E Cataldo
email: ecataldo@id.uff.br

C Soize

A stochastic model of voice generation and the corresponding solution for the inverse problem using Artificial Neural Network for case with pathology in the vocal folds

Keywords: Voice production, jitter, stochastic biomechanical models, voice pathologies

A novel stochastic model to produce voiced sounds is proposed and, mainly, the corresponding identification of some model parameters using an Artificial Neural Network (ANN). The procedure described in this paper is about an intermediate step, which has as final objective to identify pathologies in the vocal folds through the voice of patients, that is, through a non-invasive method. The proposed model presented here uses the source-filter Fant theory and three main novelties are presented: a new mathematical model to produce voice obtained from the unification of two other deterministic one mass-spring-damper models obtained from the literature; a stochastic model that can generate and control the level of jitter resulting even in hoarse voice signals and/or with pathological characteristics but using a simpler model than those usually discussed in the literature; and the most important novelty, the identification of parameters of the proposed model, from experimental voice signals, using an ANN, particularly in a pathological case. The proposed neural network-based identification method requires a construction of a database from which an ANN can be trained to learn the nonlinear relationship between the parameters of the stochastic model and some relevant quantities of interest. The corresponding inverse stochastic problem is then solved in two cases: for one utterance corresponding to a normal voice and for another utterance corresponding to a pathological case corresponding to a nodulus in the vocal folds, helping to validate the model.

Introduction

Voice has a fundamental importance in the transmission of knowledge, feelings and emotions. It has also an important role in the culture of a people, such as singing and acting and also it is the main work tool for many agents, such as singers, speakers, teachers, and others. The phonation occurs when a column of air, expelled from the lungs, passes through the vocal folds. Then, pulses of air are generated forming a (quasi-)periodic acoustic pressure signal called the glottal signal, which is further filtered and amplified by the vocal tract and finally, radiated by the mouth generating the voice. The glottal signal is not exactly periodic due to small random deviations in relation to a mean value of the glottal time interval called jitter. This phenomenon has some practical applications as to help in the identification of pathologies from the vocal folds, identification of voice aging, voice recognition, speaker recognition, and other [START_REF] Wilcox | Age and changes in vocal jitter[END_REF]Li et al., 2010;[START_REF] Mendonza | Classification of Vocal Aging Using Parameters Extracted From the Glottal Signal[END_REF][START_REF] Fraile | Physical simulation of laryngeal disorders using a multiple-mass vocal fold model[END_REF]. In general, values of relative jitter between 0.1% and 1.04% indicate normal voice, that is, a voice that is not symptomatic of a pathology [START_REF] Wong | Observation of perturbations in a lumped-element model of the vocal folds with application to some pathological cases[END_REF][START_REF] Manfredi | A multipurpose user-friendly tool for voice analysis: Application to pathological adult voices[END_REF]. Some stochastic models of jitter have already been proposed taking into account only mathematical expressions of the glottal signal without considering a mechanical model for the vocal folds (Schoentgen et al., 1997(Schoentgen et al., , 2001;;[START_REF] Vasilakis | Spectral jitter modeling and estimation[END_REF]. Other authors have recently described a mechanical model for the vocal folds considering the generation of jitter (Cataldo et al., 2012;Cataldo andSoize, 2016, 2018). However these models consider the coupling between the vocal folds and the vocal tract causing a relative model complexity that induces a significant computational cost. In addition, for addressing the problem related to the statistical inverse identification of stochastic models the proposed methods require solving a statistical inverse problem classically formulated as a stochastic optimization problem, which may be computationally expensive, because it is necessary to run many times the computational model related to the direct problem.

It is important to say that the work presented here is a continuation and an extension of the paper previously published in Journal of Biomedical Signal Processing and Control [START_REF] Cataldo | Jitter generation in voice signals produced by a two-mass stochastic mechanical model[END_REF] but with a new methodoly and also new results.

In this work, a simplified model is proposed for the generation of jitter, considering the unification of two deterministic models proposed by [START_REF] Qureshi | A one-mass physical model of the vocal folds with seesaw-like oscillations[END_REF] and also by [START_REF] Titze | Parametrization of the glottal area, glottal flow, and vocal fold contact area[END_REF][START_REF] Titze | The physics of small-amplitude oscillation of the vocal folds[END_REF] with posterior modifications [START_REF] Lucero | A theoretical study of the hysteresis phenomenon at vocal fold oscillation onset-offset[END_REF]Lucero et al., 2001), disregarding a coupling equation between the vocal tract and the vocal folds, and considering the stiffness associated to the vocal folds as a stochastic process following the ideas proposed by Cataldo andSoize (2016,2018). With simplified stochastic model it is possible to obtain very good intelligible synthesis of voiced sounds, including jitter, characterizing normal voices but also hoarse voices or voices indicating pathologies due to the high level of jitter. To validate the model, a stochastic inverse problem is solved through an Artificial Neural Network. The statistical inverse problem under consideration consists in finding the values of the parameters of the stochastic model to produce voice for some given observed quantities of interest of the voice signals. An initial database is first generated from forward numerical simulations of the computational model. A processed database can then be deduced by conditioning the initial database in order to improve the performance of the trained ANN and therefore the efficiency of the neural network-based identification method. A multilayer ANN can then be designed to learn the nonlinear relationship between the parameters (network outputs) and the quantities of interest (network inputs). As a consequence, the proposed ANN-based identification strategy is computationally cheap, easy to implement, and use.

With this model experimentally validated, through Artificial Neural Networks, it is possible to generate the random phenomenon that is present in all voice signals. By numerical simulation, a big dataset can be generated for different voice signals, with different levels of jitter, and for different kinds of pathologies. Such a big dataset can be used for training an Artificial Neural Network. Finally, it should be noted that simple models have been used for the vocal folds, even nowadays, to better understand their movement as, for example, in the recently published paper by [START_REF] Lucero | Phonation threshold pressure at large asymmetries of the vocal folds[END_REF].

In this moment and in this work, it is very important to say that the discussion is an intermediate step to reach a final objective that is to help the pre-diagnosis of pathologies related to the vocal tract and the vocal folds in a noninvasive method. The idea is to find a set of parameters from the model, which can generate voices with pathological characteristics, even if only a specific group of pathologies is considered. Then, given a voice signal, the inverse problem is solved and if the parameters obtained belong to the range of the initial group of parameters, it will indicate that a possible pathology is present to the speaker of that voice signal.

In this paper, two experimental cases are considered, one of them corresponding to a normal voice and another from an experimental signal with pathological characteristics, the case of a nodulus in the vocal folds.

A proposed unified deterministic model for the vocal folds

The complete model presented here is based on the source-filter Fant theory (Fant, 1981), illustrated in Fig. 1. The voice signal generated, s(t), is given by the convolution of the glottal signal u g (t), the corresponding impulse response function h f (t) of the filter that models the vocal tract, and the radiation by the mouth for which the impulse response function is r(t). Then, Eq. ( 1) can be written as,

s(t) = (r * (h f * u g ))(t) , ( 1
)
where * is the convolution product. In the frequency domain, Eq. ( 2) can be written as,

s(ω) = r(ω) h f (ω) u g (ω) , (2)
where . means the Fourier Transform. This equation is well discussed in the literature about voice synthesis [START_REF] Rabiner | Theory and Applications of Digital Speech Processing[END_REF][START_REF] Prasad | Back end Tools for Speech Synthesis in Speech Processing[END_REF]. With this formulation, there is no coupling between the vocal folds and the vocal tract, simplifying the model. In this paper, u g (t) is constructed using the proposed model, h f (t) is one coming from the literature and detailed later, and r(t) is a first order high-pass FIR (finite impulse response) filter, such as suggested in [START_REF] Rabiner | Theory and Applications of Digital Speech Processing[END_REF]. Before discussing the stochastic model, the also original deterministic model is constructed based on two other models from the literature, which generate the glottal signal. The final idea is to join the best characteristics of each model to construct the proposed unified deterministic model. The sketch considered is the one proposed by [START_REF] Qureshi | A one-mass physical model of the vocal folds with seesaw-like oscillations[END_REF] and reproduced in Fig. 2. The movement of each vocal fold is given by a rotary motion about its support point p 0 . A single mass-spring-damper system is attached to the glottis at its entrance. The model is assumed to be symmetric about its central line so that the left side of the vocal fold is the same as its right one. The glottal entry displacement is represented by x 1 (t). The characteristic of a nonlinear damping is introduced in the model, according to [START_REF] Laje | Continuous model for vocal fold oscillations to study the effect of feedback[END_REF], and the equation of the motion for a single vocal fold is described by [START_REF] Lucero | A lumped mucosal wave model of the vocal folds revisited: recent extensions and oscillation hysteresis[END_REF]:

