Serge Romaric Tembo

Tung The

Didier Nguyen

El-Baz

Distributed Iterative Solution of Numerical Simulation Problems on Infiniband and Ethernet Clusters via the P2PSAP Self-Adaptive Protocol

Keywords: distributed computing, cluster computing, Infiniband, micro-protocols, task parallel model, asynchronous algorithms

The distributed iterative solution of numerical simulation problems on Infiniband or Ethernet Clusters via the P2PDC environment is studied. The P2PDC decentralized environment is dedicated to task parallel applications. It has been designed for the solution of large scale numerical simulation problems via distributed iterative algorithms. The P2PDC environment is based on the P2PSAP self-adaptive communication protocol. New functionalities of the P2PSAP communication protocol aimed at using Infiniband clusters are presented. A series of computational results is presented and analyzed.

I. INTRODUCTION

In order to obtain good efficiency of High Performance Computing (HPC) applications, new transport protocols have to be designed. Existing transport protocols like TCP [START_REF]Transmission Control Protocol (TCP)[END_REF] and UDP [START_REF]User Datagram Protocol (UDP)[END_REF] were originally designed to provide ordered and reliable transmission to the application or datagram service, respectively and are no longer adapted to real-time and distributed computing applications. We also note that TCP and UDP cannot reconfigure their own structure. Moreover, the message-based transport protocol seems better suited to HPC applications than the classical stream-based communication.

Recently, new transport protocols have been standardized like SCTP [START_REF]Stream Control Transmission Protocol (SCTP)[END_REF] and DCCP [START_REF] Kohler | Datagram Congestion Control Protocol (DCCP)[END_REF]. Nevertheless, these protocols still do not offer the modularity needed in the context of HPC.

The P2PSAP self-adaptive communication protocol [START_REF] Baz | A self-adaptive communication protocol with application to high performance peer-to-peer distributed computing[END_REF] and P2PDC distributed computing environment [START_REF] Cornea | Performance prediction in a decentralized environment for peer-to-peer computing[END_REF] have originally been designed for peer-to-peer HPC applications. The P2PSAP self-adaptive communication protocol is based on the Cactus framework whereby composite protocols and micro-protocols are combined in order to build a desired communication protocol [START_REF] Hiltunen | The Cactus approach to building configurable middleware services[END_REF]. In this paper, we concentrate on new functionalities of the P2PSAP communication protocol aimed at using Infiniband clusters. The impact of the underlying network on the distributed solution of numerical simulation problems is also studied. We consider distributed synchronous or asynchronous iterative algorithms. We recall that asynchronous iterations are relative to a very general model whereby computations are carried out in parallel without order nor synchronization (see [START_REF] Bertsekas | Parallel and Distributed Computation: Numerical Methods[END_REF]). They permit one to consider distributed iterative computations where machines go at their own pace according to their intrinsic characteristics and computational load.

The paper is structured as follows. Related work is presented in Section II. Section III is devoted to new features of the P2PSAP protocol aimed at using Infiniband networks. Section IV deals with distributed iterative algorithms for the solution of a numerical simulation problem and displays experimental results. Section V deals with conclusions and future work.

II. RELATED WORK

MPICH Madeleine [START_REF] Aumage | MPICH/Madeleine: a true multi-protocol MPI for high performance networks[END_REF] has been designed for testbeds with several types of networks. Nevertheless, the modification of internal transport protocol mechanisms in function of elements of context like iterative schemes of computation and network topology is not allowed with MPICH Madeleine. P2P-MPI [START_REF] Genaud | A Peer-to-Peer framework for message passing parallel programs[END_REF] is a framework aimed at developing messagepassing programs in large scale distributed networks of computers. P2P-MPI is developed in Java and makes use of Java TCP sockets to implement the MPJ (Message Passing for Java) communication library. P2P-MPI uses a single super-node in order to manage machine registration and discovery that may become a bottleneck. P2P-MPI implements a fault tolerance approach using machine replication that may not be efficient and appropriate to connected problems, since the number of machines involved in the computation will multiply greatly.

The P2PDC decentralized environment was originally designed for peer-to-peer distributed computing. It is particularly dedicated to task parallel applications. The environment P2PDC is intended in particular to scientists who want to solve numerical simulation problems via distributed iterative methods that lead to direct and frequent data exchanges between machines like synchronous or asynchronous iterative methods [START_REF] Bertsekas | Parallel and Distributed Computation: Numerical Methods[END_REF]. The P2PDC environment relies on the P2PSAP selfadaptive communication protocol that implements a reduced set of communication operations (P2Psend, P2Preceive and P2Pwait) in order to facilitate programming [START_REF] Baz | A self-adaptive communication protocol with application to high performance peer-to-peer distributed computing[END_REF]. The programmer cares only about the choice of distributed scheme of computation, e.g., synchronous or asynchronous, that he wants to be implemented and does not care about the communication mode between any two machines. The P2PSAP communication protocol chooses dynamically the most appropriate communication mode between any two machines according to decision taken at application level like scheme of computation and elements of context like network topology at transport level. The P2PSAP protocol can also reconfigure itself dynamically to match the very nature of a given algorithm like in the case of the distributed solution of nonstationary problems whereby some synchronizations are needed at every time steps even if an asynchronous scheme of computation has been selected to solve each subproblem. Reference is made to [START_REF] Nguyen | High performance peer-to-peer distributed computing with application to obstacle problem[END_REF] and [START_REF] Cornea | Performance prediction in a decentralized environment for peer-to-peer computing[END_REF] for studies related to the centralized and decentralized versions of the P2PDC environment, respectively.

III. P2PSAP AND INFINIBAND

The P2PSAP communication protocol has a socket interface and two channels: a control channel and a data channel (see Figure 1). The control channel manages session opening and closure; it captures context information and (re)configures the data channel at opening or operation time; it is also responsible for coordination between machines during reconfiguration process. The data channel is a stack of composite protocols based on the Cactus framework (see [START_REF] Hiltunen | The Cactus approach to building configurable middleware services[END_REF]). The data channel transfers data packets between machines. The stack of protocols is globaly divided into two layers: a transport layer which is an extension called CTP* of the Configurable Transport Protocol (CTP) [START_REF] Wong | A configurable and extensible transport protocol[END_REF] and a physical layer that was first designed for IP/Ethernet networks and thanks to the work presented in this paper can encompass also Infiniband networks (see Figure 1). Thus, our contribution concerns the design and insertion of an Infiniband physical layer inside the stack of protocols. We aim also at facilitating self-adaptation of the P2PSAP protocol according to the iterative scheme of computation considered, e.g., synchronous or asynchronous, the location of machines, i.e., in a cluster or in different clusters and the type of network used, e.g., Ethernet or Infiniband. According to the Cactus point of view, the added Infiniband physical layer is a composite protocol that implements communication on the underlying Infiniband network.

We recall that a machine is connected to an Infiniband network via a Host Channel Adapter (HCA) that can have several ports identified by a Local ID (LID) and a Global ID (GID). The transport, network and data link layers are implemented in the silicon of the HCA. The HCA uses internal Direct Memory Access (DMA) engine in order to transfer or retrieve data directly to or from the main memory of a machine. During the application, several communication requests are created that contain information like address of the communication buffer and data size. Each request is transmitted to the appropriate Send Queue (SQ) or Receive Queue (RQ) via the Infiniband Verbs interface. The HCA executes the requests according to the order they are posted and stores the results called Completion Queue Elements (CQE) in the appropriate Completion Queue (CQ).

The insertion of a new layer in the data channel communication stack is not independant of the control channel since it configures and manages the data channel. The control channel detects the type of underlying network at session opening. If the underlying network is Infiniband, then the resources (LID, GID, SQ, RQ, CQ and communication buffers) that are called Queue Pair (QP) context in Infiniband specifications (see [START_REF]Infiniband architecture specification[END_REF]) and that are used by the Infiniband composite protocol IB of the data channel are created once for the entire communication duration by the control channel at data channel opening since each consumed Infiniband Verb has a nonnegligible overhead; this is particularly true if the Verb necessitates a kernel transition of the operating system like the memory region registration Verb that is used to allocate memory to communication buffer. The data channel is an Infiniband Reliable Connected (RC) channel in the case of synchronous communication or an Infiniband Unreliable Connected (UC) channel in the case of asynchronous communication since asynchronous iterations can tolerate message loss. We note that the creation of communication resources is not sufficient to establish a RC or an UC Infiniband channel. Each machine must also know some information regarding the remote QP context. This information includes LID, GID, QP Number (QPN) in addition to initial packet sequence number which is randomly generated by control channel. The exchange of this information is done via the control channel prior to any communication operation of the data channel. As soon as a machine obtains information regarding the remote QP context, the machine will enter the Ready To Receive (RTR) state and then the Ready To Send (RTS) state. During the configuration of data channel, the control channel passes the RTS QP Context to the IB Layer.

The stack of protocols relative to the Infiniband network is as follows: CTP*/IB/Verbs/Infiniband (see Figure 2), where CTP* takes into account Synchronous and Asynchronous micro-protocols. IB is the composite protocol that implements Infiniband communications using resources initially created by the control channel. CTP* contains only basic micro-protocols: TransportDrive that adds port identifiers to each CTP* segment, Resize for segmentation/reassembly and Synchronous or Asynchronous. Functionalities like congestion control, flow control, and error recovery required for synchronous communication are implemented by the HCA in order to reduce CPU protocol overhead. The micro-protocol composition of CTP* depends on the context, i.e., the type of iterative scheme, e.g., synchronous or asynchronous, the location of machines, e.g., intra or inter cluster and the type of underlying network, e.g., Infiniband or Ethernet. Table I shows micro-protocols composition of CTP* in the different contexts.

Unlike in the Ethernet asynchronous context, the Ethernet synchronous context requires reliability in order to ensure that application is not going to be blocked by message loss. This is ensured via SequencedSegment, Retransmit, RTTEstimation, PositiveAck and DuplicateAck micro-protocols. Moreover, this context requires an order delivery property which is ensured via the ReliableFIFO micro-protocol. In order to behave fairly with other flows, congestion control is also required in the case with inter clusters communications whereby we must have reliable flows, this is ensured via Window Congestion Control (denoted by WindCongCtrl) and TCP NewReno Congestion Advoidance (TCPNewRenoCA) microprotocols. In the Ethernet asynchronous context with inter clusters communications whereby we have unreliable flows, congestion control is ensured via DCCPAck, DCCP Window Congestion Control (DCCPWindCCtrl) and TCP Congestion Avoidance (TCPCongAvoid) micro-protocols.

