Dominik Pisarski
email: dpisar@ippt.gov.pl

Optimal Control of Structures Subjected to Traveling Load

Keywords: Optimal control, structural control, semi-active control, vibration control, moving load

The problem of the optimal semi-active control of a structure subjected to a moving load is studied. The control is realized by a change of damping of the structure's supports. The objective is to provide a smooth passage for vehicles and extend the time needed for the safety service of the carrying structures. In contrast to the previous works of the author, in this paper, the model used takes into account time-varying passage speeds, which allows a broader application, in particular, to robotics. The study of the optimal control problem produces a practical condition that justifies whether, for a given set of parameters, the controlled system can outperform its passively damped equivalent. For the optimization, an efficient method of parametrized switching times is developed and tested via numerical example. The designed optimal control is examined on a real test stand. The experiments are carried out for three different passage scenarios. In terms of the assumed metrics the proposed method outperforms the passive case by over 40 percent.

Introduction

Problems of structures subjected to loads traveling with high velocity are of special interest to engineers. Numerous analytical and numerical solutions are being applied to solve the problems of transportation and robotic systems with singleor multi-point interactions, such as train-track, vehicle-bridge, or effector-guideway. The high vibration levels caused by the continually increasing speeds and load carrying capacity requirements challenge scientists to seek new solutions. Constructing new railway tracks or bridges which would have a sufficiently higher load carrying capacity or ability to resist dynamic strains frequently faces difficulties of an economic nature. Similarly, increasing the mass of a structure which results from a static strengthening is often challenging for technological reasons. To cope with undesired vibration effects, a variety of control systems acting on both the structures and the suspension of the traveling loads have been proposed and put into practice.

A common objective in structural control is to enhance the stability of a system subjected to impulsive or periodic excitation. The first group of control methods, referred to as active methods, is based on force actuators. An active control method to control the beam vibrations via linear force actuators is presented, for example, in [START_REF] Frischgesel | Regelung eines elastischen fahrwegs inter verwendung eines variablen beobachters[END_REF]. A controlled piezoelectric layer was considered in Baz (1997). An actively controlled beam subjected to a harmonic excitation was presented in Pietrzakowski (2001) whereas an actively controlled string system was studied in [START_REF] Tan | Active wave control of the axially moving string[END_REF]. Similarly, the control of the shape of the railway tracks can be listed among the active methods in structural control. [START_REF] Flont | Adaptive railway truck with improved dynamic response[END_REF], the authors developed an approach that uses active smart sleepers that enable the track to shift up and down. The objective was to minimize the deflection of the track. In [START_REF] Stancioiu | Optimal vibration control of beams subjected to a mass moving at constant speed[END_REF], the authors suggested an active control method to suppress the vibration of bridges. The controller was based on a Linear Quadratic Regulator.

A recent trend is to replace force actuators with semi-active magneto-rheological dampers. This solution is usually less efficient. However, it has attracted the interest of engineers due to its significantly lower power consumption. It is also safer in the case of a control system failure. Unlike active systems, the semi-active ones based on controlled dampers can always be switched to a passive mode, providing system stability. One of the first concepts of semi-active control in mechanical systems was proposed in [START_REF] Karnopp | Vibration control using semi-active force generators[END_REF], who presented the idea of stabilizing an oscillator with one degree of freedom moving upon uneven ground. The "Skyhook" algorithm developed by the authors is today one of the most widely used in suspension control systems for vehicles. The idea was initially designed to improve the comfort of the passengers. Later, a similar control method was adapted to an oscillator moving upon carrying structures. Extensive results were demonstrated in [START_REF] Chen | Smart suspension systems for bridge-friendly vehicles[END_REF]. Controlled dampers are incorporated also for seismic isolation. Interesting results can be found in [START_REF] Ruangrassamee | Control of nonlinear bridge response with pounding effect by variable dampers[END_REF]. In [START_REF] Fulin | Theoretical and experimental research on a new system of semi-active structural control with variable stiffness and damping[END_REF], the authors proposed controling both the stiffness and the damping parameters. The control decision led to the maximum dissipation of energy.

The use of semi-active supports for a structure subjected to a moving load was first proposed in [START_REF] Bajer | Active control of beams under moving load[END_REF], who demonstrated, by means of numerical simulations, that for a wide range of travel velocities, switching damping strategies outperform standard passive solutions. The idea was later extended in Pisarski andBajer (2010, 2011). By means of numerical experiments it was shown that the metric corresponding to the total deflection of the load trajectory from the desired straight line was reduced by up to 50 percent. The model used in those studies took into account neither a varying speed of passage nor the inertial forces associated with the moving mass, which are crucial to capturing the dynamics of systems used in robotic technologies.

This paper aims at designing a practical, semi-active control method dedicated to the applications to large scale structures like bridges and overpasses subjected to traveling trains as well as to robotic guideways subjected to effectors performing technological processes, for example, cutting, bonding or painting. The control will be realized by a change of damping of the structure's supports. The control objective will be to provide a smooth passage for vehicles and extend the time needed for the safety service of the carrying structures. For this purpose, we will formulate and solve a finite time bilinear optimal control problem. Initially, the total deflection along the traveling load trajectory will be assumed as the metric to be optimized. Three additional metrics corresponding to the structural vibration amplitudes will be later used to examine the performance of the designed method. With respect to the previous research, the present paper contributes to the following directions. The model takes into account a time varying speed of the traveling load, which is essential to control robotic guideways where the effector is subjected to acceleration and deceleration. The study of the optimal control problem introduces a sufficient condition for the existence of optimally switched controls. This condition provides a simple test for optimality that is of particular importance here since semi-active controlled systems often fail to outperform the corresponding passive cases. For the optimization, an efficient method of parametrized switching times is developed and tested via numerical examples. Finally, for the first time, the designed control strategy is validated on an experimental test stand.

The proposed control will be designed in an open-loop structure. An open-loop control system is, in general, less robust with respect to system parameters variation and external disturbances compared with a closed-loop control system (comprehensive comparison for the vibrating systems was presented in [START_REF] Casciati | Technology of Semiactive Devices and Applications in Vibration Mitigation[END_REF]. Various methods can be employed to identify the mechanical parameters of the structure (see, for instance, [START_REF] Zhang | Simultaneous identification of moving vehicles and bridge damages considering road rough surface[END_REF]. Nevertheless, precise estimation of the velocity of the moving load is not always feasible. Real world applications of the proposed control may require the collection of several solutions for different profiles of passage velocities. In the cases where the speed profile cannot be predicted (as for vehicles passing bridges), the developed method can be implemented in the Model Predictive Controller allowing the adaptation of the optimal solutions to the measured parameters (see, for example,

l u 2 k 1 k 2 u m k m u 1 w 0 0 EI, µ a m a 2 a 1 x m v(t)
x m (t)

Figure 1. A span supported with a set of controlled dampers and subjected to a mass traveling at a time-varying speed v(t). [START_REF] Bordons | A generalized predictive controller for a wide class of industrial processes[END_REF]. Thanks to the closed-loop structure, the Model Predictive Controller can also improve the robustness with respect to external disturbances. The rest of the paper is structured as follows. Section 2 introduces the mathematical model of the carrying structure subjected to the traveling load. In Section 3, the optimal control problem for straight line load passage is formulated and the necessary optimality condition is given. Solution method based on parametrized switching times is introduced in Section 4. In Section 5, the performance of the designed control method is validated by means of experimental tests. Conclusions and future outlook are provided in Section 6.

Mathematical model

In this paper, we will consider a carrying structure supported with a set of controlled dampers, as illustrated in Figure 1. For the model of the span, we will use the Euler-Bernoulli beam equation, which is widely applied to thin elastic bodies subjected to small deflections [START_REF] Gere | Mechanics of materials[END_REF]. A beam of total length l is characterized by its bending stiffness EI and density per unit length µ. Controlled supports are located at the positions a i . For each of the supports, we assume a controlled damping u i and/or a fixed stiffness k i . The beam is subjected to a mass m traveling at a time-varying velocity v(t) that is assumed to be given. The position of the moving mass at time t is denoted by x m (t) and computed by using

x m (t) = t 0 v(t) dt .
(1)

The dimensions of the moving mass are assumed to be relatively small compared to the beam. As a result, at the time t, the interaction between the mass and the beam will be represented by a point contact and described by the Dirac delta function δ(x -x m (t)). Similarly, we will use δ(x -a i ) to introduce the reaction of the support located at the position a i . Let w(x, t) be the transverse deflection of the beam at the spatial coordinate x and time t. Then, the system is governed by

µ ∂ 2 w(x, t) ∂t 2 + EI ∂ 4 w(x, t) ∂x 4 = - m i=1 k i w(a i , t) + u i ∂w(a i , t) ∂t δ(x -a i ) -m g + ∂ 2 w(x m (t), t) ∂t 2 δ(x -x m (t)) .
(2)

The endpoint supports impose the following boundary conditions:

w(0, t) = 0 , w(l, t) = 0 , ∂ 2 w ∂x 2 |x=0 = 0 , ∂ 2 w ∂x 2 | x=l = 0 . (3)
For the initial condition, we assume w(x, 0) = 0 , ẇ(x, 0) = 0 .

