Meiling Yue

Zeina Al Masry

Samir Jemei

Noureddine Zerhouni

An online prognostics-based health management strategy for fuel cell hybrid electric vehicles

Keywords: decision fusion, health management, fuel cell, hybrid electric vehicle, prognostics

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

obtained under constant operating conditions. Although they are devoted to estimating the health state of the fuel cell, the real driving conditions have not been considered. Moreover, prognostics itself cannot reach the goal of improving durability, i.e., post-prognostics decisions and mitigation actions are lacking [START_REF] Yue | Review on health-conscious energy management strategies for fuel cell hybrid electric vehicles: Degradation models and strategies[END_REF].

To improve the durability of fuel cells in HEV applications, efforts have been made to develop health-conscious energy management strategies (EMSs) [START_REF] Yue | Review on health-conscious energy management strategies for fuel cell hybrid electric vehicles: Degradation models and strategies[END_REF]. A deterministic dynamic programming strategy and a rule-based strategy have been developed in [START_REF] Kandidayeni | Investigating the impact of ageing and thermal management of a fuel cell system on energy management strategies[END_REF] to minimize the cost and at the same time, respect the operation limits to avoid degradation. Liu et al. have developed a multi-objective EMS to achieve optimal cost and fuel cell lifetime based on predicted vehicle speed and battery state of charge [START_REF] Liu | Research on a multi-objective hierarchical prediction energy management strategy for range extended fuel cell vehicles[END_REF]. Moreover, an equivalent consumption minimization strategy has been developed in [START_REF] Li | Online adaptive equivalent consumption minimization strategy for fuel cell hybrid electric vehicle considering power sources degradation[END_REF], which considers the fuel cell degradation in the objective function. However, existing researches usually consider the fuel cell degradation by setting constraints or using fitting degradation models in the strategies, which are less accurate and cannot signify the real health state of the system. The developed fuel cell prognostics technologies are less applied to the energy management of HEV applications. Without an exact understanding of the current health state, the actions taken by the controller may be not appropriate and may even cause more damages to the system's health, shortening its lifetime [START_REF] Han | Investigation of fcvs durability under driving cycles using a model-based approach[END_REF].

To go further from prognostics to health management for fuel cell HEV applications, a prognostics-enabled decision-making (PDM) process has been proposed in [START_REF] Yue | Health-conscious energy management for fuel cell hybrid electric vehicles based on prognostics-enabled decisionmaking[END_REF], which uses the data provided by the prognostics step to reconfigure the mission of a fuel cell HEV depending on its health state. The decision is made by allocating confidence factors to different degradation states and fusing the parameters of fuzzy logic controllers based on the confidence factors to find the optimal power distribution between the fuel cell and the energy storage system in real time.

However, the confidence factors in [START_REF] Yue | Health-conscious energy management for fuel cell hybrid electric vehicles based on prognostics-enabled decisionmaking[END_REF] are obtained through a classification method based on a Gaussian-shaped fuzzy inference system, while only the median value of RUL predictions is used. Therefore, the classification results are less convinced if RUL uncertainties are not considered. To improve this deficiency, the uncertainties of the prognostic results are examined based on the particle filtering prognostics method in this paper. 58 To develop the prognostics-based health management strategy, 59 a probability calculation method is then introduced to obtain 60 the probability values allocated to different degradation states, 61 and a decision fusion approach is applied to calculate the con-62 fidence factors to the offline-optimized fuzzy logic controllers 63 and to determine the parameters of the online operating con-64 troller.

65

The reminder of the rest of this paper is as follows: Section 2 66 introduces an online prognostics method for the fuel cell based 67 on particle filtering with an analysis of the sources of uncer-68 tainties. Section 3 presents the development of the proposed 69 prognostics-based EMS for a fuel cell HEV, while the simu-70 lation results are compared and discussed in Section 4 before 71 concluding. 72 2. Online prognostics based on particle filtering

73

The online particle filtering prognostics method has been 74 proposed in this section to be combined with the health man-75 agement strategy of the studied fuel cell HEV. It allows one to 76 track and to continuously estimate the state of health of the fuel 77 cell, and furthermore, to predict the RUL. Particle filtering is essentially a nonlinear Bayesian tracking 81 approach that the unknown states from noisy observations 82 can be estimated with available prior knowledge. In a typical 83 Bayesian tracking problem, two models are required to de-84 scribe the system: state model and measurement model.

87 x k = f (x k-1 , u k , w k ) ↔ p(x k |x k-1 ) (1)
where the initial distribution of the states x k is modelled as 88 p(x 0 ) and the state transition is based on a probabilistic model 89 p(x k |x k-1 ). u k is the command input and w k is process white 90 noise, non-necessarily Gaussian [START_REF] Jouin | Particle filter-based prognostics: Review, discussion and perspectives[END_REF].

92

Measurement model:

93 z k = h(x k-1 , v k ) ↔ p(z k |x k ) (2)
where the measurement z k is of a marginal distribution p(z k |x k ) 94 and v k is the measurement white noise, non-necessarily Gaus-95 sian [START_REF] Jouin | Particle filter-based prognostics: Review, discussion and perspectives[END_REF].

96

The state estimation is realized by recursively estimating the 97 state probability distribution function p(x k |z 1:k ) using a predic-98 tion step and an update step: 99 Prediction step:

100 p(x k | z 1:k-1 ) = p(x k | x k-1 )p(x k-1 | z 1:k-1 )dx k-1 (3)
Update step:

101 p(x k | z 1:k ) = p(z k | x k )p(x k | z 1:k-1 ) p(z k | z 1:k-1 ) (4)
where

p(z k | z 1:k-1 ) = p(z k | x k )p(x k | z 1:k-1 )dx k (5)
In fact, the above Bayesian tracking equations are conceptual. To analytically solve this problem, particle filtering is widely applied to give approximate solutions owing to its ability to solve nonlinear and non-Gaussian problems. Particle filtering is implemented based on sequential importance sampling (SIS) technique, which is a Monte-Carlo method to calculate p(x k | z 1:k ) by sampling the states into particles and allocating weights to the particles. Details could be found in [START_REF] Arulampalam | A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking[END_REF]. The steps for implementing SIS-type particle filtering are described as follows:

1. In the first step (k = 1), n particles with an initial distribution p(x 0 ) are generated;

2. Propagate the n particles from x k-1 to the next state x k using (1) to calculate the probability density function;

3. The weights associated to the particles are calculated by the likelihood of the state to the latest measurement z k :

L(z k |x k , σ v k ) = 1 √ 2πσ v k exp(- (z k -x k ) 2 2σ 2 v k ) (6)
4. In order to avoid the dying away of the particles, a resampling process is performed to exclude small-weighted particles by setting a given weight limit and replicate those large-weighted particles;

5. The rebuilt posterior probability density function by resampling is, therefore, recursively transferred to the following iteration as its prior probability density function;

6.

Step 2 to 5 are repeated until there is no more measurement.

Prognostics require the state model to output new state estimates when the measurement is no longer available, and then to predict the future health state and the RULs. To adapt particle filtering for prognostics purpose, it needs a prediction phase in addition to the above-described state estimation process. When there is no measurement available, i.e., the prediction is started, the state x k is propagated to x k+1 without calculating the likelihood of (6) and the parameters of the model are propagated with the noises.

RUL estimation

RUL is a term that indicates how long the system could remain in use, it is calculated based on a pre-defined end-of-life (EOL) threshold λ. The estimated probability density function of the RUL at time k is:

p(RUL ≤ l | z 0:k ) = p(x k+l ≥ λ | z 0:k ) (7)
It is updated based on new z k , k = k p + 1, ..., k p + l. The RUL uncertainty can be represented by the final distribution of the particles obtained when the failure threshold is reached [START_REF] Jouin | Prognostics of pem fuel cell in a particle filtering framework[END_REF].

The pseudo-code of implementing the particle filtering prognostics method is presented in Algorithm 1.

