Panagiotis Piteros
email: panagiotis.piteros@supelec.fr

Laetitia Thirion-Lefevre
email: laetitia.thirion@supelec.frregis.guinvarch@supelec.fr

Régis Guinvarc'

Marc Lambert
email: marc.lambert@lss.supelec.fr

Cognitive kriging metamodels for forest characterization and target detection

The idea of applying the cognitive radar principle to the radar observations of forests is presented. It implies an adaptive design of experiments (DOE) that will allow us to construct probabilistic surrogate models with a known level of uncertainty. These models reduce significantly the computational cost, which is mandatory when running many numerical simulations in various configurations.

I. INTRODUCTION

A. Context

Radar waveforms are usually designed without taking into account the effect of the environment. Cognitive radar is a new framework of radar systems introduced by S. Haykin [START_REF] Haykin | Cognitive radar: a way of the future[END_REF]. His idea is that the waveform emitted by the radar at time t + 1 will be dependent on the radar return at time t and of the analysis that has been done of this return. This principle can be useful for forest observations, where a further analysis shows that the emitted waveform s e (t) and the waveform s t (t) transmitted through the forest, can actually be quite different. As illustrated in Fig. 1, the environment, characterized by its transfer function H, can filter the waveform that finally reaches the target:

s t (t) = s e (t) • H (t) (1)
The signal received later by the radar will be altered a second time, in a typical monostatic configuration. Thus, to evaluate these alterations, the transfer function has to be estimated based on our knowledge of the environment. Two applications are considered: target detection and forest parameters retrieval.

In the first case, the goal is to compensate for the effect of the forest in order to maximize the intensity backscattered by the target and received by the sensor. For the second application, the objective is to use the capability to modify the radar waveform in order to probe a part of or the whole forest.

In this paper, we first recall the work we performed on a forest of trunks with a kriging metamodel for forest descriptive parameters retrieval. Then we turn to the new technique we propose to implement this work to the case of a forest of branches and trunks and also to support new functions. One of our objectives is to estimate the impact of uncertainties on forest descriptive parameters on the transfer function.

B. Previous results and limitations

We presented in [START_REF] Vasko | Metamodel-based adaptive use of a coherent polarimetric backscattering simulator for the characterization of forested areas at low frequencies[END_REF] a new method for forest characteristics inversion based on a functional surrogate model, used instead of COSMO [START_REF] Thirion | Capabilities of a forest coherent scattering model applied to radiometry, interferometry, and polarimetry at P-and L-band[END_REF] a scattering model dedicated to forest, but too complex to be used for inversion process. This surrogate model was developed to simulate the polarimetric backscattering coefficients as COSMO would do, but faster. However, it can only handle a few input parameters, because of its structure, leading to a strong simplification of the forest under study. In this particular case, only the trunks were considered for the simulation and their dimensions were derived from the age of the trees using allometric equations [START_REF] Saleh | A forest geometric description of a maritime pine forest suitable for discrete microwave models[END_REF]. The ground moisture was also taken into account and with the age of the trees, they represented the two input parameters we needed. Considering a set of polarimetric backscattering coefficients obtained for a given radar configuration and derived from the backscattered fields, we addressed the inverse problem of determining the forest age and the ground moisture by using the surrogate model. We produced error maps as in Fig. 2, that illustrate the uncertainty we obtained on the retrieved values of the age and moisture. The combination of the different polarimetric channels allowed us to obtained a very good estimate of these two parameters, despite the low values of the cross-polarization due to the forest description.

Our objective here is first to extend this method to a complete forest of trunks and branches of different sizes and then to take into account additional parameters as the vegetation moisture, that were assumed to be constant in the previous example. Nevertheless, it is not straightforward to achieve this goal, as the number of input parameters is actually strongly limited by the two-level kriging technique we previously implemented. So to overcome this difficulty, we decided to rather apply a scalar technique. This will allow us to consider a more detailed description of the forest, and also new functionalities as the sensitivity analysis and the propagation of uncertainties. This technique is detailed in the next section. Fig. 2. Example of an error map: the point to retrieve is indicated by a white star at (30 years, 0.55). The black area refers to all the retrieved values leading to an error less than 1dB on the backscattering coefficients. In this example, all the polarization channels have been considered. The retrieved age is estimated between 35 and 42 years, and the retrieved volumetric moisture ranges between 0.42 and 0.65.

II. SCALAR KRIGING-BASED METAMODEL

When dealing with expensive-to-evaluate functions, approximation methods based on probabilistic surrogate models have significant advantages. Mainly they require fewer function evaluations. The surrogate models, also called metamodels, can help us explain the undelying simulation model's behavior and predict the expected output f (x) for combinations of input values x that have not yet been simulated. In order to make up for the lack of knowledge on the I/O function, the system is treated as a "black-box"1 and our metamodels are based on Gaussian random Processes (GP). Apart from prediction, they can be used also for validation of the simulation model, Sensitivity Analysis (SA) and optimization [START_REF] Kleijnen | Design and Analysis of Simulation Experiments[END_REF]. In our work we are going to focus on the so-called kriging metamodels.

Kriging is a probabilistic method originated from Geostatistics designed for interpolating multidimensional functions that shows certain spatial correlation [START_REF] Kleijnen | Kriging metamodeling in simulation: A review[END_REF]. Its metamodels are fitted to data that are obtained for large experimental areas, so they can be regarded as global rather than local. They provide a cheap and accurate approximation of the objective function together with an estimation of the potential error.

Kriging's main idea is to model the I/O function by a GP in order to provide a probabilistic framework2 to account for the uncertainty stemming from the lack of information on the system [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]. The GP is fully characterized by its mean, which is assumed to be constant, and its covariance, with the latter reflecting the correlation among the observation points.

The kriging predictor is the so-called Best Linear Unbiased Predictor (BLUP) [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF] of the forward function in the vector space generated by the observations. Its performance depends on both the choice of the input values, where the observation are to be done, and the choice of the covariance. Taking into account the fact that the spatial correlation of the observation points and hence the covariance is unknown, in order to improve the predictor's performance an adaptive sampling procedure will be performed. Also a flexible covariance function will be chosen, so that we can "tune" it to best fit our data. In Fig. 3

A. Construction of the Metamodel and Initial Prediction

Kriging is an exact interpolation method, so it needs the results of several simulation runs of the model in order to predict the function under study. The adaptive procedure requires as a first step an initial prediction based on the results of a sample adequately spread in the input space. For the construction of this initial sample, a Latin Hypercube (LH) maximin design is chosen. The resulting set contains n vectors X n = {x i } n i=1 which are mapped to the vector of the n simulation results f Xn = (f (x 1 ), . . . , f (x n ))

T . The next step is to define an appropriate covariance function, which is assumed stationary for computational reasons, that will reflect the spatial correlation between the input samples. The Matérn covariance function is chosen, because it's flexible and it offers the possibility to adjust its regularity and the prediction's smoothness with just a single parameter [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]. As the covariance is assumed stationary, a distance function that will reflect the spatial correlation has to be introduced. This distance has to be anisotropic since the model inputs are of different kind and their domains can vary a lot. We define it as:

h ij ≡ h (x i , x j ) = d k=1 x i,k -x j,k ρ k 2 (2)
where x i,k , x j,k are the k-th components of the x i and x j vectors respectively and ρ k the scaling factor for the k-th component of the input vectors that reflects the sensitivity of the model to that component. The Matérn stationary covariance has now the form:

k (h ij ) = σ 2 0 Γ (ν) • 2 ν-1 • 2 √ νh ij ν K ν 2 √ νh ij (3)
where K ν is the modified Bessel function of the second kind, ν is the parameter that controls the regularity of the covariance function and σ 2 0 = k (0). Since the covariance is unknown, we have to estimate its parameters ρ k , σ 2 0 and ν so that they will better fit our model i.e. our observations. This is done by Restricted Maximum Likelihood Estimation (RMLE) [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF].

To proceed to the initial prediction of the I/O function at a point x / ∈ X n we need to compute the n×n covariance matrix of the sample space:

K = [k (x i , x j )] , i, j = 1, . . . , n,
and also the prediction point's covariance vector:

k (x) = (k (x, x 1 ) , . . . , k (x, x n ))
T After these we are ready to compute the so-called kriging coefficients (or weights) λ i (x) and the Lagrange multiplier µ (x) for the prediction point x by solving the following linear system (in Lagrangian form):

K 1 1 T 0 • λ (x) µ (x) = k (x) 1
The prediction for the I/O function is then given by (see [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]):

f (x) = λ (x) T f Xn (4)
The kriging property of exact interpolation results that

∀x i ∈ X n , it is f (x i ) = f (x i ) while for all the other x ∈ X it is f (x) f (x).

B. Estimation of the Uncertainty

A major advantange of the kriging metamodeling procedure is that it provides the prediction together with an estimation of its uncertainty. This prediction unertainty is given by the variance of the prediction error:

ε (x) = f (x) -f (x) (5)
since this error cannot be evaluated without running simulations at the prediction points. Thus, assuming it has a constant and unknown mean its variance is used instead. This variance is actually the Mean Squared Error of the prediction given by the formula:

MSE = σ2 (x) = = E f (x) -f (x) 2 = = E λ (x) T f Xn -f (x) 2 = = k (0) -λ (x) T k (x) -µ (x) (6)
The disadvantage of that measure is that there is no direct relation between the variance and the input samples, since it's only the covariance function that links them indirectly via the hyperparameters. And since this uncertainty will be our tool in improving the initial prediction its disadvantage can cause an underestimation of the prediction error and a slow improvement for our prediction during the adaptive sampling.

