Raphaël Dumas

Eric Jacquelin

Stiffness of a wobbling mass models analysed by a smooth orthogonal decomposition of the skin movement relative to the underlying bone

Keywords: Soft tissue motion, marker-cluster translations, vibration modes, covariance matrix, eigenvalues f: sampling frequency U: velocity covariance matrix Ψ : smooth orthogonal vector , : smooth orthogonal value, diagonal matrix of eigenvalues circular frequency, K KK stiffness matrix, stiffness coefficient M, M: mass, mass matrix identity matrix : basis vector (i.e. marker-cluster geometrical transformation)

The so-called soft tissue artefacts and wobbling masses have both been widely studied in biomechanics, however most of the time separately, from either a kinematics or a dynamics point of view. As such, the estimation of the stiffness of the springs connecting the wobbling masses to the rigid-body model of the lower limb, based on the in vivo displacements of the skin relative to the underling bone, has not been performed yet. For this estimation, the displacements of the skin markers in the bone-embedded coordinate systems are viewed as a proxy for the wobbling mass movement.

The present study applied a structural vibration analysis method called smooth orthogonal decomposition to estimate this stiffness from retrospective simultaneous measurements of skin and intra-cortical pin markers during running, walking, cutting and hopping.

For the translations about the three axes of the bone-embedded coordinate systems, the estimated stiffness coefficients (i.e. between 2.3 kN/m and 55.5 kN/m) as well as the corresponding forces representing the connection between bone and skin (i.e. up to 400 N) and corresponding frequencies (i.e. in the band 10-30 Hz) were in agreement with the literature. Consistently with the STA descriptions, the estimated stiffness coefficients were found subject-and task-specific.

Introduction

The movement of the skin, muscles, and fat relative to the underlying bone is a well-known phenomenon. It has been described, from a kinematics point of view, as the soft tissue artefact (STA). Indeed, this relative movement has a deleterious effect on the joint kinematics estimated from skin markers and motion capture systems [START_REF] Leardini | Human movement analysis using stereophotogrammetry. Part 3. Soft tissue artifact assessment and compensation[END_REF][START_REF] Peters | Quantification of soft tissue artifact in lower limb human motion analysis: A systematic review[END_REF]. At the same time, from a dynamics point of view, the soft tissue motion, modelled as wobbling masses connected to the rigid-body model of the lower limb, is also recognised to have an effect on the joint kinetics (i.e. energy dissipation, torque reduction) [START_REF] Challis | Soft Tissue Motion Influences Skeletal Loads During Impacts[END_REF][START_REF] Gruber | A comparative study of impact dynamics: wobbling mass model versus rigid body models[END_REF] during motor tasks involving impacts with the ground.

One key parameter of these wobbling mass models is the stiffness of the springs connecting them to the rigid-bodies. Most of the models of the literature include linear or non-linear springs attached to a wobbling mass that can translate (and eventually rotate) with respect to the bone [START_REF] Alonso | Motion data processing and wobbling mass modelling in the inverse dynamics of skeletal models[END_REF][START_REF] Gittoes | Soft tissue contributions to impact forces simulated using a four-segment wobbling mass model of forefoot-heel landings[END_REF][START_REF] Gruber | A comparative study of impact dynamics: wobbling mass model versus rigid body models[END_REF][START_REF] Günther | Dealing with skin motion and wobbling masses in inverse dynamics[END_REF][START_REF] Mclean | Development and Validation of a 3-D Model to Predict Knee Joint Loading During Dynamic Movement[END_REF][START_REF] Pain | Wobbling Mass Influence on Impact Ground Reaction Forces: A Simulation Model Sensitivity Analysis[END_REF][START_REF] Wilson | Determination of subject-specific model parameters for visco-elastic elements[END_REF]. Identification of the parameters of these wobbling mass models, based on the ground reaction forces, as well as sensitivity analyses have been widely performed [START_REF] Alonso | Motion data processing and wobbling mass modelling in the inverse dynamics of skeletal models[END_REF][START_REF] Gittoes | Sensitivity of loading to the timing of joint kinematic strategies in simulated forefoot impact landings[END_REF][START_REF] Pain | Wobbling Mass Influence on Impact Ground Reaction Forces: A Simulation Model Sensitivity Analysis[END_REF][START_REF] Wilson | Determination of subject-specific model parameters for visco-elastic elements[END_REF]. However, to the best of the author's knowledge, the estimation of the stiffness parameters from the displacements of the skin relative to the underling bone measured in vivo by intra-cortical pins has not been performed yet. For this estimation, the displacements of the skin markers in the bone-embedded coordinate systems are viewed as a proxy for the wobbling mass movement.

The objective of this study was to estimate the stiffness matrix of a wobbling mass model, defined as a cluster of lumped masses undergoing translations about the three axes of the bone-embedded coordinate system, by applying a structural vibration analysis method, called smooth orthogonal decomposition [START_REF] Chelidze | Smooth orthogonal decomposition-based vibration mode identification[END_REF], to the simultaneous measurements of skin and intra-cortical pin markers [START_REF] Benoit | Effect of skin movement artifact on knee kinematics during gait and cutting motions measured in vivo[END_REF][START_REF] Reinschmidt | Effect of skin movement on the analysis of skeletal knee joint motion during running[END_REF]. In this method, the displacement of the skin markers relative to the underlying bone was modelled as the free undamped vibrations of a dynamical system for which the stiffness matrix can be straightforwardly identified.

Material and methods

Smooth orthogonal decomposition of the skin movement relative to the underlying bone

The STA vector,

  j i k v
, was defined to represent the displacement that the skin marker j (j=1:m i ) associated with the segment i (i = 1 for shank and i = 2 for thigh) underwent relative to a relevant bone-embedded coordinate system and a reference position at each discrete time k (k=1:n) during the analysed motor task (Dumas et al., 2014a). The STA of all markers on the segment i were represented using the STA field,  

i k V :     j ii kk      

Vv

(1).

A sample covariance matrix was computed from this STA field known at every sampled instants of time:

    1 T i i i k k n          S V V
(2).

This sample covariance matrix has been used in previous studies to compute, by a proper orthogonal decomposition, the main components of the STA during a running task (Dumas et al., 2014a, b). In the smooth orthogonal decomposition, a differential operator was further introduced:

1 1 0 0 0 1 1 0 0 0 0 1 1 f                 D (3),
where f was the sampling frequency.

