D Zeka

P.-A Guidault

D Néron

M Guiton

G Enchery

Model reduction for non-linear structural problems with multiple contact interfaces: application to the modeling of fowt mooring lines

Model reduction for non-linear structural problems with multiple contact interfaces:

application to the modeling of fowt mooring lines D. Zeka a , P.-A. Guidault a , D. Néron a , M. Guiton b , G. Enchery c a. Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS -Laboratoire de Mécanique Paris-Saclay, 91190 Gif-sur-Yvette, France {donald.zeka, pierre-alain.guidault, david.neron}@ens-paris-saclay.fr b. IFP Energies nouvelles, Rond-Point de l'échangeur de Solaize -BP 3, 69360 Solaize, France c. IFP Energies nouvelles, 1-4 Avenue de Bois Préau, 92852 Rueil-Malmaison Cedex, France {martin.guiton, guillaume.enchery}@ipfen.fr

The context of this work concerns the simulation of the loading on a part of a mooring line of a floating offshore wind turbine in order to accurately predict its fatigue life. Mooring lines are usually constituted by spiral strand wire ropes, which consist in a complex assembly of steel wires wrapped together in a twisted helical assembly of different wire layers around a single core wire. Due to their peculiar architecture, tension and bending loadings induce complex frictional phenomena between the wires that may induce fretting fatigue damage. A direct finite element analysis of the wire mechanics in order to predict fatigue life is computationally costly, since such kind of problem involves simulating the complex evolution of contact and friction conditions between the wires of the wire rope [START_REF] Bussolati | Modèle multi-échelle de la fatigue des lignes d'ancrage câblées pour l'éolien offshore flottant[END_REF][START_REF] Bussolati | Robust contact and friction model for the fatigue estimate of a wire rope in the mooring line of a Floating Offshore Wind Turbine[END_REF]. In order to overcome these difficulties, we will focus on model reduction methods but also on domain decomposition methods in order to efficiently deal with the multiscale content of a non-linear problem with multiple contact interfaces.

The main challenges of the problem rely on the fact that the application of model reduction for structural problems involving a large number of frictional contact interactions is a challenging issue: in fact a reduced-order model may not efficiently capture and represent singular and propagating phenomena at the contact interfaces, such as long frictional cracks, macrosliding, sliding fronts and contact dettachment between the wires. In literature one can cite a posteriori approaches with reduced basis enrichment such as the non-negative matrix factorization method [START_REF] Balajewicz | Projection-based model reduction for contact problems[END_REF] or the cone-projected greedy algorithm [START_REF] Benaceur | A reduced basis method for parametrized variational inequalities applied to contact mechanics[END_REF]. A priori methods, which build the reduced basis on the fly throughout the computation, rely on the Proper Generalized Decomposition (PGD) [START_REF] Nouy | A priori model reduction through proper generalized decomposition for solving time dependent partial differential equations[END_REF] in the framework of the LATIN non-incremental nonlinear solver [START_REF] Ladevèze | Nonlinear Computational Structural Mechanics[END_REF].

In this work, the proposed strategy aims at solving the frictional contact between the spiral strand wires in a robust and efficient way. It combines the following techniques : 1) Domain Decomposition Methods (DDM), in particular the mixed DDM based on the LATIN method [START_REF] Bussolati | Robust contact and friction model for the fatigue estimate of a wire rope in the mooring line of a Floating Offshore Wind Turbine[END_REF]; 2) A priori model reduction based on the PGD. This approach will be investigated in order to reduce the computational cost of the analyses.

In this context, the first step is to investigate the potential reducibility of the addressed problem. An a posteriori SVD analysis has been performed on the solution of a tension and bending calculation on a metric cable section developed in [START_REF] Bussolati | Modèle multi-échelle de la fatigue des lignes d'ancrage câblées pour l'éolien offshore flottant[END_REF][START_REF] Bussolati | Robust contact and friction model for the fatigue estimate of a wire rope in the mooring line of a Floating Offshore Wind Turbine[END_REF] subjected to small-amplitude oscillating loads around a pre-loaded state. A general good potential for reducibility for the different contact quantities (normal contact forces, frictional forces, inter-wire slip) is observed, due to the small variations in contact conditions induced by the particular loading. The outermost layers result to be the most critical especially for frictional forces, where bending effects are more pronounced and the contact conditions are more variable. Space modes depict a multi-scale behaviour with first global structural modes and subsequent localized modes around the sliding zones. This suggests that a multiscale computational strategy may be beneficial.

Thereafter academic examples which are representative of the physical problem at hand have been investigated with the proposed approach. A one-dimensional benchmark model of a wire of the wire rope, subjected to traction and pressure and in contact with a frictional interface, has been analyzed to highlight how the introduction of domain decomposition, allowing for the separation of sticking and sliding zones, allows for more efficient creation of local reduced bases in order to be more accurate in areas where it is needed. The analysis also shows that domains with similar contact conditions present similar modes, which suggests the idea of sharing reduced bases in order to improve efficiency. The introduction of a proper coarse-scale problem for the DDM strategy allows to improve performances.

Finally, the LATIN-based multiscale mixed DDM with PGD [START_REF] Ladevèze | A multiscale computational approach for contact problems[END_REF][START_REF] Ladevèze | The LATIN multiscale computational method and the Proper Generalized Decomposition[END_REF] is applied to two-dimensional benchmark contact problems, representative of a layer of wires of the wire. It is shown to what extent the DDM coarse-scale problem and the introduction of the PGD, which builds reduced basis per subdomain, can be efficient in the case of multiple contact interfaces.


Salma Maataoui
email: salma.maataoui@edu.umi.ac.ma

Ghita Bencheikh
email: gbencheikh@cesi.fr

Predictive Maintenance in the Industrial Sector: A CRISP-DM Approach for Developing Accurate Machine Failure Prediction Models

Keywords: Predictive maintenance, Data driven model, CRISP-DM, Industry 4.0

In production systems, avoiding repeated failures is crucial for reducing costs and preventing downtime. Industry 4.0 technologies have enabled companies to collect and analyze real-time data from machines, which helps in identifying and preventing potential problems. By using metrics like MTBF and MTTR and analyzing past failures, we can develop predictive models to prevent future failures. This paper explores the use of CRISP-DM methodology in the industrial sector to ensure the accurate prediction of machine failures. Specifically, we examine the application of this methodology in developing predictive models for cutting machines. The results demonstrate that CRISP-DM methodology is effective in developing models that can accurately predict potential failures and prevent them from occurring. The findings have implications for companies looking to implement predictive maintenance strategies in their production systems, highlighting the importance of using data-driven approaches to improve reliability and reduce downtime. Overall, our study highlights the importance of leveraging industry 4.0 technologies and CRISP-DM methodology for optimal performance of production systems in the industrial sector.