(4)

The left hand side of (2) consists of the standard terms of the Euler-Bernoulli equation corresponding to the potential and inertial forces of the beam. The first two terms of the right hand side stand for the reactions of the controlled supports. The last terms correspond to the excitation due to the moving load. For this excitation, we take into account both gravity and the inertial force. The latter is often ignored for large scale structures. For systems where the masses of the span and the moving load are comparable (for instance, maglev trains, robotics), the inertial force of the moving mass plays a key role in the dynamics (see [START_REF] Dyniewicz | New feature of a solution of a Timoshenko beam carrying the moving mass particle[END_REF].

ODE representation

It can be verified (see [START_REF] Fryba | Vibration of solids and structures under moving loads[END_REF]) that for the assumed boundary conditions (3) the set of eigenfunctions for (2) is given by θj(x) = sin( jπx/l) , j = 1, 2, ... .

By using the separation of space and time variables the solution to (2) can be represented by the following Fourier series:

w(x, t) = 2 l ∞ j=1 Vj(t) θj (x) . (6)
In this work, we will rely on the approximated solutions by taking j from 1 to n = 10. The assumed 10 modes are sufficient to render the dynamics of the structure while keeping the size of the system eligible for efficient optimisation.

Throughout the rest of the paper, we will use the first order ordinary differential representation of the dynamics (2). Firstly, we introduce the state vector y ∈ R 2n defined as

y(t) = [V 1 (t), V1 (t), V 2 (t), V2 (t), ..., V n (t), Vn (t)] T . (7)
Next, we introduce the following auxiliary matrices Ā(t), B, C(t) and the vector F (t):

( Ā) jk =                                                  1 , (j, k) = (2s -1, 2s), s = 1, 2, ..., n - EI(j/2) 4 π 4 µl 4 , (j, k) = (2s, 2s -1), s = 1, 2, .., n - 2 µl m i=1 k i sin ((j + 1)/2)πa i l sin (k/2)πa i l - 2m µl (k/2)π l sin ((j + 1)/2)πx m (t) l cos (k/2)πx m (t) l v(t) + 2m µl (k/2) 2 π 2 l 2 sin ((j + 1)/2)πx m (t) l sin (k/2)πx m (t) l (v(t)) 2 , (j, k) = (2s -1, 2s ′ ), s, s ′ = 1, 2, ..., n - 4m µl (k/2)π l sin (j/2)πx m (t) l cos (k/2)πx m (t) l v(t) , (j, k) = (2s, 2s ′ ), s, s ′ = 1, 2, ..., n 0, else , ( Bi ) jk =    - 2 µl sin (j/2)πa i l sin (k/2)πa i l , (j, k) = (2s, 2s ′ ), s, s ′ = 1, 2, ..., n 0 , else , ( C) jk =      2m µl sin (j/2)πx m (t) l sin (k/2)πx m (t) l , (j, k) = (2s, 2s ′ ), s, s ′ = 1, 2, ..., n 0 , else , ( F ) j =    - mg µ sin (j/2)πx m (t) l , j = 2s, s = 1, 2, ..., n 0 , else . (8)
Now let the matrices A(t), B(t), and the vector F (t) be defined by

A(t) = (I + C(t)) -1 Ā(t) , B(t) = (I + C(t)) -1 B , F (t) = (I + C(t)) -1 F (t) . (9)
Here I stands for the identity matrix. Then, it can be verified (see the Appendix) that the system dynamics given by ( 2)-( 4) can be represented by the following system of the ordinary differential equations:

ẏ = A(t) y + m i=1 u i B i (t) y + F (t) , y(0) = 0 . ( 10
)
For the controls, we assume that each one is bounded by two positive values, corresponding to the minimum and maximum admissible damping coefficients, i.e.:

u

∈ U = [u min , u max ] m ⊂ R m + . ( 11
)
3 The optimal control problem Let T stands for the time of the moving load passage that is computed by using

l = T 0 v(t) dt . (12)
Throughout this paper, we will aim to find the controls that arrange that the passages of the load traveling over a structure are performed as close as possible to desired straight line trajectory w d (x m (t), t) = 0 for all t ∈ [0, T ]. For that purpose, we shall now introduce an integral objective corresponding to the total vertical displacement of the moving load from the desired trajectory:

J = T 0 (w(x m (t), t) -w d (x m (t), t)) 2 dt = T 0 w 2 (x m (t), t) dt . ( 13
)
Also of importance is the fact that the objective (13) captures the extremal amplitudes of the carrying structure, since we can expect that the maximum deflection lies on the trajectory w(x m (t), t) (see [START_REF] Dyniewicz | Space-time finite element approach to general description of a moving inertial load[END_REF].

To rewrite our metric in state-space coordinates, let us define the vector

q(t) = 2 l [θ 1 (x m (t)), 0, θ 2 (x m (t)), 0, ..., θ n (x m (t)), 0] T . ( 14
)
By introducing the matrix Q(t) = q(t) q T (t) the objective ( 13) is represented by

J = T 0 y T Q(t) y dt . ( 15
)
The problem of an optimal passage for the system (10) can be written as follows:

Find u * = argmin u∈U J = T 0 y T Q(t) y dt Subject to ẏ = A(t) y + m i=1 u i B i (t) y + F (t) , y(0) = 0 . (16)

First order necessary optimality condition

To design an effective solution method for the problem ( 16), it is essential to first investigate the structure of the optimal controls. The problem is classified as a finite horizon bilinear optimal control problem. It has been extensively studied (see [START_REF] Mohler | Bilinear control processes[END_REF]) that if the objective function is not explicitly dependent on the control and the set of admissible controls is bounded from above and below, then the solution to a bilinear optimal control problem is given by controls of the bang-bang type. This result follows directly from the Pontryagin Maximum Principle [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]. By introducing the adjoint state p and the Hamiltonian

H(y, p, u) =p T A(t) y + m i=1 u i B i (t) y + F (t) -y T Q(t) y (17)
the Pontryagin Maximum Principle states that the optimal controls for the problem ( 16) satisfy

u * = argmax u∈U H(y, p, u) . ( 18
)
Here the adjoint state is given by

ṗ = - ∂H ∂y , p(T ) = 0 . (19)
From ( 18) we immediately obtain the optimal controls

u * i = u max , p T B i (t) y ≥ 0 u min , p T B i (t) y < 0 . ( 20
)
Naturally, to determine the trajectories of u * , one has to solve the following Two-Point Boundary Value Problem:

ẏ = A(t) y + m i=1 u * i B i (t) y + F (t) , y(0) = 0 , ṗ = -A T (t) p - m i=1 u * i B T i (t) p + 2 Q(t) y , p(T ) = 0 . (21)

Existence of switched optimal controls

The solution given by ( 20) defines the bang-bang structure of the optimal controls. However, it does not provide any information on the number of switches. It does not even justify whether we can expect any switches. Note that our optimal control problem is defined on a time horizon dependent on the speed of the moving load. In the case of a fast passage, this horizon becomes relatively short. There are many vibrating systems where the short time-horizon semi-active optimal policy is to set the damping to the maximum admissible value. The reader can consider a semi-active controlled oscillator subjected to a harmonic force as a good example of such a one. For a wide range of initial conditions, the minimization of energy-like functions leads to the constant maximum admissible damping, unless the time horizon is sufficiently long. Our goal now is to provide a condition that, for a given set of system parameters, justifies whether the optimal solution (20) exhibits a switched structure or is constant at the maximum admissible value. This condition is intended to serve as a test before performing the optimization procedures.

For simplicity, let us consider the system (10) with only one control, i.e.,

ẏ = A(t) y + u B(t) y + F (t) , y(0) = 0 . ( 22
)
The corresponding adjoint state is

ṗ = -A T (t) p -u B T (t) p + 2 Q(t) y , p(T ) = 0 . ( 23
)
The proposition given below establishes a sufficient condition for the existence of a switched control u * = u max that results in a more beneficial value of the objective function ( 15).

Proposition 1. Let y umax (t) and p umax (t) be the solutions of the state ( 22) and the adjoint state ( 23) when the following constant control function is given:

u(t) = u max , ∀t ∈ [0, T ]. If there exists an interval [t 1 , t 2 ] ⊆ [0, T ] such that for all t ∈ [t 1 , t 2 ]
we have p T umax B(t)y umax < 0, then there also exists a control u * ∈ U, u * = u max such that J(u * ) < J(u max ).

Proof. Let u * = u max + δu. Then, we can write the differential of the objective function as

J(u max + δu) -J(u max ) =δJ(u max )(δu) +r J (u max , δu) , (24)
where δJ(u max )(δu) is the first variation of J(u max ) and r J (u max , δu) = o(δu), i.e., r J (u max , δu)/ δu → 0 as δu → 0. For a sufficiently small δu the sign of the right hand side of ( 24) depends on the sign of the variation. Therefore, we need to prove that δJ(u max )(δu) < 0.

