Sophie Sakka
email: sophie.sakka@irccyn.ec-nantes.fr

Damien Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Ruina Ma

Fouad Bennis
email: fouad.bennis@ec-nantes.fr

Predictive model of the human muscle fatigue: application to repetitive push-pull tasks with light external load

Keywords: Predictive model, Fatigue, Push/pull operations, Repetitive tasks, simulation

Predictive model of the human muscle fatigue: application to repetitive push-pull tasks with light external load

Introduction

A better understanding on how the human body deals with muscle fatigue would allow better estimates of working abilities and enhance production.

Such understanding could also help in occupational health management, such as reducing work-related musculo-skeletal disorders (MSDs) [START_REF] Hoozemans | Pushing and pulling in relation to musculoskeletal disorders: a review of risk factors[END_REF][START_REF] Niosh | Work practices guide for manual lifting[END_REF][START_REF] Snook | A study of three preven[END_REF][START_REF] Klein | Assessment of workers' compensation claims for back strains/sprains[END_REF].

While the role of muscle fatigue in MSDs is not well understood, it has been established that when muscles fatigue the torque production strategy changes [START_REF] White | Clinical biomechanics of the spine[END_REF]. For instance, the body has to use muscle fibers that are more vulnerable to fatigue or use different muscle groups that are less efficient to maintain the level of torque demanded. If a fatiguing operation goes on without sufficient rest, such adaptations may increase the risk of MSDs.

Muscle fatigue should, ideally, be predicted to aid in designing or evaluating repetitive industrial operations. Such a predictive model would allow optimizing working and resting phases, and could enhance long term health.

Yet, such fatigue prediction models are difficult to develop. It is well known that muscle fatigue is a complex phenomenon with multiple sources, of which the relative contributions remain unclear. For example, fatigue origins may be mechanical (peripheral fatigue: muscles are not able to respond to the input signal), neurological (central fatigue: the signal sent to the muscles is noisy or reduced), psychological, and/or other sources. As a consequence, predictive muscle fatigue models remain limited to restrictive assumptions and situations and focus on limited parts of the body.

Several models have been previously proposed for muscle fatigue prediction [START_REF] Liu | A dynamical model of muscle activation, fatigue, and recovery[END_REF][START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF][START_REF] Xia | A theoretical approach for modeling peripheral muscle fatigue and recovery[END_REF]. Most models, though, have focused on static situations: a desired torque at a fixed body configuration, which is an oversimplification of workplace tasks. In reality, the ability of torque production varies with body configuration due to the length-tension relationship. This paper proposes a more general muscle fatigue model, extending the static case to quasi-static situations, in which the subject's posture can be changing, hence a varying torque capability. It is worth noting that this new modeling approach cannot yet be considered as dynamic as the torque capability is assumed to depend primarily on the joint configuration alone. In dynamic situations, this capability depends on both joint configuration and joint motion (i.e. the force-velocity relationship in muscle force production). Such data will be required to establish a fully dynamic fatigue model in the future.

To develop and analyze a quasi-static muscle fatigue model, we simulate arm motion during a repetitive flexion-extension task. Even though this task is common in industrial applications [START_REF] Auburn | Guide d'ergonomie: la manutention[END_REF] and daily life activities [START_REF] Kumar | Musculoskeletal risk factors in cleaning occupation: a literature review[END_REF], few studies have examined it with a varying body configuration [START_REF] Mital | Isokinetic pull strengths in the vertical plane: effects of speed and arm angle[END_REF][START_REF] Badi | Effect of one-handed pushing and pulling strength at different handle heights in vertical direction[END_REF][START_REF] Bonato | Changes in the surface emg signal and the biomechanics of motion during a repeti-25 tive lifting task[END_REF]. In this paper we first set the framework and assumptions for modeling. The arm configuration data is then simulated and used to set up the model. Finally, the implication of the study findings is discussed.

2 Problem setting for push/pull operations 2.1 Definition of Push/Pull operations A push/pull operation generally defines both an action of pushing during which the force generated by the hand is oriented away from the body and an action of pulling during which the force is oriented toward the body.

In daily life and in industrial environments, push/pull operations are very common, such as in objects displacement including up and down lifting or left and right shifting, drilling operations, buttons pushing, and so on. These actions of pushing and/or pulling are an important source of MSD [START_REF] Hoozemans | Pushing and pulling in relation to musculoskeletal disorders: a review of risk factors[END_REF][START_REF] White | Clinical biomechanics of the spine[END_REF]. Indeed, NIOSH reported that about 20 % of overuse injuries were associated to push/pull operations [START_REF] Niosh | Work practices guide for manual lifting[END_REF]. Moreover, almost 8 % of back pain and 9 % of sprains and strains at the back are associated with push/pull operations [START_REF] Snook | A study of three preven[END_REF][START_REF] Klein | Assessment of workers' compensation claims for back strains/sprains[END_REF]. In addition, heavy loads acting on the lumbar discs were detected when performing these activities [START_REF] White | Clinical biomechanics of the spine[END_REF], which could also contribute to MSDs. The International Standard ISO 11228-2 [START_REF] Desbrosses | Efforts de tirer-pousser : normalisation, application et prévention[END_REF] recently proposed a first initiative to limit the professional required forces in push/pull operations with the aim of reducing or prevent MSD associated with this activity. Push/pull operations may be divided into three kinds, depending on the groups of muscles in which action is required to perform the motion: updown; left-right or back-and-forth movements. In this paper, we will focus on the back-and-forth movements, which corresponds to the majority of studied tasks in the literature and the most used push/pull operations in daily life.

Another key component of a push/pull operation is the external load.

While the whole body is involved in all push/pull operations, it is assumed here that, for the case of light external loads, the main groups of muscles are in the arm. Thus we focus on the arm movement with a light load push-pull task, while the movements of the rest of the body are neglected.

Working assumptions

We will set some working assumptions to simplify the study and address the problem of fatigue modeling in the frame of repetitive drilling operations.

The study is made from the point of view of external forces. Specifically, the local effects of each working muscle will not be addressed, and instead the effects at the joint level (shoulder and elbow) will be considered for pushing or pulling. Assumptions made include: A1 Only light external loads are involved, and only the arm muscles are activated when pushing and pulling. Other muscles of the body will be neglected.

A2

The arm movement will be limited to two joints: the shoulder and the elbow. The forearm, the wrist joint, and the hand will be considered as a single rigid body. This assumption is considered valid for negligible movements of the wrist, such as in drilling operations using back and forth movements.

A3 The group of muscles used for pushing will be considered inactive when the group of muscles used for pulling is active and vice versa. This assumption is equivalent to ignoring muscle co-contraction, or more precisely the use of antagonistic muscles does not contribute to their fatigue.

The push-pull task used here is composed of two distinct phases. The pushing phase is generated by two groups of muscles: one for shoulder flexion (deltoids) and one for elbow extension (triceps). The pulling phase is generated by the two groups of muscles: one for shoulder extension (sub scapula) and one for elbow flexion (biceps). The model of fatigue will be applied to each active group of muscles.

Cyclic motion

A cyclic motion involves a repetitive movement where the beginning and ending positions, velocities and accelerations are identical. Let us consider a repetitive push/pull cycle with a regular periodicity noted T , such as the one illustrated in Fig. 1. Let T push and T pull denote respectively durations of the pushing and pulling phases durations with T = T push + T pull .

Let ℓ denote the number of completed cycles of the repetitive motion since the beginning of the work, and let t push and t pull denote the overall times counted from the beginning of the work respectively for pushing and for pulling. At the end of the work, we will have ℓT = t f -t 0 where t 0 and t f are respectively the starting and ending instants. If the work started with a pushing phase such as illustrated in Fig. 1 For an instant t ∈ [t 0 , t f ], we have t = t push + t pull . Two situations may occur, depending if t is in a pushing or a pulling phase. If t is in a pushing phase, then t push = t -t pull = t -ℓ T pull . The corresponding function for t push may be expressed using ℓ = (t f -t 0 )/T .

t push = t - t f -t 0 T T pull (1)
If t is in a pulling phase, then we have t pull = t -t push = t -(ℓ + 1) • T push , which may be expressed with the following form.

t pull = t - t f -t 0 T + 1 T push (2)
7 tions

Formulation of general muscle fatigue behavior

Using the assumptions in section 2.2, each group of muscles and its fatigue may be studied and its fatigue evaluated separately from the other groups.

The fatigue model proposed in a previous study [START_REF] Ma | A new simple dynamic muscle fatigue model and its validation[END_REF] is adopted and defined as the following differential equation.

dΓ cem dt = -k • Γ joint Γ MVC • Γ cem (3)
where k denotes a constant fatigue parameter and Γ cem is the current maximal capacity for a group of muscles to generate a joint torque, and is considered as a characteristic value to evaluate the muscles fatigue. Γ MVC (Maximal Voluntary Contraction) is the initial value of Γ cem , measured at the beginning of the experiment. Its value depends on the operator's anthropometry and body configuration as will be detailed in section 3.2.

Γ joint denotes the desired torque vector at the considered joint. It is composed of a portion coming from the body movements Γ i and a portion coming from the external load Γ i,ext introduced as separate variables:

Γ joint = Γ i + Γ i,ext i = 1, . . . , n (4)
with n being the number of joints observed in the motion (n = 2 when only considering the s). Γ i (i = 1..n) are calculated from the values of θ, θ and θ using the Lagrangian formalism [START_REF] Khalil | Modeling and control of manipulators[END_REF].

Γ i = d dt ∂L ∂ θi - ∂L ∂θ i i = 1, . . . , n (5)
where L = E -U is the Lagrangian, E being the kinetic energy and U being the potential energy of the complete system (see [START_REF] Ma | Modélisation de la fatigue musculaire dynamique et son application pour l'analyse ergonomique[END_REF] for the development applied to the current problem). Γ i,ext is directly related to the desired external loads grouped in M load . Then Γ joint = Γ joint (θ, θ, θ, M load ).

Equation ( 3) may be re-written in the following form.

Γcem (t) Γ cem (t) = -k • Γ joint θ(t), θ(t), θ(t), M load Γ MVC θ(t) (6)
⇒ Γ cem (t) = C • e -k t t 0 Γ joint (θ(u), θ(u), θ(u),M load ) Γ MVC (θ(u)) du (7)
where C is an integration constant.

Formulation of joint capacities

One of the key parameters of the fatigue model ( 7) is the maximal torque Γ MVC . In static situations, in which a force is applied by the operator onto the environment with no body movement, the value of Γ MVC varies according to its current body configuration. A predictive model to calculate Γ MVC was proposed by Chaffin et al. [START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF] and is summarized in Table 1 which gives the predictive functions for Γ MVC at the elbow (index e) and shoulder (index s)

joints. Two parameters are considered as inputs: the joint angle value θ and an adjustment gain G that accounts for the gender of the operator. Several observations may be made based on the functions in Table 1: 1. Joint capacity behaves differently for flexion and for extension movements;

2. Predictive models for flexion are more complex than for extension; The model inputs are the measured shoulder and elbow constant joint angles, with the value 0 matching a straight arm in a classical standing configuration. From the joint angles, the respective shoulder and elbow respective capacities can be estimated, assuming this motion is relatively slow.

The obtained joint capacity values will then be used for the fatigue model [START_REF] Frey-Law | Knee and elbow 3d strength surfaces: peak torque-angle-velocity relationships[END_REF].

It is worth noting that static strength data are being used here to model a dynamic task. While some dynamic strength data have been reported

previously, such as in Frey et al. [START_REF] Frey-Law | Knee and elbow 3d strength surfaces: peak torque-angle-velocity relationships[END_REF], comprehensive dynamic strength data for the shoulder are not available. For simplicity, we used the same approach for both the elbow and the shoulder joints, based on Chaffin's predictive functions. As a consequence, the proposed model is considered "quasi-static".