Algorithm 1 Particle filtering prognostics Initialize x i 0 , w 0 i , v 0 i , u i 0 with uniform distribution Time step k = 1 while x i k > λ and k ≤ k p , do for i = 1, ..., n, do SIS Generate particles

x i k ∼ p(x i k | x i k-1 , w k-1 i , u i k-1 ) Allocate weight ω i k = L(z i k | x i k , v i k ) end for Normalize weight ω i k = ω i k / n i=1 ω i k
Calculate the cumulative sum of the normalized weights

Q i k | n i=1 = Cumsum(ω i k | n i=1 ) for i = 1, ..., n, do Resampling j = 1 Draw a random value u i ∼ U[0, 1) while Q i k < u i , do j = j + 1 end while Update state x j k = x j k Update noises w k-1 j = w k-1 j , v k-1 j = v k-1 j Update parameters u j k = u j k end for k = k + 1 end while for i = 1, ..., n, do Start prognostics from k p while x i k > λ, do Propagate particles to the next time step x i k = f (x i k-1 , w k-1 i , u i k-1 ) k = k + 1 end while Estimate RUL i k = (k -k p ) • ∆t end for 2.2. Online prognostics implementation 45 2.2.1. Model selection 46
The degradation trend of the fuel cell degradation dataset 47 could be caught by various empirical models [START_REF] Jouin | Prognostics of pem fuel cell in a particle filtering framework[END_REF][START_REF] Zhang | An unscented kalman filter based approach for the health-monitoring and prognostics of a polymer electrolyte membrane fuel cell[END_REF][START_REF] Jouin | Prognostics of proton exchange membrane fuel cell stack in a particle filtering framework including characterization disturbances and voltage recovery[END_REF]. In 48 order to include more possibilities of variation, the 2nd order 49 exponential degradation model is chosen to track the fuel cell's 50 performance and perform prognostics: Then, the state space model could be written as (9):

51 V(t) = b 1 • exp(b 2 • t) + b 3 • exp(b 4 • t) (8)
54 x 1,k = x 1,k-1 • exp(b 2 ) + w 1,k , w 1,k ∼ N(0, σ 2 w1 ) x 2,k = x 2,k-1 • exp(b 4 ) + w 2,k , w 2,k ∼ N(0, σ 2 w2 ) (9)
where x 1,k and x 2,k are two first-order independent Markov pro-55 cesses and the present voltage state can be obtained from x 1,k 56 and x 2,k :

57 x k = b 1 • x 1,k + b 3 • x 2,k (10)
where b 1 , b 2 , b 3 and b 4 are also states to be transited. The measurement model is then written as:

z k = x k + v k , v k ∼ N(0, σ 2 v ) (11)
where v k are supposed to be included in the measurement signal as the measurement noise. As the initial distributions of the parameters are unknown, uniform distributions is applied [START_REF] An | A tutorial for model-based prognostics algorithms based on matlab code[END_REF].

The process noise variance is refined as σ 2 w = 10 -4 and same process noise is assigned to the unknown parameters (b 1 , b 2 , b 3 and b 4 ). The measurement noise v k is not considered as it is supposed to be contained in the input signals. Moreover, the number of particles is fixed as n = 2000.

Data preprocessing

During the system's online operation, voltage disturbances are inevitable. The original signal is too noisy to implement prognostics. Thus, the data should be preprocessed to remove the peaks and filter the noises. To this end, a kernel smoother is applied. The estimated point f (t j ) is calculated based on the weighted moving average:

f (t j ) = n i=1 s i • u(t j ) n i=1 s i (12)
where

s = K( t j -t h ) (13)
h is the bandwidth and K is a Gaussian Kernel function written as:

K(t) = e -t 2 2 √ 2π (14)
Besides, during online operation, the signal measured at each time step is too voluminous, and using these data points to estimate the states requires huge storage memory, up to gigabytes to store the states, estimated parameters, and particle weights.

This would not be actually allowed in industrial applications.

Besides, too many points for state estimation will lead to overfitting problems, resulting in the divergence of the model [START_REF] Jouin | Combined predictions for prognostics and predictive control of transportation pemfc[END_REF].

Therefore, the measured signal should be reduced. Moreover, based on the hypotheses that the degradation of the fuel cell is only due to the ageing effects and no sudden damage happens during its operation, the degradation will not progress largely on short time scales, time scales longer than an hour should be suitable for industrial application [START_REF] Jouin | Estimating the endof-life of pem fuel cells: Guidelines and metrics[END_REF][START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF]. Therefore, in this study, to track the state of the system and to perform prognostics, the data is reduced at one point per hour. The estimated voltage drawn in Figure 1 is the successive positions of the top of the particles distribution [START_REF] Jouin | Prognostics of pem fuel cell in a particle filtering framework[END_REF]. According analytical model for the degrading fuel cell system. As 52 the system is dynamic and sensible to the operation condi-53 tions, it is nonviable to find a model that covers all possible 54 variables.

Prognostics result demonstration

55

• Input uncertainties: It refers to the sensors that are used 56 to monitor the system. The measurements of sensors come 57 with a variety of noises due to electrical interference, dig-58 itization error, sensor bias, dead-band, backlash and re-59 sponse non-linearity, etc [START_REF] Saha | Uncertainty management for diagnostics and prognostics of batteries using bayesian techniques[END_REF].

Prognostics-based health management strategy

A health management strategy is proposed in this section, in which the power distribution between the fuel cell and the battery is performed using fuzzy logic control (FLC), which is optimized by genetic algorithm (GA), and a PDM process considering RUL uncertainty is designed to redistribute the power between different power sources in the studied fuel cell HEV.

System description

The studied fuel cel HEV is build in MATLAB R /Simulink R environment and the structure is shown in Figure 3. A proportional-integral (PI) speed controller is used to track the reference speed in the propulsion system, which is modelled based on the literature [START_REF] Depature | Ieee vts motor vehicles challenge 2017 -energy management of a fuel cell/battery vehicle[END_REF]. The power train consists of a 16kW PEM fuel cell stack and a 80V, 40Ah lithium-ion battery pack, which are used to provide the demanded power profile. All components are connected by current and voltage signals using a feedback loop. An EMS is then designed to split the power between the fuel cell and the battery based on the speed of the vehicle and the battery's state of charge (SOC). In order to be health-conscious, it is designed also depending on the fuel cell health state. The models of the fuel cell and the battery are detailed as follows.

Fuel cell modelling

The fuel cell in the studied vehicle is regarded as a voltage source based on the following current-controlled model, which is deduced from its polarization curve given in [START_REF] Depature | Ieee vts motor vehicles challenge 2017 -energy management of a fuel cell/battery vehicle[END_REF]. The fitting result is written as:

V f c = f (i f c ) = α 1 • i 3 f c + α 2 • i 2 f c + α 3 • i f c + α 4 ( 15
)
The fitting factors are α 1 = -6.7791e -07, α 2 = 0.00044927, α 3 = -0.11913 and α 4 = 59.124. The performance degradation of the fuel cell is considered on 49 the stack level, which can be indicated by the loss of its stack 50 voltage no matter what degradation is happening on the cell or 51 the component level. Here, the degraded voltage of the fuel cell 52 is modelled based on its operation points and on/off switches 53 [START_REF] Chen | Lifetime prediction and the economic lifetime of proton exchange membrane fuel cells[END_REF][START_REF] Fletcher | An energy management strategy to concurrently optimise fuel consumption and pem fuel cell lifetime in a hybrid vehicle[END_REF], written as:

54 V f c-degrade = V ini • D f c-degrade = V ini • (D f c + D on/o f f ) (16)
where

55 D f c = δ 0 (1 + α(P f c -P nom ) 2 P 2 nom )dt (17)
D on/o f f =        ∆ switch , if P f c,t ≥ 0 ∧ P f c,t-1 < 0 0, otherwise (18)
In ( 17) and ( 18), V ini is the initial available fuel cell supplied 56 voltage, P nom is the nominal fuel cell power and P nom = 6000W, 57 which is regarded as the maximum efficiency operation point, 58 α and δ 0 are the load coefficients and ∆ switch is the voltage loss 59 coefficient for fuel cell's once switch. The coefficients are fitted 60 based on the historical datasets presented in [START_REF] Gouriveau | Ieee phm 2014 data challenge: Outline, experiments, scoring of results, winners[END_REF]. An electrical equivalent circuit model is used to represent 63 the battery's behaviour, which describes the battery operating 64 characteristics using a RC circuit network. The output battery 65 voltage is calculated by:

66            u C c = - u C c R c C c + i bat C c V bat = u 0 -u C c -i bat R s (19)
where u 0 , R s , R c and C c are estimated based on experimental 67 data [START_REF] Depature | Ieee vts motor vehicles challenge 2017 -energy management of a fuel cell/battery vehicle[END_REF]. As there is no sensor that could read SOC directly, 68 SOC is estimated here by ampere-hour counting (Coulomb 69 counting) method, expressed as:

70 S OC t = S OC t 0 + η t t 0 i τ Q max dτ (20)
where η denotes the coulombic efficiency (η = 0.95 when 1 charging and η = 1 when discharging), i τ is the battery cur-2 rent at time instant τ (i τ > 0 when charging and i τ < 0 when 3 discharging), and Q max is the maximum available capacity of 4 the battery.