By introducing the Hamiltonian

H(y, p, u) = p T (A(t) y + u B(t) y + F (t)) -y T Q(t) y (25)
the objective function ( 15) can be represented by

J = T 0 p T ẏ -H dt . (26)
An infinitesimal change δu causes variations of the functions δy, δ ẏ, δp. This results in the following variation of the objective function:

δJ(u)(δu) = T 0 - ∂H ∂u δu - ∂H ∂y T δy dt + T 0 p T δ ẏ + ẏ - ∂H ∂p T δp dt . ( 27
)
Here the last term vanishes, since

ẏ = ∂H ∂p . ( 28
)
Under the assumption

δ ẏ = d dt (δy) (29)
an integration by parts yields

δJ(u)(δu) = T 0 - ∂H ∂u δu dt - T 0 ṗ + ∂H ∂y T δy dt + p T δy T 0 .
(30)

Prepared using sagej.cls

Here the second and the last term vanish, since we have

ṗ = - ∂H ∂y , p(T ) = 0 , δy(0) = 0 . (31) Thus, δJ(u)(δu) = - T 0 p T B(t) y δu dt . (32)
Now, let the variation of the control be given by

δu =    0, t ∈ [0, t 1 ) ǫ < 0, t ∈ [t 1 , t 2 ] 0, t ∈ (t 2 , T ] . (33)
Then u * ∈ U. For such a control, we conclude that

∀t ∈ [t 1 , t 2 ] p T umax B(t) y umax < 0 =⇒ δJ(u max )(δu) = - t2 t1 p T umax B(t) y umax ǫ dt < 0 (34)
which completes the proof.

Proposition 1 can be easily generalized to a system with multiple inputs.

Solution method: Optimal switching times approach

As demonstrated in the previous section, the optimality condition for the problem ( 16) consists of a Two Point Boundary Value Problem. To solve this problem, shooting [START_REF] Stoer | Introduction to numerical analysis[END_REF] and relaxation [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF] methods can be applied. In the shooting method, the idea would be to find such an initial condition for the adjoint state that has its zero terminal condition fulfilled. The relaxation method implements another approach. The time domain is represented as a set of points creating a mesh. The system's dynamics is represented by finite difference equations. An iterative procedure is to adjust all the state and adjoint state values on the mesh to bring them into successively closer agreement with the finitedifference equations together with the boundary conditions. In many cases, shooting and relaxation methods are combined. Both methods exhibit good performance in the case of low dimensional problems, except when the solutions are highly oscillatory or not smooth.

In the case of more complex systems, we can expect difficulties in obtaining accurate, switching-shaped, numerical solutions. Increasing the precision of the calculations is associated with a higher dimensional optimization problem, which results in the rapid extension of the time required for the computations. There is a need to use a more efficient numerical algorithm for computing the optimal switching controls. For this purpose, it is very intuitive to parametrize the switching times and reformulate the problem (16). The objective function will be optimized with respect to new parameters: the switching times. This approach transforms the optimal control problem to a nonlinear programming problem, where gradientbased methods can be applied. To obtain the gradient, we will use fundamental facts from the calculus of variations as well as the properties of the Dirac delta function. In what follows, a complete computational algorithm will be given. A numerical example will examine the convergence and the computational burden of the algorithm.

For simplicity, let us first consider the system (10) driven by only one control consisting of one switch from the minimum to the maximum admissible value

u = u min + (u max -u min ) U(t -τ ) , τ ∈ [0, T ] . (35)
Here, U stands for the unit step function. The dynamics is now represented by ẏ = A(t) y

+ (u min + (u max -u min ) U(t -τ )) B(t) y +F (t) , y(0) = 0 . ( 36
)
By defining the Hamiltonian

H(y, p, τ ) = p T (A(t) y + F (t)) +p T (u min + (u max -u min ) U(t -τ )) B(t) y -y T Q(t) y (37)
the objective function (15) takes the form

J = T 0 p T ẏ -H dt . ( 38
)
An infinitesimal change dτ causes variations of δy, δ ẏ, δp. This results in the following variation of the objective function:

δJ(u)(δτ ) = T 0 - ∂H ∂τ δτ - ∂H ∂y T δy dt + T 0 p T δ ẏ + ẏ - ∂H ∂p T δp dt . (39)
By implementing steps in analogy to ( 28)-( 31), the variation (39) is represented by

δJ(u)(δτ ) = - T 0 ∂H ∂τ dτ dt . ( 40
)
This implies that the derivative of the objective function with respect to the switching time is

∂J ∂τ = - T 0 ∂H ∂τ dt . ( 41
)
Insertion of the Hamiltonian (37) results in

∂J ∂τ = -(u max -u min ) T 0 p T B(t) y ∂ U(t -τ ) ∂τ dt . (42)
Finally, we get

∂J ∂τ =(u max -u min ) T 0 p T B(t) y δ(t -τ ) dt =(u max -u min )p T (τ )B(τ )y(τ ) . (43)
For a control with the reversed switching action, i.e.,

u = u max + (u min -u max ) U(t -τ ) , τ ∈ [0, T ] . (44)
the reader can verify that the gradient of the objective function with respect to the switching time τ is given by

∂J ∂ τ = (u min -u max )p T (τ )B(τ )y(τ ) . ( 45
)
Before we develop the computational algorithm, let us first introduce the relevant settings. For each of the controls, we assume z switching actions from the minimum to the maximum admissible value [u min ] → [u max ] and z switching actions from the maximum to the minimum admissible value [u max ] → [u min ]. The switching times will be collected into two matrices: τ = [τ i,j ] m×z , τ = [τ i,j ] m×z . Here τ and τ consist of the times of the actions [u min ] → [u max ] and [u max ] → [u min ], respectively. {τ i,j } and {τ i,j } are increasing sequences with respect to j, where for every pair (i, j) we have τ i,j ∈ [0, T ) and τi,j ∈ (0, T ]. Moreover, we assume that τ i,j < τi,j for all i, j. In the algorithm, we will use the following dynamics for the state:

ẏ = A(t) y + u min m i=1 B i (t) y +(u max -u min ) m i=1 z j=1 (U(t -τ i,j ) -U(t -τi,j )) B i (t) y +F (t) , y(0) = 0 (46)
and for the adjoint state,

ṗ = -A T (t) p -u min m i=1 B T i (t) p -(u max -u min ) m i=1 z j=1 (U(t -τ i,j )-U(t -τi,j ))B T i (t) p +2 Q(t) y , p(T ) = 0 . (47)
The computational algorithm based on the steepest descent gradient method consists of the following steps: in the optimization process Step 1 Set α to be a small positive number and guess the initial matrices [τ i,j ] and [τ i,j ].

τ 3,1 τ 2,1 τ1,1 τ 1,1 τ 4,1 u max u max u max u max τ3,1 τ2,1 τ4,1 u 1 (t)
Step 2 Solve the state equation ( 46) by substituting [τ i,j ] and [τ i,j ].

Step 3 By backward integration solve the adjoint state equation (47) by substituting [τ i,j ], [τ i,j ] and y.

Step 4 By using ( 43) and ( 45) compute the derivatives for all components of the switching time matrices.

Step 5 Update the switching times by using the formulas

τ + i,j = τ i,j -α ∂J ∂τ i,j , τ + i,j = τi,j -α ∂J ∂ τi,j .
Optionally, a line search method can be implemented here to provide the optimal decrease of the value of the objective function.

Step 6 Check whether the switching times τ i,j or τi,j extend their limited values 0 or T , respectively. If so, then set these times to the appropriate infinium or supremum of the set [0, T ] and then go to Step 2.

Step 7 Check if the length of any of the intervals [τ i,j , τi,j ] approaches zero. If so, discard those switching times, re-size the matrices [τ i,j ], [τ i,j ], and go back to Step 2. Step 8. Repeat Steps 2-7 until a terminal condition (based, for example, on the norm of the gradient) is fulfilled.

Numerical example

Before we proceed to the experimental validation of the control method designed, let us first briefly investigate the computational aspects of the algorithm developed. Namely, we are interested in the performance and the convergence of the gradient-based procedure. The optimization will be executed for the system parameters that correspond to the test stand presented in the next section.

The optimization algorithm was implemented in the MATLAB programming language. For the integration of the state and adjoint state dynamics, the 4th order Runge-Kutta scheme was employed. At each of the optimization loops, the parameter α (see Step 5 in the algorithm) was taken so that the updated switching times could move only one sample time forward or backward. The procedure was terminated when each of the switching times started oscillating between two neighboring samples. In the example, we consider four controls associated with four controlled dampers. For each control we assume two switching actions. The first action, denoted by τ i,1 (i stands here for the damper i), switches the damping from u min to u max . The second one, denoted by τi,1 , switches from u max to u min . The time interval [0, T ] was discretized into 2000 time samples where samples no. 1 and no. 2000 correspond to t = 0 and t = T , respectively. The optimization was initiated with the following values: sample no. 200 for the switches τ i,1 , and sample no. 1800 for the switches τi,1 . After 931 iterations (126 seconds performed with the use of an Intel Quad-Core processor) the optimization was terminated. The evolution of the switching times during the optimization process is demonstrated in Figure 2. The last switching time that met the terminal condition was τ 4,1 . The optimal switching times are denoted by bullets (the numerical values are summarized in Table 1). The structures of the optimal controls are illustrated in the top of Figure 2. The practical interpretation of the results is that damper no. 1 is supposed to operate permanently at its maximum value. The remaining dampers are supposed to be switched twice within the considered period.