Formulation of the push/pull fatigue in static situations

The model ( 7) is applied separately for each muscles group, for pushing and pulling.

Γ push cem (t push ) = Γ push MVC • e -k Γ push MVC t push 0 Γ push joint du Γ pull cem (t pull ) = Γ pull MVC • e -k Γ pull MVC t pull 0 Γ pull joint du (8)
Because of Assumption A3, the model of fatigue of the arm for a push/pull operation is represented by piecewise continuous functions, in which each muscle group has increasing fatigue when working and remains at the same level of fatigue when relaxing (Fig. 2). Recovery is not taken into account in this initial model development, which means that the subject should be less tired than predicted by the proposed model. Still, this model of fatigue leads to major observations on how human beings deal with fatigue independently from recovery. Depending on whether the current time t is in a pushing or a pulling phase, the model takes the two cases into account as follows.

Γ cem (t) =        Γ push cem (t push ), t ∈ pushing phase Γ pull cem (t pull ), t ∈ pulling phase (9)
When substituting (1) and ( 2) into ( 9), the following model is obtained.

Γ cem (t) =        Γ push cem t - t f -t 0 T T pull , t ∈ pushing phase Γ pull cem t - Push Pull Γ cem

Push muscle groups

Pull muscle groups It is important to note here that the two functions Γ push cem and Γ pull cem are continuous. On the contrary, Γ cem (t) is not a continuous function as illustrated in Fig. 2, which illustrates the model of fatigue [START_REF] Klein | Assessment of workers' compensation claims for back strains/sprains[END_REF] and the previous comments for one joint animated by one pushing and one pulling groups of muscles during a repetitive push/pull task. Starting with the pushing phase, only the group of muscles dedicated to pushing is activated, while the pulling one is at rest (constant value of fatigue characterized by Γ cem ). When in the pulling phase, only the group of muscles dedicated to pulling is at work, and the pushing one is at rest.

Formulation of push/pull fatigue in quasi-static situations

Quasi-static situations denote the case when the arm is in motion and Γ MVC varies with joint angles as defined in the section 3.2. These values do not remain constant but evolve with time together with the evolution of the arm joint angles: Γ MVC = Γ MVC (θ(t)). Γ MVC cannot be taken out of the integral in [START_REF] Frey-Law | Knee and elbow 3d strength surfaces: peak torque-angle-velocity relationships[END_REF]. In this case, the integration constant C is set by considering Γ cem (0) = Γ MVC (θ 0 ) where θ 0 = θ(t 0 ). The fatigue model in quasi-static situations is then expressed as follows.

Γ cem (t) = Γ MVC (θ 0 ) • e -k t 0 Γ joint (θ(u), θ(u), θ(u),M load ) Γ MVC (θ(u)) du (11)
Similarly to the previous static case, the quasi-static fatigue model ( 11) is applied successively to the pushing and pulling groups of muscles during the push/pull operation. Recovery during the resting phase is not considered.

Γ push cem (t push ) = Γ push MVC (θ 0 ) • e -k t push 0 Γ push joint (θ(u), θ(u), θ(u),M load ) Γ push MVC (θ(u)) du Γ pull cem (t pull ) = Γ pull MVC (θ 0 ) • e -k t pull 0 Γ pull joint (θ(u), θ(u), θ(u),M load ) Γ pull MVC (θ(u)) du (12)
Then the fatigue model for push/pull operations is obtained in quasistatic situations, similarly to static situations.

Γ cem (t) =        Γ push cem t - t f -t 0 T T pull , t ∈ pushing phase Γ pull cem t - t f -t 0 T + 1 T push , t ∈ pulling phase 13
and forth arm movements 4.1 Description of the push/pull task

The tasks were chosen according to Ma's work [START_REF] Ma | Multiobjective optimisation method for posture prediction and analysis with consideration of fatigue effect and its application case[END_REF], in order to compare the

Modeling of the operator's arm

The geometric model of the arm is summarized by the modified Denavit-

Hartenberg parameters [START_REF] Denavit | A kinematic notation for lower-pair mechanisms based on matrices[END_REF] given in Table 2, matching the kinematics representation in Fig. 4. The motion only occurs in the sagittal plane (x 0 , z 0 ), so the shoulder and elbow joints are represented by simple rotational joints and all the rotations angles are calculated along the y 0 lateral axis.

Joint σ d α r θ Shoulder 0 0 π/2 0 θ s Elbow 0 h u 0 0 θ e
Table 2: DH parameters of the operator's arm

x 2 x 0 , x 1 z 2 z 0 y 0 , z 1 h u h f

Trajectory generation

The hand trajectory in the task space was described using polynomial interpolation. The hand velocity and acceleration were both null at the beginning and end of each pushing and pulling phases. Beginning and ending positions denoted respectively by P 0 and P f . We obtain the following trajectory. 3 . The joint angles, angular velocities and angular accelerations were obtained from the hand trajectory profile using the inverse kinematics of the arm, leading to the joint profiles illustrated in Fig. 5 for one push/pull cycle.

P (t) = P 0 + p(t) • (P f -P 0 ) , t 0 ≤ t ≤ t f with p(t) = 3 (t/t f ) 2 -2 (t/t f )
It is worth noting that defining the kinematics in the joint space is much easier when applying the model to a real human arm, as the center of rotation of the human arm may be difficult to locate and is needed to set the inverse kinematic model of the human arm. Nevertheless, the joints profiles obtained from the definition of the kinematics in the task space offer more correspondence with the human real trajectories used to perform the described drilling operation, which means that defining the kinematics in the From the described kinematics, the joint torques at the shoulder, Γ s , and at the elbow, Γ e , can be calculated. Their respective evolution over time is illustrated in Fig. 6 for one push/pull cycle. The motion is set for an horizontal displacement of the operator's hand from P 0 = [0.4, 0.1] to

P f = [0.6, 0.1] [m]
which matches the case illustrated in Fig. 3(a).

Evolution of joint capacity Γ MVC

As mentioned in section 3.2, Chaffin's work [START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF] allows for determining the constant joint capacity Γ MVC for the shoulder and the elbow according to the arm configuration and depending on the type of desired external load (pushing or pulling). We have dissociated two working phases for a complete push/pull cycle: the pushing phase, in which the shoulder is in flexion and the elbow in extension, and the pulling phase: shoulder in extension, elbow in flexion. The evolution of the shoulder and elbow Γ MVC (θ s (t), θ e (t)) for a complete push/pull operation is illustrated in Fig. 7. Once again, a discontinuity is observed at the change between pushing and pulling phases. In this case, the discontinuity results because in our mathematical model only one Γ M V C(t) was needed to formulate push and pull (not to be confused with the individual muscle Γ MVC , which is continuous).

Muscle fatigue

The fatigue parameter k is assumed constant for a given operator and at a given joint, for any performed motion. The following mean values for k, from a previous study [START_REF] Ma | Multiobjective optimisation method for posture prediction and analysis with consideration of fatigue effect and its application case[END_REF] were used in this framework: k shoulder = 0.17 and k elbow = 0.24 [START_REF] Ma | Modélisation de la fatigue musculaire dynamique et son application pour l'analyse ergonomique[END_REF]. This setting of k was obtained by an identification process, and was based on anthropomorphic data, maximal torques in flexion and in extension, and body dynamics representation.

Using the expressions of the shoulder and elbow joint torques extracted from Eq. ( 5) and the matching joint capacities Γ MVC (t), the fatigue model can be applied to these two joints for a complete push/pull cycle. Figure 8 illustrates the resulting fatigue predicted at the elbow joint for three push/pull cycles. As the external load remains low, the decrease in joint torque capacity Γ MVC (t) is very slow and may not be easily observed in Fig. 8(a), so a zoom-in on the pulling phases was realized and is shown in Fig. 8(b). Here, the increase in fatigue can be seen. Similar results were obtained for the pushing phases.

From Fig. 8, we can see that the evolution of the fatigue (i.e., decrease in joint torque capacity Γ cem ) is not continuous at the transition between pushing and pulling phases. The main reason for this discontinuity is that a The evolution of the fatigue within a pushing or a pulling phase mainly depends on the values taken by the torque Γ joint . The resulting behavior is then different for a pushing phase or for a pulling phase as Γ joint is different according to the considered phase.

The effect of fatigue for such light loads should be evident after a long period of push/pull operations. The model was thus simulated for 37 minutes of exercise, and Figure 9 illustrates the resulting fatigue predicted at the shoulder and elbow joints. For these two graphs, the bottom curve represents the oscillations of the desired torque Γ joint , which remains always the same, and the top curve represents the oscillations of the available torque Γ cem , which decreases with the increasing muscle fatigue. For both joints, the fatigue in pushing operation was more important than during the pulling operation. This is explained both by a less important joint capacity Γ cem and a higher external load for the pushing phases. Using such graphs to predict potential MSD risks, such risk would seem most important when the two parts join (i.e. at around 26 minutes for the elbow joint and at around 35 minutes for the shoulder joint in this case). In this example too, MSD risk would seem higher at the elbow joint. Still, as the recovery model is not yet included in this approach, and thus in reality any risks would likely involve longer periods of activity. In this paper, the joint capacity Γ MVC was introduced as a variable to build a model of fatigue in repetitive push/pull operations. To highlight the difference between the use of Γ MVC as constant or variable (i.e. depending on the posture), we compare the fatigue graphs obtained in the two situations. The minimum value of Γ MVC obtained for all the simulated configurations was set as the reference constant value; this value will produce the greatest muscle fatigue. Figure 10 shows the level of fatigue in the situation when Γ MVC is set as a constant. The difference between the curves obtained by considering a variable value (figure 9) and a constant value (figure 10) is very small in this framework. The level of fatigue is slightly greater in the case when Γ MVC is constant, because the worst situation was considered. An increased risk at the elbow appears after 23 minutes instead of 25 minutes, which is almost identical. So, it seems that the first approximation of a constant value for the joint capacity is adequate to evaluate the muscle fatigue, if including a slight allowance for the prediction of MSD. However, the current work setting may have an influence on this observation, and a general sensitivity analysis should be conducted to establish a more general conclusion that using a constant Γ MVC is adequate to evaluate the muscle fatigue. workplace implications. While the differences between the two push/pull task configurations used in the simulation are seemingly small, the end effect on fatigue can be dramatic (i.e. for the push muscle group at the elbow joint). Thus by configuring workplace tasks carefully, it may be possible to reduce the workload and/or fatigue for the most vulnerable muscles and reduce MSDs.

24

In this article, a quasi-static model of muscle fatigue was proposed as an extension of a previous model of static fatigue, by incorporating a variable joint capacity Γ MVC as a function of the operator's posture. The model was applied to simulate repetitive push/pull operations with light external loads that may be observed in an industrial framework, such as a classical drilling operation. From the simulation of fatigue during one task configuration, we found that fatigue of the elbow appears faster than the shoulder. This result is in agreement with the observation that most arm MSDs appear at the elbow joint. Indeed, the muscle strength of the elbow is lower than at the shoulder. However, for the second task configuration, we found minimal fatigue for the elbow. This last observation demonstrates that it is possible to reduce MSDs by optimizing workplace tasks.

