Riccardo Petrolo

Valeria Loscrì

Nathalie Mitton

Confident-based Adaptable Connected objects discovery to HArmonize smart City Applications

Keywords: Internet of Things, Wireless Sensor Network, Service Discovery

With the advent of the Internet of Things (IoT) and its need to boost cooperation between different objects in order to improve the quality and the completeness of the produced information, it is important to understand and to evaluate data which comes out and goes into each thing. In this paper we propose CACHACA, a ranking mechanism for Sensor Networks that facilitate the discovery of services provided by each network element. By running the proposed algorithm, it is possible to evaluate and classify the neighborhood and the available services for each node. Performances of CACHACA has been first evaluated through extensive simulations and them stressed when facing a realistic environment through experimentations run on the FIT IoT-LAB testbed. Achieved results demonstrate its effectiveness in the discovery of services process with regards to traditional approaches.

I. INTRODUCTION

In the last decades, wireless Sensor Networks (SNs) have been broadly investigated by both academic and industry fields. This tremendous attention gave birth to numerous applications in different domains (e.g., event detection, environment monitoring, etc.). However, those applications are typically purpose-built and therefore based on multiple architectures and standards making then a highly fragmented scenario.

The advent of the Internet of Things (IoT) [START_REF] Ashton | That 'Internet of Things' Thing[END_REF] and its promise to connect anyone from anyplace and anytime to anythings, highlighted the need of interoperability between different things. As a result, various solutions have been proposed in literature to homogenize interfaces. IETF introduced 6LoWPAN [START_REF] Kushalnagar | Transmission of IPv6 Packets over IEEE 802[END_REF], which defines mechanisms to fragment and compress the header of IPv6 datagrams into IEEE 802.15.4 frames, enabling the integration of the physical world with computer networks. In [START_REF] Guinard | Towards physical mashups in the Web of Things[END_REF], Guinard et al. define the Web of Things (WoT) by combining REST principles into embedded devices and making therefore possible the mash-up of physical and virtual worlds. Thanks to its ability to transform everything as a service, also the Cloud Computing is called to play a key role in the IoT revolution; the Sensing-as-a-service model introduced in [START_REF] Perera | Sensing as a service model for smart cities supported by Internet of Things[END_REF], is at the base of the Cloud of Things (CoT) that we presented in [START_REF] Petrolo | Towards a Smart City based on Cloud of Things, a survey on the smart city vision and paradigms[END_REF], [START_REF]Towards a Smart City based on Cloud of Things[END_REF]; CoT aims to better use distributed resources, putting them together and thus enabling the horizontal integration of various things. On the other hand, CoT and in general Cloud services, have to face latency and intermittent connectivity issues. Recently, fog computing [START_REF] Stojmenovic | Fog computing: A cloud to the ground support for smart things and machine-to-machine networks[END_REF] represents an interesting approach to provide low latency, location awareness, and QoS for real-time applications. In all these paradigms and technologies, SNs keep playing a primary role, since they provide the major hardware infrastructure; at the same time, it is obvious that an evolution of the traditional application-specific design of WSN towards shared system design is needed and that the service model could represent a good enabler. In this context, with a huge number of resources, it is important to discover the ones (and only the ones) that are the more suitable for the application subject.

In this paper, we propose a Confident-based Adaptable Connected objects discovery to HArmonize smart City Applications (CACHACA), a ranking mechanism for Sensor Networks that facilitates the discovery of services provided by each network element. By running CACHACA, it is possible to evaluate and classify the neighborhood and the available services for each node. In order to estimate the pertinence of neighbors and services, we leverage on the flexibility of the fuzzy logic and on its capacity to handle imprecise and incomplete data. CACHACA can be used in order to obtain:

• a complete information, by combining different data sources that offer different services;

• an accurate information, by combining different data sources that offer the same services.

The main novelties introduced by CACHACA rely on the adoption of the Sensing-as-a-Service model [START_REF] Perera | Sensing as a service model for smart cities supported by Internet of Things[END_REF], which allows each network element to be seen as a service provider and the possibility to rank those services. Performances of CACHACA has been first evaluated through extensive simulations and them stressed when facing a realistic environment through experimentations run on the FIT IoT-LAB testbed. Achieved results demonstrate its effectiveness in the discovery of services process with regards to traditional approaches.

The remainder of the paper is organized as follows. Section II reviews the related literature. CACHACA is introduced in Section III and its performances are evaluated in Section V. Section VI concludes the paper and provides hints for future work.

II. RELATED WORK

Within the IoT context, one of the most important challenges is to find appropriate services that satisfy user requirements; the community refers to this challenge as service discovery [START_REF] Perera | Context Aware Computing for The Internet of Things: A Survey[END_REF]. Generally, IoT services are published into registers that can be queried by users in order to obtain a list of candidate services.

Those registers are typically available as end-points of IoT platforms. In the last years, several solutions have been proposed in literature in order to manage Sensor Networks. In this context, the use of semantic in IoT is recognized as one of the most important functionality to connect objects together [START_REF] Perera | Context Aware Computing for The Internet of Things: A Survey[END_REF]. In [START_REF] Hauswirth | Making Internet-Connected Objects readily useful[END_REF], authors point out on the need of a semantic representation to understand data which comes out and goes into the things interfaces; this "data exchange layer" could influence discovery and routing approaches and it can be crucial to enable scalability.

The benefits of semantic annotation are widely explained in [START_REF]): Study on Semantic support for M2M Data[END_REF]; to summarize the most representative are (i) re-use of Machine-to-Machine (M2M) data by many applications; (ii) "write-once run-anywhere" applications; (iii) easy adaptation in case of failures/changes of the available sensor sets.

In order to constitute semantic information in our context, two main options arise: either to use standardized data types, like the one defined by the IPSO Alliance in [START_REF] Shelby | The IPSO Application Framework draft-ipso-app-framework-04[END_REF], or rely on ontologies, like the SSN ontology proposed in [START_REF] Compton | The SSN ontology of the W3C semantic sensor network incubator group[END_REF].

Niu et al. [START_REF] Niu | Context-Aware Service Ranking in Wireless Sensor Networks[END_REF] proposed a context-aware service ranking approach by aggregating the user rating and WSN service context but do not consider a single device but rather the whole network. Durmus et al. [START_REF] Durmus | Service Knowledge Discovery in Smart Machine Networks[END_REF] proposed a discovery protocol based on semantic representation of services; the mechanism operates in the network layer and can directly run SPARQL queries on top of those devices. Anyway, this approach is not suitable for SN context due to the lack of resources. Finally, [START_REF] Wang | A ranking method for sensor services based on estimation of service access cost[END_REF] introduced a ranking strategy by estimating the cost of accessing sensor services using properties of the sensor nodes as well as relevant contextual information extracted from the service access process. In our approach, we do not consider the cost of the service, but instead, we evaluate its quality by considering some physical aspects (e.g., RSSI) related to the service provider node and features of the service itself.

III. CONFIDENT-BASED ADAPTABLE CONNECTED OBJECTS DISCOVERY TO HARMONIZE SMART CITY APPLICATIONS

A. Assumptions and metrics

CACHACA distinguishes three different network elements (see Figure 1): 1) a node that has communication capabilities, and therefore able to communicate within other elements. If a node is equipped with some sensors (circles) we refer to it as full node; then it is capable to measure physical events, for example providing the temperature of a room, or the availability of a parking spot; 2) a relay is a node with communication capabilities; 3) a gateway that is a node in charge of gathering and managing data produced by sensors, and at the same time, enhanced to act as an end-point for the communication with the Internet or with other local devices. In this work we consider a gateway just as a service provider, like a node.

We assume that a Neighbor Discovery mechanism is running on each node u to allow u to discover other nodes v in communication range. So, at a frequency f , each node receives information about its neighborhood that it stores in a Neighbor Table (NT). Note that the exact format of NT is implementation-specific, but according to [START_REF] Vilajosana | Minimal 6TiSCH Configuration[END_REF] it should contain, at least, the following for each neighbor v of u: -numTx: number of transmitted packets to v; -numTxAck: number of packets acknowledged by v; -numRx: number of packets received from v; -Timestamp of the last frame received from v; -Connectivity statistics (e.g., RSSI, LQI), which can be used to determine the quality of the link.