M ẍ1 (t) + B (1 + η x 2 1 (t)) ẋ1 (t) + K x 1 (t) = p g (t; x 1 (t), ẋ1 (t)) . (3)
where M, B, and K are the mass, the damping, and the stiffness, per area unit, and η is the nonlinear coefficient damping. The mean glottal pressure p g is obtained using the Bernoulli law, and after some simplifications given by Eq. ( 4) [START_REF] Titze | The physics of small-amplitude oscillation of the vocal folds[END_REF][START_REF] Lucero | A theoretical study of the hysteresis phenomenon at vocal fold oscillation onset-offset[END_REF]:

p g (t; x 1 (t), ẋ1 (t)) = 2 τ p sub (t) ẋ1 (t) k t (x 01 + x 1 (t)) , (4)
where x 01 is the initial glottal displacement, pre-phonation, in relation to the mean point of oscillation, in the entry, τ is a short glottal delay time, k t is the coefficient of transglottal pressure, and p sub (t) is the subglottal pressure. According to [START_REF] Qureshi | A one-mass physical model of the vocal folds with seesaw-like oscillations[END_REF], the glottal flow u g (t) is given by Eq. ( 5):

u g (t) = 2p sub (t) k t ρ area(t) , ( 5
)
where ρ is the air density and area(t) is the area at the exit of the glottis and given by

area(t) = 2 (x 02 + x 2 (t)) , x 2 (t) > -x 02 2 x 02 , otherwise (6)
in which x 02 is the initial glottal displacement, pre-phonation, in relation to the mean point of oscillation, in the exit, is the width of the vocal folds, and

x 2 (t) = -(x 1 (t) -x 0 ) , (7)
where the mean position of the oscillation is x 0 , which is specified by the horizontal position of the fulcrum point p 0 .

In this text, without loosing generality, the same values are considered for x 0 , x 01 and x 02 . After generating u g (t), the convolution with the filter (vocal tract and the mouth) should be performed to produce the sound. The next step is the construction of the corresponding stochastic model to generate the glottal pulse, which will be then a stochastic process, based upon the unified deterministic model proposed here.

Proposed stochastic model

The objective is to vary instantaneously the frequency of the voice signal to generate jitter. As the mass is fixed, the stiffness K is considered as a stochastic process {K(t), t ∈ R} following some ideas proposed by [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF] with the corresponding changes, because the model created here does not consider the coupling between the vocal tract and the vocal folds. The consideration is that jitter is generated due to the variation of the stiffness of the vocal folds, giving possible additional information to the biomechanics of the vocal folds. In [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF]) a more complex model was considered, taking into account a coupling equation between the vocal folds and the vocal tract. The idea is to reproduce similar results using the same consideration of the stochastic model associated with the stiffness and to show that it is possible to generate jitter and very good intelligible voice sounds. The details about the construction of the stochastic model associated with the stiffness will not be explained here, because they can be obtained in [START_REF] Cataldo | Stochastic mechanical model of vocal folds for producing jitter and for identifying pathologies through real voices[END_REF]. Only the most important characteristics will be listed.

Let E be the mathematical expectation. The stochastic process {K(t), t ∈ R} is constructed according to the properties defined as follows.

(i) For all t in R, it is assumed that 0 < K 0 ≤ K(t) almost surely, where K 0 is a positive constant independent of t. (ii) {K(t), t ∈ R} is a non-Gaussian stationary stochastic process such that E{K(t) 2 } < +∞ for all t (second-order stochastic process), for which its mean function (that is independent of t) is written as E{K(t)} = K > k 0 > 0, and which is assumed to be mean-square continuous in order to guaranty the existence of a power spectral measure.

A representation of the stochastic process K(t) is chosen as,

K(t) = K 0 + (K -K 0 )(z + Z(t)) 2 . ( 8
)
in which z is a positive real number and where {Z(t), t ∈ R} is a Gaussian second-order real-valued stochastic process, centered, mean-square continuous, stationary and ergodic, physically realizable, such that Z = h * N ∞ , where N ∞ is the centered Gaussian white noise (generalized stochastic process) whose power spectral density function is written, for all real ω, as S N (ω) = 1/(2π), and h = F -1 {H} is the inverse Fourier transform of the complex-valued frequency response function ω → H(ω) chosen, for all real ω, as H(ω) = a/(b + iω), in which a and b are positive constants, so that stochastic process Z can be constructed as the asymptotic stationary random solution of the following linear Itô stochastic differential equation,

dZ(t) = -b Z(t) dt + a dW(t) , t > 0 , (9)
with the initial condition Z(0) = 0, where {W(t), t ≥ 0} is the real-valued normalized Wiener stochastic process. The power spectral density function of stochastic process Z is then given by Eq. ( 10):

S Z (ω) =| H(ω) | 2 = a 2 2π(ω 2 + b 2 ) . ( 10
)
The level of jitter will mainly be controlled by a (and b), which should satisfy the condition a 2 < 4b [START_REF] Cataldo | Jitter generation in voice signals produced by a two-mass stochastic mechanical model[END_REF]. It could be intoduced a filter with a high order and then with more parameters to be identified. However, it has been chosen a filter of first order with two parameters. The simulations, discussed later, show that this filter chosen is enough to generate jitter. Following Eq. ( 3), the displacement x 1 (t) of the vocal folds becomes the stochastic process X 1 (t). The dynamics of the vocal folds is then given by the following nonlinear stochastic differential equation,

M Ẍ1 (t) + B (1 + η X 2 1 (t)) Ẋ1 (t) + K(t) X 1 (t) = p g (t; X 1 (t), Ẋ1 (t)) . (11)
All the other equations related to the unified deterministic model should be rewritten substituting x 1 (t) and x 2 (t) by X 1 (t) and X 2 (t), and also p g by P g . The realizations of the stochastic process {K(t), t ∈ R} are obtained using Eqs. ( 8) and ( 9). Consequently, the realizations of the stochastic process X 1 (t) are computed by solving Eq. ( 11) and the realizations of the stochastic process X 2 (t) are deduced. Finally, realizations of the stochastic process U g (t), corresponding to the glottal signal, are computed. The voice signal is generated through the convolution given by Eq. ( 1). The voice signals generated will indicate the presence of jitter and to quantify its level some measures can be used.

Jitter measures

Let T g be the random variable associated with the duration of the glottal cycle, which is defined as the duration between two successive times, the first one corresponding to the instant the vocal folds (glottis) open and the second one the instant when they close completely. To calculate T g from U g (t), it is used an algorithm based on an implementation of the RAPT pitch tracker [START_REF] Talkin | A robust algorithm for pitch tracking (rapt)[END_REF].

For each glottal cycle j, and each realization θ j of the random variable T g , the duration denoted by T g (θ j ) can be associated with. Considering that the set {T g (θ j ), j = 1, . . . , N} constitutes N realizations of random variable T g (corresponding to all the glottal cycles of the voice signal), jitter can be measured by the following equations [START_REF] Mongia | Estimation and Statistical Analysis of Human Voice Parameters to Investigate the Influence of Psychological Stress and to Determine the Vocal Tract Transfer Function of an Individual[END_REF].

(i) The absolute jitter, denoted by Jit abs , is defined by

Jit abs = 1 N -1 N-1 j=1 |T g (θ j+1 ) -T g (θ j )| . (12)
(ii) The relative jitter, denoted by Jit rel , is defined by

Jit rel = 1 N -1 N-1 j=1 |T g (θ j ) -T g (θ j+1 )| 1 N -1 N-1 j=1 T g (θ j ) . ( 13
)
In general, values of Jit rel from 0.1% up to 1.04% are associated to non pathological characteristics for the voice.