The increased number of switching actions resulted in a minor improvement of the value of the objective function. The problem was also solved assuming four and six switches. The results are summarized in Table 2. We observe a 2 percent improvement of the objective value while the number of required iterations doubles. It is worth noting that in the case of 6 switches, some of them were canceled when performing the algorithm. Aiming at practical simplicity, the experimental test will be performed under the assumption of two switches for each of the controls.

Experimental validation

In this section, the designed control method will be examined by means of experimental results. We will investigate the performance for different passage speeds. For that purpose, we will evaluate four different metrics, corresponding to the state of the carrying structure and the trajectory of the moving load. In terms of these metrics, we will compare the system driven by optimal switched controls with one that operates permanently at the maximum admissible damping.

The test stand

A real view of the test stand is shown in Figure 4. The supporting structure was made of an aluminum truss frame. The carrying structure is a guideway supported with two springs that reduce the static deflection. For the traveling load we use a carriage powered by an electric motor. For the controlled devices we use four magneto-rheological rotary dampers equipped with encoders. The locations for the dampers are: a 1 = 1/5 l, a 2 = 2/5 l, a 3 = 3/5 l and a 4 = 4/5 l. During the passage, the carriage first rapidly accelerates from zero to a given constant speed v max (see Figure 5). Then, this speed remains constant until the last stage of the passage (corresponding to approximately 4/5 of the length of the guideway), when the motor starts its braking process. For both the acceleration and the deceleration, we use the maximum admissible values of approximately 6 m/s 2 . The assumed speed trajectory is typical for many manufacturing processes engaging robotic technology.

In order to design the parameters for the test stand, a set of numerical simulations were first performed. Custom software simulates the dynamic interactions between the traveling carriage and the whole structure. Various combinations of the support rails and guideways were considered. For the notable deflection of the beam, the pivotally mounted guideway without the supporting rail was selected. This results in the maximum transverse displacement being within the range of ±30 mm for a mass of 5 kg traveling at a speed of 3 m/s. The carriage is attached to the guideway with a special type of conical rollers, which protect it from jamming in the case of traveling over large deflections. The carriage is powered by a stepper motor via a toothed belt girded over a gearbox. Our setup enables accelerating a mass of 6 kg to a speed of 5 m/s and stopping it within 4 m. The major parameters of the test stand are given in Table 3.

The control system is based on a PC equipped with an I/O data acquisition cards and rotary magneto-rheological dampers. The computed optimal switched control signals are first magnified by an amplifier to provide an operating current of 0.1-1 A (the nominal current range for the employed dampers: LORD TFD steering units). These signals sent to the dampers result in a damping coefficient of approximately 200-2000 Ns/m. The control process is triggered by the motor encoder's indicating the start of the passage. The control signals are sent to the dampers every 5 milliseconds. All switches are shifted backwards in time by 30 milliseconds to compensate for the response time of the dampers. During the process, the encoders incorporated into the dampers measure the local transverse deflections, which are then collected by the acquisition card.

To validate the parameters of the model used in the optimization (see the model given by (58)), some numerical simulations were compared with the experimental results. The comparative data was obtained for of the speed range of v max = 1 -5 m/s, where a constant damping of 2000 Ns/m was assumed. To reconstruct the experimental trajectories, the first four terms of the series (50) were taken into account. The limitation to four terms is due to the number of available local measurements w(a i , t) provided by the encoders incorporated in the dampers. To investigate how this truncation may impact the shapes of the moving load trajectories, some numerical simulations were performed taking into account four, six, eight and ten modes of the series (50). It was observed that for the assumed structural parameters and the traveling speed of v max = 1 -5 m/s, the dynamics is smooth and the moving load trajectories can be fairly represented by the first four modes. An increased number of terms in the series did not produce any significant changes. For a high traveling speed (v max > 10 m/s), the trajectories started to involve angular shapes, in particular, at the points where the moving load was passing by a damper performing a switching action. For such cases, one should consider using the first seven to ten modes in the numerical model and the corresponding number of sensors for the experimental study. The designed optimization procedure was implemented with the first ten modes in the numerical model so that it may be used for a wide range of traveling speeds. The same number of modes was assumed for the comparative results presented below.

The experimental and numerical shapes of the moving load trajectories are compared in Figures 678. For each of the cases, the experimental result coincides with its corresponding model-based simulation. In each case, the relative error for the maximum deflection is less than 5 percent.

Experimental results

The experiments were conducted using the following parameters: the mass of the carriage was m = 5 kg; the different passage speeds had maximum values of v max = 1 m/s, v max = 3 m/s, and v max = 5 m/s. The performance of the control method will be examined by the following metrics: Here J 1 is the optimized objective (13). All implemented optimal controls are based on this metric. For comparative results, we will verify the values of J 2 , denoting the maximum deflection over the trajectory of the carriage, as well as of J 3 and J 4 , corresponding, respectively, to the total and the maximum deflection of the guideway. Despite the fact that the experiments exhibited good replicability (with a maximum difference of 4 percent for the assumed metrics), each of the passages was repeated five times, and the values of the metrics were averaged. For each passage, we applied two damping strategies. The first one, referred to as the optimal passive strategy, employed constant damping, where all controls were set to the maximum admissible value u max . We shall demonstrate that among the admissible set of damping coefficients [u min , u max ], the value u max provides the best performance of passive damping. The second strategy, referred to as the optimal switched strategy, was based on the optimal switched solutions.

J 1 = T 0 w 2 (x m (t), t) dt , J 2 = max t∈[0,T ] |w(x m (t), t)| , J 3 = l 0 T 0 w 2 (x, t) dx dt , J 4 = max x∈[0,l],t∈[0,T ] |w(x, t)| .
The typical shape of the deflected beam subjected to a moving load is shown in Figure 9. In this example, the passage at speed v max = 3 m/s was performed under the passive strategy. In this case, the maximum deflection takes place at t ≈ 0.8 T . By increasing the traveling speed, we can observe that the maximum deflection shifts toward the end of the passage.

The first set of experiments was performed to establish the optimal passive strategy to serve as a comparative result to justify the effectiveness of the designed optimal strategy. In Figure 10, the moving load trajectories are presented in the case of v max = 3 m/s and constant damping of u = 200 Ns/m, u = 500 Ns/m, u = 1000 Ns/m, u = 1500 Ns/m, and u = 2000 Ns/m. We can clearly observe that for the straight line passage the best performance is exhibited in the case of the maximum admissible value u max = 2000 Ns/m, and this case was taken for the optimal passive strategy.

The comparison of the moving load trajectories under the optimal passive and the optimal switched strategies in the case of v max = 3 m/s is shown in Figure 11 (for the optimal controls in this case see Figure 12). Under the optimal passive damping, the maximum deflection exceeds 28 millimeters. Under the the optimal switched control, this deflection is reduced by 9 percent. For comparison of the passive strategies in the case of v max = 5 m/s see Figure 13. As in the previous case, for the optimal passive strategy we assume u max = 2000 Ns/m. The moving load trajectories under the optimal passive and the optimal switched strategies are depicted in Figure 14. The corresponding optimal controls are depicted in Figure 15. Again, the optimal switched strategy outperforms the best passive case. In terms of the maximum deflection, the improvement is now 20 percent. For the passages at speed v max < 3 m/s, we can observe a tendency of a loss of efficiency of the designed control method. This is due to the fact that for low travel velocity, the system operates quasi-statically, and the effect of switched damping becomes negligible. The trajectories of the passive and optimally controlled systems at the speed v max = 1 m/s are almost identical. On the other hand, in the case of fast passages, we can notice a very high efficiency. The experimental stand does not enable reaching v max > 5 m/s, but the numerical simulation shows that at the speed v max = 10 m/s, by using the optimal switched strategy, the maximum deflection can be reduced by over 45 percent and the total deflection by over 70 percent.

The set of values for the metrics defined in (48) is presented in Tables 4567. We can clearly observe the correlation between all metrics. In particular, the maximum deflection along the trajectory of the traveling carriage J 2 is almost identical with the maximum deflection of the guideway J 4 . The total deflections represented by J 1 and J 3 also coincide. Thus, the selection of (13) to be optimized is justified.

Regarding the structures of the optimal controls (see Figures 12,15) one can notice two practical features. The first one is that in both cases the damper placed on the left-hand side operated at the maximum damping for almost all the time. A similar conclusion was made based on the numerical simulations. Thus, in the practical design, one can consider replacing a semi-active damper located on the side where the load starts its travel, with a passive one. The second feature of the structures of the optimal controls is the pattern that each damper needs to be switched to the maximum admissible value just before being approached by the load, while the switch back to the minimum value is supposed to be done after that passage. In other words, the optimal trajectory of the traveling load is provided when the maximum damping follows the load. The last statement need to be treated roughly since each traveling speed requires slightly different timing for dampers' activation and deactivation. Nevertheless, the second feature suggests the possibility of approximating the optimal policy by using a simple and practical state-feedback control law. To do so, one can consider introducing the threshold values for vertical deflection (or velocity). Based on the fact that the deflection (or velocity) reaches the highest absolute values at the position under the traveling load, the switching policy employing the threshold values could activate and deactivate the dampers at required The key to reaching the desired performance would not only be in the selection of the threshold values but also in the design of a structure of the state-feedback control. The presented idea is out of the scope of this paper and is dedicated for future works.