At the same time, we suppose that each node uses a standardized format (e.g., IPSO [START_REF] Shelby | The IPSO Application Framework draft-ipso-app-framework-04[END_REF]) for describing its services (i.e., temperature, light, humidity, etc.). Each service is combined with other complementary information such as: -freshness of the information; can be real-time or temporized; -provider: to specify whether the service is directly provided by the node itself or by a neighbor. The above parameters can be used in order to define relationship of a node with the neighborhood. In this sense, in this work we use some of the above parameters in order to introduce two additional functions, the PHYSICAL CONFI-DENCE (ϕ) -based on the RSSI and Timestamp -and the SERVICE CONFIDENCE (ω) that is computed based on the service information. RSSI (Received Signal Strength Indicator) represents the measured power of a received radio signal; it is widely used in different standards (e.g., IEEE 802.11). According to [START_REF] Veris | Veris Aerospond Wireless Sensors: Received Signal Strength Indicator (RSSI)[END_REF], the RSSI is reported as an integer ranging from -100 dBm to 0 dBm; in this work we normalize it as a value from 0 to 100.

B. Fuzzy logic

In order to compute the physical confidence, we use a rule-based fuzzy inference system [START_REF] Zadeh | Outline of a New Approach to the Analysis of Complex Systems and Decision Processes[END_REF]. A fuzzy logic system can be developed in three steps: -Definition of fuzzy sets (fuzzification). In this first round non-fuzzy inputs (i.e., numbers) are converted into fuzzy sets by using membership functions (e.g., triangular, trapezoid, singleton, bell, or some other type of function).

-Definition of fuzzy rules. Expressed as statements like "IF ... THEN ...", the fuzzy rules summarize the relationship between the fuzzy sets and the output variable.

-Defuzzification. This last stage is used to convert the fuzzy output back into a value that can be later used to make decisions.

C. Physical confidence computation

The physical confidence is computed based on fuzzy logic rules applied to RSSI and timestamp collected by the neighbor discovery protocol in a local and distributed way by each node u for each of its neighbors v. We thus consider three fuzzy sets based on the RSSI values : BAD, GOOD, and EX-CELLENT as displayed by Figure 2 shows the diagrammatic representation of the RSSI that is computed using a trapezoidal membership function. The other parameter used for the estimation of ϕ is the Timestamp; in this case, we consider the difference ∆t (Eq. 1) between the instant at which the computation process is executed (t now ) and the Timestamp t timestamp stored into the NT.

∆t = t now -t timestamp (1)
Once ∆t is obtained, we consider again three fuzzy sets: BAD, GOOD, and EXCELLENT. Since we supposed that the application is time-constrained, we favor small values of ∆t (Fig. 3); therefore a node that provides services in real-time will be highly compared to one with higher values of ∆t. After completing the fuzzification process, we apply the fuzzy rules to obtain the physical confidence. Table I shows the definition of the rules in CACHACA. An example can be observed as: "IF RSSI is Excellent AND ∆t is Excellent THEN ϕ is Excellent". It is worth noting that we give more importance to the time parameter. Indeed, when the RSSI is Good and ∆t is Excellent, neighbors are still noted as Excellent; this is because a communication can be completed even with lower RSSI, on the other end, if the neighbor is not often active, it is important to classify it as Bad.

D. Service confidence computation

The service confidence (ω) is computed by each node considering one of its mono-modal services per time (e.g., temperature); in this case we use just the Freshness feature. As shown in Table II, ω is considered Excellent when it is possible to access in real-time to the values of the services. IV. CACHACA Yet, a node is now able to characterize the different confidence values for each of its neighbors periodically, for each packet received. Algo. 1 describes how the physical confidence is updated by u upon reception of a new packet from v. u checks whether v is already stored into its NT, if so, it updates its NT with the RSSI and the Timestamp and then it computes ϕ for each node present in its NT. If not, a new entry will be added, with the ID of v, the RSSI and the Timestamp; at this point u computes ϕ for each neighbor by applying the fuzzy logic rules above presented. add v in NT with associated RSSI and Timestamp; 5: end if 6: ∀ w in NT, update ϕ(w) following Table I.

In order to discover efficiently the different services available, nodes advertise their services and the associated confidence periodically and can relay the information about a service offered by a neighbor. The format of the frame is shown in Fig. 4; the Service uses 10 bytes, while the confidence can be transmitted by using only 1 byte. Considering that the length of the MAC frame of IEEE 802.15.4 can be maximum 127 bytes, and subtracting 31 bytes of header and 2 bytes of footer, in one message we could advertise up to 7 services. Upon reception of such a message from a neighbor v, a node u can thus upgrade entry of v in its NT with these values as shown in Table III; for each neighbor, the information stored will be: the offered Service, the ID, the two confidences (physical and service), the RSSI, and the Timestamp. This Neighbor Table is used to evaluate the neighborhood and to rank the neighbors. Furthermore, the NT can be cleaned by removing deprecated entries for space saving purposes. When the information about a service is relayed, the ω confidence is also function of the physical confidence. Table IV shows how ω transmitted by the relay node is influenced by ϕ and ω of the Neighbor that provides the service; the best value that we can obtain is Good and it is verified when the ω N eighbor is Excellent and ϕ N eighbor is Excellent or, at at least, Good; while, even when the ω N eighbor is Excellent, ω will be Bad if the ϕ N eighbor is Bad. Algo. 2 shows the process of advertisement of a service by a relay node; this can be done just when the ω is Good.

if ((ϕ(w)) = (Excellent)) || ((ϕ(w)) = (Good)) then 8:
compute ω(w);

end if 12: end if V. PERFORMANCE EVALUATION
To evaluate the performance of CACHACA, we use Contiki-OS 1 and its simulation tool Cooja; Table V summarizes the principal parameters. Among the others (e.g., TinyOS, Riot2 ) we choose Contiki because its good assessment by the community, its completeness and re-usability; with Contiki indeed, it is possible to run simulations and then re-use the code to flash real devices. We consider an area of 200x200 m 2 , in which M network elements are randomly positioned; M is the sum of R relays and N full nodes equipped with 1 sensor. Values of M, R and N and what they stand for depend of the scenario under evaluation as detailed later. We use the following metrics to assess the performances of CACHACA:

• service avg represents the average number of services discovered by each node;

• neighbor avg is the average number of neighbors discovered by each node;

• packets avg is the average number of packets transmitted by each node;

• ω avg is the average value of the service confidence ω computed by each node;

• ϕ avg is the average value of the physical confidence ϕ computed by each node.

We performed the simulations in five different scenarios (Table VI). In all scenarios, N is set to 5; each node advertises its own service periodically. In the first Scenario, we have just the 5 nodes running, while in the second and third scenarios we introduce some relay nodes. In the last 2 Scenarios, we consider that relay nodes can move inside the area with an average speed of 1 m/s. This set of Scenarios can be used to describe a generic smart city use case (e.g., smart building). A number of different sensors is available in distinct rooms; those sensors can offer services like temperature, luminosity, and so on; other devices (attached for instance to the smart-phones of employed) act as relay for the sensors' services. We chose to use only 5 Full nodes and evaluate the number of relay necessary to discover all the potential services. Moreover, we vary the number of relay between 10 and 15 because we want to study the behavior when the network is not highly dense and therefore avoiding to compare CACHACA with the Broadcast scenario that suffers from crowded cases. Fig. 5 shows the average number of packets sent per node. We can observe that the Packet avg is higher in the first Scenario; while this number decreases with relay nodes. This is because relay nodes broadcast packets only if the service offered by the neighbor has ω = Excellent and the ϕ of the neighbor is at least Good. Introducing mobility makes Packets avg increase because of the higher possibility to meet nodes and therefore for relay to broadcast services. For the sake of equity and fairness, we have also considered that relay nodes broadcast a service immediately when it is discovered, without taking account of the quality (dashed lines in the Figure ). In this case, we can observe that the number of messages increases intensely and so also the quality of the channel and the energy consumption will be negatively affected. In the first Scenario, and in general when there is no mobility, the performance are bad, this because nodes and relay are randomly deployed on the field and therefore it is possible that they are not in communication range. With mobility (Scenario5), the performance improves; each node is capable to discover about 40% of the available services and more than 50% of neighbors. In the last investigation (Figure 7), we consider the behavior of ϕ and ω in function of the Number of Nodes. We can observe that the average ω computed by each node for the discovered services is Good; this means that each node can discover more than 40% of the services provided with Good confidence. Regarding the Physical Confidence, we can note that it increases when mobility comes in play, but at the same time, ϕ decreases with a higher number of nodes, because of possible interference. Anyway, it is important to highlight that even when ϕ is Bad, a communication can happen.

