domain: math info

text:
Paolo Zappalà
email: paolo.zappala@orange.com

Marianna Belotti
email: marianna.belotti@caissedesdepots.fr

Maria Potop-Butucaru
email: maria.potop-butucaru@lip6.fr

Stefano Secci
email: stefano.secci@cnam.fr

Analyse de la fiabilité des blockchains via la théorie des jeux

Introduction

Distributed Ledger Technologies (DLTs) allow sharing a ledger of transactions among multiple users forming a peer-to-peer (P2P) network. DLTs characterized by a block architecture are called "blockchains". They enable its users to transfer cryptoassets in a decentralized manner by means of modular protocols adopted by the users themselves. Beyond the traditional blockchain architectures (i.e., layer-1 protocols), the literature proposes other protocols that respectively define and regulate interactions in an overlaying network (layer-2 protocols) and interactions between different blockchains (cross-chain protocols). In a Blockchain system players can be classified in three different categories as stated in [START_REF] Amitanand | BAR fault tolerance for cooperative services[END_REF]: (i) players who follow the prescribed protocol i.e., altruistic, (ii) those who act in order to maximise their own benefit i.e., rational, and (iii) players who may deviate from the prescribed protocol in an irrational way, i.e. Byzantine.

Interactions among users are modeled with game theory, which is used to design incentive mechanisms aiming at preventing any possible deviation from a prescribed protocol that blockchain users need to follow. Robustness of protocols governing DLTs (e.g., consensus protocols, communication protocols and storage protocols) has been addressed in several recent works. Most of the game theoretical models adopted to design secure and robust blockchain protocols, surveyed in [START_REF] Liu | A Survey on Blockchain: A Game Theoretical Perspective[END_REF][START_REF] Wang | A survey on consensus mechanisms and mining management in blockchain networks[END_REF], (i) address protocols characterizing specific blockchain implementations, (ii) analyze miners' behaviours in the consensus phase and (iii) adopt Nash Equilibria as solution concept.

In the literature, analysis of systems robustness with respect to participating actors can be classified according to the agents' nature [START_REF] Amitanand | BAR fault tolerance for cooperative services[END_REF]. Concerning rational agents, the robustness analysis includes the study of the equilibria and the evaluation of their properties. The most studied and adopted solution concept in the literature is Nash Equilibrium, i.e., a strategy profile in which no player has interest in individually deviating from her own strategy. Authors in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF] define some properties to characterise strategy profiles: (i) practical strategy profiles equilibria are those which exclude weakly dominated strategies, (ii) k-resilient equilibria are those strategy profiles such that if there is no coalition with at most k players having an incentive to deviate from the prescribed protocol. In order to analyse robustness with respect to Byzantine agents, authors in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF] introduce the concept of t-immunity, i.e., no player gets a lower outcome if there are at most t Byzantine players that can play any possible strategy.

Our contribution. This paper presents a game theoretical framework aiming at characterizing blockchain protocols, modeled as games, in terms of robustness, i.e. resilience to rational deviations and immunity to Byzantine behaviors. Robustness analysis of blockchain protocols were performed before in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF] by adopting the concept of mechanism (i.e., a pair game-prescribed strategy). In order to characterize the robustness of a distributed system authors in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF] introduce the notions of (i) kresilience, (ii) practicality and (iii) t-immunity. More precisely, k-resilience and practicality analyze the robustness with respect to rational agents, while t-immunity deals with Byzantine agents. In this paper we use the concept of mechanism proposed in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF] to model different types of blockchain protocols and we define a set of properties to be satisfied in terms of robustness. Since the property of t-immunity is often impossible to be satisfied by practical systems [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF], we introduce the concept of t-weak-immunity. A mechanism is t-weak-immune if any altruistic player receives no worse payoff than the initial state, no matter how any set of t players deviate from the prescribed protocol. We further extend the framework in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF] by proving some necessary and sufficient conditions for a mechanism to be optimal resilient and t-weak-immune. In order to make the method scalable to any modular protocol, we define a new operator for mechanism composition and prove that it preserves the robustness properties of the individual games. Using our framework we studied the properties of a set of layer-1, layer-2 and cross-chain protocols: Tendermint [START_REF] Kwon | Tendermint: Consensus without mining[END_REF], Bitcoin [START_REF] Nakamoto | A peer-to-peer electronic cash system[END_REF], Lightning Network protocol [START_REF] Poon | The bitcoin lightning network: Scalable off-chain instant payments[END_REF], the side-chain protocol [START_REF] Ranchal | Platypus: Offchain Protocol Without Synchrony[END_REF] and the very first implementation of a cross-chain swap protocol proposed in [START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] and formalized in [START_REF] Herlihy | Atomic cross-chain swaps[END_REF]. Thanks to the analysis of protocol robustness we spotted the weakness of the Lightning Network protocol to Byzantine behaviour and therefore we propose and further analyze an alternative version of the protocol. Our results are reported in Table 1 and in [START_REF] Zappalà | Game theoretical framework for analyzing Blockchains Robustness[END_REF]. An earlier version with partial results was published in [START_REF] Zappalà | Brief Announcement: Game Theoretical Framework for Analyzing Blockchains Robustness[END_REF].

Table 1: Immunity and resilience properties for Tendermint [START_REF] Kwon | Tendermint: Consensus without mining[END_REF], Bitcoin [START_REF] Nakamoto | A peer-to-peer electronic cash system[END_REF], Lightning Network [START_REF] Poon | The bitcoin lightning network: Scalable off-chain instant payments[END_REF], a sidechain protocol [START_REF] Ranchal | Platypus: Offchain Protocol Without Synchrony[END_REF] and a cross-chain swap protocol [START_REF] Herlihy | Atomic cross-chain swaps[END_REF][START_REF] Nolan | Re: Alt chains and atomic transfers[END_REF] with respect to the number of rational deviating agents (k ) and the number of Byzantine deviating agents (t) where n is the total number of players in the game.

Protocol k-Resilience t-Immunity t-Weak Immunity

Game theoretical framework

Mechanisms and Robustness. Given a distributed systems protocol, players can either decide to follow or not the prescribed instructions. The aim of our model is to understand whether the players are incentivized to follow or deviate from the prescribed protocol given the presence of some rational or Byzantine agents. In the following (i) we recall and extend the game theoretical framework based on the concept of mechanism (introduced in [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF]) and its properties, (ii) we define new properties on protocol robustness and (iii) we study properties interdependence.

