Vincent Zoma
email: vincentzoma3@gmail.com

Actors in the road transit of goods from Mali and Niger via Burkina Faso and regional integration of WAEMU

Keywords: Transport, regional integration, Burkina Faso, Mali, Niger, WAEMU

The international road transport of goods, which accounts for 90% of trade in West Africa, should be an essential sector for the strengthening of regional integration in the area of the West African Economic Monetary Union (WAEMU). This article therefore aims to analyze the roles of the main actors in the road transit of goods from Mali and Niger via Burkina Faso in relation to the process of regional integration. This research is essentially based on qualitative interviews with the main actors in the field and a quantitative survey carried out among road drivers in Mali and Niger involved in the transit of goods through Burkina Faso. Overall, the analysis shows that multi-sector integration institutions, the West and Central African Ports Management Association, chambers of commerce and shippers' councils are institutional actors that have missions to promote regional integration.

Truck drivers, for their part, are real players in regional integration from below. However, the security or control forces are above all those which, through their actions in the transport sector, constitute obstacles to the strengthening of the regional integration process.

INTRODUCTION

To be competitive, several countries in the world have opted to unite within the framework of regional integration. Thus, Africa is not on the sidelines of these groupings. African societies have always organized their space to ensure exchanges between them. In recent decades, African states in general and those of the West African Economic and Monetary Union (WAEMU) in particular, have made regional integration one of the main drivers of their economic progress. This integration aims to solve their development problems. WAEMU states have also recognized the importance of the road transport sector, which accounts for 90% of trade (WAEMU, 2017a), as a key sector for their development. Indeed, in many discourses it is not uncommon to hear: "transport and (therefore) development" (J. [START_REF] Debrie | From sentimentality to the landlocked State: circulation and coastal openings of the interior territories of West Africa[END_REF]. They also recognized that this sector is essential for the reinforcement of their integration thanks especially to the exchanges between the landlocked countries and the coastal countries.

For landlocked countries, T. Noyoulewa (2009) points out that given their landlocked situation, the foreign trade of these countries is influenced by the cost of transporting their goods. While transport costs account for 10% to 15% of the cost price of goods in countries with a maritime coastline, it is around 60% in certain countries such as Burkina Faso. Landlocked countries (Burkina Faso, Mali and Niger) are particularly affected by the problems of freight transport. The lack of direct access to the sea distances them from the world market and makes them dependent on neighboring countries. However, despite its situation as a landlocked country, Burkina Faso is a road transit country for the transport of goods from Mali and Niger. The transit of goods from these countries via Burkina Faso is therefore important in the context where all these countries are members of WAEMU.

Therefore, this article aims to analyze the roles of the main actors in the road transit of goods from these two countries via Burkina Faso in relation to the strengthening of WAEMU integration.

METHODOLOGIC APPROACH

The study combines the documentary and primary data collected mainly in three border municipalities concerned by the road transit of goods from Mali and Niger via Burkina Faso (map n°1).

Map 1: Mali and Niger transit data collection sites via Burkina Faso

The communes of Koloko, Faramana and Kantchari are entry and exit points for the road transit of goods from Mali and Niger via Burkina Faso. Surveys were carried out in these municipalities which house customs, police and gendarmerie checkpoints. The surveys (the sampling of which is based on data from the transport ministries) concern 5% of the total workforce (1,079) drivers of road freight transit via Burkina Faso from Mali (345) and Niger (734). Interviews were also conducted with the heads of the police, gendarmerie and customs brigades of these municipalities. We also spoke with the unions of road drivers, carriers, agents from the services in charge of the international transport of goods from Burkina Faso, Mali and Niger; as well as with the heads of the transport, infrastructure and land use planning departments of WAEMU to better understand the roles of the various actors in the integration process.

RESULTS AND DISCUSSIONS

This analysis addresses respectively international and regional organizations, national institutional actors and noninstitutional actors in the countries concerned.

International and regional organizations and international freight transport

Several actors are involved in the institutional chain of international freight transport at the global, continental and regional levels. These are multi-sectorial bodies that work globally and some transnational structures. Thus, at the global level, the World Trade Organization (WTO) deals with the rules governing trade between countries. The main function of the WTO is to promote smooth, predictable and free trade as much as possible (H. [START_REF] Hammouda | Transport and regional integration in Africa[END_REF]& WTO, 2018). At the level of the United Nations, the World Maritime Organization also intervenes in the international transport of goods. Its role is to establish and harmonize the regulations of the States in the fields of safety and security of ships, by setting construction standards and pollution prevention from ships.

Still at the supranational level, multinational firms such as Bolloré Africa Logistics and hydrocarbon companies are players that operate in several countries. After having signed agreements with governments, these firms work in the supply chain, particularly in certain ports or in the supply of goods to countries or in the operation or sale of fuel used for the international transport of goods (J. [START_REF] Lombard | Transport corridors in Africa and territorial integration in question[END_REF]. Alongside them are very often banks and insurance companies. Indeed, according to K. [START_REF] Adegbe | The role of the Burkinabe council of shippers in the transport of imported goods[END_REF] the banker and the marine insurer intervene in the transaction for the financing or settlement of the commercial transaction. Insurance companies provide coverage for the risk associated with the movement of goods. Among all the actors who intervene at the supranational levels, the multi-sectoral integration organizations are the true promoters of regional integration. At the level of the African continent, the African Union (AU) aims at the integration of the continent from the regional economic communities. In West Africa, ECOWAS and WAEMU were created to promote regional integration so that they work for the facilitation of exchanges in the transport of goods with in particular the promotion of free movement, the adoption of texts for international road transport, the creation of juxtaposed checkpoints and above all the establishment of an observatory of abnormal practices in the hallways.

Beyond these multi-sectorial integration bodies, there are some transnational associations working in the international transport of goods. For example, the Maritime Organization of West and Central Africa is a structure that works in the international transport of goods. It is responsible for promoting cost-effective, secure maritime transport services, working to build the capacity of the actors, sustainable financing of the maritime transport industry as well as the facilitation of transport in transit to landlocked Member States. One of these specialized structures is Union of African Shippers Councils.

Apart from this organization, it is necessary to take into account the Port Management Association of West and Central Africa. This (intergovernmental) institution plays a role in the pooling of skills and reflections on the future of ports on the West African coast : " its main objectives mark the political will for consultation, harmonization, integration, in a competitive industry,, underpinned by the sovereign discourses of independent states" (J. Lombard and al., 2014, p.248). Our interviews with shippers from Burkina Faso, Mali and Niger also enabled us to learn of the existence of an association of shippers from landlocked countries: Association of Transport Sector Actors in Landlocked Countries of West and Central Africa (ATPSALC/WCA). It aims to establish a consultation framework on transport and logistics for the promotion of foreign trade in landlocked countries of West and Central Africa (ATPSL/AOC, 2008). This association would be an asset for the defense of the interests of landlocked countries vis-à-vis coastal countries. Periodic meetings are held between the members for a synergy of action and for more efficiency. But beyond this asset, an analysis of the establishment of this association leads us to think that the creation of this association could reveal the shortcomings of regional integration in the international freight transport sector. If all the member states of the economic integration communities applied the regional conventions, the reasons that prevailed for the creation of this association will no longer be relevant. This association can then, on the one hand, be considered as proof of the weak involvement of the various actors in the strengthening of regional integration. But on the other hand, it can be considered as an asset for regional integration because it aims to work towards compliance with conventions on the facilitation of international transport of goods and to offer a framework for meetings and exchanges between actors from landlocked countries. Thus, at the global or regional level, multi-sectorial integration organizations such as the AU, ECOWAS, WAEMU and the Port Management Association of West and Central Africa are especially true promoters of regional integration in international freight transport. The analysis of the roles of road transit actors from Mali and Niger via Burkina Faso must also take into account the main actors operating in these countries.

Multiple institutional actors at the national scales

At the WAEMU level, ministries of transport, chambers of commerce, shippers' councils, transporters' unions and control forces are institutional actors working in the road transit of the Mali and Niger via Burkina Faso.

Ministries of transport: weak players in regional integration

In a vast landlocked country like Mali or Niger, the competitiveness of their economy depends on the efficiency of the transport system. In this context, the ministries in charge of transport of these two countries with regard to their specifications have essentially the mission of implementing national policies in the field of transport. In Mali, according to the AFDB (2014), the main responsibilities of the ministry in charge of transport in the road transport sector are, among others:

-The development of types of transport, the development and implementation of measures aimed at ensuring the internal and external opening up of the country; -The design, construction and maintenance of roads;

-The development and control of the application of regulations in the fields of topography and cartography;

-Research and experimentation in the field of public works;

-The development and implementation of rules relating to traffic and road safety, etc.

In Niger, the Ministry of Transport in the road sector is mainly responsible for:

-Designing and implementing policy and developing the national transport plan;

-Designing and building basic infrastructure;

-Developing and apply transport regulations and control the operation of transport infrastructure (roads, dry ports, bus stations, etc.);

-Check the qualification and control the educational establishments for driving (the case of driving schools), repair and automotive expertise; -Mobilize sufficient human, financial and material resources for the implementation of transport policies and strategies;

-Design and implement a national program for the modernization of means of transport as well as their rational management (M.T., 2017b).

The main terms of reference of these ministries of transport indicate that they make very little reference to a mission to facilitate the transport sector in connection with regional integration. However, States ratify conventions at the regional level. In addition to the ministries in charge of transport, the role of chambers of commerce and shippers' councils in the international transport of goods should be addressed.

Chambers of commerce and shippers' councils: players facilitating regional transport

Chambers of commerce and shippers' councils are involved in the international transport of goods. The Chambers of Commerce are the national guarantor of the Inter-State Road Transit Convention (TRIE). The TRIE is one of the key conventions for the implementation of regional integration in the WAEMU space. As such, the Chamber of Commerce receives and manages the TRIE Guarantee Fund.