B. Experimentation results

In order to face CACHACA to a realistic environment, we ran experimentation on FIT (Future Internet of Things) IoT-lab 3 ; a very large scale infrastructure facility suitable for testing small wireless sensor devices and heterogeneous communicating objects over large scale. We used the Rennes site, and we performed experimentation (parameters available in Table VII) using Scenarios 1, 2 and 3 (no mobility). Figure 8 shows the Services and the Neighbors discovered. We can observe that in this case, CACHACA has performance similar to the ideal scenario (dashed lines); when the Number of Nodes is 10, each network element discovers almost all the Services and about 70% of the neighbors; the same trend is maintained when the Number of Nodes is 15. Those results are in line with the ones obtained running simulation with Cooja; therefore we can conclude that when we increase the Number of Nodes the efficiency of our proposal is higher.

Regarding the Physical and Service confidences (Fig. 9), we can observe that both parameters have better performance when the network is sparse; this is because, the services are directly provided by the direct neighbor, without the intervention of relay; ϕ decreases with the Number of Nodes, because of more interference. The trend obtained in this analysis is once again complementary to the one obtained with the simulator.

VI. CONCLUSION

With the massive number of deployed things and the need of interoperability between those devices, it is important to understand which "service" each node can offer and to evaluate it. In this paper we have proposed CACHACA, a ranking mechanism for Sensor Networks that facilitates the discovery of services provided by each network element. CACHACA could run on different Smart City use cases. Smart Building which different nodes are installed. Those nodes are equipped with various sensors in order to measure physical events, e.g., environmental conditions, room occupancy, building structure. Thanks to CACHACA, every node will be aware about the services available in its neighborhood; therefore they will be able to cooperate in order to achieve common application tasks, e.g., room environmental conditions, office occupancy. Smart Street in which bus stops, cars, buses, and parking spots of a city are equipped with nodes running CACHACA. Thanks to the service discovery and to the ranking mechanism, a car can recognize the best node which offers parking facility and use it. Moreover, a bus could advertise the availability of free places and so on.

We evaluated performance under different settings both with simulation and experimentation. Results show that CACHACA performs better in terms of Packets sent while obtaining good results compared to the ideal scenario in terms of Services and Neighbors discovered.

In the future, we plan to study more sophisticated criteria for Service and Physical Confidences and to create a real testbed that validates the Smart Building scenario.

Fig. 1 :

 1 Fig. 1: Network elements: Node (square), sensor (circle), and gateway (rectangle).

Fig. 2 :

 2 Fig. 2: Diagrammatic representation of RSSI.

Fig. 3 :

 3 Fig. 3: Diagrammatic representation of ∆t.

Algorithm 1

 1 Physical confidence update -Run on node u upon reception of packet from node v. 1: if v ∈ NT then 2: update RSSI and Timestamp values for v in NT; 3: else 4:

Fig. 4 :

 4 Fig. 4: Frame format with Service and Service confidence.

Fig. 5 :

 5 Fig. 5: Number of Elements vs. Packets avg sent.

Fig. 6

 6 Fig.6indicates the average number of Neighbors and discovered by each element network in function of the Number of Nodes. In the first Scenario, and in general when there is no mobility, the performance are bad, this because nodes and relay are randomly deployed on the field and therefore it is possible that they are not in communication range. With mobility (Scenario5), the performance improves; each node is capable to discover about 40% of the available services and more than 50% of neighbors.

Fig. 6 :

 6 Fig. 6: Number of Nodes vs. Services and Neighbors Discovered.

Fig. 7 :

 7 Fig. 7: Number of Nodes vs. Physical and Service Confidence.

Fig. 8 :

 8 Fig. 8: Number of Nodes vs. Service and Neighbor Discovered (FIT IoT-lab).

Fig. 9 :

 9 Fig. 9: Number of Nodes vs. Physical and Service Confidence (FIT IoT-lab).

TABLE I :

 I Rule based fuzzy inference.

	RSSI	∆t	ϕ
	Excellent	Excellent	Excellent
	Good	Excellent	Excellent
	Excellent	Good	Good
	Good	Good	Good
	Bad	Excellent	Good
	Excellent	Bad	Bad
	Good	Bad	Bad
	Bad	Good	Bad
	Bad	Bad	Bad

TABLE II :

 II Service confidence computation for a Full node.

	P rovider	F reshness	ω
	sensor	real-time	Excellent
	sensor	temporized	Bad

TABLE III :

 III Neighbor Table.

	ID	Service	ω	ϕ	RSSI	T imestamp
	1	temp	excellent	good	80	1431108000
	30	light	good	good	50	1431108008
	2	temp	excellent	excellent	90	1431108007
	...

TABLE IV :

 IV Service confidence computation for a Relay node.

		ϕ N eighbor	ω N eighbor	ω
		Excellent	Excellent	Good
		Good	Excellent	Good
		Good	Bad	Bad
		Bad	Bad	Bad
		Bad	Excellent	Bad
	2:	update NT(RSSI, Timestamp) for NT.ID;
	3: else
	4:	store v in NT with associated RSSI and Timestamp;
	5: end if
	6: ∀w ∈ NT do update ϕ(w) with Table I
	7:

Algorithm 2 Service confidence computation for a Node. -Run at node u upon reception of a packet from v

1: if v ∈ NT then

TABLE V :

 V Simulator parameters.

	P arameter	V alue
	Nodes radio chip	CC 2420
	Nodes flash memory	1 MB
	Simulation seed	random
	Simulation runs ∀ scenario	10

TABLE VI :

 VI Simulator scenarios.

TABLE VII :

 VII Experimentation parameters.

	P arameter	V alue
	Nodes type	WSN 430
	Nodes radio chip	TI CC 2420 @ 2.4 GHz
	Nodes flash memory	1 MB

http://www.contiki-os.org

http://tinyos.net, http://www.riot-os.org

This work is partially supported by CPER DATA and by the European Community in the framework of the FP7 VITAL project.


Mathilde Granke
email: mathilde.granke@gmail.com

Quentin Grimal
email: quentin.grimal@upmc.fr

Amena Saïed
email: amena.saied@upmc.fr

Pierre Nauleau
email: pierre.nauleau@upmc.fr

Françoise Peyrin
email: peyrin@esrf.fr

Pascal Laugier
email: pascal.laugier@upmc.fr

Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women

Keywords:

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

Introduction

Bones of different individuals not only have different sizes and shapes, but also different material properties. These characteristics entirely determine the elastic response of a bone to a given mechanical loading. The elastic properties of cortical bone tissue, which has a hierarchical organization, must be described in a multiscale framework: the structure and mechanical properties at one hierarchical level determine the properties of the subsequent one. The mesoscale designates the intermediate scale between the microscale (lamellar structures) and the macroscale (organ level). More precisely, the characteristic size of a mesoscopic volume will be larger than 1.5 mm [START_REF] Grimal | A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties[END_REF] and smaller than the thickness of the cortical shell. The mesoscale elastic properties are of first interest because they depend on tissue properties at all small-scale hierarchical levels and they have a direct influence on the macroscopic mechanical response of bones. The observed intra-individual [START_REF] Orias | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF] and interindividual [START_REF] Ashman | A continuous wave technique for the measurement of the elastic properties of cortical bone[END_REF][START_REF] Rudy | Anatomic variation in the elastic inhomogeneity and anisotropy of human femoral cortical bone tissue is consistent across multiple donors[END_REF] variations of mesoscale elasticity are footprints of the remodeling process and the structure-function adaptation mechanisms of bone. This calls for a clear understanding of the variables that govern bone mesoscopic elasticity variations.

