Yassine Belghaddar

Abderrahmane Seriai

Ahlame Begdouri

Carole Delenne

Yassine Bel-Ghaddar

Nanee Chahinian

Mustapha Derras

Combining model-driven engineering and sewerage networks: towards a generic representation

Keywords: Meta-modeling, Sewerage Network, Model-driven engineering, Data sources

come

Introduction

By the time a sewerage network is set up, its graph, where the nodes represent the manholes or inlet grates and the edges represent the pipes, is mapped for the 1 first time. Later on, several operations such as reparations and expansions may occur in the field to respond to the scalable needs of the citizens. These actions are generally internally reported in several formats, representations and through models, which are often owned by the operators working on the network. Consequently, the combination of data from different sources and eras raises problems of consistency (contradictory information) and requires the establishment of a methodological framework for collecting, centralizing, updating and data archiving in order to facilitate information sharing and communication between the managers. In our vision of data fusion, we consider the uncertainties related to each type of collected information in order, for example, to anticipate and react promptly to potential dysfunctions or to quantify their impact on the results of a numerical simulation of flows in the sewerage network. In this context, and as a first step of our work, we propose in this paper a meta-model for aggregation, control and analysis of data sources related to sewerage networks, before elaborating adapted algorithms to merge heterogeneous multi-sources data. The paper is structured as follows, section 2 introduces the context of our work. We explain the motivation behind this work in section 3 and we present the state of the art and related works in section 4. Section 5 describes our meta-model and its specific viewpoints. Section 6 concludes this paper.

Context

Sewerage networks and management challenges

Sewerage system is a network for collecting and transporting wastewater and storm water to a treatment plant, also called combined sewer system. When a network collects these two types separately, it is called separated sewer system. To set up a sewerage system and make its progress in a territory, different institutional and operational actors are involved. The ministry in charge of sanitation or the ministry in charge of local communities, define sanitation policies and strategies as well as the regulatory framework at national level. The local authorities ensure the respect of regulations related to the quality of sanitation services. Finally, the contracting authorities (municipalities or state agencies) are responsible for the development of services, their quality and sustainability. For the implementation, monitoring and control of these services, they call for actors such as service operators, local associations, and development partners (founders, NGOs, and design offices).The sanitation supply is not only limited to infrastructure installation. Maintenance tasks such as reparation, expansion, damage anticipation and scheduling of interventions are all necessary actions to ensure a permanent and transparent services. Planning is an important task for decision making. It allows to develop a vision of needs in space and time, to quantify and prioritize them in order, among others, to direct funding towards the most necessary investments and at reasonable costs.

On a territory, infrastructures are large and must be managed in a collaborative way by the diverse involved actors. Urbanization and the concentration of populations in cities engender the increase of the dimensions of sewerage networks. For example, in France, this heritage consists of approximately 337,000 km of collectors [ASTEE2015]. The 2012 reform "DT-DICT" as part of the network detection process indicates that France is covered by more than four million kilometres of networks, one third of which are aerial and two thirds are buried or underwater.

The improvement of underground networks, particularly for sewerage networks, has several advantages, especially the impact on public health and environment preservation through the protection of water resources against pollution. The administrative management and the techniques of interventions play a key role in these challenges since they help to reduce damage costs induced from services interruptions and floods.

The expenses associated with the management of these infrastructures are high, particularly repairs, since the components of these networks are subject to degradation and damage caused by several factors: age, environment, etc. Furthermore, the costs of urgent and unexpected operations are far higher than the ones anticipated [ASTEE2015].

In this context, improving knowledge on the state of these networks, mostly underknown, becomes a priority. Indeed, digital technologies such as Geographic Information Systems (GIS) and Computerised maintenance management systems (CMMS) bring great added value and are officially increasingly adopted. For example, the regulations associated to the environment code in France require stakeholders to have digital and precise cartography for sensitive underground networks since January 1st, 2019 in urban units and from January 1st, 2026 in other cases.

These solutions are particularly useful for making spatial and geographic data available to the various actors, to facilitate their communication for optimal decision-making as well as to improve the administrative, economic, and financial management of this type of networks and of interventions to take place near these underground networks.

Sewerage network representation

A sewerage network is represented by a graph composed of nodes and edges. Nodes represent manholes, equipment, repairs, etc. the edges represent pipes. Each of the nodes and edges has a set of properties in the form of attributes such as, diameters of the pipes, types of materials and positions of the inspection areas for the objects. In recent years, storage and data management solutions for sewerage networks have evolved. Currently, most managers use Geographic Information Systems to create, edit, view, and analyze these data. The data structures and formats supported by these systems are diverse: relational databases, Shapefiles, GeoJSON and CSV files, etc. Moreover, to study the impact of some parameters, such as the discharge rate of consumers into the networks, specialists use hydraulic simulation software.

Although the applications are various, the digital representation of the data remains almost identical in the different solutions: • Spatial data that are represented by geometric shapes and their relationships: points and lines (figure 1).

• Atributes that are listed in attribute tables where each record is associated with a network object (figure 2).

Problematic

When using GIS solutions, data processing includes acquisition, digitization, import, export, and visualization of geographical data. The acquisition can be carried out directly in the field allowing real time data collection and mapping.

In addition, advanced data processing may allow considering multi-source data, spatial analysis through interactive queries and maps overlays.

Although the use of digital maps is increasingly adopted, there are still large communities in the world where maps and geographic data are still analog, making their use and update difficult. The detection and digital mapping of buried networks by semi-automatic or automatic approaches is a real scientific and technological challenge. Therefore, there is a large conceptual, technical and semantic gap between the analog and digital mapping models.

The attributes and characteristics associated with the various objects constituting a network are not all available at a given time. This is partly explained by the fact that the networks undergo expansions and repairs but not properly tracked and reported, or through the interventions at different stages by actors, other than the operators who ensure the continuous functioning of the supply services. However, these attributes may be reported elsewhere, for example, in public databases, calls of tender, repair reports or even in press articles reporting damages.

In addition, since information and communications technology are easy to reach and use, operators currently have access to several sources from which they can collect useful data before interventions in the field, such as images, analogue maps, reports of interventions, sensors, etc. The heterogeneity of the sources makes the extraction of relevant information and its combination a complex and time-consuming task.

On the other hand, imperfections may be found in these data sets and sources, namely inconsistency (abandoned pipelines which still appear on the maps), missing attribute values for some objects, uncertain and sometimes contradictory values. All of these aspects represent various obstacles to operators when merging the data.

Combining multi-source data also requires a unified data model to allow the centralization, updating, archiving, and monitoring of these data. Indeed, we have analyzed digital databases related to sewerage networks to understand the semantics of their data, their relationships and determine their differences. Since the associated data models, when they exist, are rarely available to the public, we proceeded by inferring them from data. In our study, we have used the data provided by reliable sources, particularly, the French open data repository [ope2015]. Among the suppliers are the urban community of the South-East of Toulouse Sicoval, Data Angers, and the region of Pays de la Loire.

As a result of studying these different sources, we identified the following constraints:

• The data models adopted by operators are different. Therefore, exchanging and reusing data is difficult.

• The models do not comply with computer design and modelling rules and standards.

• The attributes provided by the stakeholders are related to their fields of activity. For example, a company specialized in hydraulic modelling provides precise information on the flow of water in a pipeline, while this same information is generally missing in the data coming from another entity expert in the field of structures repairing.

• The history of interventions, necessary for anticipating repairs, is rarely considered in these models.

Thus, a generic model for business data and data sources is needed to overcome the conceptual and semantic gap between existing digital mapping models and to implement an optimal data fusion approach.

4 State of art 4.1 Meta-models

Several definitions have been proposed. According to [Object Management Group2006], a meta-model is a model used to model modelling itself. A Meta-model is a model that defines the structure of a modelling language [Da Silva2015]. The basic idea of a meta-model is to identify the general concepts in a given problem domain and the relations used to describe models [Gascueña et al.2012]. This generality, which we also seek to satisfy in our proposal, is one of the most important axis that have made the meta-modelling, i.e. the creation of metamodels, one of the most important approaches for modelling. Instances of a meta-model are models that must satisfy the meta-model specifications. They enable target systems to be modelled in a consistent and homogeneous manner.

Monitoring activities is one of the concepts where meta-models are used. For example, a meta-model for properties associated with software during execution is presented in [Bertolino et al.2011] to ensure the quality of software and its dynamic adaptation after deployment. Indeed, these properties provide a means to assess and improve the resilience of software through the adaptation and anticipation of abnormal situations. The instances of this meta-model are in this case a model with monitoring properties adapted to the target software. In [La Fosse et al.2020], the authors propose a meta-model for the monitoring of cyber-physical systems (CPS), particularly sensor and actuator networks which require valid data and good coordination between sensors and actuators for their operation.

Sewerage networks business modelling: related works

To help decision makers in collecting the data necessary for interventions and to diversify their data sources, some solutions have been published. For example: in [Commandre et al.2017], the authors apply deep neural networks to detect the position of manholes, visible on the ground, from a high-resolution image. In [Chen and Cohn2011], to create the cartography of underground networks, researchers use Bayesian fusion techniques to combine hypotheses extracted from the Ground Penetration Radar (GPR), the spatial location of surveyed manholes, as well as the expectations from the statutory records. In [Chahinian et al.2016], to assess whether big data can be used to reduce uncertainty in storm water modelling, the authors propose a process based on text analysis for extracting attributes about the objects of sewerage networks from data available on the web. However, none of these approaches have been submitted along with a data model.

The work in [Abdelbaki and Zerouali2012] is an attempt to design a business data model for sewerage networks to build the digital map of the sanitation network for a municipality in Algeria and to contribute to its efficient management. The authors propose, from the inventory of the various available data sources, a conceptual data model on which the necessary objects for the management and their relationships are listed.