I. INTRODUCTION

Predictive maintenance (PdM) has become an indispensable tool for managing machinery health in various industries due to the advent of Industry 4.0 and the widespread use of advanced technologies like intelligent automation, machine learning, and artificial intelligence [START_REF] Carvalho | A systematic literature review of machine learning methods applied to predictive maintenance[END_REF], [START_REF] Bousdekis | Predictive maintenance in the 4th industrial revolution: Benefits, business opportunities, and managerial implications[END_REF]. PdM is even regarded as the driving force behind the fourth industrial revolution [START_REF] Achouch | On predictive maintenance in industry 4.0: Overview, models, and challenges[END_REF], and considered as a high priority topic in industry [START_REF] Zhai | Enabling predictive maintenance integrated production scheduling by operation-specific health prognostics with generative deep learning[END_REF]. As modern machines become more complex and the demand for optimal performance increases, PdM has emerged as a proactive maintenance approach that aims to identify and address potential equipment issues before they become major problems [START_REF] Mobley | An introduction to predictive maintenance[END_REF]. PdM leverages data analytics, sensors, and other cutting-edge technologies to enable companies to optimize their maintenance schedules, minimize downtime, and reduce cost and duration associated with repairs and replacements. Furthermore, PdM helps companies avoid costly shutdowns, enhance safety, and maintain quality standards [START_REF] Lindström | Towards intelligent and sustainable production: combining and integrating online predictive maintenance and continuous quality control[END_REF]- [START_REF] Dalzochio | Machine learning and reasoning for predictive maintenance in industry 4.0: Current status and challenges[END_REF]. PdM is gaining even more attention from researchers since artificial intelligence (AI) is involved, offering more evolutionary perspectives [START_REF] Nguyen | A review of artificial intelligence methods for engineering prognostics and health management with implementation guidelines[END_REF]. The incorporation of Machine Learning (ML) has transformed decision-making for both individuals and organizations, enabling it to identify illnesses, enhance productivity, optimize transport routes, predict weather, detect fraudulent activities, and much more [START_REF] Hong | State-of-the-art on research and applications of machine learning in the building life cycle[END_REF]. By allowing machines to learn from examples and experiences, ML uncovers hidden insights in data [START_REF] Thrun | Learning to learn[END_REF]. Through data analysis, identifying patterns, and developing predictive logic, machines can predict outcomes without any explicit programming [START_REF] Thrun | Learning to learn[END_REF], [START_REF] Mahesh | Machine learning algorithms-a review[END_REF]. PdM can benefit greatly from the application of ML techniques in several ways:

• Improved accuracy: ML algorithms can learn from historical data and make predictions with a high level of accuracy. This can help organizations to reduce the risk of unplanned downtime and improve the reliability of their equipment.

• Real-time monitoring: ML algorithms can be applied to real-time sensor data to detect anomalies and patterns that indicate a need for maintenance. This can help organizations to address maintenance issues before they become more serious and lead to downtime. • Cost savings: By predicting maintenance needs more accurately, organizations can reduce their maintenance costs by avoiding unnecessary maintenance or replacing equipment before it fails. • Improved asset management: Machine learning can help organizations to better manage their assets by identifying trends and patterns that can help to optimize maintenance schedules, reduce the risk of equipment failure, and extend the lifespan of their equipment.

Over the past few decades, there has been a significant increase in the use of data mining to aid decision-making [START_REF] Tufféry | Data mining and statistics for decision making[END_REF] in manufacturing industries. Kumar and al. propose in their paper referenced [START_REF] Sahoo | Deep learning applications in manufacturing operations: a review of trends and ways forward[END_REF] a review on the current state of data mining technique, focusing on modern manufacturing methods. Data mining, also known as knowledge discovery in databases, is the process of discovering patterns and knowl-edge from large datasets. To ensure consistent outcomes from data mining projects, organizations use standardized processes such as Knowledge Discovery in Databases (KDD), Sample, Explore, Modify, Model, and Assess (SEMMA), and Cross-Industry Process for Data Mining (CRISP-DM) for managing data mining projects [START_REF] Plotnikova | Applying the crisp-dm data mining process in the financial services industry: Elicitation of adaptation requirements[END_REF], [START_REF] Shafique | A comparative study of data mining process models (kdd, crisp-dm and semma)[END_REF]. However, not every process is suitable for all machine learning objectives [START_REF] Dåderman | Evaluating frameworks for implementing machine learning in signal processing: A comparative study of crispdm, semma and kdd[END_REF]. Several studies have been conducted to compare these methods and highlight their strengths and limitations [START_REF] Shafique | A comparative study of data mining process models (kdd, crisp-dm and semma)[END_REF]- [START_REF] Azevedo | Kdd, semma and crisp-dm: a parallel overview[END_REF]. Based on these studies, we have selected the CRISP-DM method.

CRISP-DM is widely used in various domains, mostly for finance [START_REF] Plotnikova | Applying the crisp-dm data mining process in the financial services industry: Elicitation of adaptation requirements[END_REF], [START_REF] Da Rocha | Identifying bank frauds using crisp-dm and decision trees[END_REF], healthcare [START_REF] Asamoah | Adapting crisp-dm process for social network analytics: Application to healthcare[END_REF], [START_REF] Azadeh-Fard | Variations of length of stay: A case study using control charts in the crisp-dm framework[END_REF] and marketing [START_REF] Moro | Using data mining for bank direct marketing: An application of the crisp-dm methodology[END_REF], [START_REF] Peker | Transactional data-based customer segmentation applying crisp-dm methodology: A systematic review[END_REF]. A bibliographic study of the different methods and application areas of the CRISP-DM method is presented in the article [START_REF] Schröer | A systematic literature review on applying crisp-dm process model[END_REF]. CRISP-DM has also been successfully applied in manufacturing context [START_REF] Fisher | Considerations, challenges and opportunities when developing data-driven models for process manufacturing systems[END_REF]. The authors of the papers [START_REF] Huber | Dmme: Data mining methodology for engineering applications-a holistic extension to the crisp-dm model[END_REF]- [START_REF] Wiemer | Data mining methodology for engineering applications (dmme)-a holistic extension to the crispdm model[END_REF] have developed a framework to introduce big data analytic into manufacturing systems. Tripathi and al. [START_REF] Tripathi | Ensuring the robustness and reliability of datadriven knowledge discovery models in production and manufacturing[END_REF] provide a detailed review of CRISP-DM in industries. However, very few studies use this approach for purpose to predict failures in manufacturing systems [START_REF] Harding | Data mining in manufacturing: a review[END_REF].

In this study, we propose an application of the generic deep learning CRISP-DM to quantify the current and predict the future degradation of machine by means of health indicators. The approach is tested on a real case study of a cutting machine that operates continuously for 8 hours a day generating a 12 months data retrieved directly from the Cutting Assembly Optimization system (CAO) of the machine.

This paper is organized into four sections starting by an introduction. In section 2, the CRISP-DM method is elaborated. Section 3 presents the case study, business need, data acquisition and pre-processing, and the application of the machine-learning model. Finally, section 4 summarizes the paper's major findings.