At the mesoscale, bone can be described as a two-phase composite material: a dense mineralized matrix and a soft phase, hereinafter referred to as vascular porosity [START_REF] Cowin | Bone poroelasticity[END_REF], which consists of Haversian canals and resorption cavities containing fluids and soft tissues. The porosity has been established to be an important determinant of the bone mesoscopic elastic properties [START_REF] Wachter | Correlation of bone mineral density with strength and microstructural parameters of cortical bone in vitro[END_REF][START_REF] Rho | Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nanoindentation[END_REF][START_REF] Dong | The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity[END_REF]. On the other hand, one would expect that variations of the mineralized matrix properties strongly affect the mesoscopic elasticity because the matrix occupies about 85% of the cortical bone volume. However, the actual influence of matrix properties variations on mesoscale elasticity is still a matter of debate in the literature. Changes in matrix mineralization have been shown to be correlated with the mesoscopic mechanical properties variations when the data were combined from eighteen species [START_REF] Currey | The effect of porosity and mineral content on the Young's modulus of elasticity of compact bone[END_REF], but not when only human data were considered [START_REF] Mccalden | Age related changes in the tensile properties of cortical bone[END_REF]. Rho et al. [START_REF] Rho | Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nanoindentation[END_REF] found that the matrix elasticity (probed with nanoindentation) was significantly correlated to the mesoscopic axial Young's modulus. Since both vascular porosity and matrix properties determine mesoscale elasticity, it is not possible to draw general conclusions unless both porosity and matrix properties are measured on the same samples. To our knowledge, only Rho et al. [START_REF] Rho | Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nanoindentation[END_REF] investigated to what extent the changes in porosity and matrix elasticity contribute to the variations of the mesoscopic elasticity. They found a significant correlation of both variables with the mesoscopic elasticity variations. Unfortunately, the elastic properties and the porosity were assessed on different specimens and along the bone axis direction only. Human cortical bone possesses anisotropic elastic properties which are often approximated by transversely isotropic or orthotropic properties both at the microscale [START_REF] Currey | The effect of porous microstructure on the anisotropy of bonelike tissue: a counterexample[END_REF] and mesoscale [START_REF] Ashman | A continuous wave technique for the measurement of the elastic properties of cortical bone[END_REF][START_REF] Yoon | Ultrasonic wave propagation in human cortical bone -II. Measurements of elastic properties and microhardness[END_REF]. The preferential orientation of the pores and the mineralized fibrils are such that the relationships between matrix properties, porosity and mesoscale elasticity may be significantly different in the axial, radial and tangential directions of bone.

The objective of this work was to assess the relative contributions of vascular porosity and mineralized matrix elasticity to the mesoscopic elasticity variations in mature human cortical bone. To this purpose, experiments were designed following two requirements, which constitute the originality of the work. First, the bone matrix elasticity (reflected in acoustical impedance values) and porosity, as well as the mesoscopic elasticity, were measured on the same samples. Second, elasticity measurements at both the micro and the mesoscale were performed in three orthogonal directions. Finally, the experimental results were compared with the predictions of a micromechanical model to question the assumption that cortical bone can be modeled as a homogeneous transversally isotropic matrix pervaded by cylindrical pores.

Material and methods

Bone sample preparation

Fresh bone specimens were prepared from a collection of ten left femurs of female cadavers (mean donor age 81 years, range 66-98 years). Femurs were removed during multi-organ collection and stored at -20°C. Ethical approval for the collection of samples was granted by the Human Ethics Committee of the Centre du don des Corps at the University Paris Descartes (Paris, France). The tissue donors or their legal guardians provided informed written consent to give their tissue for investigation, in accord with legal clauses stated in the French Code of Public Health. A cross-section of thickness approximately 7 mm was cut in the mid-diaphysis of each femur. In order to maximize the variability of bone properties, parallelepiped-shaped samples were harvested from different anatomical quadrants (lateral, medial, posterior) of each cross-section. No sample was extracted in areas where the cortical thickness was less than 4 mm. This led to a set of twenty-one samples (nominally 5 x 5 x 7 mm 3 ): three samples from two of the femurs, two samples from seven other femurs and one sample from the remaining femur. The samples faces were oriented according to the radial (axis 1), circumferential (axis 2), and axial (axis 3) directions defined by the anatomic shape of the femoral diaphysis [START_REF] Orias | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF]. The samples were defatted for 12 hours in a chemical bath of diethylether and methanol (1:1). The parallelism of the opposite faces was controlled with a 50 µm admitted error. The six faces of each sample were polished with a hard synthetic cloth using 3 µm polycrystalline diamond abrasive particles followed by a 0.05 µm aluminum oxide suspension (Metadi Supreme and Masterprep, Buehler ® GmbH, Düsseldorf, Germany). After preparation, the samples were stored in gauze soaked in saline solution at 4°C for no more than 48 hours prior to measurements.

Assessment of mesoscale elasticity

Mesoscale elasticity was determined using a well-established method based on the measurements of ultrasonic bulk wave velocities and sample apparent mass density. The method, which has been extensively described elsewhere [START_REF] Orias | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF][START_REF] Yoon | Ultrasonic wave propagation in human cortical bone -II. Measurements of elastic properties and microhardness[END_REF], is the only existing method which provides measurements of the shear and longitudinal elastic properties in the different directions of a same bone material volume. In contrast, mechanical methods (traction, torsion, three-point bending, etc.) usually require to prepare one sample for the measurement of each property. Given the ultrasonic bulk wave velocities v and apparent density ρ, the diagonal terms Cii of the mesoscopic elastic tensor are calculated from:

v v C v v C v v C i v C ii ii ρ ρ ρ ρ ρ ρ ρ = = = = = = = = (1)
where C11, C22, and C33 are the so called longitudinal elastic coefficients which represent the stiffness in a traction-compression mode, and C44, C55, C66 are the shear coefficients.

Velocity vji denotes the velocity of a bulk wave propagating in direction i with particles motion in the j-direction. For longitudinal waves, i = j, and for shear waves, i ≠ j. Samples were measured undrained in ambient conditions. The apparent mass density of each sample was assessed by dividing its mass by its volume; geometrical dimensions were measured with a digital caliper (accuracy: ± 0.02 mm) and mass with a laboratory scale (accuracy: ± 0.1 mg). The ultrasonic (US) wave velocities were evaluated using a pulse transmission method with a pair of frequency matched transducers in contact with the sample surface.

Longitudinal waves and shear waves were measured using 2.25 MHz and 1 MHz transducers (respectively, V105RM and V152RM, Panametrics, Inc., Waltham, MA). Since the longitudinal and shear wave velocities in bone are significantly different (~ 3700 m/s and 1700 m/s, respectively), the use of different frequencies for these two propagation modes allowed obtaining a similar wavelength, of the order of 1.7 mm. Hence, the resulting wavelength, which defines the probing scale, guaranteed to retrieve the bone mesoscopic elasticity (i.e. at a scale much larger than the vascular pores). The received signal was acquired using an oscilloscope (TDS 2012, Tektronix Inc., Beaverton, OR) and postprocessed with a custom MatLab program (The Mathworks Inc., Natick, MA). The time delay, Δt, for wave transmission through the specimen was obtained as the difference between the arrival time of the US pulse with the sample inserted and the arrival time of a reference signal (transducers in contact for the longitudinal waves, Plexiglas plate inserted between the transducers for the shear waves). Each longitudinal coefficient was calculated after averaging the velocities measured in ten successive acquisitions with intermediate repositioning; each shear coefficient was obtained after averaging the two shear wave velocities from which it could be calculated (equation ( 1)).

The accuracy of the elastic coefficients evaluation was determined from measurements on a homogeneous calibrated pure polycrystalline (99.95%) copper plate (Goodfellow SARL, Lille, France) and was found to be 2.1% and 0.9% for the longitudinal and shear elastic coefficients, respectively. Measurement errors were assessed by repeating longitudinal and shear waves velocity measurements on two human bone specimens for five consecutive days with intermediate repositioning. The reproducibility was 3.2% and 4.7% for the mesoscopic longitudinal and shear elastic coefficients, respectively. Finally, our measurements were verified to be bulk wave velocities and not bar wave velocities [START_REF] Ashman | A continuous wave technique for the measurement of the elastic properties of cortical bone[END_REF]. For this, longitudinal wave velocities were measured in eight artificial composite bone samples (Sawbones, Pacific Research Laboratory Inc, Vashon WA) of dimensions 10 x 20 x d mm 3 (thickness x cross-sectional dimension), the lateral dimension d varying from 2 to 10 mm.

The same velocity was measured for all the Sawbone samples (2907 ± 11 m/s). The value corresponds to the tabulated bulk velocity for this material (2890 m/s). Thus, the velocities measured in this study, even for the smallest samples, were proved to be bulk wave velocities and equation ( 1) can be applied to derive the elastic coefficients.