At the initiative of the Aquitaine region and a public interest group (Planning and risk management), the Commission of Data Validation for Spatialized Information (COVADIS) has published a data standard for drinking water and sewerage networks intended for French municipalities [COVADIS2019]. The committee presented a class diagram describing the minimum and necessary data to be used by the actors participating in the management of these networks (municipalities, PEIC1 , delegates public services, etc.) for the purpose of a simple data exchange between them.

Since this standard describes the minimum necessary, but sufficient, data for the management of the water networks, and since it is also a standard to be adopted at a nationwide level, we adopt it in our work as a business data model. We present in the following the subpart of the COVADIS class diagram related to the sewerage networks (Figure 3).

It is composed of 4 main classes: Node, Pipeline, Reparation and Meta-data whose attributes and related possible values are listed:

• Nodes: represented geometrically by points, they illustrate apparatus (valve, counter etc.) or manholes.

• Pipelines: represented geometrically by lines, they are classified into several categories: wastewater, rainwater etc. Each of the pipes has two end Nodes.

• Repairs: geometrically represented by points, they refer to interventions made in Nodes or Pipes.

• Metadata: are data used to qualify the information of the classes Nodes and Pipes. Namely the name of the source, the date of the last update, the reliability of the year of installation and the quality of the geolocation within respect to the 2012 decree [Leg2012], which defines 3 precision classes: less than 40mm, in the range 40mm and 1.5m and greater than 1.50.

Sewerage networks and Big Data

Nowadays, we are witnessing an intensive production of data. Every day, a huge amount of data is produced by companies, on social networks, during transactions or through sensors, that conventional computer tools can no longer process and analyse. The research work around these masses of data, also called Big Data, is a response to these obstacles. On the other hand, and despite the tiny amount of the available data on sewerage networks compared to the Big Data, they share two important characteristics:

• The multitude of data sources. • The heterogeneity of the data.

The research works in the domain of the underground networks is mainly confidential [Hafsi et al.2017]. Therefore, the number of publications related to Big Data is more important compared to sewerage networks. To our knowledge, there is no data model for data sources of underground networks. To fill this gap, and since these two characters of the multitude and heterogeneity of the sources have already been examined in Big Data (see for example [Dong and Sri-vastava2013] or [Boury-Brisset2013]).We have chosen to draw inspiration from the solutions proposed in this field.

The available Big Data systems and platforms are not identical, as they come from multiple providers whose vision is not uniform. In [Erraissi et al.2018] using model-driven engineering, the authors propose a platform-independent metamodel to describe the structures of data sources involved in feeding these large volumes of data, thus allowing programmers to create applications compatible with various products. Figure 4 illustrates this proposition which divides the data sources into 3 broad categories:

• Structured: data whose set of possible values are determined and known in advance, such as relational databases.

• Semi-structured: data that have not been organized into a specialized repository. However, they contain meta-data information, which help their exploitation, for example e-mails.

• Unstructured: data represented or stored without a predefined format.

contribution

Meta-model for sewerage networks data sources

Our target is to propose a generic model for data sources, with the aim of using fusion approaches to combine their data. Therefore, on the one hand, our solution should encompass the available approaches for collecting data. On the other hand, the data sources are diverse and may change over time. An exhaustive modelling of data sources and their possible relationships is not a generic solution. Figure 5 illustrates our meta-model. For a better understanding, four viewpoints of this meta-model are presented in the following paragraphs.

Data sources viewpoint

We summarise in Figure 6 the data sources viewpoint where the main entity source characterises any entity capable of providing data, information or knowledge about sewerage networks. Interpretation of the structuring aspect of data sources is as follows:

• Unstructured sources: whose formats require significant pre-processing before extracting business data about a network. For example, the location of sewer manholes from images.

• SemiStructured sources: whose formats require simple pre-processing before extracting business data about a network. For example, parsing CSV or XML files.

• Structured sources: represent relational databases that directly provide business data about a network. Generally, these data are provided by the official managers of sewerage networks.

Attributes viewpoint

The attributes and their relationships with data sources are highlighted in this viewpoint (Figure 7). Each attribute is an entity identified by a name and possesses a String value representing a semantic data. The aggregation of attributes by data sources is encapsulated within the "TWithAttributes" entity. Since it represents, in the case of structured sources, the different attributes within the table of a relational database. Thus, the entity "Table " is connected to this entity. As for semi-structured and unstructured sources, it includes the pre-processing operations, defined by instances of this meta-model, to extract attribute values about sewerage networks. Moreover, data source path is handled within "TWithPath" entity.

Confidences viewpoint

We modelled data imperfections in this viewpoint (Figure 8) by allowing confidence attributes to each source, table (representing an object of the sewerage network) and to each attribute characterizing this table (the object). This means that:

• Each source has a confidence value that indicates the reliability or the certainty of the information it provides. This metric can be modelled by • For the components or objects of sewerage networks, this value represents the uncertainty regarding their existence. Indeed, it is possible, for example, for a pipe or a manhole to be represented on a map by mistake.

• As for attributes, the confidence is related to the confidence of the data sources providing them.

Moreover, objects and attributes default confidence values are those associated with their sources. However, this does not imply not having different confidence values later on. For example, an approach identifying manhole covers from images would define the existence confidence of the detected object as the precision of its detection. Meanwhile, the attribute position of the detected object may have a different value, since the detection and the computation of its position are two separate operations. To keep track of the previous data fusion operations, the confidence history is stored too.

Business model viewpoint

In our meta-modelling, we distinguish Business data, characterizing the sewerage networks components, from data about the sources, from which business data are extracted, such as images, documents, calls for tenders, etc.

Figure 9, illustrates this business viewpoint where we adopted the COVADIS (4.2) standard classes that inherit the properties of the Table entity. For genericity purposes, other business models may easily replace it, provided that the appropriate connections are respected.

Conclusion

In this work, we proposed a meta-model for data sources related to sewerage networks inspired from the field of Big Data. Our proposition considered the three important aspects of i) structuration of the data sources, ii) associated confidences and iii) genericity related to the business domain. Currently, we are implementing our meta-model in Moose, a platform of software and data analysis, in order to identify a use case along with a suitable workflow to instantiate this meta-model and validate its functioning through a series of data queries.

Figure 1 :

 1 Figure 1: Example of spatial representation of sewerage networks.

Figure 2 :

 2 Figure 2: Example of attributes table.

Figure 3 :

 3 Figure 3: COVADIS sewerage networks business model [COVADIS2019].

Figure 4 :

 4 Figure4: A meta-model for Big data data sources[Erraissi et al.2018]

Figure 6 :

 6 Figure 6: Data sources viewpoint.

Figure 7 :

 7 Figure 7: Attributes viewpoint.

Figure 8 :

 8 Figure 8: Confidences viewpoint.

Figure 9 :

 9 Figure 9: Business model viewpoint

Public Establishment for Inter-municipal Cooperation.


Clémence Petit

Jean-Marc Tulliani

Solene Tadier

Sylvain Meille

Jérome Chevalier

Paola Palmero
email: paola.palmero@polito.it

Solène Tadier

Jérôme Chevalier

Novel calcium phosphate/PCL graded samples: Design and development in view of biomedical applications

Keywords: Calcium phosphate, Tissue engineering, Scaffold, Biodegradation

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

Introduction

Calcium phosphates (CaP) have been extensively studied for bone tissue engineering scaffolds [START_REF] Bohner | Calcium from ceramics orthophosphates to calcium phosphate in medicine: cements[END_REF]. Among the CaP crystalline phases, two of them have received most attention: hydroxyapatite (Ca5(PO4)3(OH), HA) and β-tricalcium phosphate (β-Ca3(PO4)2, β-TCP). They are biocompatible, osteoconductive and have a chemical composition close to the mineral part of bone [START_REF] Bohner | Calcium from ceramics orthophosphates to calcium phosphate in medicine: cements[END_REF][START_REF] Legeros | Properties of osteoconductive biomaterials: calcium phosphates[END_REF][START_REF] Rey | Bone mineral: update on chemical composition and structure[END_REF][START_REF] Wagoner-Johnson | A review of the mechanical behavior of CaP and CaP/polymer composites for applications in bone replacement and repair[END_REF]. β-TCP is considered to be bioresorbable in physiological conditions and shows higher solubility in aqueous media than HA [START_REF] Bohner | Calcium from ceramics orthophosphates to calcium phosphate in medicine: cements[END_REF][START_REF] Beekmans | The hydroxylapatite-bone interface: 10 years after implant installation[END_REF][START_REF] Lu | The biodegradation mechanism of calcium phosphate biomaterials in bone[END_REF]. This latter is characterized by the lowest solubility constant among CaP phases, thus to be considered non-resorbable under in vivo conditions [START_REF] Bohner | Calcium from ceramics orthophosphates to calcium phosphate in medicine: cements[END_REF][START_REF] Jarcho | Synthesis and fabrication of β-tricalcium phosphate (whitlockite) ceramics for potential prosthetic applications[END_REF][START_REF] Miranda | Mechanical properties of calcium phosphate scaffolds fabricated by robocasting[END_REF]. Therefore, biphasic calcium phosphate (BCP) materials consisting of HA and β-TCP with different HA/β-TCP ratios have been widely developed [START_REF] Bouler | Biphasic calcium phosphates: influence of three synthesis parameters on the HA/b-TCP ratio[END_REF][START_REF] Legeros | Biphasic calcium phosphate bioceramics: preparation, properties and application[END_REF] to optimize material's properties and resorption rates.