As an alternative, we can use the jackknife estimator σ2

Jack of σ2 to locate the new observation points that have to be added to our database. This estimation method, similar to the Leave One Out Cross-Validation (LOOCV) 3 , is based on the evaluation of the reduced predictions for every prediction point, by leaving out one observation each time [START_REF] Kleijnen | Design and Analysis of Simulation Experiments[END_REF]. It is more expensive computationally than the kriging variance, but also more efficient in improving the initial prediction. Jackknife estimation consists of the following steps: 1) Evaluating the reduced predictions:

f (-i) (x) = λ (-i) (x) T f (-i) Xn
2) Defining and calculating the pseudovalues:

f (-i) (x) = n f (x) -(n -1) f (-i) (x) , i = 1, . . . , n
3) Calculating the modified prediction:

fJack (x) = 1 n n i=1 f (-i) (x)
4) Estimating the jackknife variance:

σ2 Jack (x) = 1 n (n -1) n i=1 f (-i) (x) -fJack (x) 2 (7)
where the index (-i) implies that the i-th observation is being ignored during the calculations.

C. Adaptive Sampling and Final Prediction

When constructing a metamodel, the goal is to generate an input database which will be a kind of a discrete representation of the I/O function. This can significantly contribute in a better prediction accuracy, but it cannot take place with the traditional fixed sampling designs. The best way to do so, is to adaptively implement the initial sample space. The basic idea behind the adaptive sampling is to locate where should the evaluation of the model be carried out optimally to improve our knowledge on the forward function, based on the previous observations and predictions 4 . This will result in a stepwise uncertainty reduction. The main tool for the uncertainty reduction is the estimation of the variance, whether it is the kriging one [START_REF] Kleijnen | Kriging metamodeling in simulation: A review[END_REF] or the jackknife [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]. We will choose as a new sample point, the point where the prediction is poor i.e. the estimated variance is high. In order to avoid new points that are very close to the already existing ones 5 , we use the Euclidean distance in the input space and we select each new point as follows:

x n+1 = arg max x∈X min i=1,...,n x -x i • σ2 (x) (8)
This way we ensure both a high variance and a high minimum distance from the already observed points. We continue selecting new points and reducing the prediction uncertainty by iterating the procedure described above, until a terminal criterion is met. The terminal criterion will be the total number of simulations to run at a first stage and then as an extension, a desired level of accuracy. For each new point we have to re-evaluate both the kriging prediction by ( 4) and its variance.

As soon as the optimal input space X is complete the final prediction will be:

f (x) = λ (x) T f X (9)

D. Example

A simple example of the procedure described above is presented, in order to illustrate the capabilities and the results of the method and also to provide a visualisation of them. We generated a forest consisting only of trunks using COSMO and by setting constant all the forest and radar parameters apart from the frequency. We then simulated the polarimetric backscattering coefficient σ pq of the forest, where p, q ∈ {V, H} define the incident and scattered polarization respectively. We started with an input space consisting of 7 points, 5 chosen by the LH design and 2 on the border of the frequency band 6 and we adaptively added 10 extra points. The result of the initial prediction is presented in Fig. 4 (top), where we can see the forward function, the prediction and its confidence intervals. In order to improve this initial approximation, we added to the database one point each time using [START_REF] Stein | Interpolation of Spatial Data: Some Theory for Kriging[END_REF] exactly where the uncertainty shown in Fig. 4 (bottom) was high. The final prediction shown in Fig. 5 (top) is much closer to the forward function and the confidence intervals are very small. The final absolute error is much smaller than the initial one, reaching a maximum of about 0.25 dB as seen in Fig. 5 (bottom).

During this elementary example, the σ pq coefficients under all polarizations were approximated. The electric field shows many oscillations to be adequately interpolated, while at the same time all the backscattering coefficients are of a smoother nature. The results for all coefficients were similar and so we have just presented the ones corresponding to the HH polarization. The method can be easily extended to include more inputs for the metamodel. However as the inputs increase, the number of simulation points and the approximation error also increase as expected by the curse of dimensionality.

III. CONCLUSION AND PERSPECTIVES

We proposed to apply a scalar kriging technique to build a new surrogate model of COSMO in order to fast but accurately simulate the backscattering by whole forests. This work relies on a previous study too simple to be of great interest for real applications. We were actually restricted by the heavy structure of the functional two-level kriging technique we applied. Here the objective is to fast take into account the effect of the forest when designing a radar waveform for FoPEN for instance. This new surrogate model will have more input parameters and more functionalities. In the frame of cognitive radar, it is necessary to easily and fast propagate the uncertainties on the input parameters. This new technique we are currently implementing has been developed to reach these goals and the first results on a complete forest will be further presented.

Fig. 1 .

 1 Fig.1. A radar emits the signal se (t) which the forest modifies to st (t) before it reaches the target.

Fig. 4 .

 4 Fig.[START_REF] Saleh | A forest geometric description of a maritime pine forest suitable for discrete microwave models[END_REF]. The initial prediction: (top) with its confidence intervals, (middle) positions of observation points and prediction points (bottom) with the corresponding relative error and Jackknife variance multiplied by the minimum distance as seen in (8).

Fig. 5 .

 5 Fig.5. The final prediction (to be compared with Fig.4): (top) with the very limited confidence intervals, (middle) positions of observation points and prediction points (bottom) with its corresponding error evaluated for each prediction point before and after the uncertainty reduction.

  we can see a general description of the adaptive sampling procedure that we are going to perform.

	Start		-Initial Database
			?
			Model Analysis
			?
			New point selection
			?
	Stop	Yes	Stopping Criterion	No

Fig.

3

. The general plan of our adaptive sampling procedure.

  ).

	0
								Confidence intervals
	-5							Forward function
								Approximate function
	-10							Added points
								Initial Observation points
	-15
	-20
	-25
	-30
		0.4	0.6		0.8	1	1.2	1.4	1.6	1.8	2
		0
	Sigma Values (dB)	-20 -10							Sigma HH Predictions Observations
		-30
			0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	2
						Frequency (GHz)
									Initial absolute error
	Error (dB)	0.5 1							Final absolute error
		0	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
						Frequency (GHz)

In problems like these, no closed-form formulation or gradient information of the objective function are available. The only information that we have are the measurements of system performance, which can be updated as the metamodel can suggest the location of an additional point at which the objective function should be evaluated.

According to J.P.C. Kleijnen, the deviations of the simulation output f from its mean µ form a random process. Relying on this interpretation, a random metamodel can be used for a deterministic simulation model and f is assumed to be a sample path of a second-order GP.

Jackknife is actually an approximation of the bootstrapping estimation method, but it is preferred as it is easier to compute and its samples are more similar to the original data.

This reflects the main idea of the cognitive radar principle described in[START_REF] Haykin | Cognitive radar: a way of the future[END_REF].

The jackknife variance, because of the leave-one-point-out reduced predictions, can be unusually high close to some of the observation points.

[START_REF] Kleijnen | Kriging metamodeling in simulation: A review[END_REF] Since kriging is an interpolation method it cannot extrapolate and we have to evaluate the I/O function on the border of the input space also.


Panagiotis Piteros

Laetitia Thirion-Lefevre

Régis Guinvarc'h

Marc Lambert

R Guinvarc'h Supélec

/ Sondra

Cognitive radar applied to forest and kriging-based metamodel

Keywords: Cognitive Radar, Forest, Metamodel Adaptive DOE, Kriging, Prediction Uncertainty

INTRODUCTION 1.Context

Radar waveforms are usually designed without taking into account the effect of the environment. Cognitive radar is a new framework of radar systems introduced by S. Haykin [START_REF] Haykin | Cognitive radar: a way of the future[END_REF]. His idea is that the waveform emitted by the radar at time t + 1 will be dependent on the radar return at time t and of the analysis that has been done of this return. A further analysis shows that the emitted waveform s e (t) and the waveform illuminating the target s t (t), can actually be quite different for complex environments. As illustrated in Fig. 1, the environment, characterized by its transfer function H, can filter the waveform s t (t) = s e (t) • H (t). The signal received by the radar will be altered a second time, in a typical monostatic configuration. Thus, to evaluate these alterations, the transfer function has to be estimated based on our knowledge of the environment. Two applications are considered: target detection and forest parameters retrieval. In the first case, the goal is to compensate for the effect of the forest in order to maximize the intensity backscattered by the target and received by the sensor. For the second application, the objective is to use the capability to modify the radar waveform in order to probe a part of or the whole forest.

In this paper, we first recall the work we performed on a forest of trunks with a kriging metamodel for forest descriptive parameters retrieval. Then we turn to the new technique we propose to implement this work to the case of a forest of branches and trunks and also to support new functions. One of our objectives is to estimate the impact of uncertainties on forest descriptive parameters on the transfer function.

Fig. 1: A radar emits the signal s e (t) which the forest modifies to s t (t) before it reaches the target.

Previous results and limitations

We presented in [START_REF] Vasko | Metamodel-based adaptive use of a coherent polarimetric backscattering simulator for the characterization of forested areas at low frequencies[END_REF] a new method for forest characteristics inversion based on a functional surrogate model, used instead of COSMO [START_REF] Thirion | Capabilities of a forest coherent scattering model applied to radiometry, interferometry, and polarimetry at P-and L-band[END_REF] a scattering model dedicated to forest, but too complex to be used for inversion process. This surrogate model was developed to simulate the polarimetric backscattering coefficients as COSMO would do, but faster. However, it can only handle a few input parameters, because of its structure, leading to a strong simplification of the forest under study. In this particular case, only the trunks were considered for the simulation and their dimensions were derived from the age of the trees using allometric equations [START_REF] Saleh | A forest geometric description of a maritime pine forest suitable for discrete microwave models[END_REF]. The ground moisture was also taken into account and with the age of the trees, they represented the two input parameters we needed. Considering a set of polarimetric backscattering coefficients obtained for a given radar configuration and derived from the backscattered fields, we addressed the inverse problem of determining the forest age and the ground moisture by using the surrogate model. We produced error maps as in Fig. 2, that illustrate the uncertainty we obtained on the retrieved values of the age and moisture. The combination of the different polarimetric channels allowed us to obtained a very good estimate of these two parameters, despite the low values of the cross-polarization due to the forest description.