Let us consider a game in normal form Γ = N, S , u where players find themselves in an initial state, i.e., before starting the application of the protocol. For the sake of simplicity we assign u i (σ) = 0 for every σ ∈ S when the player i is indifferent between the outcome of the strategy profile σ and the initial state one. Analogously, we assign positive utility, u i (σ) > 0, when the outcome of σ corresponds to the final state provided by the protocol and negative utility, u i (σ) < 0, when the outcome of σ is worse than the initial state one. The values of u i , for all i ∈ N , correspond to the marginal utility with respect to the initial state. Every decision-making problem is modeled by a game Γ = N, S , u , which shows all the possible strategies available to the players, including following the prescribed protocol and all its possible deviations. A specific protocol consists of a strategy profile σ = (σ 1 , . . . , σ n ) ∈ S and it is denoted by a pair (Γ, σ), called mechanism [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF]. Every player i is advised to play strategy σ i ∈ S i i.e., the recommended strategy σ is the prescribed protocol. Evaluating the robustness to deviations of a distributed protocol corresponds to identifying the properties of the mechanism (Γ, σ). Players can decide to deviate for two different reasons. On one hand, they can cooperate in order to find a strategy profile that provides a better outcome than the one given by the protocol. On the other hand, some players can behave maliciously for no specific reason and harm the altruistic ones. These two behaviours are prevented, according to [START_REF] Abraham | Distributed Computing Meets Game Theory: Robust Mechanisms for Rational Secret Sharing and Multiparty Computation[END_REF], if prescribed protocols are respectively (i) k-resilient and/or (ii) t-immune.

A mechanism (Γ, σ) is k-resilient if there is no coalition of at most k players having an incentive to simultaneously change strategy to get a better outcome. Formally, a strategy profile σ ∈ S is a k-resilient equilibrium if for all C ⊆ N with 1 ≤ |C| ≤ k, all τ C ∈ S C and all i ∈ C, we have

u i (σ C , σ -C ) ≥ u i (τ C , σ -C
). The concept of k-resilience denotes the tendency of a set of k players to cooperate to move to an equilibrium that differs from the prescribed one. Hence k-resilience generalizes the concept of Nash equilibrium.

A mechanism (Γ, σ) is t-immune if, given at most t players choosing any strategy different from the prescribed one, the other players receive at least the utility they would get if everyone followed the protocol. Formally, a strategy profile σ ∈ S is t-immune if for all T ⊆ N with |T | ≤ t, all τ T ∈ S T and all i ∈ N \ T , we have

u i (σ -T , τ T ) ≥ u i (σ).
The property of t-immunity is very strong and hardly satisfiable since it requires that the protocol provides the best outcome no matter how a set of t players deviates. We therefore introduce a weaker version of the property; t-weak-immunity. This new property guarantees that non deviating players receive at least the utility value of the initial state (i.e., players receive a positive outcome).

Definition 1 (t-weak-immunity). A mechanism (Γ, σ) is t-weak-immune if for all T ⊆ N : |T | ≤ t, all τ T ∈ S T and all i ∈ N \ T , we have u i (σ -T , τ T ) ≥ 0.

A player that joins a t-weak-immune mechanism will not suffer any loss (i.e., outcome with negative utility) if there are at most t deviating players in the game. We say that a mechanism is weak immune if it is t-weak-immune for all t ∈ N .

Composition of Games and Mechanism. Blockchains systems are complex protocols designed in a modular way. In order to study the robustness of such complex protocols, we need to analyze the individual modules and infer the properties of the system by composition. For this scope we introduce the new notion of composition of games that, to the best of our knowledge, has never been defined in the literature. Given two different games A and B, the composition of games is defined by the operator , hence A B denotes the composition of game A and B. Given two games that are played separately and independently, the composition corresponds to players picking a strategy from each game and receiving as utility the sum of the utilities of the two games. The following propositions allow us to (i) model the building blocks of complex protocols, (ii) study the properties of the subsequent mechanisms and (iii) deduce the properties of the composed protocol through the composition of mechanisms.

Concerning the solutions of the composition of games, we prove that Nash equilibria can be identified by selecting equilibria within the single games. It is not possible to create or destroy Nash equilibrium strategies by composing independent games. Theorem 1 (composition nash equilibria). Let A = N, S A , u A and B = N, S B , u B be two games in normal form representation. Then, {(σ Ai , σ Bi )} is a Nash equilibrium for A B if and only if {σ Ai } and {σ Bi } are Nash equilibria respectively for A and B.

Concerning robustness properties for composition of games, we can state the following results on resiliency and weak immunity for two composed games. The results can be generalized for the composition of multiple games.

Theorem 2 (resiliency). Let A = N, S A , u A and B = N, S B , u B be two games and let (A, σ A ) and (B, σ B ) be two mechanisms respectively k-resilient and k -resilient. Then, (A B, {σ Ai , σ Bi }) is a min(k, k )-resilient mechanism.

Theorem 3 (weak immunity). Let A = N, S A , u A and B = N, S B , u B be two games and let (A, σ A ) and (B, σ B ) be two mechanisms respectively t-weak-immune and t -weak-immune. Then, (A B, {σ Ai , σ Bi }) is a min(t, t )-weak-immune mechanism.

Definition 2 (

 2 Games Composition). Given A = N, S A , u A and B = N, S B , u B two games in normal form with the same set of players N , two different sets of strategies S A = {S Ai : i ∈ N } and S B = {S Bi : i ∈ N } and two different utility functions: u A : S A → R N and u B : S B → R N then, it is possible to define a new game C = A B, called composition of A and B, characterized as follows: C = N, S C , u C , where N is the set of the players, S C := {(s Ai , s Bi ), s Ai ∈ S Ai , s Bi ∈ S Bi , ∀i ∈ N } is the set of strategies and u C ({(σ Ai , σ Bi )}) := u A ({σ Ai }) + u B ({σ Bi }) is the utility function.


Fabien Grzeskowiak

David Gonon

Daniel Dugas

Diego Paez-Granados

Jen Jen Chung

Juan Nieto

Roland Siegwart

Aude Billard

Marie Babel

Julien Pettré

Crowd against the machine: A simulation-based benchmark tool to evaluate and compare robot capabilities to navigate a human crowd

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

I. INTRODUCTION

A large number of mobile robots are conceived with the intention of being operated in public spaces, likely to be frequented by crowds. Assessing robots' ability to navigate through the crowd in a way that is efficient and safe is thus of paramount importance. However, such assessment is difficult for several reasons. Firstly, a crowd is a dynamic and complex environment, the state of which depends on a very large number of parameters. A meaningful evaluation should cover as much of this parameter space as possible, this guarantees that the robot behaviour has been studied in most of the situations it may encounter. However, this crowd complexity compounds the already high-dimensional parameter space related to the robot behaviour.

Ideally, we should study and test various robot behaviours in the same situations. This would enable comparing efficiency and safety levels offered by various navigation techniques. Unfortunately, this results in a combinatorial explosion. The risks linked to the participants and the cost of conducting real-world experiments are our two last major problems. This explains why research work dedicated to robot navigation in crowds are limited to tests with a limited number of participants that barely constitute a crowd. To mitigate this issue, our paper explores solutions to provide the community with a crowd-robot navigation benchmark tool. Our work addresses three aspect of this problem: i) to define reference navigation scenarios, ii) to provide simulation algorithms and a framework for the simulation itself, iii) to introduce metrics for evaluating efficiency and safety levels.

