olfactory stimuli: setting the stage for further ecotoxicological studies[END_REF][START_REF] Wu | Cadmium neurotoxicity to a freshwater planarian[END_REF]). Planaria can be slow moving and take several seconds to be initially detected by the algorithm. Since the video recordings were performed outside the incubator in the same room, we recorded the ambient air temperature (°C) to control for the potential confounding effects on the movement of the planaria.

(d) Statistical analyses of activity data

We used a Gaussian double hierarchical generalized linear model to quantify the effect of cadmium exposure on the activity of planaria [START_REF] Cleasby | Quantifying the predictability of behaviour: statistical approaches for the study of betweenindividual variation in the within-individual variance[END_REF]. This model allows for fixed and random effects to be specified for the mean level of the model that predicts the population average and interindividual variation in activity levels of the planaria, and also in the residual part of the model that permits residual variation to vary with fixed effect predictors and to vary among individuals [START_REF] Cleasby | Quantifying the predictability of behaviour: statistical approaches for the study of betweenindividual variation in the within-individual variance[END_REF][START_REF] Mitchell | Towards powerful experimental and statistical approaches to study intraindividual variability in labile traits[END_REF]. By contrast, most models assume a single residual variance that does not vary (the familiar constant variance assumption). We included a random intercept effect, and a random slope effect with respect to days elapsed within a phase (1-7); these effects capture individual variation in mean activity at day 1, and individual variation in how activity changes through time. We also included the treatment (control or cadmium exposed) as a categorical variable, an interaction between day and treatment group, the length of the planaria (in mm), the temperature (°C) and the batch number (categorical) as fixed effects. Before fitting the model, all continuous variables were mean centred. In the residual part of the model, we included the same fixed effect used in the average activity part, to determine if the predictability of planaria behaviour differs between treatment groups and phases. Additionally, we included a random intercept effect, which permits the model to capture individual differences in residual variation (rIIV). The model was specified with an unstructured correlation matrix, estimating all potential correlations between the intercepts, slopes and rIIV [START_REF] Mitchell | Understanding the unexplained: the magnitude and correlates of individual differences in residual variance[END_REF][START_REF] Cleasby | Quantifying the predictability of behaviour: statistical approaches for the study of betweenindividual variation in the within-individual variance[END_REF][START_REF] Mitchell | Towards powerful experimental and statistical approaches to study intraindividual variability in labile traits[END_REF][START_REF] Hertel | A guide for studying among-individual behavioral variation from movement data in the wild[END_REF] and thus addressing any possible non-independence of the repeated measures within and across phases. The coefficient of variation of predictability (CV P ) was also used to compare the predictability between treatment groups and phases [START_REF] Cleasby | Quantifying the predictability of behaviour: statistical approaches for the study of betweenindividual variation in the within-individual variance[END_REF]. We ensured that the residuals at both the population and the individual level followed a normal distribution using histograms, boxplots and qq-plots to meet the model assumptions. Although activity rates of animals are often skewed, we found this was not the case here, and that logtransforming activity measurements negatively skewed the residuals, violating distribution assumptions. The zero activity values present in the data (9.6% of the total data) did not skew the residual distribution.

When needed, and to visualize the effect of a variable on the mean activity of planaria, or their mean residual variance at the population level, marginal effects were calculated by holding all parameters in the model constant except for the variable of interest [START_REF] Austin | An introduction to propensity score methods for reducing the effects of confounding in observational studies[END_REF].

All statistical analyses and modelling were performed using R v. 4.0.2 [START_REF]brms: an R package for Bayesian multilevel models using Stan[END_REF]. All linear models were fitted using the brms package with default priors [68]. Estimates are provided with their 95% credible intervals (95% CI). The model was run on 4 chains of 5000 iterations each with 500 iterations discarded as warmup and a thinning interval of 2. Diagnostic checks were also performed to ensure that the model converged appropriately ( R = 1, bulk effective sample size and tail effective sample size greater than 1000, random intercepts, and random slopes following a normal distribution [68]). All variance estimates reported in this study are expressed on the log scale. We used the posterior predicted values from the model to estimate differences between treatment means and variances, by calculating a difference value at each iteration to yield a distribution from which we obtained the mean difference and credible interval.

Results

We obtained a total of 1610 activity measurements, arising from 115 planaria. Over the course of the experiment, three planaria died in the control group and two in the treatment group and were excluded from the analysis. No tumours were observed during the experiment.

(a) Effect of cadmium on the average distance travelled by planaria at the population level

During the exposure phase the planaria exposed to cadmium were less active compared to the control planaria, showing an royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 291: 20232666 average difference of 41.5 mm (95% CI: -0.1,82.4 mm) over the whole phase. We observed an increase in the activity of planaria in both groups during the exposure phase of the experiment, but that increase was substantially smaller in exposed planaria compared to control planaria (mean increase of 5.2 mm per day, 95% CI: -1.72,12.29 mm per day in cadmium exposed planaria compared to 26.5 mm per day, 95% CI: 17.5,35.7 mm per day in the control planaria). The water changes on the eighth day, at the start of the recovery phase, caused a drop in the activity in all groups to levels similar to the first day of the exposure phase suggesting a short-term stress response. During this second phase planaria in both groups increased their activity at nearly the same rate (19.1 mm per day, 95% CI: 11.9,26.7 for exposed planaria compared to 25.4 mm per day, 95% CI: 16.5,34.9 mm per day for control planaria; figure 2).