The composite protocol IB that we have carried out essentially implements two operations of communication:

-the PUSH operation which sends a CTP* segment on the Infiniband network (see Figure 3); -the POP operation which retrieves a CTP* segment from the network and delivers it to the upper layer (see Figure 4).

Sending a message via Infiniband consists in building and posting a Send Work Request (SWR). The physical memory addresses contained in the Work Queue Element (WQE) must be registered by the consumer. Here, the consumer is the IB composite protocol which performs a mapping between the socket addresses known by CTP* sessions and QP contexts

Eth IB Eth TransportDrive X X X X X X X X Resize X X X X X X X X Synchronous X X X X Asynchronous X X X X SequencedSegment X X PositiveAck X X Retransmit X X RTTEstimation X X ReliableFIFO X X DuplicateAck X WindCongCtrl X TCPNewRenoCA X DCCPAck X DCCPWindCCtrl X TCPCongAvoid X
known by HCA. However, waiting for a message to be generated by CTP* before registering a memory region will add a delay to message processing before the transmission over the network. The Resize micro-protocol is introduced in CTP* sessions in order to give a precise value of the maximum size of a CTP* segment to the IB layer, making possible static registration.

At data channel establishment, the control channel registers a memory region that serves as communication buffers. This pinned region is divided into two parts: a send buffer and a receive buffer. Figure 3 displays how data are stored in the buffer in order to send a segment. In the sequel, we detail the management of free spaces in the buffer. The send buffer is divided into 1000 blocks with 8128 bytes. Each block can contain only one CTP* segment with 8128 bytes. For each CTP* segment to send, the Infiniband composite protocol IB looks for a free block in the send buffer in order to store the data; the data are then written in the block. The address of the block and the size of the segment are included in the SWR posted to the SQ. A thread continuously polls the CQ. When a Completion Queue Element (CQE) containing the identifier (ID) of a SWR is retrieved from CQ, the block that was used by this SWR can be reused without any risk of inconsistency between two successive writings on this block. A relation between the SWR ID and the block position in the send buffer allows the process which continuously polls the CQ to find out the block and set it free by inserting its position at the end of the list of free blocks.

Note that registration overhead is reduced by reusing blocks of the send buffer. The number of blocks is limited to 1000 in order to guarantee a reasonable memory consumption. A linked list is used to store the position of free blocks in order to accelerate the search for free blocks. The IB layer uses always the block whose position is at the head of the linked list to post a SWR, the head is then moved to the next item; this position is inserted at the end of the list when the SWR Completion is retrieved from the CQ. So, the IB layer always keeps track of the position of free blocks in the send buffer and does not waste time to find out a free block. Figure 3 illustrates the process:

-the search for a free block (step 1); -writing the data segment in the free block, building and posting a SWR (step 2); -retrieving the result of a SWR from CQ and releasing the block used (step 3).

Figure 4 displays how IB retrieves data from the receive buffer. The receive buffer is initially divided into 1000 blocks with 8128 bytes (see step 1). At data channel establishment, the control channel posts 1000 RWR (Receive Work Request) to RQ. Each RWR contains a block address. When the HCA receives the data, it consumes the first WQE in the RQ, writes the data in the block and places a CQE in the CQ. When IB retrieves a CQE, it calculates the position of the block from the ID of Receive WQE (RWQE) consumed (step 2). The data of the segment contained in the block that was found are copied to another memory location. This allows the reuse of the address of this block in another RWR built and posted immediately (step 3) simultaneously with the delivering of the segment to CTP* after a demultiplexing operation required to find out the appropriate session at higher level, i.e., the CTP* session (step 4).

IV. EXPERIMENTAL RESULTS

We consider the obstacle problem which occurs in mechanics and finance and can be formulated as follows. ∂Ω. The discretization of the obstacle problem leads to the following fixed point problem.

       F ind u * such that A.u * -f ≥ 0, u * ≥ φ everywhere in domain Ω, (A.u * -f )(φ -u * ) =
F ind u * ∈ V such that u * = F (u * ), (1)
where V is an Hilbert space and the mapping F : v → F (v) is a fixed point mapping from V into V . We consider the distributed solution of fixed point problem (1) via the projected Richardson method combined with several iterative schemes of computation; reference made to [START_REF] Nguyen | High performance peer-to-peer distributed computing with application to obstacle problem[END_REF] for the mathematical formulation of synchronous and asynchronous projected Richardson methods. The convergence of asynchronous projected Richardson method has been established in [START_REF] Spiteri | Parallel asynchronous Richardson method for the solution of obstacle problem[END_REF]. The choice of scheme of computation has important consequences on the efficiency of the distributed iterative algorithm. The interest of asynchronous iterations for various problems including boundary value problems and optimization has been shown in [START_REF] Bertsekas | Parallel and Distributed Computation: Numerical Methods[END_REF], [START_REF] Baudet | Asynchronous iterative methods for multiprocessors[END_REF], [START_REF] Baz | M-functions and parallel asynchronous algorithms[END_REF], [START_REF] Baz | Asynchronous gradient algorithms for a class of convex separable network flow problems[END_REF], [START_REF] Baz | Asynchronous iterations with flexible communication: contracting operators[END_REF], [START_REF] Chau | MPI implementation of parallel sub-domain methods for linear and nonlinear convectiondiffusion problems[END_REF].

We display now and analyze experimental results. We concentrate on distributed synchronous and asynchronous iterative algorithms carried out via P2PDC and P2PSAP for a 3D obstacle problem and parallelepiped-based domain. We consider a cubic domain with 256 3 points. Experiments have been carried out on the Graphene cluster of the Grid'5000 testbed [START_REF]Grid5000 platform[END_REF]. Machine characteristics and the sequential computational time are given in Table II. Table III displays the computational time of synchronous and asynchronous distributed algorithms on cluster Graphene with up to 64 machines and 1 Gb/s Ethernet or 20 Gb/s Infiniband network. Figure 5 shows the efficiency of synchronous and asynchronous distributed algorithms on Infiniband or Ethernet network; the different iterative methods are denoted by synIBA, asynIBA, synETH and asynETH, respectively. Experimental results show that synchronous iterative algorithms carried out on Ethernet network do not scale up well. Their efficiency with 64 machines of the same cluster is close to 30 % (see Figure 5). The new functionalities of P2PSAP aimed at using Infiniband networks permit one to improve the efficiency of synchronous iterative algorithms (the increment is up to 30%). This is mainly due to the reduction of synchronization overhead and fastest communication with Infiniband. Infiniband has low latency and high bandwidth; the Infiniband Adapter has also capability to bypass kernel which off-loads some protocol processing from the CPU; this saves CPU cycles to the benefit of computation. We note also that the combination of asynchronous iterative algorithms with P2PDC is very efficient due to the lack of synchronization overhead and idle time resulting from synchronization. The implementation of these algorithms on Infiniband network permit one to further improve their efficiency.

V. CONCLUSION AND FUTURE WORK

We have proposed extensions to the P2PSAP communication protocol in order to natively use Infiniband network for HPC applications. These new features have permitted us to improve the efficiency of all distributed iterative algorithms that we have carried out, i.e., synchronous and asynchronous algorithms. In particular, we note that the combination of asynchronous iterative algorithms with P2PDC using Infiniband is very efficient: the efficiency is close to 82% with 64 machines.

We are presently working on improvements to the Infiniband functionalities of the P2PSAP protocol in order to accelerate data transmission by bypassing fragmentation/reassembly at CTP* level to finally have only one send request per message instead of one request per CTP* segment. Leaving this functionality to HCA will reduce CPU time wasted in protocol handling. We also plan to deal with heterogeneity. More precisely, we will manage communications between Ethernet clusters and Infiniband clusters and between distants Infiniband clusters connected via an IP/Ethernet network. ACKNOWLEDGMENT Part of this study has been made possible with the support of ANR-07-CIS7-011 grant. Experiments presented in this paper were carried out using the Grid'5000 experimental testbed.

Fig. 1 .

 1 Fig. 1. P2PSAP protocol architecture.

Fig. 2 .

 2 Fig. 2. Data channel communication stack.

Fig. 3 .

 3 Fig. 3. PUSH operation.

  0 everywhere in Ω, B.C., where φ ∈ R 2 (or R 3 ) is an open set, A is an elliptic operator, φ a given function and B.C. are the boundary conditions on

Fig. 4 .

 4 Fig. 4. POP operation.

TABLE I TRANSPORT

 I LAYER OF P2PSAP, COMPOSITION OF MICRO-PROTOCOLS,

	Context		synchronous		asynchronous
			intra	inter	intra	inter
	Micro-protocols	IB	Eth	IB	Eth	IB

TABLE II MACHINE

 II CHARACTERISTICS AND SEQUENTIAL COMPUTATIONAL TIME

	Site	Cluster	Processor	Mem.	Time
	Nancy	Graphene	Intel Xeon 2.53 GHz	16 Gb	9523 s


Eduardo Campos Mercado

Frédéric Comby

Vincent Creuze
email: vincent.creuze@lirmm.fr

Elaba Jorge Antonio Torres Muñoz

Dolores Antonio

Olivier Strauss

Jesus Monroy Anieva

E Campos
email: ecampos@upp.edu.mx

J Torres

E Antonio

J Monroy

AUV pipeline following by artificial vision

Keywords: Underwater vehicle, PD Controller, Control, Underwater vision, Real-time experiments

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I. INTRODUCTION

The development of Remotely Operated Vehicles (ROV) and Autonomous Underwater Vehicles (AUV) has known a great growth due to their wide application. Nowadays these vehicles are used to perform marine exploration tasks, hydrographic surveys, monitoring, among others (for example [START_REF] Stewart | A Passive Sonar System Based on an Autonomous Underwater Vehicle[END_REF], [START_REF] Jaulin | Bertholom Localization of an AUV using setmembership methods[END_REF], [START_REF] Kleeman | Advanced Sonar and Odometry Error Modeling for Simultaneous Localisation and Map Building[END_REF]). A classical example is the inspection of marine structures, which is currently performed by ROV type submarines equipped with a camera, allowing the pilot to control the movements of the vehicle with respect to the object of interest. However, this task is not easy and requires trained operators to handle the vehicle. A better option for this kind of tasks is the use of AUV submarines. In this case, a sensing and control strategy is implemented, so as to autonomously control the movements of the vehicle. Given the nonlinear dynamics and the difficulty to identify the hydrodynamic parameters involved in vehicle dynamics, these topics still remain of interest for research (see [START_REF] Newman | Marine Hydrodynamics[END_REF], [START_REF] Lamb | Hydrodynamics[END_REF], [START_REF] Fossen | Handbook of Marine Craft Hydrodynamics and Motion Control[END_REF]).