5

In this study, the battery used in the fuel cell HEV is assumed

6
to have a much longer lifetime than the fuel cell so that the 7 battery degradation cost and the interaction between the battery 8 and the fuel cell are not considered.

45

GA is well adaptive to the optimization of FLCs, in which all 46 the parameters of MFs are coded in one chromosome and they 47 are selected, crossed and mutated to find an optimal solution to a defined objective function. Here, the objective function is a weighted polynomial function including fuel cell degraded voltage D f c-degrade , SOC range ∆S OC and hydrogen fuel consumption m H 2 :

Ob jFun = -w 1 • D f c-degrade -w 2 • ∆S OC -w 2 • m H 2 (21)
with

∆S OC = f (1 + (S OC bat -S OC init ) 2 S OC 2 init )dt (22)
where w 1 , w 2 and w 3 are the weighting factors. The optimization is implemented on a repeated WLTC Class 2 driving cycle and the optimized MFs under different degradation states (D f c-degrade is 0%, 3%, 6% and 9% of the initial stack voltage value) have been plotted in Figure 4. The first line is the MFs of the input SOC, the second line is MFs of the optimised fuel cell reference current and the last line is the evolution of the values of the objective function (calculated by ( 21) and ( 22)).

Prognostics-based health management based on PDM

A PDM process is proposed in this section to fulfill the health management for the long term operation of the studied fuel cell HEV. A decision fusion algorithm is applied to consider the prognostics uncertainty when determining the parameters of the online FLC.

Proposed PDM process

A PDM process is to select system actions based on the predictions of future system health states [START_REF] Balaban | A modeling framework for prognostic decision making and its application to uav mission planning[END_REF]. Speaking about PDM in HEV applications, PDM process turns out to be a part of EMS. The proposed prognostics-based health management strategy uses the predictive nature of the prognostics process to obtain the current health state of the fuel cell and conducts energy management though FLC, which is automatically adapted to the current health state.

According to Section 2, the prognostics algorithm is able to estimate the future states based on the previous degradation trend at each desired time instant. However, the degradation of power sources is a long-term procedure and it is redundant to predict their RULs on short time scales, which will lead to huge computation burdens and insufficient memory. Some researchers propose to define the time instants in order to indicate when to start performing the prognostics. According to [START_REF] Jouin | Estimating the endof-life of pem fuel cells: Guidelines and metrics[END_REF], prognostics can be triggered by monitoring the health indicator. When it comes to implementing EMS, the key point is to select control actions that are suitable for the current health state. The above described health-conscious fuzzy logic control strategy tuned by offline GA optimization may be not feasible and impossible to execute in real time as the FLC cannot be tuned at each time instant and at each degradation state. Therefore, a solution is proposed in this paper, as shown in Figure 5. The idea is to optimize a limited number of FLCs under different degradation states and when RULs are obtained through the online prognostics server, corresponding probabilities to these Suppose that during the online prognostics process, RUL min = 0 hour and RUL max = RUL CI=50% + t λ hours, different widths of bins (WOB) are defined and the corresponding probability values of the RUL prediction results in Figure 2 are calculated and listed in Table 1.

The calculation of probability is a necessary step for decision making. Online prognostics at the module level is used to assess in real time the fuel cell's health state and estimate its RUL, while during the decision-making process, the system must be auto-reconfigured depending on the health state to operate with the expected performance. The probability indicates the possibility of the fuel cell falling in a certain degradation state, which should be considered when determining the confidence factors of FLCs that have been optimized offline under certain degradation states, described in the next part of this section.

Decision fusion based on probability

The parameters of the online used FLC could be obtained Using the basic probability assignments calculated from the online prognostics process, one can combine the evidence from different degradation states and calculate the belief measure value that takes into account all possible states according to Dempster-Shafer theory:

Bel(A) = B⊆A m(B) (23)
To understand this equation, it is done in a geometrical way.

As seen in Figure 7, they are the MFs of "i f c =high" extracted from Figure 4.

To calculate the belief measure value of each side, one should longitudinally compare the range of the four S1 and four S2 of the MFs. As indicated by [START_REF] Jouin | Combined predictions for prognostics and predictive control of transportation pemfc[END_REF], a combined belief measure value is obtained if the range of one side is the subset of the range of another side. For example, S2 of the MF of "i f c =high" in FLC1 has a larger range "a" that contains the range of S2 in FLC2 and FLC3, "b" and "c", respectively, so that the belief measure value of it is determined as P1+P2+P3. To be noted, as x 43 >

x 13 , the range "a" does not contain range "d", therefore, the probability P4 is not taken into account. Other belief measure values for each side of each MFs are obtained using the same method. Then, to obtain the basic belief assignment m(A) for the MF 42 of one case, a Möbius transformation is used given the belief 43 measure value of each side [START_REF] Liu | Classic Works of the Dempster-Shafer Theory of Belief Functions: An Introduction[END_REF]:

44 m(A) = B⊆A (-1) |A-B| Bel(B) (24)
This equation is simplified as [START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF] by neglecting the term 45 (-1) |A-B| . This is because the cardinality of A-B is uncountable 46 and insignificant [START_REF] Zhou | Online energy management strategy of fuel cell hybrid electric vehicles based on data fusion approach[END_REF].

47 m(A) = B⊆A Bel(B) (25)
As an MF consists of two sides, the assigned probability to 48 the MF is calculated by the weighted arithmetic mean of the 49 side belief measure values [START_REF] Yue | Health-conscious energy management for fuel cell hybrid electric vehicles based on prognostics-enabled decisionmaking[END_REF]. Then, the parameters of the 50 online used FLC could be determined and adapted to the current health state.

Result discussion

Comparison analysis

The health management strategy is coded in the EMS module of the simulated fuel cell HEV model (Figure 3). Further analysis of how the WOBs of probability calculation will have an influence on the system performance is discussed by defining different WOBs. More simulations are executed with WOB = 0.5 • RUL max , 0.4 • RUL max , 0.3 • RUL max , 0.2 • RUL max and 0.1•RUL max , respectively. The prognostics process is triggered with the fuel cell degradation interval ∆D f c-degrade equal to 3%, as it has been proved to have the best performance, presented in Section 4.2.

The comparison of life improvement is listed in Table 3. It shows that the strategy with WOB of probability calculation that equals 0.4 • RUL max has reached the best performance. The durability of the fuel cell has been improved by 95.4%. However, the EMS with WOB of 0.5 can only improve the lifetime by 19.5%. This is because the FLC formulated based on the probability calculation with WOB = 0.5 • RUL max cannot take into account the high degradation state. Besides, the EMS with smaller WOBs (WOB = 0.1 • RUL max , WOB = 0.2 • RUL max ) are less satisfied due to the uncertainty in the prognostics results. Once the uncertainty is accumulated to a high level, the refined FLC cannot generate the best results. Once again, the hourly consumed hydrogen are compared. The strategy that reached the best lifetime improvement can also avoid high hydrogen consumption. According to the above analysis, when selecting the parameters, i.e. the prognostics occurrence frequency and the WOB of probability calculation, to define the online prognostics-based health management strategy, it is important to improve the prediction performance of the prognostics method and to define the necessary points for reconfiguration according to the characteristics of the system.

Conclusion

  AbbreviationsEMS energy management strategy FLC fuzzy logic controller GA genetic algorithm HEV hybrid electric vehicle MF membership function PDM prognostics-enabled decision-making PEM proton exchange membrane PHM prognostics and health management S OC state of charge S OH state of health WOB width of bin Symbols i bat battery current

where b 1 ,

 1 b 2 , b 3 and b 4 are the model parameters, b 1 and b 3 52 determine the initial state and b 2 and b 4 define the degradation 53 rate.

