Tiphaine Phe-Neau
email: tiphaine.phe-neau@lip6.fr

Marcelo Dias De Amorim

Vania Conan
email: vania.conan@fr.thalesgroup.com

Fine-Grained Intercontact Characterization in Disruption-Tolerant Networks

So far, efforts attempting to characterize the spatiotemporal nature of disruption-tolerant networks (DTN) have relied on the dual notion of contacts and intercontacts. A contact happens when two nodes are within communication range of each other. An intercontact is simply defined as the dual of a contact, i.e., when two nodes are not in communication range of each other. We refer to this model as "binary". Although the binary characterization allows understanding the main interaction properties of the network, it is not sufficient to capture a plethora of situations beyond the binary hypothesis. In this paper, we investigate the structural properties of the network when nodes are not in contact but do have a contemporaneous path connecting them. We first introduce the notion of nary intercontact and, to defend its adoption, we quantify the proportion of nodes bearing this new intercontact notion in well-known datasets available to the community. Surprisingly, we observe that most pairs of nodes are nearby (within a few hops) for significant amounts of time when not directly in contact. Finally, we compare the impact of our proposal with the classic intercontact definition and give incentives toward using the n-ary characterization to leverage new communication opportunities.

I. INTRODUCTION

With the growing penetration of intelligent devices like smartphones, tablets, or portable game stations into people's everyday life, disruption-tolerant networking (DTN) emerged as a groundbreaking communication paradigm in the modern networking landscape [START_REF] Fall | A delay-tolerant network architecture for challenged internets[END_REF]. To satisfy the communication needs among those devices in an opportunistic way, many new stimulating approaches have been proposed in the literature [START_REF] Hui | Bubble rap: social-based forwarding in delay tolerant networks[END_REF], [START_REF] Lindgren | Probabilistic routing in intermittently connected networks[END_REF], [START_REF] Spyropoulos | Spray and wait: an efficient routing scheme for intermittently connected mobile networks[END_REF], [START_REF] Whitbeck | Hymad: Hybrid dtn-manet routing for dense and highly dynamic wireless networks[END_REF]. A common substrate to these approaches is to rely on the complementary concepts of contacts and intercontacts.

The contact notion has a factual definition, it is when two nodes are within direct communication range of each other. An intercontact is defined as the complementary of a contact, i.e., simply when two nodes are not in contact. As we will see in this paper, such a simplistic definition for intercontacts (henceforth mentioned as binary intercontacts) ends up being a melting pot for any attempt to benefit from the geographic proximity of users.

Fig. 1 represents a network snapshot illustrating our concerns. From A's point of view, it has two nodes in contact (nodes B and C). With the binary intercontact definition, all four remaining nodes are considered in "intercontact" mode. In such a situation, most DTN approaches infer the impossibility of exchanging messages via multi-hop paths and often calls for a "wait" period until it meets the destination or find someone

A B C D E F G
nodes in contact with A unreachable nodes in "pathless intercontact" nodes in "binary intercontact" nodes in "favorable intercontact"

Fig. 1. Node A neighborhood "contact and intercontacts": A is in contact with his two direct neighbors, all four remaining nodes are in binary intercontact. However, we can reach the lower pair of nodes in intercontact via a contemporaneous path, they are in a favorable intercontact state.

else that knows the destination better (based on some other criterion). By denying the inherent ad hoc network part in DTN, we cannot pull the best of both worlds. In our example, we observe that ad hoc communication would succeed if A had data to transfer to either D or E. Contemporaneous paths may exist between nodes not in contact. We refer to this property as favorable intercontact. A natural question that arises is: should node A use multi-hop communications to send data to D and E or wait until it gets within direct communication range with them?

In this paper, we tackle the fundamental problem of knowing whether a pair of nodes shows sufficient sociability to spend a significant share of time in the vicinity of each other. To this end, we investigate several real-world datasets available to the research community and provide insights into fine details of intercontact periods. We formalize our proposal by defining the notion of n-ary intercontacts, where n stands for the distance separating two nodes.

Our work has its foundations on several previous contributions of the literature. As outlined many times, user mobility patterns are not random [START_REF] Musolesi | Mobility Models for Systems Evaluation. A Survey[END_REF]. People have a tendency to form communities and then display "favorable" intercontact properties. Gaito et al. based their study on workplaces as they also felt the force of social patterns in DTN [START_REF] Gaito | Opportunistic forwarding in workplaces[END_REF]. There are induced relationships between human-carried devices. Other works, such as the ones of Whitbeck et al. [START_REF] Whitbeck | Hymad: Hybrid dtn-manet routing for dense and highly dynamic wireless networks[END_REF] and Borrel et al. [START_REF] Borrel | Understanding the wireless and mobile network space: a routing-centered classification[END_REF] propose to classify opportunistic networking into several categories depending on the expected group formation when nodes move around. Our work is complementary to the aforementioned ones as it makes a step further by analyzing node proximity in a pairwise way and by identifying "hidden intercontact possibilities" that are often underestimated by traditional forwarding solutions.

We make several interesting observations in our study. Firstly, we confirm that the binary contact-intercontact idea is too rough to efficiently capture potential communication opportunities while nodes are not in contact. Secondly, for a significant amount of pairs, we observe that nodes spend as much time at 1 and 2-hop distances. Thirdly, for some datasets, about half of the traditional intercontact time is in fact a connected. We do believe that our results will motivate protocol designers to first check close neighborhood before adopting a wait strategy.

In summary, the contributions of this paper are:

• New ternary vision for accurate DTN understanding through n-ary intercontact characterization. • Empirical analysis to show that the binary contact assumption is not enough via an evaluation of binary vs. n-ary intercontact behaviors in existing datasets. • Extended comparison between binary and pathless intercontact distributions.

II. N-ARY INTERCONTACT: DEFINITION AND OBJECTIVES

In this section, we provide the necessary background as well as an intuitive explanation of our DTN characterization.

A. Pairwise visualization

Considering a high level abstraction for intercontact erases punctual events happening between given nodes (for instance, people eating at the same place, at the same time, every week day). Yet, we could use them for efficient forwarding. To maintain traces of these events, we choose to perform pairwise analysis. Bearing in mind the inaccuracy of the binary intercontact definition, we thought of n-ary intercontact as a reflector of human sociostructure as well as ad hoc communication opportunities. We divide it into two intuitive notions: favorable and pathless intercontacts. Definition 1. Favorable intercontact. An intercontact is considered as "favorable" with parameter n when there is a contemporaneous shortest path of length n ∈ [2; ∞[ separating the two nodes under consideration.

Note that the parameter n is a key ingredient in our study. As we will see later, this characterization provides richer information to help decide which forwarding strategy a protocol should use. Depending on the time spent at a certain distance, we can also derive other parameters such as the surrounding stability of a node. Favorable intercontact can also be seen as extended ad hoc communication opportunities. Definition 2. Pathless intercontact. In opposition to favorable situations, "pathless" intercontact indicates the lack of endto-end paths between a pair of nodes, i.e., n = ∞. This granularity reveals real situations in the DTN context and can influence a node's resolution to use asynchronous transmission scheme from delaying transmission to specific routing approaches.

B. A motivating example

In Fig. 2, we generated synthetic traces using the Random Trip mobility model and compared conventional intercontact with our n-ary vision for one pair of nodes. When using the binary definition for intercontact (Fig. 2(a)), the two nodes under consideration spend 10% of their time in contact (i.e., 1 hop) and 90% of their time in intercontact. In Fig. 2(b), we show the results using the n-ary definition. Observe that, even though while in intercontact, nodes stay at a 2-hop distance during 10% of the experiment duration, at a 3-hop distance for 16%, and at a 4-hop distance for 5% (and also some little percentage at other distances). Pathless intercontacts account for less than 50% of the time (compared to the 90% observed by the traditional definition). In practice, protocol designers could benefit from favorable intercontacts as a mean to use multi-hop transfers (likely reducing delay) instead of playing only with the "wait" parameter.