(b) Interindividual variation in the response of planaria to cadmium

During both phases of the experiment, individuals with higher initial activity tended to have higher increases in activity over time. The distance travelled by a planaria during a given day was on average larger when an individual had a high initial activity, albeit with some uncertainties around the effect sizes (figure 3).

The variance in the initial activity predicted by the model (i.e. the intercept variance predicted at day 1) was similar between the two groups during the exposure phase but larger in the control group during the recovery phase. During the exposure phase the variance was σ 2 = 9.6 (95% CI: 9.1,10.1) for the control group and σ 2 = 9.2 (95% CI: 8.6,9.6) for the cadmium exposed group, with an average difference between the two groups of 0.5 (95% CI: -0.3,1.2). During the recovery phase the variance was σ 2 = 9.5 (95% CI: 9.1,10.0) for the control group and σ 2 = 8.7 (95% CI: 8.1,9.3) for the cadmium exposed group, with an average difference between the two groups of 0.8 (95% CI: 0.1,1.5) (figure 3).

We found little evidence that the variance in the daily change in activity of individual planaria (i.e. slope variance) differed between the two phases of the experiment. During the exposure phase the variance was σ 2 = 4.6 (95% CI: -1.1,6.2) for the control group and σ 2 = 6.2 (95% CI: 4.3,7.1) for the cadmium exposed group, with an average difference of -0.4 (95% CI: -5.8,3.9). During the recovery phase the slope variance was σ 2 = 4.7 (95% CI: 0.8,5.9) for the control group and σ 2 = 5.4 (95% CI: 3.3,6.4) for the cadmium group, with an average difference of 0.8 (95% CI: -1.3,3.0) (figure 3).

(c) Effect of cadmium on the residual intraindividual variability

At the population level, the average residual variance slightly decreased over time in both control and cadmium exposed groups, with a small difference during the exposure phase which disappeared during the recovery phase. Planaria length had a considerably stronger effect, whereby residual variance increased with length (figure 4). There were significant interindividual differences in residual variation (inverse predictability) as given by CV P values, but treatment differences were not observed. Individual variation in predictability was similar in the cadmium group (CV P = 0.29, 95% CI: 0.19,0.42) and control group (CV P = 0.24, 95% CI: 0.12,0.38) during the exposure phase. Similarly, the predictability was relatively similar between cadmium exposed (CV P = 0.25, 95% CI: 0.14,0.36) and control group (CV P = 0.37, 95% CI: 0.23,0.52) planaria during the recovery phase.

We found that the individual predictability was strongly correlated to the initial activity of planaria in the control group for both phases. Planaria with high initial activity levels tended to have lower predictability (= higher variation) in activity. This correlation was less pronounced for planaria exposed to cadmium during the recovery phase of the experiment (figure 5).

Discussion

To the best of our knowledge this study is the first to apply experimental designs and advanced statistical modelling royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 291: 20232666 tools from behavioural ecology to study the effect of sublethal cadmium exposure, a cancer defence activator and subsequent recovery in freshwater planaria. Mortality during the experiment was relatively low, and overall, based on their activity levels, the planaria were able to recover from the toxic exposure. However, the behaviour of the planaria was altered. Planaria in contact with cadmium were less active compared to the control group, probably because of the direct damage caused by cadmium and/or because the energy invested in repairing those damages cannot be invested in movement. This decrease in activity was also observed during short-term behavioural trials on planaria [START_REF] Wu | Cadmium neurotoxicity to a freshwater planarian[END_REF][START_REF] Majid | Interactive toxicity of copper and cadmium in regenerating and adult planarians[END_REF]. During the recovery phase, the planaria exposed to cadmium partially recovered, as evidenced by an increase in activity compared to the exposure phase. It is not impossible that, considering their impressive regeneration abilities, planaria would fully recover if given enough time [START_REF] Reddien | Fundamentals of planarian regeneration[END_REF].