Figure 1

 1 Figure 1 shows an example of the online estimation and prognostics result of the fuel cell degradation at prediction time t λ = 500th hour. The fuel cell is assumed to reach its EOL at a predefined threshold (D f c-degrade = 10%). The predicted RUL is [84 143 284] hours with a confidence interval (CI) of [5% 50% 95%] whereas the true RUL is 159 hours.

60••

  Measurement uncertainties: All measurements are sub-61 ject to uncertainties, which is a quantification of the doubt 62 about the measurement results. Noisy measurements may 63 due to electrical, mechanical, or even thermal condition variations. External uncertainties: It contains the uncertainties coming from the operation conditions and environment. During online operation, the operation conditions are dynamic and random and the future loading conditions and environment always remain unknown. It is another inevitable source of uncertainties. When it comes to the particle filtering prognostics, the uncertainties of the learning phase is provided by all the abovementioned uncertainties and can be indicated by the distribution of the particles. Then, the uncertainties of the prediction phase account for all the previously quantified uncertainties, which are used to predict the future states and their uncertainties, as well as the RULs and their uncertainties. The future states and their uncertainties are calculated by propagation using the prediction model, while the calculation of the RULs and their uncertainties depend also on the threshold function, which is used to indicate the EOL.

Figure 3 :

 3 Figure 3: Structure of the studied fuel cell HEV

93. 2 . 20 The 34 1. 35 2.

 2203435 Energy management based on GA-optimized FLC 10 EMS is a strategy to be designed in hybrid vehicles, which 11 defines the amount of power that must be produced by differ-12 ent power sources. The objectives of EMS could be consump-13 tion minimization, power source degradation mitigation, driv-14 ing performance improvement, etc. Sometimes, these objec-15 tives may be conflicted with each other. In this study, the ob-16 jective of the proposed EMS controller is to find a trade-off be-17 tween increasing the system lifetime and saving the hydrogen baseline strategy is a fuzzy rule-based strategy as it can 21 easily be implemented in real time without huge computation 22 burdens. The input of the FLC is the SOC of the battery and 23 the output is the reference current of the fuel cell. The FLC is 24 designed by considering the following objectives: 1. The range of SOC of the battery should be limited to 26 ensure that the battery pack can provide enough power 27 for transient power demand and at the same time, remain 28 enough capability for regenerative braking; 29 2. The fuel cell is preferred to work around its highest effi-30 ciency point, P nom , to save hydrogen consumption; 31In this respect, the SOC of the battery is controlled to 75% as 32 its optimal operation point and the corresponding output is the 33 optimal i f c . The control rules are written as: If the SOC is low, then the fuel cell runs at i f c-high ; If the SOC is medium, then the fuel cell runs at i f c-med ; 36 3. If the SOC is high, then the fuel cell runs at i f c-low . 37 Details of the FLC implementation could be found in the au-38 thors' previous work[START_REF] Yue | Health-conscious energy management for fuel cell hybrid electric vehicles based on prognostics-enabled decisionmaking[END_REF].

40

  Although the FLC is easy to implement in real time, it can hardly reach the optimal solution if the MFs are designed with-42 out combining any optimization procedure[START_REF] Yue | Review on health-conscious energy management strategies for fuel cell hybrid electric vehicles: Degradation models and strategies[END_REF]. To encounter 43 with this problem, in this study, the parameters of the MFs of 44 the output i f c are tuned by offline GA optimization method.

Figure 4 :Figure 5 :

 45 Figure 4: Optimization results under different degradation states (V f c-degrade = 0%/3%/6%/9% • V init from left to right)

Figure 6 :

 6 Figure 6: Probability calculation based on RUL uncertainty

  through a decision fusion method based on probability calculation. The implementation of decision fusion is in light of the Dempster-Shafer theory. The Dempster-Shafer theory is a commonly used mathematical decision fusion method introduced by Dempster and Shafer [32, 33]. Different from the Bayesian method, Dempster-Shafer theory justifies the use of combined evidence from different measures. To combine Dempster-Shafer theory to our problem, the use of Dempster-Shafer theory to fuse offline optimization results is firstly justified. Numbering the four optimized FLCs in Figure 4 as FLC1, FLC2, FLC3, and FLC4, one may know that FLC1 is optimized as the optimal solution when D f c-degrade = 0, however, we don't know if it can also generate good results when D f c-degrade = 3%. If the range of the MFs of FLC1 contains the range of the MFs of FLC2, it means that FLC1 can work well even when D f c-degrade = 3%. To include this situation, when generating the online used FLC, we need to consider not only the FLC parameters for each degradation state but also the relationship between the different FLCs.

  Figure 7: Optimized 'high' I f c MFs under different degradation cases

  The simulation is run on a repeated WLTC class 2 driving cycle up to thousands of hours. At first, two scenarios have been proposed to justify the effectiveness of the proposed strategy. In the first test, the FLC optimized under no degradation state (D f c-degrade = 0%) is used all along with the simulation until the fuel cell reaches its EOL. Then in the second test, the parameters of the online used FLC are refined according to the prognostics results considering uncertainty. The WOB is defined as 0.3 •RUL max and the prognostics is triggered once D f c-degrade falls by 1%. The simulation stops when the fuel cell reaches the same EOL value. The evolution of the power distribution is plotted in Figure 8, in which the fuel cell operates smoothly and the battery provides all the dynamics in the demanded power.

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Power distribution between fuel cell and battery

  42

1

  This paper has contributed to developing an enhanced on-2 line prognostics-based health management strategy consider-3 ing prognostics uncertainty for fuel cell HEVs. The strategy 4 is realised based on a GA-optimized FLC. A PDM process has 5 been proposed, which refines the parameters of the FLC us-6 ing a decision fusion method based on RUL probability cal-7 culation. Results showed that by taking into consideration the 8 health states, the proposed prognostics-based health manage-9 ment strategy has effectively stabilized the fuel cell operation 10 and maintained the battery SOC, which helped to mitigate the 11 degradation of the power sources by up to 95% without sacri-12 ficing the hydrogen consumption. Moreover, variations on the 13 prognostics occurrence frequency and the calculation of proba-14 bility have also been examined and results showed that proper 15 parameters of the strategy could help to improve the EMS per-16 formance. For the perspectives, further researches on hybrid 17 system dynamic degradation monitoring and experimental val-18 idation are expected in the future.

Acknowledgement 20 This

 20 work was supported by the French regional project 21 PHyTie [grant number 2016Y-04574] and the EIPHI Graduate 22 School [grant number ANR-17-EURE-0002].

  ), the distribution of the predicted RULs is represented by 42 the distribution of the particles when reaching failure threshold, 43 as shown in Figure2.In fact, implementing online prognostics needs to contend 46 with multiple sources of uncertainties. Generally, it is impos-47 sible to eliminate all the uncertainties no matter what kind of 48 prognostics method is used. The sources of prognostics uncer-49 tainties are discussed in details: It is due to the lack of an accurate 51

		52
									Measured stack voltage
		50							PF prediction Start prediction Confidence Interval
	Stack voltage(V)	46 48	Threshold
		44
		42
		0	100	200	300	400		500	600	700	800
							Time(hour)
					Figure 1: Fuel cell prognostics result
	to (744
		350
		300
	Number of particles	100 150 200 250
		50
		0
		-100	0	100	200	300	400	500	600	700	800
							RUL(hour)
					Figure 2: Predicted RUL distribution
	2.3. Prognostics uncertainties				45

50

• Model uncertainties:

Table 1 :

 1 Probability calculation under different widths of bins (WOB)

	WOB = 0.5 • RUL max			P1 0.9879					P2 0.0121
	WOB = 0.4 • RUL max			P1 0.3208			P2 0.6727		P3 0.0065
	WOB = 0.3 • RUL max		P1 0			P2 0.9407			P3 0.0558		P4 0.0035
	WOB = 0.2 • RUL max		P1 0.3208	P2 0.6199	P3 0.0528	P4 0.0050	P5 0.0015
	WOB = 0.1 • RUL max	P1 0	P2 0.3208	P3 0.4158	P4 0.2041	P5 0.0473	P6 0.0055	P7 0.0030	P8 0.0020	P9 0.0010	P10 0.0005

  44 rence, more scenarios with prognostics happening at different 45 frequencies are designed, while the WOB remains at the same.46 The prognostics process is triggered by monitoring the fuel cell 47 degradation with the degradation interval ∆D f c-degrade equal 48