As for shippers, they are represented by companies on shippers' councils. These are the Burkinabe Council of Shippers, the Nigerien Council of Transport Users and in Mali the Malian Council of Shippers and the Malian Council of Road Carriers. According to our interviews with the managers of these structures, their explanations corroborate those of H. [START_REF] Cissé | Terms of reference for the construction of a dry port in Kayes and Sikasso in Mali[END_REF], M. [START_REF] Sounouvou | in Logistics Transport[END_REF] and USAID (2010) according to which, the mission of the shippers' councils consists, among other things, of researching, studying and implementing all measures making it possible to improve the efficiency of transport. The establishment of these structures should enable the various economic operators in the transport sector to better assess their needs and problems and to work effectively to resolve these problems in conjunction with the administrative authorities.

Overall, the main objective of setting up these structures is to allow shippers to group together, to combine their energies in order to build power, a negotiating force to obtain at the lowest cost and in the best possible delays of efficient transport services. They aim to promote and encourage an integrated maritime development program in the region. Chambers of commerce and shippers' councils participate also to the facilitation of regional transport because they house some representatives of the Organizations for the fight against Abnormal Practices (OAP) on the corridors of the regional space.

Like the chambers of commerce and structures of shippers who work for the interests of shippers in landlocked countries, we are interested in trade unions in the goods transit sector from Mali and Niger via Burkina Faso.

Absence of trade union coordination: a handicap for regional integration

The main trade unions operating in the road transport of goods in Burkina Faso, Mali and Niger are partly made up of carriers. These are the Union of Goods Transporters of Niger (UGTN), the Organization of Road Transporters of Faso (ORTRAF) and the National Union of Road, Urban-Interurban and International Transporters of Mali (NURUIIT-Mali), etc. In addition to carriers, there are also road driver unions. These are the Union of Road Drivers of Burkina Faso (URDB), the National Union of Road Drivers and Drivers of Mali (NURDD-Mali), the National Union for the Coordination of Road Drivers and Carriers (NUCRDC), the National Syndicate of Road Drivers of Niger (NSRDN), etc.

The road transport unions are responsible for finding goods for trucks, to represent and defend the interests of all transporters in their country (O. Sigué, 2015). Thus, when it makes an inventory of the available freight and distributes it, the syndicate plays an intermediary role. However, there is no union of trucking unions at the regional level to defend the interests of their members. Each union has a national competence, even if the various actors at the national level often meet during the meetings organized by WAEMU.

Despite the holding of these meetings, all the unions of carriers, road drivers and even freight forwarders confirm the persistence of the non-application of transport facilitation agreements in the regional space. However, the establishment of coordination at the regional level would facilitate the exercise of the profession of international freight drivers because currently these intermediary actors only have national competence. In addition to this shortcoming, transport unions and even drivers have practices contrary to the promotion of regional integration. Instead of defending the interests of their members, these unions levy mandatory taxes on drivers at each passage.

The absence of regional coordination in this sector favors these abusive and compulsory levies which are in no way assets for the strengthening of regional integration. The Directive N°08/20005/CM/WAEMU and Decision No. 15/2005/CM/WAEMU of December 16, 2005 indicate that the only forces authorized to carry out checks on inter-State roads are Customs, the Police, the Gendarmerie, Waters and Forests and the sanitary, phytosanitary and zoo-sanitary control services (WAEMU, 2006). The taxes imposed by the truck drivers' unions are deplorable not only by their illegal character but above all by their compulsory character; because union membership cannot be compulsory. The latter, who should be working to defend the interests of drivers, are themselves a real problem.

They thus hinder the ongoing regional integration process.

Apart from all these structures at the level of the countries which are actors in the road transit of goods from Mali and Niger via Burkina Faso, it is necessary to analyze among the institutional actors the role of the control forces.

Control Forces and Obstacles to Regional Integration

According to Directive No. 08/20005/CM/WAEMU and Decision No. 15/2005/CM/WAEMU of December 16, 2005, the only forces authorized to carry out checks on the inter-state highways in the region are customs officers, police, gendarmerie, water and forests, sanitary, phytosanitary and zoo-sanitary control services. Their presence on the corridors should make it possible to ensure the safety of transport players and to ensure that these players comply with the various regulations in force in the regional or national space. But, in general, these actors in practice constitute real obstacles to regional integration. According to a study by IMF and al. (2008), the customs of the landlocked countries of the WAEMU remain focused solely on their mission of collecting duties and taxes.

Their mission of trade facilitation (which is closely related to regional integration) is therefore much neglected. USAID ( 2004) reveals that the two major customs problems are corruption and lack of clarity. At customs stops in the interior of the country (Mali for example) and at border crossings, customs officers collect state fees which are fair as well as their negotiated "facilitation fees" which vary around 11,500 FCFA per transaction. Before arriving at the border, a trucker can pass up to five customs posts inside the country. At each stop, customs officers pretend to check documents that may or may not exist in order to collect their facilitation fees. When we asked the question to the heads of customs posts in the three border communes of Burkina Faso with Mali and Niger, they refuted these facts. Yet 80% of our respondents complain of harassment by the security forces, citing customs, the police and the gendarmerie.

In addition, in the localities crossed, drivers are very often obliged to pay parking or territorial crossing taxes and tolls. It should be pointed out that with the exception of toll charges which are a contribution for the construction and maintenance of roads, the other two taxes are illegal. During our interviews with the truck drivers' unions, some drivers and their apprentices told us that some municipalities force them to pay taxes (1000 FCFA for example for parking in a municipality in Burkina Faso).

In general, according to the 26th report of the OAP (WAEMU, 2017b), on average, on a corridor of the WAEMU space, a driver with a truck in good standing is checked 20 times per trip, i.e. (07) times more than the community standard which is 3 checkpoints per corridor, i.e. two (02) checkpoints over 100 km, i.e. 1 checkpoint over every 50 km. Still according to this report, the average of illicit collections along a corridor of the WAEMU space is 26,562 FCFA. One of the consequences of road harassment in the field of road transit is undoubtedly the increase in the cost transport which does not contribute to the strengthening of integration. Regarding landlocked countries, B. Steck (2015, p.455-456) "it is urgent, when it is officially stated that the cost of transport can represent up to

Ouagadougou,2015, 338 pages. Noyoulewa, T. A. Road Transport and Integration of WAEMU States.

After analyzing the role of these institutional actors, it is important to take into account certain non-institutional actors who intervene in road transit from Mali and Niger via Burkina Faso.

Road drivers, non-institutional actors, real actors of "regional integration from below"

Freight forwarders, coxers, carriers and drivers are non-institutional actors of road transit in landlocked WAEMU countries. The freight forwarder is a professional responsible for carrying out the formalities allowing the goods to cross the border of the country. As for the coxer, he is also a non-professional road transport intermediary. He is responsible for finding the goods for the drivers or connecting the driver with the owner (the shipper). As far as the carrier is concerned, he is responsible for physically moving the goods from point A to point B. All these noninstitutional actors are then important for the international transport of goods. They contribute to transnational exchanges. Between different countries. They are essential for the strengthening of regional integration even if in practice, the analysis of their role, and interviews with the unions of these actors, they are above all not currently concerned for the regional objective. Alongside these non-institutional actors, it is important to take into account road drivers because they are the ones most often encountered on corridors across all WAEMU countries. Through their multiple trips on the corridors of the regional space and by staying in several localities, these drivers are real actors of regional integration. Indeed, one of the important elements in the analysis of regional integration is undoubtedly languages. In the context of international transport, languages are powerful tools for regional integration because, in addition to their ability to bring people together, they promote commercial exchanges between populations. According to the results of our surveys, two main international languages (French and English) are used in road transit from Mali and Niger via Burkina Faso (Graph 1).

Graph 1: International languages spoken by drivers

Source: fieldwork, V. Zoma

Graph 1 reveals that 87.30% of drivers involved in road transit from Mali and Niger through Burkina Faso have French as their first international language, followed by English (12.70%). These results can be explained by the fact that 85.5% of drivers in transit in Burkina Faso come from WAEMU member countries which share French as an official language. The 14.5% are nationals who are only members of ECOWAS (outside WAEMU). Those who speak English come from Ghana which a country is hosting a port used by Mali and Niger. These results

show that there is a close relationship between languages and the ongoing regional integration process in the WAEMU space (where French is the first international language spoken). This situation is justified beyond the traditional process of regional integration (process qualified as integration from the "top" or through regional institutions), by the colonial heritage maintained by the political authorities of the new independent States. Ghanaian drivers speak English because Ghana is also a former British colony. As for the rate of 7.30% of drivers who do not speak any international language, the low levels of education of drivers (according to our study, 61.8% of drivers have a low level of education) could be an explanation. But beyond international languages, we are entitled to wonder whether local languages are not real factors of regional integration (graph n°2).

Graph2: National languages spoken by drivers in Burkina Faso

Source: fieldwork, V. Zoma 45.50% of drivers speak Dioula and 45.50% speak national languages not spoken in Burkina Faso.

Graph 3: National languages spoken by non-Burkinabe drivers

Source: fieldwork, V. Zoma

Graph n°3 shows that among those who do not speak languages known in Burkina Faso, 36.40% speak Hausa followed by Bambara (25.5%), l'Asanté (14.50%) and Ewe (12.7%) in fourth position. These languages are spoken respectively in Niger, Mali, Ghana and Togo. The ports of Ghana and Togo are used to open up Mali and Niger so that the probability of meeting drivers from or to these countries is high. Thus, the results presented through graphs n°2 and n°3 make it possible to conjecture that language is an essential factor of regional integration. Indeed, Dioula is the national language of Burkina Faso spoken by the drivers involved in road transit from Mali and Niger via Burkina Faso. Also, we simultaneously observe that this group of drivers has more than 20 years of experience in the profession (according to our study). This is partly explained by the fact that, when the driver 45,50% 21,80% 45,50% 0% 1,80% 3,60% 0,00% 10,00% 20,00% 30,00% 40,00% 50,00% Non-Burkinabe national language Mooré Dioula Goulmancema Fulfuldé Others Percentage Language spends several years in the profession, the probability that he understands the local languages is high. Dioula is also the most spoken language because it is a commercial language which facilitates trade within the region. The example of Dioula proves that local languages are a factor of regional integration. Our results corroborate those of P. Somé (2010) according to I. [START_REF] Igué | shared development spaces[END_REF]. The author relates that: Dioula has some similarities with English in terms of commerce and communication. There was a time when we didn't want to hear about it, then another when we can't live without it. The lesson to be learned is that we cannot integrate peoples, their cultures and activities without renunciation or sacrifice. Language is the first vector of integration. In the case of West Africa, we integrate at the top, in English or French, convenient languages for writing texts. However, ''integration from below'' goes through local languages (I. Igué, 2010, p.77).