Our objective here is first to extend this method to a complete forest of trunks and branches of different sizes and then to take into account additional parameters as the vegetation moisture, that were assumed to be constant in the previous example. Nevertheless, it is not straightforward to achieve this goal, as the number of input parameters is actually strongly limited by the two-level kriging technique we previously implemented. So to overcome this difficulty, we decided to rather apply a scalar technique. This will allow us to consider a more detailed description of the forest, and also new functionalities as the sensitivity analysis and the propagation of uncertainties. This technique is detailed in the next section. Fig. 2: Example of an error map: the point to retrieve is indicated by a white star at (30 years, 0.55). The black area refers to all the retrieved values leading to an error less than 1dB on the backscattering coefficients. In this example, all the polarization channels have been considered. The retrieved age is estimated between 35 and 42 years, and the retrieved volumetric moisture ranges between 0.42 and 0.65.

SCALAR KRIGING-BASED METAMODEL

Kriging is a probabilistic method originated from Geostatistics designed for interpolating multidimensional functions that shows certain spatial correlation. Its metamodels are fitted to data that are obtained for large experimental areas, so they can be regarded as global rather than local. They provide a cheap and accurate approximation of the objective function together with an estimation of the potential error. Kriging's main idea is to model the I/O function by a Gaussian random Process (GP) in order to provide a probabilistic framework1 to account for the uncertainty stemming from the lack of information on the system [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]. The GP is fully characterized by its mean, which is assumed to be constant, and its covariance, with the latter reflecting the correlation among the observation points.

The kriging predictor is the so-called Best Linear Unbiased Predictor (BLUP) of the forward function in the vector space generated by the observations. Its performance depends on both the choice of the input values, where the observation are to be done, and the choice of the covariance. Taking into account the fact that the spatial correlation of the observation points and hence the covariance is unknown, in order to improve the predictor's performance an adaptive sampling procedure will be performed. Also a flexible covariance function will be chosen, so that we can "tune" it to best fit our data. In Fig. 3 we can see a general description of the adaptive sampling procedure that we are going to perform. Fig. 3: The general plan of our adaptive sampling procedure.

Initial Prediction and Estimation of Uncertainty

Kriging is an exact interpolation method, so it needs the results of several simulation runs of the model in order to predict the function under study. The adaptive procedure requires as a first step an initial prediction based on the results of a sample adequately spread in the input space. In order to do that, we will generate a Latin Hypercube (LH) maximin design. The resulting set contains n vectors X n = {x i } n i=1 which is mapped to the vector of the n observations f Xn = (f (x 1 ), . . . , f (x n ))

T .

The next step is to define an appropriate covariance function, which is assumed stationary for computational reasons, that will reflect the spatial correlation between the input samples. The Matérn covariance function is chosen, because it's flexible and it offers the possibility to adjust its regularity and the prediction's smoothness with just a single parameter [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]. This stationary covariance has the form:

k (x i , x j ) ≡ k (h) = σ 2 0 Γ (ν) • 2 ν-1 • 2 √ νh ν K ν 2 √ νh ( 1
)
where h is the inhomogeneous distance between the observation points x i , x j , K ν is the modified Bessel function of the second kind, ν is the parameter that controls the regularity of the covariance function and σ2 0 = k (0). Because of the fact that the covariance is unknown, we will estimate its parameters by Restricted Maximum Likelihood Estimation (RMLE) so that they will better fit our model i.e. our observations.

To proceed to the prediction of the I/O function at a point x / ∈ X n we need to compute the n × n covariance matrix K of the sample space and also the prediction point's covariance vector k (x):

K = [k (x i , x j )] , i, j = 1, . . . , n, k (x) = (k (x, x 1 ) , . . . , k (x, x n )) T
After these we are ready to compute the so-called kriging coefficients (or weights) λ i (x) and the Lagrange multiplier µ (x) for the prediction point x by solving the following linear system (in Lagrangian form)

K 1 1 T 0 • λ (x) µ (x) = k (x) 1
The initial prediction for the I/O function is then given by (see [START_REF] Villemonteix | An informational approach to the global optimization of expensive-to-evaluate functions[END_REF]):

f (x) = λ (x) T f Xn (2)
The kriging property of exact interpolation results that

∀x i ∈ X n , it is f (x i ) = f (x i ) while for all the other x ∈ X it is f (x) ≃ f (x).
A major advantange of the kriging metamodeling procedure is that it provides the prediction together with an estimation of the its uncertainty. This prediction unertainty is given by the variance of the prediction error 2 (assuming a constant and unknown mean) which is:

σ2 (x) := E f (x) -f (x) 2 = E λ (x) T f Xn -f (x) 2 = k (0) -λ (x) T k (x) -µ (x).
The disadvantage of that measure is that there is no direct relation between the variance and the input samples, since it's only the covariance function that links them indirectly via the hyperparameters. So, it underestimates the prediction error and it slowly improves our prediction during the adaptive sampling. As an alternative, we can use the jackknife estimation σ2 Jack of σ2 to locate the new observation points that have to be added to our database. This estimation method, similar to the Leave One Out Cross-Validation (LOOCV), is based on the evaluation of the reduced predictions for every prediction point, by leaving out one observation each time [START_REF] Jack | Design and Analysis of Simulation Experiments[END_REF]. It is more expensive computationally than the kriging variance, but also more effecient in improving the initial prediction.

Adaptive Sampling and Final Prediction

When constructing a metamodel, the goal is to generate an input database which will be a kind of a discrete represantation of the I/O function. This can significantly contribute in a better prediction accuracy, but it cannot take place with the traditional fixed sampling designs. The best way to do so, is to adaptively implement the initial sample space. The basic idea behind the adaptive sampling is to locate where should the evaluation of the model be carried out optimally to improve our knowledge on the forward function, based on the previous observations and predictions 3 . This will result in a stepwise uncertainty reduction. The main tool for the uncertainty reduction is the estimation of the variance, whether it is the kriging one or the jackknife. We will choose as a new sample point, the point where the prediction is poor i.e. the estimated variance is high. In order to avoid new points that are very close to the already existing ones 4 , we use the Euclidean distance in the input space and we select each new point as follows 5 :

x n+1 = arg max x∈X min i=1,...,n x -x i • σ2 (x)
We continue selecting new points and reducing the prediction uncertainty by iterating the procedure described above, until a terminal criterion is met. The terminal criterion will be the total number of simulations to run at a first stage and then as an extension, a desired level of accuracy. For each new point we have to re-evaluate both the kriging prediction by ( 2) and its variance. As soon as the optimal input space X is complete the final prediction will be:

f (x) = λ (x) T f X (3)

CONLUSION AND PERSPECTIVES

We proposed to apply a scalar kriging technique to build a new surrogate model of COSMO in order to fast but accurately simulate the backscattering by whole forests. This work relies on a previous study we briefly recalled that shows the interest of developing such a method, in particular in the frame of forest parameters retrieval. However, for this first trial, the description of the forest was too simple to be of great interest for real applications. We were actually restricted by the heavy structure of the functional two-level kriging technique we applied. Here the objective is to fast take into account the effect of the forest when designing a radar waveform for FoPEN for instance. This new surrogate model will have more input parameters and more functionalities. In the frame of cognitive radar, it is necessary to easily and fast propagate the uncertainties on the input parameters. This new technique we are currently implementing has been developed to reach these goals and the first results on a complete forest will be further presented.

J.P.C. Kleijnen's interpretation that the deviations of the simulation output f from its mean µ form a random process can explain why we use a random metamodel for a deterministic simulation model. f is assumed to be a sample path of a second-order Gaussian random process F .

The prediction error cannot be evaluated without running simulations at the prediction points and hence its variance is used instead.

This reflects the main idea of the cognitive radar principle described in[START_REF] Haykin | Cognitive radar: a way of the future[END_REF].

The jackknife variance, because of the leave-one-point-out reduced predictions, can be unusually high close to some of the observation points.

In order to ensure both a high variance and a high minimum distance from the already observed points


Priscilla Léon

Emmanuel Chartier-Kastler

Morgan Rouprêt

Vanina Ambrogi

Pierre Mozer

MD Véronique Phé
email: veronique.phe@gmail.com

Long-term functional outcomes after artificial urinary sphincter (AMS 800) implantation in men with stress urinary incontinence

Keywords: Urinary incontinence, urinary sphincter, survival, outcomes

Conclusion:

In the long term (>10 years) the AMS 800 can offer a high rate of continence to men suffering from SUI caused by sphincter deficiency, with a tolerable rate of explantation and revision. Patient summary. On the very long run (more than 10 years) AUS is able to offer high continence rate in men suffering from SUI after prostate surgery. The revision rate of the device increases with time.

Abstract

Background. Artificial urinary sphincter (AUS) is the gold standard surgical treatment of male stress urinary incontinence (SUI) with sphincter deficiency since the 80'.

Objective. To evaluate long-term functional outcomes of AUSs and to determine how many men required explantation because of SUI caused by sphincter deficiency after prostate surgery.

Design, setting and participants: Men who had undergone placement of an AMS 800 between 1984 and 1992 to relieve SUI caused by sphincter deficiency after prostate surgery were included.

Intervention. A peribulbar AUS was surgically inserted.

Outcome measurements and statistical analysis. Continence, defined as no need for pads, was assessed at the end of the follow up. Kaplan-Meier survival curves estimated the survival rate of the device without needing explantation or revision.