II. BACKGROUND TO CRISP-DM

The CRoss-Industry Standard Process for Data Mining (CRISP-DM) is widely recognized as the industry-standard process model for implementing data mining projects [START_REF] Plotnikova | Applying the crisp-dm data mining process in the financial services industry: Elicitation of adaptation requirements[END_REF], [START_REF] Schröer | A systematic literature review on applying crisp-dm process model[END_REF]. The authors of the papers [START_REF] Huber | Dmme: Data mining methodology for engineering applications-a holistic extension to the crisp-dm model[END_REF], [START_REF] Wiemer | Data mining methodology for engineering applications (dmme)-a holistic extension to the crispdm model[END_REF] aimed to enhance the CRoss-Industry Standard Process for Data Mining (CRISP-DM), which is a publicly available standard for carrying out data mining projects. This framework comprises six stages: Business Understanding, Data Understanding, Data Preparation, Modeling, Evaluation, and Deployment, with a particular focus on the first two stages (Fig. 1). The Business Understanding stage aims to comprehend the project's objectives and requirements from a business perspective, while the Data Understanding stage involves data collection [START_REF] Pete | Crisp-dm 1.0-step-by-step data mining guide[END_REF]. To bridge the gap between Business Understanding and Data Understanding, the authors introduced intermediate steps. These steps involve transforming business objectives into technical tasks, selecting the data required to accomplish these tasks, and identifying suitable measurement equipment and methodology. These additional steps establish a direct connection between organizational goals and the technical implementation of PdM, and serve as examples of strategic maintenance implementation in a company.

A. Description of CRISP-DM steps

The guide of CRISP-DM [START_REF] Pete | Crisp-dm 1.0-step-by-step data mining guide[END_REF] describes the main idea, tasks and output of these phases shortly, bellow the summary:

• Business understanding: This initial phase focuses on understanding the project objectives and requirements from a business perspective, then converting this knowledge into a data mining problem definition and a preliminary plan designed to achieve the objectives. • Data understanding: During this phase, data is collected, examined, and explored to identify potential data quality issues, interesting subsets, and possible hypotheses. • Data preparation: Involves building the final data-set for modeling by selecting tables, records, attributes, cleansing and transforming data to suit modeling tools. This data preparation is likely to occur multiple times during the process. • Modeling: Selects and applies various modeling techniques with calibrated parameters to achieve optimal values. Some modeling techniques may require specific data forms, hence the re-execution of the data preparation phase.

• Evaluation: Ensures that the constructed models fulfill intended business objectives and comprehensively reviews the previous steps. One of the key objectives is to determine whether any crucial business issues may have been overlooked during the modeling phase. • Deployment: Incorporates the model into decisionmaking processes, ranging from generating a report to implementing a repeatable data mining process, depending on the project's requirements.

III. NUMERICAL EXPERIMENT

A. Cutting machine description

The Crimp Center 64 is a state-of-the-art crimping machine with up to four processing stations. The cutting machine is capable of performing high-quality crimping, sealing, twisting and tinning of wires ranging from 0.13 to 6 mm² (26 -10 AWG) at an impressive maximum feed rate of 12 m/s correspondent to 39.4 ft/s (table I). It boasts exceptional performance, allowing for fast changeovers, high productivity and short set-up times. The machine is userfriendly, equipped with a modern software interface and a touchscreen for easy operation. It also has easy network integration, making it accessible to a range of users. A wide range of accessories and options are available, giving it versatility and adaptability to various applications. The Crimp Center 64 is built for durability, with dynamic and powerful servo drives combined with an intelligent control system that ensures high production rates to meet even the most demanding schedules (Fig. 2). All data, including wire data, crimp data, and seal data, can be saved and retrieved for future use. At a maximum feed rate of 12 m/s (39.4 ft/s), the Crimp Center 64 allows high-quality crimping, sealing, and tinning of wires from 0.13 to 6 mm² (26 -10 AWG) at maximum productivity (Table 1).

B. Application

This section covers the implementation of the CRISP-DM methodology to predict machine failures in a wire cutting machine that operates for 8 hours a day. The methodology is implemented on an Intel Core i5 machine with 8GB of RAM and programmed in Python 3.10.

1) Business understanding: The project aims to enhance the maintenance KPIs of a manufacturing plant by focusing on two primary objectives.

• to increase the Mean Time Between Failure (MTBF) of a wire cutting machine. • to achieve a shorter Mean Time To Repair (MTTR). The wire cutting process involved several operations, including receiving wire spools, cutting wires, performing crimping operations, and conducting quality checks. This resulted in the creation of an analyzable database that is now available for the necessary analyses. The database includes information such as the machine type, number of goods produced, and downtime (measured by mean time between failures (MTBF) and mean time to repair (MTTR). All historical data is systematically recorded in the system. This study specifically pertains to a cutting machine during the year 2022, from January to December. The dataset contains 549 raw and 20 columns. The MTBF column becomes our target variable. Fig. 4 presents the data visualization. 3) Modeling: Selecting the "best model" is a critical step in machine learning, as it determines the accuracy and efficiency of the outcome. This process involves evaluating various models and selecting the one that performs the best in terms of predictive power, generalization ability, and computational efficiency. A well-informed and careful selection of the best model can greatly enhance the success of the machine-learning task. We have conducted 3 tests in the initial dataset using 3 different methods, respectively Random Forest, Decision Tree, and K-Neighbors, and obtained the results presented in Table II. It is evident from the evaluation metrics that the Random Forest model outperforms all other models with perfect scores. On the other hand, the Decision Tree model performs less effectively than the ensemble method. The K-Nearest Neighbors model has the lowest performance, leading to the conclusion that clustering methods might not be the best approach for identifying maintenance failures.

Therefore, we will use the Random Forest Classifier model for classification. This machine-learning algorithm combines multiple decision trees to perform classification. It consists of creating a large number of decision trees and using their collective predictions to classify data accurately [START_REF] Liu | New machine learning algorithm: Random forest[END_REF]. The output of each tree is combined through a voting process to produce the final prediction. The test conducted to measure the performance of the model using a dataset of samples showed excellent results, with an accuracy score of 0.8276 indicating a strong performance.

Hyperparameters are parameters that are not learned directly by estimators, but are passed to the estimator class constructor in scikit-learn. Examples include C, kernel, and gamma for Support Vector Classifier, and alpha for Lasso. To achieve the best cross-validation score, it is recommended to search for the optimal hyperparameters. To determine the best parameter setting for our model, we utilize an optimization technique called Grid Search. This method involves testing a range of parameters and comparing their performance (as shown in Figure 6). After the retaining of the model, we were able to achieve a higher model score of 0.862.

IV. CONCLUSION

This paper presents the application of the CRISP-DM framework in a study case aimed at improving the key performance indicators (KPIs) of a manufacturing plant. The process involved various stages, including business understanding, data understanding and preparation, modeling, and evaluation and deployment. The study utilized historical data recorded in the Cutting and Assembly Optimization system (CAO). The selected machine learning algorithm was the Random Forest Classifier model, which achieved an accuracy of 0.8276. The model was further optimized using hyper parameter optimization through Grid Search, resulting in a model score of 0.862. The successful application of CRISP-DM in this study case demonstrates the effectiveness of this framework in data-driven decision-making and problem-solving.

Fig. 1 .

 1 Fig.1. The CRISP-DM process[START_REF] Plotnikova | Applying the crisp-dm data mining process in the financial services industry: Elicitation of adaptation requirements[END_REF]

Fig. 2 .