The contact, favorable intercontact, and pathless intercon- [START_REF] Goodman | The familiar stranger: anxiety, comfort, and play in public places[END_REF]. Identifying prominent intercontact patterns could point out nodes with additional properties to leverage. We could also use our improved intercontact patterns to study induced communities evolution, their motion and stability. Intercontact patterns reflect more than a distance to another node, they disclose inherent sociostructure information and above all, new relevant DTN characteristics.

tact

III. DATASETS

We observed in the previous section that the n-ary intercontact concept does provide further insights in the case of a synthetic trace. But what about real situations? To answer this question, we choose several well-known DTN datasets available to the research community. Infocom05 is a dataset obtained in a conference scenario [START_REF] Chaintreau | Impact of human mobility on opportunistic forwarding algorithms[END_REF]. It involved 41 iMotes that captured contacts between participants during around 5 days of the conference. We focus on a 12-hour period of the second day, as it presents a higher network activity. In the Infocom05 experiment, each iMote performs a scan every 120 seconds. Infocom06 is another conference-based dataset with 78 iMotes for 3 days [START_REF] Chaintreau | Impact of human mobility on opportunistic forwarding algorithms[END_REF]. We consider the second day to avoid the biases of the beginning and end of the experiment. Other parameters remain the same as in Infocom05. Rollernet involved 62 iMotes during a 3-hour Rollerblade tour in Paris [START_REF] Tournoux | The accordion phenomenon: Analysis, characterization, and impact on dtn routing[END_REF]. Leguay et al. set a shorter scanning granularity of 15 seconds. This dataset represent a highly dynamic situation with inherent strong social relationships. Fig. 3. In a timeline, we observe the mutual shortest path length between two given nodes in a network. At the beginning, they stay at a 4-hop distance for a few instants then have no more contemporaneous path between them. They are in pathless intercontact. At 108,000 seconds, they are back in contact and drift away to a distance 2 until there is no path. For the longer part of this timeline, nodes bear favorable intercontact properties.

KAIST and NewYork come from an experiment performed by NCSU researchers [START_REF] Rhee | On the levy-walk nature of human mobility[END_REF]. Unlike Infocom05 or Rollernet, they do not represent contact traces but GPS coordinates. We converted them via movement simulation and assigned every node with a 10-meter wireless range transmission (to emulate a Bluetooth situation, like in the datasets above). KAIST has 92 nodes for campus-based movements and NewYork has 39 nodes in a city wide measurement.

IV. RESULTS

In this section, we evaluate our n-ary intercontact definition on the real-life DTN traces presented in the previous section. We first analyze the proportion of nodes bearing favorable intercontacts then observe the temporal characteristics of the parameters.

A. Timelines

We obtained timelines indicating the evolution of shortest paths between pairs of nodes in the network. Fig. 3 shows an example (a segment for the Infocom05 dataset). From 114,000 to 116,000 seconds, we can see nodes getting closer. Their shortest path length shrinking from 4-hop distance to 2-hop and finally dwelling at a 1-hop distance (in contact). We also observe the opposite phenomenon between 109,000 and 111,000 seconds, nodes are drifting away from distance 2 to distance 4. For all pair of nodes, these timelines embed interesting behavioral patterns.

B. Binary or n-ary intercontact?

Based on our timelines, we represent the fraction of time two nodes stay in contact and the cumulated time they spend in favorable intercontact. We present these behaviors in Fig. 4. Each dot reflects the behavior of a pair of nodes. The closer the points are to the top left corner, the more they experience Rollernet KAIST Fig. 4. Pairwise behavior according to the n-ary characterization. Each dot represents a pair of nodes. On the x-axis, we have the fraction of time they dwell in contact. On the y-axis, the fraction of time they observe favorable intercontacts. Note that, for the same density of contact, we can obtain a wide difference in favorable intercontact percentage.

favorable intercontacts and fewer direct contacts. This indicates that these nodes are frequently nearby (i.e., with endto-end paths to one another) but hardly experience a direct contact. Points strictly on the x-axis show pairs of nodes that are either in contact or in pure pathless intercontact, following the traditional DTN definition. Note that, as a significant share of the points fall in between, it means that, in practice, a non-negligible share of the pairs do show frequent favorable intercontacts.

We observe that a large portion of nodes display a significant fraction of time with end-to-end transmission capacities endorsed by contact and favorable intercontact. Nodes bearing these transmission capacities for up to 10% of their timeline (around 75 minutes except for Rollernet which is 6 minutes) represent: 70% of Infocom05 nodes, 29.7% of Infocom06, 5.8% of KAIST and 60.1% of Rollernet. In terms of duration, for their respective experiment span, more than 76% of Infocom05 nodes, 51% of Infocom06, 19% of KAIST and 83% of Rollernet nodes have more than 20 minutes of end-to-end transmission possibilities.

The importance of favorable intercontacts in these datasets quantifies how inaccurate the binary intercontact characterization is. Its use ends up in massive transmission opportunities losses. In the next section, we notice how our ternary vision gathers oblivious sociostructure information.

C. Sociostructure and favorable intercontact opportunities

Pairs of nodes can be in contact or linked via one or more relay nodes on the shortest path between them. From Fig. 5 to Fig. 8, we present what we call an aggregated network sociostructure. There, we plotted (in layered mode) the number of connected pairs for each shortest distance. This means that layer 2 shows the amount of pairs connected via a 2-hop path, layer 3 via 3-hop paths, and so on. The bottom layer symbolizes the amount of pair of nodes in contact (i.e., 1 hop). For the sake of clarity, we also show representative values in corresponding tables underneath each plot. The first line indicates the number of pairs in average for each distance. We also display the maximum value for each distance. It is important to recall that such transmission opportunities are powerful as they only involve few relays that could reduce significantly end-to-end delays. Note for example the significant share observed for 2-hop paths.

In Fig. 5, we observe several peaks of connected pairs with their extended contact neighborhood. Given that Infocom05 is a conference-based measurement, we can correlate them with morning arrivals, lunch, afternoon break, and end of sessions. An unexpected visual observation is how 2-hop Fig. 7. Rollernet being a specific sport event with a tight population, most transmission opportunities come from direct contacts (1 hop). However, during the revolving phase of the accordion phenomenon (density peaks), we observe a growth in the number of 2+-hop paths.

favorable intercontacts overcome contact opportunities during these peaks of high density. These environments of highly connected crowds ignite favorable intercontacts. As a result, in such a scenario, favorable intercontact-based transmissions should be more helpful than direct contact transmissions or real DTN schemes. For Infocom06, time divisions are less clear. However, we maintain our observation concerning 2-hop distance connected pairs (see Fig. 6). As a rule, 2+-hop favorable intercontact overthrows contact possibilities. Infocom06 is a quite dense dataset with a large amount of nodes at regular pedestrian speeds in a smaller surface than for other datasets we considered. This suggests that such environments, similar to many urban settings, are prone to have impressive extended end-toend communication possibilities.

In Fig. 7, we observe Rollernet's inherent accordion phenomenon, i.e., the sequential stretching and shrinking of the crowd due to urban obstacles preventing the crowd from moving forward [START_REF] Tournoux | The accordion phenomenon: Analysis, characterization, and impact on dtn routing[END_REF]. Rollernet has a dynamic setting with a compulsory path. Nodes do not have as much movement liberty as they have in other datasets. Contacts are prominent in Rollernet, but still favorable intercontacts are noticeable in peaks of density.

Considering KAIST, contact opportunities may seem more frequent than favorable intercontacts. But, unlike previous datasets where connectivity quickly decreases with distance on average, we discover comparable connectivity between nodes for distance 2 to 5. KAIST environment being a campus-based measurement, it has a sparser density with a lot of movement freedom. Students tend to stay where other students are, like in restaurants, buildings, libraries but they may not share stronger relationships than just being student in the same university so they do not form close groups. This is a possible explanation of how extended favorable intercontact state lingers here. In any case, peaks indicate denser areas with more communications possibilities. We visually notice how contact opportunities represent only a minor part of end-to-end transmission opportunities.