Results: Fifty-seven consecutive patients were included (median age 69 years; IQR: 64-72).

Median duration of follow-up was 15 years (IQR: 8. 25-19.75). At the end of follow-up, 25 patients (43.8%) still had their primary AUS. Explantation of an AUS was done in nine men because of erosion (n=7) and infection (n=2). Survival rates, without AUS explantation, were 87, 87, 80, and 80% at 5, 10, 15, and 20 years, respectively. Survival rates, without AUS revision, were 59, 28, 15, and 5% at 5, 10, 15, and 20 years, respectively. At the end of the follow-up, in intention-to-treat analysis, 77.2% of patients were continent.

Introduction

Artificial urinary sphincters (AUSs) have been used since 1972 for the treatment of urinary incontinence (UI) [START_REF] Scott | Treatment of urinary incontinence by an implantable prosthetic urinary sphincter[END_REF]. It has become the gold-standard treatment for males with UI caused by sphincter deficiency [START_REF] Lucas | EAU Guidelines on Surgical Treatment of Urinary Incontinence[END_REF]. The AUS improves continence and quality of life [START_REF] Haab | Quality of life and continence assessment of the artificial urinary sphincter in men with minimum 3.5 years of followup[END_REF]. The device has largely unchanged since its introduction by Scott et al. [START_REF] Scott | Treatment of urinary incontinence by an implantable prosthetic urinary sphincter[END_REF], and the number of AUSs implanted each year is increasing [START_REF] Phé | Trends in the landscape of artificial urinary sphincter implantation in men and women in France over the past decade[END_REF][START_REF] Lee | Temporal trends in adoption of and indications for the artificial urinary sphincter[END_REF]. The AUS is supposed to have an average lifetime of 10 years [START_REF] Venn | The long-term outcome of artificial urinary sphincters[END_REF][START_REF] Fulford | The fate of the "modern" artificial urinary sphincter with a follow-up of more than 10 years[END_REF]; however, few studies have reported on the longer-term (>10 years ) performance of AUSs [START_REF] Fulford | The fate of the "modern" artificial urinary sphincter with a follow-up of more than 10 years[END_REF][START_REF] Venn | The long-term outcome of artificial urinary sphincters[END_REF]. The major postoperative complication of AUSs in the longer term is needing the device to be explanted because of erosion or infection. In addition, mechanical failure, which occurs in 2.0-13.8% of devices [START_REF] Van Der Aa | The Artificial Urinary Sphincter After a Quarter of a Century: A Critical Systematic Review of Its Use in Male Non-neurogenic Incontinence[END_REF] leads to subsequent revision of the AUS.

Our goal was to report the functional results of AUSs implanted 20 years ago and to determine the mechanical-survival rate of the AMS800 in men suffering debilitating UI caused by sphincter deficiency after prostate surgery (retropubic radical prostatectomy, transurethral resection of the prostate, open prostatectomy).

Materials and methods

Population

Between 1984 and 1992, AUSs (AMS 800, American Medical Systems, Minnetonka, MN, USA) have been used to treat stress urinary incontinence (SUI) in non-neurogenic male patients in our department. The pre-implantation work-up comprised a clinical interview, analysis of voiding diaries (time and voided volumes, pad changes, incontinence episodes), a 1-hour pad test, a clinical examination, a cystoscopy and urodynamic assessment (cystomanometry and urethral-closure pressure).

Surgery

All patients had sterile urine at the time of surgery. Antibio-prophylaxis (amoxicillinclavulanic acid) was given before surgery and was maintained until removal of the urethral catheter at day 2 after surgery. Surgery was performed by two experienced surgeons.

All patients were placed in the lithotomy position. After placing a 14-Fr urethral catheter, a median perineal incision was made. The bulbocavernosus muscles were separated and the bulbar urethra was isolated along 20 mm. The circumference of the urethra was measured, and then the cuff, which had been previously purged, was placed in position. The cuff tube was passed through the inguinal incision into the subcutaneous tissue. An inguinal incision was made on the side of the dominant hand of the patient. The front fascia of the rectus muscle was incised and a pre-peritoneal space was created to place the balloon, which was filled with 20 mL of isotonic product (the cuff had been previously pressurized and the balloon was inflated to 22 mL). The fascia was then closed. The pump was placed in the ipsilateral scrotum. Connections between the balloon and the pump, and between the cuff and the pump, were then carried out ensuring that air was absent from the system. Once the device had been tested, it was deactivated.

Perioperative and late surgical complications were reported according to Martin et al.'s criteria. [START_REF] Mitropoulos | Reporting and Grading of Complications After Urologic Surgical Procedures: An ad hoc EAU Guidelines Panel Assessment and Recommendations[END_REF] Early postoperative complications (<30 days) were recorded according to the Clavien-Dindo classification. [START_REF] Mitropoulos | Reporting and Grading of Complications After Urologic Surgical Procedures: An ad hoc EAU Guidelines Panel Assessment and Recommendations[END_REF] The device was activated after 6 weeks. Ease of activation was defined as immediate comprehension of the functioning of the device and the ability to manipulate it by the patient.

Assessment of postoperative continence

The patients were assessed at implantation, at AUS activation at 6 weeks post-surgery, and at follow-up visits at 3, 6, 12, 18, and 24 months after the procedure, and then annually. A clinical examination, pump manipulation, an X-ray examination, and post-voiding residual were assessed. Continence was defined as no need to use pads. Subjective satisfaction was assessed by asking the patient.

Survival of the device

Explantation and revision of the device were analyzed separately. Explantation was defined as complete removal of the proven or potentially infected AUS. Mechanical failure was defined as a fluid leak caused by rupture or perforation of any part, insufficient pressure, or reservoir migration. Non-mechanical problems were defined as cuff erosion and infection that led to explantation. The rates of explantation, revision, and deactivation were recorded.

Statistical analyses

Statistical analyses were performed using R (Free Software Foundation's GNU project). The AUS survival rate without needing explantation (AUS in place and functioning) and the survival rate of the AUS without revision (AUS in place and functioning) were evaluated from the date of surgery to the last follow-up visit, using the Kaplan-Meier method.

Univariate analyses used Cox's model of proportional hazards (PROS PHREG) to determine the risk factors for explantation and revision. Hazard ratios were defined with a confidence interval of 95%. A p-value of <0.05 was considered statistically significant. The study was conducted after the patients' written informed consent was obtained and the study was approved by the local ethics committee.

Results

Patients' characteristics

A total of 57 consecutive patients were included in this study. The median age of patients at the time of surgery was 69 years (IQR: 64-72). Median body-mass index was 25.3 (IQR: 23.9-27.6). The characteristics of the men enrolled are listed in Table 1. Among the 57 patients, 18 (31.6%) had undergone previous surgery for UI, six patients had received radiation therapy before implantation of an AUS, and one patient received an AUS because of prostate cancer. The median 1-h pad test-weight was 70 g (IQR: 38-100).

Surgery

Median surgical time was 85 min (IQR: 66-100). The characteristics of the implanted device are listed in Table 1.

Postoperative course

The median length of hospital stay after surgery was 6 days (IQR: 7-8.7). The AUS was activated 6 weeks later. All patients had easy activation of the device. Early postoperative complications are listed in Table 2. There were five minor complications (grade I-II) and one major complication (grade III-IV). No patient died during the postoperative course.

Clinical outcomes

At the end of the follow-up, 49 patients were still alive. Median follow-up time, including men with and without the device, was 15 years (IQR: 8.25-19.75). Figure 1 shows the outcomes of the AUSs for the 57 patients. No patient was lost to follow-up. Twenty-five patients (43.8%) still had their native AUS in place at the end of follow-up. Of the total patients, 47 (82.4%) had an activated AUS at the last follow-up.

Deactivation. At the last follow-up, four patients had a deactivated device because of dementia, erosion without an explantation, regular acute urinary retention caused by gross hematuria secondary to prostate cancer recurrence, and a massive stroke, making it impossible to use the device. One patient had deactivated the device, which allowed him to be continent.

Continence. At the last follow-up, in intention-to-treat analysis, 77.2% of patients were continent. The subjective satisfaction rate reported by patients at the last visit was 89.5%.

Infections. One infection occurred at 13 years after the initial implantation of the AUS and another infection occurred 3.5 years after.

Erosion. Incidences of erosion occurred at the median time of 6.25 years (IQR: 18 months-22 years). There were two peaks of occurrence: one within the first 4 years and the second at over 12 years. Overall, there were eight erosions. Erosion occurred after initial implantation of the AUS in five patients, after a revision of the AUS because of mechanical failure in 1 patient, after two revisions of the AUS in one patient, and after a third revision for one patient. One patient did not undergo explantation of his AUS (only deactivation) because of bad general-health condition.

Explantation. Explantation of the AUS was performed in nine patients because of erosion (n = 7) or infection that caused erosion (n = 2). The 5-, 10-, 15-, and 20-year survival rates without explantation of an AUS, with it in place and functioning, was 87%, 87%, 80%, and 80%, respectively (Figure 2). In univariate analyses, being aged >69 years, having a body-mass index >25, prior radiation therapy, and the number of revisions were not significant risk factors for explantation.

Revision. The median number of revisions was one (IQR: 0-2). The 5-, 10-, 15-, and 20-year survival rates without revision of the AUS, while it was in place and functioning, was 59%, 28%, 15%, and 5%, respectively (Figure 3). In univariate analyses, being aged >69 years, a body-mass index >25, and prior radiation therapy were not significant risk factors for revision.

Discussion

To our knowledge, our study reports the longest follow-up of men implanted with an AUS.

