Romain Delannoy

Vincent Tognetti

Emmanuel Richaud

Experimental and theoretical insights on the thermal oxidation of epoxy-amine networks

Keywords:

This paper reports an investigation of the oxidation of three epoxy-amine networks obtained from DGEBA cured with three sorts of aliphatic hardeners ethylene diamine (EDA), diethylene triamine (DETA), and triethylene tetraamine (TETA). Oxidation was monitored by FTIR with an innovative discussion of CH 2 consumption. Methylene groups held by hardeners groups were shown to be more sensitive to oxidation, which was confirmed by a discussion of Bond Dissociation Energies (BDE) computed by a machine-learning approach.

Introduction

Epoxies are increasingly used in industrial applications with strong requirements in terms of durability such as the design of lightened composite parts for engines in aeronautic field or waterproofing of concrete substrates in civil engineering [START_REF] May | Epoxy resins. Chemistry and Technology[END_REF]. The lack of feedback on such materials together with the complexity of ageing conditions (and the difficulty to correlate them) with an accelerated ageing test makes necessary to predict their lifetime by a non-empirical approach based on the determination of ageing rates.

Epoxy ageing is already well documented in literature. It is known they can undergo a (reversible) physical ageing by water permeation [START_REF] Morel | Structure-water absorption relationships for amine-cured epoxy resins[END_REF][START_REF] Dang | Effects of curing program on mechanical behavior and water absorption of DGEBA/TETa epoxy network[END_REF][START_REF] Xiao | Water absorption and desorption in an epoxy resin with degradation[END_REF] and/or structural relaxation [START_REF] Montserrat | Structural relaxation in fully cured epoxy resins[END_REF], or (irreversible) chemical ageing by radio-oxidation, photo-oxidation, or thermal oxidation [START_REF] Djouani | Degradation of epoxy coatings under gamma irradiation[END_REF][START_REF] Zahra | Thermo-oxidative aging of epoxy coating systems[END_REF][START_REF] Longiéras | Degradation of epoxy resins under high energy electron beam irradiation: radio-oxidation[END_REF][START_REF] Celina | A perspective on the inherent oxidation sensitivity of epoxy materials[END_REF]. In any case, but more generally for oxidative ageing, the prediction of lifetime is based on the use of a kinetic model. Such models are easily implemented with a thorough knowledge of the chemical reactions involved in degradation. For example, complete modelling describing the reactivity of tertiary C-H groups has been proposed to study the degradation of polypropylene [START_REF] Bertin | Polypropylene degradation: theoretical and experimental investigations[END_REF], and more complex kinetic models describing the reactivity of allylic C-H groups were proposed for polyisoprene [START_REF] Colin | Kinetic modelling of the thermal oxidation of polyisoprene elastomers. part 1: unvulcanized unstabilized polyisoprene[END_REF] and polychloroprene [START_REF] Le Gac | Oxidation of unvulcanized, unstabilized polychloroprene: a kinetic study[END_REF].

The level of difficulty increases when it comes to understanding the ageing of epoxy-amine networks. The wide variety of epoxide prepolymers and hardeners that they are made of allows for an almost infinite number of mixtures differing by their viscosity, reactivity, and final properties, to suit and meet industrial requirements. Even in the simplest epoxy-diamine binary mixtures, the complex molecular structure presents many possible reactive groups, making the implementation of a representative kinetic model more intricate.

Previous works have already discussed on the nature of stable carbonyl products formed during oxidation to distinguish the nature of reactive sites involved in their formation [START_REF] Delozanne | Thermal oxidation of aromatic epoxy-diamine networks[END_REF]. However, several difficulties arise mainly due to the complex overlap of FTIR signals attributed to stable products in a relatively narrow wavenumber range, combined with the speculative aspect of radical mechanism in solid structures [START_REF] Celina | Carbonyl identification and quantification uncertainties for oxidative polymer degradation[END_REF].

The aim of the present study is to illustrate a possible solution to identify reactive sites in epoxy-amine networks by using the combination of two complementary tools:

-An identification of reactive groups based on a thorough analysis of C-H stretching wavenumber range in FTIR. -A discussion on the Bond Dissociation Energies of reactive groups (as bonds weaknesses may be the "driving force" of radical oxidation) with a theoretical approach based on a convenient machine-learning tool, which will first be validated by computational QC (quantum chemistry) calculations.