D. Pathless intercontact time characterization

Until now, we have focused on the detection of favorable intercontacts, which are interesting for their native communication opportunities. However, understanding pathless intercontacts also bring interesting network indications. We will next focus on this aspect.

Chaintreau et al. performed intercontact time characterization for various existing datasets [START_REF] Chaintreau | The diameter of opportunistic mobile networks[END_REF]. They used the traditional binary contact vision where nodes not in contact are in intercontact. We perform the same analysis but with pathless intercontact only.

In Fig. 9, we plot the CCDF of intercontact times, i.e., the probability for an intercontact to last for at least t seconds. Dotted lines represent the distribution for binary intercontact vision whereas solid ones correspond to pathless intercontact. The overall aspect of the distributions remains the same with a general deviation to the left for the pathless intercontact CCDFs. This phenomenon is logical as we extended the contact notion, i.e., it reduces pathless intercontact durations when compared to initial binary intercontact. The contact extension only occurs in periods where the network is dense. As a result, pathless intercontact distribution is not an exact translation of binary intercontact one but keeps an overall similarity. This also means that our definition does not change the intercontact properties our community used to have when studying DTN patterns. So, we can still leverage on the existing forwarding strategies background like techniques based on power law opportunities with adjusted parameters [START_REF] Chaintreau | The diameter of opportunistic mobile networks[END_REF]. In other words, our definition does not discard all the existing literature on DTN schemes; instead, it enhances it by providing new beneficial network information via extended contacts.

V. SUMMARY AND OUTLOOK

In this paper, we questioned the completeness of the binary intercontact vision where nodes are either in contact or in mere intercontact. We advocate for a new ternary vision in DTN characterization via a fine-grained intercontact definition. Our contribution, the n-ary intercontact, proposes to enhance the intercontact notion. We expose two intuitive concepts: favorable and pathless intercontacts. On a node-centered vision, favorable intercontacts bring new communication opportunities with the awareness of end-to-end paths. Whereas, pathless intercontacts include the previous binary intercontact abstraction and preserves any existing approaches relying on their distribution.

Our n-ary intercontact characterization is very compliant to DTN applications. It grasps what binary intercontact vision misses: the underlying MANET nature in DTN. It explicits new communication opportunities. Here, we intentionally introduced a simple definition for our community to integrate it in their every day approaches. We have shown for example that in Rollernet, Infocom05 and Infocom06 more than 50% of the pairs of nodes display favorable intercontact opportunities.

We hope this work will bring incentives for our community to develop new communication solutions for DTNs. We think the binary intercontact hypothesis prevents disruption-tolerant approaches from achieving their optimal efficiency when more information about the network can only help. In a near future, we plan to propose hybrid protocols considering MANET and DTN transmission opportunities alongside to leverage communication opportunities revealed by the n-ary intercontact characterization.

  Binary intercontact: nodes are in contact for 10% of the time and in traditional intercontact for 90% of the time. N -ary vision. Like in Fig.2(a), nodes spend 10% the time in contact but, in reality, they dwell at a distance 2 for around 10%, at a distance 3 for 16%. Real intercontact (i.e., pathless intercontact) deprived of multi-hop path represent only 50% of the timeline (∞).

Fig. 2 .

 2 Fig. 2. Example of time-distance distribution for a random trip-based dataset.

Fig. 9 .

 9 Fig. 9. Aggregated distribution of binary and pathless intercontact times. Pathless intercontact keeps the same overall distribution as binary intercontact with a small deviation on the bottom left. In the NewYork dataset, both distributions line up as we have almost no favorable intercontact. (log-scale on both axes.)

  triplet is the new ternary classification we suggest toward more detailed DTN characterization.

	C. Potential usages
	MANET, DTN, or hybrid solutions. With the impressive
	literature on ad hoc networks, rejecting the MANET-DTN
	correlation would be a waste. With a more accurate inter-
	contact understanding, a node may decide to use MANET
	communications instead of a pure DTN algorithm or even
	discriminate between wireless interfaces like Wi-Fi, Bluetooth,
	3G or LTE. Any protocol would be able to choose the relevant
	strategy depending on its current status. Ideally, we can think
	of new hybrid approaches using ad hoc and disruption-tolerant
	approaches together.
	Interesting forwarding situations. For routing concerns,
	sensing a node's vicinity up to a 2 or 3-hop distance sounds
	appealing. These situations embody end-to-end transmission
	opportunities via few close relays and therefore, low costs
	concerning delivery delays. We also observe higher distance

paths. However we can question path stability and delays inferred by the store and wait process. Path of higher distances are less frequent than distance 2 or 3 paths. Moreover, a node can easily obtain his neighborhood up to a distance 2 or 3 running a simple link state protocol. Oblivious communities detection. An important class of DTN movements are human-driven. Their topology relates to individuals with phenomenon like Milgram's Familiar Stranger

  KAIST is a campus-wide experiment. Students are likely to spread around alleys and buildings leading to dense areas at some points in time. Therefore, we have similar favorable intercontact proportion from distance 2 to 5. Opportunistic strategies clearly have room to spread in such a setting.

				KAIST
					6+
					5
					4
					3
					2
					contact
				3 seconds)
	KAIST	contact	2		4	5	6+
	average	25.57	19.75 17.72 15.4 11.05 7.37
	maximum	74	62	81	84	62	41
	Fig. 8.

ACKNOWLEDGMENT

This work is partially supported by the ANR project CROWD under contract ANR-08-VERS-006. The authors would like to thank John Whitbeck for its help preparing the manuscript.


J Molimard
email: molimard@emse.fr

Implementation of the direct evaluation of strains in a frequency-based image analysis code for random patterns

Keywords: strain measurement, digital image correlation, composite fabrics

A new approach for decoding displacements from surfaces encoded with random patterns has been developed and validated. The procedure is based on phase analysis of little zones of interest.

 on the grid method to random patterns for the direct measurement of strains. First metrological results are encouraging: resolution is proportional to strain level, being 9% of the nominal value, for a spatial resolution of 9 pixels (ZOI 64×64 pixels²). Random noise have to be carefully controlled. A numerical example shows the relevance of the approach. Then, first application on a carbon fiber reinforced composite is developed. Fabric intertwining is studied using a tensile test. Over-strain are clearly visible, and results connect well with previous studies.

INTRODUCTION

Digital image correlation is one of the most diffused image processing technique among experimental mechanics community [START_REF] Sutton | Determination of displacements using an improved digital image correlation method[END_REF]. The system has been extended to different cases and firstly on warp surfaces (stereocorrelation) [START_REF] Luo | Accurate measurement of three-dimensional deformations in deformable and rigid bodies using computer vision[END_REF][START_REF] Garcia | A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming[END_REF]. But basically, one of the most interesting problems in DIC is the sub-pixel detection. Usually, authors use either correlation peak interpolation either subpixel deformation of the target sub-image. The deformation hypothesis can be a rigid body motion or a complete transformation function of the zone of interest (ZOI) [START_REF] Wattrisse | Analysis of strain localization during tensile tests by digital image correlation[END_REF]. Recently, Hild [START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF] proposed an original work, based on a correlation algorithm coupled with Finite Element based transformation functions, allowing for strongly regularized displacement fields with solid mechanics assumptions.

In all these cases, the assumptions on the interpolation level are believed to have a direct influence on metrological properties of the correlation core [START_REF] Bornert | Assessment of Digital Image Correlation measurement errors: Methodology and results[END_REF].

Beside the classical DIC approach, the grid method, even if less used, has rather comparable features: it is basically an image processing technique allowing for in-plane displacement fields. In the previous case, images are random patterns; in the latter, surface signature is a periodic grid. In this case, displacements derive from a spatial phase extraction [START_REF] Surrel | Moiré and grid methods in optics[END_REF]. The choice of surface encoding should orient the user toward one method or the other, but in fact, the nature of the pattern is not crucial: DIC has been successfully applied to periodic pattern for a very long time [START_REF] Allais | Experimental characterization of the local strain field in a heterogeneous elastoplastic material[END_REF], and a random pattern can be seen has the superimposition of many frequencies. This last remark have been made recently by some authors [START_REF] Poilâne | Measurement of in-plane displacement fields by a spectral phase algorithm applied to numerical speckle photography for microtensile test[END_REF][START_REF] Vanlanduit | A digital image correlation method for fatigue test experiments[END_REF][START_REF] Molimard | Frequency-based image analysis of random patterns: a first attempt[END_REF] and will be developed hereafter. Random pattern are of more practical use than periodic pattern for two basic reasons: first, periodic pattern generation and transfer is not as easy as someone could think. Second, it is almost impossible to generate a periodic pattern on a non-flat surface -and to develop a 3D surface grid method, even if industrial demand for measurement on real structures is high.