We found a continence rate of 77.2% at the last follow-up and, overall, the patients' satisfaction was 89.5%. These results agree with rates reported by authors in studies that had 10-year periods of follow up [START_REF] Venn | The long-term outcome of artificial urinary sphincters[END_REF][START_REF] Fulford | The fate of the "modern" artificial urinary sphincter with a follow-up of more than 10 years[END_REF]. AUS remains the gold-standard treatment for male UI caused by sphincter deficiency [START_REF] Lucas | EAU Guidelines on Surgical Treatment of Urinary Incontinence[END_REF]; it has good functional outcomes and an acceptable morbidity. Table 3 summarizes the main functional outcomes for AUS in studies that have reported a >5-year follow-up [START_REF] Haab | Quality of life and continence assessment of the artificial urinary sphincter in men with minimum 3.5 years of followup[END_REF][START_REF] Venn | The long-term outcome of artificial urinary sphincters[END_REF][START_REF] Fulford | The fate of the "modern" artificial urinary sphincter with a follow-up of more than 10 years[END_REF][START_REF] Elliott | Mayo Clinic long-term analysis of the functional durability of the AMS 800 artificial urinary sphincter: a review of 323 cases[END_REF][START_REF] Gousse | Artificial urinary sphincter for postradical prostatectomy urinary incontinence:long-term subjective results[END_REF][START_REF] Raj | Outcomes following revisions and secondary implantation of the artificial urinary sphincter[END_REF][START_REF] Montague | Long-term continence and patient satisfaction after artificial sphincter implantation for urinary incontinence after prostatectomy[END_REF][START_REF] Kim | Long-Term Durability and Functional Outcomes Among Patients With Artificial Urinary Sphincters: A 10-Year Retrospective Review From the University of Michigan[END_REF][START_REF] Vg | Comparison of the Long-Term Outcomes Between Incontinent Men and Women Treated With Artificial Urinary Sphincter[END_REF][START_REF] Gülpınar | Functional Outcomes and Long-term Durability of Artificial Urinary Sphincter Application: Review of 56 Patients With Long-term Follow-up[END_REF][START_REF] Bordenave | Long-term results of the treatment of urinary incontinence with bulbar implantation of artificial urinary sphincter in men: a single-center experience[END_REF].

As also reported in the literature, we found that overall satisfaction after implantation of an AUS was high, regardless of the presence of leaks or revisions [START_REF] Dalkin | A National Survey Of Urinary And Health Related Quality Of Life Outcomes In Men With An Artificial Urinary Sphincter For Post-Radical Prostatectomy Incontinence[END_REF]. The hospital stay 20 years ago was longer for this type of surgery, reflecting medical care at that time. It is import to note that the main indication for implanting of an AUS at that period was UI after benign prostate hyperplasia whereas, nowadays, radical prostatectomy is the main indication.

In our study, survival rate without explantation of an AUS at 5, 10, 15, and 20 years was 87%, 87%, 80%, and 80%, respectively, which is in line with previous reports [START_REF] Haab | Quality of life and continence assessment of the artificial urinary sphincter in men with minimum 3.5 years of followup[END_REF][START_REF] Venn | The long-term outcome of artificial urinary sphincters[END_REF][START_REF] Fulford | The fate of the "modern" artificial urinary sphincter with a follow-up of more than 10 years[END_REF][START_REF] Elliott | Mayo Clinic long-term analysis of the functional durability of the AMS 800 artificial urinary sphincter: a review of 323 cases[END_REF][START_REF] Gousse | Artificial urinary sphincter for postradical prostatectomy urinary incontinence:long-term subjective results[END_REF][START_REF] Raj | Outcomes following revisions and secondary implantation of the artificial urinary sphincter[END_REF][START_REF] Montague | Long-term continence and patient satisfaction after artificial sphincter implantation for urinary incontinence after prostatectomy[END_REF][START_REF] Kim | Long-Term Durability and Functional Outcomes Among Patients With Artificial Urinary Sphincters: A 10-Year Retrospective Review From the University of Michigan[END_REF][START_REF] Vg | Comparison of the Long-Term Outcomes Between Incontinent Men and Women Treated With Artificial Urinary Sphincter[END_REF][START_REF] Gülpınar | Functional Outcomes and Long-term Durability of Artificial Urinary Sphincter Application: Review of 56 Patients With Long-term Follow-up[END_REF][START_REF] Bordenave | Long-term results of the treatment of urinary incontinence with bulbar implantation of artificial urinary sphincter in men: a single-center experience[END_REF]. The AUS is supposed to have an average lifetime of 10 years [START_REF] Venn | The long-term outcome of artificial urinary sphincters[END_REF][START_REF] Fulford | The fate of the "modern" artificial urinary sphincter with a follow-up of more than 10 years[END_REF]. The major postoperative complication of the AUS in the long term is the need for explantation of the device because of erosion or infection. Some papers have not reported erosion and infection separately; in our study, erosion rate was 14% and infection rate was 3.5%. In another study, pooled analysis of the mean rate of erosion and infection after implantation of an AUS in male non-neurogenic patients was 8.5% (range: 3.3-27.8%) [START_REF] Van Der Aa | The Artificial Urinary Sphincter After a Quarter of a Century: A Critical Systematic Review of Its Use in Male Non-neurogenic Incontinence[END_REF].

Infection and erosion generally occur within 2 years after implantation, although much later cases have been reported [START_REF] Duncan | Late erosion. A new complication of artificial urinary sphincters[END_REF]. It is not clear whether coating the device diminishes erosion and infection rates. The AMS 800 device is now available in some countries with a rifampicin and minocycline coating on the cuff and pump components, but not on the pressure-regulating balloon and its tubing. Surgical experience is also probably of great importance in preventing early infection and erosion. Sandhu et al. recently suggested that the learning curve for this kind of surgery does not reach a plateau, even after >200 surgical procedures [START_REF] Sandhu | The surgical learning curve for artificial urinary sphincter procedures compared to typical surgeon experience[END_REF]. Walsh et al. reported an increased risk of infection and erosion in patients that had undergone irradiation. However, placement of a urethral catheter without due consideration of the AUS cuff may also account for cases of late erosion [START_REF] Walsh I K | Artificial urinary sphincter implantation in the irradiated patient: safety, efficacy and satisfaction[END_REF]. The putative increased risk of erosion with prior or subsequent radiotherapy has never been proven in welldesigned and powered cohort studies using homogeneous patient populations. In our study, radiotherapy was not identified as a significant risk factor for explantation or revision, contrary to previous reports [START_REF] Raj | Outcomes following revisions and secondary implantation of the artificial urinary sphincter[END_REF][START_REF] Manunta | Artificial sphincter insertion after radiotherapy: is it worthwhile?[END_REF].

We reported an intention-to-treat revision rate of 40.3%. Furthermore, the 5-, 10-, 15-, and 20-year survival rates of the AUs without needing revision, and while it was still in place and functioning, dramatically decreased with time. Mechanical failure-rates varied between 2.0% and 13.8%, with failures reported from 11 months to 68.1 months postinsertion [START_REF] Van Der Aa | The Artificial Urinary Sphincter After a Quarter of a Century: A Critical Systematic Review of Its Use in Male Non-neurogenic Incontinence[END_REF]. Due to the retrospective nature of this study, the mechanical-failure rate might be underestimated.

However, global reintervention rate provides an overview of complications that occur following AUS implantation and the need invasive treatment: these should be regarded as an important end-point for comparisons with other surgical alternatives.

In a pooled analysis of 549 patients (10 studies) with non-neurogenic SUI, the mean reintervention rate was computed as 26.0% (range: 14.8-44.8%) [START_REF] Van Der Aa | The Artificial Urinary Sphincter After a Quarter of a Century: A Critical Systematic Review of Its Use in Male Non-neurogenic Incontinence[END_REF]; in addition, the timing of re-interventions was mainly influenced by the underlying cause. The patients needed a mean of 1.5 re-interventions. This result is in accordance with our study, where any recurrence of urinary leakage was attributed to urethral atrophy. Urethral atrophy is a well-known late complication after implantation of an AUS, and is typically presumed when recurrence of SUI occurs during follow-up with a functioning AUS [START_REF] Elliott | Mayo Clinic long-term analysis of the functional durability of the AMS 800 artificial urinary sphincter: a review of 323 cases[END_REF][START_REF] Dalkin | A National Survey Of Urinary And Health Related Quality Of Life Outcomes In Men With An Artificial Urinary Sphincter For Post-Radical Prostatectomy Incontinence[END_REF][START_REF] Rocha | A Prospective Study Evaluating the Efficacy of the Artificial Sphincter AMS 800 for the Treatment of Postradical Prostatectomy Urinary Incontinence and the Correlation Between Preoperative Urodynamic and Surgical Outcomes[END_REF][START_REF] Petrou | Artificial urethral sphincter for incontinence[END_REF]. Urethral-tissue hypoxia has been suggested to be the main pathophysiologic mechanism [START_REF] Martins | Artificial Urinary Sphincter in Patients Following Major Pelvic Surgery and/or Radiotherapy: Are They Less Favorable Candidates?[END_REF].

Determining the rate of urethral atrophy is a complex issue because it is potentially influenced by follow-up duration, device pressure, implantation technique, and the patient's characteristics. In the only prospective study, reported by Trigo Rocha et al., two patients developed urethral atrophy after a mean of 29.4 months [START_REF] Rocha | A Prospective Study Evaluating the Efficacy of the Artificial Sphincter AMS 800 for the Treatment of Postradical Prostatectomy Urinary Incontinence and the Correlation Between Preoperative Urodynamic and Surgical Outcomes[END_REF]. A history of radiation therapy has been proposed as a risk factor for urethral atrophy. However, current knowledge on urethral atrophy in non-neurogenic patients remains limited and is based on weak retrospective data.