Materials and methods

Epoxy-amine thermoset materials

Epoxy-amine systems were prepared by blending DGEBA (D.E.R. 332, CAS 1675-54-35 supplied by Sigma Aldrich, epoxide equivalent weight EEW 171-175 g/eq) with one of three aliphatic amine hardeners: ethylene diamine EDA (CAS 107-15-3, Sigma Aldrich, amine hardener equivalent weight AHEW 15.0 g/eq), diethylene triamine DETA (CAS 11-40-0, Sigma Aldrich, AHEW 20.6 g/eq), or triethylene tetraamine TETA (CAS 112-24-3, Sigma Aldrich, AHEW 24.4 g/eq) (Fig. 1). The two components were mixed in stoichiometric proportions to obtain ideal networks with a well-defined chemical structure. The three networks will further be called DGEBA-EDA, DGEBA-DETA, and DGEBA-TETA.

After mixing, the blends were set to thicken for up to 3 h in closed vials to avoid hardener evaporation. Thermoset filmsof thickness 20-60 μm (measured with a manual micrometre) with an experimental uncertainty of 5 μmwere then manufactured using a heating hydropress (Gibitre), and cured for 1 h at 50 • C, followed by 1 h at 90 • C. Postcuring for 30 min at 170 • C under vacuum was finally carried out to ensure complete curing of the networks (by heating over the expected glass transition of the polymer). This protocol was followed very rigorously to acquire near-ideal molecular structure of the networks. Its reliability was checked by a comparison of the glass transition values with those coming from the prediction method given by Bellenger et al. [START_REF] Bellenger | Effect of structure on glass transition temperature of amine crosslinked epoxies[END_REF] (see Suppl. Info. 1).

Differential scanning calorimetry (DSC) was performed at 10 • C min -1 -using a DSC Q10 provided by TA Instrumentsto confirm the absence of any residual exothermic curing peak and to characterize the glass transition of unaged samples (Fig. 2). Glass transition temperatures of unaged samples were identified on the second heating ramp at 143 • C, 147 • C, and 149 • C for DGEBA-EDA, DGEBA-DETA, and DGEBA-TETA, respectively. FTIR measurements were also carried out to confirm the disappearance of epoxideswith characteristic peak [START_REF] González | Applications of FTIR on epoxy resinsidentification, monitoring the curing process, phase separation and water uptake[END_REF] at 915 cm -1 -and the absence of oxidation of the unaged sampleswith characteristic peaks between 1650 and 1800 cm -1 .

Thermal ageing protocol

To understand their influence on oxidizability, several temperature conditions were studied. As such, networks were aged at 80, 100, 120, 140 and 160 • C under air in ventilated ovens. Samples were then periodically removed for FTIR measurements. Since glass transition of networks under investigation are very close (about 145 • C), those ageing temperatures were selected so as to compare their ageing either in glassy or rubbery state.

Experimental measurements

FTIR spectroscopy in transmission mode was carried out using a Frontier spectrophotometer (PerkinElmer) by averaging 8 scans at resolution 4 cm -1 between 400 and 4000 cm -1 . Absorbances were measured by subtracting baseline and defined as the maximum height of characteristic peaks, on up to 3 samples to check for repeatability of measurements. Beer-Lambert law was used to calculate concentrations of amide groups, with molar attenuation coefficient [START_REF] Celina | Carbonyl identification and quantification uncertainties for oxidative polymer degradation[END_REF] for its characteristic peak at 1660 cm -1 of 400 L mol -1 cm -1 . As determining an accurate coefficient is rather challenging, this approximate value was chosen to maximize the calculated concentrations for an estimation of the oxidation process advancement. As a common coefficient is used for the three epoxy-amine networks, a different value does not change the observations drawn in this paper.