This differential operator allowed the computation of the other covariance matrix standing for the velocities of the skin markers:

      1 1 T T i i i k k n           U D V V D (4).
The smooth orthogonal modes were solutions of the eigen-problem:

ll i i i i i          S Ψ λ U Ψ (5).
with

l i Ψ (l = 1:3m i ) the smooth orthogonal vectors and     2 1 l l i i    
the circular frequencies related to the smooth orthogonal values. Assuming that the STA field was the results of free undamped vibrations of a cluster of lumped masses (i.e. each markers having a same mass M i /m i ), the smooth orthogonal modes are good estimates of the linear normal modes. Therefore, the dynamics of the marker-cluster, as observed from the STA field, was characterized by:

  1 l l i i i i i           K Ψ M Ψ λ (6),
with K i the stiffness matrix, i λ a diagonal matrix composed of the smooth orthogonal eigenvalues and i i i M m  ME the mass matrix (i.e. with E the identity matrix of dimension 3m i ×3m i ).

Therefore, the stiffness matrix was given by:

  1 1 l l i i i i i i M m            K Ψ λ Ψ (7).
According to the recent descriptions of the STA [START_REF] Andersen | A linear soft tissue artefact model for human movement analysis: proof of concept using in vivo data[END_REF][START_REF] Benoit | Surface marker cluster translation, rotation, scaling and deformation: Their contribution to soft tissue artefact and impact on knee joint kinematics[END_REF][START_REF] Dumas | What Portion of the Soft Tissue Artefact Requires Compensation When Estimating Joint Kinematics[END_REF][START_REF] Grimpampi | Metrics for Describing Soft-Tissue Artefact and Its Effect on Pose, Size, and Shape of Marker Clusters[END_REF] and to the wobbling mass models reported in the literature [START_REF] Alonso | Motion data processing and wobbling mass modelling in the inverse dynamics of skeletal models[END_REF][START_REF] Bélaise | Effect of wobbling mass modeling on joint dynamics during human movements with impacts[END_REF][START_REF] Challis | Soft Tissue Motion Influences Skeletal Loads During Impacts[END_REF][START_REF] Gittoes | Sensitivity of loading to the timing of joint kinematic strategies in simulated forefoot impact landings[END_REF][START_REF] Gruber | A comparative study of impact dynamics: wobbling mass model versus rigid body models[END_REF][START_REF] Günther | Dealing with skin motion and wobbling masses in inverse dynamics[END_REF][START_REF] Mclean | Development and Validation of a 3-D Model to Predict Knee Joint Loading During Dynamic Movement[END_REF][START_REF] Wilson | Determination of subject-specific model parameters for visco-elastic elements[END_REF], it was useful to retrieve the stiffness matrix corresponding only to the modes defining the marker-cluster geometrical rigid transformations and more specifically to the marker-cluster translations. This stiffness matrix was given by:

T l l i i i i          K Φ K Φ (8),
where l i Φ (l = 1:3) were the unitary basis vectors built a priori (Dumas et al., 2014a) to define the 3 translations of the marker-cluster about the axes of the bone-embedded coordinate system. These basis vectors were:

1 2 3 1 0 0 1 0 , , and
0 1 0 0 i i i i i i m m m                                                          Φ Φ Φ (9).
Finally, the force vector representing the connection between the bone and skin, applied to each marker j at each discrete time k along each axes of the bone-embedded coordinate system, were given by:

        l i T ll i i i i i i i ak k k k                F K V Φ K Φ V (10).
Note that the last factors of Eq.10 matched the definition of the amplitude of the STA (Dumas et al., 2014a), that is to say the projection of the STA field on a given mode:

      T l l i i i a k k  Φ V (11).
As the basis vectors l i Φ (l = 1:3) represented the translations of the marker-cluster, each marker j had the same projected displacement and, therefore, the same force. Moreover,

    l i l i i ak vk m
 directly corresponded to the projected displacement of the centroid of the marker-cluster about the relevant axis of the bone-embedded coordinate system (i.e. l = 1 for X axis, l = 2 for Y axis, and l = 3 for Z axis). The basis vectors l i Φ (l = 1:3) were also orthogonal and, therefore, the stiffness matrix defining the marker-cluster translations was diagonal (i.e.

 

l i i diag K  K
).

Experimental data

The retrospective data used in this study included the right thigh and shank movements from five trials of a running task (i.e. stance phase, from ground contact to take-off) performed by three able-bodied male subjects [START_REF] Reinschmidt | Effect of skin movement on the analysis of skeletal knee joint motion during running[END_REF], and from one trial of walking and cutting tasks (i.e. stance phase, from ground contact to take-off) and hopping task (i.e. during 0.67 s after ground contact) performed by one able-bodied male subject [START_REF] Benoit | Effect of skin movement artifact on knee kinematics during gait and cutting motions measured in vivo[END_REF].

Clusters of markers were attached to intra-cortical pins inserted into the lateral tibial and femoral epicondyles. Both intra-cortical pin markers and skin markers (i.e. between 4 and 6 by segments) were tracked using either three high-speed cameras at 200 Hz [START_REF] Reinschmidt | Effect of skin movement on the analysis of skeletal knee joint motion during running[END_REF] or tracked using four infrared cameras at 120 Hz [START_REF] Benoit | Effect of skin movement artifact on knee kinematics during gait and cutting motions measured in vivo[END_REF]. Unfiltered raw data were used. The bone-embedded coordinate systems were defined using radiostereometry analysis [START_REF] Lafortune | Three-dimensional kinematics of the human knee during walking[END_REF], with X axis defined anterior, Y axis proximal, and Z axis lateral.

The subjects' demographics and experimental setup are summarised in Table 1. The wobbling masses M i were estimated with 4.8 % and 12.3 % of the total body mass [START_REF] Dumas | Adjustments to McConville et al. and Young et al. body segment inertial parameters[END_REF] and 77.65 % and 90.3 % (i.e. skin, adipose tissue, and muscle [START_REF] Clarys | Anthropometric Prediction of Component Tissue Masses in the Minor Limb Segments of the Human Body[END_REF])) of the segment mass for the shank and thigh, respectively.

Results

Figure 1 represents the stiffness coefficients (i.e. median, quartiles, minimum and maximum for the five running trials of subjects R1, R2 and R3) for the translation of the maker-cluster about the X, Y and Z axes of the bone-embedded coordinate systems for the different subjects and motor tasks. The stiffness appeared generally higher for the shank (i.e. between 10.2 kN/m and 55.5 kN/m) than for the thigh (i.e. between 2.3 kN/m and 11.1 kN/m). The stiffness was higher about the X axis than about the Y and Z axes. The stiffness was also higher for the three subjects (R1, R2, and R3) performing the running task than for the subject (R1) performing walking, cutting and hopping tasks, except for the thigh during the cutting and hopping tasks.