 2 Fig. 2. Cutting machine Crimp Center 64

•

  As-Is State: Business defined the root of low Overall Equipment Effectiveness (OEE) is the availability of cutting machines. • To Be State: Machine Learning based model that determines the failures. • Success: Reducing the downtime of cutting machines by 50%, the first 6 months and 100%, the second 6 months. • Entity: Improve Mean Time Between failures (MTBF) and Mean Time To Repair (MTTR). • Evaluation Metric: Accuracy. 2) Data Understanding and Data Preparation: The maintenance interventions were comprehensively recorded directly in the Cutting and Assembly Optimization system (CAO).

Fig. 3 .

 3 Fig. 3. Visualization of target variable

Fig. 4 .

 4 Fig. 4. Flowchart of the Random Forest model 4) Evaluation and Deployment: The final work package focused on evaluating the results in a small series of experiments. After preparing the datasets for training, we separated the target from the features and split the data into two parts: 80% for training and 20% for validation. To evaluate the accuracy of our model, we conducted an accuracy test, the results of which are shown in Fig. 5. An epoch in machine learning

Fig. 5 .

 5 Fig. 5. Accuracy and loss of the model

Fig. 6 .

 6 Fig. 6. Flowchart of the Grid Search algorithm

Authorized licensed use limited to: UNIVERSITE DE LILLE. Downloaded on September 14,2023 at 08:00:46 UTC from IEEE Xplore. Restrictions apply.


Alexandre T Da Silva
email: alexandre.teixeira-da-silva@inria.fr

Thibaut Métivet

Victor Romero

Mélina Skouras

Florence Bertails-Descoubes

Untangling the physics of self-locking in tight knots

Keywords: tight knot, self-locking, elastic rod, frictional contact, numerical simulation

Introduction

Thanks to the simplicity in their form, knots are commonly used in day-to-day lives like in shoelaces as a most basic example. Just by tying a rope in a certain configuration, we are able to firmly attach structures thanks to some self-locking effect of the rope. Found in activities such as hiking, fishing or knitting, knot tight tying shows utility in multiple scales and forms. However, despite their usefulness and apparent simplicity, knots have only recently been deeply studied in the fields of physics and mechanics. Although there are multiple mathematical models to describe the topology of different kinds of knots, a mechanical model capable of capturing some core behaviors, like self-locking, is still absent.

A first model was developed by Audoly et al. [START_REF] Audoly | Elastic knots[END_REF], addressing the mechanics of knots in in loose configurations. This approach is derived from an inextensible Kirchhoff rod model that is tied in a simple overhand configuration at the limit of a vanishing thickness. Because it assumes the rod to be inextensible, the model diverges in tight knots configurations. Since the core utility of knots are found in their tights forms, it is fundamentally important to understand how each physical and geometrical parameter interact with one another. Baek et al. [START_REF] Baek | Technical brief: Finite element modeling of tight elastic knots[END_REF] and Durville [START_REF] Durville | Contact-friction modeling within elastic beam assemblies: an application to knot tightening[END_REF] have explored the tight knot configuration with volumetric effects, and in particular the deformation of the cross-section of the rod. While these approaches allow, for the first time, to capture the geometry and forces of tight knots, they are restricted to a particular choice of physical parameters, and none of them studies the mechanism of self-locking.

Our work proposed here aims at expanding previous studies to multiple sets of parameters in order to characterise some core behaviours still poorly understood, in particular the self-locking property in tight knots.

Figure 1: On the left, multiple snapshots of the rod centerline curvature are plotted to exhibit the self-locking behavior. The knot is first tightened and then loosened; As the tightened state surpasses the critic locking value of the end-to-end shortening ē, the centerline curvature remains constant, which indicates a locking effect. On the right, snapshots of the knot are shown to illustrate the state of deformation of the knot alongside the centerline curvature. We note that given the locking of the knot, the structure buckles itself with the imposed displacements on both ends of the rod.

Contribution

We analyse the self-locking mechanism in tight knots using an open-source, full 3D FEM simulator [START_REF] Schneider | Poly-spline finite-element method[END_REF]. Our protocol is based on two steps: first, a thin rod is tied in an overhand knot form and tightened by imposing the displacement on both ends; then, by imposing a displacements towards the center of the knot to loosen it, the tensile force is relaxed.

Given a set of physical and geometrical features, namely the Young Modulus (E), Poisson's ratio (ν), friction coefficient (µ), diameter of the rod (D), length of the straight rod without the knot (L 0 ), and end-to-end shortening ē = (L -L 0 )/D, we find a critical dimensionless locking parameter beyond which self-locking indeed occurs, as illustrated in Figure 1. In contrast, below this threshold, no locking appear and the knot gets untied (see Figure 2). By combining simulations and experiments, we are able to validate our numerical approach and further leverage it to explore systematically the respective impacts of geometrical and mechanical parameters involved in the self-locking phenomena.

Figure 2 :

 2 Figure 2: Since the knot is not tight enough, it gets untied and recovers its initial form.


Caroline Hache

Mercedes Baugnies

Eric Tortochot

Noémie Olympio

Caroline Hache -Decap

Le cas particulier de l'enseignement de l'anglais en CAP

est destinée au dépôt

-Circulation des savoirs comme espace partagé de significations entre divers acteurs impliqués (Wenger, 1998;Morrissette, 2013;[START_REF] Mottier Lopez | L'évaluation formative des apprentissages des élèves: entre innovations, échecs et possibles renouveaux par des recherches participatives[END_REF] -La co-conception d'une ingénierie didactique (Artigue, 2002) dans un dispositif de formation -La co-conception avec des parties prenantes dans une organisation (Visser, 2002;Schön, 1997) -La construction de consensus sur l'évaluation [START_REF] Morrissette | Vers un cadre d'analyse interactionniste des pratiques professionnelles[END_REF] -L'évaluation et les compétences au lycée professionnel : un laboratoire d'innovation pédagogique (Buisson-Fenet & Rey, 2016) -L'enseignement général contribuant à l'évaluation du développement des compétences transversales [START_REF] Calvez | La voie professionnelle scolaire : viser l'excellence[END_REF] -L'évaluation comme fonction de soutien [START_REF] Mottier Lopez | L'évaluation formative des apprentissages des élèves: entre innovations, échecs et possibles renouveaux par des recherches participatives[END_REF] Quelle influence a la recherche collaborative sur la conception d'une ingénierie pédagogique autour d'une évaluation diagnostique par des enseignants de lycée professionnel ?