Finally, one should note that DIC is a genuine large-strain approach because it is based on recorrelation of decorrelated informations whereas Grid technique is a genuine small perturbation method because the phase difference is merging initial and final state of the investigated object, and because phase information is fairly more sensitive than amplitude. Consequently, it could be of great interest to adapt advances made in the context of grid techniques to random patterns, and to compare the results to those obtained by classical image correlation technique. We noted here the following specific "phase culture" items: camera distortion can be evaluated with a single grid image [START_REF] Coudert | Étalonnage transverse d'un objectif de caméra par la méthode de la grille, colloque Photomécanique[END_REF], shape and shape variation can be detected in very good conditions using a video-projector and a camera [START_REF] Molimard | Frequency-based image analysis of random patterns: an alternative way to classical stereocorrelation[END_REF]. Last, derivation can be derived analytically and the grid technique becomes sensitive to strain rather than displacements [START_REF] Badulescu | A Procedure for Accurate One-Dimensional Strain Measurement Using the Grid Method[END_REF]. This item should bring to such method a breakthrough compared to DIC techniques; it will be developed in the following.

2D DISPLACEMENT METHOD USING FREQUENCY-BASED ANALYSIS

Digital Image Correlation Principle

Assuming a reference image im0 described by f(x,y), a deformed image im1 of im0 after a small strain is described by g(x,y) by the following equation:

g  x , y= f  x- x , y- y b  x , y (1)
where δx and δy are the components of the displacement of im1 and b(x,y) the noise measurement.

A way to find δx and δy is to maximize the function h defined by:

h  x , y= g * f  x , y= ∫ -∞ ∞ ∫ -∞ ∞ g  ,  f -r ,  -s d  d  (2)
where * denotes the cross-correlation product. The obtained r and s correspond to the maximal probability of displacement (δx, δy). This method can be applied in Fourier space using Fast Fourier Transform function, noted FFT2D. Equation 2 becomes:

g * f = FFT2D -1  FFT2D g  FFT2D  f  (3)
where the overline denotes the complex conjugate, and  f  x , y = f  L x -x , L y -y  where L x and L y are the dimension of the domain. Equation 2 and 3 are used at a local scale, on typical 32×32 zone of interest (ZOI). The work is repeated all over the map, giving a final displacement chart. Now, classical digital image correlation performs the cross-product either in Fourier or real space.

More refined approachs exist by the way: for example, the signal could be normalized respect to the mean local intensity, and/or the mean local contrast [START_REF] Bornert | Assessment of Digital Image Correlation measurement errors: Methodology and results[END_REF]. The cross-correlation peak is commonly interpolated in order to reach a sub-pixel displacement accuracy. The interpolation function is not the same for all the implementations. It could be for example a Gaussian or polynomial function.

Basically, this choice doesn't have a strong theoretical basis, and authors have an empirical approach. We will propose hereafter an alternative to this weak point in the image correlation approach.

Sub-Pixel algorithm

The sub-pixel algorithm is based on the phase estimation of each ZOI. Because Fourier Transform requires continuity, in particular at the boundaries, the ZOI is windowed using a bi-triangular function. So far, the algorithm is an extension of the windowed Fourier Transform (WFT) algorithm proposed by Surrel [START_REF] Surrel | Additive noise effect in digital phase detection[END_REF]. As shown Fig. 1, in the frequency domain, each couple of frequencies is characterized by an amplitude and a phase. This phase is proportional to the displacement normal to the corresponding fringe direction. Note that the phase is defined only if the amplitude is higher than zero. Then, in absence of any phase jump, displacements can be related to any defined phase using the relationship:

{ ⋮    i ⋮ } = [ ⋮ ⋮ 2  p  i cos i 2  p  i sin  i ⋮ ⋮ ]  A {  x  y } (4)
Displacements can then be derived from eq. 4 using the pseudo-inverse of A. This operation is possible if det  A t A≠0 . In practice, this means that at least two phases along two different directions exist.

{  x  y } = A t A -1 A t { ⋮    i ⋮ } (5)
Practical problem of this approach is that the signal to noise ratio is weak for each couple of frequencies. Then, the quality of the measurement is obtained by averaging all the available information throw the pseudo-inverse function.

One should note also that a phase jump can occur in the Fourier domain. Even if a specific treatment should be developed, it sounds better to first use a pixel correlation algorithm. This ensures that the two ZOIs will be as superimposed as possible and that the fringe order is zero for any point and any couple of frequencies. No deformation of the ZOI is proposed here, considering that target applications will be in the small transformation domain.

Characterization

The method has been characterized using a simulation of a rigid body translation. A single ZOI is generated and translated from -0.5 to 0.5 pixel, in presence of noise or not. The ZOI has been oversampled 10 times. 10 translation cases equally spaced are studied, and for each displacements 60 pairs of ZOI are sampled. Typical results are shown in fig. 2 for a 32×32 region of interest, 12bit camera, with a 31 gray level white noise or not. One should note first that the error has the same amplitude as with image correlation techniques (Table 1), but the bias is fairly lower in this case.

This should be one major advantage of this method.

Spatial resolution of the method is estimated using the following procedure: two independent noise distributions are added to the same speckle image. Then, displacements between the two situations are calculated, giving a displacement error map. The autocorrelation function of this error map is only affected by the displacement extraction procedure, so its size characterize the spatial resolution of the displacement extraction procedure. For the 32×32 window, effective spatial resolution is a diameter of 9 pixels (at 50% attenuation). This result should be surprising, but it can be partially explained by the use of the bi-triangular window: in 1D (the classical WFT algorithm) spatial resolution is half the window size. Then, it is worth noting that these values are very interesting compared with others zero-order image correlation algorithms.

DIRECT DERIVATION IMPLEMENTATION

Basic principle

Recently, Badulescu and al. [START_REF] Badulescu | A Procedure for Accurate One-Dimensional Strain Measurement Using the Grid Method[END_REF] proposed a new derivation procedure for grid techniques. The technique is twofold: first, it consists in deriving the wavelet used for the phase extraction. Authors claim that this approach leads to better results than deriving the displacement map using a classical low-pass filter coupled with a least-square derivation (called hereafter the classical procedure). In the phase extraction procedure, the phase is defined as:

   x , y=arg  R  iJ  =arg  ∫ -∞ ∞ ∫ -∞ ∞ s  , ×g -x , -y×exp  -i 2  P  cossin   d  d   (6)
where s(ζ, η) is the signal and g(ζ-x, η-y) is a windowing Gaussian function. Note that this function is different than the one used previously for displacement measurement. If considering the derivation

of U θ along  x , ∂   ∂ x =  ∂ J  ∂ x × R  - ∂ R  ∂ x × J  J  2  R  2  (7) ∂ U  ∂ x =- 2  P    ∂  ∂ x  (8)
Last, the derivation of J θ or R θ along x θ consists in deriving g(ζ, η) i.e.:

∂ J  ∂ x = ∫ -∞ ∞ ∫ -∞ ∞  x- 2   4 × s  , ×g-x , -y×exp  -i 2  P  cossin   d  d  (9)
Second, reference and deformed phase gradient maps are numerically superimposed to avoid noise propagation due to local variations in the grid signature. In each situation, authors show that the results are better if noise intensity is reduced using temporal averaging. Depending on the conditions, the resolution can be one order of magnitude lower than the classical procedure for a given spatial resolution.