1 F. Grzeskowiak, M. Babel and J. Pettré are with Univ Rennes Inria CNRS IRISA, Rennes, France. julien.pettre@inria.fr 2 D. Gonon, D. Paez-Granados and A. Billard are with EPFL, Lausanne, Switzerland. aude.billard@epfl.ch 3 D. Dugas, J. J. Chung, J. Nieto and R. Siegwart are with ETH, Zürich, Switzerland. jenjen.chung@mavt.ethz.ch Simulation-based evaluation offers valuable insights. Nevertheless, it cannot replace the validity of real tests because, if robot simulation cannot already be considered perfectly accurate, the simulation of human behaviour is even more complex. It is impossible to simulate all the behaviour that a crowd can exhibit, in particular, its reactions to a robot. However, crowd simulation is useful to create a synthetic test environment. It can be a preliminary and complementary tool to real tests that evaluate the evolution of a robot in an environment that is representative in many aspects to a real crowd.

At the very least, simulation-based testing addresses the four problems outlined above: it can automatically explore large situational parameter spaces, it eases comparisons and is, without question, safe. Moreover, simulation brings the major interest of allowing one to easily define a set of standard situations, which are defined by the configuration of the crowd and the task of the robot. These can be precisely replicated from one series of tests to another, which can be elaborated and shared at the scale of a whole community. It therefore makes simulation-based testing an ideal candidate to be used as a benchmark tool to fairly evaluate and compare the behaviour of robots in a crowd.

Our article presents the following contributions: (i) a simulation platform, open and accessible to the community, allowing the simulation of robot motion among a moving crowd, (ii) a preliminary set of standard situations, representative of common ones to which a robot is exposed when navigating in a crowded environment, (iii) a set of metrics to evaluate the test results with regard to the robot's efficiency, the disturbance caused to the crowd's navigation, as well as the related risks of collisions, (iv) a demonstration of this tool's capabilities to be used as a benchmark.

II. RELATED WORK

The development of new mobile robots intended to move within crowds requires the evaluation of their navigation capabilities in terms of efficiency and safety. Simulators facilitate such evaluation by providing interfaces through which a robotic component can manipulate and perceive an abstract environment that models a real phenomenon (in our case, a crowded environment). In the following, we review some existing crowd simulators by investigating which technical concepts and metrics they implement that make them significant for robotic navigation in human crowds.

A contribution of [START_REF] Luo | Porca: Modeling and planning for autonomous driving among many pedestrians[END_REF] is a probabilistic planner that updates its belief on pedestrians' goal positions and assumes that their decision making is based on PORCA, a variant of ORCA [START_REF] Van Den | Reciprocal n-body collision avoidance[END_REF] (using a different cost function modeling pedestrians' impatience). Thus, the simulator is suitable to demonstrate exactly this planner's performance as the planner's underlying assumptions about pedestrians' decision making match exactly how the simulation controls them. The metrics for the evaluation are the robot's collision rate, travel time, length of acceleration/deceleration periods, and success rate (to reach the goal). [START_REF] Vasquez | Inverse reinforcement learning algorithms and features for robot navigation in crowds: An experimental comparison[END_REF] compare how a robot using different learning-based controllers performs when navigating among pedestrians in the PedSim1 simulator. They measure the robot's path length and smoothness as well as pedestrian comfort by the number of proxemic intrusions by the robot (proximity at particular angles). [START_REF] Chen | Crowd-robot interaction: Crowd-aware robot navigation with attention-based deep reinforcement learning[END_REF] evaluate their navigation technique for the robot in simulations with agents that apply ORCA [START_REF] Van Den | Reciprocal n-body collision avoidance[END_REF]. They report the robot's success rate (to reach the goal without a collision), collision rate, time, and reward (referring to their framework of reinforcement learning). [START_REF] Kuderer | Featurebased prediction of trajectories for socially compliant navigation[END_REF] propose an approach that uses inverse reinforcement learning and predicts collective trajectories by assuming they optimize an underlying reward function. To show that their approach works as expected, they perform simulations wherein such a reward function governs the agents' behaviour, and the approach then learns this function to make reliable predictions. [START_REF] Trautman | Robot navigation in dense human crowds: Statistical models and experimental studies of human-robot cooperation[END_REF], [START_REF] Trautman | Unfreezing the robot: Navigation in dense, interacting crowds[END_REF] call their experiments simulations wherein they replay crowd motions from a recorded dataset and replace just one pedestrian by the robot who then needs to plan its own path (whereas others do not react to it). They measure the robot's minimum distance to pedestrians and the robot's path length. Similarly, [START_REF] Bevilacqua | Reactive planning for assistive robots[END_REF] use the same dataset and crowd replay to simulate how the robot reacts to pedestrians (while they do not react to it). Webots2 is an open-source simulator offering animated human characters, albeit looking and walking rather like puppets. [START_REF] Fraichard | From crowd simulation to robot navigation in crowds[END_REF] perform crowd simulations which incorporate heterogeneous navigation methods and limited perception to investigate the difficulties that one needs to tackle to enable robots to navigate in crowds. In Webots, they simulate how a crowd with limited field of view and different control laws performs, measuring the agents' number of collisions and time to reach their goals. Some work aiming towards robotic navigation in crowds considers as the key task to understand and accurately model pedestrian decision making. One could argue that a robot just needs to replicate the human decision process to navigate among pedestrians like a regular pedestrian. They typically do not use simulation to evaluate their methods. Rather, they (e.g. [START_REF] Turnwald | Understanding human avoidance behavior: interaction-aware decision making based on game theory[END_REF]) measure the similarity between the trajectories which their methods generate for the robot in a given situation to the motion of a real human subject in the same situation (also like in [START_REF] Trautman | Robot navigation in dense human crowds: Statistical models and experimental studies of human-robot cooperation[END_REF], [START_REF] Trautman | Unfreezing the robot: Navigation in dense, interacting crowds[END_REF]). The evaluation in [START_REF] Turnwald | Human-like motion planning based on game theoretic decision making[END_REF] lets human participants interact with a virtual pedestrian and judge later on whether the virtual pedestrian was following another human participant's decisions or an automatic game theoretic planner. This varied Turing test could show whether Fig. 1. The CrowdBot simulator architecture: Part of the tool is an application based on Unity (https://unity.com). This application can generate scenarios, it simulates how the crowd moves and renders it in a 3D environment. It also handle physics simulation in order to compute collisions between the crowd and the robot. Finally, it handles the robot dynamics and sensing. A second part of the tool is the python module which controls the Unity application. This module conducts the simulation with instructions and data sent to the application through sockets, which then sends back simulation data. This module helps in the generation of datasets of crowd and robot trajectories, which can be analyzed using the benchmark tools. Finally, the module propose an API for ROS and PythonRobotics. One last component is the set of configuration files which gives the parameters of the simulation.

or not the decisions made in the background by the planner or the second participant are indistinguishable.