The mini submarine used in this work, called Lirmia 2, offers the capability of being used either as a ROV or as an AUV, depending on the performed task. In the present case, we will use it as an AUV to track a pipeline using an artificial vision algorithm. One of the most widely used technique for the identification of ducts is based on the application of the Hough's transformed, with which straight lines can be identified, corresponding to the edges of the duct. The drawback of this method is that it is not possible to identify the lines corresponding to the edges of the pipe when this latter is formed by very long curved lines (see [START_REF] Ortiz | A vision system for an underwater cable tracker[END_REF], [START_REF] Muhammad Asif | An Active Contour for Underwater Target Tracking and Navigation[END_REF]). For this reason, we have implemented another artificial vision algorithm robust to the different phenomena induced by the underwater image acquisition process as well as by the disposition of duct.

The control strategy implemented in our vehicle is based on a PD control technique with gravity compensation, taking into account the restoration forces and moments generated by buoyancy and weight of the vehicle. This paper is organized as follows: In section II, we briefly describe the dynamic model of the mini-submarine Lirmia 2. The artificial vision algorithm is presented in section III. In section IV, we describe the control strategy. The real-time experiments are presented in section V. Finally, concluding remarks and future works are given in section VI.

II. DYNAMICAL MODEL

The Lirmia 2 submarine is depicted in Figure 1, with its body fixed frame (O b , x b , y b , z b ). The center (O b ) of this latter frame corresponds to the center of gravity of the vehicle, and its axes are aligned with the main axes of symmetry of the vehicle. The motion in the horizontal plane is referred as surge (along x b axis) and sway (along y b axis), while heave represents the vertical motion (along z b axis). Roll, pitch, and yaw, denoted (φ, θ, ψ), are the Euler angles describing the orientation of the vehicle's body fixed frame with respect to the earth-fixed frame (O I , x I , y I , z I ), while (x, y, z) denote the coordinates the center of the body-fixed frame in the earth fixed frame. The propulsion system consists in six thrusters, as depicted in Figure 2, which generate the rotational and translational motion. Concerning the rotational motion of this vehicle, yaw control is performed through differential speed control of the thrusters 3, 4, 5 and 6. Pitch control is obtained similarly using thrusters 1 and 2, whereas the roll motion is unactuated. On the other hand, the translational motion of the z axis is regulated by decreasing or increasing the combined speed of thrusters 1 and 2; similarly, the translational motions along the x b and y b axes are obtained by using thrusters 3, 4, 5, 6 and by controlling the yaw angle.

The dynamics of the vehicle expressed in the body-fixed frame can be written in a vectorial setting according to Fossen [START_REF] Fossen | Guidance and Control of Ocean Vehicles[END_REF]:

M ν + C(ν)ν + D(ν)ν + g(η) = τ + w e (1)
η = J(η)ν (2)
where M ∈ R 6×6 is the inertia matrix, C(ν) ∈ R 6×6 is the coriolis-centripetal matrix, D(ν) ∈ R 6×6 represents the hydrodynamic damping matrix, g(η) ∈ R 6×1 describes the vector of gravitational/buoyancy forces and moments,

τ = (τ 1 , τ 2 ) T = ((τ X , τ Y , τ Z ), (τ K , τ M , τ N )) T ∈ R 6×1
is the vector of control inputs; w e ∈ R 6×1 is the vector of disturbances; ν = (ν 1 , ν 2 ) T = ((u, v, w), (p, q, r)) T ∈ R 6×1 denotes the linear and angular velocity vector in the bodyfixed frame; η = (η 1 , η 2 ) T = ((x, y, z), (φ, θ, ψ)) T ∈ R 6×1 is the position and attitude vector decomposed in the earth-fixed frame, and J (η) ∈ R 6×6 is the transformation matrix between body and earth-fixed frames (more details, see [START_REF] Goldstein | Classical Mechanics. Addison Wesley Series in Physics[END_REF], [START_REF] Marsden | Elementary Classical Analysis[END_REF]).

A. Gravity/Buoyancy forces and torques

According to Archimedes' principle, the buoyancy force f B applies on the center of buoyancy and acts in the opposite direction of vehicle weight f W . This leads to:

f B = - 0 0 ρg∇ f W = 0 0 mg (3)
where ρ represents the fluid density, g the gravitational acceleration, ∇ the displaced fluid volume and m the mass of the vehicle. Now, the weight and buoyancy forces can be transformed to the body fixed coordinates system by:

F B = J 1 (η 2 ) -1 f B F W = J 1 (η 2 ) -1 f W (4)
Considering that W = mg and B = ρg∇ and using the zyx-convention for navigation and control application (see, [START_REF] Fossen | Marine control systems guidance, navigation, and control of shipd, rigs and underwater vehicles[END_REF]), then the matrix transformation is J 1 (η 2 ) = R z,ψ R y,θ R x,φ . Therefore, the forces with respect to the body fixed frame are written as:

F B = Bsin(θ) -Bcos(θ)sin(φ) -Bcos(θ)cos(φ) F W = -W sin(θ) W cos(θ)sin(φ) W cos(θ)cos(φ) (5)
Thus, the restoring forces acting on the vehicle are

f g =F B +F W , this is f g = (B -W )sin(θ) (W -B)cos(θ)sin(φ) (W -B)cos(θ)cos(φ) (6)
On the other hand, the restoring moments are described by the following equation

m g = r w × F W + r b × F B ( 7
)
where r w = [x w , y w , z w ] T and r b = [x b , y b , z b ] T represent respectively the positions of the center of gravity of the vehicle (CG) and of the center of buoyancy (CB). Based on the design of the vehicle and in order to reduce computations, the origin of the coordinate system fixed to the body is placed in the center of gravity, so that r w = [0, 0, 0] T ; while the center of buoyancy is r b = [0, 0, -z b ] T . For practical reasons, the buoyancy force is greater than the weight, that is W -B = -f b, but -f b should be smaller than the force produced by the thrusters. Then from equation ( 6) and ( 7), we have:

g(η) = f g m g =        f b sin(θ) -f b cos(θ)sin(φ) -f b cos(θ)cos(φ) -z b Bcos(θ)sin(φ) -z b Bsin(θ) 0        (8)

B. Forces and torques generated by the thrusters

Figure 2 shows the forces generated by the thrusters acting on the mini submarine, these are described relative to the bodyfixed coordinate system, as summarizing and using the notation of [START_REF]Nomenclature for Treating the Motion of a Submerged Body Through a Fluid[END_REF], we have that

f = 0 0 f 1 ; f = 0 0 f 2 ; f = f 3 0 0 f = f 4 0 0 ; f = f 5 0 0 ; f = f 6 0 0
τ  = τ X τ Y τ Z = f 3 + f 4 + f 5 + f 6 0 f 1 + f 2 (9)
and the body-fixed torques generated by the above forces, are defined as

τ 2 = 6 i=1 l i × fi (10)
where l i = (l ix , l iy , l iz ) is the position vector of the force fi ∀ i = 1, .., 6, with respect to the body-fixed reference frame. Then the torques generated by the thrusters are described as

τ  = τ K τ M τ N = 0 l 1x (f 1 + f 2 ) l 3y (f 3 -f 5 ) + l 4y (f 4 -f 6 ) (11)
Now considering that two thrusters are connected to the same driver, we assumed that f 1 =f 2 , f 3 =f 4 and f 5 =f 6 , then

τ =        2f 3 + 2f 5 0 2f 1 0 2l 1x f 1 (f 3 -f 5 )(l 3y + l 4y )        (12)

III. ARTIFICIAL VISION ALGORITHM

Assuming that the camera is fixed to the vehicle, the aim of our vision algorithm is to estimate the relative angular position (ψ R ) of the camera with respect to the pipeline. This algorithm has to be robust to ill knowledge of the pipeline model and has to be of low complexity to be integrated in the onboard calculator. Our approach is mainly based on the algorithm proposed by Zingaretti [START_REF] Zingaretti | Robust real-time detection of an underwater pipeline[END_REF]. This method analyses the cumulative profiles to select candidate contour points in the edge maps. In this paper, we introduce two changes to the algorithm, namely the introduction of a Gaussian filter in order to reduce the noise produced primarily by changes in lighting and irregularities on the pipeline and an optical flow estimation to determine the direction and magnitude of movement in the image plane. These changes improve the robustness of the algorithm towards sudden movement of the vehicle. All the steps of the vision algorithm are illustrated in Figure 4. First Gaussian filter is applied in order to reduce the noise produced primarily by changes in lighting and irregularities on the pipeline. Then, a Canny edge detector is used to produce a binary image by assigning ∀ h ∈ H i , w ∈ W i : I(h, w) = 1 (white color) to the pixels that belong to an edge, and I(w, h) = 0 (black color) for the others, where I(h, w) is the value of the pixel located at position (h, w), H i the image height, and W i the image width. This is illustrated in Figure 3 (a). The next step is to divide the height of the image in B i regions, so the image is split into B i blocks with a height of H i /B i and a length of W i , as depicted in Figure 3 (a). Subsequently that the image has been divided, the horizontal profile of each region is computed and stored in a vector of W i elements, as shown in Figure 3 (b). Usually these profiles have values larger at the edges of the duct. In some cases, mainly due to noise in the image or phenomena like algae or rocks there will be peaks that do not correspond to the duct edges. Dividing images into B i blocks allows to isolate outlier edges in the next steps.