Results

Appearance of oxidized products

Formation of oxidized groups were followed by FTIR, and spectra of the three epoxy-amine systems aged at 120 • C in the 1625-1800 cm -1 wavenumber range are stacked in Fig. 3. Significant increases in absorbance can be identified during the ageing with maxima around 1660 cm -1 and 1740 cm -1 , respectively attributed to the formation of amide and other carbonyl groups [START_REF] Celina | Carbonyl identification and quantification uncertainties for oxidative polymer degradation[END_REF]. The latter is assumed to stem from one of various C=O bonds species such as imides, carboxylic acids and esters [START_REF] Delozanne | Thermal oxidation of aromatic epoxy-diamine networks[END_REF][START_REF] Mailhot | Study of the degradation of an epoxy/amine resin, 2 kinetics and depth-profiles[END_REF], but will not be further discussed here as it is not the main product of this particular oxidation. A small shift is observed for the amide group compared to our previous observations [START_REF] Delozanne | Thermal oxidation of aromatic epoxy-diamine networks[END_REF], possibly due to electronic effects induced by -SO 2 -Ar-groups.

As the molecular structure of the oxidized polymer is expected to present a blend of oxidized functions with different chemical environments, it is rather difficult to assess an accurate molar attenuation coefficient for each function [START_REF] Celina | Carbonyl identification and quantification uncertainties for oxidative polymer degradation[END_REF]. Nonetheless, assuming the chosen coefficient values are acceptable to estimate the concentrations, it appears that amide groups are the main products of the thermal oxidation. Amide concentrations for the three systems at 120 • C under air are plotted over ageing time in Fig. 4. The Gaussian deconvolutions of the spectral region ranging from 1600 to 1800 cm -1 for these spectra show no apparent overlapping of the two peaks (at 1660 and 1740 cm -1 ) that may alter the maxima values [see Suppl. Info. 2]. Let us precise that the peak centred at 1740 cm -1 was not deconvoluted anymore.

Curves at all temperatures exhibit a sigmoidal shape (less obvious at higher temperatures). First there is a slight induction period, then the auto-accelerated oxidation occurs. Lastly a noticeable plateau can be observed during which the amide groups productionor to be more exact, the appearance of its characteristic FTIR peakappears to be slowed down.

The exact origin of this plateau remains unknown and open to interpretations as the oxidation mechanism in polymers is complex and implies many competing phenomena that may affect measurements. At first, as it will be developed in the following part, the consumption of oxidizable site will slow down the oxidation. The formation of this plateau may also be induced not only by the formation of low-mass volatile compounds [START_REF] Giron | High temperature polymer degradation: rapid IR flowthrough method for volatile quantification[END_REF], but also as imide groups are accumulated within the network, as its bands overlap with those of amide groups [START_REF] Delozanne | Thermal oxidation of aromatic epoxy-diamine networks[END_REF].

Although they present slight differences in molecular structure (with the presence of secondary N-H bonds in DETA and TETA hardeners), one can observe that DGEBA TETA oxidizes slightly faster than DGEBA-DETA, itself slightly faster than DGEBA-EDA (see also Supp. Info. 5). This will be discussed in the following in terms of concentration in reactive sites.

C-H bonds disappearance during oxidation

The wavenumber range between 2750 and 3000 cm -1 was also studied to follow the changes in C-H bonds during the thermal ageing. FTIR spectra for aged networks at 120 • C are stacked in Fig. 5. Several maxima at 2830 cm -1 , 2872 cm -1 , 2933 cm -1 and 2966 cm -1 , and two small shoulders around 2890 cm -1 and 2910 cm -1 are observed. As it is rather difficult to properly allocate each peak in such a small wavenumber range, a proposition for their general assignment [START_REF] Stuart | Infrared Spectroscopy: Fundamentals and Applications[END_REF] is given in Table 1. A discussion on peak allocation by comparison with other DGEBA-amine networks presenting various hardener molecular structures (linear, aromatic, aliphatic cyclic, etc.) be found in S.I. In the following, it will be considered that 2830 cm -1 corresponds to "hardener" methylene. Interestingly, the ratio of absorbances at 2830 to 2966 one (methyl of DGEBA group) increases in line with the relative concentrations [N-CH 2 ] hardener /[-CH 3 ] DGEBA in the three system under investigation.