Figure 2 illustrates the forces (i.e. median for the five running trials of subjects R1, R2 and R3) representing the connection between the bone and skin masses about the X, Y, and Z axes of the bone-embedded coordinate systems for the different subjects and motor tasks. The forces were generally two times higher for the shank (i.e. up to 400 N during running) than for the thigh (i.e. up to 100 N during running). The forces were also higher for the three subjects (R1, R2, and R3) performing the running task than for the subject (R1) performing walking, cutting and hopping tasks. The forces were mainly about the X and Y axes, but remarkably about the X axis of the tibia-embedded coordinate systems during the hopping task. Some similarities in the pattern of the forces during the running tasks could be observed.

Discussion

This study applied a structural vibration analysis method called smooth orthogonal decomposition to estimate the stiffness matrix of a wobbling mass model from simultaneous measurements of skin and intra-cortical pin markers. The wobbling mass model consisted of a cluster of lumped masses (or, equivalently, of a concentrated mass at the cluster centroid) undergoing translations about the three axes of the bone-embedded coordinate system. Yet, Eqs. 8 and 10, allows to define stiffness matrix i K , and force vector F i , corresponding to any basis vectors l i Φ defining the rigid or non-rigid marker-cluster geometrical transformations or any other change of configuration in the skin marker positions. In this case, the stiffness matrix may not be diagonal and the force applied to each markers of the segment can be different. For instance, in the literature, rotations about the axes of the bone-embedded coordinate system have been also considered in some wobbling mass models [START_REF] Gruber | A comparative study of impact dynamics: wobbling mass model versus rigid body models[END_REF][START_REF] Schmitt | Human leg impact: energy dissipation of wobbling masses[END_REF]. Other possible changes of configuration are the smooth orthogonal vectors presented in this study being the translation of the centroid of the marker-cluster) is useful to implement the forces representing the connection between bone and skin in a dynamic model. These can be representative of the forces between the rigid-body and the wobbling mass to be introduced in the inverse dynamic computation [START_REF] Alonso | Motion data processing and wobbling mass modelling in the inverse dynamics of skeletal models[END_REF][START_REF] Günther | Dealing with skin motion and wobbling masses in inverse dynamics[END_REF]. In the practical case, without the measurements of intra-cortical pin markers, the translation of the centroid of the marker-cluster can be estimated within a multibody kinematics optimisation [START_REF] Bélaise | Effect of wobbling mass modeling on joint dynamics during human movements with impacts[END_REF][START_REF] Cerveri | Kinematical models to reduce the effect of skin artifacts on marker-based human motion estimation[END_REF][START_REF] Richard | Feasibility of incorporating a soft tissue artefact model in multi-body optimisation[END_REF].

From the skin and intra-cortical pin data used in the present study, the estimated values of the stiffness coefficients and forces were found to be different between the different motor tasks.

For instance, for the Y axis of the tibia-embedded coordinate system, the stiffness coefficients were 11.7 kN/m for walking, 21.4 kN/m for cutting, 18.4 kN/m for hopping, and between 32.4 kN/m and 49.7 kN/m for running. For the Y axis of the femur-embedded coordinate system, the stiffness coefficients were 0.4 kN/m for walking, 0.5 kN/m for cutting, 0.7 kN/m for hopping, and between 0.6 kN/m and 0.9 kN/m for running. An intra-and inter-variability also appeared across the three subjects performing the running task although some similar patterns of force was observed. As STA is well known to be subject-and task-specific [START_REF] Leardini | Human movement analysis using stereophotogrammetry. Part 3. Soft tissue artifact assessment and compensation[END_REF][START_REF] Peters | Quantification of soft tissue artifact in lower limb human motion analysis: A systematic review[END_REF], it was expected to obtain such differences.

Moreover, they could be due to the differences in impact forces and muscle activities (however not known in the present study) according to the concept of muscle tuning [START_REF] Nigg | Impact Forces and Muscle Tuning: A New Paradigm[END_REF].

The stiffness coefficients estimated in the present study (Fig. 1) are overall comparable to the other linear parameters reported in the literature to connect the wobbling masses to the rigidbodies. A range from 17.5 kN/m to 27 kN/m [START_REF] Gittoes | Sensitivity of loading to the timing of joint kinematic strategies in simulated forefoot impact landings[END_REF][START_REF] Gittoes | Soft tissue contributions to impact forces simulated using a four-segment wobbling mass model of forefoot-heel landings[END_REF] for the shank and from 12 kN/m [START_REF] Gittoes | Sensitivity of loading to the timing of joint kinematic strategies in simulated forefoot impact landings[END_REF] to 64 kN/m [START_REF] Mclean | Development and Validation of a 3-D Model to Predict Knee Joint Loading During Dynamic Movement[END_REF] for the thigh was reported. The forces estimated in the present study (Fig. 2) are also in agreement with the literature. By estimating the acceleration (multiplied by the wobbling mass M i ) of the centroid of the marker-cluster with respect to the position of other skin makers placed at joint level, [START_REF] Schmitt | Human leg impact: energy dissipation of wobbling masses[END_REF] reported maximal vertical forces of 409 N and 684 N and maximal horizontal forces of 111 N and 390 N for the shank and thigh respectively during running. Using accelerometers placed on the muscle belly of tibialis anterior, lateral gastrocnemius, vastus lateralis, and biceps femoris long head, [START_REF] Wakeling | Muscle activity reduces soft-tissue resonance at heel-strike during walking[END_REF] similarly reported vertical forces between 24 N and 196 N. Moreover, using the scalar equation 2

(2 ) [START_REF] Challis | Soft Tissue Motion Influences Skeletal Loads During Impacts[END_REF] estimated a maximal vertical force of 315 N for the shank. Considering the wobbling masses of 2-3 Kg for the shank and 7-10 Kg for the thigh (Table 1), it can be recognized (i.e.