Simulation

In this section, simulations are performed using the proposed stochastic model. The variation of the parameter a (see Eq. ( 10)) is considered. The subglottal pressure pattern, more realistic [START_REF] Lucero | A lumped mucosal wave model of the vocal folds revisited: recent extensions and oscillation hysteresis[END_REF] is given by Figure 3 shows the graph of the function t → p sub (t). In general, the frequency response function h f (ω) corre- sponding to the digital filter (the vocal tract) is characterized by having only poles associated with the resonance frequencies (the formants) related to voiced sounds. Function h f considered here takes into account the speech-forming frequencies and the effects of soft-wall, friction, thermal conduction losses and radiation on lips, which will be given by the bandwidths associated with the formant frequencies. Table 2 shows the values (in Hertz) for the first four resonance frequencies of the vocal tract, the formants, for five vowels, and also losses in the vocal tract represented by the bandwidths (Bw) of the formant frequencies based on the findings of important to perform a convergence analysis, mainly for the solution of the Itô stochastic differential equation used to generate realizations of K(t). Let K = E{K(t)} be the mean value and K 2 = E{K(t) 2 } be the second-order moment of K(t), which are classically estimated using the asymptotically stationary and ergodic solution. After performing tests, it was concluded that the convergence for K(t) is warranty from 3 × 10 5 time steps.

p sub (t) =                0, i f 0 ≤ t ≤ t 0 or T f -t 0 ≤ t ≤ T f p m sin( π(t-τ) 0.280 ), i f t 0 ≤ t ≤ t 0 + 0.140 p m , i f t 0 + 0.140 ≤ t ≤ T f -(t 0 + 0.140) p m sin( -π(t+t 0 -T f ) 0.280 ), i f T f -0.140 -t 0 ≤ t ≤ T f -t 0 , (14)

Constructing the database

The database is constructed using the stochastic model. Values of a, b and K are varied and for each given triplet (a, b, K) in the model, a voice signal is generated, and Jit Rel , Jit Abs , and the value of the fundamental frequency are extracted form it. All the other parameters of the model are fixed.

The fundamental frequency varies from low values up to higher values to cover a reasonable range of frequencies. The main parameter related to the variation of the fundamental frequency is K, because the mass is fixed. The parameter a is the main responsible for the level of jitter. Then, it is important to vary this parameter from a = 0 (the deterministic case) up to a value corresponding to a pathology not much severe or the case corresponding to a hoarse voice. The third parameter, b, is directly related to the convergence of the solution of the Itô Differential Equation, although it also has some contribution to the level of jitter. For this parameter b, a convenient value is chosen so that the solution converges and variations around this value are considered. It is important to say that for each triplet (a, b, K), different voice signals (realizations) can be generated because they are related to stochastic process {K(t)} t . The strategy adopted is then to generate a group of signals related to each triplet (a, b, K). For each voice signal of this group the parameters Jit Rel , Jit Abs , and the mean value of the fundamental frequency, denoted by F 0 are obtained. Then, mean values of these parameters are evaluated for all the group and these mean values are associated with each triplet (a, b, K).

In this work, the variations of the parameters that are considered are as follows: a varies from 0 up to 240, with a step of 30; b varies from 800 000 up to 1 200 000, with a step of 50 000; and K varies from 2 000 000 up to 6 000 000, with a step of 200 000 . With these values, the measures obtained from the voice signal have the following variations: the mean value of the random variable associated with the fundamental frequency varies from 103.34 Hz up to 180.89 Hz, the absolute jitter varies from 0 up to 3.4389 × 10 -4 , and the relative jitter varies from 0 up to 4.1370 %.

Methodology and configuration of the ANN used

A sensitivity analysis of the network target data with respect to the network input data has been performed for the databases. Figure 6 shows a classical estimate of the matrix of the correlation coefficients between each of the parameters a, b and K and each of the observed Jit rel , Jit abs and F 0 . generated by the computational model. It can be observed that parameter a is highly correlated with the values of jitter and the mean value of the stiffness is highly correlated with the values of the mean of the fundamental frequency. The parameter b has very small correlation with the values of jitter and also with the mean value of the fundamental frequency. All these observations are expected.

To solve the inverse problem, a multilayer feed forward static neural network is used with three neurons in the input layer, corresponding to the values of a, b, and K, and three neurons in the output layer, corresponding to the values of absolute jitter (Jit abs ), relative jitter (Jit rel ), and the mean value of the fundamental frequency (F 0 ). Several ANN configurations have been tested, in relation to the number of hidden layers and the number of the neurons in each one of them. Sigmoid hidden neurons are used in the hidden layers, while linear output neurons are used in the output layer. The input vectors and target vectors have been randomly divided into three distinct sets for training, validation, and testing, with 70% of the complete dataset assigned to the training set, 15% to the validation set, and 15% to the test set. The values of the input and target vectors were preprocessed and mapped into the normalized range [-1, 1] before presenting to the neural network for training. After training, the network output vectors are then transformed back to the original scale (units) of the network target vectors. The performances of the trained neural networks have been also evaluated by computing the normalized mean-square error between the network outputs and the corresponding targets. The values of the relative jitter, the absolute jitter, and the mean value of the fundamental frequency are used as inputs, and the values of parameters a, b, and K are used as outputs. Several hidden layers have been considered, by starting with one hidden layer. However, the differences for the mean-square error (MSE) have not been so significant when more hidden layers or more neurons in each hidden layer have been used. Then, the ANN considered has only one hidden layer containing four neurons. The performance is illustrated in Fig. perfinverseproblem: As expected, the best identification corresponds to parameters a and K. For parameter b, the identification is not very good due to the fact that its correlation with the inputs are not so high. However, its value does not need to be very precise because this parameter is directly related to the convergence of the solution for the Itô equation and some differences in its identified values will not modify significantly the corresponding voice signals.

5.3. First example: solving the inverse stochastic problem with a real voice signal without pathological characteristics Now, an experimental signal is considered. It is an utterance of a vowel /e/ produced by a man with the following extracted parameters: absolute jitter = 2.5131 × 10 -5 , relative jitter = 0.3091%, and the mean value of the fundamental frequency = 123.1086 Hz. The experimental voice used in this case is called expnormalused and can be accessed in the link:

www.dropbox.com/sh/eu7xvwd84dvadbv/AACqHVQ6rdkmap6pG8Uqcy17a?dl=0 . Solving the inverse problem, through the trained ANN, the values obtained are: a = 119.37, b = 1.01 × 10 6 , and K = 2.83 × 10 6 N/m. Using these values in the computational model yields: absolute jitter= 2.0750 × 10 -5 , relative

Conclusions

A neural network-based identification method has been presented for solving the statistical inverse problem related to the identification of parameters of a stochastic model for producing voice. A database has been generated using the simulations of the proposed stochastic model. A sensitivity analysis of the target data with respect to the input data has been performed. Different configurations of multilayer feedforward neural networks have been tested and the one that has given the minimum value of the mean square error (MSE) has been selected. Two experimental examples have been used to identify the parameters of the model through the defined ANN. One of them corresponding to a healthy voice and other one corresponding to a voice from a patient with nodulus in the vocal folds.

The results obtained showed that it is possible to use an artificial neural network to solve the problem with a small computational cost instead of using the stochastic optimization problem that we had previously been used. In this first essay to solve the inverse problem, it was considered a simpler mathematical model, which does not consider the coupling between the vocal tract and the vocal folds. Only three input parameters of the model have been considered for three measures from the voice signals.

It should be highlighted that only two voice signals were used to validate the model and the only parameter considered was jitter. The initial idea was to show that the consideration about the modelling for the stochastic process associated to the stiffness of the vocal folds was coherent and could generate jitter. In addition, parameters were identified, showing that it is possible to generate stochastic process corresponding to experimental voices. Not only normal voices but also with pathological characteristics.

It is important to note that the main objective of this research is to identify pathologies through the voice and then it is important to have a low computational cost because the application should be used in real time.

Acknowledgments

This work was supported by CNPq.

Conflict of interest statement

The authors do not disclose any financial and personal relationships with other people or organizations that could inappropriately influence their work.

