Christian Le Bas
email: christian.lebas@univ-lyon2.fr

In this paper we survey the literature dealing with the category of prolific inventor. We set out some elements regarding nature, scale, significance and impact of the mobility of this population of prolific inventors. In particular the paper suggests an analysis that measures the effects on mobility on individual inventive productivity and the value of invention. We call "prolificness" the capacity to accumulate knowledge and experience through mobility (that is to say through their capital of contacts and interactions).

The first goal of this paper is to survey the literature dealing with the category of prolific inventor. It is a piece of a larger research project that aims to assess the mobility of this population of prolific by measuring its effect on the individual inventive productivity and the value of invention (see footnote 1). As a consequence we survey the literature on inventor mobility 2 as well. In the last part we give some insight on what we call "The Economics of prolificness".

Literature Survey on prolific inventors

 analysing the statistical distribution of a bulk of patents from a sample of 7392 inventors who received 9 patents or more under the time period 1975-1984. He observes the frequency distribution of patent output per inventor revealing "an

approximately logarithmic decline". He performs a patent citations analysis on a random sample of 45 prolific inventors and finds that there is no statistically significant differences as far as the average citations across the range of inventor patent outputs. This is interpreted as follows: the value of patent (patent quality) does not decrease as the quantity of patent per inventor increases. This point is particularly important since it underlines that there is decreasing returns as far as the productivity of prolific inventors is concerned. Thirdly, the [START_REF] Narin | Inventive productivity[END_REF] seminal paper on "highly prolific inventors". They investigate 4 companies in the area of semiconductors and they perform an "inventor name unification" (3000 inventors). Every inventor is given credit for the whole invention regardless the number of co-inventors (Narin and Breitzman, 1995: 510). They emphasises the key role of a few researchers that "seems to be a law of nature": "One, two or three individuals are really driving their laboratory….companies should make effort to retain and nurture these key contributors". This study constitutes really the first modern study on prolific inventors even if it is on a limited sample of inventors and patents 3 . b. The previous studies from the LEFI research team (University Lyon 2). Latham, Le [START_REF] Latham | The prolific inventor as persistent inventor: an empirical study[END_REF] using date from USPTO have untaken to measure the scale and assess the scope of prolific inventors from four countries (France, Japan, Germany, U-K). They find (see table 1) that Japan and Germany are the two countries having the larger proportion of prolific inventors (more than 9 patents over the period of time . They show that one consequence of the persistent productivity of prolific inventor is that the proportion (in comparison of the total amount of patents) of patents including in a prolific inventor is very large, between 40 and 94 % along the different countries (see table 2) 4 . c. The Trajtenberg inventors patent data set [START_REF] Trajtenberg | The names game: Using Inventors Patent Data in Economic Research[END_REF](Trajtenberg, et 2007)). Trajtenberg has undertaken to build up a large data set on the inventors. Knowing that in patent document we can observe their name, first name and address, there are a lot of difficulties and traps. He suggests a two stages methodology for matching the names of inventor using the SOUNDEX coding. A first assessment from the NBER Patent Data File [START_REF] Kim | Creating a Longitudinal Research Worker-Establishment Matched Dataset from Patent Data: Description and Application to understanding International Knowledge Flows[END_REF] have conduct a study on US patents and inventors with the Tratjenberg' method onto two industries: pharmaceutical and semiconductor. They add dissertation abstract to inventor data for mapping and explaining the international knowledge flows. They find out an increase in the extent to which US innovators access researchers with foreign R&D experience.

The PATVAL survey gives the number of times the inventor changes employment after the one of the patent: one time 14, 22 %, two times 5,15 %, three times 1,53 %, more 0,78 %. It means roughly speaking 20% of employers on a time period of 10 years.

Of course the time periods differ across the studies but the main finding of that story is that employer mobility is, if not large, very significant8 . The two authors Tratjenberg (2004) and [START_REF] Hoisl | Tracing mobile inventors-The causality between inventor mobility and inventor productivity[END_REF] finally consider as a very significant finding that the inventor creating more value for invention are more mobile, but [START_REF] Hoisl | Tracing mobile inventors-The causality between inventor mobility and inventor productivity[END_REF] only tests the causality productivity of inventor AE inventor mobility, and find that more productive inventors (prolific inventors?) are not more mobile10 .

c. Theoretical background

In the Economics of Knowledge, the state of the Art as far as mobility is concerned, is well encapsulated by Teece (1982: 45): "The transfer of key individuals may suffice when the knowledge to be transferred related to the particulars of a separable routines,…only a limited range of capabilities can be transferred … . More often than not, the transfer of productive expertise requires the transfer of organizational as well as individual knowledge". Inventor mobility is key mechanism to transfer tacit knowledge between firms and then a mean for technological diffusion [START_REF] Almeida | Localization of knowledge and the mobility of engineers I regional networks[END_REF][START_REF] Rosenkopf | Overcoming Local Search through Alliances and Mobility[END_REF][START_REF] Stolpe | Determinants of Knowledge Diffusion as Evidenced in Patent Data: The Case of Liquid Crystal Display Technology[END_REF] or personnel rotation as a mechanism of knowledge [START_REF] Kane | Knowledge transfer between groups via personnel rotation: Effects of social identity and knowledge quality[END_REF] 11 . Transferring

A study from [START_REF] Song | When Is Mobility More Likely to Facilitate Interfirm Knowledge Transfer[END_REF] on the patenting activities of engineers shows that mobility is a special type of learning at the firm level: learning-by-hiring. Through the hiring of inventors the firm can get access to the inventor capital of contacts [START_REF] Breschi | Mobility and Social Networks: Localised Knowledge Spillovers Revisited[END_REF].

In the frame of the standard Microeconomics it is acknowledged that individual mobility is an important source of knowledge externalities [START_REF] Moen | Is mobility of technical personnel a source of R&D spillovers? Department of Economics Discussion[END_REF], in other terms "Knowledge flows are localized to the extent labour mobility also is" [START_REF] Breschi | Mobility and Social Networks: Localised Knowledge Spillovers Revisited[END_REF].

But there is another face to the coin: mobility is a part of human capital formation, through mobility individual increases his/her knowledge capital ("learning by moving"). This explains the importance of strategic mobility.

d. New context for inventor mobility: the frame of strategic mobility High tech firms actively encourage defection among competitors' technological personnel (Kim et Marschke, 2004): "If you have trouble with the competition, simply raid its talent".

This pushes the firms to patent in order to protect their Intellectual Property against leakage through inventor mobility. [START_REF] Gilson | The Legal Infrastructure of High Technology Industrial Districts: Silicon Valley, Route 128, and Covenants Not to Compete[END_REF] explain (partly of course) the greater success of Silicon Valley as compared to Route 128 by the fact that while California and Massachusetts respect the Trademark Act protecting the loss of trade secrets through mobility, Califormia prohibits (Massachusetts enforce) "post-employment restrictive covenants" (also known as "employee non-compete agreements", non-compete restricts a departing employee from accepting new employment with a competitor for a specified time and geographical jurisdiction, [START_REF] Fleming | Managing Creativity in Small Worlds[END_REF]. [START_REF] Fleming | Managing Creativity in Small Worlds[END_REF] show that there is a greater mobility within and toward US regions that do not enforce non-compete agreements (noncompete restrict a departing employee from accepting new employment with a competitor). In this context mobility is pulled by rival firms and is not totally governed by the inventor rational decision. There are some grounds for thinking that strategic mobility has increased in the recent period of time (at least in the USA).