				51
							EMS without prognostics
				50			prognostics-based EMS
		Stack voltage(V)	46 47 48 49
				45
				44
				0	200	400	600	800	1000
						Time(hour)
				Figure 11: Evolution of the measured fuel cell stack voltage
				1
			0.8
		Battery SOC(V)	0.4 0.6
			0.2			EMS without prognostics
							prognostics-based EMS
				0
				0	200	400	600	800	1000
						Time(hour)
				Figure 12: Evolution of the measured battery SOC
	1	to 1%, 2%, 3%, 4% and 5%, while the WOB remaining as
		0.3 • RUL max .
	8
	9	it doesn't mean that frequent prognostics can lead to better
	10	performance of the fuel cell durability. Frequent prognostics
	11	(∆D f c = 1%, 2%) have even worse performance than less fre-
	12	quent prognostics (∆D f c = 3%, 4%, 5%). This is because when
	13	the prognostics horizon is large, the prognostics accuracy can-
	14	not be guaranteed due to the lack of learning data. When the
		uncertainty accumulates, the probability calculation will also
	16	be influenced so that one cannot reach the optimal parameters
	17	of the online used FLC. However, when the frequency becomes
	18	lower, i.e., fewer prognostics are implemented, the performance
	19	becomes worse, as well. This is due to the reason that the degra-	4.2. Discussion on prognostics occurrence frequency
		dation of the fuel cell hasn't been fully considered in the strat-

43

To further investigate the influence of prognostics occur-

2

The comparison of life improvement is listed in Table

2

. 3 Compared to the EMS without prognostics, the proposed 4 prognostics-based health management strategy improves the 5 lifetime of the fuel cell with a minimum value of 42.6% and 6 the strategy with ∆D f c = 3% has reached the longest lifetime, 7 84.6% longer than the EMS without prognostics. Obviously, 20 egy. If we compare the hourly hydrogen consumption, the result 21 does not vary a lot as our objective function is defined to find a 22 trade-off between degradation mitigation and saving consump-23 tion. The EMS without prognostics consumes less because the 24 SOC of the battery drops to a lower value.

Table 2 :

 2 Lifetime improvement comparison with different prognostics occurrence frequency

	Prognostics	frequency	Lifetime im-	m H 2 (g/hour)
	(WOB= 0.3 • RUL max )	provement
	EMS without prognostics	-	41.2
	∆Dfc=1%		25.5%	42.0
	∆Dfc=2%		58.2%	41.9
	∆Dfc=3%		84.6%	41.9
	∆Dfc=4%		80.0%	41.9
	∆Dfc=5%		54.7%	41.7
	4.3. Discussion on WOB of probability calculation

Table 3 :

 3 Lifetime improvement comparison with different WOBs

	WOB (∆Dfc=3%)	Lifetime im-	m H 2 (g/hour)
		provement
	EMS without prognostics	-	41.2
	WOB = 0.1 • RUL max	61.7%	41.8
	WOB = 0.2 • RUL max	60.8%	41.8
	WOB = 0.3 • RUL max	87.3%	41.9
	WOB = 0.4 • RUL max	95.4%	41.9
	WOB = 0.5 • RUL max	19.5%	41.6


Pierre-Sylvain Marcheix
email: psmarcheix@orange.fr

Céline Cuenca

Guillaume Vergnenegre

Christian Mabit

Jérémy Hardy

Jean-Louis Charissoux

Factors influencing the mid-term radiological and functional outcomes of 41 post-fracture bipolar radial head arthroplasty cases at a mean follow-up of 87 months

Keywords: IV ; systematic retrospective analysis radial head fracture, cemented implant, bipolar, functional evaluation, X-ray evaluation, mid-term outcomes

Introduction

Fractures of the radial head (RH) make up one-third of all elbow fractures [START_REF] Vannabouathong | Interventions for displaced radial head fractures: network meta-analysis of randomized trials[END_REF] and mainly affected a younger population with a mean age of 45 years [START_REF] Van Riet | Associated injuries complicating radial head fractures: a demographic study[END_REF][START_REF] Vannabouathong | Radial head arthroplasty: fixed-stem implants are not all equal-a systematic review and meta-analysis[END_REF]. In instances where a comminuted RH fracture cannot be adequately fixed, arthroplasty is recommended [START_REF] Harrington | The functional outcome with metallic radial head implants in the treatment of unstable elbow fractures: a long-term review[END_REF][START_REF] Chen | Elbow Fractures with Instability: Evaluation and Treatment Strategies[END_REF][START_REF] Monica | Radial head arthroplasty[END_REF][START_REF] Laun | Primary cemented bipolar radial head prostheses for acute elbow injuries with comminuted radial head fractures: midterm results of 37 patients[END_REF] to ensure elbow stability and healing of associated ligament injuries [START_REF] Vannabouathong | Radial head arthroplasty: fixed-stem implants are not all equal-a systematic review and meta-analysis[END_REF][START_REF] Laun | Primary cemented bipolar radial head prostheses for acute elbow injuries with comminuted radial head fractures: midterm results of 37 patients[END_REF][START_REF] Beingessner | The effect of radial head excision and arthroplasty on elbow kinematics and stability[END_REF]. In fact, RH fractures rarely occur in isolation; in 75% of cases, they occur simultaneously with other elbow fractures or injuries to periarticular tissues [START_REF] Van Riet | Associated injuries complicating radial head fractures: a demographic study[END_REF][START_REF] Laun | Primary cemented bipolar radial head prostheses for acute elbow injuries with comminuted radial head fractures: midterm results of 37 patients[END_REF][START_REF] King | Metallic radial head arthroplasty improves valgus stability of the elbow[END_REF]. At this point, the surgeon must decide which type of implant to use. There are two competing biomechanical theories that come into play when choosing the implant. Monoblock implants stabilize the damaged elbow the best [START_REF] Chanlalit | Stress shielding around radial head prostheses[END_REF][START_REF] Chanlalit | Radiocapitellar stability: the effect of soft tissue integrity on bipolar versus monopolar radial head prostheses[END_REF][START_REF] Schneeberger | Coronoid process and radial head as posterolateral rotatory stabilizers of the elbow[END_REF]. On the other hand, bipolar implants-thanks to their mobile radial cups-distribute the loads better, thereby reducing high joint stresses [START_REF] Judet | A floating prosthesis for radialhead fractures[END_REF][START_REF] Sershon | Radial head replacement with a bipolar system: an average 10-year follow-up[END_REF]. Thus bipolar implants stabilize the joint while reducing the excessive pressure on the lateral condyle, and therefore the risks of arthritic degeneration in the humeroulnar and humeroradial joints [START_REF] Chytas | Capitellar erosion after radial head arthroplasty: A comparative biomechanical study of operated radial head fractures on cadaveric specimens[END_REF].

We hypothesized that the radiological and functional outcomes of bipolar mobile cup RH arthroplasty is not influenced by the length of follow-up but by the correct positioning of the implant intraoperatively and by the presence of associated bone or ligament injuries. The primary endpoint of this study was the functional outcomes at the longest follow-up represented by the MEPS and DASH scores relative to the radiological position of the RH arthroplasty. The secondary endpoints at the longest follow-up were:

-Functional outcomes (MEPS and DASH scores) and grip strength in the operated side relative to the contralateral side -Patient satisfaction -Pain in the operated elbow evaluated on a visual analog scale (VAS)

Materials and Methods

Study overview

This was a multisurgeon retrospective study of cases done at our orthopedics and trauma surgery department between May 1998 and December 2016. The inclusion criteria were primary RH arthroplasty in the context of acute elbow trauma, at least 1 year of follow-up, patient at least 18 years of age, patients available for final clinical and radiographic assessment. The exclusion criteria were secondary RH arthroplasty (more than 15 days after the initial fracture event) and implant inserted in the context of total elbow replacement. All the patients included in our statistical analysis were reviewed in person by a single examiner for the final assessment. This study was approved by our institutional review board on January 30, 2018 (number 243-2018-09).

Surgical procedure and postoperative course

Several surgeons were involved in these cases. The implant was a Judet-type RH prosthesis with floating cup (CRF®, Tornier, France) and a long, cemented stem. The procedure was done under general or regional anesthesia. The patient's position was supine with the operated arm on an arm board and a tourniquet in place. The surgical approach was selected based on the presence of associated injuries. It could be either posterolateral (Cadenat approach), lateral (Kaplan approach), posterior when ulnar fracture fixation was required (Boyd approach) or a combination of approaches (lateral + posteroulnar, lateral + medial).