The example of the Dioula shows that language would be a key factor in regional integration from below in the WAEMU space.

CONCLUSION

It emerges from this analysis that, in addition to the multi-sectorial integration institutions and the West and Central African Ports Management Association, which have a mission to promote regional integration at the supranational levels, most of the national institutional actors involved in road transit, work in one way or another to strengthen regional integration. This is done through certain transport facilitation missions, the exercise of their professions and especially thanks to languages. Truck drivers who regularly travel through the countries of the region are above all key players in regional integration. They thus participate in the structuring of the regional space thanks to languages but also and above all thanks to the goods they transport. However, the study reveals that the ministries in charge of transport and a large part of the intermediary actors give low priority to regional integration. The security or control forces, together with the transporters' and drivers' unions, are currently the main obstacles to strengthening regional integration in road freight transit. However, the transport sector should contribute to strengthening regional integration, particularly for the three WAEMU landlocked countries of Burkina Faso, Mali and Niger.


This work is part of an artistic-research residency where composer Aaron Einbond seeks to apply audio descriptor analysis and corpus-based synthesis techniques to the spatial manipulation of instrumental radiation patterns for projection with a compact spherical loudspeaker array. Starting from a database of 3D directivity patterns of orchestral instruments, measured with spherical microphone arrays in anechoic conditions, we wish to derive spatial descriptors in order to classify the corpus. This paper investigates the use of spherical cross-correlation as a similarity measure between radiation patterns. Considering two directivity patterns f and g as bandlimited, square integrable functions on the 2-sphere, their correlation can be computed from their spherical harmonic spectra via a spatial inverse discrete Fourier transform. The magnitudes of these Fourier coefficients provide a rotation-invariant representation of the functions on the sphere. One can therefore search for the transformation matrix m, in the 3D rotation group SO(3), which maximizes the cross-correlation, i.e. which offers the optimal spherical shape matching between f and g. The mathematical foundations of these tools are well established in the literature ; however, their practical use in the field of acoustics remains limited and challenging. In this study, we apply these techniques to both simulated and measured radiation data, attempting to answer a number of practical questions : How does the similarity measure behave when f and g are not rotated cousins ? How can we adapt the cross-correlation formalism established for complex-valued harmonics to real-valued harmonics, as the latter are predominantly used in the field of Ambisonics ? Can we compute the correlation of spherical spectra of different bandwidths ? What is the impact of the finite sampling distribution used for integration on the SO(3) space ? And most importantly, is the cross-correlation an efficient measure for the classification of 3D radiation patterns ?

Introduction

Spherical correlation is a tool that allows the evaluation of similarity between radiation data on the sphere based on its spherical harmonics expansion. This is well discussed in the mathematical literature and widely used for 3D shapematching in the field of image processing. The state of the art is introduced in [START_REF] Kostelec | FFTs on the Rotation Group[END_REF], building on the foundation of the spatial Fourier transform [START_REF] Driscoll | Computing Fourier Transforms and Convolutions on the 2-Sphere[END_REF]. However, so far, the practical use of spherical correlation in the field of acoustics remains limited and challenging.

Guillon [START_REF] Guillon | Individualisation des indices spectraux pour la synthèse binaurale : recherche et exploitation des similarités inter-individuelles pour l'adaptation ou la reconstruction de HRTF[END_REF][START_REF] Guillon | Head-Related Transfer Function reconstruction from sparse measurements considering a priori knowledge from database analysis : a pattern recognition approach[END_REF] uses spherical correlation to compute similarities between spatial frequency response surfaces (SFRS), and later to clusterize a dataset of SFRS. This work is probably the most detailed usage of spherical correlation in spatial audio thus far. Moreau [START_REF] Moreau | Étude et réalisation d'outils avancés d'encodage spatial pour la technique de spatialisation sonore Higher Order Ambisonics : microphone 3D et contrôle de distance[END_REF][START_REF] Moreau | 3D Sound Field Recording with Higher Order Ambisonics -Objective Measurements and Validation of Spherical Microphone[END_REF] employs spatial correlation to assess qualitatively a directivity model of spherical microphone arrays against measured data. However, his work does not operate in the spherical harmonics domain and does not apply rotational matching. Similar to our study, Hohl and Zotter [START_REF] Hohl | Similarity of musical instrument radiation-patterns in pitch and partial[END_REF][START_REF] Zotter | Analysis and Synthesis of Sound-Radiation with Spherical Arrays[END_REF] use the spherical correlation coefficient as a similarity measure of radiation patterns of musical instruments. They examine whether different partials at the same frequency, but originating from different played pitches, exhibit similar radiation on a given instrument. This is promising work, but their short paper does not provide much detail.

Our study is both motivated and facilitated by the availability of a large database of anechoic measurements of acoustic instrumental radiation data produced at the Technische Universität (TU) Berlin [START_REF] Weinzierl | A Database of Anechoic Microphone Array Measurements of Musical Instruments -Recordings, Directivities, and Audio Features[END_REF][START_REF] Shabtai | Generation and analysis of an acoustic radiation pattern database for forty-one musical instruments[END_REF]. With radiation patterns for 41 modern and historical orchestral instruments, the database suggests several questions : how do the radiation patterns of partials and registers of a given instrument correlate with each other, how do they correlate with those of different instruments, and how could these correlations be used to classify and navigate the database ?

2 Theoretical background 2.1 Cross-correlation on the sphere Given two shapes f and g, how do we find the rotation to align them best ? Let f be a square-integrable function on the unit sphere f ∈ L 2 (S 2 ). In the simple case, let us assume that g is a rotated version of f , i.e. f = Λ R (g) for some 3D rotation R. We denote Λ R the rotational operator defined such that ∀Ω ∈ S 2 Λ :

L 2 (S 2 ) → L 2 (S 2 ) Λ R (g) (Ω) = g R -1 (Ω) . (1)
We wish to find the rotation R. In the more general case, given the two patterns f and g, we wish to find the rotation that best aligns the two shapes on the sphere. This can be accomplished by evaluating the cross-correlation between the two functions

C R (f, g) = S 2 f (Ω) Λ R (g)(Ω) dΩ , (2)
and finding the rotation R that maximizes the above integral [START_REF] Kostelec | FFTs on the Rotation Group[END_REF]. This has a number of useful applications, e.g. in the field of 3-D shape-matching, also known as shape registration [START_REF] Kazhdan | Harmonic 3D Shape Matching[END_REF][START_REF] Kazhdan | Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors[END_REF][START_REF] Kazhdan | Symmetry Descriptors and 3D Shape Matching[END_REF][START_REF] Gutman | Shape Registration with Spherical Cross Correlation[END_REF][START_REF] Sorgi | Normalized Cross-Correlation for Spherical Images[END_REF][START_REF] Shen | Efficient Registration of 3D SPHARM Surfaces[END_REF]. How do we maximize Equation 2 ? Evaluating C R (f, g) for all possible rotations in the spatial domain is a timeconsuming task. Instead, we undertake it in the spatial Fourier domain. Since f and g are square-integrable on the sphere, we can write their Fourier expansions [START_REF] Driscoll | Computing Fourier Transforms and Convolutions on the 2-Sphere[END_REF] :

f (Ω) = N n=0 m=+n m=-n f m n Y m n (Ω) . (3)
Here we have further assumed that f and g are bandlimited, with their bandwidth B = (N + 1), N being the maximum order of the Fourier expansion. Y m n (Ω) are the spherical harmonic functions, and f m n are the Fourier coefficients of f . A well-known property of the (Euclidian) Fourier transform is that a translation in the time domain is interpreted as a phase shift in the frequency domain. For the Fourier transform in S 2 , this property means that the magnitudes of the Fourier coefficients are invariant under rotation : writing f n (Ω) for the n th frequency component of f

f n (Ω) = m=+n m=-n f m n Y m n (Ω) , (4)
the quantity |f n (Ω)| is invariant under rotation. Furthermore, the spherical harmonic basis functions of each order n transform among themselves under rotation according to

Λ R (Y m n (Ω)) = m =+n m =-n Y m n (Ω) D n mm (R) , (5)
where D n mm (R) is the Wigner-D function (see section 7). Equation 5is valid for complex-valued spherical harmonics. When considering real-valued spherical harmonics, a similar result holds, however involving the Wigner-d function instead of Wigner-D ; details can be found for instance in [START_REF] Ritchie | Fast Computation, Rotation, and Comparison of Low Resolution Spherical Harmonic Molecular Surfaces[END_REF] (Equations ( 16) to [START_REF] Moreau | Étude et réalisation d'outils avancés d'encodage spatial pour la technique de spatialisation sonore Higher Order Ambisonics : microphone 3D et contrôle de distance[END_REF]). Now, using relations 3, 4, and 5, it is possible to simplify 2 into :

C R (f, g) = N n=0 |m|≤n |m |≤n f -m n g m n (-1) m-m D n mm (R) . (6)
A complete demonstration can be found for instance in [START_REF] Kostelec | FFTs on the Rotation Group[END_REF]. Equation 6 allows us efficiently to compute the crosscorrelation by combining the Fourier coefficients of f and g. From this, we can now search for which rotation R maximizes C R (f, g).

In equations 5 and 6, the Fourier coefficients are rotated by means of the explicit formulae with the Wigner-D functions (or Wigner-d for real-valued harmonics) ; in our practical implementation, we rather use recurrence relations [START_REF] Ivanic | Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion[END_REF] as this appears to be computationally more efficient.