Conclusion and future extensions

The problem of the optimal control of a structure subjected to a moving load has been studied. For the assumed objective of a straight line passage, the optimal controls are of the bang-bang type. A simple test based on the integration of the state and adjoint dynamics justifies whether, for a given set of parameters, a switched control strategy can outperform a passively damped system. The developed method of parametrized switching times can efficiently handle the posed problem.

A numerical study has shown that even under two switches per control, the optimal solutions exhibit high performance.

The validation of the optimal controls was investigated on a real test stand. The experiments were performed for three different passage speeds. In each case, the controlled system outperformed its passively damped equivalent. With respect to the optimized metric, the improvement reached over 40 percent. The observation made in Pisarski andBajer (2010, 2011) that an increased speed of travel results in a better efficiency of the switched control strategy has also been verified. The experimental test stand does not allow for passages with speeds over 5 m/s; however, numerical simulations demonstrated that for a moving load traveling at 10 m/s, the value of the objective function can be reduced by over 70 percent.

On-going research of the present author focuses on the problem of the real-time adaptation of the optimal controls according to the measured passage speed. This problem is of special importance for transport where the precise velocity profiles of the vehicles cannot be predicted.

Figure 2 .Figure 3 .

 23 Figure 2. Evolution of the switching times during the optimization process. The optimal values are indicated by bullets. The structures of the optimal controls are given at the top.

Figure 4 .

 4 Figure 4. Real view of the test stand (on the left) and the rotary controlled damper supporting the guideway (on the right).

Figure 5 .

 5 Figure 5. The shape of the velocity of the moving load used in the experiments.

Figure 6 .

 6 Figure 6. Moving load trajectories in the case of vmax = 1 m/s. Comparison of the model-based and experimental results.

Figure 7 .

 7 Figure 7. Moving load trajectories in the case of vmax = 3 m/s. Comparison of the model-based and experimental results.

Figure 8 .

 8 Figure 8. Moving load trajectories in the case of vmax = 5 m/s. Comparison of the model-based and experimental results.

Figure 9 .Figure 10 .Figure 11 .Figure 12 .

 9101112 Figure 9. Guideway deflection in space and time in the case of vmax = 3 m/s under the passive strategy.

Figure 13 .Figure 14 .Figure 15 .

 131415 Figure 13. Moving load trajectories in the case of passive damping and vmax = 5 m/s. The best performance is provided by the maximum admissible damping umax = 2000 Ns/m.

Table 1 .

 1 The optimal switching times for the numerical example.Damper Switch from u min to u max [Time sample] Switch from u max to u min [Time sample]

	1	0	2000
	2	240	1808
	3	630	1678
	4	966	1958

Table 2 .

 2 The values of the objective function and the number of iterations performed. The objective values are normalized to the case of two switches.

	No. of switches Objective value No. of iterations
	2	1.0000	931
	4	0.9824	1546
	6	0.9798	2012

Table 3 .

 3 Parameters of the test stand

	Guideway length (l)	4 m
	Guideway stiffness (EI)	801 Nm 2
	Guideway density per unit length (µ) 2.3 kg/m
	Damping range (u min -u max )	200-2000 Ns/m
	Spring stiffness (k 1 , k 2 )	1000 N/m
	Mass of carriage (m)	0.7-10 kg
	Maximum motor torque	21 Nm
	Maximum speed of carriage	6 m/s
	Maximum acceleration of carriage	6 m/s 2

Table 4 .

 4 The values of the metric J1 normalized to the optimal passive cases.

	v max m/s Optimal passive strategy Optimal switched strategy
	1	1.0000	0.8771
	3	1.0000	0.7756
	5	1.0000	0.5909

Table 5 .

 5 The values of the metric J2 normalized to the optimal passive cases.

	v max m/s Optimal passive strategy Optimal switched strategy
	1	1.0000	0.9512
	3	1.0000	0.9123
	5	1.0000	0.8056

Table 6 .

 6 The values of the metric J3 normalized to the optimal passive cases.

	v max m/s Optimal passive strategy Optimal switched strategy
	1	1.0000	0.8522
	3	1.0000	0.7689
	5	1.0000	0.5630

Table 7 .

 7 The values of the metric J4 normalized to the optimal passive cases.

	v max m/s Optimal passive strategy Optimal switched strategy
	1	1.0000	0.9479
	3	1.0000	0.9108
	5	1.0000	0.7974

Prepared using sagej.cls

Journal Title XX(X)

Appendix

Let us demonstrate how the system given by the partial differential equation ( 2) can be represented by the set of ordinary differential equations of the form (10). For that purpose, we first recall the assumed set of the eigenfunctions θj(x) = sin( jπx/l) , j = 1, 2, ... .

By taking j = 1, 2, ..., n we look for the approximated solution to (2) given by

By inserting (50) into (2), we obtain

Now each term of ( 51) is multiplied by

and then integrated with respect to x over the interval [0, l] . This results in the following weak formulation:

It is worth noting that (53) must hold for every j = 1, 2, ... . Now we can use the orthogonality condition for the θ functions, i.e.:

Here, δj ,j stands for the Kronecker delta. By using the sifting property of the Dirac delta function, we can also give explicit formulas for the terms:

Thus, (53) can be written as follows:

It still remains to compute the second time derivative of

) . An explicit formula for this derivative is

Here, we use the notation φj (x m (t)) = cos( jπx m (t)/l). Regarding only those terms in (56) where the Kronecker deltas are not equal to zero, we can finally rewrite (2):

The initial condition is now given by V j (0) = 0 , Vj (0) = 0 , j = 1, 2, ...

The reader can easily verify that by using the system matrices and the vector given by ( 9) the systems ( 10) and ( 58) are equivalent.

Funding

The research leading to these results has received funding from the Foundation for Polish Science under grant agreement HOMING PLUS/2013-8/11.


Paul Laffont-Lozes

Didier Laureillard

Paul Loubet

Robin Stephan

Myriam Chiaruzzi

Edouard Clemmer

Aurelie Martin

Claire Roger

Laurent Muller

Pierre- Géraud Claret

Radjiv Goulabchand

Clarisse Roux

Jean-Philippe Lavigne

Albert Sotto

Romaric Larcher
email: romaric.larcher@chu-nimes.fr

Effect of Tocilizumab onMortality in Patients with SARS-CoV-2 Pneumonia Caused by Delta or Omicron Variants: A Propensity-Matched Analysis inNimes UniversityHospital, France

Keywords: COVID-19, variant of concern, interleukin-6 receptor antagonist, early administration, mortality rates

We aimed to assess the factors associated with mortality in patients treated with tocilizumab for a SARS-CoV-2 pneumonia due to the delta or omicron variants of concern (VOC) and detect an effect of tocilizumab on mortality. We conducted a prospective cohort study in a tertiary hospital from 1 August 2021 to 31 March 2022 including patients with severe COVID-19, treated with tocilizumab. Factors associated with mortality were assessed in a Cox model; then, the 60-day mortality rates of COVID-19 patients treated with standard of care (SoC) +/ tocilizumab were compared after 1:1 propensity score matching. The mortality rate was 22% (N = 26/118) and was similar between delta and omicron cases (p = 0.6). The factors independently associated with mortality were age (HR 1.06; 95% CI (1.02-1.11), p = 0.002), Charlson index (HR 1.33; 95% CI (1.11-1.6), p = 0.002), WHO-CPS (HR 2.56; 95% CI (1.07-6.22) p = 0.03), and tocilizumab infusion within the first 48 h following hospital admission (HR 0.37, 95% CI (0.14-0.97), p = 0.04). No significant differences in mortality between the tocilizumab plus SoC and SoC alone groups (p = 0.5) were highlighted. However, the patients treated with tocilizumab within the 48 h following hospital admission had better survival (p = 0.04). In conclusion, our results suggested a protective effect on mortality of the early administration of tocilizumab in patients with severe COVID-19 regardless of the VOC involved.

Introduction

Soon after the beginning of the pandemic, the pathogenicity of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was reported to be related to a "Cytokine storm" [START_REF] Moore | Cytokine Release Syndrome in Severe COVID-19[END_REF]. This deregulation in cytokine secretion was mainly characterized by an extensive expression of interleukin-6 (IL-6) and tumor necrosis factor alpha (TNF-a) which injure the lungs and other organs to a lesser extent [START_REF] Moore | Cytokine Release Syndrome in Severe COVID-19[END_REF]2]. Unsurprisingly, corticosteroids were the first immunomodulatory medications to demonstrate a mortality benefit among patients with coronavirus disease 2019 (COVID- [START_REF] Katz | The Use of High-Dose Corticosteroids Versus Low-Dose Corticosteroids with and without Tocilizumab in COVID-19 Acute Respiratory Distress Syndrome[END_REF] and are now a cornerstone of COVID-19 treatment in patients receiving respiratory support [START_REF]The RECOVERY Collaborative Group Dexamethasone in Hospitalized Patients with COVID-19[END_REF].