Important work has been also dedicated to the development of biodegradable polymer/CaP composites. Impregnation of CaP with a polymer can improve the mechanical properties, in comparison with CaP alone. The most commonly used biodegradable polymers for bone tissue engineering are aliphatic polyesters, especially poly(lactic acid) (PLA), poly(glycolic acid) (PGA) and poly(ε-caprolactone) (PCL) [START_REF] Peroglio | Toughening of bio-ceramics scaffolds by polymer coating[END_REF][START_REF] Gentile | An overview of poly(lactic-co-glycolic) acid (PLGA)-based biomaterials for bone tissue engineering[END_REF]. The latter is a semicrystalline and soft polymer. It has similar biocompatibility to PLA and PGA [START_REF] Burkersroda | Why degradable polymers undergo surface erosion or bulk erosion[END_REF] but a much lower degradation rate [START_REF] Lowry | Polycaprolactone/glass bioabsorbable implant in a rabbit humerus fracture model[END_REF][START_REF] Castilla-Cortázar | Hydrolytic and enzymatic degradation of a poly(ε-caprolactone) network[END_REF].

It is now widely accepted that a high volume fraction of interconnected macroporosity is needed to favour bone colonization and vascularization of the implants through circulation of fluids and cells into the scaffolds [START_REF] Bignon | Effect of micro-and macroporosity of bone substitutes on their mechanical properties and cellular response[END_REF][START_REF] Hing | Microporosity enhances bioactivity of synthetic bone graft substitutes[END_REF][START_REF] Liu | Influence of porosity and pore size on the compressive strength of porous hydroxyapatite ceramic[END_REF]. The suitable pore size range is still discussed in literature [START_REF] Karageorgiou | Porosity of 3D biomaterial scaffolds and osteogenesis[END_REF][START_REF] Bohner | Characterization and distribution of mechanically competent mineralized tissue in micropores of β-tricalcium phosphate bone substitutes[END_REF]: macropores with a size in the range 100-500 μm are reported by most authors [START_REF] Alshaaer | Fabrication of porous bioceramics for bone tissue applications using luffa cylindrical fibres (LCF) as template[END_REF][START_REF] Lan Levengood | Multiscale osteointegration as a new paradigm for the design of calcium phosphate scaffolds for bone regeneration[END_REF][START_REF] Kim | Improvement of the compressive strength of a cuttlefish bone-derived porous hydroxyapatite scaffold via polycaprolactone coating[END_REF]. The presence of smaller pores with a size ranging from 1 to 30 µm, is also important to favour bone cell attachment [START_REF] Alshaaer | Fabrication of porous bioceramics for bone tissue applications using luffa cylindrical fibres (LCF) as template[END_REF]. The shortcomings of such porous macrostructures are their poor mechanical properties, which limit their use to non-load bearing applications. For large bone defects, it can be interesting to combine sufficient mechanical resistance with proper biological functions. Different solutions have already been suggested in the literature to develop multifunctional implants: porous coating of dense implants [START_REF] Kim | Improvement of the compressive strength of a cuttlefish bone-derived porous hydroxyapatite scaffold via polycaprolactone coating[END_REF] as well as ceramic composites combining a strong inert phase (i.e., alumina or zirconia) with a bioactive CaP phase [START_REF] Brzezińska-Miecznik | The effect of calcination and zirconia addition on HAp hot pressed materials[END_REF]. More recently, functionally graded materials (FGM) characterized by a gradation in composition have been developed. HA/metals such as HA/titanium [START_REF] Batin | Mechanical properties of Ti/HA functionally graded materials for hard tissue replacement[END_REF][START_REF] Chu | Optimal design and fabrication of hydroxyapatite/Ti asymmetrical functionally graded biomaterial[END_REF] were first investigated, but they showed poor interfacial bonding at metal/ceramic interface. Other works focused on HA/zirconia/alumina FGMs [START_REF] Afzal | Functionally graded hydroxyapatitealumina-zirconia biocomposite: Synergy of toughness and biocompatibility[END_REF][START_REF] Guo | Laminated and functionally graded hydroxyapatite/yttria stabilised tetragonal zirconia composite fabricated by Spark Plasma Sintering[END_REF], but the significant difference in mechanical properties between CaP phases and alumina or zirconia leaded to weak interfaces. Porosity-graded structures, based on CaP materials, have been developed as well. Wang et al. [START_REF] Wang | Preparation and evaluation of a biomimetic scaffold with porosity gradients in vitro[END_REF] fabricated a three-layer FGM in which an outer dense layer is connected to a porous core (volume fraction of pores of 60vol%) by an intermediate porous layer (volume fraction of pores of 30 vol%). HA two-layer scaffolds were designed by Lee et al. [START_REF] Lee | The production of porous hydroxyapatite scaffolds with graded porosity by sequential freeze-casting[END_REF] and fabricated by freeze-casting of HA slurries with different solid loadings. Chamary et al. [START_REF] Chamary | Bio-inspired hydroxyapatite dual coreshell structure for bone substitutes[END_REF] fabricated HA two-layer scaffolds by combining the sacrificial template process (to produce a porous core with spherical pores) and freeze-drying (to produce the porous shell with elliptically shaped pores). This paper reports the design, fabrication and characterization of an innovative multi-layer scaffold based on a BCP dense core, a porous BCP intermediate layer and a porous BCP/PCL outer layer. This new graded scaffold combines a gradient of both composition and porosity, with the aim of obtaining sufficient mechanical resistance in the bulk and resorption rate on the surface. As a key point of innovation, we propose here the combination -in a simple but effective way -of an external polymer layer to an inner ceramic core. Even if out of the scope of the current work, the polymer-based outer layer can supply additional functionalities to the scaffold as it can act as a carrier for cells and drug delivery into the body. Entrapment of drugs as antibiotics in PCL/CaP composites have already been tested [START_REF] Lewis | Colloidal processing of ceramics[END_REF]. Therefore, such innovative scaffold has a potential for use in the biomedical field, but can also be viewed as a model system towards the design and fabrication of further innovative architectures with targeted multifunctionalities. The graded structure realized is then characterized in terms of phase composition and microstructure as well as biodegradation evolution.

Materials and methods

Materials

Two kinds of commercial calcium phosphate powders were used as starting materials: a HA powder (Plasma Biotal, Captal S BM192, batch P270S), and a β-TCP powder (Plasma Biotal, Captal R, batch P364S). The as-received HA powder has a purity of 95%, its specific surface area is 0.85 m 2 /g and its average particle size is 2.76 μm. The as-received β-TCP powder has a purity of 99%, its specific surface area is 1.2 m 2 /g and an average particle size of 1.84 µm. To reduce agglomerate size, HA powder was ball-milled for 1 hour, by using zirconia spheres (2 mm diameter) with a powder-to-spheres ratio of 1:5.

A polycarboxylate type dispersant (Duramax D-3021) was added into the calcium phosphate aqueous suspension to obtain well-dispersed and stable suspensions. The optimal concentration was set at 0.64 wt.% with respect to the dry powder, according to a previous experimentation [START_REF] Lombardi | Processing of a natural hydroxyapatite powder: from powder optimization to porous bodies development[END_REF].

Agar (Sigma-Aldrich, A7049, ash >2%), a polysaccharide constituted by chains of glycosyl units, was employed as gelling agent (0.5 wt% with respect to water content into the aqueous suspensions).

Polyethylene spheres (PE, supplied by Clariant Italia SpA), sieved in the 224-355 μm range, were used as pore formers for the intermediate layer.

PCL pellets were supplied by Sigma-Aldrich (Mw=80,000). Acetone (Sigma Aldrich) was used as solvent. Sodium chloride NaCl particles (Sigma Aldrich) with a size range 200-400 µm were used as pore formers for the external layer.

Powder characterization

As-received powders were submitted to X-Ray Diffraction (XRD) analysis. Spectra were recorded on a Philips PW 3800 apparatus operating with Cu Kα radiation (0.154 nm) and acquired in the range 5-70° 2θ, with a step size of 0.026° and an acquisition time per step of 5 /s.

The morphology of the two starting powders was observed by Field Emission Scanning

Electron Microscopy (FESEM, Zeiss Merlin). The mean grain sizes of the particles were determined by image analysis (public domain Java-based ImageJ software [START_REF] Ahn | Production of highly porous triphasic calcium phosphate scaffolds with excellent in vitro bioactivity using vacuum-assisted foaming of ceramic suspension (VFC) technique[END_REF]).

The particle size distribution of starting and ball-milled powders and were determined by laser granulometry (Fritsch, Laser Particle Sizer Analisette 22).

Thermogravimetric and Differential Thermal Analyses (TG-DTA, Netzsch STA 409C) were carried out up to 1400°C (heating and cooling rates of 10°C/min, under static air).

Powders were submitted to different calcination treatments in the range 1000-1300°C (without isothermal step at the maximum temperature), with heating and cooling rates of 20°C/min. A high cooling rate was used to retain high temperature phases at room temperature [START_REF] Brown | Sintering of biphasic calcium phosphate[END_REF]. The calcined powders were submitted to XRD and the peaks were indexed through the JCPDS files of HA (09-0432), β-TCP (09-0169) and α-TCP (09-0438). When different phases were identified in a pattern, their weight fractions were estimated by calculating the proportion of the relative intensity of their most intense peak [START_REF] Ahn | Production of highly porous triphasic calcium phosphate scaffolds with excellent in vitro bioactivity using vacuum-assisted foaming of ceramic suspension (VFC) technique[END_REF], which corresponds to (211), (0210) and (170) planes for HA, β-TCP and α-TCP, respectively.

Preparation of the powder batches

First, three batches of powders were prepared from HA and β-TCP powders, each batch being used for the different layers of the final scaffolds. For each batch, the required amounts of powders were weighed and ball-milled with zirconia balls. The composition of each batch is given in Table 1. In the bottom part, the specimens at the different steps of the process are graphically shown.