l l l i i i i F M f v   ,

2

(2 )

l l i i i K f M  
) that the vibration frequencies were in a band 10-30 Hz. This appears in agreement, but in the lower end of the range, with the literature [START_REF] Lafortune | Tibial shock measured with bone and skin mounted transducers[END_REF][START_REF] Pain | The influence of soft tissue movement on ground reaction forces, joint torques and joint reaction forces in drop landings[END_REF][START_REF] Schmitt | Human leg impact: energy dissipation of wobbling masses[END_REF][START_REF] Wakeling | Muscle activity reduces soft-tissue resonance at heel-strike during walking[END_REF] Using the same walking, cutting, and hopping data from [START_REF] Benoit | Effect of skin movement artifact on knee kinematics during gait and cutting motions measured in vivo[END_REF] as in the present study, [START_REF] Andersen | A linear soft tissue artefact model for human movement analysis: proof of concept using in vivo data[END_REF] applied a principal component analysis and demonstrated that the displacements of the skin markers relative to the underling bone can be represented by a linear combination of a low number of components. Similarly, using the same running data from [START_REF] Reinschmidt | Effect of skin movement on the analysis of skeletal knee joint motion during running[END_REF], Dumas et al. (2014b) applied a proper orthogonal decomposition and demonstrated that the displacements of the skin markers relative to the underling bone can be represented by a linear combination of a low number of modes. In addition, these main components and modes corresponded to the translation and rotation of the marker-cluster. Therefore, it appeared logical to also associate these components and modes to the wobbling masses vibration. This association between the main modes of the STA and the wobbling masses vibration could be obtained, in the present study, by the smooth orthogonal decomposition. As stated before, this association considers the displacements of the skin markers as a proxy for the wobbling mass movement. As explained by [START_REF] Challis | Soft Tissue Motion Influences Skeletal Loads During Impacts[END_REF], "skin-marker motion must be associated with the motion of the underlying soft tissues because skin has low stiffness compared with the stiffness of the muscle-tendon complexes". However, modelling the dynamics of the continuous deformation of the soft tissues (i.e., skin, muscles, and fat) by the discrete movement of a rigid wobbling mass remains challenging. For instance, the proportion of the soft tissues mass involved in the wobbling mass is an open question. For a better insight, the analysis of the relative movement between the skin and the bone should be completed by the analysis of the ground reaction forces (not known in the present study). The recent studies which simultaneously track skin markers trajectories and external forces to estimate joint kinematics and segment masses [START_REF] Bonnet | A constrained extended Kalman filter for the optimal estimate of kinematics and kinetics of a sagittal symmetric exercise[END_REF][START_REF] Jackson | Residual Elimination Algorithm Enhancements to Improve Foot Motion Tracking During Forward Dynamic Simulations of Gait[END_REF] may be adapted to thoroughly identify the parameters of the wobbling mass models. Additionally, the deformable modelling of the lower limb segments [START_REF] Clark | Are fixed limb inertial models valid for dynamic simulations of human movement[END_REF][START_REF] Halloran | Concurrent musculoskeletal dynamics and finite element analysis predicts altered gait patterns to reduce foot tissue loading[END_REF][START_REF] Stelletta | Modeling of the thigh: a 3D deformable approach considering muscle interactions[END_REF] may help in understanding the dynamics effects of the soft tissue deformation (e.g. mass redistribution, energy dissipation, intra-segmental forces,) and help in verifying if the displacements of the skin markers can be representative of them.

This study had some limitations. First, as mentioned above, the wobbling mass model is only built from the relative movement between the skin and the bone (i.e., STA). Second, the STA vector could be biased by the loosening and bending of the intra-cortical pins [START_REF] Ramsey | Biomechanics of the knee: methodological considerations in the in vivo kinematic analysis of the tibiofemoral and patellofemoral joint[END_REF]. Third, this study was based on retrospective data [START_REF] Benoit | Effect of skin movement artifact on knee kinematics during gait and cutting motions measured in vivo[END_REF][START_REF] Reinschmidt | Effect of skin movement on the analysis of skeletal knee joint motion during running[END_REF] and the detailed anthropometry of the three subjects performing the running task as well as the ground reaction forces and muscle activities were not available.

Fourth, the study was based on motion capture data (i.e. at 120Hz and 200Hz) while accelerometry may be considered more appropriate [START_REF] Coza | Quantification of Soft-Tissue Vibrations in Running: Accelerometry versus High-Speed Motion Capture[END_REF]. Nevertheless, as the vibration frequencies were in the band 10-30 Hz, the Nyquist-Shannon sample-rate criterion was satisfied. Fifth, the determination of the stiffness matrix using the smooth orthogonal decomposition assumed that the STA field was the results of the free undamped vibrations of a cluster of lumped masses while damping effects are not negligible in the soft tissue vibrations [START_REF] Khassetarash | Damping and energy dissipation in soft tissue vibrations during running[END_REF][START_REF] Schmitt | Human leg impact: energy dissipation of wobbling masses[END_REF][START_REF] Wakeling | Muscle activity reduces soft-tissue resonance at heel-strike during walking[END_REF].

Nevertheless, benchmark simulations demonstrated that the smooth orthogonal decomposition was also reliable in the case of forced damped vibrations [START_REF] Chelidze | Smooth orthogonal decomposition-based vibration mode identification[END_REF].

In conclusion, the smooth orthogonal decomposition applied to the measured displacements of the skin markers relative to the underling bone allowed to estimate the stiffness of a wobbling mass model translating about the three axes of the bone-embedded coordinate systems. The estimated stiffness coefficients (i.e. between 2.3 kN/m and 55.5 kN/m) as well as the corresponding forces (i.e. up to 400 N) and frequencies (i.e. in the band 10-30 Hz) were in agreement with the literature. The estimated stiffness coefficients were found subject-and task-specific.

Table and figure captions

Table 1: Subjects' demographics and experimental conditions

Ψ

  as well as the proper orthogonal vectors previously proposed to describe the main components of the STA(Dumas et al., 2014b;[START_REF] Dumas | What Portion of the Soft Tissue Artefact Requires Compensation When Estimating Joint Kinematics[END_REF]. Though, the smooth orthogonal vectors, together with the circular frequencies l i  , are the only estimates which are fully representative of the linear normal modes of the wobbling mass model. Thus, the stiffness matrix K i can be determined through an appropriate eigen-decomposition where the inverse of the eigenvalues   frequencies[START_REF] Chelidze | Smooth orthogonal decomposition-based vibration mode identification[END_REF]. When having determined the stiffness matrix K corresponding to the translations of the marker-cluser, the introduction of the amplitude (i.e. with

Figure 1 :Figure 2 :

 12 Figure 1: Boxplot for the stiffness coefficients about the X, Y and Z axes of the bone-

Figure 1 Figure 2

 12 Figure 1

Table 1

 1

							Shank (i=1)	Thigh (i=2)
	Subject	Age (years)	Mass (kg)	Height (m)	Motor task	f (Hz)	m 1	M 1 * (kg)	m 2	M 2 # (kg)
	R1, R2, and R3	25.7 ± 2.1 85.5 ± 9.6	186.7 ± 9.6	Running	200	6	3.19	5	9.50
					Walking,
	B1	22	63	175	cutting, and	120	4	2.35	4	7.00
					hopping
	* (4.8 × 77.65)% of averaged or subject's mass
	# (12.3 × 90.3)% of averaged or subject's mass

Acknowledgements

The authors would like to thank A. J. van den Bogert of Cleveland State University for kindly providing the unfiltered running data. The authors are also grateful to D. Benoit for having made available the unfiltered walking, cutting and hopping data for this special issue of the Journal of Biomechanics on the soft tissue artefact.