50-MHz Scanning Acoustic Microscopy

A custom scanning acoustic microscope (SAM), operating with a spherically focused 50-MHz transducer (V605, Valpey Fisher, Hopkinton, USA), was used to probe the acoustic impedance normal to the samples surfaces according to the measurement procedure extensively detailed in previous studies [START_REF] Raum | Bone microstructure and elastic tissue properties are reflected in QUS axial transmission measurements[END_REF][START_REF] Saïed | Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-MHz scanning acoustic microscopy and its relation to porosity in human cortical bone[END_REF]. The acoustic impedance (Z), which is modeled as the square root of the product of the local mass density and elastic coefficient in the beam direction, has been shown to be a surrogate measurement of the bone matrix elasticity at the microscale [START_REF] Preininger | Spatial-temporal mapping of bone structural and elastic properties in a sheep model following osteotomy[END_REF]. This is also reflected by the good agreement between the impedance and the Young's modulus of bone matrix as obtained from site-matched SAM and nanoindentation measurements in bone [START_REF] Hofmann | Assessment of composition and anisotropic elastic properties of secondary osteon lamellae[END_REF][START_REF] Rupin | Assessment of Microelastic Properties of Bone Using Scanning Acoustic Microscopy: A Face-to-face Comparison with Nanoindentation[END_REF] (although some discrepancies appear presumably due to the assumptions made on the Poisson ratio). Calibrated impedance maps were obtained with a lateral resolution of 30 µm for all six faces of each sample. The Zmaps were segmented allowing the separation of vascular porosity and bone matrix as previously reported [START_REF] Raum | Sitematched assessment of structural and tissue properties of cortical bone using scanning acoustic microscopy and synchrotron radiation µCT[END_REF].

The acoustic impedance of the matrix was determined from the segmented maps for each face of the twenty-one samples. Note that the small pores (Volkmann's canal, osteocyte lacunae) could not be resolved so that they contributed to the probed bulk matrix properties.

Matrix impedance in the probing direction, denoted i Z ˆ(i=1,2,3), was defined as the average of the impedance values of the matrix pixels in two opposite faces of normal ni (i=1,2,3).

The reproducibility of the assessment of Z ˆ, obtained after imaging the face of the same bone four times on different days, was found to be 1.4%.

The 2D cross-sectional porosity was calculated from the segmented Z-maps in the 1-2 plane, i.e. perpendicular to the bone axis (Fig. 1), as the ratio of the pores area to the total bone surface. Porosity is usually assumed to vary only slightly across sample thickness.

This assumption is reasonably met with the typical sample thickness of 7 mm, given that

(1) the Haversian canals are roughly aligned with the bone axis and (2) the osteon length is 4 mm on average in human femoral mid-diaphysis [START_REF] Cooper | Three-dimensional microcomputed tomography imaging of basic multicellular unit-related resorption spaces in human cortical bone[END_REF]. However in our experience, large resorption cavities visible on a cross-sectional surface can introduce a significant bias in the estimation of volumetric porosity from surface porosity. To overcome this limitation, we estimated the volumetric porosity of each sample (denoted Por) as the average value of the cross-sectional porosities assessed on the two opposite faces in the 1-2 planes.

Synchrotron Radiation Microtomography (SR-µCT)

To comfort our assumption that Por is a good surrogate for the volumetric porosity, a subset of specimens was imaged using 3D SR-µCT. SR-µCT measurements were performed on the imaging beamline ID19 at the ESRF (European Synchrotron Radiation Facility, Grenoble, France). The beam energy was tuned to 27 keV by using a (Si111) double crystal monochromator. A full set of 2D radiographic images was recorded using a CDD detector (FReLoN camera; ESRF Detector group) by rotating the sample in 1999 steps within a 360° range of rotation in about 35 minutes. We selected a pixel size of 5.4µm

on the detector providing a 3D reconstructed image volume with a measured spatial resolution of about 10 µm. Due to time limitations at the ESRF facilities, only ten of the twenty-one samples were imaged. After the 3D tomographic reconstruction and the conversion of the linear attenuation coefficients to degree of mineralization values expressed in g/cm 3 of hydroxyapatite (HA) crystals [START_REF] Nuzzo | Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography[END_REF], the 3D-porosity was derived from the segmented SRµCT images, following a fixed threshold set to 0.7 g HA/cm 3 .

Micromechanical model

Micromechanical models are useful as a means of testing how changes of the bone microscale properties affect its mesoscopic behavior. The modeled behavior depends in particular on hypothesized organizational patterns and elastic symmetry of the model material phases. In this work, a model of cortical bone mesoscopic elasticity based on asymptotic homogenization (AH) was used (source code available online [START_REF] Grimal | Cortical bone homogenization[END_REF]). This micromechanical method was chosen for its stability, even at high porosities. The theory was described in details in the case of matrix isotropy in Parnell and Grimal [START_REF] Parnell | The influence of mesoscale porosity on cortical bone anisotropy. Investigations via asymptotic homogenization[END_REF]. The model hypothesizes that cortical bone can be regarded as a homogeneous transversely isotropic (TI) matrix pervaded by cylindrical pores, which are periodically distributed within the matrix material, specifically on a hexagonal lattice (Fig. 1). Here, the plane normal to the pores (1-2 plane) is the plane of isotropy for the matrix. The representation leads to transversely isotropic elasticity at the mesoscale (isotropy in the 1-2 plane), which is a reasonable approximation in human femoral mid-diaphysis [START_REF] Orias | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF][START_REF] Yoon | Ultrasonic wave propagation in human cortical bone -II. Measurements of elastic properties and microhardness[END_REF]. Given an elastic tensor m c describing the matrix elasticity, an elastic tensor p c describing the elasticity of the material within the pores, and the volume fraction of pores, a homogenized elastic tensor C * at the mesoscale is calculated. The elastic tensor of the bone matrix was identical for all samples. Its coefficients were determined by minimizing the L2-norm of the relative error between the experimental (C) and homogenized (C * ) mesoscopic elasticity values over the twenty-one samples. Hence m c is the tensor which minimizes the objective function defined as:

∑∑ = =         - = 21 1 6 1 2 ; * ; ; 0 ) , , ( ) ( k i k ii k p m k ii k ii m C Por c c C C c H ( 2
)
where Pork refers to the estimate of porosity of sample k assessed from impedance maps, and

k ii C ; and * ;k ii C
to its experimental and homogenized elastic coefficients. Since the samples were kept moist during the measurements, the material in pores (undrained) was assumed to behave like bulk water, that is, bulk modulus and Poisson ratio were set to 2.3

GPa [START_REF] Anderson | Mechanics of fluids[END_REF] and 0.4999 (quasi-incompressible), respectively, from which the terms of p c can be calculated.

Statistics

The distribution normality and variance equality were confirmed using Shapiro-Wilk and Bartlett's tests respectively. One-way analysis of variance (ANOVA) followed by post-hoc comparisons using Tukey's HSD test were performed to evaluate the differences in the different directions for the longitudinal and shear elastic coefficients and for the mean acoustic impedance. Note that the influence of the anatomical quadrant on the elasticity was not investigated due to the small number of samples (posterior (n=2), lateral (n=9), and medial (n=10)). Adjusted R 2 (adj-R 2 ) from single linear and stepwise multiple regression analyses were used to characterize the relative contributions of the vascular porosity (Por)

and bone matrix mean impedance in the different directions ) ( i Z to the mesoscopic elastic coefficients (Cii). After the determination of the optimal matrix properties m c in the AH model (equation 2), the agreement between the experimental and homogenized elastic coefficients as obtained from the AH model was deduced from the linear regression parameters (adj-R 2 and root mean square error (RMSE)). All statistical results were considered significant for p-values less than 0.05. Statistics were made using the MatLaB Statistics Toolbox (The Mathworks Inc., Natick, MA, USA) and JMP (SAS Institute Inc., Cary, NC).

Results

We evaluated the anisotropic elastic properties of the samples at two scales. At the mesoscale, ANOVA showed that the samples exhibited a strong elastic anisotropy which was reflected in the longitudinal elastic coefficients (F = 98, p<10 -5 ) as well as in the shear elastic coefficients (F = 26, p<10 -5 ). Precisely, we observed (Tukey HSD) C33 > C11 (not different from C22) and C66 < C44 (not different from C55). At the microscale, the bone matrix also exhibited anisotropy (F = 96, p< 10 -5 ), which was reflected in a significant higher impedance value along the bone axis compared to the two transverse directions 1 Ẑ and 2 Ẑ , which did not significantly differ. The average values of the mesoscopic elastic coefficients and the bone matrix mean impedance are summarized in Table 1. The p-values of the Tukey tests are given in Figure 2.