Figure 2 :

 2 Figure 2: Schematic representation of fatigue in pushing and pulling groups of muscles (Γ cem ) at a joint during a repetitive push/pull operation in quasistatic situations

  proposed quasi-static model to the static one. The arm of a male operator of stature 188 [cm] and body mass 90 kg was modeled to simulate the performance of a repetitive push/pull task. The model of the task included the use of a tool positioned at the extremity of the arm and with mass of 2 [kg]. The initial position of the hand was defined in the sagittal plane by P 0 = [P x,0 , P z,0 ] [m] and the final position by P f = [P x,f , P z,f ] [m]. The position {0, 0} corresponds to the fixed shoulder position (the origin of the reference frame). The operator generates a 20 [N] pushing force and a 10 [N] pulling force while tracking an horizontal line, back and forth. The push/pull cycle lasted 10 [s], with an equivalent time distribution for pushing and for pulling: T push = T pull = 5 [s]. This kind of task represents an operation of classical horizontal drilling. It is illustrated in Fig. 3 for two different amplitudes of hand horizontal displacements: either with an amplitude of 20 [cm] (Fig. 3(a)), or an amplitude of 10 [cm] (Fig. 3(b)). In this figure, the left red circle at (0, 0) represents the fixed shoulder joint; the two segments, arm and forearm, are represented by the blue solid lines linked at the elbow joint represented by a moving red circle. The hand is at the end of the kinematic chain, represented by blue circles. The trajectories of the elbow are different for the two tasks. We can see in Fig. 3(a) that a 0.4 to 0.6 [m] horizontal displacement of the hand generates an elbow elevation of 10 [cm] and advancement of 7 [cm], while the 0.3 to 0.4 [m] horizontal displacement of the hand illustrated in Fig. 3(b) generates an elbow elevation of 3 [cm] and advancement of 3 [cm]. The hand has a 10 cm amplitude

Figure 3 :

 3 Figure 3: Arm movement during a drilling operation. T push = T pull = 5 [s], M push load = 20 [N], M pull load = 10 [N]

Figure 4 :

 4 Figure 4: Kinematic representation of the operator's arm

Figure 5 :

 5 Figure 5: Shoulder and elbow joint angles, angular velocities and angular accelerations over time using trajectory generation in task space. The first phase is a pushing phase

Figure 6 :

 6 Figure 6: Shoulder and elbow joint torques Γ joint for one operating cycle. The first phase is a pushing phase

Figure 7 :

 7 Figure 7: Evolution of Γ MVC (t) at the shoulder and elbow joints for a push/pull cycle

  Zoom-in on pulling phases

Figure 8 :

 8 Figure 8: Simulation of the decrease of muscle capacity Γ cem at the elbow joint during repetitive push/pull operations for three cycles. The first phase is a pushing phase

Figure 9 :

 9 Figure 9: Simulation of the decrease of muscle capacity at the shoulder and elbow joints during repetitive push/pull operations, and with the hand moving from P 0 = [0.4, 0.1] to P f = [0.6, 0.1]. The bottom curve represents the oscillations of the desired torque Γ joint and the top curve represents the oscillations of the available torque Γ cem with respect to the push and pull actions

Figure 10 :

 10 Figure 10: Simulation of the decrease in muscular capacity of the elbow and shoulder when Γ MVC is a constant. The bottom curve represents the oscillations of the desired torque Γ joint and the top curve represents the oscillations of the available torque Γ cem with respect to the push and pull actions

4. 6 . 2

 62 Effect of two push/pull tasks The hand movement amplitude was previously considered to be 20 [cm], moving from P 0 = [0.4, 0.1] to P f = [0.6, 0.1] [m]. If the initial and final positions are modified to P 0 = [0.3, 0.1] and P f = [0.4, 0.1] [m], as depicted in Fig. 3(b), the resulting fatigue curves are shown in Fig. 11.

Figure 11 :

 11 Figure 11: Evolution of fatigue for the hand moving from P 0 = [0.3, 0.1] to P f = [0.4, 0.1]. The bottom curve represents the oscillations of the desired torque Γ joint and the top curve is the oscillations of the available torque Γ cem with respect to the push and pull actions

Figure 12 :

 12 Figure 12: Evolution of the input torques at the elbow and shoulder for the hand moving from P 0 = [0.3, 0.1] to P f = [0.4, 0.1]

  , then at the end of each pushing

	Activity of		Push muscle group
	muscle groups	Pull muscle group
	Active state
	Inactive state
			t
	Push	Pull
	Figure 1: Representation of the group muscles activity during push/pull
	operations
	phase we have t push = (ℓ + 1)T push and at the end of each pulling phase we
	have t pull = ℓT pull .

Table 1 :

 1

	3. Only the shoulder extension model does not depend on the configura-
	tion of another joint.
	Joint/movement	Joint capacity (Γ MVC )	Male	G	Female
	Elbow flexion	336.29 + 1.544θ e -0.0085θ 2 e -0.5θ s G 0.1913 0.1005
	Elbow extension	264.153 + 0.575θ e -0.425θ s G	0.2126 0.1153
	Shoulder flexion	227.338 + 0.525θ e -0.296θ s G	0.2854 0.1495
	Shoulder extension	204.562 + 0.099θ s G	0.4957 0.2485

Chaffin's model for joint capacity in static situations

[START_REF] Chaffin | Occupational Biomechanics, 3rd Edition[END_REF]

t f -t 0 T + 1 T push , t ∈ pulling phase[START_REF] Klein | Assessment of workers' compensation claims for back strains/sprains[END_REF]


Hermine Biermé

Agnès Desolneux

ON THE PERIMETER OF EXCURSION SETS OF SHOT NOISE RANDOM FIELDS

Keywords: AMS 2000 subject classifications: Primary 60G60, 60E10, 26B30, 28A75; secondary 60G10, 60F05, 60E07 shot noise, excursion set, stationary process, Poisson process, characteristic function, functions of bounded variation, coarea formula

In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension n ≥ 1. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.

1. Introduction. We will consider here a shot noise random field which is a real-valued random field given on R n by (1.1)

X(x) = i∈I g(x -x i , m i ), x ∈ R n ,
where g is a given (deterministic) measurable function (it will be called the kernel function of the shot noise), the {x i } i∈I are the points of a homogeneous Poisson point process on R n of intensity λ, the {m i } i∈I are called the marks and they are independent copies of a random variable m, also all independent of {x i } i∈I . Such a random field is a very common model in Physics and Telecommunications, where it has many applications [START_REF] Baccelli | On a coverage process ranging from the Boolean model to the Poisson-Voronoi tessellation with applications to wireless communications[END_REF][START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]. It is a natural generalization of shot noise processes (n = 1), introduced by [START_REF] Rice | Mathematical analysis of random noise[END_REF] to model shot effect noise in electronic devices. More recently, it has also become a widely used model in image processing, mainly for applications in texture synthesis and analysis [START_REF] Galerne | Gabor noise by example[END_REF]. Geometric characteristics of random surfaces is an important subject of modern probability research, linked with extremal theory [START_REF] Adler | Random fields and geometry[END_REF][START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] and based on the study of random fields excursion sets. The excursion set of level t of the random field X in an open subset U of R n is defined by E X (t, U ) := {x ∈ U such that X(x) > t}.

Most of the results are obtained for stationary Gaussian random fields but recent works have allowed to drop the Gaussian assumption. In particular, in [START_REF] Adler | High level excursion set geometry for non-Gaussian infinitely divisible random fields[END_REF], asymptotics for the distribution of critical points and Euler characteristics are obtained for a large class of infinite divisible stationary random fields with suitable regularity assumptions. Central limit theorem for volumes of excursion sets have also been considered in [START_REF] Bulinski | Central limit theorems for the excursion set volumes of weakly dependent random fields[END_REF] for general stationary random fields, including some shot noise and Gaussian random fields. In this paper, we will be interested in the "perimeter" of excursion sets of shot noise random fields (we will give the precise definition of it in the first section), measured as the n -1 dimensional Hausdorff measure of its boundary: L X (t, U ) := H n-1 (∂E X (t, U ) ∩ U ).

In dimension n = 1, the "perimeter" of an excursion set is the number of crossings of the considered level, in dimension n = 2 it measures the length of the boundary of the excursion set, in dimension n = 3 it measures its surface area, and so on. The shot noise random field is not necessarily smooth or differentiable (this happens for instance when the kernel function is an indicator function) and therefore we cannot use the framework of Adler et. al ([2] or [START_REF] Adler | High level excursion set geometry for non-Gaussian infinitely divisible random fields[END_REF]) to compute its geometric features (such as the Euler characteristic or intrinsic volumes). Even in the case where the shot noise random field is smooth, it may not satisfy the regularity conditions on the marginal or conditional probability densities of the field and its gradient ( [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF]).

As a consequence, we will adopt a different viewpoint, and we will study the excursion sets via the whole function t → L X (t, U ), and not only its value for a particular fixed t. A convenient functional framework to define and compute these perimeters is the framework of functions of bounded variation. Our main tool will be the so-called coarea formula that relates the integral of the perimeter of all excursion sets to the integral of the differential of the function. The coarea formula has been already used in a similar situation by Wschebor [START_REF] Wschebor | Formule de Rice en dimension d[END_REF] and by Zähle in [START_REF] Zähle | A general Rice formula, Palm measures, and horizontal-window conditioning for random fields[END_REF] to obtain a general Rice formula for continuous random fields in dimension d. It has also been used by Azaïs and Wschebor in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] to have a proof of the theorem that computes the expected number of crossings of a random field as a function of some of its marginal and conditional probability densities. See also Adler and Taylor [START_REF] Adler | Random fields and geometry[END_REF] p.283 for a discussion about this theorem. The coarea formula approach is in some sense a weak approach since we will obtain a formula for almost every level t (and not for a specific value of t). But the advantage is that we will be able to make explicit computations in a situation where the methods of [START_REF] Adler | Random fields and geometry[END_REF] or [START_REF] Adler | High level excursion set geometry for non-Gaussian infinitely divisible random fields[END_REF] cannot be applied. Let us also mention that in our previous paper [START_REF] Biermé | A Fourier approach for the level crossings of shot noise processes with jumps[END_REF] we started using the same approach in dimension n = 1, but we worked under a piecewise regularity assumption more restrictive than functions of bounded variation. We propose here a much more general setting, allowed for any dimension n ≥ 1. Note also that our results on coarea formula only rely on a functional assumption and not on a distribution assumption. In particular, they are valid for some processes with jumps and allow to recover partially some recent results in dimension n = 1 on Rice formula for the number of crossings for the sum of a smooth process and a pure jump process [START_REF] Dalmao | Rice formula for processes with jumps and applications[END_REF] or for piecewise deterministic Markov processes [START_REF] Borovkov | On level crossings for a general class of piecewise-deterministic Markov processes[END_REF]. However, we focus here on shot noise random fields for which we set convenient assumptions to ensure the bounded variation, derive explicit computation and obtain asymptotic regime as the underlying intensity tends to infinity.

The paper is organized as follows: we start in Section 2 to give some notations and properties of the functions of bounded variation. Then, in Section 3, we define precisely the shot noise random field and give an explicit computation of the perimeter of its excursion sets. We illustrate our results with some examples. Finally in Section 4, we derive two different asymptotic behavior for the perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity.

Coarea formula.

2.1. The framework of functions of bounded variation. For sake of completeness, we first recall here some definitions and properties of functions of bounded variation. We will mainly use the framework and notations of Ambrosio, Fusco and Pallara in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]. We shall also sometimes refer to Evans and Gariepy [START_REF] Evans | Measure theory and fine properties of functions[END_REF]. In all the following L n will denote the n-dimensional Lebesgue measure in R n , and H k will denote the k-dimensional Hausdorff measure (we will most of the time use it with k = n -1 and keep L n for k = n). When there is no ambiguity we will simply denote dx instead of L n (dx) for the Lebesgue measure in integrals. We will also use the notation µ∠A to denote the restriction of a measure µ to a set A.

Let U be an open subset of R n . A real-valued function f ∈ L 1 (U ) is said to be a function of bounded variation in U if

V (f, U ) := sup U f divϕ dx | ϕ ∈ C 1 c (U, R n ), ϕ ∞ ≤ 1 < +∞,
where C 1 c (U, R n ) denotes the set of continuously differentiable R n -valued functions with compact support in U . We will denote by BV (U ) the space of functions of bounded variation in U . An equivalent definition ([3] p.117-120) of f ∈ BV (U ) is that f ∈ L 1 (U ) is such that its distributional derivative (i.e. its derivative in the sense of distributions) is representable by a finite Radon measure in U , i.e.