Normalized cross-correlation

The normalized cross-correlation [START_REF] Huhle | Normalized Cross-Correlation using SOFT[END_REF][START_REF] Sorgi | Normalized Cross-Correlation for Spherical Images[END_REF][START_REF] Guillon | Head-Related Transfer Function reconstruction from sparse measurements considering a priori knowledge from database analysis : a pattern recognition approach[END_REF][START_REF] Huhle | Normalized Cross-Correlation using SOFT[END_REF] is simply a variant of Equation 2 normalized by the energy of f and g, written

C R (f, g) = S 2 f (Ω) Λ R (g)(Ω) dΩ S 2 |f (Ω)| 2 dΩ • S 2 |g(Ω)| 2 dΩ . (7)
The numerator of this expression has already been developed in Equation 6. The denominator can be easily calculated thanks to Parseval's identity :

S 2 |f (Ω)| 2 dΩ = N n=0 m=+n m=-n |f m n | 2 .
3 Searching for the optimal rotation R A rotation in R 3 can be equivalently represented by : (a) a 3×3 rotation matrix, (b) a set of 3 Euler angles (or alternatively 3 Tait-Bryan angles), (c) a unit quaternion, also known as a versor, or (d) an axis-angle representation (i.e. a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis). The different properties -such as compactness, numerical stability, computational cost, singularity or gimbal lock, etc. -of these formalisms can be easily found in the literature [START_REF] Diebel | Representing Attitude : Euler Angles, Unit Quaternions, and Rotation Vectors[END_REF], as well as conversion formulae from one representation to another. We denote as SO(3) the 3D rotation group, also known as the special orthogonal group. SO(3) contains all rotations R about the origin of three-dimensional Euclidean space :

SO(3) = R ∈ R 3 | RR T = I 3 and det(R) = +1 ,
where I 3 is the 3 × 3 identity matrix. In other words, SO(3) is the subgroup of orthogonal matrices with determinant +1.

We need to explore the SO(3) space in order to determine argmax

R ∈ SO(3)
C R (f, g) that maximizes Equations 6 or 7. One possibility is to use a gradient descent technique. Gradient descent has been formalized on the SO(3) rotation group in [START_REF] Stein | Mathematical Models of Binary Spherical-Motion Encoders[END_REF][START_REF] Chirikjian | Rotational matching problems[END_REF]. The procedure requires approximation of the directional derivatives, with an adjustable small step size, in order to converge to the nearest local maximum. It might also be necessary to restart the algorithm from several initial values. Choosing the optimal step size or the initial value is not a trivial task. Therefore we have decided not to use a gradient descent, but instead to proceed with brute force sampling in SO(3).

Sampling the rotation group SO(3)

The rotation group SO(3) can be parameterized by quaternions, Euler angles, or orthogonal matrices. Sampling these different parameter spaces is not a trivial task, and they can generate sampling distributions of SO(3) with various properties. To solve our problem, we typically need a uniform and deterministic sampling scheme. The definition of "uniform" is subject to interpretation, but essentially the sampling grid should ensure both global coverage and local separation. This problem has been studied for instance in [START_REF] Mitchell | Sampling Rotation Groups by Successive Orthogonal Images[END_REF][START_REF] Yershova | Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration[END_REF][START_REF] Kuffner | Effective sampling and distance metrics for 3D rigid body path planning[END_REF]. An extensive analysis is beyond the scope of this article.

Parametrization by Euler angles

For the sake of simplicity, we will follow [START_REF] Kostelec | FFTs on the Rotation Group[END_REF][START_REF] Guillon | Individualisation des indices spectraux pour la synthèse binaurale : recherche et exploitation des similarités inter-individuelles pour l'adaptation ou la reconstruction de HRTF[END_REF] and use a regular sampling in terms of ZYZ Euler angles (α, β, γ) :

α j = γ j = 2πj 2B and β k = π (2k + 1) 4B , (8)
with 0 ≤ j, k < 2B. According to [START_REF] Kostelec | FFTs on the Rotation Group[END_REF], this sampling scheme is suitable for the analysis of bandlimited functions with bandwidth B. With this sampling, the size of the search space is 2B × 2B × 2B. Note that the "null" rotation (α = β = γ = 0) is not part of the sampling grid. Consequently, correlating a signal with itself will not yield α = β = γ = 0, but rather β = π/(4B). This is counter-intuitive but not problematic. To circumvent this, it has been proposed [START_REF] Guillon | Individualisation des indices spectraux pour la synthèse binaurale : recherche et exploitation des similarités inter-individuelles pour l'adaptation ou la reconstruction de HRTF[END_REF] first to rotate one of the patterns, f or g, by -π/(4B) around the y axis.

We should also note that the "true" rotation R might not be on the sampling grid. Therefore, we might only find an approximate solution R that should be close to R. The notion of "distance" in SO(3) will be discussed in section 3.1.3.

Angle and axis representation

Any 3D rotation can be represented by an angle-axis representation. This formalism is useful for visualizing the projective space in R 3 [START_REF] Yershova | Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration[END_REF] : each rotation is drawn as a vector with direction n (the axis of the rotation) and a magnitude corresponding to ϑ (the rotation angle). As can be observed graphically in Figure 1, the Hopf fibration sampling proposed in [START_REF] Yershova | Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration[END_REF] is highly uniform. Yet, for the remainder of this article, we will use the basic Euler sampling (Equation 8) for its simplicity and scalability (with respect to B). 3) sampling grids using the angle-axis representation. (a) Regular sampling of Euler angles (as in Equation 8) with B = 4, leading to (2B) 3 = 512 elements ; (b) Uniform incremental sampling using the Hopf fibration [START_REF] Yershova | Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration[END_REF] (with 576 sampling points). The color represents the magnitude of the rotation (from blue to red).

F I G U R E 1 -Visualization of SO(

Distance measure of two rotation matrices

Various functions for measuring distance between 3D rotations have been proposed in the literature. The usual (angular) distance between two rotation matrices R, Q ∈ SO(3) is given by [START_REF] Mitchell | Sampling Rotation Groups by Successive Orthogonal Images[END_REF][START_REF] Hielscher | Fast Summation of Functions on the Rotation Group[END_REF][START_REF] Deboy | Acoustic center and orientation analysis of sound-radiation recorded with a surrounding spherical microphone array[END_REF]

d 1 (R, Q) = acos 1
2 tr QR T -1 . This metric measures the angle of rotation needed to map the transformation R to the transformation Q, or equivalently the angle of rotation associated to the transformation QR -1 . Alternatively, we can interpret d 1 (R, Q) as a scaled Froebenius norm, since

d 2 (R, Q) = R -Q F = 2 √ 2 sin d1(R,Q) 2
, where

• F is the Frobenius norm defined as R F = tr (RR T ).

Huynh [START_REF] Huynh | Metrics for 3D Rotations : Comparison and Analysis[END_REF] presents a detailed analysis of various metrics, and demonstrates that many of them are functionally equivalent. Huynh concludes that the following metric, based on quaternions, is both spatially and computationally more efficient :

d(R, Q) = 1 -|q R • q Q |
, where q R is the unit quaternion corresponding to matrix R, and • denotes the inner product. This metric will be used for the remainder of this article.

Note that, while a distance metric in SO( 3) is useful, it must be handled with care in the context of this paper : consider for example an axis-symmetric pattern f , for example a cardioid in the y direction. Rotate f around the z-axis with R 0 ≡ (α 0 , β 0 , γ 0 ) = (π, 0, 0). Rotate f around the x-axis with R 1 ≡ (α 1 , β 1 , γ 1 ) = (0, 0, π). Both scenarios result in the exact same pattern g ; however, the two rotations differ, and their distance is d(R 0 , R 1 ) = 1 !

Numerical simulations 3.2.1 Impact of SO(3) sampling

As discussed in the previous paragraphs, the choice of the sampling grid for SO(3) might have an influence on the accuracy and efficiency of the maximization problem 7. For the sake of simplicity, we consider only the regular Euler sampling of equation 8. However, we investigate the use of "oversampling" i.e. we use a smaller step size in discretizing the Euler angles (α, β, γ). Instead of using 2B samples for each parameter, we use N ≥ 2B. This seems relevant as our scenario involves spatial functions with relatively low bandwidth (B = 5 if we consider the radiation patterns available in the TU Berlin database) compared to other authors (Kostelec [START_REF] Kostelec | FFTs on the Rotation Group[END_REF] typically presents results with B = 128). With such low bandwidth, the angular step size is very large (π/B = 36 • ), and therefore high angular misalignment might occur. Of course, oversampling the search grid results in higher computation time, but that is not the focus of this paper.

We run the following numerical simulation : (a) generate a random spatial function f with a limited bandwidth B, (b) generate a random rotation matrix [START_REF] Arvo | Random Rotation Matrices[END_REF] R in SO(3) and compute g = Λ R (f ), (c) sample SO(3) and search for the rotation matrix R that maximizes the cross-correlation, (d) compute the cross-correlation after rotational matching, i.e. the cross-correlation between g and Λ R(f ), (e) repeat the simulation for various samplings of SO(3), varying the oversampling factor N /(2B), (f) repeat the simulation for various bandwidths B. For each test case, we perform 10000 Monte-Carlo runs. The results are presented in Figure 2. For each simulated bandwidth B, it can be observed that oversampling the search space improves accuracy, allowing for a higher cross-correlation (closer to 1), and a smaller distance (closer to 0) between the expected and estimated rotation matrices.

F I G U R E 2 -Impact of oversampling the SO(3) search space.