Three methods are alternatively used to perform the profile analysis depending on the state of previous analysis. Their goal is to provide the coordinates of a pair of points associated to the duct edge. These three methods are called: Initialisation, N arrow and Broad search. The Initialisation method (IN I) is used in the detection of the pipe when the position of the pipe is not available on previous image. The N arrowsearch (N S) is the faster and most robust procedure of the three, as it exploits the information coming from previous image processing, this method is used when the position of the pipe was found in the previous image. Finally, the Broadsearch (BS) method is used when the N S method fails, mainly as a consequence of abrupt horizontal and vertical oscillations during the pipeline following.

The IN I method consist in choosing a pair of coordinates for each region, corresponding to the edge of the right and left side of the pipeline, the first point provided by the method IN I corresponds to the first peak found watching graphs from left to right, and the first peak viewing graphs from right to left.

The N S method replaces the first one after that duct detection was performed successfully. This second method consists in looking for potential points forming the boundary of the pipeline in an area near to the coordinates of the previous points. We propose to use the optical flow, in order to estimate the displacement of the duct edges. The Lucas-Kanade (LK) algorithm [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] is implemented in order to estimate the optical flow; the basic idea of this algorithm is based on two assumptions: brightness constancy and small displacements, it means we assume that the brightness of a pixel does not change as it is tracked from image to image, and that the magnitude of displacement between two images remains small. It leads to the well known optical flow equation :

I(w, h, t) = I(w + △w, h + △h, t + △t) ( 13
)
where I(w, h, t) is the brightness of the previous image, I(w + △w, h + △h, t + △t) is the brightness of the current image, △w, △h are the incremental changes of w and h, respectively, corresponding to the increment in time △t. Then, using the LK algorithm we obtain the values of △w and △h, which determine the search area for the N S method, this allows the artificial vision algorithm to be more robust to sudden movements of the vehicle.

The BS method is very similar to the IN I method, with the advantage that additional information is known for the pipeline detection. This method considers the distance, measured in pixels, between the points of the duct edge defined in the previous image. Once all the points are selected, we use the least squares method for curve fitting and we obtain the equations that represent the location of the points corresponding to the left and right edge of the duct. We also compute a linear correlation coefficient (ccl), which indicates the correlation between the points found and the line obtained by the curve fitting. Therefore, the value of ccl is used as an evaluation criterion for deciding which of the three methods (IN I, N S, BS) has to be applied. Finally, if ccl ≥ 0.8, we estimate the ψ R value from the equations of the straight lines representing the edges of the duct. The resulting artificial vision algorithm is described in Figure 4.

IV. CONTROL STRATEGY

For the design of the controller, it is common to assume that the hydrodynamic parameters involved in the dynamical model of the underwater vehicle are unknown. Indeed, they depend on effects and properties that are difficult to model or estimate, like added mass, skin friction, vortex shedding, characteristics of fluid, etc. Therefore, we propose to use a Proportional Derivative (PD) controller. Since in our case we know the gravity vector g(η), we apply a gravity compensation within the PD controller. Although there are many other control strategies for underwater vehicles, in this paper, we have chosen a P D+g(η) controller as the main objective is to estimate the ability of the vehicle to perform vision based autonomous pipeline following. Considering the nonlinear system given by equation ( 1) and ( 2), we propose the following control input:

τ = g(η) -J T (η)τ P D ( 14
)
with

τ P D = K p e(t) + K d de(t) dt ( 15
)
where e = ηη d represents the state error, and the gains of the controller are denoted K p = K T p > 0 and

K d = K T d > 0.
Then, introducing equation ( 14) into (1), the closed loop system can be written:

M ν + [C(ν) + D(ν)]ν = -J T (η)τ P D + w e ( 16
)
now considering that η d = cte,gives:

M ν + [C(ν) + D(ν)]ν = -J T (η)[K p e + K d η] + w e ( 17
)
introducing equation ( 2) into (17) and introducing K dd = J T (η)K d J (η), we obtain:

M ν + [C(ν) + D(ν) + K dd ]ν + J T (η)K p e = w e (18)
A Lyapunov candidate function for system (18) is

V = 1 2 ν T M ν + 1 2 e T K p e (19)
and since in our applications the vehicle will only be allowed to move at low speed, we assume that M = M T > 0. This suggests that V is globally positive definite function. Now, the time derivative of this function is expressed as:

V = ν T M ν + ėT K p e (20)
since η d = cte, then ėT = ηT =ν T J T (η), therefore:

V = ν T [M ν + J T (η)K p e] (21)
Now introducing equation (18) into the above equation, we have:

V = ν T [w e -[C(ν) + D(ν) + K dd ]ν] (22)
As C(ν) is a Skew-symmetric matrix, then ν T C(ν)ν = 0, ∀ ν, and we assume that w e ≈ 0, therefore:

V = -ν T [D(ν) + K dd ]ν (23)
We can notice that V =0 when ν=0. Now we can use LaSalle's theorem in order to prove that closed loop system is globally asymptotically stable, therefore:

Ω = [(ν, e) ∈ R n : V (ν, e) = ] (24)
then

Ω = [e ∈ R n , ν =  ∈ R n ] (25)
From equations ( 18) and (25), we can notice that: (ν, e) = (, ) is the only initial condition in Ω for which (e, ν)∈ Ω for all t≥0. This means that the equilibrium point is globally asymptotically stable according to LaSalle's theorem.

In the case where the integral action is included, it is also possible to prove the local asymptotic stability, see [START_REF] Arimoto | Satability and Robustness of PID feedback Control for Robot Manipulators of Sensory Capability[END_REF].

V. REAL-TIME EXPERIMENTS

The goal of the real-time experiments is to observe the behavior of the vehicle, test the implemented artificial vision algorithm and analyze the response of the closed loop system. Figure 5 shows the prototype that we have developed for pipeline following. For this, the experiments have been performed with a constant thrust, this is τ X = 4N . This suggests that the vehicle´s speed along the x b axis is constant (u = cte). The weight and the buoyancy force of the vehicle are approximately 186.3N and 191.2N , respectively. Considering the design of the vehicle and its low speed, we have considered that the pitch and roll angle remain close to the equilibrium point (φ = 0, θ = 0) without a control action. Therefore, the control strategy is focused on immersion (z) and yaw motion (ψ). The values used for the controller were kp z = 130, kd z = 80, kp ψ = 1.9, kd ψ = 4.1 and f b = -4.9N . To estimate the performance and the robustness of the system, a pipe has been placed on the bottom of a pool, drawing a quasi circular loop. The diameter of this loop was about 7 meters, as can be seen on Figure 5. The embedded system of "Lirmia 2" is shown on Figure 6. It consists in an embedded computer with an Intel Atom Z550 2GHz CPU and a 1GB DDR2-533 RAM memory. This embedded system also includes an inertial measurement unit (UM6 Orientation Sensor, CH Robotics), a Logitech webcam Pro 9000, and a pressure sensor. The computer's operating system is Windows XP embedded. Using Visual C++, the computer processes the data from the sensors, and then computes and sends the control inputs to the actuators. The force to keep the vehicle close to z d is shown on Figure 8. One can notice that the initial force is bigger than during the rest of the experiment. This effect is produced when the vehicle is close to the surface since the thrusters are not immersed enough and then need to turn faster in order to produce the same force as for the rest of the experiment. Figure 9 introduces the performance of the relative angle ψ R between the pipeline and the vehicle when applying the proposed control law. In our case, the goal is to keep the vehicle aligned with the pipeline while the AUV is moving forward. This suggests that the desired angle between the vehicle and the pipeline (ψ Rd ) should be equal to 0. One observes that the ψ R angle remains bounded (-22 • to 12 • ). Notice, that this behavior is generated because the vehicle is turning around the pipeline formed by straight ducts which draws a quasi circular loop. Indeed, this induces sudden changes in the ψ R 's measurement. This effect could be compared to frequent disturbances. Therefore the control action induces fast changes in the heading of the vehicle in order to remain aligned with the pipeline.

The torque τ N (yaw) generated by the lateral thrusters along the axis (Oz b ) is shown on Figure 10. Notice that this torque is saturated in order to prevent damages on the thrusters. Most of the time the torque was positive because the pipeline was drawing a circular path, as can be seen on Figure 5. Figure 11 shows the pitch angle response when the translational velocity u is not null. One observes that before t = 10s the pitch angle is close to zero as u = 0. After this time, the pitch angle is close to 4 deg, since the vehicle starts moving forward. Notice that the oscillations are induced by the movement and the immersion dynamic. This 4 deg angle cannot be avoided as the vehicle is non holonomic. In fact state variables z and θ are coupled (see, equations ( 9) and ( 11)). Figure 12 shows the unactuated roll angle response when the vehicle starts moving forward. In this case, the roll angle is close to zero before t = 10s. After this time, the roll angle is close to 2.5 deg due to unmodeled hydrodynamic effects and the oscillations are smaller than for the pitch angle.

VI. CONCLUSION AND FUTURE WORK

This paper described the design and the experimental testing of pipeline following performed by an autonomous underwater vehicle. A new prototype with an embedded control system has been developed and presented. This vehicle performs real time embedded image processing in order to implement an artificial vision algorithm. The obtained results show the good ability of the vehicle to follow the pipeline, without losing it. In a next future, we will implement another control strategy aiming to improve the performances along z and ψ. Experiments in presence of external disturbances and with less visibility will be soon conducted in natural environment.

Figure 1 :

 1 Figure 1: The Lirmia 2 vehicle, its body fixed frame (O b , x b , y b , z b ), and the earth-fixed frame (OI , xI , yI , zI ).

Figure 2 :

 2 Figure 2: Forces fi ∀i = [1...6] generated by the six thrusters of the vehicle. Position of the center of gravity, CG.

Figure 3 :

 3 Figure 3: The Image produced by Gaussian filter and Canny's method is shown in Figure (a) and the horizontal profile of each region are shown in Figure (b).

Figure 4 :

 4 Figure 4: Description of the artificial vision algorithm.

Figure 5 :

 5 Figure 5: Lirmia 2 prototype and the experimental setup (pipe lying on the bottom of a pool).

Figure 6 :

 6 Figure 6: Hardware architecture of the Lirmia 2.

Figure 7

 7 Figure 7 shows (in red) the vehicle´s trajectory along the z axis (depth). One can notice that the vehicle remains close to the reference (in blue) (z d =0.25 m) with small oscillations induced by the translation movement.