Cadmium induced tumours were reported high prevalence within 14 days in one laboratory strain of G. tigrina exposed to concentrations within the range of what we used in our study [START_REF] Voura | Planarians as models of cadmiuminduced neoplasia provide measurable benchmarks for mechanistic studies[END_REF]. No such tumours were observed in our experiments. Thus, it is possible that planaria inhabiting urban stormwater ponds are exposed to pollution which applies selective pressures and are thus relatively resistant to a transient sublethal exposure to cancer risk factors. In addition, G. tigrina is a species native to North America that was introduced to Australia (and many other areas around the world [START_REF] Stocchino | The invasive alien freshwater flatworm Girardia tigrina (Girard, 1850) (Platyhelminthes, Tricladida) in Western Europe: new insights into its morphology, karyology and reproductive biology[END_REF][START_REF] Hay | Contributions to the Biology of freshwater planarians (Turbellaria) from the royalsocietypublishing.org/journal[END_REF]). Metal pollution is known to increase the colonization success of some aquatic invasive species. For example, exposure to copper decreases the species richness of native sessile marine species while having little effects on the species richness of invasive species [START_REF] Crooks | Aquatic pollution increases the relative success of invasive species[END_REF]. Similarly invasive freshwater bivalves tend to be more resistant to heavy metal pollution compared to native ones (for example zinc [START_REF] Bielen | Differences in tolerance to anthropogenic stress between invasive and native bivalves[END_REF]). Our results clearly indicate how an exposure to a transient We observed that both the rate at which the activity increased over time and the rIIV of planaria were correlated with the predicted initial activity at the start of a phase. This pattern linking high activity and rIIV has been observed in different species. For example, wild elephants (Loxodonta africanus) and barn owls (Tyto alba) with high average activity also have a high rIIV [START_REF] Hertel | A guide for studying among-individual behavioral variation from movement data in the wild[END_REF][START_REF] Cain | Movement predictability of individual barn owls facilitates estimation of home range size and survival[END_REF], and bolder sticklebacks (Gasterosteus aculeatus), which explore their space extensively around their shelter, will be less predictable compared to shy individuals [START_REF] Jolles | Personality, plasticity and predictability in sticklebacks: bold fish are less plastic and more predictable than shy fish[END_REF]. In the case of planaria, an interpretation of these results is that high activity is energetically costly and difficult to consistently maintain over long period of times, thus explaining the high residual variance in the control group. When exposed to cadmium, damaged planaria are not able to perform short bursts of activity (the mean activity and the variance in their initial activity was lower than the control group) and their activity is more consistent over time. The effect of a contaminant on the predictability of behaviour seems to dependant on the molecule that planaria are exposed to. For example, the predictability of the behaviour of hermit crabs (Pagurus bernhardus) is similar between individuals exposed to copper and that of control crabs [START_REF] White | How do anthropogenic contaminants (ACs) affect behaviour? Multi-level analysis of the effects of copper on boldness in hermit crabs[END_REF], yet individuals of this same species exposed to microplastics have reduced predictability compared to nonexposed individuals [START_REF] Nanninga | Microplastic exposure increases predictability of predator avoidance strategies in hermit crabs[END_REF]. We also observed that the average residual variance increases with planarian length. As the size of the planaria increases, and, therefore, their ability to move large distances increased, they become less predictable. Thus, experiments investigating the effect of a pollutant on the activity of wild planaria should either use planaria of the same size, to reduce the introduction of this additional residual variance but with the cost of having a sample that is less representative of wild populations, or explicitly include planaria length using a double hierarchical mixed effect model to account for this effect.

The increase in activity in all groups, albeit to a lesser extent in the cadmium exposed planaria, over the course of the experiment suggests a lack of habituation to light exposure [START_REF] Prados | Short-and long-term habituation of photonegative and exploratory responses in the flatworm planaria (Dugesia)[END_REF][START_REF] Westerman | Somatic inheritance of habituation of responses to light in planarians[END_REF] and planaria maintained their photonegative behaviour. The effect of light exposure on the behaviour of planaria has often been studied for a relatively short period of time (e.g. [START_REF] Martinez | Behavioral responses of three freshwater planaria species to light, visual and olfactory stimuli: setting the stage for further ecotoxicological studies[END_REF][START_REF] Paskin | Planarian phototactic assay reveals differential behavioral responses based on wavelength[END_REF]) and studies investigating how planaria respond to repeated light exposure over a period of time of more than a few days are more limited. Our results contrast with previously royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 291: 20232666 published studies. For example, Prados et al. [START_REF] Prados | Short-and long-term habituation of photonegative and exploratory responses in the flatworm planaria (Dugesia)[END_REF] and Westerman [START_REF] Westerman | Somatic inheritance of habituation of responses to light in planarians[END_REF] found that the average distance travelled by G. tigrina repeatedly exposed to light decreased over time. A potential explanation for these differences is that it is likely that those two studies used planaria strains cultivated in the laboratory for extended period of times with food provided ad libitum multiple times per week (often beef liver or egg yolk) [START_REF] Merryman | Culturing planarians in the laboratory[END_REF], thus reducing the selection pressure for predatory behaviour. In the wild, planaria actively hunt for a wide range of invertebrate preys with varying temporal and spatial densities [START_REF] Pickavance | The diet of the immigrant planarian Dugesia tigrina (Girard): II. Food in the wild and comparison with some British species[END_REF]. Thus, it is possible that, as the duration of their starvation period increases, planaria actively seek food sources. It is not possible to have a definitive answer with the experimental design we used and further studies will be required to understand behavioural variation between planaria strains. Our results highlight the importance of comparing the ecology and behaviour of laboratory raised strains of planaria to those of wild populations of the same species. In this study, which is a proof of concept, we only used one concentration of cadmium which simulated a sublethal exposure. Our approaches could be replicated across multiple cadmium concentrations and modelled using a double hierarchical generalized linear model to investigate the shape of the behavioural response of planaria in relation to increasing exposure to cancer risk factors, data which is lacking for nearly all wildlife species. It is possible that as the concentration increases differences in predictability appear between treatment groups, highlighting individuals more resistant to the exposure. This approach could also be applied to a range of other risk factors likely to activate anti-cancer defences and that are predicted to have an impact on animal behaviour [START_REF] Dujon | Cancer risk landscapes: a framework to study cancer in ecosystems[END_REF][START_REF] Vittecoq | Cancer: a missing link in ecosystem functioning?[END_REF]. In addition, it may be interesting to apply this approach using a combination of both tumour initiators (such as cadmium) and promotors, which could increase the speed at which tumourous processes damage the host (as observed in certain species of planaria) [START_REF] Hall | Neoplastic transformation in the planarian: I. Cocarcinogenesis and histopathology[END_REF]. Finally, a direct measure of damage caused by the exposure to the chemical, for example damage to DNA [START_REF] Figueroa-González | Strategies for the evaluation of DNA damage and repair mechanisms in cancer[END_REF], the production of oxygen reactive species [START_REF] Knakievicz | Planarians as invertebrate bioindicators in freshwater environmental quality: the biomarkers approach[END_REF] or the expression of genes [START_REF] Plusquin | Physiological and molecular characterisation of cadmium stress in Schmidtea mediterranea[END_REF] associated with the exposure to cadmium could be included in the model to provide mechanistic insight on the effect of cadmium on the behaviour of the animal. The inclusion of damage measurements would be particularly useful to investigate how the damage-repair ability of individuals, for example in relation to ageing or reproduction, affects the variance and correlates with behavioural traits and the likelihood of tumours appearing (e.g. as observed in humans [START_REF] Huang | Single-cell analysis of somatic mutations in human bronchial epithelial cells in relation to aging and smoking[END_REF]). Overall, our results highlight the importance of considering the effect of the exposure to cancer risk factors on the behaviour of animals.