Consequently, despite favorable outcomes and satisfaction rates after implantation, the AUS continues to be associated with the risk of local complications (i.e., atrophy, erosion, and infection) or mechanical failure. Thus, regular revisions and/or explantations are mandatory in at least 30% of AUS devices. New and more sophisticated AUS devices have been recently developed to improve function, occlusive efficiency, and biocompatibility relative to the AUS's current design [START_REF] Phé | Newer and novel artificial urinary sphincters (AUS): the development of alternatives to the current AUS device[END_REF]. The present study also underlines the necessity of only implanting an AUS in cognitively and motor-"valid" patients: our population included one case of dementia and one case of brain stroke. These health issues were directly responsible for the patients' inability to handle the pump, which led to needing to deactivate the AUS and a relapse of incontinence.

New AUS devices are awaited to decrease the incidence of mechanical failure and to improve urethral protection [START_REF] Phé | Newer and novel artificial urinary sphincters (AUS): the development of alternatives to the current AUS device[END_REF]. Alternatively, slings or adjustable continence therapies (proACT) may be of great interest as there is no device, as yet, able to manipulate voidance.

However, these may create some degree of obstruction and more long-term follow-up studies are needed [START_REF] Lucas | EAU Guidelines on Surgical Treatment of Urinary Incontinence[END_REF].

Conclusion

The AUS provided satisfactory long-term functional results for more than 10 years among men with SUI caused by sphincter deficiency. Not surprisingly, the device needed revision between 5 and 10 years. However, it is worth noting that >70% of the men remained continent in the long run. Table 1. Characteristics of the patients and of the implanted artificial urinary sphincters.

Tables

Table 2.

Early postoperative complications according to the Clavien-Dindo classification. Table 3. Main outcomes of artificial urinary sphincters over a 5-year follow-up period taken from the literature.

Take Home Message

The AUS provided satisfactory long-term functional results for more than 10 years among men with SUI caused by sphincter deficiency with >70% of the men remaining continent. Not surprisingly, the device needed revision between 5 and 10 years.

*Take Home Message

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Explantation-free survival curve for men implanted with an artificial urinary

Figure 3 .

 3 Figure 3. Revision-free survival curve for men implanted with an artificial urinary sphincter.

figure 1 2 3

 123 figure 1 Click here to download high resolution image

Table 1 . Characteristics of patients and of the implanted artificial urinary sphincters

 1

	N= 57 patients


Claire Pellot-Barakat

Muriel Lefort
email: muriel.lefort@imed.jussieu.fr

Linda Chami
email: linda.chami@yahoo.com

Mickaël Labit
email: labit.mickael@gmail.com

Frédérique Frouin
email: frederique.frouin@inserm.fr

Olivier Lucidarme
email: olivier.lucidarme@psl.aphp.fr

Automatic assessment of Shear Wave Elastography quality and measurement reliability in the liver

Keywords: Elasticity imaging, Reliability, Reproducibility, Observer variability, Shear Wave Elastography, Quality criteria, Liver fibrosis

Introduction

Liver biopsy is currently considered as the gold standard for the diagnosis, staging, and monitoring of liver fibrosis [START_REF] Afdhal | Evaluation of liver fibrosis: a concise review[END_REF]. However besides being an invasive procedure, liver biopsy can be inaccurate due to fibrotic liver tissue heterogeneity, leading to sampling errors and measurement variability [START_REF] Bedossa | Intraobserver and interobserver variations in liver biopsy interpretation in patients with chronic hepatitis C[END_REF] [START_REF] Muller | Quantitative viscoelasticity mapping of human liver using supersonic shear imaging: preliminary in vivo feasibility study[END_REF]. Noninvasive, reproducible and reliable methods are thus greatly needed to overcome the limitations of liver biopsy.

Imaging techniques have been investigated for the diagnosis of liver fibrosis. Conventional imaging techniques such as MRI, CT and ultrasound imaging are useful for biopsy guidance, but are unable to diagnose early stages of fibrosis [START_REF] Klatt | In vivo determination of hepatic stiffness using steady-state free precession magnetic resonance elastography[END_REF]. Since tissue stiffening seems to be a particularly relevant biomarker for the staging of liver fibrosis, new elasticity imaging techniques as MR elastography [START_REF] Klatt | In vivo determination of hepatic stiffness using steady-state free precession magnetic resonance elastography[END_REF] [START_REF] Bensamoun | Measurement of liver stiffness with two imaging techniques: magnetic resonance elastography and ultrasound elastometry[END_REF], US transient elastography (Fibroscan) [START_REF] Sandrin | Transient elastography: a new noninvasive method for assessment of hepatic fibrosis[END_REF] [START_REF] Fraquelli | Reproducibility of transient elastography in the evaluation of liver fibrosis in patients with chronic liver disease[END_REF] [START_REF] Castera | Transient elastography and other noninvasive tests to assess hepatic fibrosis in patients with viral hepatitis[END_REF] and ARFI have been developed [START_REF] Palmeri | Quantifying hepatic shear modulus in vivo using acoustic radiation force[END_REF] [START_REF] Fahey | In vivo visualization of abdominal malignancies with acoustic radiation force elastography[END_REF] using different methodologies. The supersonic shear wave elastography (SWE) technique [START_REF] Bercoff | Supersonic shear imaging: a new technique for soft tissue elasticity mapping[END_REF] [START_REF] Tanter | Quantitative assessment of breast lesion viscoelasticity: initial clinical results using supersonic shear imaging[END_REF] has recently been proposed. SWE is a 2-D real-time approach that combines the advantages of the ARFI remote palpation and the ultrafast echographic imaging approach of transient elastography. SWE has shown to be highly reproducible for breast masses [START_REF] Cosgrove | Shear wave elastography for breast masses is highly reproducible[END_REF]. The intra-and inter-observer reproducibility of SWE stiffness measurements in the liver has already been evaluated in several studies with healthy volunteers (Ferraioli et al., 2012a) [START_REF] Hudson | Inter-and intra-operator reliability and repeatability of shear wave elastography in the liver: a study in healthy volunteers[END_REF] showing good accuracy of SWE measurements. Studies have also compared SWE with transient elastography for assessing liver fibrosis (Ferraioli et al., 2012b) [START_REF] Poynard | Liver fibrosis evaluation using real-time shear wave elastography: applicability and diagnostic performance using methods without a gold standard[END_REF].

The goal of this work was to study the conditions of reproducibility and reliability of SWE measurements in clinical practice and to propose recommendations on how to interpret SWE measurements.

Material and Methods a) Patient population

Patients who underwent a liver elastography exam in the Radiology Department at our hospital between 02/2012 and 12/2013 were prospectively included in the study without exclusion criteria such as etiology or bodyshape (BMI, liver subcutaneous fat thickness, etc). The protocol was approved by the Institutional Review Board. Altogether, thirty-one patients (12 female, 19 male; mean age: 56±14 yrs) were recruited for the study. Patients were addressed for liver examination for various reasons including liver chronic disease hepatitis (12/31), liver damage following chemotherapy (5/31), liver transplant follow up (6/31) and benign focal lesions (8/31). Informed consent was obtained from all patients.

b) Data acquisition

All exams were performed using the Aixplorer TM ultrasound system (SuperSonic Imagine S.A., Aixen-Provence, France) with a SuperCurved™ SC6-1 transducer. Patients had been fasting 6 hours prior to the scheduled exam. Patients were placed in the supine position and were asked to hold their breath during the exam in order to minimize patient variables. The protocol consisted of recording three SWE video clips of 10 seconds for each patient repositioning the probe between each acquisition.

In order to evaluate the reproducibility and reliability of the SWE estimation, operator dependent measurement stages were examined and studied separately. These steps include 1) positioning the transducer between the patient ribs, 2) placing the insonation window, 3) freezing a stable frame and 4) selecting a region of interest (ROI) in which the average elasticity is calculated. The first 2 steps consisting of placing the transducer and insonation region on the right lobe of the liver fully require the expertise of the radiologist and are labeled "acquisition dependent" steps. The last 2 steps of selecting a stable frame and a representative ROI could possibly be processed afterwards by another operator or using an automatic algorithm providing that the expert criteria for selecting a ROI can be explicitly expressed and reproduced. These steps are referred to as the "quantification dependent" steps. Since discrepancies can come not only from the positioning of the probe but also from the choice of the ROI, these steps were evaluated separately in order to estimate the contribution of each step to potential inaccuracies.

c) Elasticity measurements

Four different SWE measurements were obtained for each video clip. The first measurement (Exp) was completed at the time of the exam by the radiologist considered as the expert operator. The measurement was performed by manually selecting a measurement frame (fixed frame) and a ROI using the system quantification tool (Q-box). The three other measurements were performed in postprocessing in the lab by a second operator (ML, technologist with 2 years of experience with reading SWE images) as well as using automatic algorithms. This second operator, was trained by the expert radiologist for the frame and the ROI placement: the most homogeneous and stable frame was chosen, and an elliptical region of interest (ROI) was positioned into the most homogeneous area, placed away from borders, vessels and possible artifacts such as areas with increased stiffness due to physiological phenomenon or transducer placement or pressure. The mean elasticity in that ROI provided the Trained Operator measurement (TO). Third, an automatic selection of the most stable frame and ROI in the clip according to temporal and spatial homogeneity criteria was performed. For that purpose, a lab-made software (AAE standing for Automatic Analysis of Elasticity maps) was used to automatically extract the frame that showed the most similarities with previous and following frames.

The most representative ROI inside that frame was then automatically determined by extracting a region as large as possible with a standard deviation as small as possible among all the possible ROIs.