Comparing two patterns of different bandwidths

The cross-correlation function in equation 6 assumes that the two patterns f and g have the same bandwidth. In practice, it might be desirable also to compare patterns with different bandwidths. We run the following numerical simulation : (a) generate a random spatial function f with a limited bandwidth B f = 11, (b) generate a random rotation matrix R and compute g = Λ R (f ), (c) reduce the bandwidth of g to B g ≤ B f by zeroing the higher order components, (d) compute the crosscorrelation after rotational matching, i.e. the cross-correlation between g and Λ R(f ) (e) repeat the simulation for various truncation bandwidth B g . For each test case, we perform 10000 Monte-Carlo runs. The results are presented in Figure 3. We observe that it is possible to compare two rotated cousins even if their bandwidths differ : when B f -B g increases, the cross-correlation decreases, which is expected, but the patterns are correctly aligned since the distance between rotation matrices d(R, R) is low. When the mismatch between B f and B g becomes very large (B f -B g ) > 6, the accuracy of the rotational matching decreases significantly.

F I G U R E 3 -Comparing two patterns of different bandwidths.

Comparing two arbitrary patterns

In the most general case, we are interested in comparing two arbitrary radiation patterns, not only rotated cousins. Such a scenario generally yields |C R(f, g)| < 1, but the rotational alignment can still be effective. In Figure 4, we simulate two arbitrary patterns, having different bandwidth, shape, and orientation of their "main lobe". We observe that f is properly aligned with g. The resulting correlation coefficient is C R(f, g) ≈ 0.45.

Visualization with multidimensional scaling

With an approach to pairwise spherical correlation in place, we can now examine applications to classify and navigate F I G U R E 4 -Example of rotational matching. Left : visualization on the sphere. Right : restriction on the horizontal plane. larger collections of 3D radiation patterns. One approach is to apply multidimensional scaling (MDS). MDS is used to translate information about the pairwise distances (or dissimilarities) among a set of n objects individuals into a configuration of n points mapped into an abstract Cartesian space. MDS is therefore a means of visualizing the level of similarity of samples within a dataset.

We begin with a proof of concept using the functions Y m n (Ω) as our dataset. First we compute the normalized crosscorrelation between all pairs Y m1 n1 (Ω) and Y m2 n2 (Ω). From this similarity matrix, we perform MDS and visualize the information map, choosing a dimensionality of 2 for simplicity of visualization and interpretation. The results in Figure 5 F I G U R E 5 -Left : normalized cross-correlation between all pairs Y m1 n1 (Ω) and Y m2 n2 (Ω). Right : resulting 2D MDS.

for maximum order N = 5 show that, within each order, the functions are highly correlated. For order N = 1, the three functions are even fully correlated, which is expected as these spherical harmonics are dipoles along the x, y, and z axes (i.e. they are exactly rotated cousins). The map clearly exhibits six clusters of samples, each cluster corresponding to a given order N . Orders 0, 1, and 2 are relatively close, while 3, 4 and 5 are maximally distant from the lower orders and from each other.

Application to measured data

We now apply the approaches presented in the previous sections to measured radiation patterns of musical instruments. Our source is the acoustic instrumental radiation database, containing 41 modern and historical orchestral instruments, made available by the TU Berlin [START_REF] Weinzierl | A Database of Anechoic Microphone Array Measurements of Musical Instruments -Recordings, Directivities, and Audio Features[END_REF][START_REF] Shabtai | Generation and analysis of an acoustic radiation pattern database for forty-one musical instruments[END_REF].

Radiation patterns of partials

To test the claim in [START_REF] Hohl | Similarity of musical instrument radiation-patterns in pitch and partial[END_REF] that instrumental partials with the same frequency have correlated radiation patterns, we compute a similarity matrix containing the pairwise correlation between partials of different played pitches. The matrix is organized in blocks each containing a chromatic scale, with each successive block representing a different partial for each played note. With this matrix arrangement, partials with matching frequency are located on the secondary diagonals. Figure 6 illustrates examples of results for clarinet and cello. It can be observed that different partials at the same frequency exhibit strongly correlated radiation, regardless of the played pitch from which they originate. Other findings are also in line with [START_REF] Hohl | Similarity of musical instrument radiation-patterns in pitch and partial[END_REF], however the results are less obvious. This might be partly explained by the lower resolution of TU Berlin database, leading to a lower bound for the spatial aliasing frequency.

Clarinet_modern_et_ff Cello_modern_et_ff

F I U R E 6 -First 6 partials of the clarinet in B-flat for pitches ranging from C4 to C5 (left) and of the cello for ranging from E2 to E3 (right).

MDS applied to TU Berlin data

Another for this work is to classify the TU Berlin database subsequent with based synthesis techniques : by organizing instrumental samples according to their spatial characteristics, the resulting low-dimensional can be used, with other audio descriptors, to navigate the sample corpus. We applied MDS as presented in section 4 to measured radiation patterns different instruments for various played pitches. An example is presented in Figure 7. The color code corresponds to the three categories of instruments proposed by Shabtai [START_REF] Shabtai | Generation and analysis of an acoustic radiation pattern database for forty-one musical instruments[END_REF]. In red : instruments with one expected radiation point ; this mainly includes the brass instruments. In blue : instruments with several expected radiation points, such as woodwind instruments, with sound radiated by the bell, the fingering holes, and the mouthpiece. In black : instruments with a full body radiating sound, such as string instruments. We observe that MDS based on spherical correlation allows us efficiently to segregate and organize the instruments, in accordance with their expected category.

Conclusion

In this paper, we discussed the use of cross-correlation on the sphere as a similarity measure for the classification of 3D radiation patterns. We showed that this tool can facilitate thorough analysis of directivity pattern similarities across partials of one instrument or between different instruments. However, spherical correlation can only capture rotational resemblance ; it might therefore be necessary also to consider energetic criteria when comparing radiation patterns. Future work will investigate other spatial descriptors for the classification of the 3D database of orchestral instruments. A further direction will be transformations of radiation patterns based on interpolation between correlated and rotated pairs, which we hypothesize will give more consistent results than interpolation between unrotated patterns, to be applied in the context of regression for a machine learning algorithm.

  U R E 7 -MDS applied to radiation patterns of the pitch D2 for various brass, woodwind, and string instruments.

Acknowledgment This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101019164 (ERC MusAI -Music and Artificial Intelligence : Building Critical Interdisciplinary Studies).

Annex : Wigner-D function

Considering Euler angles with the ZYZ convention, the rotation matrix R can be expressed as a rotation of α about the z axis, followed by a rotation β about the y axis, and finally a rotation γ about the z axis : R zyz (α,

and 0 ≤ α, γ < 2π and 0 ≤ β ≤ π. With this convention, the Wigner-D function D n mm (R), required to rotate complexvalued spherical harmonics, is written

where d n mm (β) is the Wigner-d function [START_REF] Risbo | Fourier transform summation of Legendre series and D-functions[END_REF].


Ali El

Olivier Brun

Application de la théorie des jeux à l'optimisation du placement et chaînage des VNFs

Keywords: Service Function Chain, Virtual Network Function, Slice as a Service, Game Theory, Optimization

La technologie Network Function Virtualization (VNF) vise à simplifier le déploiement des services réseau de manière flexible, automatisée et indépendante du fournisseur. Le déploiement de ces services nécessite l'allocation des Virtual Network Function -Forwarding Graphs (VNF-FGs), ce qui implique de placer et d'enchaîner les VNFs en fonction des requêtes des VNF-FGs. Dans ce travail, nous considérons le problème d'allocation hors-ligne des VNF-FGs pour améliorer l'utilisation des ressources du réseau afin de répondre à un maximum de demandes, et réduire les coûts. Etant donné une fonction coût non linéaire associée à chaque ressource réseau, nous formulons le problème comme un problème de routage non linéaire à chemin unique dans un graphe étendu. Ensuite, nous proposons un algorithme heuristique de routage inspiré de la théorie des jeux pour le résoudre. Nous montrons que cet algorithme converge et établissons son rapport d'approximation dans un certain nombre de cas. Les résultats expérimentaux obtenus pour différentes topologies de réseau et différentes fonctions de coût montrent que cet algorithme fournit des solutions de très bonne qualité avec des temps de calcul assez modestes.

Introduction

According to ETSI, a VNF Forwarding Graph (VNF-FG) is a directed graph of an ordered set of Virtual Network Functions (VNFs) that compose an end-to-end service [START_REF] Nfv | Network Functions Virtualisation (NFV): Architectural framework[END_REF]. The logical service graph described by a VNF-FG is decoupled from the network and has to be mapped to existing network capabilities. This implies not only fulfilling of the service requirements in terms of Quality of Service (QoS), but also considering the constraints of the underlying infrastructure. Two types of decisions have to be made: (a) where to run the VNFs, and (b) how to interconnect them in the physical network, taking into account specific VNFs ordering requirements. This problem is known as the VNF Forwarding Graph Embedding (VNF-FGE) problem and has been studied in the online context where requests to create network services are submitted one after the other [START_REF] Quang | Single and multi-domain adaptive allocation algorithms for VNF forwarding graph embedding[END_REF][START_REF] Amine | Shortening the deployment time of SFCs by adaptively querying resource providers[END_REF], and in the offline context where multiple VNF-FGs have to be embedded in the physical network at the same time [START_REF] Zhong | Cost-aware service function chaining with reliability guarantees in NFV-enabled inter-dc network[END_REF][START_REF] Guo | Cost-aware placement and chaining of service function chain with VNF instance sharing[END_REF].

Despite the previous efforts on the VNF-FGE problem, little considered the re-optimization of resources utilization on the physical infrastructure with non-linear cost functions. Thus, in this work, we study the offline joint routing and VNF placement problem with non-linear cost functions to reduce total costs and improve resource utilization under a physical infrastructure capacity constraints. We formulate the problem as a non-linear single-path routing problem in an extended graph. To solve this problem, we adapt a singlepath routing algorithm inspired by Game Theory which was proposed in [START_REF] Brun | A penalized best-response algorithm for non-linear single-path routing problems[END_REF]. Experimental results obtained for different network topologies and different cost functions show that this algorithm provides very good quality solutions with a rather modest computation time.