Conflict of interest

The authors do not have any financial or personal relationships with other people or organizations that would have inappropriately influence this study.


Clémence Baudin
email: clemence.baudin@ifsttar.fr

Marie Lefevre

Bernard Laumon

Anne-Sophie Sophie Evrard

Marie Lefèvre

The effects of aircraft noise exposure on psychological distress: the results of the DEBATS study in France

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

INTRODUCTION

Many studies have evidenced adverse effects of exposure to aircraft noise on health, such as annoyance, cardiovascular diseases, sleep disorders or altered cognitive performance [START_REF] Babisch | Annoyance due to aircraft noise has increased over the years-results of the HYENA study[END_REF]- [START_REF] Stansfeld | Aircraft and road traffic noise and children's cognition and health: a cross-national study[END_REF]. Some studies have investigated low birth weights, obesity or diabetes [START_REF] Pyko | Exposure to traffic noise and markers of obesity[END_REF], [START_REF] Matsui | Association between the rates of low birth-weight and/or preterm infants and aircraft noise exposure[END_REF], but these effects remain discussed because of divergent results. It's also the case for psychological distress: some studies showed a significant association between aircraft noise exposure and psychological disorders [START_REF] Miyakita | POPULATION-BASED QUESTIONNAIRE SURVEY ON HEALTH EFFECTS OF AIRCRAFT NOISE ON RESIDENTS LIVING[END_REF]- [START_REF] Van Den Berg | The Relation between Self-Reported Worry and Annoyance from Air and Road Traffic[END_REF], and some other did not [START_REF] Reijneveld | The impact of the Amsterdam aircraft disaster on reported annoyance by aircraft noise and on psychiatric disorders[END_REF]- [START_REF] Sygna | Road traffic noise, sleep and mental health[END_REF]. DEBATS (Discussion on the health effects of aircraft noise) is the first research program to investigate the health effects of aircraft noise exposure in France. One of the specific objectives of the study was to evaluate the effects of aircraft noise exposure on psychological distress in people living in the vicinity of French airports.

METHODS

Participants

1,244 individuals older than 18 and living in the vicinity of three French airports (Paris-Charles de Gaulle, Lyon-Saint-Exupéry and Toulouse-Blagnac) were randomly selected to participate in the longitudinal study of DEBATS. The study population was stratified on aircraft noise contours. These contours are based on the day-evening-night equivalent level (Lden) defined in four categories, <50, 50-54, 55-59 and ≥60 dB(A). All participants responded to a questionnaire administered by an interviewer at their place of residence. The questionnaire collected in particular socio-demographic information, the personal medical history of the participants and lifestyle factors such as smoking and alcohol consumption.

Psychological distress

Psychological distress was evaluated within this questionnaire by two ways:

-with a single question about the existence of depression's symptoms: "During the past 12 months, was there ever a time when you felt sad or depressed: extremely, very, moderately, slightly, not at all?". It was then categorized into two classes for the statistical analyses: extremely or very versus moderately, slightly or not at all; -with the General Heath Questionnaire, version in 12 items (GHQ-12) [START_REF] Golderberg | A User's Guide to the General Health Questionnaire[END_REF], which evaluates psychological disorders covering four domains: mood, behavior, and current or recent feelings, in the period of recent weeks. A GHQ-score was then calculated and treated either as a continuous variable, or as a dichotomous variable, with a 2/3 threshold.

Confounders

Several factors which could affect psychological distress were obtained from the questionnaire: age (six categories: 18-34; 35-44; 45-54; 55-64; 65-75; >75 years old), country of birth (France-born/foreign-born), gender, occupational activity (yes/no), alcohol consumption (no/light/moderate/heavy), smoking (non-smoker/ex-smoker/occasional smoker/daily smoker), number of stressful life events (0/1/≥2), and income (<2,300€/month ; 2,300€ to 4,000€/month ; ≥4,000€/month).

Aircraft noise exposure assessment

Exposure to aircraft noise was estimated with a 1-dB(A) resolution at the place of residence of the participants with noise maps produced by the French Civil Aviation Authority with the INM (Integrated Noise Model) [START_REF] He | Integrated Noise Model (INM) Version 7.0 User's Guide[END_REF]. Four noise indicators were derived and used in the statistical analyses: Lden, LAeq24h, LAeq6h-22h, and Lnight.

Statistical analysis

Logistic and linear regression models were estimated in order to study the relationship between aircraft noise exposure and psychological distress. They were adjusted for the confounding factors previously defined. Subgroups analyses were also performed separately for women and men.

RESULTS

About 13% of the participants felt extremely or very sad or depressed during the past 12 months: 18% of women and 7% of men. According to the GHQ-12, 22% of the participants had psychological disorders: 25% of women and 17% of men. No significant association was shown between aircraft noise exposure and psychological distress, evaluated with the single question about the existence of depression's symptoms or with the GHQ-12, and whatever the noise indicator. When analyses were carried out separately for women and men, this association remained non-significant.

DISCUSSION

In France, the prevalence of depression's symptoms varies between 5% and 13% according to different studies [START_REF] Alonso | Prevalence of mental disorders in Europe: results from the European Study of the Epidemiology of Mental Disorders (ESEMeD) project[END_REF], [START_REF]rale%C2%A0-images-et-r%C3%A9alit%C3%A9s-smpg[END_REF], [START_REF] Sapinho | Mesure de l'épisode dépressif majeur en population générale: apports et limites des outils[END_REF]. The estimation highly depends on the scale used to evaluate these symptoms. Indeed, a lot of different validated tools are available for the evaluation of depression's symptoms. They differ from each other by the completion time, the collection mode, the association with other measurements, etc. No prevalence has yet been published for the French population using the GHQ-12.