Step 1: BCP1 dense core by gel-casting

Ceramic slurries with an initial solid loading of 69 wt% were prepared from BCP1 powder and dispersed under magnetic stirring for 24 h. Agar was dissolved in deionized water (2 wt%), at 90°C for 10 minutes, to ensure a complete dissolution. The solution was then cooled down to 60°C and added to the dispersed ceramic slurries, at the same temperature. After agar addition, the BCP1 slurries reached a final solid loading of 62 wt%, while the agar concentration decreased to 0.5 wt% (as respect to water). The suspensions were cast into cylindrical Plexiglas moulds (12 mm internal diameter, 20 mm high). Casting was carried out under vacuum (about 10 -2 Pa) to remove air bubbles entrapped into the ceramic suspensions. Gelation of the samples occurred during slow cooling to room temperature. Samples were then slowly dried into humidity controlled chambers, in which the high initial relative humidity (RH > 90%) was slowly decreased after demoulding of the samples.

Step 2: BCP2 porous intermediate layer by gel-casting/sacrificial template

For the ceramic porous layer, ceramic slurries were prepared from BCP2 batch and dispersed under magnetic stirring for 48 hours. A fixed amount of sieved PE spheres was added to the around the dense BCP1 core, which was already partially dried for one week. This procedure was set up on the ground of the different shrinkage behaviour of the two layers. In fact, during drying, BCP2 layer undergoes to a lower shrinkage as respect to BCP2, due to the presence of the PE spheres. When simultaneously dried, an undesired separation between the two layers occurred, thus requiring the two-steps fabrication process.

After drying, the bilayer samples were submitted to different heating steps of two hours each at 230°C, 300°C, 450°C, and 600°C, to gradually burn out the organic phase (PE and agar) without collapse of the ceramic bodies. This thermal cycle was set up on the ground of TG-DTA curves of PE spheres, as reported in a previous study [START_REF] Lombardi | Gel-casting of dense and porous ceramics by using a natural gelatine[END_REF]. The thermogravimetric curves showed that PE burn out was accomplished at 550°C, through different decomposition steps at the previously mentioned temperatures. The samples were then sintered at 1300°C for 3h (heating and cooling rate of 1°C/min), being this thermal cycle defined on the ground of dilatometric analyses, as described in 2.4.4.

Step 3: PCL/BCP3 porous external layer by salt leaching

First, 1.5 g of PCL pellets were dissolved in 10 mL acetone for 6 hours under vigorous magnetic stirring. The required amount of the BCP3 powder was then added (40 wt% relatively to PCL content). The mixture was stirred for 24 hours for a complete homogenization. NaCl particles were added in the mixture at a weight ratio of 9:1 (NaCl:PCL) under stirring. After homogenization of the NaCl particles in the mixture, this latter was poured around the BCP1/BCP2 sintered samples in plastic moulds (30 mm diameter, 20 mm high). Before the pouring step, the sintered BCP1/BCP2 ceramics were dipped into PCL solution (150 g.L -1 of PCL dissolved in acetone) for 5 seconds. This coating step was added to obtain a better adhesion between the BCP2 and PCL/BCP3 layers. The three-layer samples were then air-dried for 24 hours at room temperature for complete evaporation of acetone. They were demoulded and immersed in deionized water for 4 days to leach out NaCl crystals. The water was refreshed every 12 hours.

Fabrication and characterization of single pellets

In order to characterize properly the different layers, single cylindrical specimens "representing" each layer of the final scaffold (i.e., dense BCP1, porous BCP2 and porous PCL/BCP3) were fabricated with the processes described above (i.e., gel-casting, gel-casting with PE spheres and solvent casting/salt leaching respectively).

In the case of BCP1, gelcast bars were prepared and submitted to dilatometry (Netzsch 402E).

The following conditions were applied: peak temperature of 1300°C, heating and cooling rate of 1°C/min, soaking time of 3 h. On the ground of this analysis, both BCP1 and BCP2 were sintered according to this thermal cycle.

The phase composition of the single samples was analysed by XRD, under the same conditions described for the powders.

The densities were estimated from weight and geometrical measurements and the relative densities were calculated considering the densities of fully dense samples and pure phases: 3.15 g.cm -3 for HA, 3.07 g.cm -3 for β-TCP and 1.15 g.cm -3 for PCL.

The microstructures were observed under SEM with a voltage of 12 kV. Before observation, the samples were mirror polished and metallized with a chromium coating. The grain size was determined by linear intercept method on different SEM images using Image J software. The results of these characterizations are presented in section 3.3.

In vitro dissolution test of the three-layer scaffolds 2.5.1. Immersion tests

The three-layer scaffolds were immersed in a 0.

Characterization of the scaffolds

The scaffolds were characterized before and after immersion. At each time-point, they were removed from the solutions, dried at 37°C until constant weight, weighed, and measured.

Variation of weight and apparent density was calculated with the equation ( 1):

𝛥𝑋 = 100 ( 𝑋 𝑖𝑛 -𝑋 𝑓𝑖𝑛 𝑋 𝑖𝑛 ) ( 1
)
where ΔX is the weight loss (or apparent density) in percent, Xin and Xfin are the initial and final weight (or apparent density) of the scaffolds.

Evolution of the phase composition was analysed by XRD (Brucker D8 Advance, equipped with a LynxEye detector) with Cu Kα radiation (λ = 0.15406 nm). Collection of data was done along the angular range 4°-65°, with a step size of 0.02° and a time per step of 0.4 s. To avoid cutting the samples, the whole surface of each scaffold was scanned in one single analysis.

Therefore, the resulting patterns provide an average composition of the three layers.

The whole microstructure of the samples was investigated by X-ray tomography with a standard The volumes were reconstructed from radiographs thanks to a commercial software coupled to the equipment. They were then analysed using ImageJ software.

Finally, the surface of each layer of the scaffolds was observed by SEM.

Results and discussion

Characterization of the as-received powders

The two powders were composed of HA and β-TCP crystalline phases respectively, as declared by the supplier and confirmed by XRD analyses (not shown).

In Fig. 2a and2b, the SEM micrographs of as-received HA and β-TCP powders are depicted, showing a not regular shape of the primary particles, characterized by a micronic size and a certain agglomeration.

Fig. 3 shows the particle size distribution of the same powders. HA presents a bimodal distribution, with d10, d50 and d90 (i.e. the agglomerate sizes corresponding to 10%, 50% and 90% of the cumulative frequency distribution) equal to 0.8, 2.5, and 14.7 m respectively. On the opposite, as-received -TCP was characterized by a narrower particle size distribution, with d10, d50 and d90 corresponding to 0.8, 1. Considering the DTA-TG curves of -TCP, we can observe a clear endothermal peak at 1270°C (Fig. 4b), while no mass loss was detected. From XRD analysis (Fig. 5b), we observe that α- A small amount of α-TCP was also detected in the HA powder calcined at 1300°C (Fig. 5a).

We can hypothesize that a small amount of the β-TCP phase resulting from the previous decomposition of the CDA/HA powder was transformed into α-TCP.

Sintering behaviour of the BCP1 bars

Dilatometric analyses were carried out on BCP1 gelcast bars. According to this characterization, the sintering cycle has been defined and applied to both BCP1 and BCP2 samples, as the bilayer structure (made by BCP1 / BCP2 layers) is sintered under the same thermal cycle (see § section 2.4.2).

As we can see in Fig. 6, the onset shrinkage temperature was observed at 1018°C. As mentioned in the previous section, this temperature corresponds to the beginning of CDA/HA decomposition, which thus overlaps with the beginning of densification. The derivative curve showes an inflection point at 1148°C, which could again be related to the partial decomposition of CDA into β-TCP, on the ground of previous DTA (Fig. 4a) and XRD results (Fig. 5a). BCP1 bars underwent a total linear shrinkage of 11.6%. The major contribution of this shrinkage (7.6%) took place during the isothermal step at 1300°C. After sintering, the samples reached a fired density of (80.1 ± 1.0)%, which is quite low in comparison with literature [START_REF] Akao | Mechanical properties of sintered hydroxyapatite for prosthetic applications[END_REF][START_REF] Halouani | Microstructure and mechanical properties of hot pressed hydroxyapatite ceramics[END_REF]. This incomplete densification can be partially explained by the gel-casting process, and precisely by the use of organic matter as gelling agent, which could give rise to a certain residual porosity in the sintered microstructure [START_REF] Koçlar | Gelcasting of alumina ceramics with gelatin and carrageenan gum and investigation of their mechanical properties[END_REF][START_REF] Bartuli | Mechanical properties of cellular ceramics obtained by gel casting: characterization and modeling[END_REF]. However, in our case, the amount of agar was quite low (0.5 wt% as respect to water). Therefore, to increase the density, the whole gel-casting process has to be improved, in particular way by increasing the solid loading of the ceramic particles in the slurries [START_REF] Chen | Fabrication and mechanical properties of β-TCP pieces by gel-casting method[END_REF].

Characterization of the single pellets

The results presented here refer to the single pellets "representing" each layer of the three-layer scaffold (i.e., dense BCP1, macroporous BCP2 and macroporous PCL/BCP3).

The relative densities of the pellets were (80.1 ± 1.0) %, (34.6 ± 1.6) % and (23.8 ± 1.8) % for BCP1, BCP2 and PCL/BCP3 samples, respectively. Despite the incomplete sintering of the allceramic samples, a gradient of porosity was obtained from the BCP1 (i.e., the internal layer of the scaffold) to the PCL/BCP3 layer (i.e., the external layer of the scaffold). The volume fraction of pores measured in the BCP2 pellets (i.e., 65.4%) was slightly higher than the volume fraction of PE spheres introduced in the suspensions during the gel-casting process (i.e., 60 vol%): the difference was due to the residual pores left by incomplete sintering. The volume fraction of pores in PCL/BCP3 pellets (i.e., 76.2 %) is also in agreement with the volume fraction of NaCl porogens added in the samples (i.e., 79%). These results show that the two processes used to prepare the porous layers are effective to obtain the expected volume fraction of pores.