The comparison, for a subset of ten samples, of the 3D-porosity obtained from the SR-µCT to the estimated porosity value (Por) allowed to validate the assessment of volumetric porosity from the segmented impedance maps. Precisely, Por and the 3D-porosity were not significantly different (paired t-test, p = 0.48) and were highly correlated as shown by the linear regression results (adjR 2 = 0.98, RMSE = 0.94%, slope not significantly different from 1) (Fig. 3). Por was found to be (mean ± sd) 13.5 ± 6.8 %, covering a wide range of values [3-27%].

A weak but significant correlation was found between all Cii, except C22, and 3 Ẑ (bone axis direction) (adj-R 2 <0.25, p=[0.01-0.04]) (Table 2). No significant correlation was found between the Cii and the matrix impedance in the radial and circumferential directions

Discussion

To our knowledge, the current work is the first to provide, for the same set of samples, measurements of the anisotropic elastic properties at two scales together with an evaluation of the cortical porosity. A set of human femoral cortical bone data, obtained on twenty-one samples from ten donors, was used to investigate the relative contributions of both the matrix elasticity and the porosity to the bone mesoscopic elasticity.

The experimental data corroborated well with previous studies, be it in respect of the mesoscopic elastic coefficients [START_REF] Orias | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF][START_REF] Ashman | A continuous wave technique for the measurement of the elastic properties of cortical bone[END_REF][START_REF] Yoon | Ultrasonic wave propagation in human cortical bone -II. Measurements of elastic properties and microhardness[END_REF], the mean acoustic impedance of the bone matrix [START_REF] Saïed | Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-MHz scanning acoustic microscopy and its relation to porosity in human cortical bone[END_REF][START_REF] Rupin | Assessment of Microelastic Properties of Bone Using Scanning Acoustic Microscopy: A Face-to-face Comparison with Nanoindentation[END_REF], or the range of the intracortical porosity [START_REF] Bousson | CT of the middiaphyseal femur: cortical bone mineral density and relation to porosity[END_REF][START_REF] Thomas | Regional variation of intracortical porosity in the midshaft of the human femur: age and sex differences[END_REF][START_REF] Cooper | Age-dependent change in the 3D structure of cortical porosity at the human femoral midshaft[END_REF].

Impedance measurements suggested that the average elastic properties of the mineralized matrix did not undergo large variations in the different samples (with coefficients of variation of the i Z ˆ all inferior to 6%). The limited variations of bone matrix elasticity reflected in Z might explain the lack of correlation between the mean acoustic impedance of the matrix and the mesoscopic elastic coefficients. A literature review reveals that such modest variations of the bone matrix properties have been observed in a number of studies.

Cross-sectional reports have shown that the mean degree of mineralization of bone does not exhibit large variations between individuals, independently of age [START_REF] Mccalden | Age related changes in the tensile properties of cortical bone[END_REF][START_REF] Boivin | The degree of mineralization of bone tissue measured by computerized quantitative contact microradiography[END_REF][START_REF] Bergot | The degree of mineralization in the human femoral shaft change with age and sex in a microradiographic study[END_REF] and gender [START_REF] Boivin | The degree of mineralization of bone tissue measured by computerized quantitative contact microradiography[END_REF]. A few studies have measured the matrix elasticity on several individuals at the same cortical bone site (femoral diaphysis and neck [START_REF] Hoffler | Heterogeneity of bone lamellar-level elastic moduli[END_REF], femoral diaphysis [START_REF] Rho | Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nanoindentation[END_REF], radius [START_REF] Saïed | Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-MHz scanning acoustic microscopy and its relation to porosity in human cortical bone[END_REF]).

Similarly, they all reported small changes in the mean value of the matrix elasticity (average of several measurements points on a surface of at least 1 mm) with coefficients of variation ranging between 3 and 10 %. Hence, although the bone matrix elasticity is known to display strong local heterogeneities (in particular between the osteonal and interstitial tissues), its mean value over a few millimeters remains relatively constant in healthy individuals. However, a selection of bone specimens in a population with known bone pathologies could result in a wider variation of matrix material properties and lead to different conclusions.

Our results demonstrate that, for an elderly population, the change in porosity is the major determinant of the variations of the anisotropic elastic coefficients at the mesoscale, at least in the femoral mid-diaphysis. To our knowledge, only one study has experimentally examined the impact of porosity variations on the elasticity of human femoral cortical bone in several directions [START_REF] Dong | The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity[END_REF]. While they also found a strong dependence of the Young's moduli and shear moduli on porosity (R 2 = [0.66 -0.72]), they observed no significant correlation between the elastic properties in the transverse direction and the porosity, in contrast to our findings.

The fact that all the mesoscopic elastic coefficients have a dependency on the porosity is supported by the theoretical results obtained with several models using different homogenization approaches [START_REF] Grimal | A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties[END_REF][START_REF] Hellmich | Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions?[END_REF][START_REF] Dong | Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method[END_REF][START_REF] Baron | Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1 MHz: a finite-difference time domain study[END_REF][START_REF] Grimal | Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale[END_REF][START_REF] Grimal | A two-parameter model of the effective elastic tensor for cortical bone[END_REF]. We compare the outcome of a homogenization model to experimental data for known values of porosities associated to a number of bone material volumes. As far as we know, only two previous studies confronted experimental results with the predictions of a micromechanical model. However, the elastic constants were not assessed on the same specimens [START_REF] Dong | Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method[END_REF], or the shear constants were lacking [START_REF] Deuerling | Specimen-specific multiscale model for the anisotropic elastic constants of human cortical bone[END_REF]. In our study, because six elastic coefficients have been measured for each sample, a large data set is available for the comparison. We found that modeling cortical bone as a two-phase composite with a transversely isotropic matrix pervaded by cylindrical pores provided a good estimate of the elasticity variations at the mesoscale, as shown by the It is noteworthy that the model was particularly efficient considering its ability to fit all experimental mesoscopic elastic coefficients with a relatively good accuracy using a unique elastic tensor for the matrix and the pores and a sample-dependent porosity. This was despite the many idealizations of the model, in particular the elastic properties of the matrix and the modeling of the pores. Universal, homogeneous, elastic properties were assigned to the bone matrix. The choice of a unique matrix was supported by the small change in the average elastic properties of the matrix, as testified by the matrix impedance data. We verified that the optimized set of TI elastic properties assigned to the bone matrix ) ( m c were physically acceptable. In fact, once converted into engineering moduli (ET = 16.5

GPa, EL = 24.0 GPa, GT = 5.8 GPa, GL = 7.3 GPa), the matrix elastic properties were found consistent with the nanoindentation values in human femoral bone available in literature [START_REF] Rho | Elastic properties of microstructural components of human bone tissue as measured by nanoindentation[END_REF][START_REF] Turner | The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurements techniques[END_REF][START_REF] Franzoso | Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation[END_REF]. Moreover, the matrix elastic coefficients ) ( m c used in our model compared well with those derived from the experimental acoustic impedance mean values using the conversion relationship between Z and m c [START_REF] Preininger | Spatial-temporal mapping of bone structural and elastic properties in a sheep model following osteotomy[END_REF]. Precisely, the elastic coefficients of the matrix as derived from the Zi (i=1,2,3) ( exp The vascular porosity was idealized as infinite cylinders of circular cross-section aligned along the bone long axis. Hence, the pores were modeled as continuous even though a discontinuous representation might seem more realistic. However, we have found that, for aspect ratios (length of the pore / diameter of the pore) larger than 5, modeling the pores as infinite cylinders yields a very good approximation (less than 1% error) of discontinuous pores with typical aspect ratio of the Haversian canal [START_REF] Granke | Elasticity and porosity in human cortical bone: models and experiments[END_REF]. Although this representation has been commonly used for modeling cortical bone [START_REF] Hellmich | Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions?[END_REF][START_REF] Dong | Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method[END_REF][START_REF] Deuerling | Specimen-specific multiscale model for the anisotropic elastic constants of human cortical bone[END_REF], it does not take into account the variability of pores shapes, size, and distribution. Considering the gradient of porosity from the endosteal to the periosteal region [START_REF] Saïed | Spatial distribution of anisotropic acoustic impedance assessed by time-resolved 50-MHz scanning acoustic microscopy and its relation to porosity in human cortical bone[END_REF][START_REF] Bousson | CT of the middiaphyseal femur: cortical bone mineral density and relation to porosity[END_REF] or the change in the pores size [START_REF] Cooper | Age-dependent change in the 3D structure of cortical porosity at the human femoral midshaft[END_REF][START_REF] Baron | Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1 MHz: a finite-difference time domain study[END_REF] may improve the predictions of the bone effective elastic properties.