Figure 1 :

 1 Figure 1: Sketch of the source-filter theory.

Figure 2 :

 2 Figure 2: Sketch of the Qureshi model.

Figure 3 :

 3 Figure 3: Graph of the subglottal-pressure function t → p sub (t).

Figure 4

 4 Figure 4 shows three cases of glottal signals considering different levels of jitter, taking into account different values of a, illustrating the generation of the jitter phenomenon with the model proposed.Table 3 shows different

Figure 6 :

 6 Figure 6: Matrix of correlation coefficients between input and output parameters.

Figure 7 :

 7 Figure 7: Performance for the ANN considered in the inverse problem.

Table 1 :

 1 in which p m is the maximum glottal pressure and t ∈ [0, t f ]. In this work, the values considered are p m = 800 Pa, t 0 = 0.01 s, and t f = 1 s. All the other parameters are fixed and their values are summarized in Tab. 1.

	Parameters	Values
	M	4.76 kg/m 2
	B	100 (N × s/m)/m 2
	K	4.2 × 10 6 (N/m)/m 2
	K 0	2 × 10 5 (N/m)/m 2
	η	5 × 10 3 /m 2
	p m	800 Pa
	k t	1.1
	τ	0.001 s
	x 0	10 -3 m
	x 01	10 -3 m
	x 02	10 -3 m
	ρ	1.15 kg/m 3
		0.014 m
	b	10 6

List of parameters and their values (M , B, K and K 0 are given per area unit).

Table 2 :

 2 Formants, in Hertz, of the vocal tract, the digital filter, for the case of vowels production and the corresponding bandwidths.

	F1	F2	F3	F4
	/a/ 860 1513 2489 3600
	/e/ 423 1899 2017 3546
	/i/ 283 2113 2800 3566
	/o/ 504 905 2624 3439
	/u/ 352 809 2394 3450
	Bw 20	25	200	50

[START_REF] Titze | Benchmarks for time-domain simulation of sound propagation in softwalled airways: steady configurations[END_REF]

. For the simulations, it is

a Jit abs Jit rel 160 2.27 × 10 -5 0.34% 200 3.25 × 10 -5 0.50% 500 1.72 × 10 -4 2.70% 700 1.88 × 10 -4 3.00% 1000 3.78 × 10 -4 6.13% Let F 0 = 1/T g be the random variable called the fundamental frequency. Considering the values of the parameter a from Tab. 3, it is possible to construct the probability density function of the random variable F 0 , estimated by using the Gaussian kernel estimation method from the nonparametric statistics [START_REF] Bowman | Applied smoothing techniques for data analysis: The kernel approach with S-Plus illustrations[END_REF], and shown in Fig. 5, for all cases of the Tab. 3. signals corresponding to the other synthesized vowels have similar levels of jitter.

Solution of the inverse problem using an artificial neural network (ANN)

In order to validate the model proposed, parameters a, b, and K are identified using experimental voice signals. The main role of K is to fit the fundamental frequency while the parameters a and b fit the values of jitter. The objective is to use an ANN to learn the nonlinear relationship between the values of the hyperparameters a, b, and K and the measures extracted from the voice signal: the jitter measures and the mean value of the fundamental frequency.


R K Abbe

N Chutani

S Passilly

C Galliou

V Gorecki

Giordano

A High-Performance Frequency Stability Compact CPT Clock Based on a Cs-Ne Microcell

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

Abstract-This paper reports on a compact table-top Cs clock based on coherent population trapping (CPT) with advanced frequency stability performance. The heart of the clock is a single buffer gas Cs-Ne microfabricated cell. Using a distributed feedback (DFB) laser resonant with the Cs D 1 line, the contrast of the CPT signal is found to be maximized around 80°C, a value for which the temperature dependence of the Cs clock frequency is canceled. Advanced techniques are implemented to actively stabilize the clock operation on a zero-light-shift point. The clock frequency stability is measured to be 3.8 × 10 -11 at 1 s and well below 10 -11 until 50 000 s. These results demonstrate the possibility to develop highperformance chip-scale atomic clocks using vapor cells containing a single buffer gas.

I. I

T  combination of coherent population trapping (CPT) physics [START_REF] Vanier | Atomic clocks based on coherent population trapping: A review[END_REF] and micro electromechanical systems (MEMS) allows the development of chip-scale atomic clocks (CSAC) [START_REF] Knappe | A microfabricated atomic clock[END_REF] exhibiting a volume of a few cubic centimeters, a power consumption of about 100 mW, and a fractional frequency stability better than 10 -11 at 1 day or even 1 week integration time [START_REF] Lutwak | The chip-scale atomic clock: Prototype evaluation[END_REF]. Such frequency references, outperforming crystal quartz oscillators over long time scales, are intended to provide excellent base timing for numerous battery-operated portable applications, such as telecommunication networks synchronization, navigation, military systems, or even power distribution and underwater sensing [START_REF]SA.45s CSAC Chip-scale atomic clock[END_REF].

The heart of a miniature atomic clock consists of a hermetically sealed microfabricated cell containing alkali vapor and a buffer gas atmosphere. On the one hand, the buffer gas is required to reduce the CPT resonance broadening resulting from depolarizing wall collisions and to eliminate Doppler broadening through Lamb-Dicke narrowing [START_REF] Dicke | The effect of collisions upon the Doppler width of spectral lines[END_REF]. On the other hand, the buffer gas causes a temperature-dependent buffer gas pressure collisional frequency shift of the clock transition. Traditionally, the temperature coefficient of the frequency shift is reduced by mixing two buffer gases, each of which shifts the clock frequency in opposite ways [START_REF] Arditi | Pressure, light and temperature shifts in optical detection of 0-0 hyperfine resonance of alkali metals[END_REF]- [START_REF] Kozlova | Temperature and pressure shifts of the Cs clock transition in the presence of buffer gases: Ne, N 2 , Ar[END_REF]. This technique is widely used in commercially available optically pumped devices. However, such a mixture is difficult to control accurately in MEMS cells because of their small size and the high temperatures involved in the anodic bonding process required for their fabrication.

Most microfabricated cells are basically realized by etching of a silicon wafer and subsequent anodic bonding of a thin glass wafer to each side [START_REF] Kitching | Miniature vapor-cell atomic-frequency references[END_REF]. Our filling method differs from traditional ones [START_REF] Liew | Microfabricated alkali atom vapor cells[END_REF], [START_REF] Knappe | Atomic vapor cells for chip-scale atomic clocks with improved long-term frequency stability[END_REF] in that the Cs vapor is generated only after complete sealing of the cell by local heating of a side-cavity Cs metallic dispenser with a highpower infrared laser source [START_REF] Douahi | Vapour microcell for chip scale atomic frequency standard[END_REF]. This method is intended to reduce interactions with highly-reactive Cs chemistry and to achieve a strong bonding quality [START_REF] Hasegawa | Microfabrication of cesium vapor cells with buffer gas for MEMS atomic clocks[END_REF]. Moreover, we recently reported the first direct observation of a quadratic dependence on temperature of the Cs clock transition around a so-called inversion temperature T inv of about 80°C in presence of a single Ne buffer gas [START_REF] Miletic | Quadratic dependence on temperature of Cs 0-0 hyperfine resonance frequency in single Ne buffer gas microfabricated vapor cell[END_REF]. The inversion temperature, found to be similar in a pulsed Cs CPT clock using centimeter-scale cells [START_REF] Kozlova | Temperature dependence cancellation of the Cs clock frequency in the presence of Ne buffer gas[END_REF], was measured to be independent of the Ne buffer gas pressure. These results emerged as significantly interesting for the development of simple-configuration high-performance miniature clocks and justified deep characterizations of CPT resonances in Cs-Ne microcells [START_REF] Boudot | Coherent population trapping resonances in Cs-Ne vapor microcells for miniature clocks applications[END_REF].

The short-term frequency stability of a passive atomic clock is improved by both reducing the detected resonance linewidth and increasing its signal-to-noise ratio. The best reported short-term frequency stabilities of millimeterscale Cs-cell-based CPT clocks are measured to be 1 to 3 × 10 -10 at 1 s when operated with a laser resonant on the D 2 line [START_REF] Knappe | A microfabricated atomic clock[END_REF] and are improved to 2 to 4 × 10 -11 [START_REF] Lutwak | The chip-scale atomic clock: Prototype evaluation[END_REF], [START_REF] Knappe | Atomic vapor cells for chip-scale atomic clocks with improved long-term frequency stability[END_REF] by using the D 1 line. For integration times typically higher than 1000 s, the clock frequency is subject to drift. This is mainly due to collisional frequency shifts and light shift phenomena. The first issue is solved by stabilizing the cell temperature close to the inversion temperature, whereas the total light shift can be canceled by fine adjustment of the RF modulation index [START_REF] Zhu | Theoretical and experimental study of light-shift in a CPT-based Rb vapor cell frequency standard[END_REF]- [START_REF] Mikhailov | Performance of a prototype atomic clock based on lin||lin coherent population trapping resonances in Rb atomic vapor[END_REF]. An additional servo loop can then be implemented to lock the clock to the point of minimum light shift [START_REF] Shah | Continuous light-shift correction in modulated CPT clocks[END_REF]- [START_REF] Mcguyer | Simple method of light shift suppression in optical pumping systems[END_REF].