In summary, existing works on robots navigating in crowds mostly just consider one respective facet of crowds in their simulations, which is likely representing the specific task they are addressing. Therefore, it seems that a more versatile simulator, which reflects and integrates such aspects, could give new impulses to similar research. In particular, the above simulators are limited to local collision avoidance, i.e. do not exhibit high-level path planning, and they mostly lack realistic gait animation and graphical rendering of pedestrians, while realistic visual rendering is important for fully visionbased navigation techniques (e.g. end-to-end learning). Further, their metrics are mostly the same, surprisingly not quantifying the crowd's efficiency in most evaluations. The metrics for safety also seem fairly simplistic with most only considering the robot's distance to pedestrians. Colliding agents' mass, velocity and kinetic energy are suitable to quantify the potential for injury as several works in humanrobot interaction have shown [START_REF] Rossi | A pre-collision control strategy for human-robot interaction based on dissipated energy in potential inelastic impacts[END_REF], [START_REF] Haddadin | On making robots understand safety: Embedding injury knowledge into control[END_REF].

III. CROWDBOT SIMULATION PLATFORM

This section describes the CrowdBot simulation architecture shown in Fig. 1.

A. Crowd simulation

The purpose of the crowd simulation component is to compute the motion of a crowd in a given environment, represented as a collection of moving points, one for each simulated individual. For this, we rely on microscopic crowd simulation algorithms, which perform this calculation based on the notion of an agent whose trajectory is a function of its goals and its interactions with the environment. Note that literature has proposed a number of numerical models of local interactions, and more specifically of collision avoidance, that determine agents' behaviours [START_REF] Helbing | Simulation of pedestrian crowds in normal and evacuation situations[END_REF] [2] [START_REF] Dutra | Gradient-based steering for vision-based crowd simulation algorithms[END_REF].

To avoid the effect of limiting crowd behaviours to one of a unique algorithm, we employ a recent crowd simulation technique which is capable of reproducing a collection of crowd simulation algorithms [START_REF] Van Toll | Generalized microscropic crowd simulation using costs in velocity space[END_REF]. The motion of each agent is driven by the definition of a cost function that the agent optimizes, defined in the agent's velocity space, which depend on the state of the agent as well as the state of one of its neighboring objects (other agents, robot, obstacles). Several existing algorithms can be reproduced by changing the definition of this cost function. In addition, agent motion is subject to the parameter settings of this function.

By choosing such a technique to simulate crowds in the CrowdBot framework, we open the possibility to include, in experimental plans, a variety of simulation algorithms as well as settings.

B. Environment and human simulation

Robot simulations are usually based on 3D rendering engines with physics simulation, in order to reproduce the robot's real environment. In this paper, we focus on mobile robots that operate on the x-y plane, defining the robot workspace on which dynamic obstacles can interact with the robot. For our crowd simulation dedicated to robotics applications, we extend the 2D representation of the crowd with animated avatars walking according to the crowd simulation output. Those avatars are defined with a visual and a collision model, used by the physics engine, to interact with the robot. We use Unity3D in order to have high definition rendering with physics simulation. The crowd simulation library UMANS [START_REF] Van Toll | Generalized microscropic crowd simulation using costs in velocity space[END_REF] is integrated in Unity thanks to a dedicated interface (API). The simulation, running in Unity, can be controlled externally thanks a ROS node, or through a dedicated python module.

C. Robot simulation

In our simulator, a mobile robot is defined as a mechanical model associated to a visual model, as well as a linear and angular velocity command as input to control the robot motion, and outputs that sense the robot' environment. The visual model is a list of 3D models of the joints of the robot associated with a skeleton of the robot, which helps defining the mechanical behaviour of the robot.

Our simulator offers various ways to define the mechanical properties of a robot. First, every robot we implemented can be considered as a kinematic rigid body. This method is useful for simple simulations when physics is not required, and can be used on every robot. The second method consist in considering the robot as a unique point-mass, or a set of connected rigid bodies. With this method, the robot is moved around using forces applies on the base of the robot. Friction limits the mobility of the robot, uniformly or nonuniformly, which is useful to define holonomic robots as well as differential drive robots or car-like robots. The third way to control a robot, dedicated to wheeled robots, is through giving velocity commands to a regulator which directly controls the torque of the simulated motors in the wheels of the robot.

The robot model also defines a set of sensors, which uses the physics engine and the graphics rendering of Unity to generate outputs. The software can simulate various standard sensors: LiDAR, RGBD camera, ultrasound sensors, odometry. Sensor models also contain parameters that can be used to fit any sensor specification. Imperfections, such as noise and missing data, can be added to any sensor for more realism. The simulation also outputs information about the simulation, such as simulation time, physics engine report on agent-robot collision, the crowd position, or crowd mask for LiDAR, which gives the ID of the agent of the crowd detected in a LiDAR scan.

D. Robot navigation plugin

We made a simple python module in order to control the simulation. The module sends data to the simulator: the simulation clock, which controls the time between simulation steps, the simulation controller, which is used to switch between scenarios or to stop the simulation, and a velocity command, directly used by the robot controller in the simulation. When a step is executed by the simulation, the module receives all available data for this step: robot sensor outputs, crowd location and collision report. We also provide a ROS node that uses this python module to control the simulation through ROS topics and ROS parameters and converts the received data into ROS topics. The module is also designed to be used with PythonRobotics [START_REF] Sakai | Pythonrobotics: a python code collection of robotics algorithms[END_REF], a python code collection for robotics algorithms, or more generally any python implementation of robotics algorithms.

E. Initialisation pipeline and outputs

The simulation require a extensive amount of parameters, stored in scenarios files. Scenarios files are read by the Unity simulator and for a given simulation, a scenario file defines the 3D environment, the camera controls (third person view), the robots to load and their initial location, the crowd initial location, the density of the crowd, the flow of the crowd, i.e the set of goals for each agent, the path planner to use for each agent, the crowd reactivity to the robot (reactive, not reactive), and the crowd simulator (UMANS) basic parameters such as preferred speed or maximum acceleration. The scenario will use the default configuration of UMANS, but it can be be easily changed to any of the methods proposed by UMANS in another file. Finally, some parameters are given as input for the python module directly or as ROS parameters if the ROS node is used: the time step of the simulation, the sleep time between steps (for real-time simulation), the ending conditions, and the number of scenarios to run. Also, the python module includes a recorder that saves JSON files of the whole simulation, which is configurable in the python module (file name and location or data to save).

F. Open source project

The benchmark tool is available freely for anyone on http://CrowdBot.eu/CrowdBot-challenge/. On this website, one can download the CrowdBot challenge software which was used to generate the dataset and the benchmark results. It is possible to request access for the source code by contacting the developers on this page. This website also links to the documentation (wiki) of the software which gives details on the functionalities, tutorials and examples, and videos.

IV. CROWD ROBOT NAVIGATION BENCHMARK

Our intention is to use the CrowdBot simulation platform as a benchmark tool to evaluate and compare robot crowd navigation capabilities in terms of safety and efficiency. This section presents both some navigation techniques we have selected from literature to be tested, as well as our evaluation protocol.