Early on, the potential benefit of IL-6 receptor antagonist monoclonal antibodies such as tocilizumab was suggested [2]. Indeed, since IL-6 was identified as one of the key cytokines involved in the COVID-19-induced cytokine storm, tocilizumab has been proposed to block the downstream signal transduction by binding IL-6 receptors. Thus, it blocks the JAK/STAT tyrosine kinase system and the Ras/mitogen activated protein kinase (MAPK)/NF-_B-IL-6 pathway, thereby limiting the cytokine storm and its life-threateningconsequences [START_REF] Zhang | Rational Use of Tocilizumab in the Treatment of Novel Coronavirus Pneumonia[END_REF].

In mid-2021, two large platform trials reported that tocilizumab reduced mortality and the use of invasive mechanical ventilation, and increased the chances of a successful hospital discharge [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF]. Subsequently, numerous studies have investigated its effect in patients with SARS-CoV-2 pneumonia caused by the alpha, beta or gamma variants of concern (VOC) [START_REF] Cassone | Predictive Factors of Clinical Outcomes in Patients with COVID-19 Treated with Tocilizumab: A Monocentric Retrospective Analysis[END_REF][START_REF] Radulescu | Treatment with Tocilizumab in Adult Patients with Moderate to Critical COVID-19 Pneumonia: A Single-Center Retrospective Study[END_REF][START_REF] Hafez | Factors Influencing Disease Stability and Response to Tocilizumab Therapy in Severe COVID-19: A Retrospective Cohort Study[END_REF][START_REF] San-Juan | Analysis of the Factors Predicting Clinical Response to Tocilizumab Therapy in Patients with Severe COVID-19[END_REF][START_REF] Nigo | RealWorld Long-Term Assessment of The Efficacy of Tocilizumab in Patients with COVID-19: Results from A Large De-Identified Multicenter Electronic Health Record Dataset in the United States[END_REF], even recently reporting that the antagonist of IL-6 receptors also improved long-term outcomes [START_REF]Writing Committee for the REMAP-CAP Investigators[END_REF]. Unfortunately, in numerous countries, several factors have limited the use of tocilizumab, such as drug shortages, high cost or late approval (December 2021 in Europe, December 2022 in the USA). Therefore, real-life data on tocilizumab treatment remain scarce, especially in patients infected with delta or omicron VOCs [START_REF] Rezaei Tolzali | Efficacy of Tocilizumab in the Treatment of COVID-19: An Umbrella Review[END_REF][START_REF] Oliynyk | Lack of Difference in Tocilizumab Efficacy in the Treatment of Severe COVID-19 Caused by Different SARS-CoV-2 Variants[END_REF][START_REF] Ullah | Assessment of Tocilizumab (Humanized Monoclonal Antibody) for Therapeutic Efficacy and Clinical Safety in Patients with Coronavirus Disease (COVID-19)[END_REF]. Moreover, its use is questionable in omicron cases. Indeed, the reduced intrinsic virulence of the omicron VOC and immunization might contribute to decrease the inflammatory response, reducing the COVID-19 severity, and accordingly reducing the benefit of tocilizumab treatment [START_REF] Wolter | Early Assessment of the Clinical Severity of the SARS-CoV-2 Omicron Variant in South Africa: A Data Linkage Study[END_REF].

This study aimed to report real-life data on tocilizumab use in patients with SARS-CoV-2 pneumonia caused by the delta or omicron VOC. The main objective was to assess factors associated with mortality in patients treated with tocilizumab plus standard of care (SoC). Then, we aimed to compare the prognosis of patients treated with tocilizumab plus SoC to those treated with SoC alone, to detect a potential effect of tocilizumab on mortality.

Results

Population

Characteristics of the study population are reported in Table 1.

Among the 1213 patients admitted for COVID-19 during the study period, 118 patients (9.7%) were treated with tocilizumab and included in the study (Figure 1). Of them, 67% were males (n = 79), and the median age and Charlson index were 69 years old (IQR, 56-77) and 1 (IQR, 0-2), respectively. At hospital admission, the World Health Organization Clinical Progression Scale (WHO-CPS) was 5 in 68 patients (58%) and 6 in 50 patients (42%). The median C-reactive protein (CRP) level was 124 mg/L (IQR, 89-165), and the pulmonary involvement on CT-Scan was classified as moderate or severe in more than 80% of the patients. A total of 101 patients (86%) were infected with delta, and 17 (14%) with omicron.

Tocilizumab was administered at a median dose of 600 mg (IQR, 600-800), and 68 patients (58%) received their first tocilizumab dose within the first 48 h of hospital stay. Only six patients (5%) had a second dose.

Outcomes

The median CRP decreased to 47 mg/L (IQR, 23-89) following the tocilizumab infusion. The decrease in CRP was shown in all patients but 9, and the WHO-CPS increased in 34 patients (29%) (Figure 2).

The median length of hospital stay was 10 days (IQR, [START_REF] Cassone | Predictive Factors of Clinical Outcomes in Patients with COVID-19 Treated with Tocilizumab: A Monocentric Retrospective Analysis[END_REF][START_REF] Radulescu | Treatment with Tocilizumab in Adult Patients with Moderate to Critical COVID-19 Pneumonia: A Single-Center Retrospective Study[END_REF][START_REF] Hafez | Factors Influencing Disease Stability and Response to Tocilizumab Therapy in Severe COVID-19: A Retrospective Cohort Study[END_REF][START_REF] San-Juan | Analysis of the Factors Predicting Clinical Response to Tocilizumab Therapy in Patients with Severe COVID-19[END_REF][START_REF] Nigo | RealWorld Long-Term Assessment of The Efficacy of Tocilizumab in Patients with COVID-19: Results from A Large De-Identified Multicenter Electronic Health Record Dataset in the United States[END_REF][START_REF]Writing Committee for the REMAP-CAP Investigators[END_REF][START_REF] Rezaei Tolzali | Efficacy of Tocilizumab in the Treatment of COVID-19: An Umbrella Review[END_REF][START_REF] Oliynyk | Lack of Difference in Tocilizumab Efficacy in the Treatment of Severe COVID-19 Caused by Different SARS-CoV-2 Variants[END_REF][START_REF] Ullah | Assessment of Tocilizumab (Humanized Monoclonal Antibody) for Therapeutic Efficacy and Clinical Safety in Patients with Coronavirus Disease (COVID-19)[END_REF]. At the end of the follow-up period of 60 days, 88 patients (75%) were discharge alive (WHO-CPS _ 3), 4 patients (3%) were still hospitalized with a WHO-CPS score of 4, 5, 6 and 9, respectively, and 26 patients (22%) had died (Table 1).

Importantly, the 60-day survival probability was at 87%, CI95% (79-95%) in patients treated with tocilizumab within the first 48 h following hospital admission, and at 66%, CI95% (54-80%) in those treated later (p = 0.009). Moreover, the 60-day survival probability was not statistically different in delta and omicron cases (77%, CI95% (69-86%) versus 82%, CI95% (66-100%), p = 0.6), as illustrated in Figure 3.

Figure Survival

Curves at day 60 for patients treated with tocilizumab within 2 days of hospital admission and patients treated with tocilizumab after 2 days (A), in omicron and delta cases treated with tocilizumab (B), in patients treated with standard of care (SoC) alone and with SoC plus tocilizumab (C), in patient treated with SoC alone and patients treated with tocilizumab plus SoC within 2 days of hospital admission and patients treated with tocilizumab plus SoC after 2 days (D).

Factors Associated with Mortality

Results of the univariate and multivariable Cox regressions are presented in Table 2.

In the multivariable analysis, age (HR 1.06; 95% CI (1.02-1.11), p = 0.002), Charlson index (HR 1.33; 95% CI (1.11-1.60), p = 0.002) and WHO-CPS score at the time of tocilizumab infusion (HR 2.58; 95% CI (1.15-1.66), p = 0.03) were independently associated with mortality. In contrast, early administration of tocilizumab had a protective effect (HR 0.37; 95% CI (0.14-0.97), p = 0.04).

Tocilizumab Treatment Effect on Mortality

Using propensity score matching in a Cox model, tocilizumab treatment was shown to have no effect on mortality (p = 0.5). However, as reported in Figure 3, the 60-day survival probability was higher in patients treated with tocilizumab within the first 48 h of their hospital stay compared to those treated later and those treated with SoC alone (87%, 95% CI (79-95%) vs. 66%, 95% CI (54-80%) vs. 75%, 95% CI (68-84%), p = 0.04).

In addition, when we included variables resulting in an imbalance between the groups after propensity score matching (namely, age, high-flow oxygen support needs and mechanical ventilation needs), the effect of tocilizumab was significant in the adjusted Cox model (HR 0.48, 95% CI (0.27-0.86), p = 0.01).

Of note, COVID-19 associated pulmonary aspergillosis (CAPA) was diagnosed in 3 patients among those patients treated with tocilizumab (3/118; 3%) and in 6 among those treated with SoC alone (6/706; 1%). All of them required mechanical ventilation.

Discussion

This study reported the results of a single-center cohort that included 118 patients with severe COVID-19 treated with tocilizumab on top of SoC and described their clinical features and outcomes in a period of the circulation of the delta and omicron variants. We found a 22% overall 60-day mortality rate, similar in both omicron and delta cases. A higher age, higher Charlson index and higher level of oxygen support were independently associated with mortality. In contrast, the early administration of tocilizumab was independently associated with better survival. Importantly, after propensity score matching, this study did not find a significant effect for tocilizumab + SoC on 60-day mortality compared to SoC alone. However, the effect of tocilizumab was significant after adjustment for variables resulting in an imbalance between the groups.