The Economics of prolificness

One basic characteristic of prolific inventor is that individually he produces more patents (productivity) and patents having more value. By prolificness we denote the twin process of high inventor productivity and great value of inventions produced by these inventors. To put it simply there is a relationship between the quantity and the quality of invention 12 . There is an accumulation of knowledge and experience behind the accumulation of patents.

Our point of view is founded upon the empirical evidence from two basic studies. We can hypothesize Prolific inventors are "technological goalkeepers" who mediate the flow of knowledge into the research organization [START_REF] Allen | Roles in technical communication networks[END_REF] 16 . In a sense they act as "Knowledge integrators" (see [START_REF] Latham | The prolific inventor as persistent inventor: an empirical study[END_REF]. This knowledge worker plays a prominent

12 By contrast Mariani (2005) has suggested the existence of a trade-off between patent quantity and patent quality at the inventor level. 13 In other terms there is no choice between quantity and quality: quantity _ quality.

14 Mariani and Romanelli (2007) claim the relationship between quantity and quality is not direct but indirect: when an inventor produces a lot of invention the probability a technological hit is increased. 15 Age, academic degree and involvement in large research team.

16 Levine (1985) adds they are recognized as sources of information, top performers, valuable to the organization in meeting its technical objectives.

role in the design, development and integration of pieces of knowledge within a department of research. In the invention team there are people with different technological and scientific specializations. Prolific inventor and his/her Engineering knowledge are basically important.

He/she increases the rate at which individuals and organizations learn and consequently achieve sustainable competitive advantages. The prolific inventors are innovation "champions". Prolific inventors through their professional mobility might be viewed as "knowledge translators" or "knowledge broker" 17 between different firms, organizations and communities as well. They help to transfer some pieces of knowledge through the different communities they overlap at one or different points of time.

At this stage, it is relevant to put in relation prolific inventors and "stars scientists" model exemplified by [START_REF] Zucker | Labor Mobility from Academe to Commerce[END_REF]. A "stars scientist" is an individual with higherquality intellectual capital (measured in terms of number of citations), a "prolific inventor" is an individual with high intellectual capital (measured in terms of number of patents with likely higher values) at the beginning of their career or through their industrial evolution. A "star scientists" makes major discoveries (Zucker andDarby, 1996, 2001). That should be confirmed/infirmed for prolific inventor. In the biotechnology sector "the labour of the most productive scientists is the main resource around which firms are built or transformed" (generalized to high-tech industries, Zucker and Darby, 2006) a. From the literature a double relationship between prolificness and mobility must be considered:

17 See Brown and Duguit (1999) who define these two terms. 18 Zucker and Darby emphasize the importance of the tacit character of the new discoveries. Knowledge being embodied in individuals this implies "bench-level" collaboration which is measured by co-authoring.

1) The relationship prolificness AE mobility. A correlation between value of patents and high rate of inventor mobility is envisaged by [START_REF] Almeida | Localization of knowledge and the mobility of engineers I regional networks[END_REF]. They observe a high rate of inter-firm mobility of skilled engineers who hold major semiconductor patents. This evidence is in favor of the idea that prolific inventors are more mobile than the others. By contrast, [START_REF] Hoisl | Tracing mobile inventors-The causality between inventor mobility and inventor productivity[END_REF] show that high productive German inventors are less inclined to move.

This result is consistent with the [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF] findings when they note that inventors in software who are very productive, have a decreasing probability to move assignees as their career progress (conversely: less productive inventors have larger probability to move in later periods of their career).

2) The relationship between mobility AE prolificness. [START_REF] Hoisl | Tracing mobile inventors-The causality between inventor mobility and inventor productivity[END_REF] show multiple moves German inventors produced patents with more value (measured by the economic value gathered through a survey). By contrast, [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF] found that the variable "cumulative moves of inventors" has no effect on the patent value measured by forward citations. In fact the authors estimate two different equations for explaining the number of citations received (a proxy for patent quality). The first indicates on a robust effect of moves between assignees on patent citations. The second contains a new variable "mean citation for the prior patents of the inventor" that plays the role of individual inventor's fixed effects. On the one hand this variable has an important significant effect, on the other there is no longer impact of moves on patent citations after controlling for this individual effect 19 . Our opinion is that this variable gives at each period of time a measure of the inventor intellectual capital.

For this reason we can suspect in fact a correlation between the variable moves and this variable since moves participate to the inventor intellectual capital formation.

In the relationship mobility/value of patent it may be there are Country effects since the institutions governing the worker mobility differ across countries. For instance [START_REF] Trajtenberg | The names game: Using Inventors Patent Data in Economic Research[END_REF] has shown the Japanese inventors are less mobile than the American one. This result is of course linked to the employment rules of the large industrial Japanese firms.

The work of [START_REF] Allison | Productivity Differences Among Scientists : Evidence for Accumulative Advantage[END_REF] on productivity differences among scientists enable us to understand why the differences in inventor performance (in terms of invention) persist if not increase. In the world of Science because of feedbacks in terms of resources or recognition the highly productive scientists maintain or increase their productivity. In the world of research on Technology a not too different mechanism seems to work: a good 19 Interestingly [START_REF] Schankerman | Software Patents, Inventors and Mobility[END_REF] note mobility "breeds" mobility: (software) inventors who have already moved in the past are more inclined to move again.

inventor will receive more resources and as a consequence will be implicated in more projects. There is the possibility that he leaves an organization for being hire by another organization. A virtuous circle (a self-reinforcing process) is at work. In the very beginning of our empirical study we observed two kinds of prolific inventors: the fast prolific inventors and the persistent prolific inventors. The first kind of inventors patent a lot on a short time period, while with the second type an inventor becomes prolific after a longer time period (of learning?). It might be the threshold of 15 patents and industry characteristics play a role in this story.

b. From the different studies it seems that the importance of countries characteristics emerges and may be more important than the sector (or technology) differences for explaining the inventor mobility:

-For Japan the style of employment management in the large firms is characterized by the persistent presence of the workers within the same firm -For Germany the existence of German employees'inventions act creates a singular situation -For USA the surge of starts-ups and the very large transfer of individuals from University to industry could affect the relationship between mobility and inventive productivity. In this country some evidence shows that the share of prolific inventors of large firms fell from 72 % to 69 % and the one of small firms rose from 12 % to 16 % from the mid 90's and earlier in the next decade (CHI Research Inc., 2004). The same study points out 1) that small firms were particularly attractive destinations for "elite" inventors20 working for public sector organizations and 2) that elite inventors tend to move into the same category of firms (large versus small).