The remaining part of experimentally determined elasticity C which is not explained by the model is due to experimental uncertainties and model assumptions. The latter comprise the assumptions regarding the pores as mentioned above and the fact that some variability of the matrix properties exists between different samples.

A first limitation of the study arises from the estimation of the sample porosity as the average value of the cross-sectional porosities assessed on the two opposite transverse faces. However, the validation of the porosity evaluation with 2D SAM on ten samples against the vascular porosity as obtained from 3D SRµCT data confirmed that Por is a good proxy for the vascular porosity. A second limitation in the study is the fact that all donors were elderly female donors (with a mean age superior to 80 years). Although the bone matrix elasticity has been shown to be independent from age and gender [START_REF] Hoffler | Age, gender, and bone lamellae elastic moduli[END_REF], aging strongly affects the range of porosity and could change the relative contributions of the matrix elasticity and the porosity to the mesoscopic elasticity in younger individuals. Thus the conclusions of this study hold true only for an aged population, which is most commonly affected by osteoporosis and bone fragility. Finally, in spite of a limited sample size (n = 21 from 10 subjects), the range of values covered by the porosity (from 3 to 27%) was wide enough to provide conclusive results.

In summary, the findings of this paper demonstrate that, in aged women, the changes in porosity prevail over those of matrix elasticity to drive the variations of the bone mesoscopic elasticity. The impact of the porosity on the elasticity is all the more important considering the increased intracortical porosity as a consequence of aging [START_REF] Mccalden | Age related changes in the tensile properties of cortical bone[END_REF][START_REF] Cooper | Age-dependent change in the 3D structure of cortical porosity at the human femoral midshaft[END_REF][START_REF] Bousson | Distribution of intracortical porosity in human midfemoral cortex by age and gender[END_REF][START_REF] Burghardt | Age-and gender-related differences in the geometric properties and biomechanical significance of intracortical porosity in the distal radius and tibia[END_REF][START_REF] Zebaze | Intracortical remodeling and porosity in the distal radius and postmortem femurs of women: a cross-sectional study[END_REF] and disease, e.g. hyperparathyroidism, osteoporosis [START_REF] Chavassieux | Insights into material and structural basis of bone fragility from diseases associated with fractures: how determinants of the biomechanical properties of bone are compromised by disease[END_REF]. In particular, Zebaze et al. [START_REF] Zebaze | Intracortical remodeling and porosity in the distal radius and postmortem femurs of women: a cross-sectional study[END_REF] showed that 84% of the bone loss occurs after the age of 65 of which 68% would be cortical bone manifested as an increase of cortical porosity. Moreover, the increase of cortical porosity, pointed out as the dominant factor occurring in elderly individuals, is known to reduce bone strength [START_REF] Yeni | The influence of bone morphology on fracture toughness of the human femur and tibia[END_REF]. A simple mechanical model was proposed to interpret the experimental data: the dependence on porosity of shear and longitudinal elastic properties in the radial, circumferential, and axial directions of bone is correctly described when idealizing bone as a two-phase material with a 'universal' (same for all bone samples) transversely isotropic matrix pervaded by cylindrical pores. results from experiments (■, o, ▲) and asymptotic homogenization model solid and dotted lines). Note that all the homogenized elastic coefficients computed from the AH model are obtained using a unique set of elastic constants for the bone matrix.

Figures captions

Tables captions

Table 1 Experimental data (mean ± std [range])

  The mesoscopic elastic coefficients were well correlated to the porosity (adj-R 2 = [0.72 -0.84], p<10-5 ). The use of a stepwise regression analysis showed no improvement of the correlation when adding the bone matrix impedance ( i Z ˆ) to the porosity to explain the mesoscopic elasticity variations between samples. The transversely isotropic elastic tensor of the matrix ( m c ) which allowed the best agreement (in the sense of equation 2) between measured and modeled mesoscopic elastic properties was found to be m c 11 = m c 22 = 26.8 GPa, m c 33 = 35.1 GPa, m c 44 = m c 55 = 7.3 GPa, elastic coefficients correlated well with the effective elastic coefficients as computed from the AH model (adj-R 2 = [0.78-0.82], p<10 -5 ) (Fig. 4). The precision of the model prediction was evaluated by means of the RMSE absolute and relative values: C11 =1.0 GPa (5.2 %), C22 = 1.2 GPa (6 %), C33 = 1.7 GPa (5.6 %), C44 = 0.3 GPa (5.5 %), C55 = 0.4 GPa (8.5 %), C66 = 0.3 GPa (7.6 %).

  strong correlations (relative RMSE = [5.2 -8.5 %]) between the experimental results and the prediction of the micromechanical model. Note, however, that the homogenized elasticity C * is not strictly independent of the mesoscale experimental data C because the matrix elasticity of the model ) ( m c was determined such that the agreement between C and C * is optimum (equation 2).

3 . 3 11 =

 3311 GPa) were in agreement with the elastic coefficients assigned in the model ( = 26.8 GPa, m c 33 = 35.1 GPa).

  Figure 1 (a) 3D reconstruction of a cortical bone volume from SR-µCT data. The samples

Figure 2

 2 Figure 2 Experimental results: (a) mesoscopic elastic coefficients (b) mean acoustic

Figure 3

 3 Figure 3Validation of the assessment of volumetric porosity from the segmented

Figure 4

 4 Figure 4 (a) Longitudinal and (b) shear mesoscopic elastic coefficients versus porosity:

Table 2

 2 Multivariate analysis regression (adjusted R 2 and RMSE): relative contributions of the vascular porosity (Por) and the matrix impedance ( i Z ˆ) to the mesoscopic elastic coefficients (Cii).

	Figure1

Click here to download high resolution image Figure2 Click here to download high resolution image Figure3 Click here to download high resolution image Figure4 Click here to download high resolution image

  .: not significant (p > 0.05); * p < 0.05; ** p < 10 -5

										C	22 C	33 C	44 C	55 C	66 C
		Z	1						n.s.	n.s.	n.s.	n.s.	n.s.	n.s.
		Z		2						n.s.	n.s.	n.s.	n.s.	n.s.	n.s.
		Z		3						0.21 * 1.96	n.s.	0.26 * 2.75	0.22 * 0.68	0.26 * 0.71	0.16 * 0.59
		Por					0.79 ** 1.01	0.76 ** 1.09	0.74 ** 1.64	0.84 ** 0.31	0.72 ** 0.44	0.78 ** 0.30
	, Por	1 Z	,	Z	2	,	Z	3	0.79 ** 1.01	0.76 ** 1.09	0.74 ** 1.64	0.84 ** 0.31	0.72 ** 0.44	0.78 ** 0.30
	n.s

Acknowledgments

This work has been conducted within the European Associated Laboratory "Ultrasound Based Assessment of Bone" (ULAB) and was supported by the ESRF Long Term Proposal MD431.


V Kravets

K Bas

T Kravets

Lyudmila Tokar

Analytical Modeling of Transient Process In Terms of One-Dimensional Problem of Dynamics With Kinematic Action

Keywords: material system, kinematics action, mathematical model, analytical solution, characteristic equation, dynamic design

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

Introduction. Analytical modeling is the essential stage of technical system dynamic design followed by computational and full-scale experiment [0, 0]. Analytical modeling of dynamic systems is based upon traditional mathematical methods of solutions of differential equation systems [0], theory of modal control [0], root-locus technique [0], and root-locus method [0].

Free motion dynamics of one-dimensional mechanical system experiences analytical study in a work by Kravets [0]; forced motion dynamics in terms of external dynamic effect was considered in a work by Kravets [0]. The paper models forced motion dynamics in terms of external kinematic effect.

Formulation of the problem. Fig. 1 demonstrates dynamic scheme of one-dimensional mechanical system.

Fig. 1. Dynamic scheme of kinematic effect problem

Here М and m are interacting masses, с is coefficient of elasticity, µ is damping coefficient.