U f (x) ∂φ ∂x l (x) dx = - U φ(x) D l f (dx) ∀φ ∈ C ∞ c (U, R), ∀l = 1, . . . , n
for some R n -valued measure Df = (D 1 f, . . . , D n f ). Its total variation is the positive Radon measure denoted by Df and defined by

Df (E) = sup P ∈P(E) k∈K Df (E k ) ,
for all measurable sets E ⊂ U , with P(E) the set of finite or countable partitions P = (E k ) k∈K of E into disjoint measurable sets E k , and where Df (E k ) denotes the Euclidean norm (in R n ) of Df (E k ). Let us note S n-1 the unit sphere of R n . According to the polar decomposition, which follows from Radon-Nikodym Theorem (see Corollary 1.29 of [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]), there exists a unique S n-1 -valued function ν f that is measurable and integrable with respect to the measure Df and such that Df = Df ν f . And we also have

V (f, U ) = Df (U ).
In particular, according to [START_REF] Evans | Measure theory and fine properties of functions[END_REF] p.91, when f : U ⊂ R n → R is Lipschitz, then it is differentiable almost everywhere, and in that case if we denote ∇f (x) = ∂f ∂x 1 (x), . . . , ∂f ∂xn (x) ∈ R n the gradient of f at a point x, and ∇f (x) its Euclidean norm, then Df = ∇f L n , Df = ∇f L n and ν f = ∇f / ∇f , so that

V (f, U ) = Df (U ) = U ∇f (x) dx.
One can define a norm on BV (U ) by

(2.1) f BV (U ) = f L 1 (U ) + Df (U ), so that (BV (U ), • BV (U ) ) is a Banach space ([3] p.121).
The framework of functions of bounded variation is of special interest to study the perimeter of a set because of the following definition and property. Let E be an L n -measurable subset of R n . Then for any open subset U ⊂ R n , we say that E is a set of finite perimeter in U if its indicator function χ E is of bounded variation in U . In that case, we define the perimeter

L E (U ) of E in U as V (χ E , U ), i.e. L E (U ) := sup E divϕ dx | ϕ ∈ C 1 c (U, R n ), ϕ ∞ ≤ 1 .
The term "perimeter" meets here its usual sense in dimension n = 2 as "the length of the boundary". Indeed, it can be shown (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] p.143) that for all sets E with piecewise C 1 boundary in U and such that H n-1 (∂E ∩ U ) is finite, then by Gauss-Green theorem the distributional derivative of χ E is Dχ E = ν E H n-1 ∠∂E, where ν E is the inner unit normal to E, and that

L E (U ) = H n-1 (∂E ∩ U ).
Let f ∈ BV (U ). For t ∈ R, we can consider the excursion set (also sometimes called "upper level set" or "superlevel") of level t of f :

E f (t, U ) := {x ∈ U such that f (x) > t}.
Then for L 1 -almost every t ∈ R, the set E f (t, U ) is of finite perimeter in U . We will denote its perimeter in U by L f (t, U ). Moreover, the function t → L f (t, U ) belongs to L 1 (R) and we have the coarea formula:

(2.2) Df (U ) = R L f (t, U ) dt.
The proof of this formula can be found in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] p.145, or also in [START_REF] Evans | Measure theory and fine properties of functions[END_REF] p.185. Now, in order to use this formula, we need to be more explicit about the pointwise properties of f and the decomposition of its distributional derivative Df . Let us recall that f is said

approximately continuous at x ∈ U ⊂ R n if (2.3) lim ρ→0 ρ -n Bρ(x) |f (y) -f (x)| dy = 0, where B ρ (x)
(x), f -(x) ∈ R and ν f (x) ∈ S n-1 such that f + (x) > f -(x) with lim ρ→0 ρ -n B + ρ (x,ν f (x)) |f (y) -f + (x)| dy = 0 and lim ρ→0 ρ -n B - ρ (x,ν f (x)) |f (y) -f -(x)| dy = 0, where B + ρ (x, ν) (resp. B - ρ (x, ν)) denotes the half-ball determined by ν ∈ S n-1 , that is {y ∈ B ρ (x), y -x, ν > 0} (resp. {y ∈ B ρ (x), y -x, ν < 0}). The set of approximate jump points is denoted by J f , it is a Borel subset of S f . Moreover, by Federer-Vol'pert theorem ([3] Theorem 3.78 p.173), since f ∈ BV (U ), the set S f is a countably H n-1 -rectifiable set with H n-1 (S f J f ) = 0 and Df ∠J f = (f + -f -)ν f H n-1 ∠J f .
By Radon-Nikodym theorem, the distributional derivative Df can be decomposed into the sum of three terms (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] p.184):

Df = ∇f L n + (f + -f -)ν f H n-1 ∠J f + D c f.
These three terms are defined in the following way:

• D a f := ∇f L n is the absolutely continuous part of the Radon measure Df with respect to the Lebesgue measure L n . And moreover, ∇f is here the approximate differential of f (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] p.165 and Theorem 3.83 p.176).

• D j f := (f + -f -)ν f H n-1 ∠J f is the jump part of Df .
• The last term D c f is the so-called Cantor part of Df , it has the property to vanish on sets which have a H n-1 finite measure.

2.2.

A general coarea formula. In our framework, we will be interested in functions that have no Cantor part in their distributional derivative (we will mainly study piecewise C 1 functions). These functions have been introduced by De Giorgi and Ambrosio to study variational problems where both volume and surface energies are involved, and they are called "special functions of bounded variation". Their set is denoted by SBV (U ), and it is a closed subset of (BV (U ), • BV (U ) ) (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF], Corollary 4.3 p.213).

As we already mentioned it in the introduction, our viewpoint will be to study the function t → L f (t, U ). The coarea formula (2.2) only provides the integral of t → L f (t, U ) on R, and we will extend it to the integral of t → h(t)L f (t, U ) on R for any bounded continuous function h. This is the aim of the following theorem.

Theorem 1. Let U be an open subset of R n and let f : U → R be a function in SBV (U ). Using the notations and definitions of the previous section, its distributional derivative is given by

Df = ∇f L n + (f + -f -)ν f H n-1 ∠J f . while Df = ∇f L n + (f + -f -)H n-1 ∠J f .
Let h : R → R be a continuous and bounded function.

Then R h(t)L f (t, U )dt = U h(f (x)) ∇f (x) dx + J f ∩U f + (y) f -(y) h(s) ds H n-1 (dy).
Proof. Let h : R → R be a continuous and bounded function. Let us first assume that there exists an ε > 0 such that h(t) ≥ ε for all t ∈ R. We define for all t ∈ R,

H(t) = t 0 h(s) ds.
Then H is a C 1 diffeomorphism from R to R. It is strictly increasing and since H (t) = h(t) for all t ∈ R is bounded, it is also Lipschitz on R. Now, let u be the function defined on U by u = H • f . Then by the chain-rule (see [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] p.164 and Theorem 3.96 p.189) we have that u ∈ SBV (U ), its jump set J u = J f , and its derivative is given by

Du = (h • f ) ∇f L n + H • f + -H • f -ν f H n-1 ∠J f .
Then by the coarea formula (2.2), we have that

R L u (s, U ) ds = Du (U ) = U h(f (x)) ∇f (x) dx + J f ∩U H(f + (y)) -H(f -(y)) H n-1 (dy) = U h(f (x)) ∇f (x) dx + J f ∩U f + (y) f -(y) h(s) ds H n-1 (dy). But for all s ∈ R, we have that E u (s, U ) = E f (H -1 (s), U ) where H -1 denotes the inverse of the C 1 diffeomorphism H.
And thus for all s ∈ R, we also have that

L u (s, U ) = L f (H -1 (s), U ).
Then by the change of variable t = H -1 (s) (see for instance [START_REF] Rudin | Real and complex analysis[END_REF] p.153), we get

R h(t)L f (t, U ) dt = R L u (s, U ) ds = U h(f (x)) ∇f (x) dx+ J f f + (y) f -(y) h(s) ds H n-1 (dy),
which is the announced formula.

In the general case, when h is not strictly positive, we simply apply the above formula to h 1 = 1 + sup(h, 0) and to h 2 = 1 + sup(-h, 0), and then we have it for h = h 1 -h 2 , which ends the proof of the theorem.

3. Shot noise random fields. Let (Ω, A, P) be a probability space and Φ

= {(x i , m i )} i∈I be a Poisson point process on R n × R d with intensity λL n ⊗ F , with F a probability measure on R d . Let g : R n × R d → R be defined such that for F -almost every m ∈ R d the function g m := g(•, m) belongs to SBV (R n ). From Section 2, it follows that for such m ∈ R d Dg m = ∇g m L n + (g + m -g - m )ν gm H n-1 ∠J gm and g m BV (R n ) = g m L 1 (R n ) + Dg m (R n ) = g m L 1 (R n ) + ∇g m L 1 (R n ) + Jg m (g + m (y) -g - m (y))H n-1 (dy) < +∞.
Under the assumption that

(3.1) R d g m L 1 (R n ) F (dm) < +∞,
one can define almost surely the shot noise random field

(3.2) X Φ = i∈I τ x i g m i , as a random field in L 1 loc (R n ), where τ x i g m i (x) := g m i (x -x i ).
In the sequel we will also consider Φ j = Φ {(x j , m j )} for j ∈ I and its associated shot noise random field

X Φ j = i∈I;i =j τ x i g m i .
Throughout the paper we make the stronger assumption that

(3.3) R d g m BV (R n ) F (dm) < +∞.
3.1. Regularity of the shot noise random fields. The aim of the following theorem is to show that the shot noise random field inherits the regularity properties of the kernel functions g m . The general idea is that if the kernel functions are special functions of bounded variation, then so is locally the shot noise. The theorem also gives the decomposition of its distributional derivative.

In the following, we will need the functional spaces L 1 loc (R n ) and SBV loc (R n ). We recall that they are defined by: a function f belongs to

L 1 loc (R n ) (resp. SBV loc (R n )) if and only if it belongs to L 1 (U ) (resp. SBV (U )) for all bounded open subset U of R n .
Theorem 2. Under Assumption (3.3), one can define almost surely (a.s.) in L 1 loc (R n ) the two shot noise random fields

X Φ := i∈I τ x i g m i and ∇X Φ := i∈I τ x i ∇g m i . Moreover, a.s. X Φ ∈ SBV loc (R n ) with DX Φ = ∇X Φ L n + (X + Φ -X - Φ )ν X Φ H n-1 ∠ J X Φ ,
and for H n-1 almost every y ∈ J X Φ , there exists a unique (x j , m j ) ∈ Φ such that y ∈ J τx j gm j = x j + J gm j and

(3.4) X + Φ (y) = τ x j g + m j (y) + X Φ j (y) and X - Φ (y) = τ x j g - m j (y) + X Φ j (y).
Proof. Let U be a bounded open set of R n . First, note that by Campbell's formula,

E i∈I τ x i g m i BV (U ) = R d R n τ y g m BV (U ) λdyF (dm) = λ R d R n τ y g m L 1 (U ) + τ y Dg m (U ) dyF (dm).
By Fubini's theorem and by the translation invariance of L n we have

R d R n τ y g m L 1 (U ) dyF (dm) = L n (U ) R d g m L 1 (R n ) F (dm).
Moreover, recalling the notation χ V to denote the indicator function of a set V , we have

R n τ y Dg m (U )dy = R n Dg m (U -y)dy = R n ×R n χ U -y (x) Dg m (dx)dy = L n (U ) Dg m (R n ),
by Fubini's theorem and the translation invariance of L n . It follows that

E i∈I τ x i g m i BV (U ) = λL n (U ) R d g m BV (R n ) F (dm) < +∞.
Hence, almost surely i∈I τ x i g m i BV (U ) < +∞. Since (SBV (U ),

• BV (U ) ) is a Banach space, it implies that X Φ = i∈I τ x i g m i is almost surely in (SBV (U ), • BV (U ) ). Now, let us identify DX Φ . First, remark that i∈I τ x i g m i BV (U ) < +∞ implies that i∈I τ x i ∇g m i L 1 (U ) + i∈I Jτ x i gm i ∩U τ x i g + m i (y) -τ x i g - m i (y) H n-1 (dy) < +∞, so that the vectorial Radon measure i∈I τ x i ∇g m i L n ∠ U + i∈I τ x i g + m i -τ x i g - m i ν τx i gm i H n-1 ∠ U ∩Jτ x i gm i
is well-defined. By uniqueness of the Radon Nikodym decomposition we get

∇X Φ L n ∠ U = i∈I τ x i ∇g m i L n ∠ U , (3.5) (X + Φ -X - Φ )ν X Φ H n-1 ∠ U ∩J X Φ = i∈I τ x i g + m i -τ x i g - m i ν τx i gm i H n-1 ∠ U ∩Jτ x i gm i . (3.6)
Note in particular that the last equality implies that

H n-1 U ∩ J X Φ ∩ ∪ i∈I J τx i gm i c = 0,
where for a set S, S c denotes the complement of S.