One basic hypothesis of the research is that knowledge (labour) markets is more important than Knowledge (non-market) spillovers. In this context it seems important to measure the extent to which the prolificness and the mobility of prolific inventor could explain firm R&D performance21 .

  regresses (negative binomial function) the number of moves across assignees (per inventor). He adds variables as age (= 1999 -year of first patent), different from: patent duration = last year -first year. The moves of inventors are correlated with "younger" inventors (sign of age = negative), inventors having more patents in Drug and medicine, inventors having more partners (large R&D team?), inventors more technologically specialised (less technologically diversified), inventors having more important patents (more citations) but the reverse in Japan, inventors US (versus Japanese). One result deserves a greater attention: it seems that more "valuable" inventors move more. But what is the causality 9 ?Hoisl (2006) observe that mobile inventors are more than four times as productive (patent per inventor divided by the age of inventor in 2002 minus 25) as non-movers (survey of 3049 German inventors). The level of education has no influence and an increase of productivity decreases the number of moves. In the post-move period inventors produce more patentable innovation that are characterized by more value (survey of 3049 German inventors), but the gains of mobility dissipate over time.

  First the paper fromGambardella et al. (2005) using the PATVAL survey (7000 patents). They note the characteristics of the inventor, his past number of patents is the more important (we underline) determinant of the private value of invention (gathered through a survey) than the characteristics of organization in which he is employed13 . Secondly the research from[START_REF] Latham | The prolific inventor as persistent inventor: an empirical study[END_REF] confirms this result. Using a data set of patents granted by the US Patent Office to French, German and British inventors over the period from 1975 to 1999 they estimating a relationship explaining the citations received by each patent. Prolific inventors tend to produce inventions having more economic value. We now need to explain why there such a relationship between quantity and quality 14 . Nevertheless some intriguing stylised facts emerge from other empirical studies. For instance[START_REF] Mariani | Stracking' or 'Picking' Patents? The Inventors' Choice Between Quality and quantity[END_REF] find with a sample of 793 European inventors drawn from the so-called PATVAL survey that individual and organisational characteristics affect the level of single inventor productivity (quantity of patent) after controlling for countries and sectors15 . By contrast these factors produce no direct effect on the value of patent measured by the number of forward citations (quality of patent). Only the scale of the research project has a small effect the value of patent on what[START_REF] Latham | The prolific inventor as persistent inventor: an empirical study[END_REF] point out as well.

  . The model of mobility of Stars scientists is from "Academe to Commerce", in others words technology transfer from University to Industry is important. Stars scientists are important in the process of technology transfer because of the value of their knowledge to the success of firms 18 . It is clear that Stars scientists and prolific inventors are two close categories of highly productive knowledge workers, the first in Science, the second in Technology. It may be some Stars scientists patent as well. As a result they could be prolific inventors too. The stars scientists are "entrepreneurial individuals". It may be prolific inventors are entrepreneurial university researcher (first defined by Etzkowit, 2003) Entrepreneurial researcher is an entrepreneur active towards technology transfer and partnership with industry. There are reasons to think that there are more numerous in the USA. Conclusion: some temporary remarks.

Table 1 .

 1 Distribution of inventors according to the number of US patents bycountry, 1975-1999 (%)

	Number of patents France	Germany	UK	Japan	TOTAL
	1	56,72	50,24	52,14	42,74	47,32
	2-9	38,76	41,83	42,03	45,89	43,59
	10-49	4,36	7,41	5,60	10,64	8,53
	50 et plus	0,16	0,53	0,23	0,73	0,56
	TOTAL	100,00	100,00	100,00	100,00	100,00
	Source: Latham, Le Bas and Touach (2006)

Table 2 .

 2 Patents including a prolific inventor by country, 1975-1999[START_REF] Latham | The prolific inventor as persistent inventor: an empirical study[END_REF] use the same data base of patents granted by the US Patent Office to French, German and British inventors over the period from 1975 to 1999 and the NBER data base. Negative binomial multiple regression models support for the hypotheses that both prolific and foreign inventors tend to be parts of larger teams of inventors 5 and that both

		France	Germany	UK	Japan
	Number of US patents
	including a prolific
	inventor	26.299	111.985	24.448	368.844
	Proportion of US patents	40.45	65.93	42.29	94.41
	including a prolific
	inventor (%)
	Source: Latham, Le Bas and Touach (2006)
	prolific and foreign inventors tend to produce inventions having more value (value being
	measured by citations).

3 A note from the

USPTO (1998)

giving the name of prolific inventors receiving utility patents from 1988 to 1997. The

Ernst et al. (2000)

study show that very productive inventors are associated to valuable patents.

4

Their results must be considered as temporary because the data set has been quickly built up. Very recently in the same vein

Harhoff and Hoisl (2006)

noted few inventors produce the lion's share of inventions within an R&D department.

. Inventor mobility: scale, determinants, theoretical background

	one country move, at the other 1 only inventor moves through 6 countries.
	allow him onto 2
	million patents and 2 inventors per patent on average to find 4,298 912 records. After
	matching it with the Trajtenberg procedure we obtain 1,565 780 inventors. As a first
	comparison with there is 58% with just one patent, and 5% with 10 patents or more 6 . Of
	course the research implemented by Trajtenberg (2007) does not deal with the population of
	prolific inventors, but represents a very rich tool for measuring and mapping it.
	2a. Scale. The data set worked out by Tratjenberg (2007) enables us to trace the mobility of
	inventors among other issues. First of all mobility of inventors across assignees: for the
	overall sample of US patent we find 216, 581 (33% of the overall population of inventor) as a
	number of inventors movers 7 . The number of mobile inventors across countries: 12 371 for

Harhoff and Hoisl (2006) observe that the dynamics of R&D team is salient feature of invention process.

This result is in accordance with those found by Latham, Le[START_REF] Latham | The prolific inventor as persistent inventor: an empirical study[END_REF].

But Tratjenberg acknowledges there is a need to consolidate assignee code.

Intra-firm mobility inside R&D network stays still important. According to Criscuolo (2005) short-term forms of mobility are less costly and effective inside MNC.

[START_REF] Turner | La recherche publique dans la production de connaissances[END_REF] studying the mobility of researchers in Physic (CNRS) found an elasticity quality of publication/ mobility of researchers = 0,3 citation more with mobility. Here the causality runs from mobility to quality.

[START_REF] Hoisl | Tracing mobile inventors-The causality between inventor mobility and inventor productivity[END_REF] a move increases productivity but an increase in productivity decreases the probability to observe a move.

This matches the knowledge "reuse"[START_REF] Langlois | Knowledge, Consumption, and endogeneous growth[END_REF], a type of increasing economies of scale at the core of economic growth.

Elite inventors have at least 10 patents in the time period(1993-95 and 2000-02) and at least one paten in each period.

Work soon undertaken by[START_REF] Redor | La mobilité des chercheurs dans le système d'innovation français[END_REF].

The research has been funded by ANR (project n° 06-APPR-002-001


Richard Alligier

Nicolas Durand

Gregory Alligier

Efficient Conflict Detection for Conflict Resolution

Keywords: conflict detection, conflict resolution, GPU

Accurate tools to detect and solve conflicts are becoming necessary to assist air traffic controllers in their task. Air traffic controllers will eventually rely on tools to test and choose alternative trajectories. Enabling such tools demands a near real-time conflict detection algorithm.

. For each aircraft, 161 alternative trajectories were considered. Detecting all the conflicts required to compare 2,721,705 pairs of trajectories for a 15 aircraft scenario and 128,308,950 pairs for a 100 aircraft scenario.

The conflict detection procedure uses predicted trajectories which are inherently entangled with uncertainties. A seamlessly way to handle these uncertainties is to bound the future positions in a sequence of volume. This is how the uncertainties are modeled in the scenarios. However, this uncertainty model makes the conflict computation more time consuming.