It is assumed that M>>m. m mass is finite and specified. 𝑥(𝑡) motion of m mass cannot effect a(t) motion of М mass. Notion of М mass is supposed as specified function of 𝑎(𝑡) time. For example: 𝑎(𝑡) = V0t where V0 is М mass velocity.

It is required to develop 𝑥(𝑡) analytical solution modeling stable transient and determining steady motion of m mass depending upon such running design parameters of mechanical system as µ and c.

Mathematical model. Continuous dynamic model with single degree of freedom is described with the help of following matrix differential equation:

‖ 𝑥 1 ẋ2 ̇‖ = ‖ 𝑎 11 𝑎 21 𝑎 12 𝑎 22 ‖ ‖ 𝑥 1 𝑥 2 ‖ + ‖ 𝑓 1 (𝑡) 𝑓 2 (𝑡) ‖ (1)
where 𝑥 1 = 𝑥̇(𝑡) ; 𝑥 2 = 𝑥(𝑡).

For mechanical system under consideration, the equation coefficients are determined as follows:

𝑎 11 = - 𝜇 𝑚 , 𝑎 12 = - 𝑐 𝑚 , (2)
𝑎 21 = 1, 𝑎 22 = 0 .
Power function for kinematic effect is identified in the form of:

𝑓 1 (𝑡) = 𝜇 𝑚 𝑎̇(𝑡) + 𝑐 𝑚 𝑎(𝑡) (3)
f2(t) = 0 .
Analytical solution. Following normalized form for analytical solution x(t) determining motion of mass m is as follows:

𝑥(𝑡) = 𝑒 𝜆 1 𝑡 𝜆 1 -𝜆 2 | 𝜆 2 - 1 - 𝑥 0 ẋ0 | + 𝑒 𝜆 1 𝑡 𝜆 1 -𝜆 2 ∑ 1 𝜆 1 𝑛 ∞ 1 f (0) (n-1) - 1 𝜆 1 -𝜆 2 ∑ 1 𝜆 1 𝑛 ∞ 1 f (t) (n-1) (4)
+ 𝑒 𝜆 2 𝑡 𝜆 2 -𝜆 1 | 𝜆 1 - 1 - 𝑥 0 ẋ0 | + 𝑒 𝜆 2 𝑡 𝜆 2 -𝜆 1 ∑ 1 𝜆 2 𝑛 ∞ 1 f (0) (n-1) - 1 𝜆 1 -𝜆 2 ∑ 1 𝜆 2 𝑛 ∞ 1 f (t) (n-1) .
Here analytical solution is represented in the form of dependence on the roots of characteristic equation: 𝜆 1 , 𝜆 2 ; specified initial disturbances 𝑥̇0 𝑥 0 ; specified external power effect within the initial time period f(0) and current one f(t).

Analytical modeling.

If external kinematic effect is specified as: 𝑎(𝑡) = 𝑉 0 𝑡 then considering 𝑎̇(𝑡) = 𝑉 0 that both function and its derivatives are:

f(t) = 𝜇 𝑚 V0 + 𝑐 𝑚 V0t , f(0) = 𝜇 𝑚 V0 , ( 5
) 𝑓 ̇ (t) = 𝑐 𝑚 V0 , 𝑓 ̇ (0) = 𝑐 𝑚 V0 , 𝑓 ̈ (t) = 0 , 𝑎(𝑡).
Hence:

∑ 1 𝜆 1 𝑛 ∞ 1 𝑓 (0) (𝑛-1) = -𝑉 0 ; ∑ 1 𝜆 2 𝑛 ∞ 1 𝑓 (0) (𝑛-1) = -𝑉 0 ; (6)
∑ 1 𝜆 1 𝑛 ∞ 1 𝑓 (𝑡) (𝑛-1) = 𝑉 0 (𝜆 2 𝑡 -1); ∑ 1 𝜆 2 𝑛 ∞ 1
𝑓 (𝑡) (𝑛-1) = 𝑉 0 (𝜆 1 𝑡 -1).

Substituting the results into general formula and performing simple transformations we obtain laconic analytical representation of transient:

𝑥(𝑡) = 𝜆 1 𝑒 𝜆 2 𝑡 -𝜆 2 𝑒 𝜆 1 𝑡 𝜆 1 -𝜆 2 𝑥 0 + 𝑒 𝜆 2 𝑡 -𝑒 𝜆 1 𝑡 𝜆 1 -𝜆 2 (V0 -𝑥 0 ̇) + V0t . ( 7
)
In case of complex roots of characteristic equation: 𝜆1,2 = α ±iβ transient is modeled by means of following function:

𝑥(𝑡) = 𝑒 𝛼𝑡 [| 𝛼-𝑥 0 1-𝑥 0 | 𝑠𝑖𝑛 𝛽𝑡 𝛽 + 𝑥 0 𝑐𝑜𝑠 𝛽𝑡] -𝑒 𝛼𝑡 𝑉 0 𝑠𝑖𝑛 𝛽𝑡 𝛽 + 𝑉 0 t . ( 8
)
According to the given different forms of transient records, following particular cases of root distribution are being modeled:

1. 𝜆 1 , 𝜆 2 = 0; 2. 𝜆 1 = 𝜆 2 = 𝛼, 𝛽 = 0; 3. 𝜆 1,2 =±iβ , 𝛼 = 0.
In the context of particular case one, transient is modeled as:

𝑥(𝑡) = 𝑒 𝜆𝑡 𝜆 (𝑥 0 ̇-V0) + 1 𝜆 (V0 -𝑥 0 ̇ + λx0) + V0t. ( 9
)
In the context of particular case two assuming 𝜆1→𝜆2 or 𝜆 1 -𝜆 2 →0, i.е. ∆𝜆 →0 and considering that:

𝑙𝑖𝑚 ∆𝜆→0 1-𝑒 ∆𝜆𝑡 ∆𝜆 = -𝑡, (10)
We obtain following transient:

𝑥(𝑡) = eλt[(𝑥 0 ̇-𝜆𝑥 0 )𝑡 + 𝑥 0 -𝑉 0 𝑡] + 𝑉 0 𝑡 (11)
Assuming that β→0 , i.е. 𝜆1=𝜆2=α and considering that:

𝑙𝑖𝑚 𝛽→0 𝑠𝑖𝑛 𝛽𝑡 𝛽 = 𝑡, 𝑙𝑖𝑚 𝛽→0 𝑐𝑜𝑠 𝛽𝑡 = 1 (12)
We obtain transient in its equivalent record:

𝑥(𝑡) = 𝑒 𝛼𝑡 [| 𝛼-𝑥 0 1-𝑥 0 | 𝑡 + 𝑥 0 -𝑉 0 𝑡] + 𝑉 0 t. ( 13
)
In the context of case three transient is described with the help of following time function:

𝑥(𝑡)=[| 0-𝑥 0 1-𝑥 0 | -𝑉 0 ]
     а а а а (15)
Roots of characteristic equations depend on coefficients of differential equations as follows:

𝜆 1 + 𝜆 2 = 𝑎 11 + 𝑎 22 , 𝜆 1 • 𝜆 2 = | 𝑎 11 𝑎 12 𝑎 21 𝑎 22 | . (16)
For the involved mechanical system running parameters are directly determined by the formulas: 𝜇 = -𝑚(𝜆 1 + 𝜆 2 ), 𝑐 = 𝑚 • 𝜆 1 • 𝜆 2 .

(17

)
In terms of complex roots we obtain: 𝜇 = -2 𝑚𝛼, c= 𝑚(𝛼 2 + 𝛽 2 ),

where α is degree of stability; and β is factor of natural frequency.

In the context of particular case one we determine:

𝜇 = -2𝑚𝜆 1 , c=0, (19)
i.е. elastic element is not available in the mechanical system.

In the context of particular case two we determine:

𝜇 = - 2

 2 𝑚𝜆, 𝑐 = 𝑚𝜆 2 (20) or 𝜇 = -2 𝑚𝛼, 𝑐 = 𝑚𝛼 2 (21)

In the context of particular case three we determine:

i.е. damping component is not available in the mechanical system.

Summary. New record of analytical solutions of linear differential equations in harmonic form is proposed. The form is applied for analytical modeling and design of one-dimensional dynamic system in terms of kinematic effect. Qualitatively different forms of transients within onedimensional mechanical system as well adequate running design parameters of elastic and damping elements have been obtained.