Problem Statement

We are given a physical network represented by a directed graph G = (V, E), where V is the set of nodes and E is the set of links. A subset D ⊆ V of the nodes have the required compute and storage resources to host VNF. These nodes will be called the function nodes in the following. They are used to execute one or more VNFs out of a set F of 𝐾 VNFs, where a function 𝑓 ∈ F is available at function nodes 𝑣 ∈ D ( 𝑓 ) ⊆ D. A function node corresponds to a logical entity representing reserved compute and storage capacity in a data center. Multiple VNFs may be deployed at the same function node, and a VNF may be replicated at different function nodes. The transmission links of the network and the function nodes are the critical resources in our problem and, in the following, we shall denote by R = E ∪ D the set of these resources.

We are also given a set W of traffic demands. Each traffic demand 𝑤 ∈ W is characterized by its source node 𝑠 𝑤 , its destination node 𝑡 𝑤 and the sequence F 𝑤 = 𝑓 𝑤 1 , 𝑓 𝑤 2 , . . . , 𝑓 𝑤 𝐾 𝑤 of network functions it must flow through in that prescribed order. The sequence is called the Service Chain associated with the traffic demand. The processing path of demand 𝑤 may therefore be decomposed into different stages, where stage 0 corresponds to the transmission of the original traffic volume from the source to VNF 𝑓 𝑤 1 (excluded), stage 1 represents the processing at VNF 𝑓 𝑤 1 followed by the transmission to VNF 𝑓 𝑤 2 , etc. We define 𝑏 𝑤 𝑘,𝑟 as the capacity required from resource 𝑟 for processing demand 𝑤 at stage 𝑘. Here, 𝑟 could be either a transmission link 𝑒 ∈ E or a function node 𝑣 ∈ D ( 𝑓 𝑤 𝑘 ). The problem at hand amounts to choosing a single path through the network for each traffic demand such that the function nodes are visited in the prescribed order and that a certain network cost function is minimized. We shall shortly discuss the network cost function to be optimized, but we would first like to emphasize that a first difficulty is related to the modeling of precedence constraints on the order in which the function nodes must be visited by each traffic request. We handle this constraint by constructing an expanded network. Due to the lack of space, we omit the discussion about expanded networks that can be found in [START_REF] Amine | Offline Re-Optimization of VNF Placement Decisions for Existing Network Slices -A Game-Theoretic Algorithm[END_REF].

Penalized Best-Response Algorithm

Suppose that we are given a set Π 𝑤 of candidate feasible paths through the expanded network for each demand 𝑤 ∈ W, a feasible solution for the joint routing and VNF placement problem is then defined as a vector 𝝅 = (𝜋 𝑤 ) ∈ Π, where 𝜋 𝑤 is the path assigned to traffic demand 𝑤 and Π = 𝑤 ∈W Π 𝑤 . The goal is to find a feasible solution 𝝅 ∈ Π that minimizes the network cost 𝐹 (𝝅) = 𝑟 ∈R 𝑦 𝑟 (𝝅) 𝜙 𝑟 (𝑦 𝑟 (𝝅)). Formally, the problem is as follows:

minimize 𝐹 (𝝅) = ∑ 𝑟 ∈R 𝑦 𝑟 (𝝅) 𝜙 𝑟 (𝑦 𝑟 (𝝅)) (OPT)
subject to:

𝜋 𝑤 ∈ Π 𝑤 , 𝑤 ∈ W. (1)
where 𝑦 𝑟 (𝝅) is the total amount of traffic flowing through resource 𝑟 and 𝜙 𝑟 () is the cost per unit of capacity of resource 𝑟 that may depend on the total traffic flowing on that resource. In order to solve the above problem, we propose a heuristic algorithm based on game theory. The idea is to view the traffic demands as the players of a non-cooperative game in which each player independently optimizes its own objective function. Starting from an arbitrary initial feasible solution, the algorithm then mimics the best-response dynamics of the game, that is, the players take turns in some order to adapt their strategy based on the most recent known strategy of the others.

Let us think of the traffic demands as the players of the game. The strategy of player 𝑤 ∈ W is the path 𝜋 𝑤 it chooses in the set Π 𝑤 , and a strategy profile of the game is a feasible solution to problem (OPT), that is, a vector 𝝅 ∈ Π. Given the strategy of the other players 𝝅 -𝑤 = (𝜋 𝑢 ) 𝑢≠𝑤 , we shall assume that the player 𝑤 seeks to solve the following problem:

minimize 𝜋 ∈Π 𝑤 𝑐 𝑤 (𝜋, 𝝅 -𝑤 ) = 𝑓 𝑤 (𝜋, 𝝅 -𝑤 ) + 𝑝 𝑤 (𝜋, 𝝅 -𝑤 ), (OPT-𝑤)
where the value 𝑓 𝑤 (𝜋, 𝝅 -𝑤 ) associated to path 𝜋 by player 𝑤 reflects the cost of this path, whereas the term 𝑝 𝑤 (𝜋, 𝝅 -𝑤 ) is a penalty term measuring the impact of player 𝑤's choice on other players. More precisely, we shall assume that the selfish and penalty objective functions of player 𝑤 are respectively as follows:

𝑓 𝑤 (𝜋, 𝝅 -𝑤 ) = ∑ 𝑟 ∈ 𝜋 𝑦 𝑤 𝑟 (𝜋) 𝜙 𝑟 𝑦 -𝑤 𝑟 + 𝑦 𝑤 𝑟 (𝜋) , (2)
𝑝 𝑤 (𝜋, 𝝅 -𝑤 ) = ∑ 𝑟 ∈R ∑ 𝑢≠𝑤 𝑦 𝑢 𝑟 (𝜋 𝑢 ) 𝜙 𝑟 𝑦 -𝑤 𝑟 + 𝑦 𝑤 𝑟 (𝜋) -𝜙 𝑟 𝑦 -𝑤 𝑟 ( 3
)
Given the strategies 𝝅 -𝑤 of the other players, the path 𝜋 minimizing 𝑐 𝑤 (𝜋, 𝝅 -𝑤 ) in problem (OPT-𝑤) is known as the best response of player 𝑤. The heuristic algorithm that we propose is an iterative algorithm that starts from an initial feasible solution 𝝅 (0) . At each iteration, the players update their strategies in a given order by computing their best responses to the strategies of the others. This algorithm has the following characteristics: (a) it converges in a finite number of steps, (b) If it ever reaches a global optimum of problem (OPT), it returns this optimal solution, (c) it computes an optimal solution of problem (OPT) for linear resource costs, that is, when the functions 𝜙 𝑟 () are constant functions 𝜙 𝑟 (𝑦) = 𝑎 𝑟 and (d) its approximation ratio is

2 1 𝑑+1 -1 -(𝑑+1)
when the functions 𝜙 𝑟 () are polynomials of degree 𝑑, that is, 𝜙 𝑟 (𝑦) = 𝑑 𝑛=0 𝑎 𝑟 ,𝑛 𝑦 𝑛 . Due to lack of space, all proofs are omitted and can be found in [START_REF] Amine | Offline Re-Optimization of VNF Placement Decisions for Existing Network Slices -A Game-Theoretic Algorithm[END_REF].

Performance Evaluation

In this section, we experimentally evaluate the performance of the penalized best-response algorithm. We first describe the different types of objective functions used to solve the penalized best-response algorithm, then we present the numerical results for the different topologies evaluated under the different types of objective functions.

Given the cost per-unit of capacity of resource 𝑟 function 𝜙 𝑟 , we denote by Φ 𝑟 = 𝑦 𝑟 𝜙 𝑟 (𝑦 𝑟 ) the cost function of resource 𝑟. We consider four types of cost functions: linear (Φ 𝑟 = 𝑎 × 𝑦 𝑟 ), increasing piecewise linear, quadratic (Φ 𝑟 = 𝑦 2 𝑟 /𝑐 2 𝑟 ) and the Kleinrock Function 𝑀/𝑀/1 (Φ 𝑟 = 𝑦 𝑟 /(𝑐 𝑟 -𝑦 𝑟 )). In contrast to linear cost functions that assume constant costs per unit capacity, non-linear cost functions allow the costs per unit capacity to increase as the traffic 𝑦 𝑟 increases. This allows to achieve a better load distribution.

In order to evaluate the performance of the penalized best-response algorithm, we use five network topologies (see Table 1) collected from the IEEE literature and from The Internet Topology Zoo [START_REF] Knight | The internet topology zoo[END_REF]. We expand the original topology with three function nodes that can host two types of VNFs out of three. Then, for each topology, we generate 100 random instances with 25 traffic demands each. We report in Tables 1 and2 the performance of the penalized Best Response (BR) algorithm for different cost functions under distinct topologies. We evaluate the algorithm performance by calculating the relative gap between the penalized BR solution and the optimal solution from Gurobi. Following the obtained results, the penalized BR algorithm provides very good quality solutions for the different cost functions with impressive computation time. Its average relative gap to the optimal solution is under 5%, and it is almost negligible for small topologies such as CESNET. The average execution time of the algorithm depends on the size of the topology and on the cost function used. Nevertheless except for the PWL cost function where the MILP solver is known to be efficient and quick at solving, the solver is significantly slower for the other more complex cost functions. In the case of the quadratic cost function, the maximum relative error of the BR algorithm over the 100 instances is 3.725%. However, if we look at the computing times in Table 2, we observe that the BR algorithm is at most 16 times faster and at worst 6.3 times faster than the solver. Here, the standard deviation has been omitted because the runtime values for the different instances for each topology were similar. It is interesting to note that for the 𝑀/𝑀/1 cost function, the solver has reached its time limit that has been set to 5 minutes, while the BR algorithm takes less than 10 sec to execute with a solution as close as 1.43% to the solver. While PWL approximations can sometimes be considered for non-linear functions, it is not evident how to choose the intervals to build an approximation for non-linear cost functions.

Conclusion

In this work, we have considered the offline VNF placement and chaining problem, assuming that a non-linear cost function is associated to each network resources. With respect to previous works, the main originality of the considered model is that the cost per unit capacity of a resource is not constant, instead it grows with its utilization rate, which is an essential feature to achieve a better load distribution. We have formulated the problem as a single-path routing problem in an extended network and adapted an existing game-theoretic algorithm to solve it. Our numerical results suggest that the algorithm provides near-optimal solutions in substantially lower computing times than the solver for non-linear cost functions.