, are computed in 30 ms and the 100 aircraft scenarios are computed in 1 s.

I. INTRODUCTION

To sustain air traffic growth, accurate tools to detect conflicts are becoming necessary to assist air traffic controllers in their task. Many uncertainty models have been proposed in the literature to make future tools realistic and trustable for their users. We can for example cite Erzberger conflict probe model as one of the pioneer approaches [START_REF] Erzberger | Conflict probing and resolution in the presence of errors[END_REF], [START_REF] Arthur | User request evaluation tool (uret). interfacility conflict probe performance assessment[END_REF] in the USA. In Europe the HIPS project [START_REF] Meckiff | PHARE : Highly interactive problem solver[END_REF], [START_REF] Price | Hips and its application to oceanic control[END_REF], representation of the conflicting zones were taking into account uncertainties and they had to compute in real time complex detection. This is also the case in approaches proposed by French CENA 1 [START_REF] Durand | Automatic aircraft conflict resolution using genetic algorithms[END_REF], [START_REF] Granger | Optimal resolution of en-route conflicts[END_REF] where aircraft are not modeled with points, instead trajectories are discretized in time and aircraft are represented by convex hulls that expand with time.

In 2017, Allignol et al. [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF] proposed a framework to compare conflict resolution algorithms that precalculated the conflicting alternative trajectories. The model used discretized 1 Centre d' Études de la Navigation Aérienne maneuver options for 161 alternative trajectories taking into account an uncertainty model able to handle various uncertainties. The number of alternative maneuvers seems high but it is quite small in comparison to all the possible combinations of parameters: different times for maneuver start and maneuver end, different types of maneuvers using speed change, altitude change, direct route, or heading changes.

The optimization methods proposed in [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF] are able to efficiently solve moderate size conflict resolutions problems in real time. However, it did not take into account the time to compute the conflict between the alternative trajectories. This time can be high in the scenarios envisioned in [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF]. For each aircraft, 161 alternative trajectories were considered. Detecting all the conflicts requires comparing 2,721,705 pairs of trajectories for a 15 aircraft scenario and 128,308,950 pairs for a 100 aircraft scenario.

In the context of calculating many alternative options in real time for applications such as conflict resolution, it seems very useful to focus on minimizing the time spent to detect potential conflicts between diverse options. This article proposes a method to reduce drastically the time spent to determine the conflicting alternative trajectories.

A. Related work

In this article, we implement a conflict detection algorithm on Graphics Processing Unit (GPU). Several articles report a GPU implementation to accelerate computation in the Air Traffic Management (ATM) domain. Tandale et al. [START_REF] Tandale | High-speed prediction of air traffic for real-time decision support[END_REF] used a GPU to accelerate the computation of trajectory predictions. A two orders of magnitude speedup was obtained. The detection of aircraft entering in a restricted airspace has been implemented in a GPU by Thompson et al. [9]. In [START_REF] De La Iglesia | Parallel trajectory synchronization for aircraft conflicts resolution[END_REF], a GPU is used to compute the conflict between two trajectories that are hours long but described with few points. The two trajectories are interpolated using a GPU and the distances are computed between the points in order to detect if the two trajectories are in conflict.

For a conflict detection problem involving n trajectories discretized in T time steps, n(n-1)

2

T distance computations are required to detect all the possible conflict between the trajectories. However the number of distance computation can be drastically reduced. Many articles rely on spatial partition to avoid testing all the possible pairs of positions. Isaacson et al. [START_REF] Isaacson | Design of a conflict detection algorithm for the center/tracon automation system[END_REF] discarded pairs according with their altitude. A spatial grid was used to reduce the number of pairs in [START_REF] Sridhar | Computationally efficient conflict detection methods for air traffic management[END_REF], [START_REF] Wieland | Using quad trees for parallelizing conflict detection in a sequential simulation[END_REF]. In [START_REF] Kuenz | Tiling the world-efficient 4d conflict detection for large scale scenarios[END_REF], [START_REF] Ruiz | Relational time-space data structure to speed up conflict detection under heavy traffic conditions[END_REF], this idea was improved by using a spatiotemporal partition to reduce even more the number of pairs to be tested. Ruiz et al. [START_REF] Ruiz | Relational time-space data structure to speed up conflict detection under heavy traffic conditions[END_REF] uses a uniform grid, and Kuenz et al. [START_REF] Kuenz | Tiling the world-efficient 4d conflict detection for large scale scenarios[END_REF] uses a 4D partitioning tree.

These articles all model the future aircraft positions as points. Hence, uncertainties on the future positions are not modeled. Some articles [START_REF] Prandini | A probabilistic framework for aircraft conflict detection[END_REF], [START_REF] Prandini | A probabilistic approach to aircraft conflict detection[END_REF] model the position as a bivariate Gaussian random variable. Hence, the position is associated with a probability distribution with a specific Gaussian shape.

In this paper, we consider the scenarios described in [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF]. We consider a conflict detection problem involving n aircraft. For each aircraft, we consider m = 161 alternative trajectories discretized in T time steps. Each position is modeled as a volume. We assume that this volume surely contains the actual future position of the aircraft. This volume is a convex polygon in the horizontal plane, and a minimum and maximum altitude in the vertical plane. This model is flexible and can take into account a large spectrum of uncertainty sources. However computing conflicts with such a model is very computationally demanding as we have to compute distances between convex polygons. We propose an almost brute-force GPU implemented algorithm to detect conflicts between a large number of alternative trajectories.

B. Outline

In part II, we describe the models involved in the conflict detection problem. Part III details the conflict detection algorithm used and part IV explains how it was parallelized. Part V details the experimentation made and part VI analyses the results. We conclude and bring some insight of further work to be done.

II. DESCRIPTION OF THE PROBLEM

In this section, we describe the models involved in the conflict detection problem addressed in this paper. We detail the trajectory prediction model used and we explain why an efficient conflict detection algorithm is required for conflict resolution.

A. Trajectory prediction model

In order to detect a conflict, we need to predict the future trajectories and check if there is a loss of separation between these predicted trajectories. Due to the erroneous nature of the predictions, the trajectory prediction model should handle uncertainties.

In our model, the time horizon is divided into steps of duration τ . The predicted trajectory is modeled as a sequence of positions, one position for one time step. However, in order to take into account the uncertainty on the predicted position, each position is actually modeled as a volume. This volume supposedly always contains the actual future position. In our model this volume is defined by a minimum altitude, a maximum altitude and a 2D convex in the horizontal plane. The Figure 1 illustrates such a volume. No assumptions are made on how these volumes are built. The trajectory prediction module only has to generate the sequence of volumes. The volume can take into account a full spectrum of uncertainties.

For instance, the size of the volume can be the result of the latency of a pilot to follow an ATC instruction, the uncertainty on the speed and the aircraft performance if a climb maneuver is involved. We can compute the predicted point-wise trajectory for different combination of values for these three uncertainty sources. Then, for each time step, we build a volume by computing the convex of the points of the different predicted point-wise trajectories. Using this procedure, we obtain the desired sequence of volumes.

Figure 2 illustrates the sequence one can obtain by applying such a procedure. In this example, we have an uncertainty on the speed, the heading change angle and the time at which this heading change is performed.