In this paper, we present a compact CPT clock prototype based on a Cs microfabricated cell filled with Ne, combined with an externally-modulated DFB laser resonant with the Cs D 1 line. Implementation of advanced techniques leads to frequency stability performance comparable to those of the best microcell-based CPT clocks. These results prove for the first time that our Cs-Ne cell The laser source is a commercially-available GaAs semiconductor DFB laser diode emitting at 894.6 nm on the Cs D 1 line. A Faraday optical isolator is placed at the output of the DFB laser to avoid optical feedback. Optical sidebands required to perform CPT spectroscopy are generated by driving an external pigtailed phase electrooptic modulator (EOM) at 9.192 GHz using a microwave frequency synthesizer inspired from [START_REF] Boudot | Simple design low noise NLTL-based frequency synthesizers for a CPT Cs clock[END_REF]. The fiber modulator ensemble is inserted into a box with insulating foam and is actively temperature-stabilized to the millikelvin level to reduce output laser intensity fluctuations. No phase noise degradation was observed between the microwave frequency synthesizer output and the 9.192-GHz optically carried signal. The 2-mm-diameter collimated laser beam is circularly polarized using a quarter-wave plate. The laser power is 17 µW before entering the atomic resonator. The atomic resonator consists of a 2-mmdiameter and 1.4-mm-long cylindrical microfabricated Cs vapor cell filled with Ne. The cell temperature T cell is stabilized close to 80°C, which corresponds to a Cs density of 4.05 × 10 12 atoms/cm 3 . Temperature fluctuations of the microcell are below the millikelvin level. A static magnetic field of a few microteslas parallel to the laser beam propagation direction is applied to split the hyperfine ground-state Zeeman transitions. The ensemble is surrounded by a two-layer mu-metal magnetic shield. The CPT resonance is monitored by detecting the laser power transmitted through the cell using a low-noise photodiode. The laser is frequency-stabilized near the center of the homogeneously broadened absorption line by modulating the DFB laser current at 60 kHz and demodulating it with the lock-in amplifier LA1. The atomic clock closedloop operation is achieved by a conventional synchronous modulation-demodulation technique (~1 kHz frequency) applied to the local oscillator frequency through the lockin amplifier LA2.

III. E I  R

A typical CPT resonance observed in the Cs-Ne cell for T cell = 80°C is reported in the inset of Fig. 1. The resonance lineshape is well fitted by a Lorentzian function. The full-width at half-maximum (FWHM) is measured to be 1.6 kHz in correct agreement with theoretical values. From [START_REF] Boudot | Coherent population trapping resonances in Cs-Ne vapor microcells for miniature clocks applications[END_REF], the hyperfine coherence relaxation rate γ 2 is calculated to be 4391 rad/s and the optical resonance homogeneous broadening Γ*/2π is 862 MHz. At 80°C, contributions to the CPT linewidth from wall-collisions and spin-exchange collisions are about equal and clearly dominant over buffer gas collisions. The contrast is measured to be about 1.25%. Fig. 2 shows both the CPT resonance contrast and the clock frequency shift versus the cell temperature. The CPT contrast is maximized in the 74°C to 81°C temperature range. A bell-shaped curve, already observed in [START_REF] Boudot | Coherent population trapping resonances in Cs-Ne vapor microcells for miniature clocks applications[END_REF], [START_REF] Knappe | Temperature dependence of coherent population trapping resonances[END_REF], [START_REF] Castagna | Investigations on continuous and pulsed interrogation for a CPT atomic clock[END_REF], is obtained. It is explained by an increase of the alkali atomic density for temperatures up to 80°C. For higher temperatures, the vapor medium becomes optically thick and causes a strong reduction of the CPT signal. In [START_REF] Boudot | Coherent population trapping resonances in Cs-Ne vapor microcells for miniature clocks applications[END_REF], with a vertical-cavity surface-emitting laser (VCSEL) emitting at 852 nm, the CPT signal maximum with tem- perature was measured to be in the 55°C to 60°C range for quite similar Cs-Ne microcells (pressure and dimensions). This can be qualitatively explained by intrinsic differences between D 1 and D 2 excitation lines hyperfine structures. The Cs D 2 line excited state is composed of 4 hyperfine levels with a frequency separation (~150 to 250 MHz) much lower than buffer-gas-induced optical broadening in microfabricated cells. Both hyperfine levels of the Cs D 1 line are well-resolved and separated by about 1.16 GHz. Also, some excited states on the Cs D 2 line couple to only one of the ground states and act as loss-channels for the CPT interaction. Consequently, optical absorption is stronger for the Cs D 2 line and the alkali vapor becomes optically thick for lower temperatures than in the D 1 line case. Moreover, we confirmed that the Cs clock frequency temperature-dependence is canceled close to 80°C in the presence of Ne. Using coefficients published in [START_REF] Kozlova | Temperature and pressure shifts of the Cs clock transition in the presence of buffer gases: Ne, N 2 , Ar[END_REF], a fit of experimental points gives a Ne pressure of 79 Torr. This cell allows the overlap of a zero-sensitivity of the clock frequency to temperature variations and a maximum of CPT contrast close to 80°C. This will lead to mutual improvements of the clock long-term and short-term frequency stability.

Combined with a greatly reduced sensitivity of the clock frequency to temperature variations, the light shift is continuously minimized using a method inspired by [START_REF] Shah | Continuous light-shift correction in modulated CPT clocks[END_REF]. For this purpose, the laser power is slowly modulated at 15 Hz (12% depth) using a voltage-controlled liquid crystal display (LCD) attenuator. This laser power modulation is converted across the cell through the light shift effect into a slight clock frequency modulation that can be easily detected at the error signal output of the lockin amplifier LA2 used to stabilize the LO frequency. This modulation signal is found to vanish close to a so-called RF power zero-light-point driving the EOM. An additional servo loop based on the lock-in amplifier LA3 is even-tually added to lock the RF power point where no light shift is present. For illustration, Fig. 3 plots both the light shift slope and the CPT resonance contrast versus the RF modulation power.

The light shift slope is measured to be negative for P RF < -2 dBm, positive for P RF > -2 dBm and can be nulled by finely adjusting the RF power to P RF ~ -2 dBm. Moreover, as shown in Fig. 3, this RF power coincides with the one maximizing the CPT resonance contrast. It corresponds well to the RF power maximizing the optical power contained in the two resonant light fields. Such behavior was also reported in [START_REF] Shah | Continuous light-shift correction in modulated CPT clocks[END_REF]. Recent results obtained with a D 2 -line VCSEL and a Cs-Ne microcell (P = 60.7 Torr) are reported for comparison [START_REF] Boudot | Coherent population trapping resonances in Cs-Ne vapor microcells for miniature clocks applications[END_REF]. In the present case, the short-term frequency stability is measured to be 3.8 × 10 -11 at 1 s. It is in agreement with a measured atomic Q-factor of 5.7 × 10 6 and a clock resonance signal-to-noise ratio of 4500 in a 1 Hz bandwidth. We measured that our laser power modulation for the light shift correction degrades the optimal short-term stability by a factor of about 1.5. The Allan deviation curve decreases well with a τ -1/2 slope until 1000 s. After 1000 s, the Allan deviation increases again with a τ 1/2 slope corresponding to a random-walk frequency noise. The frequency stability is found to be 4 × 10 -12 at 40 000 s. These results obtained with optimal laboratory-scale electronics are comparable to those presented in [START_REF] Lutwak | The chip-scale atomic clock: Prototype evaluation[END_REF], [START_REF] Knappe | Long-term stability of NIST chip-scale atomic clock physics packages[END_REF].