The postoperative care did not follow a set protocol. The need for postoperative immobilization and its type was based on the associated injuries encountered. The same goes for the need for postoperative rehabilitation, which could either be immediate or delayed, either by a physiotherapist or self-directed.

Patients

Between May 1998 and December 2016, 92 arthroplasty procedures were done with the CRF implant in 91 patients immediately following a RH fracture. There were 37 men (41%) and 54 women (59%). The left side was involved in 53 cases (58%) and the right side in 39 (42%). The fracture was in the dominant arm in 34 patients. The mean age at the time of RH arthroplasty was 53 years . At the time of the injury, 47 patients were considered as sedentary or retired (52%), 29 did light manual labor (32%) and 15 did heavy manual labor (16%).

Fracture

All 92 RH fractures were classified as Mason type 3 according to the Mason classification as modified by Broberg-Morrey [START_REF] Iannuzzi | In brief: the Mason classification of radial head fractures[END_REF]. Among these 92 fractures, 29 were considered as isolated (31.5%). These isolated fractures were treated by resection-arthroplasty, with no additional procedures. Forty-one fractures (44%) were associated with ligament or other soft tissue injuries. Eighteen fracture (20%) had other bones fractured such as the olecranon, ulnar metaphysis or shaft, humeral condyles or distal humerus, while four fractures (4%) had combined ligament and bone damage. In these cases, resection-arthroplasty was combined with:

suture repair of the lateral collateral (18 cases) and/or medial collateral ligaments (3 cases)

anterior capsulorraphy by reinsertion on the coronoid (12 cases)

fixation of coronoid fracture (7 cases)

fixation of ulna fracture (19 cases)

fixation of lateral condyle fracture (2 cases)

fixation of proximal radius fracture (1 case).

Only the patients who were reviewed in person for the final assessment were included in our statistical analysis. Our study design is summarized in Figure 1. All the patients were evaluated at the last follow-up visit by an independent surgeon. The radiological analysis was done by an independent radiologist.

Clinical assessment, satisfaction, and functional outcome scores

Pain at the final assessment was evaluated using a visual analog scale (VAS) from 0 to 10.

The patient's satisfaction was determined by asking the patient whether they were "very satisfied", "satisfied", "somewhat satisfied", "not satisfied" or "dissatisfied" with the procedure. The elbow's range of motion was measured with a goniometer. The elbow's mobility was considered as functional when flexion was greater than 130° and the extension deficit less than 30°. Elbow stability was evaluated based on subjective criteria (patient's feeling of instability) and objective criteria (examiner looking for posterolateral instability through a pivot shift) [START_REF] O'driscoll | Posterolateral rotatory instability of the elbow[END_REF]. Maximum grip strength was evaluated over three successive trial using a Jamar dynamometer. The ratio between the operated and non-operated side was calculated. The functional outcome scores used were the Mayo Clinic Performance score (MEPS) [START_REF] Cusick | Accuracy and reliability of the Mayo Elbow Performance Score[END_REF] and the Disability of Arm-Shoulder-Hand (DASH) [START_REF] Hudak | Development of an upper extremity outcome measure: the DASH (disabilities of the arm, shoulder and hand) [corrected]. The Upper Extremity Collaborative Group (UECG)[END_REF]. The MEPS was considered excellent if it was between 90 and 100, good if between 75 and 89, average if between 60 and 74, and poor if below 60.

Radiological outcomes

The radiological workup consisted of an AP view of the elbow in extension and a lateral view of the elbow in 90° flexion. These x-rays were used to look for radiolucent lines suggestive of periprosthetic loosening as described by Popovic et al. [START_REF] Popovic | Midterm results with a bipolar radial head prosthesis: radiographic evidence of loosening at the bone-cement interface[END_REF] (Figure 2a), suspended implant as evidenced by asymmetry in the humeroulnar joint (river delta sign) [START_REF] Dou | Prosthesis replacement in Mason III radial head fractures: A meta-analysis[END_REF] (Figure 2b), decentering of the radial cup (radial shaft axis does not pass through center of capitellum -equal to Storen's line) (Figure 2c), arthritic degeneration of the humeroulnar joint according to the Broberg-Morrey classification and graded as present or absent in our study [START_REF] Broberg | Results of treatment of fracture-dislocations of the elbow[END_REF] (Figure 2d), demineralization of the lateral condyle graded as present or absent in our study [START_REF] Lamas | Comminuted radial head fractures treated with pyrocarbon prosthetic replacement[END_REF] and heterotopic ossification graded as present or absent in our study [START_REF] Brooker | Ectopic ossification following total hip replacement. Incidence and a method of classification[END_REF] (Figure 2d).

Statistical analysis

Results of the quantitative variables are given as mean ± standard deviation, minimum, maximum, and median values; those of qualitative variables are expressed as counts and percentages. For the qualitative variables, the Chi 2 test was used according to the Monté Carlo method to estimate the value of a small p. To determine if two quantitative variables were related, a correlation analysis by linear regression was done and the statistical significance estimated with the Pearson test.

To compare quantitative variables, the Mann Whitney was used according to the Monté Carlo' method to estimate the values of a small p. To compare groups based on quantitative variables, the non-parametric Mann-Whitney or Kruskal Wallis test was used. In every case, the value of the small p was estimated with a Monté Carlo method with 5000 iterations and a 95% confidence interval. Significance threshold was p < 0.05.

Results

The population for analysis consisted of 41 fractures in 41 patients who were reviewed in person with a mean follow-up of 86.9 months (range, 12-228). There were 14 men (34%) and 27 women (66%). Their mean age at the time of arthroplasty was 59 years (24-81).

Among these 41 fractures, 11 were considered as isolated (27%), 20 were associated with ligament injuries (49%) and 10 were associated with both ligament and bone injuries (24%).

Complications and surgical revisions

There were 10 postoperative complications (29%). The characteristics used for comparison are shown in Table 1. There was one case of posterior interosseous nerve palsy that developed immediately postoperative; it resolved in 10 months.

Clinical outcomes at final assessment (Table 2)

The mean MEPS was 88.7 (min 61, max 100). The mean DASH score was 18.7 (min 0, max 55). Three patients had to change jobs (7%). The grip strength in the operated arm was 87%

(min 20%, max 189%) of the contralateral arm. The mean pain on VAS was 0.78 (min 0, max 5). Twenty-six patients (63%) had an average residual extension deficit of 13° (min 0, max 60°), although only three patients (7%) had non-functional range of motion. Seven patients (17%) had incomplete pronation-supination arc: one (2%) had no pronation-supination at all due to concurrent fracture of the distal ulna and radius; two patients (5%) had isolated supination deficit (50° and 45° supination, respectively), four patients (10%) had both pronation and supination deficits. One patient reported a feeling of elbow instability, but this was no associated with objective findings. No patients had objective signs of elbow instability.

Radiological outcomes at final assessment

The radiological results are report in Table 2. Figure 3 shows the radiological findings in a patient in whom the implant was positioned too high, with decentering of the radial cup, demineralization of the lateral condyle and humeroulnar osteoarthritis.

Clinical outcome as a function of follow-up time

Radiological findings relative to follow-up time

Radiolucent lines suggestive of loosening were found around 10 implants (24%). The mean follow-up for these implants was 80 months (min 17, max 218). The 31 implants with no radiolucent lines had a mean follow-up of 82 months (min 12, max 228). There was no statistical relationship between the appearance of periprosthetic radiolucent lines and the time elapsed after RH arthroplasty (p = 0.8). 3,4)

Impact of associated injuries at time of RH fracture (Tables

The presence of associated ligament or bone injuries at the time of the RH fracture significantly increased the risk of periprosthetic radiolucent lines, radiological deterioration of the humeroulnar joint and decentering of the radial cup (p = 0.03, p < 0.0001 and p = 0.03, respectively). There was no statistical relationship between the presence of associated ligament or bone injuries and the functional outcomes (MEPS, DASH, and grip strength) (p = 0.3, p = 0.3, p = 0.2, respectively)

Functional outcomes relative to radiological findings

The radiological findings of cup decentering, periprosthetic calcifications and lateral condyle demineralization had no impact on the functional outcomes at the final assessment, nor the grip strength relative to the contralateral side (Table 5).