B. Conflict detection

ATC must ensure a vertical and a horizontal separation norm between the aircraft. Two predicted trajectories are said in con-flict if there is a time at which the predicted aircraft position are neither vertically separated by NORM vertical = 1000 ft nor horizontally separated by NORM horizontal = 5 NM.

1) Conflict between two trajectories: We want to decide if two trajectories are in conflict. With our trajectory prediction model, two trajectories p and q are in conflict if there is a time step t at which the two volumes p(t) and q(t) are not vertically nor horizontally separated. Two volumes p(t) and q(t) are not separated if and only if there exist (x p , y p , z p ) ∈ p(t) and (x q , y q , z q ) ∈ q(t) such that |z pz q | < NORM vertical and (x px q )2 + (y py q ) 2 < NORM 2 horizontal . We have assumed that the actual future trajectories will be always in the predicted volumes. Thus, if the predicted trajectories are separated then the actual future trajectories will be also separated.

However, one must be aware that depending on the aircraft speed, the step duration τ must be small enough to not miss any conflict.

2) Conflict between two aircraft: For a conflict between two aircraft, we have to consider alternative trajectories for these two aircraft. Assuming that we consider m alternative trajectories for each aircraft, we have to determine which ordered pair (k, l) of alternative trajectories are in conflict. These results can be stored in a conflict matrix (C k,l ) 1≤k,l≤m . If we consider trajectories with T timesteps, this matrix can be built by testing the separation of at most m 2 T pairs of volumes.

3) Conflict between n aircraft: Even if being in conflict is a binary relationship, one aircraft can be in conflict with several aircraft. One way to solve this complex situation is to group the aircraft in conflict in a cluster. A cluster is a transitive closing on conflicting pairs of aircraft.

Because one maneuver of one aircraft might create conflicts with the other aircraft involved in the cluster, we need to know all the conflict matrices of all the pairs of aircraft in the cluster. Thus, with n aircraft, we have to compute n(n-1) 2 conflict matrices. In the end, it leads to compare n(n-1) 2 m 2 T volumes at most. This number can be huge even with a limited number of aircraft involved. For the smallest scenario in this paper with n = 15 aircraft, m = 161 alternatives trajectories and T = 150 time steps, we have 408,255,750 volumes comparisons. Because conflict detection and resolution problems have to be solved quickly, there is a need to build these conflict matrices as fast as possible.

III. CONFLICT DETECTION ALGORITHM

The conflict detection problem can be seen as a collision detection problem where the moving objects are the aircraft positions increased by half the separation norm. Detecting if moving objects will collide is a problem occurring in video games, physical simulations and robotics. This problem can be decomposed in three phases [START_REF] Ericson | Real-time collision detection[END_REF]. The broad-phase is a phase eliminating pairs of objects that are too far away too collide. Then, for the remaining pairs we can apply a cheap collision test on bounding volumes, this is called the mid-phase. Finally, when the mid-phase failed to discard the pair, a fine-grained and costly colliding test between the two objects is done during the narrow-phase. 1) Broad-Phase: When n objects are moving, n(n -1)/2 pairs of potential collision need to be tested. However using a spatial partitioning or temporal coherence, one can cull away some of these pairs. One example of spatial partitioning is the uniform grid [START_REF] Bentley | Data structures for range searching[END_REF]. The 3D space is split in cells. Objects that are not in adjacent cells are not colliding. An example of algorithm exploiting temporal coherence is the sweep and prune algorithm [START_REF] Bara | Dynamic simulation of non-penetrating rigid bodies[END_REF]. In this type of algorithm, a structure is incrementally updated at each time step. In this algorithm, the objects are sorted along the x-axis, the y-axis and the z-axis. These sorted axes are used to discard non colliding pairs. If the objects are moving slowly from one frame to another, then the sorted axes do not change between two time steps. In this case, sorts are performed in O(n) using an insertion sort for instance. 2) Mid-Phase: In order to avoid time-consuming colliding tests, objects are usually bounded by simpler bounding volumes. The idea is that deciding if two simple volumes will collide can be done with a cheap test. In general, the simpler the bounding volume is the cheaper the collision test is. However, if the bounding volume is too simple compared to the object then the bounding volume will be unnecessary large. In this case the bounding volumes might not discard a non-colliding object pair. Thus, there is a trade-off between the cost of the colliding test and how the bounding volume fits the object. Common bounding volumes are Axis-Aligned Bounding Boxes (AABBs), oriented bounding boxes and spheres. 3) Narrow-Phase: Colliding test between two objects can be done by different algorithms depending on the shape of the objects (convex or not) and if the objects can be preprocessed (rigid or not). The choice of the algorithm depends also on the envisioned application. When required by the application, some algorithm can also provide a penetration depth and contact points if the two objects are colliding.

A uniform grid algorithm was successfully applied for conflict detection in [START_REF] Ruiz | Relational time-space data structure to speed up conflict detection under heavy traffic conditions[END_REF] but the objects in this publication are typically 5 NM large. However, a uniform grid does not suit our problem. In order to not miss any collision, the grid's cells must be larger than the largest object 2 . However, some objects in our problem are quite large making the cell quite large too 3 . This drastically reduces the filtering power of the uniform grid. In the end, the conflict detection algorithm presented in this paper does not use a broad-phase algorithm. All the object pairs are considered.

A. Mid-Phase: Axis-Aligned Bounding Boxes

In our algorithm we chose Axis-Aligned Bounding Boxes (AABBs) as bounding volumes. Building an AABB of a volume can be efficiently computed, we only have to compute the minimum and maximum coordinates of the volume. As two position volumes must be separated by a norm, half the norm is added to each side of the AABB. Figures 3 and4 illustrate how the final AABB is built. First we compute the AABB (in blue) of the position volume and then the AABB is increased by half the norm.

The collision test between AABBs is cheap as it requires six floating point comparisons at most.

B. Narrow-Phase

If the test between the AABBs has failed to discard a collision then we perform a fine test between the increased position volume. A position volume is defined by a convex polygon on the horizontal plane and two values on the zaxis, a z min and a z max . If the mid-phase failed to discard the conflict, then we only have to test if the convexes on the horizontal plane are separated by NORM horizontal .

1) ISA-GJK:

The ISA-GJK algorithm [START_REF] Bergen | A fast and robust gjk implementation for collision detection of convex objects[END_REF] can be used to decide if two arbitrary convex shapes intersects. It is a variant of the GJK algorithm [START_REF] Gilbert | A fast procedure for computing the distance between complex objects in three-dimensional space[END_REF] which is widely used in robotics and computational geometric to determine the distance between two convex shapes. This is one of the algorithm implemented on GPU in the physics library Bullet [START_REF] Coumans | Bullet physics simulation[END_REF].

The GJK algorithm relies on the Minkowski sum. The Minkowski sum, denoted by the operator ⊕, of two sets of points A and B is defined by the following equation: B). The goal of the ISA-GJK algorithm is to determine if the point O is in this sum. Actually, GJK uses support functions in order to avoid computing this sum.