Figure 7 :

 7 Figure 7: Desired (in blue) and measured (in red) depth trajectory of the vehicle.

Figure 8 :

 8 Figure 8: Control input of the thrusters along z axis.

Figure 9 :

 9 Figure 9: Relative angle ψR (in red) between the AUV (Ox b ) axis and the pipeline.

Figure 10 :

 10 Figure 10: Torque N along the z b axis during the pipeline following.

Figure 11 :

 11 Figure 11: Pitch angle θ the vehicle´s trajectory

Figure 12 :

 12 Figure 12: Roll angle φ along the vehicle´s trajectory.


Philippe Weber
email: philippe.weber@cran.uhp-nancy.fr

Didier Theilliol
email: didier.theilliol@cran.uhp-nancy.fr

Philippe Poure

Shahrokh Saadate

S Saadate+

Reliability analysis in a fault tolerant control strategy

Keywords: Fault Tolerant Control, System Reliability, Active power filter, Control Reconfiguration

come

INTRODUCTION

In most conventional control systems, controllers are designed for fault-free systems without taking into account the possibility of fault occurrence. In order to overcome these limitations, modern complex systems use sophisticated controllers which are developed with fault accommodation and tolerance capabilities, in order to meet reliability and performance requirements. The Fault Tolerant Control (FTC) system is a control system that can maintain system performance closely to the desirable one and preserves stability conditions, not only when the system is in fault-free case but also in the presence of faulty component, or at least ensures degraded performances which can be accepted as a trade-off. FTC has been motivated by different goals for different applications; it could improve reliability and safety in industrial processes and safety-critical applications such as flight control and nuclear power plant operation [START_REF] Zhang | Bibliographical review on reconfigurable fault-tolerant control systems[END_REF].

Various studies on FTC are based on hardware or analytical redundancy. The hardware redundancy technique consists of switching from the failed part of the process to another achieving the same task. The analytical redundancy is an alternative to solve the FTC problem avoiding the disadvantages of the hardware redundancy such as high cost and space. In this paper, the main goal is to determine the "best" reconfiguration structure among various with or without redundant components.

Generally, Fault Tolerant Control systems are needed in order to preserve the ability of the system to achieve the objectives that has been assigned when faults or failures occurred. [START_REF] Staroswiecki | From control to supervision[END_REF] proposes a terminology on fault tolerant control problems. The main goal of FTC is to increase system's reliability. Some publications have introduced reliability analysis for fault tolerant control systems. Also [START_REF] Staroswiecki | Sensor network design for fault tolerant estimation[END_REF] have proposed a sensor reconfiguration based on physical redundancy where the reliability analysis provided some information in order to select the optimal redundant sensors. In, [START_REF] Wu | Reliability of fault tolerant control systems: Part I[END_REF], [START_REF] Wu | Reliability of fault tolerant control systems: Part II[END_REF], [START_REF] Wu | Reliability and supervisory control[END_REF] Markov models are used to dictate the system reliability where it's supposed that the sub-systems take two states intact (available) or failed (unavailable).

In this paper, a FTC strategy, based on reliability analysis dedicated to an active power filter is considered. The reliability of power electronic equipments becomes extremely important in industry applications [START_REF] Bobbio | Dependability analysis of faulttolerant systems: a literature survey[END_REF], [START_REF] Bobbio | Architectural factors influencing the reliability of fault-tolerant VLSI arrays[END_REF]. The fault mode behaviour of static converters, protection and fault tolerant control of voltage source inverter systems has been covered in a large number of papers. Most of them are focused on induction motor drive applications. According to (El [START_REF] Brouji | Study and comparison of fault tolerant shunt three-phase active filter topologies[END_REF], various shunt three phase active filter topologies or reconfiguration structure can be considered in faulty cases with a classical hardware redundancy or suitable structures. Based on the highest reliability criterion, the FTC method provides an optimal structure in order to achieve desired objectives closed as possible as nominal ones.

The paper is organized as follows. Section 2 is dedicated to define the associated standard problem of FTC. Section 3 is devoted to the three legs shunt active filter topologies in fault free and faulty cases. Section 4 concerned to the reliability computation according to the various topologies. A simulation example is considered in Section 5 to compare the various topologies. Finally, concluding remarks are given in the last section. , in the spirit of [START_REF] Staroswiecki | From control to supervision[END_REF], where:

• g γ Global objectives • C
A set of constraints given by the structure of system S , and the parameters of closed-loop system θ

> < U C g , , γ
, which has a solution that could achieve nom g γ by changing the structure, parameters and/or control law of the post-fault system (which results in the disconnection or replacement of faulty sub-systems). In some cases, no solution may exist, and then global objectives must be redefined to the degraded ones, denoted as d g γ .

Under assumptions that there exist several structures

m S ) , , 1 ( M m K =
, the problem statement is formulated by the following question: how to choose the optimal structure in the sense that for a given criterion J the chosen structure can maintain the objectives nom g γ ? An answer will be provided in the following section based on reliability analysis dedicated to an active power filter. Compared to [START_REF] Guenab | Fault tolerant control method based on costs and reliability analysis[END_REF], and due to the specific application, the reliability analysis will be studied off-line. determine the current harmonic reference to be generated by the active filter as proposed by [START_REF] References Akagi | Generalized theory of the instantaneous reactive power[END_REF] using classical active and reactive power method.

SHUNT ACTIVE FILTER TOPOLOGIES

It is composed by a grid (esi for i = {1, 2, 3}), a nonlinear load and a voltage source converter. The load is a three-phase diode rectifier feeding an (R, L) series load. The grid is supposed to be balanced with equal series resistances r s and inductances l s for each phase. The static converter is a voltage source inverter with equal series resistances r f and inductances l f for each phase. The output currents of the shunt active filter are controlled to provide reactive power and harmonic currents generated by the non-linear load to ensure filtering. The capacitor C dc of the DC filter side is an energy storage capacity. According to (El [START_REF] Brouji | Study and comparison of fault tolerant shunt three-phase active filter topologies[END_REF], two fault tolerant shunt active filter schemes are more particularly studied, i.e.

e s1 i s1 i f3 e s2 e s3 r s l s r s l s r s i s2 i s3 i c1 i c2 i c3 l c r c l c r c l c r c r f l f l f l f 1 2 3 i f2 i f1 L R l s V dc T 3 T 2 r f C dc r f T 6 T 5 T 4 r f

2) M

, (1, m = = K
. These structures use connecting and isolating devices; they allow reconfiguring the power converter topology after fault detection of one of the semi-conductor devices. These two fault tolerant filter structures are based on the classical and healthy structure presented in Fig. 1.

The first one, 1 S presented Fig. 2, is a three legs topology. The three legs are used in healthy condition. This structure includes bi-directional devices T rk (k = {1, 2, 3}) (triacs in this case to simplify control) allowing the connection between the grid (for the phase number k having a faulty semiconductor device) and the middle point O of the capacitor C dc . Under this topology, if one of the semiconductor T i or its associated switch driver is faulty, the leg including T i could be isolated and the point k will be directly connected to the middle point O by switching on the triac T rk . Consequently, the fault tolerant topology is a two legs structure with faulty phase connected to O.

e s1 i s1
i f1

e s2 e s3 r s l s r s l s r s

i s2 i s3 i c1 i c2 i c3 l c r c l c r c l c r c r f l f l f l f 1 2 3 i f2 i f3 L R l s T 3 T 2 r f T6 T5 T4 f 1 T 1 f 1 f 1 f 6 f 4 f 3 f 2 f 5 Tr3 Tr2 T r1 Vdc 2 C dc

Cdc

Vdc 2 e s1 O Fig. 2: Fault tolerant three-legs shunt active filter: 1 S .

The second structure 2 S , is based on hardware redundancy: a four-leg filter topology. As presented in Fig. 3, the fourth leg composed of devices T 7 and T 8 is a redundant leg. This leg will replace if necessary the faulty one of the three other legs. Bidirectional devices allow connecting the faulty phase number k to the leg constituted by T 7 and T 8 .

i s1 i f3 T i s2 i s3 i c1 i c2 r f l f l f 1 2 3 i f2 i f1 L R f 2 f 5 T r3 T r2 T r1 v dc C dc T 8 T 7 l s e s1 r s r c l c r c l c r c l c i c3 T 2 T 3 T 6 T 4 T 5 e s2 e s3 l s l s r s r s r f r f r f f 1 f 3 T 1 f 4 f 6
Fig. 3: Fault tolerant four-legs shunt active filter: 2 S .

The same shunt active filter control is available for three-legs and two-legs structures as illustrated in Fig. 4. The major advantage of the control law U u ∈ is to be suitable for the set of fault-free nom S

and faulty

m S ) , , 1 ( M m K = structures.
According to this control scheme, the same reference voltage V dcref can be used for the Fault tolerant fourlegs shunt active filter. The reconfigured structure is the same as in fault-free case. However, concerning the fault tolerant three-leg shunt active filter, illustrated in Fig. 2, the structure is initially composed of a three legs and becomes a two legs topology after reconfiguration. The V dcref value changes after reconfiguration simultaneously with converter reconfiguration. The new V dcref should be determined twice greater than in fault-free case.

Increasing the V dcref allows decreasing the frequency of T i commutations.

As presented in this section, according to [START_REF] Brouji | Study and comparison of fault tolerant shunt three-phase active filter topologies[END_REF], various shunt three phase active filter topologies or reconfiguration structure can be considered in faulty cases with a classical hardware redundancy or suitable structures with

MARKOV CHAIN MODEL OF RELIABILITY

Representing system failure in a probabilistic way is attractive because it naturally accounts the uncertainty. To make such representation, the process behaviour is considered as a random variable that takes its values from a finite state space corresponding to the possible process states. Deterioration can be modelled by a discrete time stochastic process. In the case of finite or countable state space, Markov processes can be represented by a graphic called Markov Chain [START_REF] Ansell | Practical methods for reliability data analysis[END_REF].