With this study we demonstrate the value of using behavioural ecology modelling tools to investigate the effect of the exposure to a sublethal cancer risk factor on the behaviour of animals. As environments are increasingly degraded, and royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 291: 20232666 cancer risk landscapes are changing, understanding how oncogenic effects perturb ecosystems is becoming increasingly important. Studies at the interface of ecology and oncology will contribute significantly to predict the negative effect that cancers can have on the biosphere.

Ethics. This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility. The data are provided on Dryad [START_REF] Klaassen | Behavioural ecology meets oncology: quantifying the recovery of animal behaviour to a transient exposure to a cancer risk factor[END_REF] and in the electronic supplementary material [START_REF] Klaassen | Behavioural ecology meets oncology: quantifying the recovery of animal behaviour to a transient exposure to a cancer risk factor[END_REF].

Figure 1 .

 1 Figure1. Hypothetical responses of three different animals in an experiment during which they are exposed to cadmium for one week and then recover in a clean environment for one additional week. During each day of the experiment each animal is subjected to a behavioural trial during which the distance travelled is measured. In all plots, the activity of animals during each behavioural trial are represented as grey dots, the blue lines represent the mean distance travelled by a planarian during the first 7 days of the experiment, and the orange lines represent the mean distance travelled by a planarian during the next 7 days. (a) As Individual 1 is damaged by cadmium (e.g. by tumoural processes), the mean distance travelled during the first half of the experiment decreases over time. Once the animal is moved into a clean environment the mean distance increases over time as it recovers. The variance of the residuals is the same for both phases of the experiment, thus the predictability is the same. (b) As Individual 2 is damaged by cadmium, the mean distance travelled during the first half of the experiment decreases over time. Once the animal is moved into a clean environment the mean distance increases over time as it recovers. The variance of the residuals also increases during the second phase of the experiment. Thus, the predictability decreases during the second phase. (c) The exposure of Individual 3 to cadmium has no effect on the mean distance travelled during each phase of the experiment; however, the variance of the residuals increases during the second phase of the experiment. The predictability decreases during the second phase. (d ) The three individuals are pooled into a statistical population which may be obtained at the end of an experiment. All three individuals have different responses to exposure to cadmium with variations in their mean (solid grey lines) but also in their residual variance. This generates a relatively complex dataset in which both the information about the mean distance travelled by an animal and the residual intraindividual variability (rIIV) must be quantified to capture changes in residual variance to obtain an accurate picture.

Figure 2 .

 2 Figure 2. Effects of the exposure to cadmium, represented as their marginal effects, on the mean activity of planaria (solid lines, with their associated 95% CI error bands) during the (a) exposure and (b) recovery phases of the experiments. The planaria exposed to cadmium were less active in both phases compared to the treatment group.

rFigure 3 .

 3 Figure 3. Effect of a sublethal exposure to cadmium followed by a recovery phase on the initial activity and change in activity (represented by black dots with their associated standard error as grey lines) on (a,b) control unexposed planaria and (c,d ) exposed planaria. The histograms on the right of the plots visualize the variance in the predicted initial activity (in mm) of planaria during the two phases of the experiments (while accounting for temperature, body size and batch effects). The histograms on the top of the plots visualize the variance in the predicted change in activity (in mm per day) of planaria during the two phases of the experiments (while accounting for temperature, body size and batch effects). Exposure to cadmium reduced the variance of the initial activity of planaria during both the exposure and recovery phases of the experiment. No difference in variance in the daily change between phases was observed.

Figure 4 .