Acknowledgment

This work was supported by the French DGA project ONSET.

TABLE 1 :

 1 Relative gap (% ± std) to the optimal solution for different topolgies.

	Formatting a submission for AlgoTel
	Topology	# Nodes # Links		PWL	Quadratic		M/M/1
	AARNET	19		46	0.71 ±0.65	1.81 ±0.85	1.43 ±1.76
	ARPANET	25		56	1.15 ±0.5	3.725 ±1.55		1.5 ±0.79
	NSFNET	13		30	0.42 ±0.49	1.67 ±1.12	0.31 ±0.32
	IBM	18		48	1.7 ±0.8	2.85 ±0.98	0.53 ±0.71
	CESNET	10		18	0.051 ±0.003 0.047 ±0.008 0.070 ±0.056
	Function Costs	AARNET BR Gurobi BR Gurobi BR Gurobi BR Gurobi BR Gurobi ARPANET NSFNET IBM CESNET
	PWL	7.1	1.2	8.09	1.72	4.93	0.72	6.83	0.9	1.96	0.07
	Quadratic	7.89	126	8.89	124	6.3	40	8.27	109	2.33	0.3
	M/M/1	8.17	300	9.84	300	7.52	300	9.17	300	2.79	300

TABLE 2 :

 2 Average computing times (seconds) for the 100 problem instances.


MD Pierre-Marie Chiaroni

MD Kévin Premat

MD Eimad Shotar

MD Mehdi Drir

Hugo Trebern

MD Adrien Beth

Baptiste Bonnet

MD Jugurtha Mathout

MD Raphaël Bonaccorsi

MD Laetitia Morardet

MD Évelyne Cormier

MD Jacques Chiras

MD, PhD Frédéric Clarençon
email: frederic.clarencon@aphp.fr

Frédéric Pr

Clarençon

Extraction of Cement Leakages and Malpositioned Spindles Complicating Percutaneous Interventions: Why, When and How?

Keywords: Cementoplasty, Vertebroplasty, Intraoperative Complications, Foreign Bodies CBCT: Cone Beam Computed Tomography -CT: Computed Tomography -IRB: Institutional Review Board -PMMA: Poly-methyl-methacrylate

Key points:

-Soft tissues cement leakages or spindles malposition are a non-rare occurrence during cementoplasty, and may cause technical failure and/or chronic pain -Most soft tissue cement fragments and malpositioned spindle can easily be extracted using simple percutaneous techniques

Abstract

Objectives

Cement leakages in soft tissues are a common occurrence during cementoplasty.

They may cause chronic pain, and thus treatment failure. Spindle malposition during reinforced cementoplasty may cause vascular, nerve or cartilage injury. Our goal was to evaluate the rate of cement leakage / spindle extraction and describe the techniques used.

Methods

This retrospective monocenter study included 104 patients who underwent reinforced cementoplasty and 3425 patients who underwent cementoplasty between 2012 and 2020. Operative reports and fluoroscopic images were reviewed to identify extraction attempts and their outcomes.

Results

Six patients (5.8%) had a malpositoned spindle, and all of them underwent spindle extraction during reinforced cementoplasty, with an 80% success rate. A total of 7 attempts were performed, using 2 different techniques. One thousand one hundred thirty patients (32%) had a cement leak in soft tissues, and 7 (0.6%) underwent cement leakage extraction during cementoplasty, with a 100% success rate. A total of 10 attempts were performed, using 3 different techniques. No major complication related to the extraction procedures occurred.

Conclusions

Spindle malpositions and soft tissue cement leakages are not uncommon. We described 5 different percutaneous techniques that were safe and effective to extract spindles and paravertebral cement fragments.

INTRODUCTION

Cementoplasty is a minimally invasive technique for symptomatic fractures or mechanically compromised bone lesions, providing pain relief, both at short and longterm follow-up, and stabilization preventing bone collapse [START_REF] Weill | Spinal metastases: indications for and results of percutaneous injection of acrylic surgical cement[END_REF][START_REF] Hulme | Vertebroplasty and Kyphoplasty: A Systematic Review of 69[END_REF]. Although the procedure is generally considered safe, complications may occur during cementoplasty [START_REF] Barragán-Campos | Percutaneous Vertebroplasty for Spinal Metastases: Complications[END_REF]. Cement leakage is the most common one, observed as frequently as in 59.7% of the cases for vertebroplasty [START_REF] Zhan | Risk Factors for Cement Leakage After Vertebroplasty or Kyphoplasty: A Meta-Analysis of Published Evidence[END_REF]. Cement leakage in soft tissues is a common issue, estimated to occur in 21% of the cases. Such soft tissues leakage may sometimes cause chronic pain, leading to treatment failure [START_REF] Zhu | Risk factors for bone cement leakage in percutaneous vertebroplasty: a retrospective study of four hundred and eighty five patients[END_REF].

Reinforced cementoplasty (percutaneous internal fixation using dedicated spindles combined with cementoplasty) provides significant pain relief, and additional resistance to secondary fracture, by inserting spindles in the target bone, compared to non-reinforced cementoplasty [START_REF] Premat | Reinforced cementoplasty using dedicated spindles in the management of unstable malignant lesions of the cervicotrochanteric region[END_REF]. Possible complications include malpositioned spindle, which may cause pain or functional impairment.

With a growing number of procedures over the last decades, developing safe techniques is of the utmost importance. While preventing the above-mentioned issues is preferable, developing strategies to remove the foreign body, whether it is cement leakage or a spindle, can help salvage a procedure gearing towards failure and thus benefit the patient [START_REF] Kaufmann | A Technique to Circumvent Subcutaneous Cement Tracts during Percutaneous Vertebroplasty[END_REF]. Several percutaneous techniques have been described to remove such foreign bodies complicating bone percutaneous interventions [START_REF] Amoretti | Foreign body extraction from soft tissue by using CT and fluoroscopic guidance: a new technique[END_REF][START_REF] Shaikh | Needle-in-Needle Technique for Percutaneous Retrieval of a Fractured Biopsy Needle during CT-Guided Biopsy of the Thoracic Spine[END_REF], using a supple biopsy forceps for instance [START_REF] Durrleman | Percutaneous removal of a soft tissue cement leakage complicating a percutaneous vertebroplasty by the mean of a supple biopsy forceps[END_REF]. However, large series focused on this topic are lacking.

The purpose of our study was to evaluate the rate of cement leakage/spindle extractions in a large cohort of cementoplasties. We also aimed to describe the different percutaneous strategies and techniques that may be used to extract such foreign bodies.

MATERIAL AND METHODS

This is a single centre, retrospective, observational study performed according to the STROBE guidelines [START_REF] Von Elm | The Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) statement: guidelines for reporting observational studies[END_REF]. Our local institutional review board (IRB) approved the study protocol (IRB approval # HJ 26 6 20a). The need for patients' informed consent was waived by our IRB. This work adheres to the World Medical Association Declaration of Helsinki. Risks and benefits of the procedures were clearly explained to the patients involved in this study, and informed consent to the procedures was obtained from all participants.

Study population

We screened all patients who had a CT scan performed in our hybrid angiosuite from January 2012 to September 2020 and all patients in our institutional database of cementoplasties. We reviewed clinical charts of all eligible patients for inclusion and exclusion criteria.

Inclusion criteria were (a) patients aged 18 or more; (b) patients who underwent cementoplasty or reinforced cementoplasty at our institution.

Exclusion criteria were (a) imaging data unavailable; (b) medical chart unavailable.

Recruitment flow chart is displayed in Supplemental Fig. 1.

Clinical and paraclinical data

We reviewed all patients' medical chart for clinical data, including age, sex, fracture aetiology. We reviewed all operative reports and images for procedural data, including procedure type, target lesions, presence of cement leak. We assessed through operative reports and fluoroscopic images whether cement extraction or spindle extraction or repositioning were performed.

Procedures' protocol

Procedures were performed under conscious sedation or general anaesthesia, either in a single plane hybrid angiosuite with a CT-scanner (Miyabi Emotion 16, Siemens) or in a biplane angiosuite. We used either 13G or 11G bone needles (Thiebaud) and poly-methyl-methacrylate (PMMA) bone cement (Biomet UK Ltd) to perform cementoplasty. We used 9G or 8G bone needles (Thiebaud) and 2.5-mm diameter spindles (material: stainless steel in accordance with the international standard AISI 316 L [Thiebaud]) (length ranging from 5 to 8 cm) for reinforced cementoplasty.

Spindle and cement extraction

Spindle extraction: the supple biopsy forceps technique

The bone needle used to insert the spindle is left in place if possible. Otherwise, a large (8G or 9G) bone needle is inserted under fluoroscopy using the same path used for the spindle insertion, and the spindle is "catheterized". Great attention should be paid to avoid pushing forward the spindle while trying to recatheterize it. A supple biopsy forceps (AN/AMHBFC-WC, Life Partners Europe) is inserted within the bone needle, until contact with the spindle. The clamp is then opened as much as possible inside the bone needle. The biopsy forceps is then pushed forward slightly and the clamp closed to grab the spindle. While maintaining the forceps' clamp closed, it is gently pulled until extraction of the spindle. Fluoroscopy is used to guide the forceps. Several attempts are often necessary, as friction between spindle and bone is important. With this technique, it is possible to reposition and reuse the spindle. The technique is illustrated in Fig. 1 and Supplemental Video 1.