-

  La recherche participative/collaborative (designbased research) (Sanchez & Monod-Ansaldi, 2015) -Des chercheurs et une équipe d'enseignants, de chefs d'établissement et d'inspecteurs -Un dispositif innovant en CAP, lycée professionnel

-

  La recherche collaborative & la mise en place de la conception d'ingénierie pédagogique d'évaluation en LP 2e Congrès international sur la formation et la profession enseignante -CRIFPE -Université de Brasilia L'évaluation : quelle perception des acteurs ? Focus groupe avec les enseignants autour de la conception des évaluations diagnostiques -Question du sens de l'évaluation -Perception négative ou positive de l'évaluation (pour l'enseignant et pour l'élève) -Nécessité d'explicitation du sens de l'évaluation aux élèves -Évolution des perceptions lors de confrontation de postures différentes La recherche collaborative & la mise en place de la conception d'ingénierie pédagogique d'évaluation en LP 2e Congrès international sur la formation et la profession enseignante -CRIFPE -Université de Brasilia Vers une approche réflexive de l'évaluation de tous les acteurs

Université de Brasilia TITRE DE LA PRÉSENTATION > TITRE DE LA PARTIE 3. Les choix méthodologiques Cadre épistémologique constructiviste Niveau Macro

  La recherche collaborative & la mise en place de la conception d'ingénierie pédagogique d'évaluation en LP 2e Congrès international sur la formation et la profession enseignante -CRIFPE -

	3.
				Terrain observé
	--	: volet quantitatif permettant de dresser un état des lieux Profils socio-démographiques et pratiques des enseignants de LP (dans et hors CAPture) Profils des élèves (sociaux et scolaires, rapport à l'école etc.)	Le lycée professionnel Les classes de CAP Journée du Plan académique de formation (PAF)
		Niveau Méso : volet qualitatif
		-	Analyse discursive d'entretiens permettant de comprendre la réalité des	Équipe
			acteurs (inspecteurs, chef d'établissement, enseignants du dispositif,
			élèves de CAP)	1 Inspecteur et 1 chargé
			Niveau Micro : volet qualitatif -Pratiques des enseignants (observations de classe,	de mission 16 enseignants d'anglais en classe de CAP
				4 enseignants-chercheurs
				(ADEF, LPL, LEST)
				5 chefs d'établissement

autoconfrontations, conception d'évaluations diagnostiques [travail avec les inspecteurs + focus groupe + conception + observation participante])

Méthodologies mixtes (perspective interdisciplinaire avec 3 échelles d'analyses) 8 Cadre épistémologique constructiviste Niveau

	7/10/2019	27/10/2019	20/11/2019	20/11/2019	03/2020
					Terrain observé
	Travail en équipe des enseignants pour coconcevoir	Analyse a priori des traces de l'activité des enseignants	Focus groupe entre l'équipe d'enseignant et un chercheur à partir de la	Travail en équipe des enseignants pour retravailler leur	Le lycée professionnel Les classes de CAP Journée du Plan académique de formation (PAF) Entretiens
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Les ancrages théoriques

Question de recherche

Méthodologies mixtes (perspective interdisciplinaire avec 3 échelles d'analyses)

Méthodologie


Claude Verdier

Liviu I Palade

A rheological model for spheroids including extra-cellular matrix

The rheology of spheroids has been studied intensively recently and it was shown that the presence of the extra-cellular matrix (ECM) can have significant effects on the overall behaviour of these biological systems. Collagen I can indeed be a proxy between cells and bring new intriguing effects, as its content increases. To investigate these effects further, a two-phase emulsion model is proposed including interactions between cells and the ECM. Starting with the single cell and collagen individual viscoelastic properties, the model can be tested against previously obtained data for spheroids. The model has interesting features and capabilities for it covers a variety of behaviours and uses fitting parameters such as collagen and cell concentration, as well as adhesion energy. It is shown that the final intercellular collagen content can be large as compared to the initial one, and that this increase in collagen content induces a larger packing of cells, together with a larger adhesion energy.

Introduction. -Spheroids are very interesting 3D biological systems, and good candidates to describe tumours [1]. Although there has been a growing interest to investigate how they react to osmotic pressures or mechanical forces [2][3][4][5][6], their mechanical properties are not yet satisfactorily understood. Indeed, cells packed together correspond to a concentrated suspension with shear-dependent viscosity or viscoelastic behaviour [7], but the extra-cellular matrix embedded into the system, or secreted by cells, makes the picture harder to describe.

In addition cells are viscoelastic, in particular cancer cells are known to exhibit different mechanical signatures like a larger deformability [8][9][10] or a glass transition depending on invasiveness or grade [11][12][13][14]. Moreover their mechanical behaviour depends largely on the environment, in particular they rigidify when in contact with a stiffer substrate, i.e. they are mechanosensitive [6,15]. They can also transform when the microenvironment is mechanically altered [16].

A few studies have focused on the effect of extra-cellular matrix (ECM) and its potential role to change the rheology of the spheroids. In particular, ECM can be considered as a proxy [17,18] acting as a porous material drained by water or the culture medium present in the spheroid [19]. But the influence of a larger collagen content may allow the building up of a new microstructure made of (a) claude.verdier@univ-grenoble-alpes.fr (corresponding author) interconnected fibers responsible for the larger mechanical moduli observed [20]. It is definitely accepted that the whole spheroid exhibits a viscoelastic pattern, due to the presence of cells embedded within the ECM, a gel-like system. In addition spheroids could be explored in more detail at larger frequencies to investigate possible poroelastic behaviors like the ones found for tendons [21].

Probably the best way to model the behaviour of such spheroids is to consider the ECM-water surrounding medium as a gel with viscous and elastic properties. Such ECM properties have been studied extensively in the literature in the linear and nonlinear regimes, showing the role of cross-linking effects [22], negative normal stresses [23], concentration-dependent shear moduli [24], nonlinear rheology following pre-stress protocols [25], and the insensitivity of Magnetic Resonance (MR) based diffusion measurements to various collagen hydrogels [26]. In addition the gel biopolymers enable cells to bind or adhere within connective tissues to allow stress transfer [27].

In this work we propose to study spheroids consisting of two phases, i.e. the ensemble of cells and the collagen matrix. We use the matrix mechanical properties as well as individual cell properties in contact with a similar environment, that is to say when they are surrounded by other cells. Then we consider a viscoelastic emulsion model, valid for large concentrations in order to predict the whole spheroid rheology in a large range of frequencies and scales. This is carried out in detail by varying the model parameters. Finally we optimize these parameters to rationalize previous results [20] obtained for spheroids containing an excess of collagen.

Materials and methods. -Collagen was prepared according to the manufacturer's protocol (Corning). Rat tail collagen I was mixed with Phosphate Buffer Saline (PBS, 1X) and NaOH (1M) was added at 4 • C until the correct pH was obtained (7.4). Then it was let to polymerise at 37°C for 30 minutes. Collagen properties were measured as described in previous work [28]. Briefly, classical rheometry tests at 37 • C using parallel plate geometry (20mm-diameter, gap size 500-800 µm) were carried out in oscillatory shear mode in the [0.05Hz-10Hz] range, using a small deformation of 1%, within the linear range. Then experiments using Atomic Force Microscopy (AFM) in force modulation mode were done following a previous method [29,30] using indentation of a flat collagen layer (in PBS) with a pyramidal AFM tip (Bruker, MLCT, half angle θ ∼ 20°). This geometry was prefered as compared to colloidal probes, since it avoids adhesion to such soft gels. An initial indentation δ 0 was applied, followed by small oscillations in the linear regime (δ << δ 0 ), thus allowing to measure the

complex shear moduli G * (ω) = 1-ν 3 δ0 tan θ F * δ * -i ω b(0)
, where G * =G'+iG", G' and G" are respectively the elastic and loss shear moduli, ν is the Poisson ratio (ν ∼ 0.5), F * and δ * are the complex force and indentation, and the last term is the drag on the surrounding liquid, with b(0) being a coefficient containing the geometry of the system as well as the fluid viscosity [29]. Using these complementary experiments, the range [0.05Hz-1kHz] was covered. Note that the hypothesis ν ∼ 0.5 involves the fact that poroelastic effects are neglected. However, this could become important for some hydrogels, in particular when analyzing relaxation curves at longer times [31]. In the present study, since fast solicitation frequencies are used, it is known that the fluid does not move relative to the microstructure if f > 10 Hz [32,33], which is mainly true in our AFM experiments. This argument is further enhanced by checking the necessary agreement between classical rheology and AFM data.