Now, the grid technique can be seen as a particular case of the frequency-based image analysis of random patterns. So far, it is possible to generalize this new strain measurement procedure. The proposed implementation is based on:

1. the superimposition of the original and deformed images, using a linear interpolation of gray levels. Note that it is impossible here to deform directly the phase gradient maps.

Deformation of the intensity map is a major difference between the two approaches. In a first step, intensity maps are submitted to a simple translation. Then, a first evaluation of the strain is used to deform the intensity map, and procedure stop when convergence is achieved.

2. the expression of phase derivative according to (7), 3. instead of using direct expressions, a calibration procedure for 4 reference numerical deformations of the original ZOI is made (two translations, a rotation, and in-plane shear), giving an in-situ sensitivity. It is believed that this procedure is less sensitive to ZOI signature.

Characterization

The method has been characterized using a simulated deformation of a single ZOI. The ZOI is submitted to uniaxial strain at different levels ranging from 2.10 -5 to 2.10 -3 . In order to characterize average behaviour, , 60 ZOI are generated for each strain level. The rigid body displacement is set to zero in order to avoid possible coupling effects. For each case, all the displacement gradients are recorded and mechanical strain tensor is reconstructed (ε xx , ε yy , ε xy , ω). The different cases are:

-ZOI varies from 32×32 and 64×64 px 2 ;

noise on intensity from 0 to 31 gray levels;

speckle radius from 2 to 16,

encoding deepness from 8 to 16 bits.

Because of the high number of parameters, we use here a linear experimental test plane. Results are presented Table 2. Note that every parameter has a good enough confidence interval, and is possible to draw the main behavior. It was necessary to add 1 st order interaction to have a good enough model. Finally, uncertainty on strain increases with the speckle size and with intensity noise, and decreases when ROI size and encoding increase. Note that speckle size and noise are strongly correlated, with a positive effect. Finally, results can be strongly improved by minimizing gray level noise using image averaging. Last, it is proved here that the use of small speckle is necessary to achieve good accuracy. Best results (64×64 px 2 ZOI, 2 px speckle radius, 0 GL noise, 16 bits encoding) gave a mean final uncertainty of 58 µm/m. In this case, the uncertainty is entirely random, with no bias. Dispersion is related to quantization error and to random process generation of the speckle pattern. Results are presented figure 4. The uncertainty increases with the load case, and finally, the relative uncertainty remains roughly constant (9 %).

Last, spatial resolution must be measured as explained before. Here, the value obtained for the 64×64 zone of interest is 9 pixels. This very low value can be explained by the use of a Gaussian windowing with σ = 10.7 pixels.

EXAMPLES

Simulated experiment

First test on the sub-pixel algorithm has been conducted on a fake displacement map provided by GDR 2519 [START_REF] Bornert | Assessment of Digital Image Correlation measurement errors: Methodology and results[END_REF]. It consists in a sine-wave displacement field, as shown figure 4 encoded on a 8-bits intensity map, without noise. Here, the displacement amplitude has been set to 0.02 pixels, and spatial wavelength to 128 pixels. Root mean square error found both in x and y directions is 0.003 pixels. The strain map (fig. 5) are focused on a limited area. Its obvious anyway that the values show a very good trend. If comparing classical derivation with the same spatial resolution, this new technique gives better results in absence of noise: 1.7 10 -4 for the proposed technique vs. 2.6 10 -4 for the classical one.

Strain field a single-ply carbon-fiber reinforced composites

Carbon fiber reinforced composites are commonly used for structural parts. The reinforcement can be delivered as UD tapes; mechanical properties are at the maximum, but handling is difficult. On the opposite, fabrics can be handled easier, but the intertwining of tows decreases their loading capacity. This effect is due to the development of local stresses. Then, it is important to characterize a weaving with a security coefficient corresponding to the local amplification of global stress.

Because the geometrical description of a fabric is difficult, experimental study through an optical full-field technique is a good solution. This work has already been done by Lee [START_REF] Lee | Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension[END_REF] using interferometry technique outlining strain concentrations. Here, the idea is to propose a simpler optical set-up using a single camera. Direct derivation is a good way to catch very localized phenomena.

The result of a feasibility test is given figure 6 and7. The specimen is a composite made of a single fabric carbon ply. The fabric unit cell is 8×8 mm 2 . Applied load is 200 N. Extraction of data is made here with a 64×64 ZOI. Displacement maps show variations in conjunction with the tow interlacing.

Strain map obtained on figure 7 had to be filtered for cosmetic reasons using a first a salt-and-paper filter, then a Gaussian low-pass filter with σ = 9 pixels. Results are in good agreement with former works on the same structure [START_REF] Lee | Digital phase-shifting grating shearography for experimental analysis of fabric composites under tension[END_REF][START_REF] Lomov | Full-field strain measurements for validation of meso-FE analysis of textile composites[END_REF]. It is shown that interlacing of tows implies periodic over-strain with band (banana) shapes. The shear strain is maximum in the resin rich region at the crossing of longitudinal and transverse tows. Here, the final spatial resolution is 13 pixels. This value is very low compared to other geometric methods, such as digital image correlation, or simply to a derivation post-processing.

CONCLUSION AND PERSPECTIVES

Strain evaluation is a very critical task for mechanical engineers. Usual geometrical methods (image correlation, grid techniques) classically need a post-processing based on a low-pass filter combined with least square fit. Recently, we proposed a frequency-based image analysis code for random patterns. This code has been extended to direct strain evaluation using a phase derivation kernel.

First results show very promising performances if the noise level is controlled. Main advantage of the method should be its final spatial resolution. Two examples illustrate the first applications of the technique. The first one is a simulated experiment; the second one is on a carbon fiber reinforced composite. The fabric interlacing produces local strain peaks, visible with the described procedure.

Table 1. Error quantification for a 32×32 window.

Figure 1 .

 1 Figure 1. Basic principle of sub-pixel algorithm.

Figure 2 .

 2 Figure 2. Quality of identified displacements for a pure translation without and with noise.

Figure 3 .

 3 Figure3. Relative error on strain for the optimal test case (Spatial resolution 9 pixels).

Figure 4 .

 4 Figure 4. In-plane displacement maps for a simulated in-plane displacement (in pixels).

Figure 5 .

 5 Figure 5. In-plane strain maps for a simulated case. a/proposed procedure b/classical approach (in m/m).

Figure 6 .

 6 Figure 6. Displacement maps obtained on a 1-fabric ply composite coupon under tension (in pixels).

Figure 7 .Figure 1 .-Figure 2 .

 712 Figure 7. Strain maps obtained on a 1-fabric ply composite coupon under tension a/εxx, b/εxy. Full scale is 10-3 (in m/m).

Figure 3 .

 3 Figure3. Relative error on strain for the optimal test case (Spatial resolution 9 pixels).

Figure 4 .Figure 5 .

 45 Figure 4. In-plane displacement maps for a simulated in-plane displacement (in pixels).

Table 2 .

 2 Error sensitivity to parameters

Table 1 .

 1 Error quantification for a 32×32 window.

		0 GL noise	31 GL noise
	Resolution (in pixels)	0.025	0.035
	Bias (in pixels)	0.0044	0.0056

Table 2 .

 2 Error sensitivity to parameters

	Value	Confidence level

ACKNOWLEDGEMENTS

Author thanks Saint-Etienne Metrope for granting this work

Full scale is 10 -3 (in m/m).