A. Navigation techniques and hypotheses

We have selected the 3 following navigation techniques: i) a "Baseline" method, which consists in having the robot going straight toward the goal ignoring the crowd, ii) the dynamic window avoidance method (DWA) [START_REF] Fox | The dynamic window approach to collision avoidance[END_REF], and iii) the reciprocal velocity obstacles method (RVO) [START_REF] Van Den | Reciprocal n-body collision avoidance[END_REF]. The reason for this choice is that the robot will respectively: i) ignore the crowd agents, ii) consider them as static obstacles, and iii) predict the short term future motion of agents. With this choice, the benchmark tool shall demonstrate its capability to reveal benefits and drawbacks on the robot navigation efficiency and safety with respect to crowd agents. If our evaluation tool is adequate to be used as a benchmark, the evaluation should satisfy the following hypotheses: H1: The Baseline method should get the best score in terms of efficiency, since the robot is ignoring the crowd. However, it should be the worst in terms of safety, since it shall move closer to crowd agents and provoke many collisions with them, not even trying to lower the collision forces with the crowd. H2: The DWA method, in comparison with Baseline, should show a lower number of collision, and thus a higher safety level. As the method considers obstacles to be static, it should show quite poor results in terms of efficiency, since it may put the robot on useless or late avoidance trajectories H3: The RVO method should show the best compromise between efficiency and safety, as it is capable of predicting agents' motion and expects contribution from agents in performing avoidance.

B. Benchmark setup

The idea of the benchmark is to propose a number of scenarios that each define the robot, the task of the robot and the activity of the crowd, with a number of parameters. The simulation platform generates trajectories for both the robot and the crowd, which are ultimately evaluated according to several metrics to compare the performances of various navigation techniques in similar conditions. In this paper, we illustrate the benchmark with a limited set of scenarios, which can be extended at will. a) Robot: We choose to illustrate the benchmark on one single robot, the TurtleBot2, a common differential drive robot. The simulated TurtleBot has a maximum velocity of 1 ms -1 and a maximum acceleration of 5 ms -2 . It is controllable with a linear and angular velocity command. The simulation provides the crowd location, the robot odometry and the static obstacles. Those data can be used as an input for the robot navigation. The virtual robot is also equipped with a set of virtual sensors that offer the possibility of testing the robot navigation robustness to limited data: two 2D LiDARs that provide 360 • coverage, a front-facing RGBD camera and 12 simulated ultrasound sensors.

b) Environment and task: the environment is defined as a corridor of 50m by 10m. The robot starts at one side of the corridor and its task is to reach the other side. The goal is reached when the robot travels 40m in the direction of the corridor. We enforces a time limit of 180 seconds to avoid infinite simulation for situations where the navigation does not find a solution. The robot is free to move in this corridor but cannot cross the borders delimited by four walls. The crowd however, can cross the walls. In the case that they traverse a wall, the agent's position is reset at the opposite one of the environment, which maintains a constant level of average density over the entire trial.

c) Crowd activity: Our scenarios are finally defined by a crowd flow direction, a crowd behavior, and a crowd density. The possible crowd flows can be in the main direction of the corridor, pointing toward the same wall as the robot (1D+) or pointing toward the opposite wall, facing the robot (1D-). The crowd flow can also be bidirectional, in the main direction of the corridor, where half of the crowd is moving in the opposite direction of the other half (1Dx). Finally, the crowd can move in the secondary direction of the corridor, crossing the robot path, either in one main direction (2D), or with two opposing flows (2Dx). We selected two main crowd simulation algorithms which are fairly common and easily configurable: "RVO", and "Social Forces". These two algorithms can be configured as reactive or not reactive to the robot. We also chose to set the parameters of RVO with 0.5s of horizon time (the crowd only reacts when close to the robot) or with 1.5s horizon time, which results in more anticipatory interactions. Finally, we selected four levels of crowd density: 50 (0.1pm -2 ), 100 (0.2pm -2 ) , 200 (0.4pm -2 ) or 350 (0.7pm -2 ) agents in the corridor. With all those parameters combined, we end up with 100 scenarios. We randomly selected the initial position of the agents of the crowd and generated the definitive scenarios files to be used by the different robot navigation techniques.

C. Evaluation metrics

For this paper we limited the study to seven metrics which can be split into three categories: path efficiency, effect on the crowd flow, and crowd proximity. The latter two can be regrouped under the topic of crowd safety. In the path efficiency metrics, we compare the situations where the robot is alone to the configuration where the robot is surrounded by a crowd. For those two configurations, we compute and compare the time taken by the robot to reach the goal (T /T cr), the length of the robot path (L/Lcr), and the variation of speed (linear and angular) of the robot during the whole scenario (J/Jcr). The metrics dealing with the effects on the crowd compare the speed of the robot neighbors (within 1m range) to the speed of the whole crowd (N BRvel. =

V N eighbors V All
). The slower the neighbors are, the smaller the metric is. In the second metric, we compare the angular velocity of the whole crowd to the velocity of the neighbors of the robot in scenarios (N BRreac. = ω All ω N eighbors ). If the neighbors rotates more than the rest of the crowd, N BRreac. will be small.

Finally, the metrics dealing with crowd proximity are first the proximity metric (P rox.) which is defined by

P rox = 1 - 1 t f inal t f inal t=0 d min (t)/R
where d min is the distance to the closest agent, and R is the range in meter. In our case, we consider that there is at least 1 person that is in a 5 meter range near the robot. The last metric is the colliding metric, which is defined by

Colliding = 1 -T Collision
T Scenario where T Scenario is the total time of the scenario, and T Collision is the cumulation of the collision instants. A score of 1 means no collisions at all.

D. Collision Assessment

We assess each collision by exploiting the simulator's report of the colliding human body part in order to compute its reflected mass m ref , which approximates the human inertia's contribution to the impact's kinetics. Each colliding body part's m ref is computed as the reflected mass of the part's corresponding segment of a model that groups the human body into a vertical articulated chain of four rigid bodies, subsuming respectively the feet, lower legs, upper legs, and remaining upper body. These segments' lengths and inertial properties are sums of the subsumed parts' values for average adults (according to [START_REF] Winter | Biomechanics and motor control of human movement[END_REF]). Following the approach in [START_REF] Khatib | Inertial properties in robotic manipulation: An objectlevel framework[END_REF], we compute for our model m ref ≈ 4, 13, 24 [kg] for the feet, lower and upper legs, respectively. Assuming the robot's mass as m rob = 20 [kg], we compute the kinetic energy which a collision would absorb (and possibly restore) as ∆E = µv 2 rel /2, where µ = (m -1 ref + m -1 rob ) -1 is the reduced mass and v rel is the relative velocity of both agents' centers towards each other. As ∆E directly quantifies the bodies' potential deformation, it provides a measure for collisions' severity [START_REF] Rossi | A pre-collision control strategy for human-robot interaction based on dissipated energy in potential inelastic impacts[END_REF].

V. RESULTS

A. Benchmark results

We represent the seven metrics described in Section IV-C on two radar charts shown in Fig. 2. Table I gives the standard deviation for each metric of the radar charts.

B. Collision Assessment Results

Fig. 3 shows how many collisions occurred with particular values of energy ∆E during all the simulations with a given robotic controller. To normalize by the simulated time, we

VI. DISCUSSION

This section takes each of the results and gives an interpretation in two levels of comparison: path efficiency and crowd safety.