For a fixed point (x j , m j ) ∈ Φ, let us remark that since X Φ and τ x j g m j are both in SBV (U ) we also have X Φ j = X Φ -τ x j g m j ∈ SBV (U ). Analogously, we get

H n-1 U ∩ J X Φ j ∩ ∪ i∈I;i =j J τx i gm i c = 0.
Note that when y ∈ J τx j gm j ∩ S c X Φ j ∩ U we obtain that y ∈ J X Φ with (3.4) satisfied. Therefore, it suffices to prove that the set of points in U that belong to ∪ j∈I (J τx j gm j ∩ S X Φ j ) is H n-1 negligible. We have

E H n-1 ∪ j∈I (J τx j gm j ∩ S X Φ j ) ∩ U ≤ E   j∈I H n-1 J τx j gm j ∩ S X Φ j ∩ U   ≤ R d R n E H n-1 (J τxgm ∩ S X Φ ∩ U ) λdxF (dm),
by Mecke's formula (see [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]). Now, using Fubini's theorem, for F -a.e. m

R n E H n-1 (J τxgm ∩ S X Φ ∩ U ) dx = E S X Φ ∩U L n (y + J gm )H n-1 (dy) , since J τxgm = x + J gm for all x ∈ R n . But L n (y + J gm ) = L n (J gm ) = 0 for all y ∈ R n ,
which implies that H n-1 ∪ j∈I J τx j gm j ∩ S X Φ j ∩ U = 0 almost surely. Finally, we conclude the proof using the fact that R n is covered by a countable union of bounded open sets so that a.s.

X Φ is in SBV loc (R n ).
Under Assumption (3.3), X Φ is well-defined a.s. as a function in SBV (R n ) but we can also consider X Φ as a real random field indexed by R n . More precisely, let us define

D Φ = x ∈ R n such that X Φ (x) = i∈I τ x i g m i (x) ∩ S c X Φ .
Note that the convergence in

L 1 loc (R n ) implies that a.s. L n (D c Φ ) = 0 so that E (L n (D c Φ )) = 0. Moreover, for U a bounded open set in R n , by Fubini's theorem, one has E (L n (D c Φ ∩ U )) = U P(x ∈ D c Φ )dx = P(0 ∈ D c Φ )L n (U ),
where the last equality comes from

P(x ∈ D c Φ ) = P(0 ∈ D c Φ )
, by stationarity of the point process {x i } i∈I . It follows that for all x ∈ R n , P(x ∈ D c Φ ) = 0 and a.s. x ∈ S c X Φ and X Φ (x) = i∈I τ x i g m i (x). The same remark may be applied to

∇X Φ since L n (S ∇X Φ ) = 0 as ∇X Φ ∈ L 1 loc (R n ).
This allows to compute the finite dimensional law of the random field {(X Φ (x), ∇X Φ (x)); x ∈ R n } itself. In particular, the shot noise random fields have the nice property that their characteristic function is explicit (see for instance [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF] Chapter 2). More precisely, in our framework, the shot noise field {(X Φ (x), ∇X Φ (x)); x ∈ R n } is stationary and therefore the joint characteristic function of X Φ (x) and ∇X Φ (x) is independent of x and is given for all u ∈ R and all v ∈ R n by (3.7)

ψ(u, v) := E(e iuX Φ (x)+i v,∇X Φ (x) ) = exp λ R d R n
(e iugm(y)+i v,∇gm(y) -1) dyF (dm) .

In the following we will also simply denote ψ(u) = ψ(u, 0) = E(e iuX Φ (x) ) the characteristic function of X Φ (x). Let us also mention that the real random variables X Φ (x), ∂X Φ ∂x 1 (x), . . . , ∂X Φ ∂xn (x) are integrable with

E (X Φ (x)) = λ R d R n g m (y)dyF (dm) and E ∂X Φ ∂x l (x) = λ R d R n ∂g m ∂x l (y)dyF (dm),
for all l = 1, . . . , n, implying that ∇X Φ is also integrable. Moreover, under the additional assumption that (3.8)

R d R n g m (y) 2 dyF (dm) < +∞ and R d R n ∇g m (y) 2 dyF (dm) < +∞,
the real random variables X Φ (x), ∂X Φ ∂x 1 (x), . . . , ∂X Φ ∂xn (x) are also square integrable with

Var (X Φ (x)) = λ R d R n g m (y) 2 dyF (dm) and Var ∂X Φ ∂x l (x) = λ R d R n ∂g m ∂x l (y) 2 dyF (dm),
for all l = 1, . . . , n.

3.2.

Perimeter of the excursion sets. We consider the excursion set of X Φ defined as previously by

E X Φ (t, U ) = {x ∈ U such that X Φ (x) > t},
as well as L X Φ (t, U ) its perimeter in U . According to Theorem 1, when h : R → R is a continuous and bounded function, one has a.s. the coarea formula

R h(t)L X Φ (t, U )dt = U h(X Φ (x)) ∇X Φ (x) dx + J X Φ ∩U X + Φ (y) X - Φ (y)
h(s) ds H n-1 (dy).

By (3.4), the jump part rewrites as

J X Φ ∩U X + Φ (y) X - Φ (y)
h(s) ds H n-1 (dy) = j∈I Jτ x j gm j ∩U τx j g + m j (y)+X Φ j (y)

τx j g - m j (y)+X Φ j (y)
h(s) ds H n-1 (dy) = j∈I Jτ x j gm j ∩U τx j g + m j (y)

τx j g - m j (y) h(s + X Φ j (y)) ds H n-1 (dy). (3.9)
We compute the expectation of the jump part of the coarea formula in the next proposition.

Proposition 1. Let h : R → R be a continuous and bounded function. Then,

E J X Φ ∩U X + Φ (y) X - Φ (y) h(s) ds H n-1 (dy) = λL n (U ) R d Jg m g + m (y) g - m (y) E (h(s + X Φ (0))) ds H n-1 (dy)F (dm).
Proof. From Mecke's formula (see [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF]), taking the expectation in (3.9), we get

E J X Φ ∩U X + Φ (y) X - Φ (y) h(s) ds H n-1 (dy) = R n ×R d Jτ xgm ∩U τxg + m (y) τxg - m (y) E (h(s + X Φ (y))) ds H n-1 (dy)λdx F (dm) = R n ×R d Jg m ∩(U -x) g + m (y) g - m (y) E (h(s + X Φ (x + y))) ds H n-1 (dy)λdx F (dm),
by translation invariance of H n-1 . Moreover, by stationarity of X Φ , for all s ∈ R, and

x, y ∈ R n , E (h(s + X Φ (x + y))) = E (h(s + X Φ (0)))
. By Fubini's theorem, integrating with respect to x, this last integral is equal to

λL n (U ) R d Jg m g + m (y) g - m (y) E (h(s + X Φ (0))) dsH n-1 (dy)F (dm),
which is the announced result.

We can now give our main result about the mean value of the perimeter of the shot noise random field. It is a direct consequence of the coarea formula of Theorem 1, when taking for h the function h(t) = e iut , and of the computation of the expectation of the jump part of the coarea formula given in Proposition 1.

Theorem 3. Let X Φ be a shot noise random field given on R n by (3.2) and such that Assumption (3.3) is satisfied. For all t ∈ R, let us denote

C X Φ (t) = E (L X Φ (t, (0, 1) n )) .
Then the function t → C X Φ (t) belongs to L 1 (R) and its Fourier transform is given for all u ∈ R, u = 0 by

C X Φ (u) = E(e iuX Φ (0) ∇X Φ (0) ) + E(e iuX Φ (0) ) λ iu R d Jg m (e iug + m (y) -e iug - m (y) ) H n-1 (dy)F (dm)
and for u = 0, we have

C X Φ (0) = E(V (X Φ , (0, 1) n )) = E( ∇X Φ (0) ) + λ R d Jg m (g + m (y) -g - m (y)) H n-1 (dy)F (dm).
A direct consequence of Theorem 3 is the following corollary that gives a Rice formula in a weak sense (i.e. for almost every level t) and includes the "jump part".

Corollary 1. Under the assumptions of Theorem 3, and if we moreover assume that the random variable X Φ (0) admits a probability density on R, denoted by t → p X Φ (0) (t), then for almost every t ∈ R we have

C X Φ (t) = E( ∇X Φ (0) |X Φ (0) = t)p X Φ (0) (t)+λ R d Jg m g + m (y) g - m (y) p X Φ (0) (t-s) dsH n-1 (dy)F (dm).
Proof. Note that since X Φ (0) admits p X Φ (0) for density we may define the positive measurable function

C(t) := E( ∇X Φ (0) |X Φ (0) = t)p X Φ (0) (t)+λ R d Jg m g + m (y) g - m (y) p X Φ (0) (t-s) dsH n-1 (dy)F (dm), for almost every t ∈ R. Moreover, R C(t)dt = E( ∇X Φ (0) ) + λ R d Jg m g + m (y) -g - m (y) H n-1 (dy)F (dm) ≤ E( ∇X Φ (0) ) + λ R d g m BV (R n ) F (dm) < +∞,
by Assumption (3.3). Then we may compute its Fourier transform and find C = C X Φ . The result follows from the injectivity of the Fourier transform.

Let us quote that sufficient conditions for X Φ (0) to admit a probability density are given in section 3.2 of [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF].

3.3. Some particular cases. In order to have explicit formulas for the mean perimeter C X Φ (t) = E (L X Φ (t, (0, 1) n )), we need to be able to compute the two terms of C X Φ (u) in the formula of Theorem 3. We will give in this section many situations in which the computations are doable. The first case is the one of piecewise constant functions g m , since in that case the first term vanishes. The second case is when E( ∇X Φ (0) 2 ) is finite, because we are then able to use the joint characteristic function of X Φ (0) and ∇X Φ (0) to have an explicit formula for the term E(e iuX Φ (0) ∇X Φ (0) ). This will be the aim of the next section.