Markov chain models a sequence of random variables { }

] ) ( ),..., ( ),... ( ) ( 1 M k m k k k s X p s X p s X p X p = = = = with 1 ) ( 1 = = ∑ = M m m k s X p (1)
In this paper, only constant failure rates are considered. Homogenous Markov chains are used for this case because of the stationarity of transition probabilities. In such model, transition probabilities are time invariant and depend only on values states:

) | ( ) | ( 1 2 1 j k i k j k i k ij s X s X p s X s X p p = = = = = = + + + (2)
In a homogeneous discrete-time MC, the transition matrix P MC between the states is defined from failure rate parameters. For instance let us consider a system (Fig. 5) with two states {1,2}, then: Given an initial distribution over states ) ( 0 X p , the probability distribution over states after k stage ) ( k X p is obtained from the Chapman-Kolmogorov equation:

      - = 1
MC K i k X p X p P 0 0 ) ( ) ( = ∏ = (4) Assuming that { } l i ... 0 ∈
represents the functioning states, system reliability is defined as:

( ) { } ∑ ∈ = = l i i k s X p k R ... 0 ) ( (5)
The use of classic MC to model deterioration in systems needs to enumerate all possible states which lead sometimes to a huge transition matrix.

RESULTS AND COMMENTS

Performances analysis

This simulation results are obtained with Saber CAD simulator for the two fault tolerant topologies. General simulation parameters are equal to:

-Grid : 230V, 50 Hz -Non-linear load : R = 0,6 Ω, L = 2,5 mH, -l c = 15µH, r c = 0,4 mΩ -Filtering capacitor : C dc = 17,6 mF.

Depending on the studied topology, hysteresis band value and the V dcref are defined in order to achieve a THD value below 5%.

Fig. 6 presents results in an open circuit case, the fault of the bottom switch of the leg number 3 is introduced at t=200 ms for three-legs shunt active filter structure 1 S . The value of the capacitor voltage reference is set to 800 V before the fault in the threelegs topology and set to 1300 V after the fault in the two-legs topology reconfiguration. The value of l f is 150µF and the hysteresis band equal to 20 A. The proposed fault tolerant system preserved the main performance features after fault compensation. More, the mean switching frequency is equal to 7 kHz before fault and equal to 9.5 kHz after fault. for the same control law U u ∈ .

Reliability analysis

Due to the fact that the number of commutations decrease when the voltage increase, consequently, the failure rates of T i decrease:

( ) ( ) 800V Ti λ 350V Ti λ ≥ .
However, the failure rate of the capacitor increase with the voltage:

( ) ( ) 700V C λ 350V C λ dc dc ≥
.

The following s represents the state of the system for the reconfigured structures with the presence of a failure on T i . It should be noted that the Fig. 9 and 10 include the commutation ability of the three and four legs shunt solutions. .

2 1 ( ) ( ) 350V Ti λ 6 - 1 × × α 3 ( ) ( ) 800V Ti λ 4 800V Cdc λ 2 × + × ( ) ( ) 350V Ti λ 6 350V Cdc λ 2 × × + × α
Fig. 9. Markov Chain for modelling the system with three legs shunt 1 S .

2 1 ( ) ( ) 350V Ti λ 6 - 1 × × α 3 ( ) ( ) 800V Ti λ 6 700V Cdc λ × + ( ) ( ) 350V Ti λ 6 700V Cdc λ × × + α

S

The reliability curves of the systems are shown in Fig. 11. With a hardware redundancy or a suitable structure, the reliability is greater than a nominal structure. Neverthless, in regard with the parameters, defined in Table 1, the three-legs shunt system is less preferment in the beginning but become more interesting in a long horizon. Consequently, according to the performance requirements, it is convenient to get the optimal structure as the threelegs topology.

Other scalar criteria such as MTTF could be evaluated to choose the structure with the highest reliability. Thank to the Markov model, the MTTF is computed and equal to 9 375 h for the initial structure, 18 162 h for the four legs shunt structure, 24 240 h for the three legs shunt structure. It should be appreciated that the analysis of the MTTF confirms the previous choice: the optimal structure is the three-legs topology (according to Table 1).

CONCLUSIONS

In this paper, a Fault Tolerant Control strategy dedicated to an active power filter which incorporates reliability analysis into the reconfigurable control structure selection has presented. Once a fault occurred, the proposed FTC strategy will switch to the suitable structure. The selected structure will guarantee an optimal performance of the reconfigured system according to the "highest" reliability in order to ensure the dependability of the active power filter system and the "human safety".

Fig. 1

 1 Fig. 1 presents a classical three-leg shunt power active filter which represents the nominal structure nom S . The nominal global objectives nom g γ

Fig. 1 :

 1 Fig.1: Classical three-leg shunt active filter: nom S .

Fig. 4 :

 4 Fig. 4 : Block diagram of the control system.

  structure with the highest reliability.

  set of the possible mutually-exclusive states of each k X . The probability distribution over these states is represented by the vector p(X k ):

[

  Fig. 5. Markov Chain for modelling deterioration model

Fig. 6 .

 6 Fig. 6. Current and voltages dynamic behaviour for 1 S

Fig. 7 .

 7 Fig. 7. Current and voltages dynamic behaviour for 2 S

Fig. 8 .

 8 Fig. 8. Markov Chain for modelling the system with the nominal structure nom S .

Fig. 10 .Fig

 10 Fig. 10. Markov Chain for modelling the system with four legs shunt 2 S .

  table presents the value of the failure rate according to the voltage and also the Mean Time To Failure (MTTF). It can be noted that the variable α represents the switched failure probability of triacs T rk . ). The state 1 s corresponds to the fault-free state. If a failure occurs on capacitor or T i , the system is out of order which corresponds to the state 2 s for the

	Failure rates 1.67 E-05 1 E-05 0.5 E-05 0.67 E-05 1 E-05 12.5 E-05 (All these values are not real parameters) ( ) 350V Ti λ ( ) 800V Ti λ ( ) 350V Cdc λ ( ) 700V Cdc λ ( ) 800V Cdc λ α Table 1: Failure rates of the elementary components MTTF (h) 60 000 100 000 200 000 150 000 100 000 8 000 Based on section 4, the fault-free structure and the two reconfigured structures are represented as Markov Chain models as illustrated in the figures 8, 9 and 10 (for simplicity t ∆ is considered equal to 1 structure reconfigured structures. The state 2 nom S (see Fig. 8.) or	3 s for the


Matthieu Peyre

Ph.D Marc Zanello

M.D Karima Mokhtari

M.D Anne-Laure Boch

Ph.D Laurent Capelle

M.D Alexandre Carpentier

Ph.D Stephane Clemenceau

M.D Carine Karachi

Ph.D Soledad Navarro

M.D Aurelien Nouet

M.D Vincent Reina

M.D Charles-Ambroise Valery

M.D Marc Sanson

Ph.D Philippe Cornu

Ph.D Michel Kalamarides
email: michel.kalamarides@psl.aphp.fr

Patterns of relapse and growth kinetics of surgery-and radiation-refractory meningiomas

Keywords: Anaplastic Meningioma, Atypical meningioma, Recurrence, Natural history, Tumor progression

Introduction

Meningiomas are the most frequent central nervous system tumors in the adult population [START_REF] Ostrom | CBTRUS statistical report: Primary brain and central nervous system tumors diagnosed in the United States in 2006-2010[END_REF].

Most meningiomas are benign (WHO grade I) but there is a significant proportion of tumors classified as WHO grade II (atypical meningioma) or WHO grade III (anaplastic/malignant meningioma), and these exhibit more aggressive clinical behavior [START_REF] Durand | WHO grade II and III meningiomas: a study of prognostic factors[END_REF] and have a higher risk of recurrence [START_REF] Aghi | Long-term recurrence rates of atypical meningiomas after gross total resection with or without postoperative adjuvant radiation[END_REF][START_REF] Sughrue | Outcome and survival following primary and repeat surgery for World Health Organization Grade III meningiomas[END_REF] with increased mortality [START_REF] Hanft | A review of malignant meningiomas: diagnosis, characteristics, and treatment[END_REF]. For several decades, the mainstay of treatment has been surgical resection for all meningioma grades, and post-operative radiotherapy in grade III tumors. But for a fraction of Grade II and most Grade III meningiomas, the current therapeutic strategies fail to achieve tumor control and are followed by significant morbidity and mortality. The lack of effective chemotherapies results in part from our limited understanding of atypical/malignant meningioma biology despite numerous molecular studies [START_REF] Goutagny | Genomic profiling reveals alternative genetic pathways of meningioma malignant progression dependent on the underlying NF2 status[END_REF]. But the assessment of the efficacy of potential innovative treatments is also limited by our incomplete knowledge of multi-recurrent meningioma natural history especially concerning tumors with malignant transformation [11,[START_REF] Krayenbuhl | De novo versus transformed atypical and anaplastic meningiomas: comparisons of clinical course, cytogenetics, cytokinetics, and outcome[END_REF]. A recent meta-analysis underlined the absence of response criteria or well-documented benchmarks regarding overall survival (OS), progression-free survival (PFS), or 6-month PFS (PFS-6) for recurrent meningioma [START_REF] Kaley | Historical benchmarks for medical therapy trials in surgery-and radiation-refractory meningioma: a RANO review[END_REF].

Here we perform a retrospective review of all surgery-and radiotherapy-refractory meningioma cases followed in our department focusing on the patterns and kinetics of tumor recurrence. We clearly identified and characterized three distinct patterns of tumor recurrence, highlighting the frequency of multi-nodular relapses and the variability of growth rates among concomitant nodules in the same patient.

Material and methods

Patient population. The aim of the study was to describe a specific patient population diagnosed with histologically-aggressive meningiomas eligible for a potential clinical trial for recurrent treatment-refractory tumors. Patients were selected from the database of the Department of Neuropathology from 1998 to 2013. Patients with Grade I meningiomas and/or long-term relapses (more than 10 years after first surgery) as well as patients with staged resections were excluded from the study. There was a total of 491 WHO Grade II meningiomas and 64 Grade III meningiomas operated upon during this period. In order to determine tumor kinetics, we selected patients with interpretable radiological follow-up, presenting at least one relapse with two distinct modalities of treatment and with a MRI at diagnosis and at surgery. The local institutional review board approved this retrospective chart review. All patients or their parents provided informed consent.