A suspended implant was associated with significantly worse functional outcomes (MEPS, DASH and grip strength, respectively: p = 0.04, 0.05, 0.04) (Table 5). In 86% of cases, a suspended implant caused an extension deficit in the elbow (Table 6). However, our statistical analysis did not reveal a relationship between a suspended RH implant and extension deficit. There was also no statistical relationship between an implant being suspended and the appearance of periprosthetic radiolucent lines or arthritic degeneration in the humeroulnar joint (Table 6).

Discussion

Our study found a strong relationship between the functional outcomes of mobile cup bipolar RH arthroplasty and the implant's positioning on radiographs. To improve the functional outcomes and reduce the risk of radiological degeneration, good implant positioning is critical [START_REF] Van Riet | Associated injuries complicating radial head fractures: a demographic study[END_REF][START_REF] Allavena | Outcomes of bipolar radial head prosthesis to treat complex radial head fractures in 22 patients with a mean follow-up of 50 months[END_REF][START_REF] Jones | Radial Head Arthroplasty Understuffing in the Setting of Longitudinal Radioulnar Dissociation: A Case Report[END_REF][START_REF] Van Glabbeek | Detrimental effects of overstuffing or understuffing with a radial head replacement in the medial collateral-ligament deficient elbow[END_REF]. The aim is to reduce the initial length of the radius. If the radius is too short, there is a risk of secondary ulnar-carpal impingement [START_REF] Jones | Radial Head Arthroplasty Understuffing in the Setting of Longitudinal Radioulnar Dissociation: A Case Report[END_REF]. Conversely, if the radius is too long, then there is a risk of causing arthritic degeneration of the lateral condyle and the humeroulnar joint [START_REF] Van Riet | Associated injuries complicating radial head fractures: a demographic study[END_REF][START_REF] Van Riet | Failure of metal radial head replacement[END_REF]. There should be no contact between the implant and the lateral condyle, both in flexion and extension [START_REF] Allavena | Outcomes of bipolar radial head prosthesis to treat complex radial head fractures in 22 patients with a mean follow-up of 50 months[END_REF]. It is also important not to overstuff the radial cup to limit the space taken up by the implanted cup. This size must be either the same as or slightly less than the original radial cup.

Our study also confirms that functional outcomes are stable in the medium term after performing RH arthroplasty with CRF® bipolar mobile cup for an acute elbow fracture. At a mean follow-up of 80 months, we found a mean MEPS of 89 and 78% good and excellent results. These functional outcomes are comparable to other published studies with the CRF® implant. The mean MEPS ranges from 87 to 96 in published studies with mostly good or excellent results (Table 7). This is confirmed by the DASH score, which was 18.7 on average in our study and is similar to the score reported by Laum et al. at 67 months' follow-up [START_REF] Laun | Primary cemented bipolar radial head prostheses for acute elbow injuries with comminuted radial head fractures: midterm results of 37 patients[END_REF].

This type of arthroplasty also helps patients regain satisfactory grip strength, which was 87% of the contralateral side. This result within the range of that reported in other studies with the same type of implant (Table 7). Thus using a CRF® bipolar radial cup with cemented stem restores good function in the operated limb, while allowing associated ligament injuries in the elbow and forearm to heal [START_REF] Laun | Primary cemented bipolar radial head prostheses for acute elbow injuries with comminuted radial head fractures: midterm results of 37 patients[END_REF][START_REF] Chytas | Capitellar erosion after radial head arthroplasty: A comparative biomechanical study of operated radial head fractures on cadaveric specimens[END_REF][START_REF] Antoni | Comparison of bipolar and monopolar radial head prostheses in elbow fracture-dislocation[END_REF].

When we looked at the radiographs made during the final assessment, 24% had radiolucent lines around the stem. This incidence ranges from 0% to 53% in other published studies [START_REF] Popovic | Midterm results with a bipolar radial head prosthesis: radiographic evidence of loosening at the bone-cement interface[END_REF].

Chanlalit et al. reported a high loosening rate (65%) independent of the type of fixation (cemented vs press-fit) for the RH implant and the stem design, although the loosening had no major clinical consequences and did not get worse over time [START_REF] Chanlalit | Stress shielding around radial head prostheses[END_REF]. In their meta-analysis, Delclaux et al. concluded that the percentage of radiolucent lines was the same whether the stem was cemented or not [START_REF] Delclaux | Complications of radial head prostheses[END_REF]. In his meta-analysis of revision RH arthroplasty, Kachooei found no superiority of one implant over the other in terms of longevity and revision rate [START_REF] Kachooei | The Rate of Radial Head Prosthesis Removal or Revision: A Systematic Review and Meta-Analysis[END_REF]. Moreover, the literature, like our study, does not find a relationship between the appearance of radiolucent lines around the stem and alteration in functional outcomes at the final assessment [START_REF] Chanlalit | Stress shielding around radial head prostheses[END_REF][START_REF] Berschback | Clinical and radiographic comparisons of two different radial head implant designs[END_REF].

Arthritic degeneration of the capitellum and demineralization of the lateral condyle has frequently been reported in the literature following RH arthroplasty [START_REF] Flinkkila | Short-to mid-term results of metallic press-fit radial head arthroplasty in unstable injuries of the elbow[END_REF]. Our study found that 54% of elbows had radiographic evidence of lateral condyle demineralization. However, a bipolar mobile cup appears to be better able to adapt its articulation with the lateral condyle, which helps to reduce the excess pressure over the lateral condyle and thereby to reduce the likelihood of arthritic degeneration [START_REF] Chytas | Capitellar erosion after radial head arthroplasty: A comparative biomechanical study of operated radial head fractures on cadaveric specimens[END_REF][START_REF] Antoni | Comparison of bipolar and monopolar radial head prostheses in elbow fracture-dislocation[END_REF][START_REF] Flinkkila | Short-to mid-term results of metallic press-fit radial head arthroplasty in unstable injuries of the elbow[END_REF]. Some authors advocate using a cementless stem to allow the implant to adapt its position relative to the lateral condyle, which would reduce the pressure on the lateral condyle. However, this is theoretical since many surgeons do not recommend using the bipolar implant in a press-fit configuration and instead recommend using a cemented stem [START_REF] Flinkkila | Short-to mid-term results of metallic press-fit radial head arthroplasty in unstable injuries of the elbow[END_REF][START_REF] Laumonerie | Mid-term outcomes of 77 modular radial head prostheses[END_REF]. When it comes to choosing between unipolar and bipolar implants, Chytas et al. concluded that unipolar metal RH implants were more likely to cause capitellum degradation and thus to trigger stiffness in the operated elbow [START_REF] Chytas | Capitellar erosion after radial head arthroplasty: A comparative biomechanical study of operated radial head fractures on cadaveric specimens[END_REF]. However, the clinical study by Marsh et al. of 33 patients who received unipolar implants found that only 22% of them had lateral condyle demineralization at a mean follow-up of 8 years, which is lower than what we found with a bipolar implant [START_REF] Marsh | Radial Head Fractures Treated with Modular Metallic Radial Head Replacement: Outcomes at a Mean Follow-up of Eight Years[END_REF]. Our study did not allow us to establish a link at the final assessment between lateral condyle demineralization and worse functional outcomes in our patients. Several other studies confirm this observation, no matter which implant was used [START_REF] Popovic | Midterm results with a bipolar radial head prosthesis: radiographic evidence of loosening at the bone-cement interface[END_REF][START_REF] Burkhart | Mid-to long-term results after bipolar radial head arthroplasty[END_REF].

As for arthritic degeneration of the humeroulnar joint, our study found a 63% incidence of osteoarthritis at a mean follow-up of 80 months. This is comparable to the 64% incidence found by Moghaddam et al, who used a monoblock modular implant [START_REF] Moghaddam | Outcome of Radial Head Arthroplasty in Comminuted Radial Head Fractures: Short and Midterm Results[END_REF]. Based on the current literature, this implant also does not appear to affect the arthritic progression of the humeroulnar joint over the long term [START_REF] Vannabouathong | Radial head arthroplasty: fixed-stem implants are not all equal-a systematic review and meta-analysis[END_REF][START_REF] Moghaddam | Outcome of Radial Head Arthroplasty in Comminuted Radial Head Fractures: Short and Midterm Results[END_REF].