 4 Figure 4. Effects of cadmium exposure on the mean residual variance of the activity of planaria (solid lines, with their associated 95% CI error bands) during (a) the exposure and (b) recovery phases of the experiments. (c) Effect of planaria length on the mean the mean residual variance of the activity of planaria.

rFigure 5 .

 5 Figure 5. Residual interindividual variability (rIIV, black dots) in function of the initial activity levels of planaria during the exposure and recovery phases of the experiments. Planaria with higher initial activity levels in the control group exhibit a higher rIIV, and are thus less predictable, often with large variations in day-today activity. Grey lines represent the 95% CI of each estimate.

Downloaded from https://royalsocietypublishing.org/ on 20 February 2024

royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 291: 20232666

Funding. This work was funded by the MAVA Foundation and the Hoffman family, a CNRS International Associated Laboratory Grant, the ANR EVOSEXCAN project (ANR-23-CE13-0007) and an Alfred Deakin Postdoctoral Research Fellowship (A.M.D.).

Declaration of AI use. We have not used AI-assisted technologies in creating this article.

Authors' contributions. H.K.: conceptualization, data curation, investigation, methodology, writing-review and editing; S.T.: conceptualization, methodology, validation, writing-review and editing; J.M.: methodology, resources, validation, writing-review and editing; J.B.: writing-review and editing; A.M.: investigation, writing-review and editing; P.A.B.: investigation, methodology, validation, visualization, writing-original draft, writing-review and editing; D.J.M.: methodology, validation, writing-review and editing; B.U.: writing-review and editing; A.S.: conceptualization, methodology, writing-review and editing; F.T.: funding acquisition, writing-original draft, writing-review and editing; A.M.D.: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, supervision, validation, visualization, writing-original draft, writing-review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration. The authors have no conflict of interest to declare.


Xing-Yi Wang

Adlan Merlo

Claire Dupont

Anne-Virginie Salsac

Dominique Barthès-Biesel

X.-Y Wang

A.-V Salsac

Identification of the mechanical properties of capsules using semi-automatic microfluidic method

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Liquid core microcapsules are widely used in pharmacology, biotechnology to protect sensitive substances from the environment by means of a thin hyperelastic membrane. When subjected to an external flow, capsules undergo large deformation and the membrane may break. It is thus important to characterize the mechanical properties of microcapsule membranes to control the release of the internal fluid. For this purpose, we propose a microfluidic method based on an automatic inverse analysis of the deformed profiles of capsules flowing in a capillary tube to measure the elastic properties of a population of microcapsules.

Methods

The three-dimensional dynamics of an initially spherical capsule flowing in a cylindrical channel, when its membrane is governed by a neo-Hookean (NH) or a general Hooke's law (GH) is solved by means of a fluid-structure interaction numerical model [1]. The capsule deformed profile and velocity are gathered in numerical databases, for different capsule-to-tube confinement ratios, relative flow strengths and material constitutive laws. The experimental deformed shape of a microcapsule flowing in a cylindrical tube is compared to those in the database. The best diffuse approximation fit, allows one to determine the unknown membrane shear modulus and the capsule size [2]. Another method consists of using microrheometry, where a capsule is subjected to a simple shear flow: the deformation is then directly linked to the flow strength for a given membrane constitutive law [3].

Results

The two methods are used to characterize a suspension of artificial capsules with a cross-linked ovalbumin membrane. The membrane shear modulus predicted by the microfluidic method decreases with deformation for GH law. For a NH law, an approximately constant shear modulus 0.043 ± 0.004 N.m -1, is obtained. The mechanical behaviour of the ovalbumin membrane can thus be modelled with the NH law. The average value of the shear modulus predicted by microrheometry is 0.039 ± 0.01 N.m -1 for the NH law, which validates the microfluid method [4].

Discussion

The microfluidic method is simple to implement and can be automatized for on-line measurements to characterize in real time the mechanical properties of a suspension of microcapsules with a 10% precision. It also allows one to determine the type of mechanical behaviour of a capsules wall, which is very important as capsules with a strain softening membrane may be prone to break-up though continuous elongation.

Acknowledgements

This work received funding from the European Research Council (ERC) (Grant agreement No. ERC-2017-COG -MultiphysMicroCaps).


Claire Dupont
email: claire.dupont@utc.fr

Florian De Vuyst

Anne-Virginie Salsac

DATA-DRIVEN KINEMATICS-CONSISTENT MODEL ORDER REDUCTION OF FLUID-STRUCTURE INTERACTION PROBLEMS: APPLICATION TO MICROCAPSULES IN FLOW

Keywords: capsule, data-driven, Reduced Order Model, Proper Orthogonal Decomposition, Dynamic Mode Decomposition

Data-driven kinematics-consistent model order reduction of fluid-structure interaction problems: application to microcapsules in flow