In a previous work [START_REF] Marcassoli | Mechanical characterization of hydroxyapatite micro/macro-porous ceramics obtained by means of innovative gelcasting process[END_REF], we tested the mechanical performance of gel-cast microporous HA samples (i.e., samples having same composition and residual porosity as BCP1) : the compressive strength determined (up to 340 MPa) was in line with the values commonly reported for cortical bone [START_REF] Öhman | Compressive behaviour of child and adult cortical bone[END_REF], suggesting a possible mechanical function of the core layer in our scaffold.

The XRD patterns of the single pellets are depicted in Fig. 7. The three patterns showed the presence of HA and β-TCP phases. In the case of the PCL/BCP3 specimen, the XRD pattern confirmed the presence of PCL thanks to its two main peaks at 2θ = 21.3° and 23.7°, corresponding to (110) and (200) crystallographic planes respectively [START_REF] Castilla-Cortázar | Hydrolytic and enzymatic degradation of a poly(ε-caprolactone) network[END_REF]. No peak corresponding to the α-TCP phase was detected, contrary to what observed for both HA and -TCP powders calcined at 1300°C (Fig. 5). In fact, for powders characterization, it was used a high cooling rate (20°C/min) in order to retain the metastable α-TCP phase and highlight the transformations occurring in the samples at high temperatures. On the contrary, the absence of α-TCP in the BCP1 and BCP2 single pellets (Fig. 7) can be explained by the low cooling rate (1°C/min) applied on purpose to enable the reversible α-to-β-TCP transformation to take place [START_REF] Brown | Sintering of biphasic calcium phosphate[END_REF]. Thus, biphasic calcium phosphate samples (containing only HA and β-TCP) could be obtained.

After fabrication, the samples exhibited a gradation of HA/β-TCP ratio from the internal to the external layer: in fact, the estimated value of β-TCP fraction gradually increased (from 42, to 53 and finally to 68 wt%) moving from the core to the outer layer. In the case of the BCP1 and BCP2 ceramic layers, the β-TCP content is coherent with the important decomposition of HA observed after the calcination treatments of the initial powder (see section 3.2). Despite this phenomenon, affecting especially the dense core, a gradation of composition through the layers is successfully achieved. The SEM micrograph of each pellet is shown in Fig. 8. The dense BCP1 polished surface (Fig. 8a) exhibited well-sintered grains with an average size of (3.2 ± 1.2) µm. The residual micrometer-sized pores left by incomplete sintering were also visible. Fig. 8b shows the fracture surface of the BCP2 specimen. The macropores had a spherical shape with an average size of (245 ± 21) µm, which is in good agreement with the size of the sieved PE spheres. In addition, we can observe the numerous interconnections present among the macropores, with size ranging from one to a few tenths of microns. Fig. 8c shows the fracture surface of the PCL/BCP3 composite. Here is clearly visible the irregular, square-shape of the macropores left by the NaCl crystals, due to the grinding and sieving procedure used to produce the granular salt. The average pore size was (229 ± 46) µm, in a good agreement with the size of NaCl granules. Fig. 8d shows a magnified image of the polymer strut between the pores, showing the presence of well-dispersed CaP particles within the PCL matrix. The size of the ceramic particles was in the range 0.5-2 µm, in good agreement with their granulometric data, thus suggesting a good dispersion of the CaP powder in the polymer solution. Residual pores having a size of few micrometers left by solvent evaporation were visible too.

Characterization of the three-layer scaffolds

In Fig. 9a, a scheme of the three-layer graded structure is overlapped with the experimentally developed scaffold, showing the good joining of the three different layers, while the stereomicroscopy image in Fig. 9b highlights the continuous and crack-free interface between the two ceramic layers. Moreover, the comparison between the scheme and the picture in Fig. 9a shows that our approach was successful to produce a three-layer scaffold with good interfaces between the layers. In particular, the polymer adhered strongly to the ceramic porous layer. This is a key point for the structures developed in this work. Each scaffold was analysed by XRD on the lateral (flat) surface of the cylindrical sample before and after immersion. Table 2 summarizes the evolution of the weight fraction of β-TCP, measured on a same sample before and after immersion. As explained in section 2.5.2, the resulting XRD patterns characterized an average of the phase composition on the whole surface (i.e., including the three layers). For this reason, the values of the weight fraction given in Table 2 cannot be directly compared to weight ratio calculated for each layer (section 3.3). For the 4 samples tested before immersion, small variations can be observed from one sample to another.

presence of local degradation pits after immersion has already been observed in the literature for biphasic CaP materials such as HA/β-TCP [START_REF] Raynaud | Calcium phosphate apatites with variable Ca/P atomic ratio III. Mechanical properties and degradation in solution of hot pressed ceramics[END_REF] or β/α-TCP [START_REF] Lin | Petal-like apatite formed on the surface of tricalcium phosphate ceramic after soaking in distilled water[END_REF]. Fig. 11c and 11d depict some representative SEM images of the surfaces of the PCL/BCP3 layer of the scaffolds immersed for 11 and 17 days scaffolds, respectively. No significant changes appeared in the morphology of the layer after 11 days of immersion with respect to the starting one (see Fig. 8c). On the opposite, the scaffold immersed for 17 days presented a rougher aspect. This characteristic has already been observed for PCL/CaP composites after immersion in different media [START_REF] Wu | Fabrication and properties of porous scaffold of magnesium phosphate/polycaprolactone biocomposite for bone tissue engineering[END_REF][START_REF] Gallo | Resorption of calcium phosphate materials: considerations on the in vitro evaluation[END_REF]. Slightly less CaP particles seemed present at the surface of the immersed scaffolds as compared with the initial state (Fig. 8d), indicating their partial dissolution. The weight losses measured prove that a dissolution phenomenon took place from the beginning of the test. Tomography and SEM strengthen this idea showing a matter loss in the bulk and at the surface.

The results from XRD and SEM evidence a preferential dissolution of β-TCP as compared with HA. It agrees with previous results reported for BCP ceramics [START_REF] Raynaud | Calcium phosphate apatites with variable Ca/P atomic ratio III. Mechanical properties and degradation in solution of hot pressed ceramics[END_REF][START_REF] Sánchez-Salcedo | In vitro structural changes in porous HA/b-TCP scaffolds in simulated body fluid[END_REF][START_REF] Shuai | Optimization of TCP/HAP ratio for better properties of calcium phosphate scaffold via selective laser sintering[END_REF] and the higher solubility of β-TCP in comparison with HA is well-known. This feature and the large dissolution observed in the external layer appear in agreement with the targeted multifunctionality of our scaffolds (i.e., a bioresorbable β-TCP-rich layer in contact with bone coupled with a HA-rich inner layer with lower bioresorption). However, the high HA decomposition in the BCP1 and BCP2 layers during sintering has changed the targeted compositional gradient. The less marked gradation in HA/β-TCP ratio between the layers has probably led to a less marked differentiated dissolution between the layers. It can be noted that a more marked gradation of HA/β-TCP ratio between the layers would be beneficial to emphasize the functional gradient character of the scaffolds. For this purpose, a better control of the composition of the sintered samples appears necessary.

3D images suggest that the largest part of the weight loss can be attributed to a dissolution of the external PCL/BCP3 layer. The low degradation rate of PCL is often mentioned in the literature [START_REF] Castilla-Cortázar | Hydrolytic and enzymatic degradation of a poly(ε-caprolactone) network[END_REF][START_REF] Peña | Long-term degradation of poly(εcaprolactone) films in biologically related fluids[END_REF]. However, numerous studies report a higher dissolution rate for PCL/CaP composites [START_REF] Gallo | Resorption of calcium phosphate materials: considerations on the in vitro evaluation[END_REF][START_REF] Nithya | Biodegradation and cytotoxicity of ciprofloxacin loaded hydroxyapatite-polycaprolactone nanocomposite film for sustainable bone implants[END_REF][START_REF] Ang | Compressive properties and degradability of poly(εcaprolatone)/hydroxyapatite composites under accelerated hydrolytic degradation[END_REF], linked to the higher dissolution rate of CaP phases in comparison with the PCL alone [START_REF] Ang | Compressive properties and degradability of poly(εcaprolatone)/hydroxyapatite composites under accelerated hydrolytic degradation[END_REF].

It is also difficult to conclude about the respective influence of the porosity (pore size and pore volume fraction) and the HA/β-TCP ratio on the overall dissolution behaviour. To obtain more information, dissolution tests could be performed on different scaffolds with only one variable parameter (i.e., either a porosity gradient only or a compositional gradient only and not both).

The signs of dissolution detected are more marked for the scaffolds after 17 days of immersion than for the other ones, suggesting an acceleration of the dissolution phenomenon after 11 days of immersion in the in vitro conditions chosen in this study. To investigate better the long term evolution of the scaffolds, a longer duration of the test would be necessary with time-points of several months [START_REF] Gallo | In vitro evolution of resorbable brushite cements: a physico-chemical, micro-structural and mechanical study[END_REF].

Conclusion

In this work, a three-layer graded scaffold was successfully obtained via gel-casting and solvent casting/particle leaching processes. The aim was to obtain a gradation of composition and porosity from a dense core to a porous external layer. In addition, an external polymer layer was strongly joined to an inner ceramic one, being a strong point of innovation in this structure as compared to previous literature. Characterization of the scaffolds demonstrated a clear gradient of porosity between the layers, whereas the gradation of phase composition was less marked than the targeted one, because of the partial HA decomposition during sintering. The fabrication process can be improved to better control the final composition of the scaffolds and to achieve a stronger gradation in composition after sintering. An in vitro dissolution test was conducted by immersion of scaffolds in TRIS solution at 37°C for durations ranging from 2 to 17 days. Different signs of dissolution were found in the three layers. The results give some indications about different behaviours in the different layers. This fits with the targeted multi-functionality of the graded samples, with a resorbable external layer in contact with bone and a core devoted to the mechanical function. However, longer dissolution tests and other in-vitro investigations on the immersed samples are needed to understand better the long-term behaviour and to deepen the dissolution mechanisms of the different layers constituting the scaffolds. As a final consideration, the proposed scaffold can be considered as a model system, opening the way to the design and fabrication of further innovative architectures with targeted multi-functionalities.