In our cohort of patients treated with tocilizumab, as has been widely reported in numerous previous studies on COVID-19 patients [START_REF] Williamson | Factors Associated with COVID-19-Related Death Using OpenSAFELY[END_REF], mortality was independently associated with higher age and a higher Charlson index. Among comorbidities, diabetes has been reported as a critical risk factor for severe SARS-CoV-2 infection and death [START_REF] Lv | Excess Diabetes Mellitus-Related Deaths during the COVID-19 Pandemic in the United States[END_REF], as highlighted by our result (death rates of 43% vs. 23%). Similarly, the level of hypoxemia assessed by PaO2/FiO2, the SpO2/FiO2 ratio or the WHO-CPS is one of the most significant factors previously reported to drive mortality in COVID-19 patients [START_REF] San-Juan | Analysis of the Factors Predicting Clinical Response to Tocilizumab Therapy in Patients with Severe COVID-19[END_REF].

Other studies [START_REF] Cassone | Predictive Factors of Clinical Outcomes in Patients with COVID-19 Treated with Tocilizumab: A Monocentric Retrospective Analysis[END_REF][START_REF] Radulescu | Treatment with Tocilizumab in Adult Patients with Moderate to Critical COVID-19 Pneumonia: A Single-Center Retrospective Study[END_REF][START_REF] San-Juan | Analysis of the Factors Predicting Clinical Response to Tocilizumab Therapy in Patients with Severe COVID-19[END_REF][START_REF] Nigo | RealWorld Long-Term Assessment of The Efficacy of Tocilizumab in Patients with COVID-19: Results from A Large De-Identified Multicenter Electronic Health Record Dataset in the United States[END_REF][START_REF] Katz | The Use of High-Dose Corticosteroids Versus Low-Dose Corticosteroids with and without Tocilizumab in COVID-19 Acute Respiratory Distress Syndrome[END_REF], including platform trials [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF], have reported a favorable effect of tocilizumab in alpha, beta or gamma cases; however, the benefit of tocilizumab in patients infected with delta, which is reported to be more virulent [START_REF] Twohig | Hospital Admission and Emergency Care Attendance Risk for SARS-CoV-2 Delta[END_REF], or in patients infected with omicron, which is, in contrast, reputed to be less virulent [START_REF] Nyberg | Comparative Analysis of the Risks of Hospitalisation and Death Associated with SARS-CoV-2 Omicron (B.1.1.529) and Delta (B.1.617.2) Variants in England: A Cohort Study[END_REF], remained disputable. To the best of our knowledge, this study was the first to report the use of tocilizumab in a large cohort of severe COVID-19 patients infected with the delta and omicron VOC, suggesting that, among the patients developing a severe form of the disease, the mortality rates were similar in the omicron and delta cases, as it has been reported with previous VOCs [START_REF] Oliynyk | Lack of Difference in Tocilizumab Efficacy in the Treatment of Severe COVID-19 Caused by Different SARS-CoV-2 Variants[END_REF]. In the same line, some authors have highlighted that the rate of severe COVID-19 may change depending on the VOC, but, in cases of severe forms, they reported that the cytokine secretion remained stable and independent from the VOC [START_REF] Korobova | Cytokine Profiling in Different SARS-CoV-2 Genetic Variants[END_REF]. Accordingly, our results suggested that tocilizumab remained an efficient option for the treatment of severe COVID-19 patients in association with corticosteroids, regardless of the VOC involved.

One striking finding of this study was that early administration of tocilizumab (within 48 h following hospital admission) was associated with a better outcome. This result was in accordance with those reported in other studies suggesting the benefit of tocilizumab administration within 2-3 days of hospital admission [START_REF] Radulescu | Treatment with Tocilizumab in Adult Patients with Moderate to Critical COVID-19 Pneumonia: A Single-Center Retrospective Study[END_REF][START_REF] San-Juan | Analysis of the Factors Predicting Clinical Response to Tocilizumab Therapy in Patients with Severe COVID-19[END_REF][START_REF] Nigo | RealWorld Long-Term Assessment of The Efficacy of Tocilizumab in Patients with COVID-19: Results from A Large De-Identified Multicenter Electronic Health Record Dataset in the United States[END_REF]. Similarly, the results of the sensitivity analyses of RECOVERY trials [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF] have also suggested that patients who received tocilizumab within seven days of symptom onset are those who benefited the most. It is worth noting that some authors [START_REF] Richier | Tocilizumab and COVID-19: Timing of Administration Assessment[END_REF] have highlighted that tocilizumab's effect on mortality is more related to its rapid administration after the onset of oxygen requirement than after symptom onset. Accordingly, the probability of any benefit of tocilizumab has been estimated at 99%, 96% and 75%, for patient receiving simple oxygen, non-invasive ventilation and mechanical ventilation, respectively [START_REF] Richier | Tocilizumab and COVID-19: Timing of Administration Assessment[END_REF]. Our results, consistent with these studies, suggest that administering tocilizumab as soon as a COVID-19 patient requires oxygen support may provide the greatest survival benefit. However, after two years of the pandemic, tocilizumab remains a debated therapeutic option for some physicians caring for COVID-19 patients. First, conflicting results [START_REF] Stone | Efficacy of Tocilizumab in Patients Hospitalized with COVID-19[END_REF][START_REF] Rosas | Tocilizumab in Hospitalized Patients with Severe COVID-19 Pneumonia[END_REF][START_REF] Salama | Tocilizumab in Patients Hospitalized with COVID-19 Pneumonia[END_REF] discouraged some of them from using it. Second, tocilizumab has been reported to be associated with an increased risk of infections [START_REF] Sandhu | Secondary Infection Risk in Patients with Severe COVID-19 Pneumonia Treated with Tocilizumab[END_REF], and especially an increased risk of CAPA [START_REF] Prattes | Risk Factors and Outcome of Pulmonary Aspergillosis in Critically Ill Coronavirus Disease 2019 Patients-A Multinational Observational Study by the European Confederation of Medical Mycology[END_REF]. However, COVID-19 itself and corticosteroids also increased this risk [START_REF] Ripa | Secondary Infections in Patients Hospitalized with COVID-19: Incidence and Predictive Factors[END_REF][START_REF] Langford | Bacterial Co-Infection and Secondary Infection in Patients with COVID-19: A Living Rapid Review and Meta-Analysis[END_REF], whereas few superinfections, especially CAPA, have been reported in large trials [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF]. In accordance with these reports, during the study period, only 12 COVID-19 patients were diagnosed with proven, probable or possible CAPA [START_REF] Koehler | Defining and Managing COVID 19-Associated Pulmonary Aspergillosis: The 2020 ECMM/ISHAM Consensus Criteria for Research and Clinical Guidance[END_REF] in our center, and among them only 3 had been treated with tocilizumab. Moreover, despite this possible increased risk of health-care associated infections, the benefits of tocilizumab use seem to outweigh the risks in severe COVID-19 patients, since it decreased all-cause mortality in the short [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF] and long term [START_REF]Writing Committee for the REMAP-CAP Investigators[END_REF]. Some data have suggested a second dose of tocilizumab in the case of weak clinical improvement [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF]. However, the patients who could benefit from a second dose remain poorly identified. In our cohort, 6 patients who had increased their oxygen requirements 12-24 h after a first dose of tocilizumab subsequently received a second dose. All of them were discharged alive. Moreover, in our study, among the 9 patients whose CRP increased after a first infusion of tocilizumab, none received a second dose, and 4 (44%) of them died. Along the same lines, Khurshid et al. reported that the COVID-19 patients who maintained higher CRP values during treatment had the worst outcomes [START_REF] Khurshid | -19: Experience in a Single-Arm Pakistani Center[END_REF]. However, further exploration is mandatory to assess the use of biomarkers, such as CRP, to trigger a second dose of tocilizumab.

Our study has both strengths and limitations. First, our conclusions are limited by the study's monocentric design, which could induce bias in the interpretation of the results and limit their generalization. However, this bias is limited by the standardization of care for COVID-19 patients in accordance to the WHO international guidelines [33]. Moreover, all patients received dexamethasone, which may synergistically interact with tocilizumab [START_REF] Albuquerque | Mortality Rates among Hospitalized Patients with COVID-19 Infection Treated with Tocilizumab and Corticosteroids: A Bayesian Reanalysis of a Previous Meta-Analysis[END_REF]. Second, the data were retrospectively collected for our control group, which could induce bias in data collection. Third, we performed a propensity score analysis and found that tocilizumab had no effect on mortality. However, the relatively small size of the cohort could have limited the detection of its effect on mortality. Indeed, based on previously published data [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF], we should have included 3902 patients in the study (alpha = 0.05 and power 90%) to detect a 4% reduction in mortality. Finally, we reported mortality rates similar to those of randomized clinical trials [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF][START_REF]Writing Committee for the REMAP-CAP Investigators[END_REF]; however, in our cohort, the patients were older [START_REF]The WHO Rapid Evidence Appraisal for COVID-19 Therapies (REACT)Working Group Association between Administration of IL-6 Antagonists and Mortality among Patients Hospitalized for COVID-19: A Meta-Analysis[END_REF] and had more severe cases (according to our institutional guidelines, tocilizumab was given to those with rapid increase in oxygen requirements), which could have increased mortality rates.