IV. C

We developed a compact prototype CPT clock based on a Cs-Ne microfabricated vapor cell with an externally modulated DFB diode laser. Advanced techniques such as the use of the D 1 line laser resonance, low-noise microwave local oscillator, clock frequency temperature-dependence cancellation and light-shift reduction techniques were implemented. The clock frequency stability is close to 1 × 10 -12 at 2000 s and 4 × 10 -12 at 50 000 s. Despite the fact that these results do not report a fully-integrated clock, they prove for the first time the potential of the Cs-Ne microcell solution to develop miniature atomic clocks with high long-term frequency stability and confirm that our original Cs microcell technology is compatible with such developments.

A

  Fig.1presents the Cs CPT clock experimental setup. The laser source is a commercially-available GaAs semiconductor DFB laser diode emitting at 894.6 nm on the Cs D 1 line. A Faraday optical isolator is placed at the output of the DFB laser to avoid optical feedback. Optical sidebands required to perform CPT spectroscopy are generated by driving an external pigtailed phase electrooptic modulator (EOM) at 9.192 GHz using a microwave frequency synthesizer inspired from[START_REF] Boudot | Simple design low noise NLTL-based frequency synthesizers for a CPT Cs clock[END_REF]. The fiber modulator ensemble is inserted into a box with insulating foam and is actively temperature-stabilized to the millikelvin level to reduce output laser intensity fluctuations. No phase noise degradation was observed between the microwave frequency synthesizer output and the 9.192-GHz optically carried signal. The 2-mm-diameter collimated laser beam is circularly polarized using a quarter-wave plate. The laser power is 17 µW before entering the atomic resonator. The atomic resonator consists of a 2-mmdiameter and 1.4-mm-long cylindrical microfabricated Cs vapor cell filled with Ne. The cell temperature T cell is stabilized close to 80°C, which corresponds to a Cs density of 4.05 × 10 12 atoms/cm 3 . Temperature fluctuations of the microcell are below the millikelvin level. A static magnetic field of a few microteslas parallel to the laser beam propagation direction is applied to split the hyperfine ground-state Zeeman transitions. The ensemble is surrounded by a two-layer mu-metal magnetic shield. The CPT resonance is monitored by detecting the laser power

Fig. 1 .

 1 Fig. 1. Schematic of the Cs-Ne microcell-based coherent population trapping (CPT) clock. MS: 10-MHz quartz-oscillator-based microwave synthesizer; EOM: electro-optic modulator; LCD: voltage-controlled liquid crystal plate attenuator; P: polarizer; HWP: half-wave plate; QWP: quarter-wave plate; LA1: lock-in-amplifier-based servo for distributed feedback (DFB) laser frequency stabilization through modulation at 60 kHz of the laser bias current; LA2: lock-in-amplifier-based servo for locking the local oscillator frequency to the CPT resonance transition, LA3: lock-in-amplifier-based servo for light shift correction (LSC).

Fig. 2 .

 2 Fig. 2. Coherent population trapping (CPT) resonance contrast (filled squares) and clock frequency shift (triangles) versus the cell temperature.The solid line is a second-order polynomial function fit to the frequency data. For a given temperature, each frequency shift is found from an extrapolation to zero intensity of the light shift curve. The CPT contrast is found to be optimized close to the clock frequency inversion temperature.

Fig. 3 .

 3 Fig. 3. Coherent population trapping (CPT) resonance contrast (filled squares) and light shift slope (triangles) versus the RF modulation power. The indicated RF power is before amplification of 24 dB. Close to P RF = -2 dBm, the CPT resonance contrast is maximized and the light shift can be cancelled. The cell temperature is 80°C. For each RF power, the frequency shift is measured for 5 to 6 different laser intensity values and is found to be well approximated by a linear function. Each triangle point corresponds to the slope of this function.

Fig. 4 .

 4 Fig. 4. Allan deviation of the Cs coherent population trapping (CPT) clock based on a Cs-Ne microcell. (circles): vertical-cavity surface-emitting laser (VCSEL) D 2 line, no light shift correction, T cell = 60°C, Ne pressure = 60.7 Torr [16]; (squares): distributed feedback (DFB) D 1 line, light shift correction (LSC), T cell ≃ 80°C, Ne pressure = 79 Torr.

Fig. 4

 4 Fig.4reports the frequency stability (Allan deviation) of the CPT clock based on a Cs-Ne microcell. It is measured by comparison with a reference hydrogen maser. Recent results obtained with a D 2 -line VCSEL and a Cs-Ne microcell (P = 60.7 Torr) are reported for comparison[START_REF] Boudot | Coherent population trapping resonances in Cs-Ne vapor microcells for miniature clocks applications[END_REF]. In the present case, the short-term frequency stability is measured to be 3.8 × 10 -11 at 1 s. It is in agreement with a measured atomic Q-factor of 5.7 × 10 6 and a clock resonance signal-to-noise ratio of 4500 in a 1 Hz bandwidth. We measured that our laser power modulation for the light shift correction degrades the optimal short-term stability by a factor of about 1.5. The Allan deviation curve decreases well with a τ -1/2 slope until 1000 s. After 1000 s, the Allan deviation increases again with a τ 1/2 slope corresponding to a random-walk frequency noise. The frequency stability is found to be 4 × 10 -12 at 40 000 s. These results obtained with optimal laboratory-scale electronics are comparable to those presented in[START_REF] Lutwak | The chip-scale atomic clock: Prototype evaluation[END_REF],[START_REF] Knappe | Long-term stability of NIST chip-scale atomic clock physics packages[END_REF].

A

The authors greatly thank E. De Clercq (LNE-SYRTE, Paris, France) for fruitful discussions and careful reading of the article. R


Valentin Massin
email: valentinmassin@free.fr

MD Damien Lami

MD, PhD Matthieu Ollivier

PhD Martine Pithioux

MD, PhD Jean-Noël Argenson

Comparative biomechanical study of five systems for fixation of the coracoid transfer during the Latarjet procedure for treatment of anterior recurrent shoulder instability

Keywords: Latarjet procedure, coracoid bone-block, glenohumeral instability, glenohumeral stabilization, fixation system, graft healing

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Thank you for reviewing this work.

As you ask me, I include references you provided in the manuscript. I also made schema to explain fixation devices (Figure 2). The manuscript has been check by English native person to ensure grammatical reliability.

As you mentioned, there is already papers comparing different fixation devices, but there is no one comparing this self-compressive screws with "new" devices as endobutton and two diameters self-compressive screws and the gold standard malleolar screws. Moreover, there is no paper about testing both traction of the conjoint tendon and pull-out of the humeral head in the same test, as we performed using a traction machine with the scapula in the following position.

COMPARATIVE BIOMECHANICAL STUDY OF FIVE SYSTEMS FOR FIXATION OF THE CORACOID TRANSFER DURING THE LATARJET PROCEDURE FOR TREATMENT OF ANTERIOR RECCURENT SHOULDER INSTABILITY I -INTRODUCTION

Glenohumeral stabilization by coracoid bone-block was described in 1954 by Latarjet [START_REF] Latarjet | Treatment of recurrent dislocation of the shoulder[END_REF].

The procedure consists, by a delto-pectoral approach, in transposing the horizontal portion of the coracoid process to the lying position at the anterior-inferior edge of the glenoid, through an incision in the tendon of the subscapular.

Stabilization is obtained by a stop effect related to the increase of the inferior-internal rim of the glenoid, associated with a sling effect of the coraco-biceps tendon when placing the upper limb in anterior elevation. Patte [START_REF] Patte | Unstable painful shoulders (author's transl)][END_REF] and Walch [START_REF] Edwards | The latarjet procedure for recurrent anterior shoulder instability: Rationale and technique[END_REF] proposed improvements to the original technique, doing a capsuloplasty on the coracoacromial ligament. Coracoid bone grafting can restore the glenoid rim defect [START_REF] Paladini | Latarjet procedure: is the coracoid enough to restore the glenoid surface?[END_REF], and this triple locking technique remains the gold standard for the treatment of anterior glenohumeral instability [START_REF] Boileau | Arthroscopic Bankart-Bristow-Latarjet (2B3) Procedure: How to Do It and Tricks To Make it Easier and Safe[END_REF][START_REF] Carli | Latarjet procedure versus open capsuloplasty in traumatic anterior shoulder dislocation: long-term clinical and functional results[END_REF].