The treatment of comminuted RH fractures requires an extensive preoperative assessment to make sure none of the associated injuries are missed. A preoperative CT scan is often needed to classify the RH fracture and to look for a coronoid fracture [START_REF] Laun | Primary cemented bipolar radial head prostheses for acute elbow injuries with comminuted radial head fractures: midterm results of 37 patients[END_REF]. Moreover, it is essential to look for associated ligament damage in the context of a terrible triad elbow injury or Essex-Lopresti injury. In fact, surgical repair of these lesions is essential to reducing the risk of poor radiological and functional outcomes after RH arthroplasty [START_REF] Vannabouathong | Radial head arthroplasty: fixed-stem implants are not all equal-a systematic review and meta-analysis[END_REF]. According to Vannabouathong et al., the type of implant used is of secondary importance; the priority is making the correct diagnosis and treating the associated ligament injuries properly [START_REF] Vannabouathong | Radial head arthroplasty: fixed-stem implants are not all equal-a systematic review and meta-analysis[END_REF].

Consequently, no published studies has been able to demonstrate that one implant is superior to the others [START_REF] Monica | Radial head arthroplasty[END_REF][START_REF] Broberg | Results of treatment of fracture-dislocations of the elbow[END_REF][START_REF] Lamas | Comminuted radial head fractures treated with pyrocarbon prosthetic replacement[END_REF][START_REF] Brooker | Ectopic ossification following total hip replacement. Incidence and a method of classification[END_REF][START_REF] Allavena | Outcomes of bipolar radial head prosthesis to treat complex radial head fractures in 22 patients with a mean follow-up of 50 months[END_REF][START_REF] Jones | Radial Head Arthroplasty Understuffing in the Setting of Longitudinal Radioulnar Dissociation: A Case Report[END_REF][START_REF] Van Glabbeek | Detrimental effects of overstuffing or understuffing with a radial head replacement in the medial collateral-ligament deficient elbow[END_REF]. As for the stability of the operated elbow, Yian et al.

showed in an in vitro study that bipolar implants restore elbow stability as well as monoblock implants, while providing better joint congruency and thereby theoretically reducing the risk of implant subluxation [START_REF] Yian | Malpositioning of radial head prostheses: an in vitro study[END_REF]. Pomianowski et al. also showed that bipolar implants were as effective against valgus instability as monoblock implants [START_REF] Pomianowski | Contribution of monoblock and bipolar radial head prostheses to valgus stability of the elbow[END_REF]. Several other studies have corroborated this observation [START_REF] Van Riet | Failure of metal radial head replacement[END_REF][START_REF] Hartzler | Radial head reconstruction in elbow fracture-dislocation: monopolar or bipolar prosthesis?[END_REF]. However, other studies have contradicted these findings. Thus, certain authors feel that bipolar implants do not restore joint stability in a satisfactory manner. Monoblock implants are more effective at addressing an unstable elbow, for example in the context of a terrible triad injury [START_REF] Jones | Radial Head Arthroplasty Understuffing in the Setting of Longitudinal Radioulnar Dissociation: A Case Report[END_REF][START_REF] Kachooei | The Rate of Radial Head Prosthesis Removal or Revision: A Systematic Review and Meta-Analysis[END_REF][START_REF] Pomianowski | Contribution of monoblock and bipolar radial head prostheses to valgus stability of the elbow[END_REF].

Our study is one of the most important published studies on the CRF ® implant, given the number of patient and length of follow-up (Table 7). To limit bias in this study, the functional outcomes were determined at the review point by an experienced physician who was not involved in the patient's surgical care. The radiographs were evaluated by an independent radiologist. Nevertheless, this study has the typically limitations associated with a retrospective study. Many patients were lost to follow-up. The functional scores were only evaluated at the final assessment; since we did not have this data preoperatively, we could not assess how function changed over time.

Conclusion

Our study found very good results when doing RH arthroplasty with a CRF® bipolar mobile cup in the context of comminuted RH fractures. Because of the elbow stability conferred by this implant, the radiological and functional outcomes were highly satisfactory in the medium term. However, to achieve good results, it is essential to properly analyze the fracture and associated injuries preoperatively. It is also critical to find any associated ligament lesions intraoperatively in order to treat them. The radiological and functional outcomes of RH arthroplasty after RH fracture are correlated with appropriate care of these associated lesions and good intraoperative positioning of the implant. Complication without surgical revision:

Figures and Tables

-Type 1 complex regional pain syndrome

-

Figures 4 , 5

 45 Figures 4, 5 and 6 show the functional scores and grip strength relative to contralateral side

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Study flow chart Figure 2: Radiographic findings 2a: Radiological appearance of periprosthetic radiolucent line 2b: Radiological appearance of suspended implant (river delta sign) 2c: Postoperative radiological appearance of non-centered radial cup in an implant that is too high 2D: Radiological appearance of early humeroulnar osteoarthritis and periprosthetic calcifications Figure 3: In Patient 31, the implant was too high with off-centered radial cup, lateral condyle demineralization and humeroulnar osteoarthritis.

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: MEPS relative to follow-up after radial head arthroplasty Figure 5: DASH score relative to follow-up after radial head arthroplasty Figure 6: Grip strength relative to contralateral side as a function of follow-up after radial head arthroplasty

Figure 2

 2 Figure 2 :

Figure 3 : 4

 34 Figure 3 :

Figure 5 :

 5 Figure 5:

Figure 6 :

 6 Figure 6:

Transient posterior interosseous nerve palsy 5

 5

Table 1 :

 1 Complications documented at final assessment

Table 2 :

 2 Clinical and radiological outcomes at final assessment

Table 3 :

 3 Impact of associated lesions on radiological findings

Table 4 :

 4 Impact of associated lesions on clinical outcomes

Table 5 :

 5 Functional outcomes based on presence of implant loosening, periprosthetic calcifications, off-centered cup, lateral condyle demineralization or incorrect implant positioning

Table 6 :

 6 Effect of implant positioning on radiological findings and extension deficit

Table 7 :

 7 Comparison of our findings with those of other published studies on the CRF

	implant

Table 1 :

 1

	Complications, n (%)

Table 3 :

 3 RH: Isolated radial head fracture (stable elbow) RH+LL: Radial head fracture + ligament lesions (unstable elbow) RH+LL+BL: Radial head fracture + ligament lesions + other bone lesions (unstable elbow)

		Periprosthetic radiolucent line	Humeroulnar joint degeneration		Cup decentered
		No	Yes	p (Chi 2 ) = 0.03	Yes	No	p (Chi 2 ) = 0.0001	Yes	No	p (Chi 2 ) = 0.03
	RH	11	0		1	10		10	1
	RH+LL	15	5		15	4		10	9
	RH+LL+BL	5	5		10	0		5

Table 4 :

 4 RH: Isolated radial head fracture (stable elbow) RH+LL: Radial head fracture + ligament lesions (unstable elbow) RH+LL+BL: Radial head fracture + ligament lesions + other bone lesions (unstable elbow)

	Variable	Number	Mean	Standard Deviation	p (Kruskal-Wallis)	95% CI
	MEPS | RH	11	91.5	14.5
	MEPS | RH + LL	20	89.2	13.3	0.3	0.280; 0.304
	MEPS | RH + LL + BL	10	84.6	14.6
	DASH | RH	11	13.2	15.9
	DASH | RH + LL	20	19.7	16.6	0.3	0.250; 0.273
	DASH | RH + LL + BL	10	22.6	14.6
	Grip strength ratio | RH	11	1	0.3
	Grip strength ratio | RH + LL	20	0.8	0.3	0.2	0.141; 0.159
	Grip strength ratio | RH + LL + BL	10	0.9	0.2

committee of our institution

Funding: none

Author contributions: PSM, CC designed the study, collected data, did the statistical analysis, wrote the manuscript GV, JH, CM, JLC provided critical review of the manuscript

Conflict of interest:

The authors have no conflict of interest to disclose.

Clinical outcomes

Mean MEPS (min-max)

Excellent result (%)

Good result (%)

Mediocre result (%)

Poor result (%)

Patient satisfaction

Very satisfied (%)

Satisfied (%)

Mean VAS for pain (min-max)

Range of motion (°)

Flexion (min-max)