Today the solution of fluid-structure interaction (FSI) problems is central to numerous engineering and bioengineering applications. High-fidelity numerical simulations are developed for computed-aided analysis. However, they are very time-consuming even when using high-performance computing facilities. To reduce the computational cost, model-order reduction methodologies have been developed. Efficient intrusive reduced order models (ROM) for FSI have been proposed e.g. in [START_REF] Quarteroni | Reduced basis methods for partial differential equations -An introduction[END_REF]. But there are far less contributions in non-intrusive ROMs dedicated to FSI. We propose a generic approach of a dynamical data-driven model-order reduction technique for three-dimensional FSI problems. We use a coupling between Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD), and a Tikhonov regularization for robust identification purposes. The method is designed to be non-intrusive: the results of the three-dimensional capsule dynamics simulated by a high-fidelity solver (called also Full Order Model (FOM)) [START_REF] Walter | Coupling of finite element and boundary integral methods for a capsule in a Stokes flow[END_REF][START_REF] Hu | Flow of a spherical capsule in a pore with circular or square cross-section[END_REF] are used as input of the ROM. The ROM methodology is thus data-driven. The ROM is consistent with the equations of kinematics: the model returns the displacement and velocity fields from a few snapshots provided by the FOM. To show the potential of the method and identify its limits, we will apply it to canonical examples of a microcapsule (liquid droplet enclosed by a thin hyperelastic membrane) subjected to an external flow (microchannel or simple shear flow) [START_REF] Dupont | Data-driven kinematics-consistent model order reduction of fluid-structure interaction problems: application to deformable microcapsules in a Stokes flow[END_REF]. In the case of a capsule flowing in a microchannel with a square section, the FOM is used to get a complete numerical training database of the capsule dynamics as a function of the parameters of the problem:

• The capillary number Ca, which measures the ratio between the viscous forces acting onto the capsule membrane and the membrane elastic forces,

• The capsule-to-tube confinement ratio a/ , when the external flow is confined.

An interpolation method is implemented to predict the solution for any values of governing parameters that are not part of the training database. We get very satisfactory results. The model is able to predict the capsule dynamics for any (a/ , Ca) couple. The temporal evolution of the difference between the 3D capsule shape predicted by the ROM and that given by the FOM is illustrated in Fig. 1 for a channel flow. The maximum difference is 0.01% in this case and never exceeds 0.3% over the entire range of parameter values. The excellent agreement is proven by the inserts showing the superposition of the two profiles in the mid-plane. Comparing the computational times, we obtain a speedup of the online stage varying between 4200 when Ca ≥ 0.05 and 52106 for the lowest tested value of Ca. It is thus possible to estimate the capsule dynamics very precisely with the developed ROM, while considerably reducing the computational time.

Figure 1 .

 1 Figure 1. Temporal evolution of the difference between the 3D reference capsule shape given by the FOM and the 3D shape predicted by the ROM (Ca = 0.13, a/ = 0.8). The inserts shows the 3D dynamics predicted by the ROM and the comparaison of the capsule shapes in the channel mid-plane (black: FOM, orange: ROM).


Chaymae El Mertahi

Nicolas Grandmaison

Claire Dupont

Rachid Jellali

Delphine Brancherie

Anne-Virginie Salsac
email: a.salsac@utc.fr

Microrheometric study of damage and rupture of capsules in

Keywords: microcapsules, simple shear flow, microrheometry, fluid-structure interactions, damage, rupture

Microcapsules, which are fluid droplets enveloped by a thin elastic membrane, are widely used in industry to protect and transport active substances, aromas, cells, etc. When subjected to an external flow, they undergo strong deformations, which can potentially lead to their breakup. Since the 1990's, capsule membrane properties have been studied until breakup subjecting the particles to various flow conditions: simple shear flow [START_REF] Chang | Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow[END_REF], hyperbolic flow [START_REF] Xie | Interfacial rheological properties of self-assembling biopolymer microcapsules[END_REF], spinning flow [START_REF] Husmann | Deformation and bursting of nonspherical polysiloxane microcapsules in a spinningdrop apparatus[END_REF], tube flow [START_REF] Wang | A microfluidic methodology to identify the mechanical properties of capsules: comparison with a microrheometric approach Flow 1[END_REF], etc. But hardly any study provided a comprehensive set of experimental data on rupture or investigated the underlying damage phenomena that eventually lead to breakup. To overcome the lack of experimental data in the literature, we have conducted a microrheometric study of microcapsules, subjecting them to simple shear flow conditions leading to their damage and rupture. Our objectives are to characterize the impact of damage on the mechanical properties of the capsule membrane and determine the occurrence of rupture depending on the parameters of the problem. Suspensions of ovalbumin microcapsules are prepared using an interfacial cross-linking technique [START_REF] Edwards-Lévy | Determination of free amino group content of serum albumin microcapsules using trinitrobenzenesulfonic acid: effect of variations in polycondensation pH[END_REF]. This technique provides spherical deformable capsules with radii ranging from a few dozens to a few hundreds of microns. To avoid any osmotic prestress in the membrane caused by differences in concentration between the internal and external fluids, capsules are suspended in a glycerol solution containing 3%(w/v) of ovalbumin. They are then sheared in a counter-rotating microrheometer (Figure 1) under increasing values of shear rate and exposure time to shear. They are visualized using a high speed camera in the mid-plane of the shear flow in order to determine whether each of the capsules entering the shear flow mid-plane has been damaged or not by hydrodynamic forces. The surface shear modulus of observed capsules is identified using the inverse analysis method of Wang et al [START_REF] Wang | A microfluidic methodology to identify the mechanical properties of capsules: comparison with a microrheometric approach Flow 1[END_REF]: it consists in comparing the measured capsule profiles with those computed by a full numerical model solving the capsule-flow interactions and in finding the best fit. For time exposures above 2 minutes, a significant decrease in surface shear modulus is observed over time for capsules of radius larger than 50 µm. The study of rupture is based on the quantification of the number of capsules present before and after shear exposure for different values of shear rates and exposure times to shear. The difference between both is assumed to be the number of ruptured capsules. The percentage of broken capsules increases with their radius. A phase diagram is obtained varying the capillary number (ratio of the viscous to the elastic characteristic forces) and the exposure time to shear. For a given exposure time to shear, increasing the capillary number increases the probability of rupture. And as the exposure time to shear increases, the capillary number required to observe capsule breakup decreases.