T24 cells (epithelial bladder cancer line) were cultured in RPMI medium with 10% FBS and 1% penicillinstreptomycin. These cells have been described before and their properties were previously found to depend on the environment, i.e. the substrate. In some cases, corrections depending on substrate stiffness need to be made [12]. Here we chose to measure T24 cell properties as if they were in contact with other cells, like in a real spheroid. The best possible way to do so is to form a spheroid for 3 days (see description below) and approach a small AFM tip close to the surface, where cells are located. In such a case, no substrate corrections were needed since the environment consists of cells themselves. Such AFM measurements were carried out with a pyramidal tip (Bruker, MLCT, half angle θ ∼ 20°) for their viscoelastic shear mod-uli G' and G" at 37 • C (same method as above). It was indeed important to find a relevant micro-environment for these cells, like in a real spheroid.

Spheroids were prepared in 15µL hanging droplets containing 5,000 T24 cells in culture medium including small amounts of collagen (initial concentration c 0 = 0.01 mg/mL or 0.03 mg/mL) as described previously [20]. This method allows cells to accumulate ECM without having too much collagen around the spheroids, which would render AFM measurements more difficult. A spheroid was formed after 3 days. Spheroids were transferred into a Petri dish for AFM measurements at 37 • C. A large tipless cantilever (Nanosensors, TL-NCL model, length 225 µm, width 38 µm [6]) was chosen for these measurements in force modulation mode, in order to make a plane-spherical contact. Viscoelastic data was obtained in the range . The other spheroids were kept for confocal microscopy imaging.

Properties of the collagen matrices. -After collecting data from classical rheology and AFM, results were plotted on the same graph, as a function of frequency. Matching of the two types of experimental results was found to work nicely as previously shown for PolyAcrylamide samples [30]. Such results are shown in Fig. 1. A typical power-law behaviour [34] was found for both G' and G" at different collagen concentrations (c=1 mg/mL, 2 mg/mL, 4 mg/mL) in the range [0.05Hz-10Hz] and both G' and G" increased faster in the high frequency range. Note that the measured values of the moduli were usually smaller as compared to cell's values [6]. Finally fitting of the data was achieved using simple power-laws as described below. Cell properties. -To investigate cell rheology, it was necessary to consider cells in the proper environment as it is known that the micro-environment plays a role on their cytoskeleton organization, therefore changes cellular mechanical properties [6,15]. Cells within a spheroid are surrounded by other cells and the ECM therefore it is convenient to use such a micro-environment [13]. Unfortunately, using AFM, it is hard to have access to the cell mechanical properties within the tissue, unlike when using other techniques such as bead tracking microrheology [35]. Here we preferred to use cell properties (T24 epithelial bladder cancer cells) obtained when in contact with similar cells, i.e. in the real spheroid. Fig. 2 below represents the G' and G" moduli obtained for cells located on the spheroid's periphery, as suggested before [13]. This is the closest representation of the micro-environment experienced by cells in a spheroid. The frequency range was chosen from 1 Hz to 600 Hz using three points per decade. The cell elastic modulus G' exhibits a frequencydependent modulus with a slope ∼ 0.17 at small frequencies (see modeling below), increasing faster at larger frequencies. As for the G" modulus, it is similar to G' but showing different slopes [12]. Two-phase model with interfacial tension. -Our starting point is Palierne's seminal work published in the early 90s [36,37]. This two-phase model designed for polymeric or viscoelastic systems requires the knowledge of complex shear moduli obtained for small deformations, i.e. in the linear regime. The model has been used mainly for two-phase polymeric systems (see for example the works on polymer blends [38]). Nevertheless it may be used for materials such as gels or cellular media exhibiting known viscoelastic properties.

Emulsion model. Here the medium, i.e. the collagen gel, has a characteristic complex modulus G * m (ω) whereas the inclusions (cells) have modulus G * i (ω). The volume fraction of the inclusions is ϕ but different inclusions may be considered with different sizes and concentrations (in case of heterogeneous sizes). Here we assumed that only cells with a typical radius R coexist, which is usually the case since cell size is roughly constant. In this case, G * (ω), the average modulus of the spheroid (i.e. cells embedded in the collagen matrix) in the case of non-diluted emulsions [36] is written as:

G * (ω) = G * m (ω) 1 + 3ϕH * (ω) 1 -2ϕH * (ω) (1)
where

H * (ω) = 4 α R [2G * m (ω) + 5G * i (ω)] + [G * i (ω) -G * m (ω)]D * (ω) 40 α R [G * m (ω) + G * i (ω)] + [2G * i (ω) + 3G * m (ω)]D * (ω) (2)
and we introduced the complex modulus D * :

D * (ω) = 16G * m (ω) + 19G * i (ω). (3)
In this formula, α is the interfacial tension and may be considered here as an interaction energy per unit area between cells and the matrix. It will be significant when cells adhere a lot to the matrix or are able to make a sufficiently large number of bonds. This may be the case for high collagen content. Note that for the case of dilute emulsions, Eq.( 1) is different and should be replaced by

G * (ω) = G * m (ω)(1 + 5 2 ϕH * (ω))
. This allows to recover the usual Einstein's formula [39] for two Newtonian fluids.

To summarize, we could use this model based on the cell properties G * i (ω) determined previously and the gel properties G * m (ω), known for different collagen I concentrations. Note that the collagen concentration c (in mg/mL) is unknown within the spheroid, and similarly ϕ, the cell volume concentration is to be determined. ϕ should be large since most of the spheroid is made of cells, and one could expect the ECM content to be possibly in the range 1-15% [5], assuming that cells also make their own matrix. The other adjustable parameters are α, the adhesion energy per unit area (in N/m), and R the cell radius, but the latter one is known since such T24 cells usually have a radius of 10 ± 1 µm.

The set of Eqs (1)-( 2)-(3) being rather hard to read, we first attempted to study a simplified problem by noticing that collagen moduli G ′ m and G ′′ m were much smaller (an order of magnitude or more) than the corresponding cell moduli G ′ i and G ′′ i . Therefore, we could neglect the viscoelastic part in Eq (1). This would imply that

D * (ω) ∼ 19 G * i (ω) and H * (ω) ∼ 1 2 therefore G * (ω) ∼ G * m (ω) 1+1.5 ϕ 1-ϕ . So G * (ω)
scales with the matrix complex modulus G * m (ω), with the prefactor 1+1.5 ϕ 1-ϕ . Note that a small collagen content like 5% gives ϕ = 0.95 therefore a prefactor of 48.5. Values of the prefactor are shown in Table 1 below. This shows that even though the collagen modulus G * m (ω) is rather small (Plateau of G' around 200 Pa), as compared to cell's moduli (plateau of G' at 500 Pa), the amplification due to spherical inclusions such as cells could lead to the high values obtained in spheroids [20]. The simplified formula also originates from analysis of the effect of the adhesion energy α. If α/R becomes very small as compared to the other moduli (and collagen moduli are neglected vs. inclusions moduli) then the function H * (ω) again becomes independent of α and close to 1/2. In what follows, calculations are made by accounting for all terms in the set of Eqs ( 1)-( 2)-( 3) without simplification. We will therefore consider the effect of all parameters and consequently check the influence of G * m (ω), ϕ, α/R upon the numerical results, while G * i (ω) remains fixed (Fig. 2).