A ⊕ B = {a + b | a ∈ A, b ∈ B} Considering that -B = {-b | b ∈ B}, A and B intersect if and only if the origin point O is in A ⊕ (-
In GJK, each shape is solely described by its support function. The input of a support function is a direction and its output is the farthest point in this direction. For any shape S, the farthest point in the direction -→ d is simply argmax

P ∈S - → d • --→ OP ,
where O is the origin. With this definition, one can notice that any shape has the same support function as its convex hull. Actually, with the support functions, computing explicitly the convex hull is not required although it might speed up the computation of the farthest point. If the considered shapes are not convex, the ISA-GJK algorithm will compute the intersection between the two shapes' convex hull. For any shapes A, B and any direction -→ d , we have:

support A⊕B ( - → d ) = support A ( - → d ) + support B ( - → d ) support -B ( - → d ) = -support B (- - → d ) support A⊕(-B) ( - → d ) = support A ( - → d ) -support B (- - → d )
With this last equation, we are able to compute the support function of the set A ⊕ (-B) by only using the support O d P Fig. 5. The black shape is A⊕(-B) and the axis (P, d) in red is a separating axis. In such a case, A and B are not colliding.

functions of A and B without computing the actual Minkowski sum. The next step consists in deciding if a shape described by its support function contains the origin O.

For the ISA-GJK algorithm, there are only two termination cases:

1) An axis separating the shape A ⊕ (-B) from the origin point O is found. In that case, the shapes A and B are not colliding. This case is illustrated by the Figure 5; 2) A simplex, a triangle in 2D, included in the shape A ⊕ (-B) contains the origin. In that case, the shapes A and B are colliding. The idea of ISA-GJK is moving a simplex towards O. This simplex is moved by replacing one of its point. Then we check if the newly introduced point defines a separating axis or if the new simplex contains the origin O.

We have described ISA-GJK, an algorithm used to determine if two convexes intersects. However, we want to decide if two convexes are closer than NORM horizontal . This can be easily done by using

A ⊕ Circle(NORM horizontal ) ⊕ (-B) instead of A ⊕ (-B)
in ISA-GJK. The result of A ⊕ Circle(NORM horizontal ) is illustrated by the Figure 6.

One advantage of ISA-GJK is its capability to work with non-convex shapes without explicitly computing convex hulls. Using ISA-GJK, we might skip the convex hull computation phase. This advantage was not used in our experiment because the shapes provided by the trajectory prediction module were convexes.

IV. PARALLELIZATION

In this paper we compare a CPU based implementation and a GPU based parallel implementation of a conflict detection algorithm. In these two implementations, the position volumes are bounded by bounding volumes, the AABBs described in subsection III-A. This helps to avoid a time consuming test as the one described in subsection III-B.

A. CPU based implementation

We need to compute n(n-1) 2 conflict matrices. Each conflict matrix is computed using Algorithm 1. The method used to decide if two position volumes intersect is the ISA-GJK method described in subsection III-B1.

Algorithm 1 Computation on CPU of the conflict matrix between aircraft i and j 1: Initialize the matrix C with false (no conflicts) 2: for each alternative trajectory k for aircraft i do for each alternative trajectory l for aircraft j do if AABB k,t and AABB l,t are not separated then if k(t) and l(t) are not separated then

B. GPU based parallel implementation

The implementation was completed using the NVIDIA Compute Unified Device Architecture (CUDA) [START_REF] Nvidia | Compute unified device architecture programming guide[END_REF], a generalpurpose parallel computing platform and programming model. This platform provides a "CUDA C/C++" compiler nvcc and tools to fully take advantage of the GPU.

In CUDA, a kernel is a function that is executed N times by N different CUDA threads. The threads might be executed in parallel depending how they are scheduled. Each thread is given a unique ThreadId. The programmer can access this value inside the kernel. Using this ThreadId, each CUDA thread can perform a specific computation on a specific data. The threads are grouped in blocks. Each block has a unique BlockId. Each thread has its own local memory and the block threads share a shared memory. All threads of all the blocks have access to a global memory.

Algorithm 2 is the algorithm dispatching the computation on the GPU. This algorithm transfers the predicted trajectories to the global memory. This data is not directly available, we need to wait the transfer to be completed, hence the line 2. At line 3, the kernel computes the AABBs The resulting AABBs are stored in the global memory. Then, a loop computes all the conflict matrices using one kernel call per conflict matrix. These matrices are first stored in the global memory. After all the computation are completed, the conflict matrices are transferred back to the host memory.

Algorithm 2 Dispatching the computation on the GPU 1: Transfer predicted trajectories to GPU device 2: Wait until end of all GPU tasks 3: Launch kernel computing AABBs 4: for i = 0 to n -1 do 5:

for j = i + 1 to n -1 do 6:
Launch kernel computing conflict matrix for aircraft i and j 7:

end for 8: end for 9: Wait until end of all GPU tasks 10: Transfer conflict matrices from GPU device to host 11: Wait until end of all GPU tasks 12: return Conflict matrices

The kernel used to compute the matrix is described in the Algorithm 3. It computes one conflict matrix. At line 6, the method used to decide if two position volumes intersect is the ISA-GJK method described in subsection III-B1.

A running thread uses registers, however thread registers are allocated from a global register pool on the GPU. The global register pool has a limited size. Using the NVIDIA Visual Profiler, a tool used to profile CUDA applications, it was found that the number of register used by this kernel was limiting the number of threads running simultaneously on the GPU. To solve this issue, we used the compilation option --maxrregcount=32 which limits the number of registers used to 32 by thread. This limitation of the register number might slow down the thread execution but more threads are Algorithm 3 Kernel computing conflict between k(t) and l(t)

1: k = "trajectory (blockId x × blockDim x + threadId x )" 2: l = "trajectory (blockId y × blockDim y + threadId y )" end if 10: end if running simultaneously. In the end, this option has reduced the computation time by 25 %.

V. EXPERIMENTS In order to compare the GPU and CPU implementations, we consider different conflict detection scenarios. We also describe the hardware used for the experiments.

We consider 11 sizes of scenarios, involving 15, 20, 25, 30, 40, 50, 60, 70, 80, 90 and 100 aircraft, with three levels of uncertainties (ε ∈ {1, 2, 3}). For each combination, 10 scenarios of aircraft converging to the center of the considered airspace volume were randomly built. For each scenario, speeds are chosen from 384 kn to 576 kn (i.e. 20 % variation around a typical speed of 480 kn). The nominal vertical speed for maneuvers is set to 600 ft min -1 . The aircraft initial positions are chosen on a 100 NM radius circle and are noised within a 20 NM-side square. The initial heading is also noised with a value chosen in [-1, 1] radians (≈ ±60 • ). In the vertical plane, aircraft are equally dispatched on 5 different levels from F L280 to F L320. A total of 11 × 10 scenarios are built. Figure 7 illustrates these dimensions.

For the each aircraft, m = 161 alternatives trajectories are considered. Each trajectory is a sequence of T = 150 position volumes with one position each τ = 10 s. These trajectories are the result of a heading change angle or Flight Level A comprehensive description of how these trajectories are built can be found in [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF].

For each scenario, three levels of uncertainties are defined. The size of the convex hulls approximating the possible positions of aircraft increases with the uncertainty level, creating more conflicts for the same scenario. A total of 11 × 3 × 10 = 330 scenarios were thus built and tested with two different approaches in the next section. These scenarios are available online at http://clusters.recherche.enac.fr. The available files might be useful for other researchers to compare their conflict detection method with ours.