45°-mirror

Figure 1 .

 1 Figure 1. Microrheometry set-up.


Xu-Qu Hu

Claire Dupont

Dominique Barthès-Biesel

Anne-Virginie Salsac
email: a.salsac@utc.fr

INTERACTION DAMAGE OF TWO CAPSULES IN SHEAR FLOW: EFFECT OF MEMBRANE PROPERTIES DIFFERENCES

Keywords: capsule, suspension, damage, boundary integral method, finite element method

Capsules are widely used in industry to protect an internal fluid against the external environment and control its release. Due to processing problems, suspensions of artificial capsules (or of cells) are rarely uniform. It is thus important to analyze the pair interaction of capsules with different physical properties in order to evaluate the stress levels in the membranes and predict potential mechanical damage that would alter the integrity of the particles. We consider two spherical capsules C 1 and C 2 , filled with the same liquid and enclosed by a very thin Neo-Hookean membrane. They have the same radius but their membranes have different surface shear modulus (G s1 and G s2 ). They are freely suspended in a simple shear flow under inertialess conditions. The capsule motion is computed by means of a fluid structure interaction model based on the boundary integral method to solve for the flow, coupled to a finite element method to compute the capsule wall deformation [START_REF] Walter | Coupling of finite element and boundary integral methods for a capsule in a Stokes flow[END_REF][START_REF] Hu | Minuet motion of a pair of capsules interacting in simple shear flow[END_REF]. [START_REF] Lac | Pairwise interaction of capsules in simple shear flow: Three-dimensional effects[END_REF][START_REF] Singh | Hydrodynamic interactions between pairs of capsules and drops in a simple shear: Effects of viscosity ratio and heterogeneous collision[END_REF][START_REF] Hu | Minuet motion of a pair of capsules interacting in simple shear flow[END_REF]. They continue their path and are eventually far enough to recover the single capsule profile that they had previously. However, an irreversible trajectory shift of their center of mass is observed, leading to self diffusion effects in the suspension. The numerical model allows us to monitor the mechanical stress level in the capsule membranes during the whole interaction process. Specifically, during the close interaction phase, the capsules undergo a peak in It is thus possible that at least one of the capsules exceeds the damage threshold [START_REF] Grandmaison | Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture[END_REF]. For a given flow strength, a capsule may therefore be damaged because of the interaction whereas it would not have been damaged if it had been alone in the flow. We will present phase diagrams that will show how the interaction damage depends on the capsule specific properties and on the flow strength. This information is useful to adjust the flow conditions to prevent damage to the weakest particles in a suspension, or to eliminate particles with properties that are below a given threshold.

  s

Figure 1 .

 1 Figure 1. Trajectories and deformation of two initially spherical capsules subjected to a simple shear. The grey arrows indicate the direction of motion for each capsule and the black arrows show the flow direction. In this case G s1 = 4G s2 . The two capsules are initially positioned in the same shear plane but on different streamlines (Fig 1). The initial distance along the flow direction is large: the capsules thus first reach the steady deformation that they would have if they were alone in the flow. Then they come closer and pass each other (Fig 1) doing a leap-frog motion in the shear plane[START_REF] Lac | Pairwise interaction of capsules in simple shear flow: Three-dimensional effects[END_REF][START_REF] Singh | Hydrodynamic interactions between pairs of capsules and drops in a simple shear: Effects of viscosity ratio and heterogeneous collision[END_REF][START_REF] Hu | Minuet motion of a pair of capsules interacting in simple shear flow[END_REF]. They continue their path and are eventually far enough to recover the single capsule profile that they had previously. However, an irreversible trajectory shift of their center of mass is observed, leading to self diffusion effects in the suspension. The numerical model allows us to monitor the mechanical stress level in the capsule membranes during the whole interaction process. Specifically, during the close interaction phase, the capsules undergo a peak in

Figure 2 .

 2 Figure 2. Evolution of the membrane energy W as a function of the centroid position along e x for both capsules (G s1 = 4G s2 ). The value of the membrane energy when the capsule is alone in the flow at the same flow strength is represented by the dashed line. The arrows indicate the direction of time evolution.