A. Path efficiency

The path efficiency can be evaluated using the radar chart on Fig. 2. For the metric L/Lcr, an indicator of path length, naturally, the score for the baseline method is 1, as the method will take the shortest path possible in the corridor (which validates H1). RVO is a lot more direct than DWA in both low density and high density crowds, which shows the tendency of DWA to take less direct paths than RVO to reach the goal (validates H2 and H3). This idea is reinforced by the J/Jcr metric which is an indicator of the variation of linear and rotational speed. Indeed, the baseline and RVO methods have a lot of erratic movements (start and stop), while DWA takes the robot dynamics into account and punishes the difference between the current velocity and desired velocity. The metric T /T cr, which is an indicator of the time taken to reach the goal, shows that the most direct method, the baseline, is the fastest in both low and high density scenarios (which validates H1). The two methods RVO and DWA are much slower in high density scenarios. Indeed, such scenarios lead to situations where the only solution for the robot is to totally stop the movements or even move back, facing the starting point instead of the goal.

B. Crowd safety

The crowd safety can be appreciated through the Proximity metric, the Colliding metric, the neighbour-related metrics ("NBR reac." and "NBR vel.") and the histograms of impact energy on Fig. 3.

First, it is interesting to notice that, on the radar chart for low density crowds, the colliding metric has a maximum value for each of the methods which means that in low density (0.1 pm -2 ) the crowd is perfectly capable of avoiding the robot without it using any specific method. In high density scenarios, the baseline has a lower score than the two other methods: the robot does not try to avoid the crowd, thus the risk of collision increases (which validates H1). Also, in high density scenarios, RVO performs better than DWA on the colliding metric (which validates H2 and H3). The crowd reaction to the robot can be evaluated through the crowd velocity metrics: "NBR reac.", and "NBR vel.". The radar charts show that in high density crowds, the robot's neighbors are more likely to change their orientation instead of their speed. In low density, the robot presence does not impact the crowd much, with the exception of the DWA "NBR reac." metric which shows that the robot's neighbors change their orientation a lot more than the rest of the crowd. We think that the robot movement generated by RVO and the baseline methods generate more linear trajectories than the DWA method (which validates H3). Linear trajectories are easier to predict by the crowd simulators we used in the benchmark (which, at best, linearly extrapolate the trajectory). The Proximity is naturally higher in high density than in low density crowds, but the robot navigation method does not seem to affect this metric. The histograms of impact energy in Fig. 3 show that the 3 considered techniques generate a similar distribution of impact energies and thus also impact forces when collisions occur. It is also visible that DWA leads to the most collisions at all energy levels. However, the larger total number results from longer simulations, as the rate of collision f c (Sec. V-B) for DWA is in between both other methods (confirming the colliding metric's ranking). Still, the Baseline collides more often than RVO on high energy, while RVO collide more often than the baseline on low energy, which makes it a good compromise between the number of collision and energy (which validates H1 and H3). Finally, the rates of collision and energy absorption f c and Q (Sec. V-B) are consistently bad for the Baseline, better for DWA, and a lot better for RVO, which validates H1, H2 and H3 and indicates that RVO leads to the lowest number and risk of collisions occurring per unit time.

VII. CONCLUSION

We have introduced a new software to simulate the navigation of robots in virtual crowds. We have demonstrated the potential of such simulator to be used as a benchmark tool to compare various navigation techniques. This paper demonstrate limited results, only sufficient to demonstrate the relevance of this solution to be used as a benchmark tool. Nevertheless, we believe that the solution we propose is of interest for the community and opens new perspective in the evaluation of robot crowd navigation capabilities. For future work, we want to first extend the metrics by which we evaluate the robot behaviour. For example, we would like to measure near-collision situations to evaluate more deeply safety aspects. Other elements than just trajectories should be considered. For example, simulation can further explore the question of sensing the crowd such as by comparing navigation results in various sensors configuration. Another direction is to extend the number of scenarios we consider. The strength of simulation is its potential to explore many situation for the robot, and is yet unexploited here. Devising scenarios of greatest interest should however be done at the community scale.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Radar charts of the benchmark. (a) We compute the seven metrics described in Section IV-C for 20 scenarios: 4 levels of density, 5 crowd flows (see Section IV-B). (b) Similarly, we compute the same radar chart for 5 scenarios with low density only (0.1pm -2 )

TABLE I STANDARD

 I DEVIATION FOR HIGH AND LOW DENSITY RADAR CHARTS

	Navigation method	T/Tcr	L/Lcr	J/Jcr	Metrics NBR reac.	NBR vel.	Prox.	Colliding
	BASELINE High density	0.190	0.00	0.238	1.022	0.070	0.132	0.132
	BASELINE Low density	0.022	0.00	0.244	1.402	0.084	0.164	0.027
	DWA High density	0.257 0.225 0.289	0.488	0.089	0.175	0.117
	DWA Low density	0.171 0.123 0.280	0.0.807	0.096	0.171	0.040
	RVO High density	0.236 0.100 0.015	1.052	0.129	0.141	0.092
	RVO Low density	0.145 0.067 0.029	1.863	0.125	0.149	0.031
	define a particular robotic controller's collision rate as f c =
	N c /T and energy rate Q = Σ∆E/T , where N c , T and Σ∆E
	denote respectively the total number of collisions, elapsed
	time and energy absorption in corresponding simulations. For
	the baseline, DWA, and RVO, we measure f c = 0.624, 0.610,
	0.311 [1/s] and Q = 2.568, 2.285, 0.901 [J/s], respectively.

http://pedsim.silmaril.org/

https://www.cyberbotics.com/

ACKNOWLEDGMENT

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 779942, CrowdBot.


P Mazerolle

• T Meresse

D Gangloff

K Kolsi

A Dupret-Bories

Vascularized lyn1ph node transfer with subn1ental free flap

Keywords:

Limb lymphoedema is common in patients who have undergone inguinal or axillary lymph node dis section. Lymphoedema seriously impacts the patient's quality of life by inducing adipogenesis, fibrosis and repeated episodes of lymphangitis and cellulitis. Following failure of compression therapies, several curative or symptomatic surgical options have been proposed over recent decades, such as liposuction or lymphovenous anastomosis.

Vascularized lymph node transfer techniques have recently been described, with promising results. Vascularized lymph node transfer with submental free flap appears to be the most reliable of these technique, associated with the lowest morbidity. The flap harvesting technique presents several specific differences compared to conventional submental free fla p. Agood knowledge of neck anatomy is essential and multimodal and multidisciplinary management is often required.

ln the light of a case report, we describe the basic submental free flap technique for vascularized lymph node transfer for the treatment of lower limb lymphoedema. This technique may be used by head and neck surgeons performing flap harvest, as well as plastic surgeon surgeons or gynaecologists ensuring long-term management of these patients.