Let us start with the piecewise constant case. When the functions g m are piecewise constant then ∇X Φ (0) = 0 a.s. and therefore we simply have

C X Φ (u) = E(e iuX Φ (0) ) λ iu R d Jg m (e iug + m (y) -e iug - m (y) ) H n-1 (dy)F (dm).
Example 1: we consider a shot-noise process X Φ in R 2 made of random shapes, i.e. we assume that n = 2, that the marks m are given by m = (β, r) with β ≥ 0, r ≥ 0 and with the distribution F given by F (dm) = F β (dβ)F r (dr) (having thus β and r independent). We also assume that the functions g m are of the form g m (x) = βχ Kr (x) for all x ∈ R 2 , where for each r, K r is a compact set of R 2 with piecewise smooth boundary and such that the mean perimeter and the mean area respectively defined by

p = R + H 1 (∂K r ) F r (dr) and a = R + L 2 (K r ) F r (dr)
are both finite. Then in that case, we have

∀u ∈ R, u = 0, C X Φ (u) = λp F β (u) -1 iu e λa( F β (u)-1) ,
where F β (u) = R + e iuβ F β (dβ). • In particular if β follows an exponential distribution of parameter µ > 0 then F β (u) = µ µ-iu , and then

C X Φ (u) = λp 1 µ -iu e λa iu µ-iu .
We recognize here (thanks to tables of Fourier transforms !) the Fourier transform of a noncentral chi-square distribution, and we thus have for a.e.

t ∈ R + , C X Φ (t) = λµpe -λa-µt I 0 (2 λµat),
where I 0 is the modified Bessel function of the first kind of order 0 that is given for all t ∈ R by I 0 (t) = 1 π π 0 e t cos θ dθ. • Another explicit and simple case is when β = 1 a.s., which implies that F β (u) = e iu . Then C X Φ (u) is the product of two Fourier transforms: the one of a Poisson distribution and the one of the indicator function of the interval [0, 1]. Therefore we get

∀k ∈ N, for a.e. t ∈ (k, k + 1), C X Φ (t) = λp (λa) k k! e -λa .
We illustrate this result on Figure 1 where we show a sample of a shot-noise process made of two indicator functions of squares. We show here on the left a sample (in the square domain (0, 2000) 2 ) of a shot noise process made of two indicator functions of squares of respective side length 60 and 200, and with respective probability 1/2; with a Poisson point process intensity λ = 2.10 -4 . In the middle we show the boundaries of some of the excursion sets. And on the right we plot the empirical distribution of the perimeter of the excursion sets as a function of the level, together with the expected values of these perimeters (red stars).

3.4. Link with directional derivatives. In the general case, when the functions g m are not piecewise constant, we need to be able to compute the term E(e iuX Φ (0) ∇X Φ (0) ). In order to have an explicit formula for it in terms of the characteristic function of the shot noise (that is given by (3.7)), we will first prove the following proposition.

Proposition 2. Let X and Y be two random variables, such that X is real-valued and Y takes values in R n , n ≥ 1. Let φ be the joint characteristic function of X and Y given by

∀u ∈ R, ∀v ∈ R n , φ(u, v) := E(e iuX+i v,Y ). Assume that E( Y 2 ) < +∞, then ∀u ∈ R, E(e iuX Y ) = -1 2πω n-1 S n-1 +∞ 0 1 t 2 (φ(u, tv) + φ(u, -tv) -2φ(u, 0)) dt H n-1 (dv),
where ω n-1 is the L n-1 measure of the unit ball of R n-1 .

Proof. We first use the well-known identity:

Y = 1 2ω n-1 S n-1 | v, Y | H n-1 (dv).
Now, for any y ∈ R, we have that

-1 π +∞ 0 1 t 2 (e ity + e -ity -2) dt = 2|y| π +∞ 0 1 t 2 (1 -cos(t)) dt = |y|.
We can use this identity for each v, Y , integrate on v ∈ S n-1 , multiply by e iuX and finally take the expectation. Then, since for any t and y in R, we have |e ity + e -ity -2|/t 2 ≤ min(4/t 2 , y 2 ), and since E( Y 2 ) < +∞, we obtain by Fubini's theorem that

E(e iuX Y ) = -1 2πω n-1 S n-1 +∞ 0 1 t 2 E(e iuX+it v,Y ) + E(e iuX-it v,Y ) -2E(e iuX ) dt H n-1 (dv),
which is the announced result.

Example 2: we give here an example of the use of Proposition 2 in the particular case of radial functions g m . To have simpler formulas, we will compute only the total variation of the field. We assume here that n = 2, that the functions g m are "cone" functions (with no jumps) given by g

m (x 1 , x 2 ) = (1 -m x 2 1 + x 2 2 )χ B(0, 1 m ) (x 1 , x 2 )
, and that the marks m ∈ R + are distributed with the Γ(3, 1) distribution. In that case, the characteristic function of ∇X Φ (0) is given by

ψ(0, v 1 , v 2 ) = exp λ 2 R + B(0, 1 m ) (e -im v 1 x 1 +v 2 x 2 √ x 2 1 +x 2 2 -1)m 2 e -m dx 1 dx 2 dm = exp λ 2 R + 2π 0 1 m
0 (e -im(v 1 cos θ+v 2 sin θ) -1)m 2 e -m r dr dθ dm .

Therefore, for any v = (v 1 , v 2 ) ∈ S 1 and any t ∈ R, we have ψ(0, tv 1 , tv 2 ) = ψ(0, t, 0) and we can further compute

ψ(0, t, 0) = exp λ 2 R + 2π 0 1 m 0 (e -imt cos θ -1)m 2 e -m r dr dθ dm = exp λ 4 R + 2π 0 (e -imt cos θ -1)e -m dθ dm = exp λ 4 2π 0 ( 1 1 + it cos θ -1) dθ = exp - λπ 2 + λ 2 π 0 1 1 + t 2 cos 2 θ dθ = exp - λπ 2 + λπ 2 √ 1 + t 2 .
Finally, by Theorem 3 and Proposition 2 we can compute the expected total variation of the shot noise random field in (0, 1) 2 , and we get This last integral is related to Bessel functions, and this is not a surprise since Bessel functions are involved in systems that have cylindrical symmetries.

E(V (X Φ , (0, 1) 2 )) = E( ∇X φ (0) ) =
Example 3: we consider a shot-noise process in R 2 made of a deterministic function with random amplitude. More precisely, we assume that n = 2 and that the marks m are in R (i.e. d = 1) with the distribution F (dm) given by the exponential distribution with parameter 1 so that F (u) = 1 1-iu . We consider the function g

(x) = g(x 1 , x 2 ) = e -x 1 χ R + (x 1 )χ [0,1] (x 2 ) and g m = m × g. Note that g ∈ SBV (R 2 ) with J g = ({0} × [0, 1]) ∪ (R + × {0}) ∪ (R + × {1}) and ∇g(x 1 , x 2 ) = -e -x 1 0 χ R + (x 1 )χ [0,1] (x 2
). It follows that the joint characteristic function of X Φ (x) and ∇X Φ (x) is given by In the middle we show the boundaries of some of the excursion sets. And on the right we plot the empirical distribution of the perimeter of some excursion sets as a function of the level (blue circles), together with the expected values of these perimeters (red curve) given by Formula (3.10). Now, we notice that here the gradient ∇g is non-zero only in the x 1 direction, and that moreover it is always non-positive in that direction. Therefore we have

ψ(u, v 1 , v 2 ) = exp λ +∞ 0 1 0 F e -y 1 (u -v 1 ) -1 dy 1 dy 2 = (1 -i(u -v 1 )) -λ .
E(e iuX Φ (0) ∇X Φ (0) ) = -E e iuX Φ (0) ∂X Φ (0) ∂x 1 = i ∂ψ ∂v 1 (u, 0, 0) = λ (1 -iu) -λ-1 .
By Theorem 3, it remains to compute the second term corresponding to the jump part to get an explicit expression for C X Φ . In this example, for u = 0, we have

1 iu R d Jg m (e iug + m (y) -e iug - m (y) ) H n-1 (dy)F (dm) = F (u) -1 iu + 2 +∞ 0 F (ue -t ) -1 iu dt.
Therefore,

C X Φ (u) = 2λ (1 -iu)) -λ-1 -2λ log(1 -iu) iu (1 -iu) -λ .
The second term corresponds to the Fourier transform of the function

f (t) = 2λ Γ(λ) χ R + (t) t 0 s λ-1 e -s (κ(λ) -log(s)) ds,
where κ is the logarithmic derivative of the Γ function and thus finally we get

(3.10) for a.e. t ∈ R + , C X Φ (t) = 2λt λ e -t Γ(λ + 1) + f (t).
This example is also illustrated by Figure 2, where we show a sample of such a shot noise random field together with the empirical and the theoretical expected length of its excursion sets.

4. Asymptotic Gaussian behaviour. We assume in this section that

R d R n g 2 m (y) dyF (dm) < +∞.
As the intensity λ of the Poisson point process goes to infinity, it is well-known (see for instance [START_REF] Heinrich | Normal convergence of multidimensional shot noise and rates of this convergence[END_REF] or [START_REF] Papoulis | High density shot noise and Gaussianity[END_REF]) that the shot noise random field converges (after normalization) to a Gaussian random field. More precisely, if we denote by X λ the shot noise field defined by Equation (3.2) with a homogeneous Poisson point process {x i } i∈I of intensity λ, then the random field Z λ defined by

(4.1) ∀x ∈ R n , Z λ (x) = X λ (x) -λ g m (y) dyF (dm) √ λ
converges, in the sense of finite dimensional distributions, to a stationary Gaussian random field B of mean 0 and covariance function E(B(x)B(0)) = g m (y -x)g m (y) dyF (dm). The aim of the following theorem is to give the asymptotic behavior of the mean perimeter of the excursion sets of Z λ as λ goes to infinity. It shows in particular that when there are no jumps, we have a finite Gaussian asymptotic whereas when there are jumps, the mean perimeters are not bounded anymore and behave like √ λ.

Theorem 4. Let Z λ be the normalized shot noise random field defined by Equation (4.1). Assume that the functions g m satisfying condition (3.3), also satisfy the following conditions:

σ 2 := g 2 m (y) dyF (dm) < +∞ , σ 2 ∇ := ∇g m (y) 2 dyF (dm) < +∞ , and
(g + m (y) -g - m (y))(|g + m (y)| + |g - m (y)|) H n-1 (dy)F (dm) < +∞ .
Then we have two different asymptotic behaviors:

(a) If there are no jumps, that is if (g + m (y) -g - m (y)) H n-1 (dy)F (dm) = 0 then as λ goes to +∞, we have for any fixed u ∈ R:

C Z λ (u) = 1 ω n-1 √ 2π e -u 2 σ 2 /2 S n-1 v, ∇g m (y) 2 dyF (dm)H n-1 (dv) + o(1). (b) If there are jumps, that is if (g + m (y) -g - m (y)) H n-1 (
dy)F (dm) > 0 then as λ goes to +∞, we have for any fixed u ∈ R:

C Z λ (u) = √ λe -u 2 σ 2 /2 ∇g m (x) dxF (dm) + (g + m (y) -g - m (y)) H n-1 (dy)F (dm) +o( √ λ).
Proof. Let us denote µ := g m (y) dyF (dm), and let us recall that

E(X λ (0)) = λµ and Var(X λ (0)) = λ g 2 m (y) dyF (dm) = λσ 2 ,
and for all v ∈ S n-1 ,

E( v, ∇X λ (0) ) = λ v, ∇g m (y) dyF (dm) =: λµ ∇ (v) and Var( v, ∇X λ (0) ) = λ v, ∇g m (y) 2 dyF (dm) =: λσ 2 ∇ (v). Since Z λ = (X λ -λµ)/ √ λ, the function C Z λ is related to the function C X λ by the rela- tionship C Z λ (u) = 1 √ λ e -iuµ √ λ C X λ ( u √ λ ).
And therefore, for u = 0, we have

C Z λ (u) = 1 √ λ E e iu X λ (0)-µλ √ λ ∇X λ (0) + √ λE(e iu X λ (0)-µλ √ λ ) e i u √ λ g + m (y) -e i u √ λ g - m (y) iu/ √ λ H n-1 (dy)F (dm),
and when u = 0, the expected total variation of Z λ is

E(V (Z λ , (0, 1) n )) = C Z λ (0) = 1 √ λ E( ∇X λ (0) ) + √ λ (g + m (y) -g - m (y)) H n-1 (dy)F (dm).
We have then two cases, depending on whether there are jumps or not.