Spindle extraction: the cementing technique

Similarly to the technique described above, the bone needle used to insert the spindle is either left in place or re-inserted in order to "catheterize" the spindle under fluoroscopy, so that at least half the spindle is within the bone needle's lumen. As for the previous technique, the operator should recatheterize the spindle very carefully to avoid pushing it more distally. PMMA bone cement is then injected very gently within the bone needle under fluoroscopy so that the cement fills the lumen, most importantly around the spindle. Cement must not leak at the distal aspect of the bone needle (in the target bone), otherwise there is a risk for the spindle and the needle to stay stuck in the bone. Afterwards, the operator waits for the bone cement to harden (around 20 mins for the Biomet V PMMA cement). The bone needle is not moved or rotated during this time interval, as the objective is the spindle, the cement and the bone needle core to be solidarized together. Once cement polymerisation is completed, the bone needle is then withdrawn with the spindle stuck inside. Neither the spindle nor the bone needle can be reused. The technique is illustrated in Fig. 2 and Supplemental Video 2.

Cement

Cement extraction: the resheating technique

For cylindrical shape cement leaks, related to a cement column within the dead space of the bone needle, released during its withdrawal, extraction can be achieved using the same bone needle. The bone needle is inserted using the same initial entry point and pushed toward the cement leak under fluoroscopy. The mandrel is removed, and the bone needle used to catheterize the cement fragment under fluoroscopic guidance, like a sheath on its sword. Upon catheterizing as much of the cement cylinder as possible (ideally up to the vertebral pedicle), the needle is angulated to bend and break the cement cylinder. The needle is then withdrawn while maintaining it angulated so that the cement stays inside the lumen. The technique is illustrated in Fig. 3 and Supplemental Video 3.

Cement extraction: the supple biopsy forceps technique

This technique has been previously described in the literature [START_REF] Durrleman | Percutaneous removal of a soft tissue cement leakage complicating a percutaneous vertebroplasty by the mean of a supple biopsy forceps[END_REF]. A supple biopsy forceps is inserted through the incision and navigated under fluoroscopy, clamp closed, toward the cement fragment. Optionally, the supple biopsy forceps can be navigated inside a rigid bone needle, which can make reaching deeper targets easier. Upon reaching the fragment, the clamp is opened, and the biopsy forceps navigated further to grab the fragment. The supple biopsy forceps is then withdrawn with the clamp kept closed, and the cement fragment is pulled alongside. Several attempts may be necessary to break the fragment free from the cement inside the patient's bone. The technique is illustrated in Supplemental video 4.

Cement extraction: the needle holder technique

The initial incision is enlarged using a surgical scalpel blade No. 11. A needle holder is inserted through the enlarged incision and moved toward the cement fragment under fluoroscopy. Its jaws are kept closed until the fragment is reached to avoid damaging the soft tissues. Upon reaching the fragment, the jaws are used to grab it.

Finally, the needle holder is removed with its jaws kept firmly shut, and the fragment is pulled alongside. Several attempts may be necessary. The technique is illustrated in Fig. 4 and Supplemental Video 5.

RESULTS

Patients' demographics and procedures characteristics

From January 2012 to September 2020, a total of 3 529 patients (1225 males and 2304 females; mean age: 67.3 years) for a total of 7 139 cemented bones. One hundred and four patients (2.9%) underwent reinforced cementoplasty, 40 males and 64 females, mean age 61.6 years, for a total of 116 cemented bones. Three thousand four hundred and twenty-five patients (97.1%) underwent cementoplasty (1 185 males and 2 240 females, mean age 67.5 years), for a total of 7 023 cemented bones.

Detailed characteristics are displayed in Supplemental Table 1.

Spindle extraction / repositioning

Out of 104 patients (40 males and 64 females, mean age 61.6 years) who underwent reinforced cementoplasty, 6 (5.8%) presented with spindle malposition. All of them underwent spindle repositioning / extraction. The cases are described in Table 1.

Two extractions (33%) were performed for an intra-/trans-articular spindle, 2 extractions (33%) were performed for an intra-muscular spindle, 1 extraction (17%) was performed for an intra-foraminal spindle, and 1 extraction (17%) was performed

DISCUSSION

Our study evaluates the rate of cement leakage at 32%, the rate of spindle malposition at 5.8%, and presents several percutaneous techniques that can be used to extract or reposition a spindle, or extract a cement fragment located in the soft tissues. We showed that spindle malposition is not rare, as extraction was performed in 5.8% of reinforced cementoplasties. Cement leakages, while much more common, underwent an extraction attempt in 0.6% of cases.

Spindle malposition can lead to poor outcome depending on the location of the spindle. In the best-case scenario, the spindle will not provide the additional stability expected from a reinforced cementoplasty. In the worst cases, the spindle may damage important structures, such as cartilage, nerves or arteries with potential disastrous consequences. To the best of our knowledge, no previous study has estimated the prevalence of spindle malposition. The prevalence was 5.8% in our study. While uncommon, this issue was not rare either, and it is essential to learn techniques and strategies to fix it and avoid complications. A CBCT / CT scan should systematically be performed before cementing the spindle to confirm proper positioning, as both percutaneous and surgical extraction will be impossible after cementing. We provide herein several techniques to reposition or retrieve the spindle.

Only one case ended with a technical failure, with a spindle positioned in the gluteus medius, probably leading the operator to be less aggressive in his attempt to extract the spindle compared to other cases. No complication related to the extraction attempt occurred in our series.

Soft tissue cement leakage is much more common, as described in the literature [START_REF] Zhu | Risk factors for bone cement leakage in percutaneous vertebroplasty: a retrospective study of four hundred and eighty five patients[END_REF].

However, its clinical impact is uncertain. It is difficult to evaluate whether lingering pain after cementoplasty is related to cement leak, as it is not uncommon after a flawless cementoplasty with no leak, and not always present even with large cement leaks. The prevalence of cement leaks was evaluated in our study at 32%, meaning it is a common occurrence. Similarly, cement leakages have been reported in about 21% of the cases in the literature [START_REF] Zhu | Risk factors for bone cement leakage in percutaneous vertebroplasty: a retrospective study of four hundred and eighty five patients[END_REF]. Despite the high incidence of these leakages, only 0.6% resulted in an extraction attempt in our study, with all cases leading to technical success. We suggest simple guidelines on when to attempt extraction: cement in soft tissues with a length > 3 cm or foraminal location or painful subcutaneous location. These guidelines are derived from our experience. However, to our knowledge, consensual indications are lacking in the literature. We described several techniques to perform the extraction of the cement with simple tools available in most angiosuites. No major complication related to the extraction procedures occurred. Of note, the extraction techniques we used may cause local pain, as they involve (re-)inserting and manipulating instruments in the paravertebral soft tissues.

However, our study could not assess this additional pain since the extractions were performed during the same procedure as the cementoplasty/reinforced cementoplasty.

Our study suffers from several limitations: (i) its retrospective nature and reliance on saved fluoroscopic loops and operative report means that we may have missed some cases of cement/spindle extraction; (ii) extraction was performed at the operator's discretion, as there are no consensual, standardized guidelines on when to do it.

While we believe the spindle extractions' indications to be rather consensual, for instance when the spindle is intra-articular, soft tissue cement fragment extraction is less consensual. Its clinical impact is unclear, with some being symptomatic while others are not; (iii) all procedures were performed in an expert institution so our results might not be generalizable to all centres; (iv) we did not evaluate the clinical outcome of the extraction, as all extractions but one were performed at the same time as the initial (reinforced) cementoplasty. Thus, the clinical impact of the extraction could not be assessed.

In summary, spindle malpositions and cement leaks during cementoplasties are not uncommon and may lead to treatment failure. Several percutaneous techniques may be safely and effectively used to retrieve such malpositioned spindles or cement leaks to improve patient outcome.

FIGURE CAPTIONS

  Spindle malposition systematically led to spindle repositioning or extraction during the procedure. Malpositions include (a) para-vascular location; (b) para-neural location;(c) intra-articular location. Two different techniques, described below, were used; chosen at the operator's discretion.

  leakages did not systematically lead to cement extraction. While extraction was ultimately the operator's decision, consensual indications at our Institution are (a) cement in soft tissues with a length > 3 cm; (b) foraminal location; (c) painful subcutaneous location. Several techniques, described below, were used; chosen at the operator's discretion.

  for a para-neurovascular spindle. All of them underwent spindle extraction or repositioning during the procedure, one requiring 2 different extraction techniques, leading to a total of 7 extraction attempts. Three attempts were made with the supple biopsy forceps technique, with 2 technical failures (success rate: 33%). Four attempts were made with the cementing technique, with no technical failure (success rate 100%). No complication related to the extraction techniques occurred.Cement extractionOut of 3 529 patients who underwent percutaneous cementoplasty or reinforced cementoplasty, for a total of 7 139 cemented bones, 1 130 (32%) presented with a cement leak in soft tissues. Out of those, 7 (0.6%) underwent cement extraction. The cases are described in Table2. Indication for extraction was a cement leakage > 3cm in 3 cases (43%), a cement leakage in the oral cavity in 2 cases of transoral vertebroplasty (29%), a cement leakage reaching the skin in 1 case (14%), and 1 extraction (14%) was performed for a foraminal location in 1 case (14%). One patient had an elective extraction procedure after first unsuccessful attempt during the cementoplasty procedure. All other extractions were performed during the initial cementoplasty. Although all patients ultimately had a successful extraction, a total of 10 extraction attempts were necessary. Two attempts were made with the resheating technique, with 1 technical failure (50% success rate). Two attempts were made with the needle holder technique, with no technical failure (100% success rate). One unsuccessful attempt was made with the cementing technique described for spindle extraction (success rate 0%). Four attempts were made with the supple biopsy forceps technique, with 1 technical failure (75% success rate). One attempt was made with the manual extraction technique, with 0 technical failure (100% success rate). No major complication related to the extraction techniques occurred.

Figure 1 A

 1 Figure 1

Figure 2 (

 2 Figure 2

Figure 3 A

 3 Figure 3

Figure 4 A

 4 Figure 4

TABLE CAPTIONS Table 1 -Spindle extraction cases

 CAPTIONS1 Description of all spindle extraction cases performed in our study population. Two attempts have been made for 1 patient.

Table 2 -Cement extraction cases

 2 Description of all cement extraction cases performed in our study population. Multiple attempts have been made for 2 patients.

Ethical approval:

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

The study received approval from our local institutional review board (IRB). Patient signed consent was waived by our local IRB.