Introduction

Lower limb lymphoedema is common after inguinal lymph node dissection, especially in patients who have received adjuvant radio therapy, as interruption of the main lymphatic drainage pathways leads to interstitialaccumulation ofprotein-rich lymph in the lower limbs. This mechanism leads to chronic inflammation with adipo genesis and fibrosis, or even repeated infections (lymphangitis or cellulitis) with a major impact on the patient's quality of life [START_REF] Beesley | Lymphedema after gynecological cancer treatment: prevalence, correlates, and supportive care needs[END_REF].

The only treatments currently proposed for lower limb lym phoedema are purely symptomatic, designed to reduce the patient's symptoms of swelling and discomfort by means of lymphatic drainage massages or compression therapy. Following failure of these treatments, several surgical options have been pro posed, such as liposuction or lymphovenous anastomoses [START_REF] Allen | Lymphedema surgery: patient selection and an overview of surgical techniques[END_REF].

Vascularized lymph node transfer techniques have recently become more popular. In particular, lymph node transfer to the ankle by means of a submental free flap has provided encouraging results (3.4). The transferred lymph nodes capture interstitial lymph, which then enters the venous circulation via the flap pedicle [START_REF] Cheng | The mechanism of vascularized lymph node transfer for lymphedema: natural lymphaticovenous drainage[END_REF]. However, dissection of this flap requires a good knowledge of the anatomy of the submental and submandibular region to ensure reliable flap harvesting, while avoiding damage to adjacent nerves and vessels or cosmetic sequelae.

Technique

Descriptive anatomy

The submental artery is a constant branch of the facial artery, arising between 5 and 6.5 centimetres from the origin of this artery, deep in the submandibular gland (Fig. 1). It travels deeply under neath the basilar edge of the mandible, above the mylohyoid muscle, then above or through the anterior belly of the digastric muscle. It terminates in the submental region, where it anasto moses with the contralateral submental artery. Ouring its course, it gives rise to several septocutaneous perforating branches also supplying the adipose tissue containing submental (la) and sub mandibular (lb) lymph nodes. Retrograde dissection of the facial Venous return is ensured by the submental vein, draining into the facial vein, according to a satellite course of the artery. The facial vein drains inconstantly into the thyrolinguofacial venous trunk and then into the internal jugular vein, via a more superficial course than that of the facial artery, generally above the posterior belly of the digastric muscle.

During dissection, particular attention must be paid to the marginal mandibular branch of the facial nerve, which forms a loop by crossing over the facial artery in the superficial submandibular region [START_REF] Righini | An original submandibular approach technique sparing the cervical branch of the facial nerve[END_REF]. The hypoglossal nerve, situated in the deep submandibular region, underneath the posterior belly of the digastric muscle, adjacent to the origin of the facial artery, must also be carefully identified.

Flap landmarks and design

It is recommended to perform preoperative neck ultrasound to identify the side with the greatest number and the largest lymph nodes in areas Ia-Ib, in contact with the facial pedicle.

The patient is placed in a conventional head and neck surgery position: head in hyperextension, slightly turned towards the contralateral side. Passage of the facial artery over the mandibular margin can be identified by palpation or Doppler ultrasound, and indicates the superior limit of dissection. Cutaneous perforators derived from the submental artery can be identified by Doppler ultrasound, particularly when a large skin paddle is required. A horizontal elliptical or spindle-shaped skin paddle centred on the submental region is raised, with a maximum transverse long axis of 15 cm and a vertical axis of 10 cm, ensuring sufficient laxity to allow one-stage closure of the flap harvest site. The superior limit of the flap must be situated flush with the basilar edge of the mandible in order to limit cosmetic sequelae. The incision is extended laterally in a skin fold as far as the mandibular angle.

Facial nerve monitoring using an electrode placed over the mentalis muscle is recommended in order to prevent any risk of injury to the marginal mandibular branch of the facial nerve.

Flap harvest

The incision extends as far as the platysma, starting in the lateral part of the skin paddle (Fig. 2). Careful dissection flush with the deep surface of the platysma muscle must be performed from above downwards in order to widely expose the submandibular region. Identification of the marginal mandibular branch of the facial nerve underneath the plane of the superficial cervical fascia may be difficult and time-consuming. However, this step is essential, as this small nerve branch must be carefully dissected so that it can be retracted to allow safe access to the submandibular region.

The anterior border of the sternocleidomastoid muscle must be released in order to identify the jugular vein and carotid artery. The posterior belly of the digastric muscle, indicating the posterior limit of the flap must then be dissected, taking care to preserve a maximum of adipose tissue within the flap (areas IIA and posterior Ib).

The facial artery and vein are then dissected over the inferior edge of the mandible. The submental pedicle arises about 1 cm underneath the basilar edge of the mandible. After identifying the submental artery, the facial pedicle can then be ligated distally to the origin of this pedicle. The submental artery is then dissected distally starting at its origin, by continuing the skin incision of the superior part of the flap. The anterior belly of the digastric muscle can be included to ensure a stronger flap. Meticulous haemostasis of the deep muscle perforating branches must be performed at this stage. All of the adipose tissue as far as the basilar edge of the mandible must be included to ensure a maximum of lymph nodes in the flap and to avoid injuring the septocutaneous perforating vessels. The inferior incision of the skin paddle of the flap is then completed as far as the suprahyoid muscle plane, completely releasing the medial extremity of the flap.

Finally, the facial artery and facial vein are dissected via a retrograde approach, in contact with the submandibular gland, allowing the inclusion of all adipose tissue of the submandibular (Ib) and subdigastric (IIa) areas. This dissection is continued as far as the internal jugular vein and external carotid artery, where the facial pedicle can be ligated flush with these vessels. Particular care must be taken when dissecting the subdigastric course of the facial artery, as the hypoglossal nerve crosses the facial artery in this region. The submental flap therefore comprises lymph node areas Ia (underneath the skin paddle), Ib (without the submandibular gland) and IIa. The submandibular gland is consequently almost totally released and only remains adherent by its inferior edge and partially by its deep surface. The donor site is then closed by sutures after placing a suction drain.

Preparation of the flap recipient site and flap placement

The preferred flap recipient site is a medial retromalleolar site with anastomoses onto the posterior tibial artery (Fig. 3). An S-shaped incision is made and large subcutaneous dissection is performed to create a compartment for insertion of the flap. The recipient vascular pedicle is readily identified in the medial retromalleolar region. End-to-end anastomoses are performed after release of the posterior tibial vessels from their sheath. Tensionfree closure is essential to avoid compromising venous drainage of the flap. A non-suction drainage system is placed at the flap recipient site. When necessary, a large calibre venous anastomosis with the long saphenous vein at the anterior part of the incision can be performed.

Postoperative care

Close clinical surveillance must be ensured for the first 48 hours according to local protocols. The main risk is that of impaired venous drainage due to excess tension in the recipient zone. In the presence of a congested or "venous" appearance of the flap, tension must be immediately decreased by releasing sutures or even revising anastomoses in the operating room. Elevation of the limb helps to promote venous return. Drainage systems are removed starting on the second postoperative day depending on the volume drained. Weight bearing can be resumed on the day after the operation. Prophylactic anticoagulation is continued for two to three weeks, until the patient has regained satisfactory mobilization. The use of a compression stocking on the operated side depends on the quality of venous return of the flap and may need to be continued until normal venous return has been restored.