Imaging analysis. All tumors volumes were measured on 3D T1 Gadolinium volumetric sequences using OsiriX Software and a semi-automated algorithm for tumor segmentation (SegmentiX) by a single investigator (M.P). This algorithm was already validated for segmentation of other intracranial lesions, including brain arteriovenous malformations [START_REF] Clarencon | Elaboration of a semi-automated algorithm for brain arteriovenous malformation segmentation: initial results[END_REF].

To validate this algorithm in meningioma measurements, we performed a manual segmentation in 10 tumors at diagnosis and compared the results with the volume determined by semi-automated segmentation leading to a mean difference of 4.2% (median: 3.1% -range: 1 -9.3%). When meningiomas recurred not as a single tumor mass, but as a series of spatially distinct tumor nodules, all tumor nodules were measured separately. Only tumors greater than 1 cm in diameter were considered in the analysis. There was a total of 271 MRI scans analyzed with mean number of 9.3 MRI scans per patient (range: 3 -24 MRI scans). A linear fit model was used to estimate meningioma growth. To estimate the slope of the growth curve over time for each tumor nodule and each relapse, we performed linear regressions of the tumor volume over time, using the volumes at the time of relapse (initial volume) and at surgery (final volume). Tumor growth rate was calculated using the following formula: (final volume -initial volume)/follow-up interval. Time to tumor progression was defined as the time necessary to achieve a 20% increase in tumor volume starting from baseline. This criteria is based on studies performed in Neurofibromatosis type 2 patients who harbor meningiomas [START_REF] Plotkin | Suggested response criteria for phase II antitumor drug studies for neurofibromatosis type 2 related vestibular schwannoma[END_REF]. This choice is based on previous definitions of tumor response and progression in early phase II trials in Neurofibromatosis type I. This also corresponds roughly to a 2 mm change in diameter of a 22 mm diameter, which would be the typical threshold for reliable clinical measurement of a minimal change in a discrete tumor such as meningioma.

Anatomopathology. All cases were retrospectively reviewed by one pathologist (K.M) to determine tumor grade according to 2007 WHO criteria.

Results

Clinical characteristics of the cohort. Twenty nine patients, 14 females (48%) and 15 males (52%), with multi-recurrent meningioma were included in the study. The mean number of relapses per patient was 2.9 (range, 1-6). All patients underwent surgery with a mean number of interventions of 3.4 (range, 1-7) and a total of 91 surgical procedures. The extent of resection was defined as Simpson 1 in 10 cases (11%), Simpson 2 in 12 cases (13%), Simpson 3 in 8 cases (9%) and Simpson 4 in 51 cases (56%). Pathological analysis found 11 Grade II meningiomas, 13 transformed Grade III meningiomas (WHO Grade I or II with subsequent malignant transformation) and 5 de novo Grade III meningiomas. Fourteen patients received postoperative conformational radiotherapy: all 5 patients with de novo Grade III tumors, 7 patients with transformed Grade III tumors (67% of all Grade III tumors) and 2 patients with Grade II meningiomas. Among patients with transformed Grade III meningiomas, 2 patients did not receive radiotherapy because they were considered palliative at the time of malignant transformation, 2 because they had a low performance status and were treated by fractioned radiosurgery and 2 because patients were lost to follow-up between surgeries. Fourteen patients had radiosurgery: ten patients had one radiosurgical procedure and 4 patients had 2 to 5 radiosurgical procedures. Radiosurgery was performed in addition to radiotherapy in ten patients. Chemotherapy was used for 5 patients (17%) without any radiological response. The different chemotherapy regimens were: 1) Bevacizumab then Vincristine/Ifosfamide/Cisplatin chemotherapy, 2) Sandostatin then Bevacizumab, 3) Bevacizumab, 4) Hydroxyurea with Bevacizumab and 5) Hydroxyurea. Mean follow-up lasted 91 months (range: 31-211 months).

At the end of follow-up, 9 (31%) patients were dead, 9 (31) patients had a Karnofsky Performance Scale (KPS) of less than 50% and 11 (38%) had a KPS of more than 50%.

Median progression free survival (PFS) was 11 months (mean: 17 -range: 1-105 months).

PFS was shorter with each subsequent relapse. Principal characteristics of each group are summarized in Table 1. Median PFS were 16 months for patients with de novo Grade II meningiomas. In patients with Grade III meningiomas, the de novo subgroup had a median PFS of 4 months compared with 7 months in patients with transformation. 1, A-D). Among the 29 patients analyzed, 9 (31%) had a « classical » relapse pattern, and 20 (69%) had a multi-nodular relapse including 6 (21%) with a « local multi-nodular » relapse and 14 (48%) with a « multinodular metastatic » relapse. In the latter group, there were two identified cases of metastatic dissemination along cerebrospinal fluid circulation pathways resulting in diffuse pachymeningeal thickening in one case, and multiple bi-hemispheric nodules in the other (Figure 1-D and 2-2). During progression, there was a mean number of 3,1 nodules per tumor.

Pathological grading. The initial pathological diagnosis was

We then looked at the precise sites of relapse. Most relapses were intracranial and inserted on the dura. No intraparenchymal relapses were identified. Subcutaneous relapses were operated upon in two cases. In the subgroup of 11 parasagittal meningiomas, the intrasinusal remnant was the site of recurrence in 7 cases (5 in the « classical » group and 2 in the « local multinodular » group). Interestingly, in the 4 cases of parasagittal meningiomas with « multinodular metastatic » pattern of relapse, the intrasinusal remnant had the lowest growth rate and was never the most aggressive tumor nodule. Finally, two cases of visceral metastases were encountered in our cohort.

We did not find any correlations between pathological grading and radiological patterns of relapses. Especially, in the group of tumors with multi-nodular metastatic relapses, we confirmed that the onset of multiple nodules and malignant transformation did not occur simultaneously. In 8 cases, the tumor retained its initial pathological grade (4 WHO Grade II and 4 WHO Grade III) while presenting with a multi-nodular relapse. In 6 cases of tumors with malignant transformation, the transition to WHO Grade III and multi-nodular seeding occurred at the same time in only 3 cases.

Kinetics of tumor growth. Volumetric analysis of tumor growth concerned 95 tumor nodules in 50 relapses. The median growth rate of tumor nodules was 11.3 cm 3 /year (mean: 45.5 cm 3 /year -range : -1.2 -1317 cm 3 /year-95% confidence interval: 16-75 cm 3 /year) (Figure 1-E). In the « classical » group, data was available for two relapses in two patients. The growth rate was quite similar between the two relapses: 10.1 vs. 14.6 cm 3 /year in the first case and 460,6 vs. 371 cm 3 /year in the second case. In the « local multi-nodular » and « multi-nodular metastatic » groups, there was no correlation between the growth rates of the multiple tumor nodules of a single relapse, resulting in various patterns of tumor growth. Mean time to tumor progression (TTP) was 6.5 months (median: 3.5 months).

In multi-nodular relapses, we observed frequently an indolent growth of several tumor nodules while one to two nodules demonstrated the most aggressive clinical and radiological behavior (Figure 2). Assessing the global tumor burden to evaluate tumor growth and response would lead to an underestimation of the growth rate, especially in patients with multi-nodular relapses. We therefore decided to select target tumors corresponding to the growing nodules requiring complementary treatment. In 12 of 14 « multi-nodular metastatic relapses », the target tumor was a metastatic nodule and did not grew from the initial site of dural insertion. We selected 57 target tumors in 50 relapses among three radiological subgroups. The mean volume of target tumors at the time of treatment was 30,0 cm 3 (median : 21.8 cm 3 ) and their mean growth rate was 47.0 cm 3 /year (median: 19.6 cm 3 /year-95% confidence interval: 26.3 -67.7 cm 3 /year). When considering only target tumors, mean TTP was 4.5 months (median: 2.3 months). Among radiological subgroups, mean TTP was lower in multi-nodular metastatic relapses (3.3 months) compared to classical relapses (5.5 months) (p=0 .3).

In this series, kinetics of tumor growth shed light on the different patterns of treatment failure.

Distant multi-nodular relapses occurring after radiotherapy are often two diffuse to be amenable to radiosurgery or gross-total resection, resulting in repeated incomplete surgeries (Figure 3). In the case of well-circumscribed tumor relapses, CSF dissemination may cause distant relapses despite local tumor control with radiotherapy (Figure 4).

Discussion

This retrospective study on surgery-and radiation-refractory meningiomas sheds new light on tumor growth patterns and tumor growth rates. Numerous chemotherapies have failed to demonstrate their efficacy [START_REF] Wen | Phase II study of imatinib mesylate for recurrent meningiomas (North American Brain Tumor Consortium study 01-08)[END_REF][START_REF] Chamberlain | Hydroxyurea for recurrent surgery and radiation refractory highgrade meningioma[END_REF][START_REF] Chamberlain | The role of chemotherapy and targeted therapy in the treatment of intracranial meningioma[END_REF], with the exception of a recent prospective study that has delivered promising results on the effects of sunitinib malate [START_REF] Kaley | Malignant progression in meningioma: documentation of a series and analysis of cytogenetic findings[END_REF]. One explanation is that current criteria could underestimate responses of some nodules needing further investigations.

Our data should help in the future establishment of precise response criteria in upcoming clinical trials. Macdonald criteria [START_REF] Macdonald | Response criteria for phase II studies of supratentorial malignant glioma[END_REF], often used to measure these tumors may not be accurate in this population, as pointed out in a recent meta-analysis [START_REF] Kaley | Historical benchmarks for medical therapy trials in surgery-and radiation-refractory meningioma: a RANO review[END_REF]. It should be reminded that this study concerns a highly selected patient population and may therefore suffer from selection bias. But to our knowledge, this study also offers for the first time a precise volumetric evaluation of tumor growth in this population. In practice, linear measurements of coronal and axial sections are often inaccurate and have considerable interand intraobserver variations [START_REF] Erasmus | Interobserver and intraobserver variability in measurement of non-small-cell carcinoma lung lesions: implications for assessment of tumor response[END_REF][START_REF] Harris | Three-dimensional volumetrics for tracking vestibular schwannoma growth in neurofibromatosis type II[END_REF][START_REF] Cai | Tumor burden in patients with neurofibromatosis types 1 and 2 and schwannomatosis: determination on whole-body MR images[END_REF].The use of semi-automated algorithms to determine tumor volume has already been performed in discrete tumors of the nervous system [START_REF] Weizman | Bashat DB Semiautomatic segmentation and follow-up of multicomponent low-grade tumors in longitudinal brain MRI studies[END_REF] and other cancers [START_REF] Dinkel | Biederer J Inter-observer reproducibility of semi-automatic tumor diameter measurement and volumetric analysis in patients with lung cancer[END_REF] and has been found to be a reliable method to monitor tumor growth in patients, interobserver-variability being reduced to about half to one-third compared to standard manual measurements.