• We first consider the case (a), when there are no jumps, which means that (g + m (y)g - m (y)) H n-1 (dy)F (dm) = 0. Then for F -almost every m, the function g m is such that its distributional derivative is given by Dg m = ∇g m L n . And since g m ∈ L 1 (R n ), we necessarily have R n ∇g m (x) dx = 0. (Indeed a way to see this is to consider the function t ∈ R → R n g m (x+tw) dx for any vector w ∈ R n . This function is constant equal to g m and therefore its derivative at t = 0 that is equal to R n w, ∇g m (x) dx is equal to 0). We then have, using the result of Proposition 2 and the change of variable t → t/ √ λ,

C Z λ (u) = 1 √ λ E e iu X λ (0)-µλ √ λ ∇X λ (0) = -e -iuµ √ λ 2πω n-1 S n-1 +∞ 0 1 t 2 (ψ λ ( u √ λ , tv √ λ ) + ψ λ ( u √ λ , - tv √ λ ) -2ψ λ ( u √ λ , 0)) dtdv, where ψ λ ( u √ λ , tv √ λ ) = E(e i u √ λ X λ (0)+i t √ λ v,∇X λ (0) ).
On the one hand, using Formula (3.7) for ψ λ , we have for any fixed t, u and v:

e -iuµ √ λ ψ λ ( u √ λ , tv √ λ ) -→ λ→+∞ e -u 2 σ 2 /2-t 2 σ 2 ∇ (v)/2 .
And on the other hand, using the fact that |e ix + e -ix -2| ≤ min(4, x 2 ) for any x real, we get

ψ λ ( u √ λ , tv √ λ ) + ψ λ ( u √ λ , - tv √ λ ) -2ψ λ ( u √ λ , 0) ≤ min(4, t 2 1 λ E( v, ∇X λ (0) 2 )).
Now since E( v, ∇X λ (0) ) = 0, we have E( v, ∇X λ (0) 2 )) = Var( v, ∇X λ (0) ) = λσ 2 ∇ (v) ≤ λσ 2

∇ . Therefore, we can use the dominated convergence theorem and obtain

C Z λ (u) -→ λ→+∞ -e -u 2 σ 2 /2 2πω n-1 S n-1 +∞ 0 1 t 2 (2e -t 2 σ 2 ∇ (v)/2 -2) dtdv = e -u 2 σ 2 /2 ω n-1 √ 2π S n-1 σ 2 ∇ (v) dv,
which ends the proof of case (a).

• For the second case (b), when there are jumps, which means that (g + m (y)-g - m (y)) H n-1 (dy)F (dm) is strictly positive, then ∇g m (x) dxF (dm) is not necessarily equal to 0 anymore. In that case, we will consider C Z λ (u)/ √ λ and we will show that it converges to a finite strictly positive limit. To begin with, we use again Proposition 2 and the change of variable t → t/λ, to have: v) .

On the other hand, we have

ψ λ ( u √ λ , tv λ ) + ψ λ ( u √ λ , - tv λ ) -2ψ λ ( u √ λ , 0) ≤ min(4, t 2 1 λ 2 E( v, ∇X λ (0) 2 )).
Now here E( v, ∇X λ (0) 2 )) = Var( v, ∇X λ (0) ) + E( v, ∇X λ (0) ) 2 = λσ 2 ∇ (v) + λ 2 µ ∇ (v) 2 . Therefore, we can again use the dominated convergence theorem and get to obtain, thanks to the hypothesis in the statement of the theorem, that Using additional assumptions on the order-three moments of X λ and ∇X λ it is possible to obtain bounds of convergence for C Z λ (u) in a way similar to the technical result of our previous paper [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF] in the framework of smooth functions g m .

Example: Assume we are considering a shot noise random field on the plane (i.e. n = 2) and such that there are no jumps. Then if we denote for i, j = 1 or 2, This shows that we have a weak convergence of C Z λ (t) to the formula for the length of level curves in the Gaussian case (that is exactly the formula obtained through Rice formula and probability density functions of Gaussian fields in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]).

γ ij = R d R 2

  Fig 1.We show here on the left a sample (in the square domain (0, 2000) 2 ) of a shot noise process made of two indicator functions of squares of respective side length 60 and 200, and with respective probability 1/2; with a Poisson point process intensity λ = 2.10 -4 . In the middle we show the boundaries of some of the excursion sets. And on the right we plot the empirical distribution of the perimeter of the excursion sets as a function of the level, together with the expected values of these perimeters (red stars).

2

 2 (ψ(0, t, 0) + ψ(0, -t, 0) -2ψ(0, 0, 0)) dt = e -

Fig 2 .

 2 Fig 2. On this figure we show a sample of the shot noise random field of example 3 (exponential in the horizontal direction, with random amplitudes). The sample here (left image) is shown on the square domain (0, 10) 2 and we have taken λ = 4.In the middle we show the boundaries of some of the excursion sets. And on the right we plot the empirical distribution of the perimeter of some excursion sets as a function of the level (blue circles), together with the expected values of these perimeters (red curve) given by Formula (3.10).

  the one hand, for any fixed t, u and v we have 2 σ 2 /2+itµ ∇ (

2

 2 (e itµ ∇ (v) + e -itµ ∇ (v) -2) dtdv + o(1) = e -u 2 σ 2 /2 2ω n-1 S n-1 |µ ∇ (v)| dv + o(1)= e -u 2 σ 2 /2 ∇g m (x) dxF (dm) + o(1).For the jump part, we use the inequalitye m (y) -g - m (y))(|g + m (y)| + |g - m (y)|)

( 4 . 2 )

 42 1 (dy)F (dm) = e -u 2 σ 2 /2 (g + m (y) -g - m (y)) H n-1 (dy)F (dm) + o(1).

γ

  11 cos 2 θ + γ 22 sin 2 θ + 2γ 12 cos θ sin θ dθ.

  is the Euclidean ball of radius ρ and centered at x. The set S f of points where this property does not hold is a L n negligible Borel set called approximate discontinuity set (see[START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] Proposition 3.64 p.160). A point x ∈ S f is called an approximate jump point of f if there exist f +

Acknowledgments. This work has been partly supported by the French National Research Agency under the grant ANR-09-BLAN-0029-01.


Mathilde Faure

Sonia Korchane

Isabelle Le Potier

Antoine Pallandre

Claude Deslouis

Anne-Marie Haghiri-Gosnet

Jean Gamby
email: jean.gamby@upmc.fr

la diffusion de documents scientifiques de niveau

Keywords: Transthyretin, Naphthalene-2,3-dicarboxaldehyde, Capillary electrophoresis, Differential pulse voltammetry

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

Investigating of labelling and detection of transthyretin

synthetic peptide derivatized with naphthalene-2,3-dicarboxaldehyde

Introduction

The naphtalene-2,3-dicarboxyaldehyde (NDA) is a fluorogenic and electroactive reagent commonly used to covalently react with amino functional group of various molecules (see in Scheme 1). This compound is particularly attractive to tag biomolecules such as peptides and proteins and therefore to create novel diagnostic strategies. The N-2-substituted-1cyanobenz-[f]-isoindole (CBI) derivative represents also another opportunity to collect competitive data with electrochemical [1] or optical detections [2,3,4] In recent years, electrochemical detection (ED) has been used with different techniques of separation, such as liquid chromatography in order to analyze amino acids neuron from Helix Aspersa neuron [5] or several aliphatic amines [6]. Capillary electrophoresis has also been coupled

with ED for detection of amino acids in brain homogenate [7], in single erythrocyte [8] and in single mouse peritoneal macrophages [9]. Electrochemical properties of CBI have been utilized to identify compounds from Planorbius corneus dopamine neuron. Detection and separation of amino acids and small peptides can also be carried out in microchips [10].

Electrochemical detection presents some advantages for multi derivatized molecules [11].

Indeed, peptides which bear more than one NH 2 group can be derivatized as often as their number of NH 2 groups. These multi derivatized amino acids present a self quenching effect and therefore are difficult to detect by fluorescence [12] whereas electrochemical response is not attenuated. This derivative was already used and reported in the literature because of its highly fluorescent and relatively stable properties [12,13]. The use of cyanure as nucleophilic in the reaction instead of other one like 2-Mercaptoethanol usually used, permits to obtain the more stable and the best fluorescence intensity for derivative [13].

O O RNH 2 N C N R CN +
Scheme 1: Derivatization reaction of primary amine by NDA.

Here we will focus on the reaction between the NDA and a small peptide of 22 amino acids (PN) likely to be useful for the diagnosis of transthyretin-related familial amyloïd CBI derivative polyneuropathy [14,15,16] (see in Scheme 2). Firstly, in a systematic approach, we study by capillary electrophoresis coupled with laser induced fluorescence (CE-LIF) the influence of some of major parameters to analytically evaluate their influence over the derivatization reaction efficiency. We believe that the following results demonstrate that we optimized the

Experimental

Chemicals and reagents

Chemicals

NDA was purchased from Interchim (Montluçon, France). Potassium cyanide, HEPES, boric acid, lysine, serine and threonine were purchased from Sigma Aldrich (Saint-Quentin Fallavier, France). Methanol and sodium hydroxide from VWR (Fontenay-sous-Bois, France) and the PN peptide was obtained from Genepep (Prades le Lez, France), histidine and tyrosine from Alfa Aesar (Karlsruhe, Allemagne).

Peptide labelling

The labelling method was inspired and adapted by previous works on NDA labelling [1,12,13]

Electrochemical measurement

Electrochemical measurements were performed using a Biologic SP-300 electrochemical analysis system with EC-lab software. Experiments were performed with a three-electrode cell configuration. The reference and the counter electrodes were a saturated calomel electrode (SCE) and a platinum grid (Pt), respectively. The working electrode was glassy carbon with an area equal to 0.071 cm². DPV parameters as modulation time (100 ms), modulation amplitude (7.5 mV), and scan rate (25 mV.s -1 ) have been optimized to obtain analytical curves. Baselines were recorded after DPV response with a blank solution.

Results and discussion

Tagging reaction optimization by capillary electrophoresis

From the literature, it is clear that the chemical and structural aspects of the biomarker are the first parameters to reach high reacting efficiency. Several authors have used NDA\CN - mixtures for amino acids (AA) and small peptides labelling [17][18][19][20]. Their reports were an inspiriting root for our work. However reaction parameters have been optimized to get the best reaction yield for our peptide (22 AA).

We obviously see in Fig. 1 the fluorogenic character of the NDA. In fact, when there is no reaction with the peptide, there is no peak. However, after the peptide derivatization, we logically detect a peak corresponding to the CBI derivative.

In order to develop an efficient derivatization method, the main parameters were fixed by our readings or studies. First, inspired by Hu et al [19], a 1:1 molar ratio of KCN and NDA was fixed for all studies, then we evaluated the yield of the reaction by measuring the peak area when NDA/PN molar ratio changes keeping all the other parameters constant. And in order to keep a pseudo-first-order reaction, we worked with a high excess of NDA compared to PN.