The present study did not found any significant association between aircraft noise exposure and psychological distress. This finding confirms the results of some studies [START_REF] Reijneveld | The impact of the Amsterdam aircraft disaster on reported annoyance by aircraft noise and on psychiatric disorders[END_REF]- [START_REF] Sygna | Road traffic noise, sleep and mental health[END_REF], [START_REF] Miyakawa | Relationship between subjective health and disturbances of daily life due to aircraft noise exposure-Questionnaire study conducted around Narita International Airport[END_REF], even if others found significant associations [START_REF] Miyakita | POPULATION-BASED QUESTIONNAIRE SURVEY ON HEALTH EFFECTS OF AIRCRAFT NOISE ON RESIDENTS LIVING[END_REF]- [START_REF] Van Den Berg | The Relation between Self-Reported Worry and Annoyance from Air and Road Traffic[END_REF].

Among the studies investigating the effect of aircraft noise on psychological disorders, different tools have also been used to evaluate these disorders. Three of them used the GHQ [START_REF] Reijneveld | The impact of the Amsterdam aircraft disaster on reported annoyance by aircraft noise and on psychiatric disorders[END_REF], [START_REF] Van Kamp | Environmental noise and mental health: evidence from the Schiphol monitoring program[END_REF], [START_REF] Miyakawa | Relationship between subjective health and disturbances of daily life due to aircraft noise exposure-Questionnaire study conducted around Narita International Airport[END_REF]: they did not find any significant association. The absence of an association observed in this study could be explained by an effect similar to the "healthy worker effect": people with psychological disorders could choose not to live in the vicinity of airports.

However, further studies are necessary in order to conclude.

Table 1 :

 1 OR of psychological distress per aircraft noise levels

		Single question on the
		existence of		GHQ-12
		depression's symptoms
		OR	95%CI	OR	95%CI
	Lden
	<50dB(A)		ref		ref
	50-54dB(A)	0.88	(0.54-1.44)	0.84	(0.56-1.25)
	55-59dB(A)	0.95	(0.59-1.53)	0.82	(0.55-1.23)
	60dB(A) and +	0.83	(0.50-1.36)	0.88	(0.59-1.31)
	LAeq 24h
	<45dB(A)		ref		ref
	45-49dB(A)	0.84	(0.50-1.42)	0.77	(0.50-1.17)
	50-54dB(A)	0.87	(0.51-1.49)	0.74	(0.48-1.14)
	55dB(A) and +	0.87	(0.53-1.42)	0.79	(0.53-1.19)
	LAeq 6h-22h
	<45dB(A)		ref		ref
	45-49dB(A)	1.01	(0.55-1.84)	0.74	(0.46-1.20)
	50-54dB(A)	0.94	(0.50-1.76)	0.79	(0.48-1.29)
	55 dB(A) and +	0.95	(0.54-1.69)	0.72	(0.46-1.13)
	Lnight
	<40dB(A)		ref		ref
	40-44dB(A)	0.66	(0.39-1.12)	0.65	(0.43-1.00)
	45-49dB(A)	0.69	(0.41-1.17)	0.77	(0.50-1.18)
	50dB(A) and +	0.90	(0.57-1.41)	0.76	(0.52-1.11)


Cecilia Gustavsson

F.-J Hambsch

Stephan Oberstedt

Stephan Pomp

Kaj Jansson

Marc Olivier Frégeau

Ali Al-Adili

Alf Göök

Franz-Josef Hambsch

The new double energy-velocity spectrometer VERDI

The new double energy-velocity spectrometer VERDI

Introduction

In the fission process fragments with a very large distribution in mass, kinetic energy and excitation energy are created. Good knowledge of the yields for each isotope produced is important for applications like nuclear reactors and radioactive waste transmutation. With the advent of accelerator driven system, fission yield dependence to excitation energy of the fissioning nuclei will become especially important [START_REF]Fission Product Yield Data for the Transmutation of Minor Actinide Nuclear Waste[END_REF].

How the mass is shared between the nascent fragments is subject to many theoretical models, but none has yet managed to produce a predictive description of good accuracy [START_REF] Regnier | [END_REF][3][4][5] and very few have even started to consider the influence of excitation energy [6]. The distribution of excitation energy between the fragments [7,8] is also a very disputed question [9][10][11][12]. In order to guide theory, it is important to provide theorists and evaluators with experimental data containing as many correlations as possible between the different observables, such as fragments mass, charge, kinetic energy and emitted particle multiplicities. Neutron multiplicities are directly related to how the excitation energy is shared at scission. VERDI provides the fission fragments' masses, kinetic energies and neutron multiplicities using the double energy and double velocity (2E-2v) method. In its purest form this method is valid for excitation energies up to the compound nucleus's neutron emission threshold. The main assumption of the 2E-2v method is that the prompt neutron emission is isotropic. It means that the average velocity is a e-mail: stephan.oberstedt@ec.europa.eu conserved and calculations of both pre-and post-neutron masses can be made through

M pre 1 = v 2 v 1 + v 2 M Cf (1) M post 1 = 2E 1 v 2 1 , (2)
where M Cf would be substituted by the mass of the appropriate compound nucleus if not spontaneous fission of 252 Cf, as in this case, is studied.

VERDI setup and upgrade

The energy measurement is achieved by 32 silicon detectors located at about 50 cm from the target. They are arranged in two arrays, each housing 16 detectors on their respective side of the target. These detectors measure the kinetic energy and the arrival (stop) time of the incoming fission products. Emission (start) time of the fragments is determined by micro channel plate (MCP) detectors. The electrons sputtered from the surface of the actinide sample are deflected onto the MCPs by electrostatic mirrors. A complete description of the setup using one MCP and the same silicon arrays as in this work was made in Ref. [13]. Significant difficulties were observed arising from the use of a single MCP, since electrons are mainly detected from one side of the sample. α-particles emitted away from the MCP will mainly lead to electron emission in that same direction and would therefore often not be detected by the single MCP. Nonetheless, it is necessary to be able to detect α-particles in both directions for calibration purposes. In order to correct for the limitations inherent to the use of a single MCP detector, a second MCP has been installed.

After this upgrade we concluded that the efficiency of coincident events (both fission products detected in opposing silicon detectors) compared to non-coincident events had improved from ∼32% to ∼44%. The latter efficiency is close to what is expected after effects of particle losses in acceleration grids and non co-linearity has been taken into account. In addition, the efficiencies on both sides are now equal, whereas a few per cent difference was seen before.

Calibration

The energy calibration mainly follows the procedure of Schmitt et al. [14] but, unlike earlier [13], it also includes information from the α-peak. The latter is required to improve the calibration for alpha and ternary particles which could be studied with VERDI in the future.