Application to the rheology of spheroids. We have at our disposal the data from a previous study where collagen was added when making spheroids [20]. Two initial concentrations c 0 were used: 0.01 mg/mL and 0.03 mg/mL. This does not mean that the local collagen content (c) is the same when the spheroids are formed. But it seems that cells are able to aggregate collagen around them in a more concentrated form, as already observed [28]. Indeed spheroids were prepared in hanging droplets containing collagen; cells used it to adhere and spheroids exhibited a round shape. The same T24 cells were used for this study. Spheroids radii were R=140.6 µm ± 17.6 µm (c 0 = 0.01 mg/mL) and R=144.8 µm ± 23.3 µm (c 0 = 0.03 mg/mL) as compared to R=120.2 µm ± 24 µm when no collagen was used. Fig. 3 shows typical confocal microscopy images of such spheroids obtained using the two initial collagen contents: c 0 = 0.01 mg/mL and 0.03 mg/mL. As shown in the red channel, the reflectance of collagen was enhanced at higher collagen content, thus allowing to check the presence of collagen inside the spheroid. Thus we postulated that the collagen content can be high within the spheroid, probably between 1 and 4 mg/mL. Cells (green channel, transfected with the LifeAct plasmid expressing actin-GFP) seemed to form nice round spheroids except for the first case where no extra collagen was added.

We will now model the different phases of the spheroid. The first one is collagen. Using the previous data in Fig. 1, it is found that the collagen behaviour versus frequency f varies according to power laws. Fits of our previous data give the respective formulae for the three contents, as also shown in Fig. 1. For c = 4 mg/mL, we find G ′ (f ) = 200 * f 0.05 + 20 * f 0.5 and G ′′ (f ) = 25 + 2 * f . For c =2 mg/mL, G ′ (f ) = 45 * f 0.05 + 4.5 * f 0.5 and G ′′ (f ) = 9 + 0.65 * f , and finally the lowest collagen content, c = 1 Remarkably, these power law exponents are independent of the concentration, so only the prefactors are different.

Similarly, we fitted the viscoelastic response of T24 cells, as shown in Fig. 2. The moduli behave as G ′ (f ) = 560 * f 0.17 + 9.5 * f 1.0 and G ′′ (f ) = 180 * f 0.04 + 68 * f 0.8 . T24 cells moduli exhibit a small slope at the lowest frequencies then the slope increases at higher ones.

Let us consider the effects of the main parameters. Regarding modulus G * i , cells are assumed to have the behaviour depicted in Fig. 2 so this modulus is fixed. Then the values of the collagen modulus G * m can be changed depending on its content (c in mg/mL) within the spheroid (see Fig. 1). Finally, the roles of the cell content (ϕ) and adhesion energy per unit area (α/R) will be studied.

Effect of collagen matrix within the spheroid.

Here we consider the case ϕ = 0.8 which corresponds to typical cell volume concentration, as seen for example in Fig. 3. For this case, we used α = 10.0 mN/m with R = 10 µm. Plots of G' and G" are shown in Fig. 4 for collagen matrix concentrations 1, 2 and 4 mg/mL. As expected, spheroids moduli increase with collagen content, from 1 to 4 mg/mL. It is found that the resulting spheroid properties are in the range of the data previously reported [20], and this will be studied further below.

Role of cell concentration. Next we studied the role of cell concentration. Usually, in spheroids, cells are closely packed but the presence of collagen, as seen in confocal microscopy (Fig. 3) suggests that the intercellular spacing can be modified. It has been estimated that the ECM content can sometimes reach ∼ 15% in cases where cells make their own ECM [5]. Therefore, we selected typical cell concentrations between ϕ = 0.8 and ϕ = 1. For this case we used the intermediate collagen content (c = 2mg/mL) and α/R=10 3 Pa. Simulations are shown in Fig. 5. It can be concluded that cell concentration has an important effect on the results, especially due to the large difference between matrix and cell's moduli.

Effect of adhesion energy between cells and the matrix.

Regarding the effect of the adhesion energy, we used a large range of α-values from 0.3, 3, 30, and 300 mN/m at a cell density ϕ = 0.9, and medium collagen content properties (c = 2mg/mL). This is shown in Fig. 6. The of α was justified by previous AFM experiments [40] investigating the work of de-adhesion ∼ 10 -15 J between cells and their receptors, on corresponding surface areas ∼ 1µm 2 , leading to adhesion energies around 1 mN/m, depending on area, number of bonds, loading rate, etc.

To summarize results from Fig. 6, it appears that the role of α/R is important, for the chosen parameters and the values of the current viscoelastic parameters. This role becomes more important at lower frequencies where it was found that higher values of the adhesion energy increased the values of G' and G", in particular enhancing the formation of a plateau (in G') at low frequencies. Increasing the adhesion energy also increased the transition frequency (crossing between G' and G"). Thus, the more important the adhesion, the more elastic the spheroids became.

Optimizing emulsion model parameters for real spheroids. Finally, real spheroid properties were considered within the [1Hz-200Hz] range as previously described using AFM microrheology in plane-sphere contact [20]. Measurements were carried out in a manner similar to the one proposed on tissues [21]. Measurements display typical slopes as a function of frequency. To fit the data, parameters were optimized as follows. We chose the best collagen concentration between available values 1, 2 or 4 mg/mL, optimized ϕ and α/R using the least squares method. Results are shown in Fig. 7. Fits were in very good agreement with experimental data. Fig. 7: Spheroids properties at c0 = 0.01 mg/mL and 0.03 mg/mL [20]. Optimized fits with parameters from Table 2. the optimal collagen concentration c = 4 mg/mL seemed to be relevant, and the cell volume fraction was in the range of that measured experimentally using confocal microscopy ϕ ∼ 0.82 and 0.90 respectively (data not shown).

Finally cells seemed to be more densily packed in the second case (c 0 = 0.03 mg/mL), meaning that collagen plays the role of an interstitial layer enhancing the microstructure as well as the viscoelastic properties. This is confirmed by a higher value of the adhesion energy α = 81 ± 3 mN/m (c 0 = 0.03 mg/L) whereas it is only 59 ± 3 mN/m at c 0 = 0.01 mg/L. Further calculations are provided in the Supplementary Information (SI) on the sensitivity of the method with respect to parameters c, ϕ and α.