The mean elasticity computed inside the selected ROI corresponded to the Best Frame (BF) measurement. Fourth, the video clip mean frame was computed and a Mean Frame (MF) measurement was performed in a representative ROI automatically selected from that mean frame using the AAE algorithm as well.

d) Data quality indexes

Preliminary studies in our lab [START_REF] Labit | Reproducibility of liver elasticity assessed by shear wave elastography[END_REF] suggested that the SWE variability of an exam is higher for hard than for soft livers. This is mostly due to the fact that fibrotic livers are not homogeneous and thus consecutive measurements in different parts of the liver lead to different elasticity values. Based on these preliminary findings, we tried to find parameters descriptive of the data that would provide an estimation of the reliability and accuracy of a given SWE measurement in a non-healthy liver. Ideally, these parameters must be as much independent as possible of the liver hardness which is the parameter that must be estimated. We defined three parameters representative of the video clip quality (two parameters directly related to the quality of the video clip and one related to the spatial homogeneity of the investigated frame) and we evaluated the SWE measurement accuracies in sub-populations of exams obtained by splitting the general population according to cut-off values of these parameters. First, using the lab-made software, the temporal stability of the clip was evaluated by comparing elasticity values of frames in a central ROI. This parameter called Temporal Variability (TV) was computed as the averaged pixel to pixel differences between corresponding central regions in adjacent frames of the clip. These central regions were automatically computed at the center of the insonation window; its axial and lateral sizes were half the dimensions of the insonation window in order to exclude border values that can be degraded.

The quality of the acquisition was also evaluated as the Percentage of Non-Filled pixels (PNF) in the selected best frame (f*) of the video clip using the AAE software. Non-filled areas showed as uncolored pixels in which shear wave velocity could not be determined by the system.

Finally, a parameter related to the homogeneity of the data defined as the spatial variability (SV) of the best frame was defined as the standard deviation of the elasticity values over the whole insonation window (and not only the central zone as for TV in order to measure the spatial heterogeneity due to border artefacts).

Detailed descriptions of the parameters can be found in Appendix A1, A2 and A3. Each parameter was first studied independently in order to evaluate its influence on the resulting measurement variability.

For each parameter (TV, PNF and SV), a threshold that allowed discriminating the clips with low variability from the ones with high variability was defined. The two quality parameters (TV and PNF)

were then combined in order to provide a single confidence index that allowed assigning a qualitative index to each video clip (high or low grade).

e) Statistical analysis

The SWE accuracies were evaluated by computing the relative differences between the mean elasticity values (E) of two measurements performed on a same patient either by a same operator but on different clips (acquisition intra-observer relative variability) or on a same clip but by different operators (quantification inter-observer relative variability).

-SWE acquisition intra-observer accuracy

The acquisition intra-observer relative variability V a (Appendix A4) was accessed by analyzing the mean variation of E between all the combinations of clips for a same patient. The acquisition relative variability was only computed for the Exp method as the expert acquired all the data. The measurement agreement between different scans in terms of Intraclass Correlation (ICC) was also computed for comparison with ICC values of SWE liver reproducibility studies reported in the literature (Ferraioli et al., 2012b), [START_REF] Hudson | Inter-and intra-operator reliability and repeatability of shear wave elastography in the liver: a study in healthy volunteers[END_REF].

-SWE quantification inter-observer accuracy

The quantification inter-observer relative variability (V q ) was studied by performing Bland-Altman comparisons of E obtained using the Exp method with those obtained using the TO, BF, and MF methods respectively (Appendix A5).The Pearson correlation coefficient r between measurements performed by the Exp and the other observers was also computed to estimate the relationships between the different quantifications.

Results a) Image acquisition

The quality of the acquired shear wave elasticity maps was very variable from one exam to the other as illustrated in Fig. 1 which shows different elasticity maps that can be encountered: high quality data and homogeneous SWE with complete filling of the elasticity map and low spatial variation (Fig. 1a), heterogeneous data with complete filling of the elasticity map but high spatial variation (Fig. 1b) and low quality data with a partial filling of the elasticity map (Fig. 1c).

Exploitable SWE measurements could not be obtained in three patients (n=3/31) because of patient obesity or anatomical inaccessibility, which hindered the propagation of shear waves. This showed as no or very little filling of the elasticity map. For these three cases, the percentage of unfilling of the elasticity map was greater than 60% and the candidate patient was excluded from the study. Acquisition failures occurred in three patients (n=3/31). There were 28 patients in the remaining population (n=28/31) included in the statistical analysis.

b) Data quality indexes

Across the whole set of clips, the values of the temporal variability (TV) parameter ranged from 0.05 to 3.4 kPa, the percentage of unfilling across the clip (PNF) ranged from 0 to 49% and the spatial variation ranged from 0.2 to 24.8 kPa. Table 1 summarizes the min, max, mean and median values of each quality parameter. From the distribution of these values, a threshold value was defined for each parameter in order to separate high from low quality clips: 1 kPa for TV, 10% for PNF and 2 kPa for SV. The thresholds were chosen so that about one fourth of the clips be classified in the low quality category.

The correlations between each parameter (TV, PNF and SV) and the expert measured elasticity (E exp )

were assessed (Fig. 2). The temporal variability (TV) and the elasticity were correlated with a Pearsons' correlation coefficient r=0.44 (p <0.001). There was no significant correlation between the percentage of non filling (PNF) and the elasticity (r=0.062). The spatial variability (SV) was the parameter the most correlated with the elasticity with a Pearsons' correlation coefficient r=0.67 (p <0.001).

c) SWE acquisition intra-observer accuracy

To compute the Exp acquisition variability, all the combinations of clips for a same patient were analyzed. We found a mean variability of acquisition between clips 𝑉 𝑎 of 22% corresponding to an ICC of 0.855. Since the clips that were compared had different quality parameters, the mean values of TV, SV and PNF of the different clips of a given patient i were computed (TV i , SV i and PNF i ) in order to get one value per patient for each parameter. Exams characterized by either a PNF i or a TV i above the criteria thresholds were classified as low grade exams (LG i ) while exams meeting the PNF i and TV i criteria were classified as high grade exams (HG i ). About one third of the clips (NC=27) belonging to 9 patients were classified as LG i and showed an acquisition variability of 32.5%. The remaining population (NC=57 corresponding to 19 patients) had a variability of 17%. The acquisition variability for different groups of patients separated according to the quality indexes (TV i , SV i and

PNF i ) is summarized in Table 2.

d) SWE quantification inter-observer variability

The mean elasticity values of the acquired elasticity maps were computed using the different methods described in sub-section (c) of the Material and Methods section. For each clip, the mean elasticity measured by the expert was compared to that measured by the trained operator (TO) and the two automatic methods (BF and MF). The plot of the mean elasticity values measured by the expert (abscissa) versus the TO, BF and MF methods (Fig. 3a) for the whole population (N=84 clips) shows that the correlation is higher for clips that have low elasticity values than high ones. This is also highlighted by the Bland-Altman plot of differences in SWE measurements performed by the different operators (Fig. 3b) which shows that absolute differences are generally smaller for low than for high elasticity values. Correlations of 0.97, 0.95 and 0.89 (p < 0.001) were found between the measurements of the expert and the TO, BF and MF methods respectively (Table 3). Characteristics of the linear fittings (intercept and slope) are also shown in Table 3. The percentages of variability between the measurements of the different observers for the general population ranged between 11.2 and 20.1 % (table 3). The variability between the two manual methods was 14.5% and the variability between the two automatic methods was 11.2%. The variability between a manual and an automatic method ranged from 15.6% to 20.1% (17.9% in average). The BF method presented a variability with Exp and TO of 15.6% and 16% and the MF method of 20.1% and 20% respectively.

The variability between the measurements of the expert and those of the different observers for subgroups of clips defined according to the thresholds of the different individual and combined quality parameters (TV, PNF and SV) was also computed (table 4). The variability between the two manual methods was 18.1% for unstable clips (TV≥1) and 13.2% for stable clips (TV<1). The variability between the two automatic methods was 20.4% and 7.8% for unstable and stable clips respectively.

The average variability for all pairs of methods was also 20.4% for unstable clips and 14.7% for stable clips. The average variability for all pairs of methods was 13.9% for spatially homogeneous clips (SV<2) and 24.0% for heterogeneous clips with SV≥2. The average variability for all pairs of methods was 13.5% for highly filled elasticity maps (PNF<10) and 24.5% for PNF≥10. Clips presenting either a PNF or a TV above the criteria thresholds were classified as low grade clips (LG). This concerned 33 clips which in average showed an inter-observer quantification variability ranging from 17.6% to 28.3% for the different pairs of observer comparisons as summarized in Table 4. The 33 LG clips belonged to 16 different patients amongst which 5 patients had all three clips classified as LG. The remaining 51 clips met the PNF and TV criteria and were classified as high grade clips (HG). They had an inter-observer quantification variability ranging from 6.9% to 16.1% (Table 4). The mean elasticity of the sub-population with low grade exams was about 30% higher than that of the subpopulation with high grade exams (11.6 kPa in average for LG and 8.5 kPa for HG).

Discussion and conclusion

The acquisition and quantification accuracies of SWE estimates of liver were assessed in a population of patients with various hepatic pathologies. The acquisition accuracy which depends on the positioning of the probe and shear window was assessed by comparing elasticity estimates obtained on consecutive clips acquired by the same expert on a same patient after repositioning the probe (intraobserver inter-clip variability). The quantification accuracy depends only on the choice of the measurement frame and area and was assessed by comparing the expert measurement with the measurement performed by a trained operator and 2 automatic methods on a same registered video clip (inter-observer intra-clip variability).