A correction for the fragment's energy loss in the backing is also made. This correction is applied for both sides of the target since the californium atoms have diffused into the 250 nm nickel backing. Since the degree of diffusion is not well known, this introduces the diffusion depth as another parameter affecting the calibration.

We are developing an iterative procedure to improve upon the Schmitt calibration since it was found that the energy spectrum of Schmitt is higher than other, newer data [15], and that it also in our analysis led to a too high value of the average pre-neutron total kinetic energy, TKE pre .

Plasma delay time

The time-of-flight calibration was corrected for plasma delay time (PDT) according to the same procedure as outlined in Eqs. ((2)-( 6)) in Ref. [13]. Using known neutron multiplicities the plasma-delay-time (PDT) correction was computed event-by-event as the difference between the measured velocity and the one derived from the measured energy. Due to the mass dependence of the energy calibration and the neutron multiplicity, the procedure was run iteratively until the obtained masses converged.

In Fig. 1 it is shown how the PDT corrected velocity scales linearly with the measured velocity. The corrected velocity can therefore be calculated by a linear function once the appropriate linear relationship has been found. Only the linear function mapping measured velocity to the calculated one is used in the final analysis.

Improved energy calibration

The procedure is iterative and makes subsequent corrections to the energy calibration. It scales the velocities to match the literature value of TKE pre = 184.1 MeV [START_REF] Wagemans | The Nuclear fission process[END_REF]. Since the pre-neutron masses were already found to be in agreement with previous data, both fragment velocities must be scaled by the same constant in order to preserve them Eq. [START_REF]Fission Product Yield Data for the Transmutation of Minor Actinide Nuclear Waste[END_REF]. By again utilising known neutron multiplicities together with the corrected velocities, the calculated pre-and post-neutron quantities reproduce the literature data. Since we get both M post and E post on an event-by-event basis we can use them to deduce new values of the four Schmitt parameters [14]. With improved energy calibration the PDT correction changes and TKE pre approaches the literature value without any scaling. Only the new values of the Schmitt parameters are used in the final analysis. Although this is ongoing work and the method still needs to be refined, some promising results have been obtained in the cases where the recalibration method works well.

Results

We compare our results to the data from Göök et al. [15] taken using a ionisation chamber, since that allow us to compare all the correlated quantities to a single experiment. Our improved energy calibration does not work equally well for all detector pairs, so we show results from a detector pair where it happens to work well. The energy and mass distributions have been symmetrised as VERDI at the moment has an instrumental asymmetry between the different sides of about 10% when it comes to detection efficiency of light and heavy fragments, respectively.

Our pre-neutron energy distribution (Fig. 2) is in excellent agreement with the data from Ref. [15], and so is the heavy fragment peak (low energy) of the post-neutron energies (Fig. 3). However, some discrepancies are present in the post-neutron light fragment peak (high energy). Both pre-neutron (Fig. 4) and post-neutron (Fig. 5) masses are in good agreement with Ref. [15], and showing a slightly higher peak-to-valley ratio indicating higher resolution.

The general good agreement validate our PDT correction as well as recalibration procedure. Since both pre-and post-neutron masses are available we can easily calculate the average neutron multiplicity ν(A). In Fig. 6 ν as a function of M pre is shown. We can see discrepancies mainly around the symmetry region where our data indicate a negative neutron multiplicity which is unphysical. No tweaking, e.g., by using a known average neutron multiplicity for 252 Cf has been made, and still a rather good agreement with the data in Ref. [15] is achieved except for the lowest (A < 100) and highest (A > 155) masses. The total ν was 3.62 which is within 4% of the recommended value of 3.759(48) for 252 Cf [START_REF] Chadwick | [END_REF].

The TKE as a function of M pre also agrees well with Ref. [15] (Fig. 7) except for masses far from symmetry, and the average TKE before neutron emission, TKE pre = 184.8 MeV, is not too far off from the 184.1(13) MeV recommended value [START_REF] Wagemans | The Nuclear fission process[END_REF] also used by Göök et al. [15] for calibration. The excellent timing properties of the MCP detectors allow for a higher mass resolution as one clearly sees in the symmetry region. Similarly VERDI measures a slightly higher TKE post (Fig. 8) with, TKE post = 182.1 MeV compared to 181.4(13) MeV from Ref. [15].

Conclusion and outlook

VERDI has now two complete arms with fully independent capabilities, thanks to the second electron mirror and MCP detector. A new calibration procedure is developed which, for the detector pairs where it works well, have successfully produced pre-and post-neutron mass and energy spectra agreeing well with other data. The neutron multiplicity shows some problems around the symmetry region which could be due to a limited M post resolution but also due to the californium sample used. The deposited californium is assumed to have diffused into the nickel backing and the analysis is affected by how far we assume that the diffusion process has gone. The same sample has been used in a 2E measurement with an ionisation chamber, and problems with the angular distribution were encountered for the backing side (A. Al-Adili, private communication). In future measurements a better sample would be preferred.

Although not shown in Fig. 6, since the data has been symmetrised, the two different arms of VERDI gives very similar and consistent neutron multiplicity indicating that we are handling the asymmetry due to the diffusion in the sample quite well.

In order to measure neutron induced rather that spontaneous fission the mid section of VERDI needs to be refurbished to accommodate a neutron beam. A more flexible placement of the mirror arrangement would certainly be convenient in order to ensure isochronous arrival of the electrons on the MCP [18], something that is not fully supported today due to geometrical constraints. Even when our method is fully developed, the calibration to 252 Cf would be necessary to perform whenever the experimental conditions change. With a new mid section a measurement of the thermal fission of 235 U would provide a final validation to our method and calibration procedures. After that more exotic nuclei could be explored and high quality correlated data could be supplied to the community.

Figure 1 .Figure 2 .

 12 Figure 1. The PDT correction to the velocity scales linearly with the velocity. The line represents a linear fit to the data.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Post-neutron emission fission fragment kinetic energy measured by VERDI compared to data from Ref. [15].

Figure 6 .Figure 7 .

 67 Figure 6. Neutron multiplicity measured by VERDI compared to data from Ref. [15].

Figure 8 .

 8 Figure 8. Post-neutron TKE measured by VERDI and compared to data from Ref. [15].

c The Authors, published by EDP Sciences. This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).

We sincerely thank W. Geerts, T. Gamboni and M. Vidali for their technical support. K.J. thanks the CHANDA project for its financial support and M.O.F. thanks the EC for awarding him a fellowship at the JRC-GEEL.