Discussion. -The rheological properties of spheroids are of great importance in order to understand how microstructural changes can affect mechanics, possibly giving rise to tumour invasion and the formation of metastases. Furthermore, their growth is also a topic of major interest. Tumours grow and exert pressure on the environment through a process limited by the surrounding medium [41]. Earlier models considered spherical growth depending on the nutrients, but more sophisticated ones now use numerical tools to evaluate hoop stresses inside and at the periphery of the growing spheroid [5,[START_REF] Stylianopoulos | Proc. Natl Acad. Sci[END_REF][START_REF] Preziosi | [END_REF]. Interesting studies have shown the effect of collagen on such processes, in particular the environment microstructure can become the signature of cancer progression and prognosis [44][45][46]. In this work, we investigated the role of the ECM on the viscoelastic properties of spheroids grown during 3 days [20], in culture medium containing an excess of collagen I (initial concentrations c 0 = 0.01 or 0.03 mg/mL). Using a model particularly well adapted for viscoelastic behaviors, we studied the possible effects of ECM density, cell concentration and adhesion. This was assessed by modeling frequency-dependent individual cell and collagen properties (Figures 12). Based on these properties, we used the two-phase emulsion model [36,37] able to recover the spheroid viscoelastic data.

Collagen has been shown to be a linker within spheroids using confocal microscopy (Fig. 3). Furthermore, we could also analyze its role on the overall spheroid's rheology as its concentration increases, using the two-phase emulsion model. It was shown that a high collagen content (c varying from 1 to 4 mg/mL) enhanced the viscoelastic properties (Fig. 4). Another parameter, cell concentration (ϕ), was seen to have a quite significant contribution to reinforce viscoelastic properties as well (Fig. 5). Indeed, the simplified Eq. ( 1) already showed the importance of cell packing. With the complete model, this effect still remains. Finally the role of the adhesion energy α (between cells and the matrix) was found to be significant, in particular when α/R becomes comparable with the other moduli, i.e. cell (G * i ) and collagen ones (G * m ). This was particularly noticeable at the lowest frequencies used in the experiments (Fig. 6). A more elastic behavior (higher elastic plateau and higher transition frequency) was observed for high values of the adhesion energy α. This was also found in works dealing with polymer emulsions [38], reporting the relevance of α at lower frequencies, when the terminal relaxation time scales with interfacial tension.

After analyzing two different spheroids [20], prepared with various initial collagen contents (c 0 = 0.01 mg/mL or 0.03 mg/mL) we used the emulsion model to optimize the parameters and came up with the best possible fits (Fig. 7 and Table 2). The results of the model were found to be in very good agreement with the experiments. This motivates further discussion in view of the parameters found. Indeed the packing of cells within the spheroid seems to be linked with the presence of collagen forming a dense meshwork, enabling cells to connect with each others, using adhesion molecules like cadherins [47]. The optimal adhesion energy increased from 59 to 81 mN/m, when increasing c 0 (see Table 2). In addition, an increase in the initial collagen content was found to enhance cell packing (higher value of ϕ) -which was rather unexpectedthis giving rise to higher viscoelastic moduli (see Figs 12). Altogether, the model, when used as an inverse method, provides good estimations of parameters ϕ, α and c. These parameters could not be measured independently, but ϕ is in reasonable agreement with confocal images, c takes on realistic values, and similarly for α.

The model presented here predicts a rich variety of viscoelastic properties and power-law dependencies. Hence it would be interesting to use it to study spheroids in details in the future, for example with various cell types or ECM leading to different microstructures [48]. As it is general enough [36], the model could also apply to multicomponent spheroids, i.e. containing cells of different nature, size or viscoelastic properties. Regarding stiffness, such invasive T24 cells are known to be soft as previously described [8,9], but applications to spheroids containing stiffer cells (muscle cells, cartilages) should be also possible since no further hypotheses are made. Finally it could be useful to monitor rheological properties during tumour growth [41][START_REF] Stylianopoulos | Proc. Natl Acad. Sci[END_REF][START_REF] Preziosi | [END_REF]. To investigate this further, the associated microstructures will have to be minutely analyzed using confocal microscopy, in order to extract the basic relationships or forces involved in such interactions, in relation with the adhesion energy α, depending on bond strength, number of bonds and loading rate [40]. Finally cells can pull on collagen to generate forces or remodel it [28]. This could lead to matrix stiffening at larger deformations, and could eventually modify spheroids properties.

Conclusions. -The rheology of spheroids has insufficiently been studied so far. Here we used previous viscoelastic data on spheroids measured with an AFM, as well as the individual response of the other components (i.e. cells and collagen). Results were combined in the framework of an emulsion model including interfacial energy. The results were in very good agreement with the experiments. They showed that the final spheroid microstructure contains dense collagen regions within the inter-cellular spacing. Finally an increased initial collagen concentration led to an enhanced compactness of the spheroids with a smaller inter-cellular spacing, and a larger adhesion energy. Thus collagen plays the role of a connecting adhesive layer between the cells and improves spheroid stability. With this model, future predictions should be possible regarding time-dependent spheroid growth.

Fig. 1 :

 1 Fig. 1: Rheological properties of collagen matrices (c = 1mg/mL, 2 mg/mL and 4mg/mL). Data collected from classical rheology [0.05Hz-10Hz] and AFM [1Hz-1kHz]. Average of N=3 measurements, error bars are Standard Errors of the Mean (SEM).

Fig. 2 :

 2 Fig. 2: Rheological data of a T24 cell, measured at the spheroid surface, as a close representation of the cell's environment. Average of N=20 measurements, error bars are Standard Errors of the Mean (SEM), c0=0.01mg/mL.

Fig. 3 :

 3 Fig. 3: Confocal microscopy images of spheroids prepared using T24 cells, at different initial collagen contents (a) c0 = 0.01 mg/mL, (b) c0 = 0.03 mg/mL. Scale bar 100 µm.

Fig. 4 :

 4 Fig. 4: Predictions of the two-phase emulsion model for 80% cell volume concentration, α/R=10 3 Pa, at collagen contents c = 1 mg/mL, 2 mg/mL and 4 mg/mL.

Fig. 5 :

 5 Fig.5: Predictions of the two-phase emulsion model for various cell concentrations ϕ = 0.8, 0.9, 0.95, 0.98, 1, using α/R=10 3 Pa, and collagen content c = 2 mg/mL.

Fig. 6 :

 6 Fig. 6: Predictions of the two-phase emulsion model for various adhesion energies α = 0.3-3-30-300 mN/m, with R = 10 µm, collagen content c = 2 mg/mL, and ϕ = 0.9.

Table 1 :

 1 Values of the prefactor

Table 2

 2 summarizes these results, at a fixed value R = 10 µm. For both cases c 0 = 0.01 and 0.03 mg/mL,

	c 0 (mg/mL) 0.01	0.03
	c (mg/mL)	4	4
	ϕ	0.88 ± 0.01 0.94 ± 0.01
	α (mN/m)	59 ± 3	81 ± 3

Table 2 :

 2 Optimal parameters (R = 10µm). ϕ and α optimized by least squares curve fitting with errors.

The authors are thankful to the Grenoble Nanoscience Foundation. C.V. is a member of LabeX Tec 21 (Invest. d'Avenir: grant No. ANR-11-LABX-0030). The authors thank D. Tsvirkun for spheroids preparation, and J. Revilloud for useful discussions.