Conflict of interest

None

Table's and figure's captions

Table 1: β-TCP weight fraction (wt%) of the three BCP powder batches Table 2: Evolution of β-TCP weight fraction in the whole scaffold before and after immersion in TRIS for each duration of immersion. These values were obtained for one scaffold per immersion time.

(a) (b)

Fig. 1 Fig. 1 :

 11 Fig. 1 depicts a schematic diagram of the different preparation steps of the graded scaffold.

Step 1 :Step 2 :Step 3 :

 123 Fabrication of inner core Gel-casting Fabrication of intermediate layer Gel-casting / Sacrificial template • Preparation of the ceramic slurry • Addition of gelling agent • M ixing • Casting into moulds • Gelation and demoulding • Partial drying • Preparation of the ceramic slurry • Addition of porogen agents (PE spheres) • Addition of the gelling agent (agar) • M ixing • Casting into moulds • Gelation, demoulding and drying • Debinding and sintering of the bilayer samples Fabrication of outer layer Solvent casting / Particle leaching • Dissolution of the polymer (PCL) in acetone • Addition of the ceramic powder • Addition of the pore formers (NaCl) • M ixing • Casting of the mixture around the previosly prepared bilayer samples •Drying suspension, to obtain a porosity of 60 vol.% in the fired intermediate layer. Agar addition was carried out following the same procedure described for BCP1 dense core. Then, BCP2 suspension was cast into cylindrical Plexiglas moulds (25 mm internal diameter, 15 mm high)

  laboratory tomograph (Phoenix v | Tome | x, General Electric), equipped with a Varian Paxscan detector (1920x1536 pixels). The following scanning conditions were used: voxel size of 13 µm, scan time of 333 ms, 1000 radiographs per scan, intensity of 80 µA and voltage of 140 kV.

7 and 4 . 1 Figure 2 :Figure 3 :Figure 4 :

 41234 Figure 2: FE-SEM micrograph of the (a) HA and (b) β-TCP starting powders. Insets: higher magnification micrographs.

  discrepancy, the β/α-TCP phase transformation can explain the DTA signal. Previous investigations attested the -TCP phase transformation into the -polymorph at temperatures above 1110°C[START_REF] Torres | Influence of Mg-doping, calcium pyrophosphate impurities and cooling rate on the allotropic α↔β tricalcium phosphate phase transformations[END_REF][START_REF] Elliott | Structure and Chemistry of the Apatites and Other Calcium Orthophosphates[END_REF][START_REF] Enderle | Influence of magnesium doping on the phase transformation temperature of β-TCP ceramics examined by Rietveld refinement[END_REF][START_REF] Ryu | An improvement in sintering property of β tricalcium phosphate by addition of calcium pyrophosphate[END_REF].

Figure 5 :

 5 Figure 5: XRD patterns of (a) HA calcined from 1000 to 1300°C, showing the formation of and β-TCP (the non-indexed peaks correspond to HA) and (b) β-TCP calcined from 1100 to 1300°C showing its conversion into α-TCP (the non-indexed peaks correspond to β-TCP)

Figure 6 :

 6 Figure 6: Linear shrinkage (solid line) and derivatives curves (dashed lines) versus

Figure 7 :

 7 Figure 7: XRD patterns of the single pellets

Figure 8 :

 8 Figure 8: FE-SEM micrographs of (a) polished and thermally etched BCP1 surface, (b) fracture surface of the BCP2 specimen, (c) fracture surface of the PCL/BCP3 composite, (d) magnified image of the polymer strut between the macropores in the PC/BCP3 composite.

Figure 9 :

 9 Figure 9: (a) Digital picture (bottom part) and schematic illustration (upper part) of the three-

Figure 11 :Figure 12 :

 1112 Figure 11: SEM micrographs of the BCP1 surface (a) from the scaffold immersed during 11 days and (b) the scaffold immersed for 17 days; of the PCL/BCP3 surface from (c) the scaffold immersed for 11 days and (d) the scaffold immersed for 17 days

Figure 1 :

 1 Figure 1: Schematic diagram illustrating the main steps of the fabrication of the graded

Figure 2 :

 2 Figure 2: FE-SEM micrographs of the (a) HA and (b) β-TCP starting powders. Insets : higher

Figure 3 :

 3 Figure 3: Frequency (a) and cumulative frequency (b) distributions of as-received and ball-

Figure 4 :

 4 Figure 4: DTA/TG curves of (a) HA and (b) β-TCP powders. The two main peaks with the

Figure 5 :

 5 Figure 5: XRD patterns of (a) HA calcined from 1000°C to 1300°C showing the formation of

Figure 6 :

 6 Figure 6: Linear shrinkage (solid line) and derivative curves (dashed lines) versus temperature

Figure 7 :

 7 Figure 7: XRD patterns of the single pellets

Figure 8 :

 8 Figure 8: FE-SEM micrographs of (a) polished and thermally etched BCP1 surface, (b) fracture

Figure 9 :

 9 Figure 9: (a) Digital picture (bottom part) and schematic illustration (upper part) of the three-

Figure 10 :Figure 11 :

 1011 Figure 10: Weight loss of the scaffolds at different time-points of the test

Figure 1 (

 1 Figure 1

Figure 7 Figure 8 (

 78 Figure 7

Figure 9 Figure 10 Figure 11

 91011 Figure 9

Table 1 :

 1 β-TCP weight fraction (wt%) of the three BCP powder batches

Layers in the final scaffold Powder batch designation β-TCP (wt%)

	Inner core	BCP1	0
	Intermediate layer	BCP2	20
	Outer layer	BCP3	70
	2.4. Fabrication of the scaffolds

Table 1

 1

Table 2

 2

Acknowledgments

This work was partly supported by COST Association in the frame of the COST Action MP1301 NEWGEN "New generation biomimetic and customized implants for bone engineering". The authors want to acknowledge Labormet Due srl, Torino, Italy, for the images of stereomicroscopy and Jérome Adrien, MATEIS, for the X-ray tomography.

However, these differences were not significant. Comparing the evolution of each sample during immersion, the weight fractions of β-TCP did not vary for immersion times of 2, 6 and 11 days. On the opposite, this weight fraction decreased significantly after 17 days of immersion, suggesting the partial dissolution of the -TCP phase. Besides, this dissolution is consistent with the weight loss observed between 11 and 17 days (Fig. 10).


Clémence Petit

Sylvain Meille

Eric Maire
email: eric.maire@insa-lyon.fr

Cellular solids studied by X-ray tomography and finite element modellinga review

Keywords: X-Ray Tomography, Cellular solids, Finite Element

I. Introduction

Cellular solids can be defined as materials with a volume fraction of porosity higher than 70%. They have been known in their natural form for a very long time (trabecular bone, wood, cork, sponges, wools...). The structure of these highly porous solids is characterized by the presence of porous cells, hence their name, surrounded by a network of solid-phase. These cells can be open or closed and their dimensions can span over several orders of magnitude, from nanometers to centimeters. The specific properties of natural cellular solids are such that they are used for a very long time in special applications where these properties are required (lightweight beams and structural parts, insulators, filters, absorbers...). Their microstructure has widely inspired engineers to produce highly porous engineering solids such as wool (glass or rock wool e.g.,) and foam (metal, polymer and ceramic foams). Over the last 20 years, interest in these materials has grown. They have numerous interesting properties qualifying them as multifunctional materials [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF][START_REF] Maire | X-ray tomography applied to the characterization of highly porous materials[END_REF] . Thanks to their low relative density and good mechanical proper-ties, they can be used as lightweight sandwich cores. Their high fracture strain, together with their damaging behavior when compressed, induces a large energy absorption capacity for packaging applications and shock absorbers. Their low thermal and acoustic conductivity is interesting for heat and sound insulation. The tailored high volume fraction of open porosity favors their use as filters and bone substitutes. However, the highly porous nature of cellular solids induces experimental difficulties in terms of 2D characterization. Different morphological parameters, such as density distribution and cell and pore size, are difficult to measure because of sample preparation problems. Nondestructive 3D images can help us to overcome this difficulty, explaining why X-ray computed tomography (XRCT) has recently become a widely used technique to study cellular materials. X-ray tomography aims at obtaining 3D images of the samples. The technique being nondestructive, it is also possible to follow the microstructure of cellular materials during in situ mechanical tests, which in turn gives us information on deformation, damage and fracture behavior. Different in situ mechanical tests have been developed so far 3 : compression, tension, fatigue, hydrostatic pressure, double torsion... The availability of 3D images of cellular materials also allows the numerical calculation of their mechanical properties through finite element (FE) modeling. Specific approaches have been developed in the last 20 years to calculate the average elastic modulus and stress concen-trations during loading of cellular materials. Combined with in situ testing, a comparison of experimental and modeled behavior is now possible. This review will present the most recent articles dealing with cellular solids studied by x-ray tomography and FE modeling.

II. X-ray tomography applied to cellular solids

A) Principle of XRCT The principle of XRCT has been described in many reviews. [START_REF] Baruchel | X-Ray Tomography in Materials Science[END_REF][START_REF] Stock | Recent advances in x-ray microtomography applied to materials[END_REF] It is derived from X-ray radiography where an X-ray beam is sent through a sample and the attenuation through the sample is measured by an appropriate detector. In XRCT, this sample rotates along an axis and a radiograph is taken for different angular positions of the sample. Reconstruction software uses all of these radiographs to recalculate a 3D image from all the 2D projections. As postprocessing, it is possible by using image analysis software to visualize slices of the sample or a 3D image. A standard laboratory tomograph allows the acquisi-tion of good quality images with a minimum resolution of tomography based on a synchrotron source are also available. These are more powerful and allow a better resolution. Tomography is able to provide 3D images of all types of cellular solids. The following sections present different examples of cellular solids observed by XRCT, reproduced from recent publications. These examples are classified regarding the nature of their solid phase.