Jean Geringer
email: geringer@emse.fr

Frédéric Bertrand Boyer

F Farizon

Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis-Highlighting the role of arthrofibrosis

Keywords: Hip replacement prosthesis, Dual mobility concept, Ultra High Molecular Weight Polyethylene, Wear mechanism, Surface roughness

Understanding the dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis-highlighting the role of arthrofibrosis

Introduction

Sir John Charnley introduced, for the first time, the use of polyethylene liner against a metallic femoral head, SS 316L [START_REF] Hernigou | The history of total hip arthroplasty, Prothèse totale de hanche: Les choix J. Puget[END_REF]. This couple of materials became the Gold standard, i.e. the reference for the hip arthroplasty. This smooth (liner)-hard (head) couple was an alternative to others, as metal on metal, Mc Kee and Watson-Farrar [START_REF] Mckee | Replacement of Arthritic Hips Prosthesis[END_REF] or ceramic on ceramic, Boutin [START_REF] Boutin | Arthroplastie totale de la hanche par prothèse en alumine frittée[END_REF].This couple of materials became the Gold standard. The head diameter chosen was 22.2 mm, more than half the normal femoral head diameter, to decrease wear rates. In 1975, to overcome the early impingement issue, causing a mean dislocation rate of 5 %, the dual mobility socket was introduced by Pr. Gilles Bousquet [START_REF] Bousquet | The ceramic coated cementless total hip arthroplasty. Basic concepts and surgical technique[END_REF][START_REF] Rambert | chirurgien et technicien[END_REF][START_REF] Lecuire | Intra-prosthetic dislocation of the Bousquet dual mobility socket[END_REF]. This concept was based on two articulating surfaces instead of one -between femoral head and UHMWPE liner, as well as between UHMWPE liner and metal back, Figure 1. The second mobility exists because of a retentive chamfer maintaining head and liner together, thus allowing increasing head diameter while following Charnley 22.2 mm low friction philosophy. *Corresponding author. Tel.: +33 477 426 688; fax: +33 477 420 157. E-mail address: geringer@emse.fr; jgeringer@voila.fr (Jean Geringer)

Ref N°: 0313, Geringer, Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis 2

The usual liner-head assembly is constituted on ly by the first mobility. This new implant enhanced both stability and survivorship because of decreasing dislocations [START_REF] Langlais | Dual mobility cemented cups have low dislocation rates in THA revisions[END_REF][START_REF] Leclercq | Results of the Evora dual mobility socket: five years followup[END_REF][START_REF] Lautridou | Survival of the cementless Bousquet dual mobility cup: Minimum 15-year follow-up of 437 total hip arthroplasties[END_REF][START_REF] Guyen | Use of a dual mobility socket to manage total hip arthroplasty instability[END_REF]. The follow-up studies of dual mobility concept exhibited similar behaviors to classical THA in prosthetic joint lifetime [START_REF] Farizon | Results with a cementless alumina-coated cup with dual mobility. A twelve-year follow-up study[END_REF][START_REF] Adam | Dual articulation retentive acetabular liners and wear: surface analysis of 40 retrieved polyethylene implants[END_REF][START_REF] Philippot | Survival of cementless dual mobility sockets: ten-year follow-up[END_REF][START_REF] Philippot | Survival of cementless dual mobility socket with a mean 17 years follow-up[END_REF][START_REF] Philippot | The use of a dual-articulation acetabular cup system to prevent dislocation after primary total hip arthroplasty: analysis of 384 cases at a mean follow-up of 15 years[END_REF]. To prevent dislocation due to impingement, the dual mobility concept seems to be the most reliable treatment [START_REF] Philippot | Prevention of dislocation in total hip revision surgery using a dual mobility design[END_REF]. The second mobility is believed to be triggered when the femoral neck hits the liner. It lowers liner rim degradations and generation of wear debris. Dual mobility concept is concerned in 30% of THA in France, and is more and more investigated for the implants in US. Considering an extra friction surface, one might expect dual mobility to produce more wear debris than the classical artificial joint; the osteolysis mechanisms might be reinforced. The main keypoints related to dual mobility are the understanding of the in-vivo biomechanical behavior, the wear rate of UHMWPE involved in the second mobility and the material transfers due to wear and corrosion between the cup and the metal back. In this study, the explant convex side was investigated for understanding the evolution of surface morphologies. The main aim was to analyze friction zones, wear orientation and total wear rates (from concave and convex sides). Finally, from these experiment results, including a specific treatment of data such as selecting the relevant roughness parameters, a tribological scenario might be determined for highlighting the main wear steps and understanding the in vivo course of the second mobility.

2-Materials and methods

Explanted prostheses, implant selection

Since 1990, all explanted prostheses are stocked for scientific and legal purposes. The cleaning protocol consists in a curing with an antiseptic product, without sterilization process to avoid changes of the mechanical properties due to water absorption. Among the thousands of explants, over 250 dual mobility explants were available. A drastic selection of explants was practiced, based on clinical criteria, to get only the prosthetic elements without other complication than pure liner wear.

Different factors have to be considered for avoiding the mismatching of the selected explants. The loosening and osteolysis because cup wear could be increased compared to an actual behavior without extra wear debris [START_REF] Dumbleton | A literature review of the association between wear rate and osteolysis in total hip arthroplasty[END_REF][START_REF] Han | Effect of polyethylene wear on osteolysis in cementless primary total hip arthroplasty: minimal 5-year follow-up study[END_REF][START_REF] Shirong | Wear behavior and wear debris distribution of UHMWPE against Si 3 N 4 ball in bidirectional sliding[END_REF][START_REF] Goodman | Wear particles, periprosthetic osteolysis and the immune system[END_REF]. Moreover loosening itself could cause extra mobility and it could confuse the wear scenarii.

septic revision because of suspicion of a non usual triggering of the dual mobility socket.

explants from peri-prosthetic fractures because of mismatching of the hip joint and mobility triggering.

explants from second revision (or more) revision to avoid combinations of different materials.

Finally, the same implant was selected (implant stem 'Profil' from the SERF company manufactured between 1988 and 1998), with the same neck length, the same head diameter (22.2 mm) and head material (Co-Cr alloy). The neck length is significant in the positioning. The metal back could be manufactured in titanium alloy Ti-6Al-4V or stainless steel 316L. As both materials could involve a particular mobility of the cup, it has been decided to consider two liner-metal back couples: UHMWPE/Ti-6Al-4V and UHMWPE/316L. The contact between the liner and metal-back is submitted to the weak forces, specific to the materials in contact. The socket mobility will be triggered according to the material in contact.

After taking into account these clinical factors, 12 explants were selected from the initial group of 250, each representing a distinct population of patients or implants. From the 12, 8 were stainless steel metal-backed explants, 4 were titanium alloy metal-backed explants. We decided to include a genuine UHMWPE cup, manufactured in 1993 for integrating the influence of the initial surface on the wear behavior. All clinical and material parameters were described on Table 1 and Table 2. The Devane score is related to the patient activity label, Table 3 [START_REF] Devane | Three-dimensional polyethylene wear of a press-fit titanium prosthesis[END_REF]. It is useful for quantifying the sedentary activity from the high level physical activity (competitive sport for example). The Brooker score is useful for quantifying the peri-prosthetic ossifications, Table 4 [START_REF] Brooker | Ectopic ossification following total hip replacement, Incidence and a method of classification[END_REF][START_REF] Boddu Siva | Heterotopic Ossification After Surface Replacement Arthroplasty and Total Hip Arthroplasty: A Randomized Study[END_REF]. It is related to the hip joint mobility according to the space between opposing bone surfaces from ossification with huge distance to ossification with ankylosis of hip joint, i.e. bone coalesce. This score will point out information about the mobility of the artificial hip joint during the in vivo joint running. Peri-prosthetic arthrofibrosis is an element that has to be mentioned in operative reports. It states that the articular capsule is thickened, limiting hip movements. Table 3: Devane score and the related activity [START_REF] Devane | Three-dimensional polyethylene wear of a press-fit titanium prosthesis[END_REF].

Table 4: Brooker score and related peri-prosthetic ossification [START_REF] Brooker | Ectopic ossification following total hip replacement, Incidence and a method of classification[END_REF][START_REF] Boddu Siva | Heterotopic Ossification After Surface Replacement Arthroplasty and Total Hip Arthroplasty: A Randomized Study[END_REF].