Case report

To illustrate this procedure, we present the case of a 49-year-old man managed by vascularized lymph node transfer with submental free flap for secondary lymphoedema of the right lower limb, one year after resection of a right inguinal melanoma associated with inguinal and pelvic lymph node dissection followed by adjuvant radiotherapy (48 Gy) with good cancer control, as he was considered to be in remission following this therapeutic sequence.

He progressively developed lymphoedema of the lower limb, first observed at the third postoperative month, which became progressively worse despite compression therapy and lifestyle changes. After presentation and discussion of the various treatment options with the patient, it was decided to perform vascularized lymph node transfer with submental free flap, according to the technique described above.

The postoperative course was uneventful despite a congested appearance of the flap immediately after the operation, although the flap maintained good capillary refill and skin trophicity. Reduction of lymphoedema was observed and experienced by the patient by the second postoperative day, with 30% reduction of lymphoedema one week after the operation, possibly enhanced by perioperative bedrest. The patient was discharged on the tenth postoperative day with unrestricted mobilization. At 5 months, the patient presented a marked reduction of limb circumference, essentially in the distal part of the lower limb, decreasing from 45 cm to 37 cm 20 cm below the knee and from 61.5 cm to 60.8 cm 20 cm above the knee (Fig. 4 A andB). The patient also reported a marked reduction of his symptoms. Lymphoscintigraphy performed 5 months postoperatively demonstrated lymphatic drainage on the medial aspect of the right ankle via the transferred lymph nodes (Fig. 4 C andD).

Discussion

Although the submental free flap has been described for a long time, harvesting of this flap with the inclusion lymph nodes must comply with a number of rules to ensure functional success of the procedure on lymphoedema, as reduction of lymphoedema is directly dependent on the number of lymph nodes present in the flap [START_REF] Gustafsson | Correlation between quantity of transferred lymph nodes and outcome in vascularized submental lymph node flap transfer for lower limb lymphedema[END_REF]. Harvesting of the submental free flap remains difficult and anatomically demanding, particular in order to preserve adjacent nerves, and must be performed by a surgeon experienced in neck surgery, regardless of his or her primary specialty.

Creation of a submental skin paddle presents a number of advantages. In addition to allowing surveillance of flap vitality, this skin paddle is often necessary to allow tension-free skin closure of the recipient site, a zone with limited elasticity often associated with marked fibrosis of the cutaneous and subcutaneous tissues as a result of chronic lymphoedema. Skin is also known to be richly endowed in lymphatic capillaries and may consequently contribute to better results in terms of drainage of the lymphoedema via the dermis and the lymph nodes of the flap. This skin paddle is generally removed at a second operation for cosmetic purposes, after resolution of the lymphoedema. In the future, flap harvest techniques without a skin paddle could be useful to minimize the cosmetic sequelae of flap donor and recipient sites.

According to two recent reviews of the literature of the various reported lymph node transfer free flaps, the advantage of the submental flap is the length of its pedicle (about 5 cm), with a large diameter artery (about 3 mm), constant lymph nodes with a mean of 3 lymph nodes included in the flap, i.e. an equivalent number of lymph nodes to that provided by an inguinal flap, the flap most commonly used at the present time in this indication. The submental flap also avoids the risk of secondary lymphoedema of the donor site, the main iatrogenic risk associated with inguinal or axillary flaps. However, harvesting of a submental free flap is associated with a risk of injury of the marginal mandibular branch of the facial nerve and cosmetic sequelae related to the visible neck scar [START_REF] Pappalardo | Vascularized lymph node transfer for treatment of extremity lymphedema: an overview of current controversies regarding donor sites, recipient sites and outcomes[END_REF][START_REF] Scaglioni | Comprehensive review of vascularized lymph node transfers for lymphedema: outcomes and complications[END_REF].

Only limited data are available in the literature concerning the intermediate and long-term results of the submental flap in this indication. According to two recent studies, the estimated success rate, defined by improvement of lymphoedema, is between 70 and 100% [START_REF] Pappalardo | Vascularized lymph node transfer for treatment of extremity lymphedema: an overview of current controversies regarding donor sites, recipient sites and outcomes[END_REF][START_REF] Scaglioni | Comprehensive review of vascularized lymph node transfers for lymphedema: outcomes and complications[END_REF]. No consensus has been reached concerning patient selection or the choice of donor and recipient sites for lymph node transfer techniques, due to the low level of proof currently available (retrospective studies based on small series) and the poorly elucidated mechanisms of action of transferred lymph nodes. Lymph node transfer in the head using a supraclavicular flap has also been described, but the results of the procedure are more uncertain due to the inconstant presence of lymph nodes in this zone (area Vb) [START_REF] Steinbacher | The surgical anatomy of the supraclavicular lymph node flap: a basis for the free vascularized lymph node transfer[END_REF]. No comparative study has been conducted between vascularized lymph node transfer and lymphovenous anastomoses, the main alternative technique for the treatment of lymphoedema.

Although this technique primarily concerns head and neck surgeons, the indication for vascularized lymph node transfer must be discussed with practitioners specialised in the management of lymphoedema, which remains multimodal in the context of a global treatment strategy [START_REF] Kung | Current concepts in the surgical management of lymphedema[END_REF]. Two-team surgery in this clinical setting also allows a gain of time by operating on the two sites concomitantly, in addition to the combined expertise of the head and neck and plastic surgeons.

Conclusion

Vascularized lymph node transfer is an increasingly popular technique, which raises great hopes for the treatment of lymphoedema. The submental free flap is now one of the preferred techniques in this setting, but it requires a good knowledge of head and neck surgery in order to ensure safe and reliable flap harvesting. It therefore constitutes a multidisciplinary procedure, in which the patient's preference and motivation play a predominant role.

Fig. 1 .

 1 Fig. 1. Surgical anatomy of the submandibular gland. a: facial artery; b: facial vein; c: submental pedicle; 1: marginal mandibular branch of the facial nerve (VII); 2: hypoglossal nerve (XII); asterisks: lymph nodes.

Fig. 2 .

 2 Fig. 2. Appearance of the flap after dissection. Asterisks: lymph nodes larger than 1 cm in areas Ib (black asterisk) and IIa (yellow asterisk). White arrow: submental pedicle.

Fig. 3 .

 3 Fig. 3. Medial retromalleolar recipient site before (left) and after (right) submental flap placement. Note the slightly congested appearance of the flap after placement despite tension-free skin closure.

Fig. 4 .

 4 Fig. 4. Appearance of the lower limb preoperatively (A) and 5 months postoperatively (B). Lymphoscintigraphy performed 5 months postoperatively showing lymphatic drainage of the right lower limb on the medial aspect of the right ankle via the transferred lymph nodes (C, D).

Acknowledgements

Fanny Cros, Pauline Jeanneton.

Disclosure of interest

The authors declare that they have no competing interest.