B) Examples for cellular polymers

Polymeric foams were the first developed cellular solids. Their large energy absorption capacity enables us to use them for packaging or for protection against shock (such as helmets). This explains the large number of recent studies on the compressive behavior of this type of foam: PVC, 6 polyurethane, [START_REF] Youssef | Finite element modeling of the actual structure of cellular material determined by X-ray tomography[END_REF] and polypropylene 8 (see Fig. 1 for 3D renderings of these materials). McDonald et al. [START_REF] Mcdonald | In situ 3D X-ray microtomography study comparing auxetic and non-auxetic polymeric foams under tension[END_REF] also studied the tensile behavior of polyurethane foam during an in situ tensile test. Polymer foams are also widely used as templates to process ceramic or metal foams [START_REF] Burteau | Impact of material processing and deformation on cell morphology and mechanical behavior of poly-urethane and nickel foams[END_REF] as illustrated in the next section. Cellular ceramics are used in a wide range of applications: filters, solid oxide fuel cells, construction materials (cellular concrete plasterboards), biomaterials. Some authors used XRCT to characterize the morphology of the samples [START_REF] Maire | Characterisation of the morphology of cellular ceramics by 3D image processing of X-ray tomography[END_REF][START_REF] Zeschky | Preceramic polymer derived cellular ceramics[END_REF] or their fracture processes under load [START_REF] Meille | Mechanical properties of porous ceramics in compression: On the transition between elastic, brittle, and cellular behavior[END_REF][START_REF] Angelo | Finite elements analysis of reticulated ceramics under compression[END_REF][START_REF] Zhang | Modeling the mechanical properties of optimally processed cordierite-mullite-alumina ceramic foams by X-ray computed tomography and finite element analysis[END_REF] (Fig. 2). Many publications deal with bone substitutes: XRCT is used to compare their mechanical properties with bones [START_REF] Lacroix | Micro-finite element models of bone tissue-engineering scaffolds[END_REF] , or to follow the interaction between bone substitutes and natural bones [START_REF] Renghini | Microstructural characterization and in vitro bioactivity of porous glass-ceramic scaffolds for bone regeneration by synchrotron radiation X-ray microtomography[END_REF][START_REF] Okanoue | Comparison of in vivo bioactivity and compressive strength of a novel superporous hydroxyapatite with beta-tricalcium phosphate[END_REF] . Figure 2(a) shows an example of a solid strut in SiC foam [START_REF] Angelo | Finite elements analysis of reticulated ceramics under compression[END_REF] produced by coating a polymer cellular template with the ceramic slurry. During the sintering step at high temperature, the polymer precursor subsequently burns and then disappears, leading to the observed hollow strut structure. Metal foams are a relatively recent class of cellular solids. As polymeric foams, their lightness and energy absorption capacity is extremely interesting but they can be used at higher temperatures. Moreover, cellular metals are stronger compared to cellular polymers and tougher compared to cellular ceramics. This combination of prop-erties is very interesting in load-bearing applications (sanddwich cores), or in applications that require shock absorption (automobile, helmet, packaging, and cushion). Several authors have thus investigated the mechanical behavior of aluminum alloy foams because of their set of interesting properties: low weight, high strength, ductility, corrosion resistance, and recyclability [START_REF] Gioux | Failure of aluminum foams under multiaxial loads[END_REF][START_REF] Jeon | Finite element simulation of the plastic collapse of closed-cell aluminum foams with X-ray computed tomography[END_REF][START_REF] Michailidis | Experimental and FEM analysis of the material response of porous metals imposed to mechanical loading[END_REF][START_REF] Veyhl | On the mesh dependence of non-linear mechanical finite element analysis[END_REF] . More recently, metallic materials based on other metals (copper alloy [START_REF] Guillén | Compressive behavior of bovine cancellous bone and bone analogous materials, microCT characterisation and FE analysis[END_REF] , nickel 10 , steel [START_REF] Caty | Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF][START_REF] Lhuissier | Quasistatic mechanical behavior of stainless steel hollow sphere foam: Macroscopic properties and damage mechanisms followed by X-ray tomogra-phy[END_REF] , and titanium [START_REF] Singh | Characterization of the deformation behavior of intermediate porosity interconnected Ti foams using micro-computed tomography and direct finite element modeling[END_REF] ) have also been investigated (Fig. 3). E) Image processing X-ray tomography is an appropriate method to observe the mesostructure of a sample, i.e., the architecture of the cell walls and the distribution between the materials and the pores. The images can also be quantitatively used to calculate important descriptors of the cellular structure: tortuosity of the different phases, density distribution, pores, and cell size distribution [START_REF] Maire | Characterisation of the morphology of cellular ceramics by 3D image processing of X-ray tomography[END_REF][START_REF] Saadatfar | Imaging of metallic foams using X-ray micro-CT[END_REF][START_REF] Van Bael | Micro-CTbased improvement of geometrical and mechanical controllability of selective laser melted Ti6Al4V porous structures[END_REF] . Figure 4 shows an example of pore size distribution measured on porous metals. Tariq et al. [START_REF] Tariq | Characteriza-tion of hierarchical pore structures in ceramics using multiscale tomography[END_REF] determined the pore distribution in cellular ceramic catalysts thanks to tomographic images and compared it with mercury porosimetry measurements. Numerous softwares (ImageJ, IMorph 30 ) have been de-veloped to improve 3D image analysis. For example, Morpho1 31 has recently been improved to enable rapid analysis of larger datasets. One advantage of XRCT is its ability to characterize large samples, but this implies a limitation in the size resolution. To overcome this limitation, local tomography allows the imaging of the solid phase of cellular materials at a higher magnification. In this mode, the sample is placed near the x-ray source. Only a small part in the middle of the sample is irradiated by the x-rays. As a consequence, finer details of the microstructure in the material can be observed, as illustrated in Fig. 5 showing the 2 mm diameter central cylinder inside a 6 mm diameter cylindrical sample [START_REF] Youssef | Finite element modeling of the actual structure of cellular material determined by X-ray tomography[END_REF] of aluminum foam [START_REF] Toda | Quantitative assessment of microstructure and its effect on compressive behavior of aluminum foams via high resolution synchrotron X-ray tomography[END_REF] The images obtained during the deformation of the samples enable us to better understand the damage mech-anisms occurring inside the material. Figure 6 shows the results obtained in an ex situ com-pression test on a hollow sphere stainless steel structure [START_REF] Caty | Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF] . The image of the deformed sample allows observing the bending of the sphere walls during compression. In Fig. 7 9 , the alignment of the struts of polyurethane foam along the loading direction during a tensile test can be observed. This relationship is based on the hypothesis that deformation in cellular materials is governed by bending. It is mostly used for periodic structures. However, the microstructure of cellular materials is usually not per-fectly periodic. This can lead to the limitation of the predictive capacity of the above models. Several approaches have been developed so far to take into account the presence of imperfections on the mechanical behavior of cellular solids [START_REF] Grenestedt | Influence of cell shape variations on elastic stiffness of closed cell cellular solids[END_REF][START_REF] Fleck | The effect of hole size upon the strength of metallic and polymeric foams[END_REF] using periodic structures and adding randomly dispersed imperfections. Nevertheless, such methods are not specific to a given structure and the representation simplifies the actual microstructure of cellular materials. Numerical models and particularly FE models adapted to XRCT volumes allow the mechanical simulation of a material taking into account a realistic microstructure. Before modeling, the tomographic image has to be binarized. The binarized image is then used to create the FE mesh. The type of element selected is important to obtain an accurate model. In studies, three types of elements have been used to date: -cubic elements, -beams or wall elements, -tetrahedral elements.