Roughness investigations

While the explants were selected, macroscopic analysis allowed highlighting a strip wear close to the liner convex side equator. The evolution of the roughness parameters was fruitful for isolating the particular wear zones and for pointing out the significant wear zones. The specific methodology dedicated to this study was established from [START_REF] Xq Xiang | Three-dimensional surface characterization for orthopaedic joint prostheses[END_REF][START_REF] Blunt | Three dimensional measurement of the surface topography of ceramic and metallic orthopaedic joint prostheses[END_REF][START_REF] Najjar | About the relevance of roughness parameters used for characterizing worn femoral heads[END_REF][START_REF] Peltonen | Topograpical parameters for specifying a three-dimensional surface[END_REF][START_REF] Blunt | The role of tribology and metrology in the latest development of bio-materials[END_REF]. Thus, in order to characterize quantitatively the wear mechanism at a microscale, the following analyses were practised: analyses of 11 zones on the convex side of each UHMWPE liner, i.e. 5 areas on the no-worn zone (latitude 2 as depicted on Figure 2), 5 areas on the worn zone (latitude 3, Figure 2) and 1 zone on the liner apex.

The profilometry analyses were investigated using a mechanical profilometer SOMICRONIC ® . The measurement interval in X and Y was 4 µm. The images resolution was 256 x 256 points. The raw data were filtered using a least squares plane and a B-spline filter; the cut-off length was of 2 mm. As the surface is spherical, a particular attention was paid on the influence of the area size on the roughness parameters.

Moreover, reproductibility was investigated on a single cup. Finally, the maximum area of measurements was chosen equal to 1 mm 2 . 11 zones of 1 mm 2 for one liner were considered; these analyses were practised on 12 explanted liners and on a reference liner (total of 141 areas). The Figure 3 presents a typical 3D area after filtering, it is related to the T1 UHMWPE convex area of the worn zone, latitude of 60-80° close to the equator (latitude of the zone 3-Figure 2b), Figure 3a. The Figure 3b is related to the apex zone, summit of the convex area (zone 2-Figure 2b). The software Matlab ® allowed drawing surface area from the filtered data. From the comparison of the images, one may notice that the peak heights of the worn zone are higher than the ones of the apex zone. Further results would exhibit the roughness parameters extracted from these topographical data. The aim of this study was to revealed the most relevant roughness parameters for distinguishing the noworn and worn zones from the convex side of the UHMWPE liner. From the filtered data, various roughness parameters were evaluated:

-the amplitude parameters: Sa, Sz, Sq, Ssk, Sku ;

-the spatial parameters Sds:, Str ; both parameters were extracted from the 3D roughness analysis;

-the Abbott-Firestone parameters: Srpk, Srpkm, Srvk, Srvkm, Srk, Smr1, Smr2, Sv1 et Sv2.

All parameters were defined in [START_REF] Najjar | About the relevance of roughness parameters used for characterizing worn femoral heads[END_REF] and in ISO standard 25178. As example, the Figure 3a shows the evolution of Sa according to the liner type and the Figure 3b.

It is worth noting that a part of the roughness parameters was relevant for distinguishing worn and noworn zones and the different explants, Sa parameter. Otherwise, Sds, the summits density, did not allow distinguishing different explants. The complete analysis would be investigated in the results part.

Tridimensional measurements, 3D CMM

A 3 Dimensional Co-ordinate Measure Machine (3D CMM), Mahr ® MMQ, was used on dual mobility liner convex sides. The main goal of these measurements was to estimate the macroscopic wear of all the UHMWPE explanted liners (convex side), thus this method allowed performing the wear volumes of the convexity. One point per degree, for each latitude, was measured. Distance between latitudes was set at 0.5 mm.

More than 10,000 points per explanted liner were acquired thanks to this methodology; it aimed at decreasing the discrepancy on the wear volume. A drawback of this mechanical device was its wider probe (1 mm diameter),

involving measured zone crushing. This experiment had to occur later than 3D roughness measurements because of the polishing effect after 3D CMM measurements. Using 3D CMM, liner outer surface was extrapolated and compared to the liner original blueprint, manufacturing tolerances included. Figure 4a presents threedimensional data from 3D CMM for one explant, convex side.

Moreover, the measurements of the inside volume were carried out on all explanted liners and on the pristine liner. The number of the measured points on the concave side, Figure 4b, is lower than the one on the convex side, due to the space bulk between the 3D CMM stylus and the inner volume of the liner, 22.2 mm. The stylus positioning on the concave side is more difficult than the one on the convex side. Thus the total volume, outside and inner wear volumes, was measured on each explanted liner.

Scanning Electron Microscopy Observations

Convex sides of three UHMWPE liners were analyzed. This last test tends to be destructive because of the metallization layer, gold deposit. A scanning electron microscope, MEB JEOL 6400, was chosen, including Electron Dispersive Spectroscopy probing in order to analyse metallic shell particle transfers to the surface of the liner. Sharp settings of the gold metallization protocol allowed to get images with a maximum magnification of 30000. Apex, worn zone and no-worn zone images were obtained.

Results and discussion

Worn volumes by CMM

The twelve explants were analyzed using 3D CMM. The pristine liner was used as reference, calibrating the measurement of wear volume. Figure 5 represents the liner volume, in white, according to the manufacturing minimal diameter. The red stripe corresponds to a zone where the measured diameter was inferior than this minimal manufacturing diameter. Apex was not measured on this experiment. As no wear stripe was found in this zone, we estimated unnecessary to add apex dimensional data. This wear stripe was viewed on all arthrofibrosis-free explants, it is typical to a specific worn zone of a dual mobility liner.

Table 5 presents 3 D convex (second mobility), concave (first mobility) and total wear volumes of dual mobility liners. The average wear volume corresponds to the mean of the minimal and maximal wear volumes that were calculated from the biggest (smallest) radius according to manufacturing data, and the 3D CMM measured volume, actual volume after explantation. Wear in mm 3 /year corresponds to the ratio between total wear volume and implant survival, i.e. implant duration. Negative values, for the concave side, probably signify that manufacturing blueprint guidelines were not precisely respected. Starting from acquired data, it had been impossible to find a significant tendency between wear volume and metal back composition, peri-prosthetic arthrofibrosis presence or patient parameters.

Roughness measurements analysis

The analysis, on Table 6, of all available 3D roughness parameters and the Abbott-Firestone parameters allows classifying the relevant ones. An ANOVA test (p < 0.05) was investigated for highlighting whether the roughness parameters were relevant or not for discriminating the no-worn zone from the worn zone on the convex side, for each implant.

The Table 6 exhibits the relevant 3D roughness parameters for differentiating the wear zone from the noworn zone. As mentioned in Figure 6, wear was approximately located upper to the equator on the convex side of the liner. It is worth noting that Ssk parameter was highlighted. The Figure 7 shows the evolution of Sa and Sdr parameters according to the liner type. These parameters allowed highlighting the influence of arthrofibrosis.

SEM observations of convex sides

2 explants and the bulk liner were analyzed using SEM, on apex, worn zone and no-worn zone. It is worth noting that the worn zone on the stainless steel metal backed liner, i.e. the red stripe identified in the previous section on CMM, presented evidence of a damaging process that could correspond to a material pulling-out or even to a ploughing mechanism. Wear mechanisms of liner, between stainless steel and titanium alloy metallic shells, seemed rather different. One could presume that UHMWPE liner is worn by abrasion and ploughing for stainless steel metal back liners when it is more an adhesion, cracking then flaking mechanism for Ti-6Al-4V metal back. Particles found on the stainless steel MB worn zone image were the size of a millimetre.

Finally, no-wear zones, whatever the metal back material, showed a polished zone, in comparison with the bulk liner image.

These observations confirmed the S a values and even more the differences between worn and no-wear zones. Highest S a values were found on stainless steel metal backed liners, i.e. the most damaged profiles, according to the stainless steel MB worn zone image. Also, it clearly appeared that this wear stripe was also found on titanium alloy metal backed liners.