Here we illustrate the high variability of tumor growth rates, underlining the necessity of performing measurements to evaluate the tumor growth rate before starting a treatment in future clinical trials. We demonstrate the high frequency of multi-nodular relapses with different growth rates for each nodule (Figure 2) and therefore advocate for thorough evaluation of tumor growth, not considering the whole tumor burden but focusing on target tumors. This is all the more true considering that growing nodules at relapse are often found at a distant metastatic site. We may postulate that multi-nodular relapses may be caused by different genetic events in the initial tumor clone, leading to different tumorigenesis pathways and different rates of tumor growth. Targeted therapies should therefore focus on targeted lesions. This study also illustrates the patterns of treatment failure that may occur despite excellent tumor control on the initial site of dural insertion as illustrated in Figures 3 and4.

The frequency of distant relapses underlines the limits of local or loco-regional treatments such as surgery, radiosurgery and radiotherapy and should lead to an additional effort to find an efficient chemotherapy against those tumors. The different growth rates of each nodule may also reflect different tumorigenesis mechanisms with different sensitivities to chemotherapy. The efficacy of future treatments should therefore not be evaluated on the global tumor burden, at the risk of hiding potential antitumor effects on specific tumor nodules. These results emphasize the fact that multi-relapsing meningiomas represent a locoregional disease of the meninge rather than an unique tumor and also explain a limited efficacy of radiosurgery with a frequent lack of coverage of the dural insertion volume, leading to multi-nodular local relapses.

The choice of an endpoint in clinical trials is another challenge in this population of meningiomas that have failed surgery and radiotherapy. PFS-6 was the most consistently recorded endpoint in a recent literature review [START_REF] Kaley | Historical benchmarks for medical therapy trials in surgery-and radiation-refractory meningioma: a RANO review[END_REF]. The PFS found in our study with de novo and transformed Grade III meningiomas are consistent with this result. But the longer PFS found in Grade II meningiomas reminds us of the clinical heterogeneity of those tumors. The inclusion of Grade II meningiomas in future clinical trials should be limited to cases with histological progression and/or PFS of less than 6 months. The variation between PFS among different relapses from the same patient and different histological subgroups may also question the use of this criteria as an endpoint for future trials. The median TTP may appear as a more reliable endpoint as it considers only growth kinetics of the concerned relapse and nodule. A mean TTP of 3.7 months favors a regular follow-up of patients with multi-recurrent meningiomas, with a regular 3 month-interval between MRI-scans, instead of waiting for clinical deterioration.

This study also demonstrates the high heterogeneity of multi-recurrent meningiomas. On the histological level, our study adds new information on de novo and transformed Grade III meningiomas. In previous reports [START_REF] Yang | Atypical and anaplastic meningiomas: prognostic implications of clinicopathological features[END_REF][START_REF] Krayenbuhl | De novo versus transformed atypical and anaplastic meningiomas: comparisons of clinical course, cytogenetics, cytokinetics, and outcome[END_REF], progressing tumors represented 54% to 70% of Grade III meningiomas. Here the high frequency (72%) of progressing tumors among Grade III meningiomas might also be explained by a selection bias. As we selected multi-recurrent tumors, de novo Grade III meningiomas leading to a fatal issue before the second operated relapse might have been excluded. In contrast to a previous study [START_REF] Krayenbuhl | De novo versus transformed atypical and anaplastic meningiomas: comparisons of clinical course, cytogenetics, cytokinetics, and outcome[END_REF], we did not find a more aggressive clinical behavior in transformed compared to de novo Grade III tumors. In regard of the small number of patients in both studies, we believe future clinical studies should be directed at answering this question. On the radiological level, our work confirmed that multirecurrent meningiomas were mostly non-skull base tumors as already pointed out in several previous studies on the relationships between meningioma location and histological aggressiveness [START_REF] Cornelius | Malignant potential of skull base versus non-skull base meningiomas: clinical series of 1,663 cases[END_REF][START_REF] Mcgovern | A comparison of World Health Organization tumor grades at recurrence in patients with nonskull base and skull base meningiomas[END_REF]. At relapse, we describe three different patterns of tumor growth. The group of « multi-nodular metastatic » tumors represents almost 50% of tumor relapses and demonstrates specific growth patterns : distant tumor nodules present increased growth rates compared to nodules located at the initial dural insertion site and true CSF dissemination may be encountered. CSF dissemination occurs in about 4% of meningiomas and preferentially in highly-pretreated patients [START_REF] Chamberlain | Cerebrospinal fluid-disseminated meningioma[END_REF]. Contrary to the patients reported by Chamberlain et al., we did not find any lepto-meningeal dissemination to the spinal cord in our cases. Extraneural metastases were found in only one case. These patterns might correspond to a specific underlying molecular mechanism that should be specifically addressed in future molecular studies. In previous works, invasive properties of meningioma cells have been linked to different migratory behaviors involving extracellular matrix modeling by tumor cells using matrix metalloproteinases [START_REF] Nordqvist | Expression of matrix metalloproteinases 2 and 9 in meningiomas associated with different degrees of brain invasiveness and edema[END_REF][START_REF] Jalali | A role for matrix remodeling proteins in invasive and malignant meningiomas[END_REF] and aquaporines [28].

In summary, surgery-and radiation-refractory meningiomas are predominantly progressing tumors with malignant transformation and are characterized by frequent multi-nodular metastatic relapses, representing a loco-regional disease amenable to systemic therapies. After the first surgery, the patient underwent radiotherapy and local control was achieved for the initial tumor bed (red tumor and curve). Two years after, the patient had an emergency craniotomy for a relapse with intratumoral hemorrage revealing a transformed Grade III meningioma. Only 9 months after the second relapse, the patient had two fast growing distant relapses (blue and purple tumors and curves) not amenable to radiosurgery. These two relapses could not be controlled despite two extensive surgeries.

  Grade I in 2 patients, Grade II in 22 patients and Grade III in 5 patients. The two patients with initial Grade I meningioma progressed to Grade III after 1.1 and 12.2 years respectively. Among the patients initially diagnosed with Grade II meningioma, 10 progressed to Grade III (transformed meningioma) after a mean follow-up of 6.1 years (range: 1.2 -14.7 years). Radiological characteristics at diagnosis. All patients harbored non-skull base meningiomas (9 convexity ; 11 parasagittal ; 3 falcorial ; 4 tentorial ; 2 pterional). Volumetric data was available for 11 patients at diagnosis. The mean tumor volume at diagnosis was 39.5 cm 3 (median: 35.6 cm 3 ). Patterns of relapse. We describe three radiological patterns of relapse in our cohort: « classical », where a new tumor nodule grows back from the original site of dural insertion, « local multi-nodular », where multiple tumor nodules grow near the site of dural insertion or on the borders of the craniotomy and « multi-nodular metastatic », where multiple tumor nodules grow at several distant sites of dural insertion (Figure

FiguresFigure 1 .

 1 Figures

Figure 2 .

 2 Figure 2. Differential growth of multiple tumor nodules originating from a single meningioma. This figure illustrates the differential growth of three distinct tumor nodules (blue and purple tumor and curves) originating from a single falcine meningioma (red tumor and curve).

Figure 3 .

 3 Figure 3. Patterns of treatment failure in surgery-and radiation refractory meningiomas: distant CSF dissemination. This figure illustrates the case of a parasagittal Grade I meningioma (red curve and tumor) with initial metastatic relapse near the insertion site of a ventricular catheter (blue curve and tumor) with malignant transformation requiring protontherapy. Despite a local control of the irradiated nodules (red and blue tumors) this treatment failed to control tumor progression with the occurrence of a remote relapse (purple curve and tumor) at the clivus.

Figure 4 .

 4 Figure 4. Patterns of treatment failure in surgery-and radiation refractory meningiomas: absence of local control. This case of a pterional Grade II meningioma illustrates the difficulty of local tumor control with repeated surgery.After the first surgery, the patient underwent radiotherapy and local control was achieved for the initial tumor bed (red tumor and curve). Two years after, the patient had an emergency craniotomy for a relapse with intratumoral hemorrage revealing a transformed Grade III meningioma. Only 9 months after the second relapse, the patient had two fast growing distant relapses (blue and purple tumors and curves) not amenable to radiosurgery. These two relapses could not be controlled despite two extensive surgeries.

Table 1 : Characteristics of the three subgroups

 1

		Grade II Meningioma	De novo Grade III	Transformed Grade III
			Meningioma	Meningioma
	Number of patients	11	5	13
	Age of discovery (year)	52 (range, 40-68)	66 (range, 52-80)	53 (range, 31-76)
	Sex :
	Male	6	3	6
	Female	5	2	7
	Relapses	2.4 (range, 2-6)	3 (range, 2-4)	3.3 (range, 2-7)
	Surgery	2.9 (range, 2-5)	2.4 (range, 1-3)	4.2 (range, 2-7)
	Radiosurgery	3	4	7
	Conformational	2	5	7
	radiotherapy
	Chemotherapy (patient -	1 (Bevacizumab then	2 (1 st : Sandostatin then	2 (1 st : Hydroxyurea
	drug)	VIP)	Bevacizumab -2 nd :	with Bevacizumab -
			Bevacizumab)	2 nd : Hydroxyurea)
	Mean PFS (months) :
				Grade II:	Grade III:
	1 st relapse -2 nd relapse	33	13	33	-
	2 nd relapase -3 rd relapse	18	5	33	14
	3 rd relapse -4 th relapse	20	4	17	6
	4 th relapse -5 th relapse	10	2	12	6
	Overall survival	99 (range, 49-211)	51 (range, 17-74)	108 (range, 31-201)

Acknowledgments

The authors would like to thank Franck Maizeroi-Eugène for his kind gift of the Segmentix plugin.