Previous reproducibility studies of SWE measurements in the liver reported by Ferraioli (Ferraioli et al., 2012b) showed ICC values of 0.95 and 0.93 for intra-observer SWE measurements performed the same day and 0.84 and 0.65 for measurements performed on different days by the expert and novice operators respectively. In [START_REF] Hudson | Inter-and intra-operator reliability and repeatability of shear wave elastography in the liver: a study in healthy volunteers[END_REF], the authors found SWE to be repeatable and reliable (ICC=0.92-0.87). In order to confront our findings to these previous studies, we evaluated the intraobserver ICC and found an agreement for same day assessments of 0.85 which is slightly below the agreements observed in (Ferraioli et al., 2012b) [START_REF] Hudson | Inter-and intra-operator reliability and repeatability of shear wave elastography in the liver: a study in healthy volunteers[END_REF]. It is important to note that these previous studies only concerned healthy subjects, with homogeneous and soft livers and generally good quality elasticity maps. We had previously observed that soft livers provide highly reproducible measurements [START_REF] Labit | Reproducibility of liver elasticity assessed by shear wave elastography[END_REF]. This was confirmed by the results of this study.

In order to provide the radiologist with an estimation of the reliability and accuracy of a given SWE measurement in clinical practice, we attempted to evaluate the variability of SWE measurements according to parameters depending on the data quality instead of the liver elasticity. Data quality parameters based on temporal stability (TV-Temporal Variability parameter) and propagation of shear waves (PNF-Percentage of Non Filling parameter) as well as a spatial homogeneity parameter (reflected by the Spatial Variation SV) were defined for that purpose and computed in all the clips. In order to evaluate whether these parameters reflected not only the pathology but also the quality of the data, correlations between these parameters and the measured elasticity were computed. The PNF parameter of the clip was found to be insignificantly correlated to the elasticity value measured in the clip. This is not surprising as the PNF should more likely be related to subcutaneous fat thickness than to liver stiffness. The spatial variability parameter was found to be correlated to the elasticity which was expected as the higher the elasticity, the higher its fluctuations. The temporal variability was moderately correlated with the elasticity. In order to provide an automatic estimation of each clip or exam quality (high or low grade), a criterion combining the TV and PNF parameters was defined. The SV parameter was excluded from the criterion as it is related to the standard deviation measurement which is already displayed in the system Q-box and which reflects the variability of the measurement due to the liver heterogeneity more than that due to the data quality (artifacts).

The intra-observer expert acquisition variability V a was higher than the inter-observer variability between all quantification methods. This is not surprising as some discrepancies arise from the positioning of the probe. The proposed qualitative data related criterion (high grade or low grade) was shown to be related to the acquisition variability. The acquisition variability was almost twice higher for exams classified as low grade than for high grade exams. The acquisition variability for high quality data is in agreement with the constructor precision of 15%.

The average correlation between the different quantification methods was excellent (ranging from 0.87 to 0.98). The agreement between methods is much higher for low elasticity values than for high ones as shown on the Bland-Altman plot in Fig. 3 where the points are less scattered in the low elasticity range than in the high one. Concerning the quantification accuracy, the variability between the two manual methods was lower than the variability between a manual and an automatic method. The BF method presented a lower variability with the manual methods than the MF method. Moreover, the variability between the two automatic methods was almost three times higher for unstable clips (TV≥1) than for stable clips (TV<1). This was expected as the mean frame is only representative of the clip when the temporal stability is high. When splitting the population according to the PNF criteria, the variability for clips that have a low PNF is less than 17% between all quantification methods while it is measurement method dependent when the PNF is higher with a range of 18.5 to 33.8 % between all quantification methods. This indicates that the choice of the area of measurement is crucial when the clips are of poor quality. This also suggests that the mean elasticity is not very pertinent for poor quality clips as it very much depends on where the measurement is performed. The SV criteria also allows to split the population of clips between the ones that have reproducible measurements (13.9% variability in average), and the ones that are less reproducible (24.0% variability in average). This parameter is also highly related to the elasticity as stated above with a mean elasticity for clips with SV≥2 (17.1kPa) more than double that of the clips with SV<2 (6.8kPa).

As for the acquisition variability, exams classified as low grade quality have a significantly higher quantification related variability (17.6 to 28.3% for all comparisons of quantification methods) than high grade exams (quantification related variability of 6.9 to 16.1%). This quality index provides a confidence grading and allows deducing whether elasticity estimates can be considered accurate and reliable or not.

This study demonstrated that elasticity values obtained with the SWE technique are accurate for high quality exams according to the defined quality criteria, and must be interpreted cautiously for low quality exams. This quality assessment does not provide a marker totally unrelated to the elasticity itself, the average elasticity of low quality exams being about 30% higher than that of high quality exams.

The goal of this study was to evaluate the accuracy of SWE measurements in diseased livers and not to state on the capability of SWE to discriminate between different stages of fibrosis. A correlation with pathology would have to be performed for that purpose. A recommendation from this study is to acquire SWE clips because they make it possible to perform additional quantifications on a more stable frame if necessary. They also allow assessing the quality of the data. If the quality is low, additional clips can be acquired which might be of better quality. Amongst the 31 patients initially included in the study, acquisition failures due to narrow inter-costal space or obesity occurred in three patients (10%) and there were five patients (16%) who had all their clips classified as low grade. The remaining 23 patients (74%) had at least one high grade clip making it possible to confidently assess the liver elasticity for most patients.

Our conclusion about the importance of taking into account the quality of the data could certainly be extended to SWE studies concerning other organs. These quality criteria could easily be implemented in a commercial machine in order to provide an estimation of the accuracy of the measurement.

Appendix

A1) The temporal variability parameter TV was expressed as:

𝑇𝑉 = 4 𝑛 𝑓 .∆𝑥.∆𝑦 ∑ 𝐸 𝑥,𝑦 𝑓 𝑓,𝑥,𝑦 (1)
where n f represents the number of frames in the clip, ∆𝑥 and ∆𝑦 are the lateral and axial sizes of the insonation window and 𝐸 𝑥,𝑦 𝑓 represents the elasticity of pixel p x,y (𝑥 = [∆𝑥/4,3∆𝑥/4 ], 𝑦 = [∆𝑦/4,3∆𝑦/4 ]) in the frame f (𝑓 = [1, 𝑛𝑓 ]).

A2) The percentage of non-filling parameter PNF was expressed as:

𝑃𝑃𝑃 = 1 ∆𝑥.∆𝑦 (𝑛𝑛𝑛𝑛𝑛𝑛 𝑝𝑝𝑥𝑛𝑝𝑝 / 𝐸 𝑥,𝑦 𝑓 * = 0) (2)
where f* represents the best frame selected by the AAE algorithm

A3)

The spatial variation parameter SV was expressed as:

𝑆𝑉 = � ∑ ∑ �𝐸 𝑥,𝑦 𝑓 * -𝐸 � 𝑓 * � 2 ∆𝑦 𝑦=1 ∆𝑥 𝑥=1 (∆𝑥.∆𝑦-1) (3)
where 𝐸 � 𝑓 * is the mean elasticity value in the frame f* .

A4)

The acquisition variability was defined as:

𝑉 𝑎 = 1 𝑁𝑁 ∑ ∑ 2�𝐸 𝑒𝑥𝑒 𝑖 (

A5)

The quantification inter-observer relative variability was defined as: Tables Table 1:

𝑉 𝑞 = 1 𝑁 ∑ ∑ 2�𝐸 𝑒𝑥𝑒 𝑖 (

  𝑖 (𝑗) represents the mean elasticity value measured in the clip 𝑗 of patient 𝑝 by an operator different than the expert (𝑜𝑛𝑝 = {TO, BF, MF}), N the total number of clips and 𝑛 𝑖 the number of clips corresponding to patient 𝑝.

  Description and distribution of the quality parameters values; n f represents the number of frames in the investigated clip, Thr represents the threshold value of the parameter, r and p represent the Pearson correlation with the elasticity measurement and its significance.Table 2: Percentage of variability between elasticity measurements of different clips of the same patient performed by the expert radiologist. The mean variability was computed for different groups of exams separated according to the individual quality parameters of the exam (TV i , SV i and PNF i ) as well as for Low Grade (LG i ) and High Grade (HG i ) exams. NC represents the total number of combinations of clips and P the number of patients in each category. Table 3: Percentages of variability and coefficient of correlation between the expert (Exp) and observer (TO, BF and MF) measurements over the whole population.Table 4: Variability between the expert and observer (TO, BF and MF) measurements according to quality criteria (TV: Temporal Variability, SV: Spatial Variation, PNF: Percentage of Non Filled pixels). The last 2 rows correspond to low quality clips (TV or PNF criteria not met) and high quality clips (TV and PNF criteria met).

Fig. 1 :

 1 Fig. 1: SWE images of right livers accessed intercostally. a) Low spatial variation (SV=0.78) with high quality elasticity map (PNF=0%); b) Heterogeneous spatial distribution (SV=4.11) with high filling (PNF =0%), c) Heterogeneous spatial distribution (SV=7.88) with poor filling (PNF=49%).

Fig. 2 :

 2 Fig. 2: Quality parameter as a function of the estimated elasticity for the 84 clips. Pearson's correlations of 0.40, 0.06 and 0.69 are found between the elasticity and TV, PNF and SV respectively.

Fig. 3

 3 Fig. 3: a) SWE estimates of the N clips (N=84 corresponding to P=28 patients) using the Trained Operator (TO), Best Frame (BF) and Mean Frame (MF) measurement methods as function the Expert measurement (Exp), b) Bland-Altman plot of differences in ratings performed by the expert and the three other operators. The blue,

Acknowledgments

The authors wish to thank the French Cancer Association ARC (Association pour la Recherche sur le Cancer) for their generous funding (grant number SFI20111203653).

red and green solid lines represent the mean of difference of ratings. The dashed lines define the limits of agreement.