Nicolas Grandmaison

Delphine Brancherie

Anne-Virginie Salsac
email: a.salsac@utc.fr

MODELLING OF DAMAGE AND STRAIN LOCALIZATION OF A CAPSULE IN FLOW †

Keywords: Capsule, Fluid-structure interaction, Damage, Strain localization, Rupture

INTRODUCTION

Capsules, which consist of a droplet surrounded by a thin elastic membrane, offer promising biomedical applications for the controlled release of active substances. However, one limitation comes from the lack of model of damage/rupture of capsules in flow. Recently, we developed a fluid-structure interaction (FSI) numerical model of a capsule in flow accounting for the diffuse damage mechanisms in the membrane [START_REF] Grandmaison | Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture[END_REF]. The aim of the present work is to extend this model to take into account the subsequent phenomenon of strain localization, corresponding to the concentration of strain and damage in zones of very small thickness over the membrane. These localization zones are precursors of the onset of a crack. In order to take into account the effect of these very localized phenomena on the material response of the capsule wall, we follow a multi-scale approach based on the Embedded Finite Element method.

MODEL AND NUMERICAL METHOD

Macro-scale FSI problem

We consider an initially spherical liquid-core capsule surrounded by an elastic wall modeled as a twodimensional membrane S. The capsule is immersed in a simple shear flow. The internal and external flows are governed by the Stokes equations which are reformulated using a boundary integral method [START_REF] Grandmaison | Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture[END_REF]. The membrane is in equilibrium at each instant. Therefore, the principle of virtual work gives a relation between the internal first Piola-Kirchhoff tension tensor P of the membrane and the external load vector q on the membrane:

for any virtual displacement ŵ , ż S 0 ∇ 0 s ŵ : P dS 0 " ż S ŵ ¨q dS , (1)
where S 0 is the reference configuration of S, and ∇ 0 s ŵ is the surface gradient of ŵ over S 0 . The motions of the fluids and the membrane are coupled by imposing the continuity of the velocities and stresses over S. The FSI problem is governed by the capillary number Ca (ratio of the fluid viscous forces to the membrane forces).

Micro-scale material problem

To take into account localized phenomena while computing the tensions P (eq. ( 1)), the deformation field is enriched locally. Following the strong discontinuity approach [START_REF] Oliver | Continuum modelling of strong discontinuities in solid mechanics using damage models[END_REF], the localization zones are represented at the micro-scale as lines Γ 0 on which the membrane displacements are discontinuous. A discontinuity Γ 0 affects the strains of the material points located in a neighbourhood S µ Ă S 0 such that the micro-scale membrane displacement field in S µ is decomposed as u µ " u `M vuw, where u is the continuous membrane macroscale displacement associated with S, the vector vuw is the displacement jump on Γ 0 , and the function M is discontinuous on Γ 0 . The jump vuw is determined from the equilibrium equation of a discontinuity:

t Γ " P `¨N " P ´¨N , (2)
where t Γ is the nominal traction vector acting on Γ 0 , P `and P ´are the values of P on both sides of Γ 0 , and N is the unit vector perpendicular to Γ 0 on S 0 . The local problem is closed using the constitutive laws of the bulk and discontinuity. For the material points of the bulk (S 0 zΓ 0 ) it is the one associated with the model given in [START_REF] Grandmaison | Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture[END_REF], which takes into account diffuse damage through the damage variable d, chosen as a scalar ranging from 0 (sound material) to the threshold d c (ď 1) corresponding to the introduction of a discontinuity Γ 0 (see [START_REF] Grandmaison | Modelling of damage of a liquid-core microcapsule in simple shear flow until rupture[END_REF] for more details). The constitutive behaviour of the discontinuities is given by a brittle cohesive law relating the traction vector t Γ to the displacement jump vuw.

Numerical method

We perform a Lagrangian tracking of the membrane. At the beginning of a given time step, the current configuration of the membrane is known. We start by solving the solid problem using the finite element method. First, we solve the local equilibrium (eq. ( 2)) to determine the displacement jumps and compute the tensions. Then, we solve the global equilibrium (eq. ( 1)) to find the external load at the nodes. Next, we solve the fluid problem by computing the velocity of the fluids at the nodes using the boundary integral formulation. Finally, the velocity is explicitly integrated to update the position of the nodes for the next time step.

RESULTS

We have studied the damage and rupture mechanisms of a capsule as a function of the value of Ca for fixed values of the material parameters. Damage occurs when Ca is greater than a critical value Ca c , and a larger threshold value Ca h of Ca delimits two failure modes. For the first failure mode (Ca c ď Ca ă Ca h ), damage develops on the flanks of the capsule around the tip of the vorticity axis while the capsule elongates, and damage leads to the initiation of rupture when Ca is larger than a limit value Ca ℓ . If Ca ă Ca ℓ , the capsule reaches a steady state of damage. For Ca ℓ ď Ca ă Ca h , rupture initiates when d " d c at the tips of the vorticity axis and the two cracks formed by the discontinuities then propagate perpendicularly to the direction of capsule elongation (Figure 1a). For the second failure mode (Ca ě Ca h ), damage develops at the apices of the short axis of the capsule while the capsule elongates, and rupture initiates when d " d c at the apices from which two cracks then propagate perpendicularly to the direction of capsule elongation (Figure 1b).

Figure 1 .

 1 Figure 1. Damage of a capsule observed in the shear plane at an instant during cracks propagation in the cases (a) Ca ℓ ď Ca ă Ca h and (b) Ca ě Ca h . The discontinuities are represented by the thick black lines.

† This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. ERC-2017-COG -MultiphysMicroCaps).