The results are effective with a recurrence in 1 to 7% of cases and a low number of complications [START_REF] Mizuno | Long-term results of the Latarjet procedure for anterior instability of the shoulder[END_REF].

Bone-block fixation is essential because poor bone contact is responsible for nonunion [START_REF] Schell | The course of bone healing is influenced by the initial shear fixation stability[END_REF] [ Fig 1]. In the Latarjet-Patte-Walch procedure, the coracoid process is fixed by two 4.5mm partial threaded cancellous AO malleolar screws.

Failure and re-intervention can occur in case of non-union, lysis of the bone-block or secondary displacement [START_REF] Willemot | Analysis of failures after the Bristow-Latarjet procedure for recurrent shoulder instability[END_REF][START_REF] Griesser | Complications and re-operations after Bristow-Latarjet shoulder stabilization: a systematic review[END_REF], and the length of the screws can cause neurological damage [START_REF] Lädermann | Injury of the suprascapular nerve during latarjet procedure: an anatomic study[END_REF].

Other fixing methods were later described to overcome these pitfalls: use of a single screw, optimization of the screw support using a washer or miniplate, use of screws with a smaller diameter, full threaded or cannulated screw as Saragaglia's mini-invasive technique [START_REF] Lateur | Results of the Latarjet coracoid bone block procedure performed by mini invasive approach[END_REF].

Recently, Lafosse [START_REF] Lafosse | The arthroscopic Latarjet procedure for the treatment of anterior shoulder instability[END_REF] described arthroscopic procedure using two screws, while Boileau [START_REF] Boileau | A guided surgical approach and novel fixation method for arthroscopic Latarjet[END_REF], described cortical button fixation. Arthroscopic procedure seems to be as reliable as open procedure regarding to the positioning of the graft [START_REF] Kordasiewicz | Comparative study of open and arthroscopic coracoid transfer for shoulder anterior instability (Latarjet)clinical results at short term follow-up[END_REF], and Kordasiewicz showed a very high graft healing rate using two cannulated screws with washers [START_REF] Kordasiewicz | Comparative study of open and arthroscopic coracoid transfer for shoulder anterior instability (Latarjet)computed tomography evaluation at a short term follow-up[END_REF].

Several studies have already compared the methods of fixing the block, whether on biomechanical models or on cadaveric models [START_REF] Alvi | Latarjet Fixation: A Cadaveric Biomechanical Study Evaluating Cortical and Cannulated Screw Fixation[END_REF][START_REF] Schmiddem | Monocortical fixation of the coracoid in the Latarjet procedure is significantly weaker than bicortical fixation[END_REF][START_REF] Shin | Biomechanical Analysis of Latarjet Screw Fixation: Comparison of Screw Types and Fixation Methods[END_REF]. The resistance of endobuttons has also been studied compared to malleolar screws [START_REF] Boileau | A guided surgical approach and novel fixation method for arthroscopic Latarjet[END_REF][START_REF] Gendre | Coracoid bone block fixation with cortical buttons: An alternative to screw fixation?[END_REF][START_REF] Provencher | Biomechanical Comparison of Screw Fixation Versus a Cortical Button and Self-tensioning Suture for the Latarjet Procedure[END_REF]. Previous works have shown that bicortical fastening systems are more resistant than the use of unicortical screws [START_REF] Schmiddem | Monocortical fixation of the coracoid in the Latarjet procedure is significantly weaker than bicortical fixation[END_REF][START_REF] Willemot | Influence of screw type and length on fixation of anterior glenoid bone grafts[END_REF].

The use of self-compressive screws such as Herbert seems to us be an interesting option to allow good contact between the bone-block and the glenoid and to promote boneunion [START_REF] Hart | Fixation strength of four headless compression screws[END_REF]. However, they have not been studied biomechanically in this application.

Given the diversity of fastening systems on the market, we wanted to compare their effectiveness under physiological conditions. The purpose of this study is to explore the strength of these systems, in order to choose the one that offers satisfactory strength, while allowing optimal contact between the bone surfaces, minimum drilling to avoid weakening the bone-block and offering solutions for precise placement of the block. Null hypothesis was that all these fixation systems are bio-equivalent on resistance to tensile strength.

The main judgement criterion was the maximum strength developed to achieve system failure.

The secondary judgement criterion was the stiffness of the assembly, calculated from the force/displacement curve of each test.

II -MATERIAL AND METHODS

Five fixation devices of coracoid bone-block were tested during a cadaveric biomechanical study : For each of the 5 fixing modes, three laboratory tests were carried out. The shoulders were randomized, with a matched allocation so that two joints from the same cadaver would not be tested with the same device.

Cadaver with a surgical or shoulder trauma history were excluded.

For each shoulder, a Latarjet procedure was performed, then the scapula was included in a resin in a testing position allowing the force vector to simulate both the humeral head pressure and coraco-biceps traction, as described in previous work [START_REF] Alvi | Latarjet Fixation: A Cadaveric Biomechanical Study Evaluating Cortical and Cannulated Screw Fixation[END_REF][START_REF] Provencher | Biomechanical Comparison of Screw Fixation Versus a Cortical Button and Self-tensioning Suture for the Latarjet Procedure[END_REF] [ Fig 3].

All screws were bicortical. The tightening was done in real conditions: with two fingers, using the dedicated screwdriver, until a tension considered sufficient by the operator was reached. Traction was increasingly performed at a rate of 10mm.min -1 with a traction machine equipped with a 1000N sensor, with an accuracy of +/-5*10 -3 N (E5566A, Instron, Norwood, USA). The data obtained were processed using BlueHill 3 software (ITW, Norwood, USA).

A failure of the assembly was noted in the event of bone-block fracture, fixing material failure, bone-block displacement by more than 3mm from its initial position [ Fig 4].

For each fixing method, mean strength and stiffness values of the 3 tests were compared to the mean of all other fixing modes using Mann-Whitney tests. Statistical tests were carried out using P-Value, graphical interface of R software (The R Foundation, Vienna, Austria).

III -RESULTS

A strength/displacement curve was obtained from each test [Fig. 5], and derivated function at the failure point was used to calculate stiffness.

Failure mode was different according to the type of fixing: single screw generally induced a displacement of the block around the screw then a tearing off, while fixing with large diameter screws (4.5mm) induces a block's fracture or a cut-out of the screw.

For all the tests performed, the mean failure threshold was 193N (SD 112N).

The strength of failure for each fixing method using 2 screws or 1 endobutton is equivalent, without any significative difference. At the opposite, testing using 1 malleolar screw and washer showed a significative weaker fixation strength, with a mean strength of 86N (SD 27,1N, p<0.01) [Table 1].

Mean stiffness of all systems was 14,6N.mm -1 (SD 4,77N.mm -1 ). Stiffness of the "endobutton group" was lower 11,6N.mm -1 (SD 1,44N.mm -1 ), in a non-significant way.

There was no difference on the secondary judgement criterion according to the others types of fixation [Table 1].

IV -DISCUSSION

Latarjet-Patte-Walch's procedure has become common because of its good clinical results [START_REF] Mizuno | Long-term results of the Latarjet procedure for anterior instability of the shoulder[END_REF]. Over successive developments and in particular since arthroscopic improvement, several fixing methods have been described and tested [START_REF] Boileau | A guided surgical approach and novel fixation method for arthroscopic Latarjet[END_REF][START_REF] Alvi | Latarjet Fixation: A Cadaveric Biomechanical Study Evaluating Cortical and Cannulated Screw Fixation[END_REF][START_REF] Provencher | Biomechanical Comparison of Screw Fixation Versus a Cortical Button and Self-tensioning Suture for the Latarjet Procedure[END_REF][START_REF] Garcia | Comparative Systematic Review of Fixation Methods of the Coracoid and Conjoined Tendon in the Anterior Glenoid to Treat Anterior Shoulder Instability[END_REF][START_REF] Weppe | A biomechanical evaluation of bicortical metal screw fixation versus absorbable interference screw fixation after coracoid transfer for anterior shoulder instability[END_REF].

In order to optimize the clinical result, the positioning of the block is essential [START_REF] Hovelius | The effect of capsular repair, bone block healing, and position on the results of the Bristow-Latarjet procedure (study III): long-term follow-up in 319 shoulders[END_REF].