The experiments were completed on a personal computer consisting of a quad-core Intel R i7-6700K equipped with 16 GB of memory and a Palit R GTX 1080 GameRock GPU. Both the GPU and the CPU are not overclocked.

VI. RESULTS

In this section we have tested our two algorithms on the scenarios described in the previous section.

A. Results for the CPU version

The conflict computation time is plotted on the Figure 8. The computation time ranges from 3 s for 15 aircraft to 149 s for 100 aircraft. The increase is consistent with the fact that n(n-1) 2 m 2 pairs of trajectories are compared. The computation time increases with the uncertainty level. When the uncertainty level increase, the position volumes increases and more pairs of trajectories are in conflict. As a consequence, more finegrained collision tests are executed. For example, using a 100 aircraft problem, when the uncertainty levels are 1, 2, 3, the numbers of conflicting pairs are respectively 4.6 × 10 6 , 7.7 × 10 6 and 8.8 × 10 6 .

B. Results for the GPU parallel version

For this version, we have measured two durations on Algorithm 2: the time to transfer the trajectory data to the GPU (first line to line 2) and the time to actually compute the conflicts (line 3 to last line). The duration of the complete algorithm is the sum of these two durations.

The transfer time is plotted on Figure 9. The duration of the trajectory data transfer to GPU is almost linear with the amount of data to transfer which is proportional to nmT .

The conflict computation time is plotted on Figure 10. As with the CPU version, the time increases proportionaly to n(n-1) 2 m 2 . However, the 100 aircraft problem that requires the comparison of 128,308,950 pairs of trajectories is handled in 1090 ms. For the 15 aircraft problem, the 2,721,705 pairs are processed in 27 ms. On average, the trajectory pairs are processed at a rate of 117,000 pairs per millisecond.

If we consider the transfer time and the conflict computation time, the GPU implementation is two order of magnitude faster. It is 90 times faster for the 15 aircraft problem and 140 times faster for the 100 aircraft problem. This difference in speed-up between the small and large problem is mainly due to the transfer time.

q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 50000 100000 150000 25 50 75 100 number of aircraft [-] conflict computation time [ms] uncertainty level q q q 1 2 3

Fig. 8. Conflict computation time using the CPU implementation. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 10 20 30 25 50 75 100 number of aircraft [-] transfer to GPU time [ms] uncertainty level q q q 1 2 3 Fig. 9. Transfer time from host to the device, the GTX1080 GPU. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 300 600 900 25 50 75 100 number of aircraft [-] conflict computation time [ms] uncertainty level q q q 1 2 3

Fig. 10. Conflict computation time using the GPU implementation.

VII. CONCLUSION AND FURTHER WORK

We have shown in this article that the time spent to detect conflicts could be reduced by two order of magnitude when we use a GPU parallel implementation. A previous paper ( [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF]) demonstrated that the optimization problem associated to conflict resolution can be solved in real time for medium size problems. This paper demonstrates that the detection phase required by the algorithms in [START_REF] Allignol | Comparing conflict resolution algorithms on large-size opensource 3dproblems[END_REF] can also be done in real time. This might enable the use of conflict resolution tools in real time.

This algorithm could be very useful to improve different levels of Air Traffic Management. At a short term level (30 min or less), it can accelerate the necessary detection phase to find alternative trajectories for conflicting situations involving many aircraft. It could also be useful in order to make tools representing dynamically conflicting zones. Air Traffic Controllers will eventually rely on tools to test alternative trajectories and it seems reasonable that these tools instantly show the effect of a chosen option. A fast detection tool able to handle complex uncertainty models can be very useful.

The next step of our research will be to focus on the trajectory prediction time which can also be parallelized and reduced. We also plan to integrate our algorithm in a ATC assistant tool. This integration should help building a reliable efficient tool able to handle realistic uncertainty models and offer real time answers to the controllers.

Fig. 1 .

 1 Fig. 1. In our trajectory prediction model, each position is a volume. This volume supposedly always contains the actual future position.

Fig. 2 .

 2 Fig. 2. The predicted trajectory is modeled as a sequence of positions. This figure shows such a sequence in the horizontal plane.

Fig. 3 .

 3 Fig. 3. The aircraft position volume is in green, its bounding box in blue, and the bounding box actually used in black.

Fig. 4 .

 4 Fig. 4. The aircraft position volume is in green, its bounding box in blue, and the bounding box actually used in black.

Fig. 6 .

 6 Fig. 6. The result of A ⊕ horizontal ) is the red shape.

4 :Initialize the time step t at 0 5 :

 45 while t < T and not C k,l do 6:

  end for 14: end for 15: return Conflict matrix C

Fig. 7 .

 7 Fig. 7. Geometry of conflict scenario generation.

  3: t = blockId z × blockDim z + threadId z 4: if AABB k,t and AABB l,t are not separated then

	5: 6:	if C k,l is false then if k(t) and l(t) are not separated then
	7:	C k,l = true
	8:	end if
	9:

More specifically, a missed collision is possible if, along one axis, the sum of the size of the two largest object is longer than two times the size of the cell.

Some objects are

NM large and the objects are all in a 200 NM-side square.

This happens if and only if O ∈ {a -b|a ∈ A, b ∈ B}.


David Mitton
email: david.mitton@ifsttar.fr

Edison Zapata

Hélène Follet

Mechanical loading at the organ level: which consequences for bones?

Keywords: bone loading, fracture risk, failure load, finite element model, bone strength

Previous studies have shown that failure loads are sensitive to loadings (e.g. direction, speed). The fracture risk could be defined by the ratio between the external loading and the failure load. The prediction of the failure load using finite element models is encouraging but the external loading considered are still simplistic.

Introduction:

The mechanical loading at the organ level can have at least two consequences. Firstly it affects the structural bone organization on the basis of the Wolf's law [START_REF] Frost | Skeletal structural adaptations to mechanical usage (SATMU): 1. Redefining Wolff's law: the bone modeling problem[END_REF]. Indeed the physiological loadings can be linked to the cortical thickness (e.g. in the proximal femur) and to the trabecular network (e.g. in the femoral head and neck). Secondly, the mechanical loading on a specific bone can induce failure. A better prediction of the fracture risk would be of interest for diagnostic purpose and to limit the consequences of an accident. Finite element models have been proposed to contribute to this goal.

Advanced finite element approaches reach a certain level of confidence during the past 10 years. The geometry can be derived from quantitative computed tomography [START_REF] Keyak | Prediction of femoral fracture load using automated finite element modeling[END_REF] or High-Resolution computed tomography to catch microarchitectural features in vivo for distal segments such as the distal radius or tibia [START_REF] Boutroy | In vivo assessment of trabecular bone microarchitecture by high-resolution peripheral quantitative computed tomography[END_REF]. The mechanical properties of the bone tissues can be derived from bone mineral density using quantitative computed tomography [START_REF] Duchemin | Prediction of mechanical properties of cortical bone by quantitative computed tomography[END_REF][START_REF] Kopperdahl | Quantitative computed tomography estimates of the mechanical properties of human vertebral trabecular bone[END_REF]. However further accurate assessment is still needed and researches are currently oriented towards this aim [START_REF] Poelert | Patient-specific finite element modeling of bones[END_REF]. These two input data (geometry and mechanical properties) are requested to build realistic bone models. Such specimen-specific models can predict the bone strength (2).