A) Meshing with cubic elements

This method consists of translating one voxel into an element of the mesh. The algorithm is very simple to implement and no further complex meshing steps are needed. This meshing technique has been used on polymer, metal or ceramic cellular materials, mainly for the determination of Young's modulus and Poisson's ratio [START_REF] Saadatfar | Mechan-ical and transport properties of polymeric foams derived from 3D images[END_REF][START_REF] Garboczi | An algorithm for computing the effective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase poisson ratios[END_REF][START_REF] Arns | Computation of linear elastic properties from micro-tomographic images: Methodology and agreement between theory and experiment[END_REF] , but also for the determination of the thermal conductivity of metal hollow sphere structures [START_REF] Fiedler | A refined finite element analysis on the thermal conductivity of perforated hollow sphere structures[END_REF] or the influence of the oxidation of the nuclear graphite on mechanical properties (Fig. 8) [START_REF] Berre | Failure analysis of the effects of porosity in thermally oxidised nuclear graphite using finite element modeling[END_REF] . The main drawback of this meshing technique is the difficulty in representing the details of the sample microstructure. The cubic elements do not correctly follow the curved surface. Different solutions exist to: increase mesh resolution (but the calculation becomes very long) as was done by Maire et al. [START_REF] Maire | X-ray tomography applied to the characterization of cellular materials. Related finite element mod-eling problems[END_REF] or Berre et al. [START_REF] Berre | Failure analysis of the effects of porosity in thermally oxidised nuclear graphite using finite element modeling[END_REF] (Fig. 8), adapt a FE program to mesh large samples [START_REF] Roberts | Elastic moduli of model random three-dimensional closedcell cellular solids[END_REF] or use the fast Fourier transform technique [START_REF] Escoda | Estimation of local stresses and elastic properties of a mortar sample by FFT computation of fields on a 3D image[END_REF] . This technique is now being increasingly replaced by meshing with noncubic elements, allowing reduced computation time. Meshing with beam or shell elements can be used for actual structures from tomographic images provided that the microstructure is composed of such a structural element. This is, for instance, very often the case in polymer foams. Different examples exist in studies: PVC foams 6 (see Fig. 9) and stainless steel hollow sphere structures [START_REF] Caty | Modeling the properties of closed-cell cellular materials from tomography images using finite shell elements[END_REF] . For the latter, the structure was scanned in its initial state and this scan was used to create a FE mesh. A compression test until a strain of 9% was performed. Experimental stresses and strains were recorded during the test. The FE mesh was made with shell elements. An elastoplastic law was used to model the behavior of the solid material constituting the shells. The parameters were adjusted to take into account the pores in the steel walls: Young's modulus of 150 GPa and Poisson ratio of 0. The third meshing technique category describes more precisely the actual structure of cellular materials. This is the most currently favored method in studies. Meshing with tetrahedral elements is carried out in two steps. First, a 2D mesh with triangular surface elements is created from the binarized tomographic image. The triangular external faces of the elements can approximate the actual surface of the cellular solids as determined by XRCT. Then, a 3D mesh is built with a tetrahedral filling the interior of the previously defined surface. One of the first studies of tetrahedral meshing was carried out on trabecular bones [START_REF] Ulrich | Finite element analysis of trabecular bone structure: A comparison of image-based meshing techniques[END_REF] : a comparison was drawn between meshing with voxel and tetrahedra. For some samples, the authors obtained more accurate results with tetrahedral meshing. Examples of tetrahedral meshes created for the three types of cellular solids are presented in Fig. 12: the precise reproduction of the microstructure is particularly highlighted in Fig. 12(b). As is already presented in Fig. 10, the results obtained with tetrahedral meshes are generally close to the experimental ones (Fig. 13). A nickel foam sample was first scanned. A tensile test was performed with a displacement rate of 0.1 mm/min. The FE mesh was built from the tomographic image. An elastoplastic law was chosen to model the behavior of nickel. The parameters used in the model were: Young's modulus of 170 GPa, Poisson ratio of 0.3 and yield strength of 25 MPa. The curves presented in Fig. 13 are the experimental data from the tensile test and the simulated data created from the FE modeling. [START_REF] Michailidis | Experimental and FEM analysis of the material response of porous metals imposed to mechanical loading[END_REF] In this example, the Young's modulus is the same in the experiment and in the FE calculation. In terms of macroscopic properties, Vesenjak et al. [START_REF] Vesenjak | Analysis of anisotropy and strain rate sensitivity of open-cell metal foam[END_REF] used the method to demonstrate the influence of the anisotropy and the strain rate sensitivity on the compressive behavior of aluminum foam. It should be noticed that the use of this technique seems to overestimate the elastic stiffness (see again Fig. 10) and the authors sometimes have to use an unrealistic elastic behavior of the solid phase to fit their experimental elastic regime. For example, Marcadon et al. [START_REF] Marcadon | Mechanical behavior of hollow-tube stackings: Experimental characterization and modeling of the role of their constitutive material behavior[END_REF] obtained overestimated Young's modulus as compared to experimental ones for the compressive behavior of Inconel hollow tube stacking. No clear explanation has yet been suggested as to why the models are stiffer than the experimental materials. In terms of local behavior, the prediction is, on the contrary, always very useful to analyze the weak points in the microstructure. Figure 14 presents the simulated stress contours of a representative cell of polyurethane foam before and after a tensile test. [START_REF] Mcdonald | In situ 3D X-ray microtomography study comparing auxetic and non-auxetic polymeric foams under tension[END_REF] The polyurethane foam was scanned in its initial state and underwent an in situ tensile test. The FE mesh was built from the tomographic image. The polyurethane behavior was considered to be elastic with Young's modulus of 45 MPa and Poisson ratio of 0.3. The FE modeling simulates correctly the elongation of the cell in the tensile direction and the contraction in the transverse direction. The zones of stress concentrations can also be predicted (visible in brighter blue and green colors on the figure).

In cellular ceramics, as with other cellular materials, FE models of XRCT images are used to estimate the average Young's modulus and the stress concentrations within the solid phase. [START_REF] Angelo | Finite elements analysis of reticulated ceramics under compression[END_REF][START_REF] Zhang | Modeling the mechanical properties of optimally processed cordierite-mullite-alumina ceramic foams by X-ray computed tomography and finite element analysis[END_REF][START_REF] Arns | Computation of linear elastic properties from micro-tomographic images: Methodology and agreement between theory and experiment[END_REF] FE simulation shows stress concentration logically in the thinnest sections of the solid walls or struts. [START_REF] Angelo | Finite elements analysis of reticulated ceramics under compression[END_REF] Some limitations are noted in these results, especially in some cases, where the estimated Young's modulus is no more accurate than the analytical solutions, such as the Gibson and Ashby model. In the study carried out by Zhang et al. [START_REF] Zhang | Modeling the mechanical properties of optimally processed cordierite-mullite-alumina ceramic foams by X-ray computed tomography and finite element analysis[END_REF] , this finding was related to the volume size, too small in some cases to be representative of the whole microstructure of a material. As a rule of thumb, a representative volume element of a cellular material should at least encompass 10 cells across the three main directions of the solid. Bone substitutes are an important application for cellular materials and have been the subject of recent work on property evaluation using XRCT and FE analyses. Such materials are often more dense than the cellular ceramics mentioned above, with a relative density varying from 50 to 75%. The FE calculation shows in this case the important influence of the pore distribution on the stress distribution [START_REF] Lacroix | Micro-finite element models of bone tissue-engineering scaffolds[END_REF] , local agglomeration of pores leading to higher stress con-centration in the solid phase.

Another issue that confirms the potential of FE simulation based on XRCT volumes is the coupling between mechanical stimuli and the biological response of cellular materials in vivo. Other techniques are possible to mesh a cellular solid, such as mixed meshes made of different elements [START_REF] Veyhl | On the mesh dependence of non-linear mechanical finite element analysis[END_REF] . First, the solid surface is meshed with triangles. Then, a volume mesh is created with hexahedra. Tetrahedra and pentahedra are added where hexahedra cannot fill the overall volume. The comparison of different meshes for metallic foam shows that mixed meshes give the most accurate results. The discrete element method was also used to simulate the mechanical behavior of entangled materials (made of fibers) [START_REF] Barbier | Numerical study of 3D-compressions of entangled materials[END_REF] . This technique consists of discretizing the fibers in segments. It then oversimplifies the description of the material but is useful when friction problems are pre-valent in the material to be analyzed, and the computation time can sometimes be reduced compared to the FE analysis.

IV. Conclusion

X-ray tomography has become one of the preferred methods of characterization for studying cellular solids. The 3D images enable us to obtain a good description of the architecture of the foams. Applied at different scales, using either a standard mode of acquisition or local tomography, the technique permits a more precise de-scription of the material microstructure.

XRCT images are often used to create meshes for FE modeling. Every class of cellular solids (polymer, ceramic, and metal) has been investigated using this combination of techniques. The subsequent FE calculations lead to results in terms of macroscopic properties that can be compared with standard mechanical tests. Many studies have reported that the prediction of this type of calculation is in good agreement with the macroscopic response of the material, although they seem to lead to a systematic overestimation of the elastic properties (Fig. 10). XRCT also allows us to follow the deformation processes of cellular solids when coupled with an in situ mechanical test. In this case, the FE prediction of the local stress concentration can also highlight the regions where stresses localize. This has been shown many times to be in good agreement with the local damage mechanism observed during in situ mechanical tests. This is illustrated by Fig. 15. This combination of techniques is then ready to be applied as a predictive tool. Either based on a tomographic image, or on a numerically generated microstructure, FE models can help engineers to select the best microstructure for a given property (then for a given application) without the need for a complex experimental investigation based on mechanical tests. This could be particularly useful in the case of multiaxial loading, which is not easy to carry out experimentally.

V. References

  X-ray tomography image of (a) a polypropylene foam 8 and (b) a polyurethane foam 7 C) Examples for cellular ceramics

2 :

 2 XRCT images of: (a) an alumina foam 14 and (b) a cross-section of a SiC foam strut 12 D) Examples for cellular metals

  (a) (b) FIG. 3: (a) Ti foams 26 and (b) CuSn12Ni2 foam[START_REF] Guillén | Compressive behavior of bovine cancellous bone and bone analogous materials, microCT characterisation and FE analysis[END_REF]

27 2. 6 )

 276 FIG. 4: Cell size distribution of four metal foams in log-normal scale (Samples A and B : AlSi7 samples, C sample : AlSi11 sample and D sample : Zn sample) 27

FIG. 5 .

 5 FIG.5. Tomographic slice of a cell wall of aluminum foam imaged using the local tomography mode (the sample is larger than the detector field of view). Fine details of the microstructure as micropores[START_REF] Toda | Quantitative assessment of microstructure and its effect on compressive behavior of aluminum foams via high resolution synchrotron X-ray tomography[END_REF] are shown.

  (a) Voxel mesh of a PU foam wih a high resolution 35 and (b) Voxel mesh of the same foam with a lower resolution 36 b) Meshing with beams and shells

3 .FIG. 10 :

 310 FIG.9: Meshing of a cell of PVC foam with beam and shell[START_REF] Fischer | Numerical simulations of mechanical properties of cellular materials using computing tomography analysis[END_REF]

12 : 16 FIG. 13 :

 121613 FIG.13: Experimental and simulated stress strain curve for a tensile test of a nickel foam[START_REF] Michailidis | Experimental and FEM analysis of the material response of porous metals imposed to mechanical loading[END_REF]