However, complications during fixation are frequent: according to Willemot, graft nonunion is responsible of most revision cases [START_REF] Willemot | Analysis of failures after the Bristow-Latarjet procedure for recurrent shoulder instability[END_REF]. The parameters influencing bone consolidation are numerous [START_REF] Einhorn | The science of fracture healing[END_REF] but one of the most important is the contact between the two bone pieces. Claes showed less healing when the gap between the two bone pieces is greater than 2mm [START_REF] Claes | Influence of size and stability of the osteotomy gap on the success of fracture healing[END_REF], that's why a 3mm displacement was considered as a failure.

Several types of screws have been marketed to solve these issues, by offering k-wires sighting instruments and self-compressive screws.

The aim of this study was to test the biomechanical resistance of these systems and compare them to the gold standard of open-air and arthroscopic Latarjet.

This study has several limitations: because it is a cadaveric work, the constraints applied to the model may differ from reality. Nevertheless, we tried to reproduce both the coracobiceps' traction force and the humeral head's pressure force. Bone quality can be changing: most of anatomical subjects used come from elderly patients, and conservation care can alter bone quality. The shoulders were randomized in the study groups to avoid a bias related to very porotic bone, but Latarjet procedure is proper to young people: it is likely that bone quality supposed to be better, so failure thresholds in real conditions may be higher.

This work is original since it compares five different bone-block' fixation, using a standardized model. The gold standard is compared to commonly used and recently developed fasterners. The coraco-biceps' tensile force is taken into account, which is generally neglected in this type of study [START_REF] Alvi | Latarjet Fixation: A Cadaveric Biomechanical Study Evaluating Cortical and Cannulated Screw Fixation[END_REF][START_REF] Willemot | Influence of screw type and length on fixation of anterior glenoid bone grafts[END_REF], despite its stabilizing role [START_REF] Kephart | Biomechanical analysis of the modified Bristow procedure for anterior shoulder instability: is the bone block necessary?[END_REF].

The methods of attaching the coracoid bone-block, using two bicortical screws or an endobutton therefore seem equivalent. While failures way vary, means are comparable which shows a homogeneity of testing. Standard deviations are relatively large, due to a low number of samples per group and likely sampling fluctuations, which are common in cadaveric studies. However, there is quite a homogeneity in order of magnitude, which are comparable to those described in the literature.

On the other hand, fixing with a single screw shows a lower failure threshold. It seems insufficient, even with bicortical screwing and distributing stresses along a washer [START_REF] Weppe | A biomechanical evaluation of bicortical metal screw fixation versus absorbable interference screw fixation after coracoid transfer for anterior shoulder instability[END_REF].

The stiffness of the assemblies is also similar: these fixing methods suffer little distortion and failure mostly result as a tearing off of the bone/screw complex, or a bone-block's fracture on the screw holes.

The mean strength values of groups using endobutton or two screws are higher than the strength applied to a shoulder in daily life activities [START_REF] Westerhoff | In vivo measurement of shoulder joint loads during activities of daily living[END_REF]. Concerning the "screw + washer" group, the rupture may occur too early, especially since the coraco-biceps tension has not been measured in vivo in the literature and remains unknown to date.

The use of a single screw is intended to reduce the risk of a small block fracture.

Nevertheless, due to its reduced stability, the choice of this system seems to us to be abandoned. It may be more appropriate to turn to smaller diameter screws than to a single one.

The use of self-compressive Herbert type screws ensures a good biomechanical resistance. As these screws are cannulated, they also provide precise positioning through use of k-wires. Alvi also showed biomechanical equivalence between cannulated and solid screws [START_REF] Alvi | Latarjet Fixation: A Cadaveric Biomechanical Study Evaluating Cortical and Cannulated Screw Fixation[END_REF].

We have shown here a biomechanical equivalence in strength and stiffness between solid screws with a 4.5 diameter and cannulated screws with 3.5 diameter. This allow the drilling of holes with a smaller diameter, which favors holding in the cancellous bone [START_REF] Steeves | How pilot-hole size affects bonescrew pullout strength in human cadaveric cancellous bone[END_REF].

Finally, as Gender previously noted, endobuttons seems to fully meet the requirements for this procedure and make it possible to safely replace use of screws [START_REF] Gendre | Coracoid bone block fixation with cortical buttons: An alternative to screw fixation?[END_REF]. The stiffness is slightly less, in a non-significant way, than the other groups. This shows the relative elasticity of this kind of assembly, as opposed to solid metal screws that support few distortion, and even if single-point fixations have a tendency to rotate around this pivot, it does not seem to impact the strength of failure.

This work should be completed by clinical studies, as radiological failures such nonunions or bone-block lyses have not always led to recurrence of instability or surgical recovery [START_REF] Griesser | Complications and re-operations after Bristow-Latarjet shoulder stabilization: a systematic review[END_REF].

V -CONCLUSION

The methods of coracoid bone-block fixation tested in this study appear to be biomechanically comparable, except the use of a single screw that shows a lower failure threshold. Systems based on two bicortical screws or an endobutton accept higher stresses than those to which the joint is usually subjected. We can consider them to be reliable, as tensile tests are reproductible. Their stiffness is comparable, which implies that these devices may don't deform under the effect of the stresses they undergo.

The use of a single screw should be abandoned at the risk of insufficient fixation including secondary displacement or increased pseudarthrosis.

The use of a small diameter screws can reduce the risk of fracture, and the use of cannulated self-compressive screws can increase the contact with the glenoid edge to maximize healing, while allowing accurate placement of the bone-block using specific instruments. Herbert 3.5mm screws can solve all these problems. However, it is essential to clinically assess our biomechanical observations and further studies must be performed. Table1

VII -ANNEXES

Click here to access/download;Table ;Table1.pdf

Figure1

Figure2

2 4 .

 4 5mm malleolar solid screws (Synthes, West Chester, USA), 1 4.5mm malleolar solid screw with washer (Synthes, West Chester, USA), 1 endobutton (Implanet, Martillac, France), 1 4mm self-compressive cannulated screw + 1 3mm selfcompressive cannulated screw (Newclip Technics, Haute-Goulaine, France), 2 3.5mm self-compressive cannulated screws (Newclip Technics, Haute-Goulaine, France) [Fig 2].

Fig. 1 :Fig. 2 :Fig. 4 :Fig. 3 :Fig. 5 :

 12435 Fig. 1: Examples of Latarjet procedures failure

Table 1 :

 1 Tests results and statistical analysis

				Main judgement criterion (Strength, in N)
		N	Mean (SD)	Median (Q25-75)	Min	Max	P*
	All samples	15	193 (112)	170 [108; 242]	55.0	424	N/A
	Endobutton	3	184 (±76.8)	176 [144 -220]	112	265	0.73
	Single screw + washer	3	86.0 (±27.1)	98.0 [76.5 -102]	55.0	105	<0.01
	3mm screw + 4mm screw	3	288 (±151)	314 [220 -369]	125	424	0.18
	2* 3.5mm screws	3	148 (±39.3)	170 [136 -171]	103	172	0.63
	2* malleolar screws	3	257 (±129)	220 [185 -310]	150	400	0.29
			Secondary judgement criterion (Stiffness, in N.mm	-1 )
		N	Mean (SD)	Median (Q25-75)	Min	Max	P*
	All samples	15	14.6 (4.77)	16.2 [11.9; 17.8]	4.24	22.6	N/A
	Endobutton	3	11.6 (±1.44)	11.6 [10.9 -12.3]	10.1	13.0	0.18
	Single screw + washer	3	14.9 (±6.94)	12.8 [11.0 -17.7]	9.23	22.6	0.95
	3mm screw + 4mm screw	3	15.9 (±3.22)	17.4 [14.8 -17.7]	12.2	18.1	0.73
	2* 3.5mm screws	3	12.7 (±7.36)	16.4 [10.3 -17.0]	4.24	17.5	0.73
	2* malleolar screws	3	18.2 (±1.99)	18.3 [17.2 -19.2]	16.2	20.1	0,14

*compared to other samples mean (Mann-Whitney's test)

*compared to other samples mean (Mann-Whitney's test) Main

judgement criterion (Strength, in N) Secondary judgement criterion (Stiffness, in N.mm -1 )