Clémence Baudin
email: clemence.baudin@ifsttar.fr

Marie Lefevre

Bernard Laumon

Anne Sophie Evrard

Anne Sophie

Evrard Self-Reported Health

Self-reported health and aircraft noise exposure: the results of the DEBATS study in France

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

12th ICBEN Congress on Noise as a Public Health Problem

Self-reported health and aircraft noise exposure: the results of the DEBATS study in France

Clémence BAUDIN 1 , Marie LEFÈVRE 1 , Bernard LAUMON 2 , Anne-Sophie EVRARD 1

INTRODUCTION

Many studies have evidenced adverse effects of exposure to aircraft noise on health, such as annoyance, cardiovascular diseases, sleep disorders or altered cognitive performance [START_REF] Babisch | Annoyance due to aircraft noise has increased over the years-results of the HYENA study[END_REF]- [START_REF] Stansfeld | Aircraft and road traffic noise and children's cognition and health: a cross-national study[END_REF]. Some studies have investigated mental disorders, low birth weights, obesity and diabetes [START_REF] Pyko | Exposure to traffic noise and markers of obesity[END_REF]- [START_REF] Miyakawa | Relationship between subjective health and disturbances of daily life due to aircraft noise exposure-Questionnaire study conducted around Narita International Airport[END_REF], but these effects remain discussed because of divergent results.

Studies dealing with the effects of aircraft noise exposure on self-reported health are very rare: only one study has been identified in the PubMed database. The authors found a worse general health reported by people living near the Schiphol airport in Amsterdam [START_REF] Franssen | Aircraft noise around a large international airport and its impact on general health and medication use[END_REF]. They have also highlighted a lack of studies on this subject, with similar measures of noise exposure and of perceived health by airports' residents to allow comparisons, and thus draw conclusions.

DEBATS (Discussion on the health effects of aircraft noise) is the first research program to investigate the health effects of aircraft noise exposure in France.

One specific objective of the study was to evaluate the effects of aircraft noise exposure on self-reported health in people living in the vicinity of French airports.

METHODS

Participants

1,244 individuals older than 18 and living in the vicinity of three French airports (Paris-Charles de Gaulle, Lyon-Saint-Exupéry and Toulouse-Blagnac) were randomly selected to participate in the longitudinal study of DEBATS. The study population was stratified on aircraft noise contours. These contours are based on the day-evening-night equivalent level (Lden) defined in four categories, <50, 50-54, 55-59 and ≥60 dB(A). All participants responded to a questionnaire administered by an interviewer at their place of residence. The questionnaire collected in particular socio-demographic information, the personal medical history of the participants and lifestyle factors such as smoking and alcohol consumption.

Self-reported health

Self-reported health was evaluated using the question: "In general, would you say that your health is excellent, good, fair, or poor?". It was then categorized into two classes for the statistical analyses: excellent or good versus fair or poor.

Confounders

Several factors which could affect self-reported health were obtained from the questionnaire: age (six categories: 18-34; 35-44; 45-54; 55-64; 65-75; >75 years old), , country of birth (France-born/foreign-born), gender, occupational activity (yes/no), alcohol consumption (no/light/moderate/heavy), smoking (non-smoker/ex-smoker/occasional smoker/daily smoker), number of stressful life events (0/1/≥2), and income (<2,300€/month ; 2,300€ to 4,000€/month ; ≥4,000€/month).

Aircraft noise exposure assessment

Exposure to aircraft noise was estimated with a 1-dB(A) resolution at the place of residence of the participants with noise maps produced by the French Civil Aviation Authority with the INM (Integrated Noise Model) [START_REF] He | Integrated Noise Model (INM) Version 7.0 User's Guide[END_REF]. Four noise indicators were derived and used in the statistical analyses: Lden, LAeq24h, LAeq6h-22h, and Lnight.

Statistical analysis

Logistic regression models were estimated in order to study the relationship between aircraft noise exposure and self-reported health. They were adjusted for the confounding factors previously defined. Subgroups analyses were also performed separately for women and men in order to assess a potential specific effect on each gender. Linearity of the relation between the dependent variable and aircraft noise exposure was tested using generalized additive models including a smooth cubic function with linear and quadratic terms for aircraft noise exposure.

RESULTS

About 15% of the participants considered that their general health was fair or poor (16% of women, and 15% of men in the study population).

No significant association was shown between aircraft noise exposure and self-reported health when both gender were considered, whatever the noise indicator. Subgroup analyses found a rise in OR of self-reported health with increasing aircraft noise exposure: OR=1.71 -95%CI = 1.05-2.80 per 10 dBA increase in noise levels for LAeq 24h, OR=1.64 -95%CI=1.06-2.55 for LAeq 6h-22h, and OR=1.55 -95%CI=1.02-2.35 for Lnight. Generalized Additive Models including a smooth cubic spline function suggested approximately a linear relationship between aircraft noise exposure and self-reported health.

DISCUSSION

The present study suggests that the more men were exposed to aircraft noise, the more they were likely to report a fair or poor general health. In 2012, almost 9% of the French population reported a fair or poor perceived health (8% in men, and 9% in women) [START_REF]L'état de santé de la population en France -RAPPORT[END_REF] 1 . This prevalence was slightly higher, 15%, in the DEBATS study.

In the study carried out among Schiphol airport's residents in Amsterdam [START_REF] Franssen | Aircraft noise around a large international airport and its impact on general health and medication use[END_REF], the prevalence of bad perceived health was 20%.

The present study suggests that aircraft noise exposure is significantly associated with a bad or a poor self-reported health in men only. It confirms the findings of Franssen et al. [START_REF] Franssen | Aircraft noise around a large international airport and its impact on general health and medication use[END_REF] who found a significant association between self-reported health and aircraft noise exposure for the whole study population with an OR=1.23 (95%CI=1.04-1.46) for a 10 dB(A) increase in Lden.

Subgroup analyses were not published separately for men and women.

In addition to the question "How is your health in general: very good, good, moderate, sometimes good and sometimes bad, bad?", Franssen et al. used a validated questionnaire in Dutch which evaluates stress symptoms with a list of different health complaints [START_REF] Dirken | Het meten van stress in industriële arbeidssituaties[END_REF].

Because of very few differences in the results based on the single question or on the validated questionnaire, the authors concluded that the method used to evaluate self-reported health had almost no impact on the findings.

The associations observed in the present study could be due to residual confounding or to declaration bias which cannot be excluded. Further studies are necessary in order to better understand this association.

http://drees.social-sante.gouv.fr/IMG/pdf/rappeds v11 16032015.pdf