Materials and Methods

Study Design and Settings

This prospective, monocentric, observational cohort study was carried out in a French University Hospital from 1 August 2021 to 31 March 2022.

In our hospital, during the study period, the bed capacity for COVID-19 patients care ranged from 9 to 81 ward-beds and from 41 to 81 ICU-beds. 4.2. Tocilizumab Treatment Tocilizumab was first approved by the European Medicines Agency (EMA) in 2009 for the treatment of rheumatoid arthritis and juvenile idiopathic arthritis. In June 2021, tocilizumab was granted an emergency use authorization (EUA) for the treatment of COVID-19 in the United States, and the EMA recommended tocilizumab for adults with COVID-19 who are receiving systemic treatment with corticosteroids and require supplemental oxygen or mechanical ventilation in December 2021. More recently, in December 2022, the Food and Drug Administration (FDA) approved tocilizumab for patients with severe COVID-19.

Taking into account the body of evidence available in the literature [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF][START_REF]The REMAP-CAP Investigators Interleukin-6 Receptor Antagonists in Critically Ill Patients with COVID-19[END_REF], the Anti-Infective Drugs Committee of the Nimes University Hospital authorized the off-label use of tocilizumab in the treatment of severe COVID-19 in June 2021. Thus, tocilizumab started to be administered at physicians' discretion to patients treated with SoC for severe or critical COVID-19 (i.e., intravenous dexamethasone 6mg once a day, prophylactic heparin and oxygen therapy to maintain SpO2 > 94%). Importantly, remdesivir was not recommended in our center, according to international [33] and national guidelines [START_REF]COVID-19 Treatment Guidelines Panel Coronavirus Disease 2019 (COVID-19) Treatment Guidelines[END_REF].

The detailed criteria for tocilizumab initiation were as follows: (1) CRP >/= 75 mg/L and an increase of oxygen needs >/= 2L/min within 48 h, or (2) lack of improvement after 48 h of SoC or (3) ICU admission </=24 h.

Tocilizumab was administered intravenously at a dose depending on bodyweight: 800 mg if weight > 90 kg; 600 mg if weight > 65 kg and >/=90 kg; 400 mg if weight > 40 kg and </= 65 kg; and 8 mg/kg if weight </= 40 kg [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF]. A second dose could be given 12-24 h later at the clinician's discretion if the patient's condition had not improved.

Contraindications for tocilizumab were untreated confirmed or suspected bacterial or fungal infection, hepatic cytolysis > 5 upper limit of normal (ULN), neutropenia < 0.5 cells <109/L and thrombopenia < 50 cells x 109/L [START_REF] Abani | Tocilizumab in Patients Admitted to Hospital with COVID-19 (RECOVERY): A Randomised, Controlled, Open-Label, Platform Trial[END_REF].

Patients

All patients hospitalized during the study period and receiving tocilizumab were screened daily for inclusion using the software of the Pharmacy Department (Pharma®, Computer Engineering, Paris, France). Then, those patients receiving tocilizumab for severe or critical COVID-19 were included in the study. Patients younger than 18 years old or pregnant, those treated with tocilizumab for a reason other than COVID-19 and those with a hospital stay of a length </= 1 day were excluded (Figure 1). Severe COVID-19 was defined as previously described [START_REF]European Centre for Disease Prevention and Control Strategies for the Surveillance of COVID-19[END_REF][START_REF] Ojha | CT in Coronavirus Disease 2019 (COVID-19): A Systematic Review of Chest CT Findings in 4410 Adult Patients[END_REF].

All consecutive adult COVID-19 patients hospitalized during the study period for severe SARS-CoV-2 pneumonia and treated with SoC were retrieved by screening the hospital database using the International Classification of Diseases [START_REF]International Classification of Diseases (ICD)[END_REF] to build a control group. Patients younger than 18 years old or pregnant, those treated with tocilizumab and those with a hospital stay of a length </= 1 day were excluded (Figure 1).

Data Collection

In patients treated with tocilizumab, their demographical data and morbidities were collected prospectively at hospital admission, and their Charlson index was calculated [START_REF] Charlson | A New Method of Classifying Prognostic Comorbidity in Longitudinal Studies: Development and Validation[END_REF]. The percentage of pulmonary involvement was assessed on a chest CT-Scan at admission and was classified as previously described [START_REF] Abdel-Tawab | A Simple Chest CT Score for Assessing the Severity of Pulmonary Involvement in COVID-19. Egypt[END_REF] by a radiologist and confirmed blindly by an infectious disease physician trained in chest CT-Scan interpretation (R.L.). SARS-CoV-2 VOC were also recorded after detection by genotyping or gene sequencing on nasopharyngeal swabs. CRP was recorded 24-48 h before and after tocilizumab infusion. The severity of respiratory failure was assessed using the WHO-CPS at admission and at the time of tocilizumab infusion [START_REF] Marshall | A Minimal Common Outcome Measure Set for COVID-19 Clinical Research[END_REF]. In the control group, the demographical data, morbidities, oxygen requirements, VOC and outcomes were collected retrospectively.

Outcomes

The lengths of the ICU and hospital stays were recorded. The respiratory status was assessed using the WHO-CPS at 28 and 60 days after hospital admission as recommended by the WHO working group on the clinical characterization and management of COVID-19 infection [START_REF] Marshall | A Minimal Common Outcome Measure Set for COVID-19 Clinical Research[END_REF]. Vital statuses at ICU and hospital discharge and 60 days after hospital admission were collected; then, mortality rates were calculated. Additionally, cases of CAPA have been retrospectively collected [START_REF] Prattes | Risk Factors and Outcome of Pulmonary Aspergillosis in Critically Ill Coronavirus Disease 2019 Patients-A Multinational Observational Study by the European Confederation of Medical Mycology[END_REF].

Statistical Analysis

Categorical data were described as numbers and percentages, and continuous data as medians with 25th and 75th percentiles (interquartile range: IQR). The population was divided into two groups according to vital status at 60 days. The categorical variables were compared by Chi-square or Fisher's exact test, and the continuous variables were compared by Student's t test or Wilcoxon's rank-sum test as appropriate.

Factors associated with 60-day mortality were assessed using univariable and multivariable cox regression model. Factors associated with 60-day mortality in the univariate analysis (cutoff of p </= 0.2) were included in the multivariable analysis. Proportional hazard assumption was assessed by inspecting the scaled Schoenfeld residuals. Results of Cox regression model were reported as hazard ratio (HR) with 95% confidence interval (95% CI).

Propensity score matching was also performed to compare COVID-19 patients treated with tocilizumab and SoC with those treated with SoC alone. Patients were matched (1:1) with the algorithm for nearest-neighbor matching without replacement, using a maximum tolerance distance between the matched subjects of 0.1 standard deviation. The confounding variables used to calculate the propensity scores were age, BMI >/=30 kg/m2, ICU admission needs, non-invasive high-flow extra-oxygen support needs, mechanical ventilation needs, previous immunosuppressive treatment (including corticosteroids) and previous chemotherapy, as well as each variable included in the Charlson index, its value and the VOC. We identified the variables resulting in an imbalance between the groups after propensity score matching by calculating the standardized mean difference; then, we included these in the subsequent Cox proportional hazards models as covariates to assess the effect of tocilizumab treatment.

Survival curves were obtained using the Kaplan-Meier method. Survival rates between patients treated with tocilizumab within two days after hospital admission and later, and between omicron and delta cases, were compared using the log-rank test. Then, survival rates between patients treated with tocilizumab + SoC (studied population) and SoC alone (control group), and between patients treated with tocilizumab within two days after hospital admission and later (studied population divided in two groups), and those treated with SoC alone (control group) were also compared using the log-rank test.

All tests were two-sided, and a P-value less than 0.05 was considered statistically significant. Analyses were performed using the R software version 4.2.0 (The R Foundation for Statistical Computing, Vienna, Austria).

Conclusions

In a cohort of severe COVID-19 patients infected with delta and omicron VOC and treated with tocilizumab in a French teaching hospital, we reported a 22% mortality. Mortality rates were similar in the omicron and delta cases, and in COVID-19 patients treated with tocilizumab plus SoC and SoC alone. However, our results suggested that the early administration of tocilizumab had a protective effect on mortality in severe COVID-19 patients regardless of the VOC involved. In contrast, a higher age, higher Charlson index and higher WHO-CPS were significantly associated with mortality. Further multicenter studies are awaited to confirm our results.

Funding:

This research received no external funding.

Institutional Review Board Statement:

The study was conducted in accordance with the Declaration of Helsinki and approved by the Institutional Review Board of Nimes University Hospital (No. 220115, date of approval: 2 October 2022).

Data Availability Statement:

The authors consent to share the collected data with others. The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation. Data will be available immediately after the main publication and indefinitely.

Informed Consent Statement:

Patient written consent was waived by the Institutional Review Board of Nimes University Hospital in accordance with the national legislation and institutional requirements because this observational study did not modify existing diagnostic or therapeutic strategies. However, patients were informed of their inclusion in the study.

Conflicts of Interest:

The authors declare no conflict of interest.