Discussion

The CMM measurements exhibited the influence of arthrofibrosis on the triggering of the second mobility, the usual mobility, between head and liner, being preserved from wear. Average wear values could be compared to Charnley cemented THA mean wear values (no dual mobility liner), i.e. from 35 mm 3 /year [START_REF] Hall | Wear in retrieved Charnley acetabular sockets[END_REF] to 94 mm 3 /year [START_REF] Hall | Wear in retrieved Charnley acetabular sockets[END_REF][START_REF] Jasty | Wear of polyethylene acetabular components in total hip arthroplasty. An analysis of one hundred and twenty-eight components retrieved at autopsy or revision operations[END_REF]. As mentioned by Saikko et al. [START_REF] Saikko | Wear comparison between a dual mobility total hip prosthesis and a physical modular design using a hip joint simulator[END_REF], from in vitro hip joint simulator tests, no significant differences were highlighted. The Figure 6 is relevant because it highlights when fibrosis was indicated the total wear volume was equal to the wear volume of the convex side (Dual mobility). From these results, one might suggest that the dual mobility does not increase the total wear volume. The most relevant 3D roughness parameters were selected from a statistical analysis. Sa and Sdr highlighted the role of arthrofibrosis, and the metal back impact on the liner surface: polishing effect on the convex side for Ti-6Al-4V metal back and abrasion for 316L metal back. The SEM images confirmed the macroscopic (CMM measurements) and microscopic (3D profilometry) analyses. The comparison with the results from the Figure 6 assessed the fact that arthrofibrosis plays the significant role. In that way, the second mobility is preferentially triggered, Sa and Sdr are low. One might suggest that the convex side is polished during the movement. When both mobilities are triggered, the profilometry analysis, from I2 to I7, highlighted huge values of Sa and Sdr. The friction is not homogeneous and random adhesive friction could be promoted.

Only the 316L metal-back allowed this typical behaviour. On the contrary the Ti-6Al-4V alloy did not allow the severe spoiling of the convex side.

The wear stripe is clearly a feature of all dual mobility liners. When a wear stripe was discovered, the implant revision operative report did not mention peri-prosthetic arthrofibrosis. This wear stripe could be attributed to a "regular" function of dual mobility explanted liners. Second mobility can be triggered either when the first is blocked in the case of arthrofibrosis, or due to adherence changes of the contact between head and concave side (lubrication conditions, microscopic structure of surfaces in contact, etc.), or also because of the movement amplitudes, causing a contact between neck and liner.

The finding of convex side wear shows the second mobility snaps intermittently, and that we can indeed call this concept dual mobility, rather than a simple anti-dislocation security. However, the discovery of an equatorial blocking leads to a reflection on a prosthetic design optimization. A second consequence of this blocking is the overuse of the first mobility, with more brutal and more frequent contacts of the neck on the liner retentive chamfer, which may lead to premature wear of this retentive chamfer and so increasing the risk of intraprosthetic dislocation. Preventing the blocking might have a beneficial effect on the total hip arthroplasty global survival. Changes on the prosthetic design have been decided since the end of the 1990s. There is no intraprosthetic dislocation that we know of concerning a dual mobility concept implanted after 1998, when usually the first intra-prosthetic dislocation occurs around 4 or 5 years of implantation.

These results highlighted that the dual mobility is not deleterious in terms of wear volume. However, it allows restoring valuable mobility even in case of arthrofibrosis. Thus this scenario would have to be confirmed with other studies on wider series of explants and in vitro analyses.

Conclusions and outlooks

The results from multi-scale analyses allowed suggesting a draft of a dual mobility function theory. The liner, when load is applied, is fixed within the metal back. Its position is roughly symmetric to the stem neck axis, it is its equilibrium. Upon contact between liner and metal back during load, comes a strain on the liner which widens near the equator, which is the area where width is maximal, resulting in a liner blocking. Creep occurs from the implant apex, maximal force application point. If hip movement overcomes the adhesion threshold between liner and metal back, then this movement results in a wear stripe by excessive friction at the equator. Liner-neck contact could contribute to trigger this movement, this event coming close to the limit of first mobility. Materials used play a big part in dual mobility function, specifically titanium has to be not indicated as a dual mobility metal back material. Different adhesion and compression cycle phenomena cause liner rotating movements, which results in circular wear stripe. It would be suitable to simulate this process in vitro to test this new theory. We suggest realizing static and dynamic hip simulator cycles. Terms of these tests are in progress.

This dual mobility function theory could allow this concept, which has proven its efficiency on implant stability to widen its indications. Dual mobility concept is recommended by the Société Française de Chirurgie Orthopédique et Traumatologique (SOFCOT) [START_REF] Philippot | Survival of cementless dual mobility socket with a mean 17 years follow-up[END_REF] for the patient older than 60 years, or at risk of dislocation. A progress on implant long term survival could benefit to younger patients who want to maintain their sport activities and keep a sufficient hip range of motion, without a risk of dislocation.

Figure 1

 1 Figure 1 Dual mobility hip implant; *: first mobility between head and cup, concave side for cup; **: second mobility between cup and metal back, convexe side for cup

  following factors were taken into account for excluding implants:

Figure 2 a

 2 Figure 2 a) Cestilène ® (UHMWPE) liner with functional dimensioning and tolerancing; b) zone 1: apex ; zone 2: no-worn zone; zone 3: worn zone

  zone of T1 explant b) apex zone of T1 explant Ref N°: 0313, Geringer, Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis 6

Figure 4

 4 Figure 4 dimensional data, from 3D CMM, of a UHMWPE explanted liner, a) outer surface (second mobility, convex side), 35 latitudes, 360 points per latitude, 12600 measured points; b) inner surface (first mobility, concave side), 7 latitudes, 120 points per latitude, 840 measured points.

  0313, Geringer, Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis

7

 7

Figure 5

 5 Figure 5 Three dimensional CMM a) Explant I6 ; b) Explant T2 ; the red stripe represents the zone where measured diameter was below the minimal manufacturing diameter.

  0313, Geringer, Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis 8

Figure 6

 6 Figure 6 Convex side, wear volumes of dual mobility liners.

9 Figure 7 a

 97 Figure 7 a) Sa values according to liner type; b) Sdr values according to liner type.

10 Figure 8

 108 Figure 8 SEM images from convex side dual mobility liners on 3 zones (apex, unworn zone, worn zone), from different types of metal backs (316L or Ti-6Al-4V) and the bulk liner.

  Ref N°: 0313, Geringer, Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis[START_REF] Farizon | Results with a cementless alumina-coated cup with dual mobility. A twelve-year follow-up study[END_REF]

Table 1 :

 1 clinical data from patients.

	Implant	Birth Date	Age	Body weight	Implantation	Devane	Brooker	Arthrofibrosis
	code		(years)	(kg)	duration (months)	score	score
	I1	1974-07-06	13	58	144	4	3	No
	I2	1954-12-09	40	61	137	4	2	No
	I3	1929-06-15	62	78	125	3	1	No
	I4	1940-04-26	52	77	123	3	1	Yes
	I5	1926-06-11	70	64	112	3	1	No
	I6	1956-01-21	38	56	109	3	1	No
	I7	1958-04-12	39	91	87	4	1	No
	I8	1956-09-18	39	74	76	4	1	Yes
	T1	1977-03-29	14	45	186	4	1	No
	T2	1951-09-30	41	75	167	5	2	No
	T3	1966-05-20	35	82	87	4	1	Yes
	T4	1932-08-04	59	60	74	4	1	Yes

Ref N°: 0313, Geringer, Understanding the Dual mobility concept for total hip arthroplasty. Investigations on a multiscale analysis 4

Table 2 :

 2 implant data from the operative report, metal back material, metal back diameter, inclination and post-operative data, strip wear approximately upper to the equator on the convex side of the cup, .

Table 5 3D

 5 CMM annual wear volumes, depending on metal back material, presence of peri-prosthetic arthrofibrosis and implant duration; Range is the difference between the maximum and minimum values of wear volume; SD is the standard deviation; MB: Metal-

back; SS: Stainless Steel; Brooker: Brooker score.

Table 6

 6 Classification of the roughness parameters in order to highlight the relevant parameter according to ANOVA test (p < 0.05)

Acknowledgements

The authors wish to acknowledge Dr. M. Dursapt and Pr. H. Zahouani for the full-time access to the 3D profilometer at ENISE. Moreover the authors thank F. Blion for performing the 3D CMM measurements.