To assess the fracture risk from a mechanical point of view, the external load applied on the bone should also be considered [START_REF] Hayes | Biomechanical considerations of hip and spine fractures in osteoporotic bone[END_REF]. Our assumption is that the mechanical loading can be a key factor to predict the fracture risk. Thus the goal of this paper is to discuss the clues coming from previous studies on human femur, vertebrae and radius to assess the importance of the external loading and to suggest future works.

Sensitivity to loadings (directions, speed, …)

It has been shown on the femur from ex vivo studies that the failure load in the stance phase configuration compared to the lateral load configuration (on the greater trochanter) leads to extremely different results. For example Duchemin et al. obtained on 40 pairs of femurs 9000 N and 2500 N for the average failure respectively in the stance and lateral loading configurations [START_REF] Duchemin | Femoral fracture load and failure energy in txo loading configurations: an in vitro study[END_REF]. The load direction sensitivity was also assessed by modeling [START_REF] Wakao | The effect of impact direction on the fracture load of osteoporotic proximal femurs[END_REF]. In addition, it was recently showed that the failure strength can be affected by the speed of loading [START_REF] Juszczyk | The human proximal femur behaves linearly elastic up to failure under physiological loading conditions[END_REF][START_REF] Laporte | Influence of the velocity on the in vitro fracture of the femoral neck for lateral compression tests[END_REF].

On vertebra, previous studies showed that the location of the external load has a major effect on the failure load. For instance, Travert et al. showed that an anterior displacement of the load by as little as 1 cm decrease the vertebral strength by 50% on average [START_REF] Travert | Sensitivity of patientspecific vertebral finite element model from low dose imaging to material properties and loading conditions[END_REF]. This result suggests considering anterior bending when estimating vertebral fracture risk with finite element models. The load location has a crucial effect.

The failure load of the radius is also dependent on the loading. Edwards et al. reported that if the radius is loaded until failure as an isolated bone (e.g. ( 13)) the failure load is 1.5 to 3 times greater in comparison to intact wrist loading configuration [START_REF] Pistoia | Estimation of distal radius failure load with micro-finite element analysis models based on threedimensional peripheral quantitative computed tomography images[END_REF].

Whatever the bone considered (femur, vertebra or radius) these studies suggested that failure load depends on the loading: orientation, speed, location and distribution.

2. What are the loadings considered in finite element model to predict failure load? When using finite element models, the geometry (macrostructure and microstructure) is taken into account with an improved accuracy. The mechanical properties are derived from indirect assessment and are still a key point. Accurate estimation of the mechanical properties is needed including elasticity and failure criterion. But on top of that, the loading conditions have to be considered. The loading conditions considered in most of the finite elements approaches are based on simple quasistatic loadings (stance configuration on the femur, axial loading on the spine, axial loading on the radius) and not on multiaxial loadings nor on high speed loadings reproducing a fall. When the same boundary conditions as the experiments are applied in the finite element models, the bone strength is predicted from 75% up to 90% [START_REF] Varga | Validation of an anatomy specific finite element model of Colles' fracture[END_REF][START_REF] Pistoia | Estimation of distal radius failure load with micro-finite element analysis models based on threedimensional peripheral quantitative computed tomography images[END_REF].

How fracture risk is estimated from a mechanical point of view?

The bone strength (failure load) is a part of the fracture risk assessment from a mechanical point of view. Indeed the fracture risk (called fracture risk index or  factor, ( 7)) is defined by the ratio between the external loading and the bone strength. The external loading can be computed using some assumptions. The real loading applied on the bone is not precisely known. Musculoskeletal models have been developed to assess the joint loadings (e.g. ( 15)) but their predictions are still hard to validate due to the lack of experimental measurements. Most of the time simplified computation of the loading considers the subject height in case of a fall from a standing height (e.g. ( 16)). The estimation of the external loading might be a limitation of these methods.

What is the prediction capacity of the finite element model approaches?

The fracture risk index was used in clinical studies to assess fractured and non-fractured groups. In a recent study, finite element models were built for 100 postmenauposal women (distal forearm fracture) and 105 controls using high-resolution peripheral quantitative computed tomography. The odd ratio obtained from the finite element model (OR 1.99) is similar to the one based on the bone mineral density measurement using the same imaging modality (OR 1.86) [START_REF] Christen | Improved fracture risk assessment based on nonlinear micro-finite element simulations from HRpQCT images at the distal radius[END_REF]. Most of these simulations are based on a linear constitutive law for the bone. It was shown that plastic parameters could improve the separation of cases and controls by one third (OR 2.66) [START_REF] Christen | Improved fracture risk assessment based on nonlinear micro-finite element simulations from HRpQCT images at the distal radius[END_REF]. Another limitation is the simplistic loading cases as mentioned in Vilayphiou et al. [START_REF] Vilayphiou | Finite element analysis performed on radius and tibia HR-pQCT images and fragility fractures at all sites in postmenopausal women[END_REF].

5. What are the loading scenarios leading to fracture? Melton et al. Osteoporosis Int studied 100 cases of distal forearm fractures [START_REF] Melton | Assessing forearm fracture risk in postmenopausal women[END_REF]. Eighty-nine % of the fractures resulted from a fall from the standing height and only 15% from an axial compression.

However in a clinical setup the loading of the finite elements model is exclusively an axial compression. In addition to the previous observations it seems important to pay attention to the external loading that can be considered for the various bones. In a large number of cases multiaxial loading seems to be more realistic.

Can multiaxial loading affect damage?

The influence of the external loading at the organ level can be related to the fracture events, but before the fracture, the bone can be damaged. Very recently bone microdamage was assessed in cancellous bone specimens from the upper extremity of the femur according to the type of the external loadings [START_REF] Wu | In vivo microdamage is an indicator of susceptibility to initiation and propagation of microdamage in human femoral trabecular bone[END_REF]. The authors showed that multiaxial loading (representing a fall or an accident) combining compression and shear caused much more damage than either simple compression or shear and caused greatest stiffness loss than either simple compression or shear.

Conclusion

This paper focused only on the effect of the mechanical loading of bone at the organ level. The physiological effect of the mechanical loadings is well known since a long time (Wolf's law), and is responsible of the shape of the bones (e.g. femoral cortical thickness and femoral trabecular network).

The mechanical loading can affect the ultimate behavior of bones. Based on previous studies the following highlights can be listed:

1 -Bone strength is sensitive to loading (orientation, position, speed, distribution) 2 -In most of the studies quasi-static loadings (axial compression, stance configuration…) are considered to predict bone strength (in finite element models) 3 -Fracture risk from a mechanical point of view is computed from simplistic loadings 4 -Up to now fracture risk is moderately predicted from a mechanical approach 5 -The loading scenarios are not well known but seem to involve more complex external loading than currently considered 6 -Multiaxial loading can affect the damage initiation Knowing that the bone structure will be adapted to sustain the main loadings, the bone "optimization" will maintain the bone strength along the main loading pattern. But the bone strength could be reduced under more complex and unusual loadings. The bone strength with regards to these "exceptional" loading cases could be a key factor for the fracture risk prediction. These "exceptional" loading cases

has to be estimated (e.g. during a fall) and to be simulated, using subject-specific bone models, to assess whether they are key aspects of the fracture risk prediction. In addition bone strength is sensitive to the loading speed and should be considered in the future finite element models.