The Fig. S1 in the supplementary material shows the importance of the first studied parameter: the molar ratio between peptide and NDA. We see a net increase in the CE peak area that is characteristic of the tagged peptide, between a NDA/PN molar ratio of 20 and 40, and the maximum peak area is reached for a ratio equal to 40. This observation shows that the reaction is quantitative for this ratio. When the NDA/PN ratio is higher than 40, the main peak area decreases, and it seems that a side peaks appearance is responsible. Indeed, the fluorogenic reagent may react very slowly in different pathways to form fluorescent side products [20]. Those side products can be seen in the Fig. 2 underlining the differences between a 40 and a 100 NDA/PN molar ratio. In fact, the intensity of the 3.5 min peak is much larger in the ratio 100 than in the ratio 40. Moreover, at 5.7 min a peak corresponding to the CBI derivative becomes shouldered when the ratio reaches 100.

Previous studies revealed that the reaction yield increases when the ionic strength of the buffer solution (borate buffer) increases, certainly due to the catalytic effect of this concentration [12]. Thus, we studied the influence of the ionic strength of the borate buffer (from 10 mM to 100 mM) on the reaction yield by keeping the NDA/PN ratio and the pH equal to 40 and 10, respectively. As expected and displayed in Fig. S2 (see supplementary material), we obtained an increase of the peak area when the ionic strength increases.

However, on the same time, the separation efficiency dramatically decreases. This can be explained by the analysis buffer characteristics. In fact, all the analysis were performed with a 40 mM HEPES buffer pH=8 as background electrolyte (BGE). On the other hand, ionic strength of derivatization solution was increasing that makes its conductivity higher than the BGE's one. This phenomenon leads to a flattening of the peaks. In this context, the best compromise between the reaction yield and the separation efficiency was the buffer ionic strength of 70 mM.

The third optimized parameter is the reaction pH. It represents another critical parameter that controls the reaction yield. De Montigny et al [13] reported that the optimum pH of the reaction depends on the amine's pKa. In this direction, Lunte et al [2] found that the optimum pH for derivatization was usually near to the analyte pKa. Taking into account the pKa range of the peptide amine function (from 8.7 to 10.7) and the increase of the fluorescence yield at alkaline pH, we chose to study the reaction on pH varying between 8.5 and 10.

As presented in Fig. S3 (see supplementary material), an increase of the separation efficiency was observed for pH ranging from 8.5 to 9.5, while a decrease of the peak area was observed in this pH range. When the pH is higher than 9.5, the separation efficiency decreased, whereas the peak area increases. In this context, the best compromise between the reaction yield and the separation efficiency was the borate buffer pH 9.

Experiments were carried out to find the optimum reaction time. From the literature we knew that this parameter is another critical one that is correlated with other parameters such as peptide or amino acid concentration, sample volume, composition of the incubating solution [17][18][19][20][21]. The conformation of the peptide is also a factor to take into account because the steric hindrance increases the reaction time. In our work, the optimum reaction time was 15 min. Moreover, the stability of the peptide labelled with NDA was studied, and we found out that it was stable for at least 5 hours in darkness. This latter condition is very important because the photobleaching phenomenon occurs quickly when the sample is light-exposed.

Finally, the sensivity of detection of the CE-LIF method was studied by calculating the limit of detection (LOD) corresponding to a signal-to-noise ratio of 3. In the optimized conditions defined above (reaction time of 15 min, NDA/PN molar ratio equal to 40, ionic strength buffer of 70 mM and pH buffer equal to 9) the LOD of labeled PN was found to be 5 µM. (NDA/PN in a molar ratio of 100). No oxidation or reduction peaks was observed for peptide alone. This confirmed that PN peptide was electroinactive in the potential window, in spite of tyrosine presence in PN sequence which was expected as an electroactive amino acid. Moreover, in this oxidative potential window NDA is also electroinactive (see in Fig. 4). Nevertheless, if we consider a reductive potential window, reduction of two aldehydes groups NDA at pH 9 were observed at -1.1 V/SCE and -1.3 V/SCE. In the case of derivatized PN peptide, an oxidation peak was located at 0.55V/ECS corresponding to CBI oxidation. As noted in Fig. 4, CBI oxidation is irreversible in agreement with the literature [11,22]. Likewise, there was only one peak and no shouldered peak for CBI. It seemed to confirm the inactivity of tyrosine residue in PN. In the literature, peptides with tyrosine residue show two waves of oxidation, one for tyrosine and one for the isoindole ring of CBI [22]. For example, Nussbaun et al explained that the peak potential of isoindole group was characteristic to the derivatized amino acid or peptide. In order to provide an answer to this hypothesis, derivatization of four amino acids contained in PN sequence were performed for lysine, serine, histidine and threonine at the same concentration for each AA (25µM).

Peaks potential for CBI-(lysine), CBI-(serine), CBI-(histidine) and CBI-(threonine) were found at 0.54 V/SCE; 0.57 V/SCE, 0.59 V/SCE and 0.58 V/SCE, respectively as shown in Fig. 5. Our results shown that lysine which was doubly derivatized was easier to oxidize than histidine and serine. Moreover, amino acids with basic chains like histidine are more easily oxidized than one with neutral or acid chains like serine. Threonine has a chain more acid than the histidine one but more basic than the serine chain.

Furthermore, peak potential is dependent on the chain length and the derivatized NH 2 number. In addition PN peak potential occurred approximately at the same peak potential found value that lysine. This seems to demonstrate that PN derivatized is relatively easily oxidized. By considering the PN sequence of 22 amino acids (see in Scheme 2), we assume that PN is obviously doubly derivatized, firstly by the threonine and secondly by the second NH 2 group of lysine to form a CBI-TSESGELHGLTTEEEFVEGIYK-CBI.

Three different protocols of derivatization were performed with DPV. The first one at pH=10 with 2.5 mM NDA/ 25 µM PN molar ratio of 100, the second one at pH=9 with 1 mM NDA/ 25 µM PN molar ratio of 40, the last one at pH=10 with ratio 1 mM NDA/ 25 µM PN molar ratio of 40. As displayed in Fig. 6, CBI derivative oxidation peak was located at 0.55 V/SCE for each protocol, whatever the scan rates. These observations underline that any chemical reaction (such as proton transfer) is involved during charge transfer reaction on CBI derivative obtained from peptide. Indeed, the oxidation process of CBI-PN is only due to a loss of electron. Otherwise, peaks intensity would have been found as dependent of the protocol used. The peak area for the first method was found to be equal to (3.8 ± 0.3).10 -5 , (3.3±0.2).10 -5 for the second, and (3.2±0.2).10 -5 for the third protocol. A comparison for the both pH at molar ratio of 40 shows a quasi identical area. Moreover, peak area of the first protocol with NDA/PN molar ratio of 100 was found as being more important than the one obtained with NDA/PN molar ratio of 40. The observed difference was due to the fact that peptide was probably doubly derivatized leading to a self quenching effect, which happened to be annoying with a fluorescence detection but not with an electrochemical detection. The peak can be decomposed into two different peaks, the first one corresponding to the doubly derivatized at 0.55 V/SCE and the second one to the simply derivatized at 0.69 V/SCE, this latter being more difficult to oxidize and having a peak potential more anodic than amino acids because of the chain length. An area of 3.5x10 -5 and 2x10 -5 were found for the doubly and simply derivatized, respectively.

Another protocol with low concentration of PN (10 µM) in borate buffer pH 10, and ratio of 40 (0.4 mM NDA/ 10 µM PN) was studied by DPV. The peak of CBI-PN was also present at 0.55 V/SCE and had an area of (1.9±0.1).10 -5 . The peak potential is therefore independent of peptide concentration. When added PN concentration increases, CBI-PN yielded increases too, while the corresponding peak potential position still remains independent of the concentration of CBI-PN. In addition, the peak area increased with concentration of CBI-PN formed. Thereby, we have demonstrated that CBI-PN was quantitative using electrochemical detection as low concentration as 10µM of peptide.

Finally, the third method at pH=10 with NDA/PN molar ratio of 40 was also studied by cyclic voltammetry leading to the same previous conclusions: no variation on peak potential position when CBI-PN increases and for scan rate variation.

For protocol at pH=10 and 1 mM NDA/ 25 µM PN molar ratio of 40, variation of peak intensity against time has been investigated. As displayed in Fig. 7, the peak intensity increased as CBI yielded was increasing. Peak intensity was followed after 3, 9 and 15 minutes of derivatization time (Fig. 7A). At the same time, an increase CBI-PN can be correlated to a decrease of the two NDA peaks reduction. As mentioned previously, NDA was electroactive through successive reduction of the two aldehydes groups (see in Fig. 7B). In the other hand, it was an original way for online monitoring of PN derivatization. A time of 3 minutes was sufficient to obtain the formation of CBI-PN. Nevertheless, peak intensity was higher for a derivatization time over 15 minutes in agreement with those obtained with fluorescence detection. Stability of the labelling protocol was also performed by electrochemistry. We observed that the CBI-PN peak was still present at least after 24 hours. After 30min, the CBI peak area represents 98% of the i value, 97% after an hour and 80% after 24hours. This confirms that the solution has to be kept away from light, then the labeling reaction is stable and the solution can be analyzed by electrochemistry 24 hours after mixing.

Conclusions

This work describes NDA derivatives of AAs and peptide for quantitative analysis.

From this assessment we decided to perform several experiments to optically and electrochemically characterize the label reaction of NDA with AAs involved in peptide PN.

The response of derivatives on glassy carbon electrode exhibits excellent electrochemical properties and adequate stability over 24 hours. The labelling reaction was studied by characterizing the resulting labelled molecules with optically and electrochemically characterizations. The measurements are coherent. We notice that the limit of CBI-PN detection is slightly lower for electrochemical measurements (0.2 µM) compared to laser induced fluorescence detection (5 µM). CBI-AAs can be detected at lower working potential, +0.5-0.7 V/ SCE, making this approach selective enough and enabling difference between simply or doubly derivatized peptide. These results have demonstrated the practical utility of detecting CBI derivatives electrochemically in micro-total analysis (µ-TAS). Indeed, electrochemical detection is particularly attractive to perform portable µ-TAS. Electrochemical detection could rather simplify the future miniaturized instrument set-up since the popular electrophoretic separations already require electrical parts. Even if it benefits from a good limit of detection, the introduction of optical detection in microfluidic analytical instrument is experimentally rather complicated or bulky.

Tables

Amino

Scheme 2 .

 2 Scheme 2. Sequence of the studied peptide (PN)

8 3. 2 .

 82 Electrochemical characterizationThe electrolyte support for all experiments is constituted by 100 mM borate buffer in methanol:water mixture (50:50 v/v). PN derivatization was followed by differential pulse voltammetry (DPV) at 25 mV s -1 . The Figures3 and 4display signal differences between 25 µM PN alone, 2.5 mM NDA and for online derivation solution constituted by 25 µM PN with 2.5 mM NDA in presence of 2.5 mM KCN in 100 mM electrolyte support at pH=9

Fig. 1 :

 1 Figure

Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 : 21 -Fig. 7 :

 23456217 Fig. 2: Effect of the NDA/PN molar ratio on the derivatized PN electropherogram: NDA/PN ratio =100 (top), NDA/PN ratio =40 (bottom). Derivatization and electrophoretic conditions are similar to these given in Fig. 1.

Table 1 :

 1 Peak potential characteristics for the derivatized 25 µM for each amino acid (AA-CBI) and for 25 µM PN-CBI versus saturated calomel electrode.

		E	Sd a	L 1/2	b
	acid	(mV)	(mV)	(mV)
	lysine histidine threonine serine PN	540 570 580 590 555 c 690 d	3 4 5 5 4 6	170 240 210 140 190 190
	[a] Standard deviation. [b] Full-widths at mid-height. [c] TTR simply derivatized. [d] TTR doubly
	derivatized.

Acknowledgements

We are indebted to Myriam Taverna for discussions and access to experimental facilities. This research was supported by the French ANR (Agence Nationale de la Recherche) in the context of the P2N "DIMIPOLE" project.