A global decrease of convoluted peaks can be seen during the thermal oxidation, with one peak at 2830 cm -1 that appears to drastically reduce, suggesting the disappearance of one specific C-H function. Peaks were deconvoluted, using a 4-peak deconvolution method (two tentative 6-peak deconvolution methods are given in S.I. Section 4), for a better understanding of the changes in C-H bonds throughout the thermal oxidation. Deconvoluted curves are presented in Fig. 5 and were obtained in two steps: linear baseline correction followed by data fitting of the gaussian peaks sum with a Python script (mainly using SciPy library for the fit optimization) (S.I. section 4). Following tentative assignment of the peaks [START_REF] Stuart | Infrared Spectroscopy: Fundamentals and Applications[END_REF], several conclusions can be drawn:

-Isopropyl CH 3 groups located between aromatic rings of DGEBA appear to be stable to the thermal oxidation of the network as peaks at 2966 cm -1 and 2872 cm -1 barely decrease with ageing. -The three types of CH 2 groups next to a heteroatom (N or O) can hardly be differentiated within the peak at 2933 cm -1 , which appears to decrease but not to the same degree as the peak at 2830 cm -1 . N-CH 2 -CH 2 groups located on the hardener side, with characteristic peak at 2830 cm -1 , mostly disappear with the oxidation of the network, suggesting they are responsible for the formation of oxidized products. As the deconvolution spectra show that the 2830 cm -1 peak maxima barely overlap with neighbouring peaks, the method was considered satisfactory for detecting consumption of hardener CH 2 during ageing.

Although the peak at 2830 cm -1 clearly disappears with ageing, and according to the spectra presented in Fig. 6, it is unclear if both amineyielding methylene groups located either within the "epoxy" or "hardener" moiety really take part in the oxidation process. In fact, since the oxidation of one methylene seems to induce some changes in the FTIR absorbance of the other -CH 2 -connected to the same nitrogen atom, it is possible that the observed 80% decrease of 2830 cm -1 absorbance corresponds to a lower ratio of consumed >N-CH 2 .

Discussion

The main goal of this section is to understand the differences in reactivity of all the oxidizable sites within the molecular structure of systems under study. As such, we will:

-Validate a method to easily assess the bond dissociation energies (BDE) of epoxy-amine networks under study by comparing results from composite QC method and easy-to-use machine-learning model for small representative molecules. -Apply this machine-learning approach for the description of all C-H bonds of interest in molecules representative of the constitutive repetitive units. -Discuss on the fate of those C-H bonds in terms of degradation mechanisms and more particularly on stable products, to assess the representativeness of accelerated ageing and later the possibility to extrapolate data to low temperature ageing. As systems under study have almost the same glass transition temperature, that possible difference only comes from an intrinsic difference in reactivity and are not due to a "mobility-reactivity" coupling. Eventually, evidence of the major reactivity of one kind of C-H compared to others would allow to conclude about the "unicity" of a propagation process ROO • + RH → ROOH + R • . A simpler kinetic model based on the reactivity of only one reactive site (RH) could then be considered, where rate constants depict the reality of reactive species (R • , ROO • , ROOH…) identically for example to the case of polyethylene [START_REF] Colin | Determination of thermal oxidation rate constants by an inverse method. application to polyethylene[END_REF].

On the use of machine learning approach for BDE calculations

A recent computational tool will be validated and used to tackle this crucial physicochemical feature. To this aim, St John et al's machine learning (ML) model, which only requires the SMILES code of the molecule, was assessed specifically for the prediction of C-H standard dissociation enthalpies on secondary and tertiary carbon atoms using the accurate CBS-QB3 [START_REF] Montgomery | A complete basis set model chemistry. VI. use of density functional geometries and frequencies[END_REF] composite method (considered here as reference) that includes a full density functional theory (DFT) 3D-optimization of the molecular geometry followed by subsequent post-Hartree-Fock energy calculations (up to the CCSD(T) level with complete basis set extrapolation), which provides the so-called "chemical accuracy" (absolute error lower than 1 kcal mol -1 in average) for atomization enthalpies. All QC calculations were performed with the Gaussian 09 package [START_REF] Frisch | Gaussian 09, Revision D.01[END_REF].

For this purpose, a representative dataset of 25 molecules with secondary carbon atoms was first considered, gathering the main chemical functions involving carbon, nitrogen, and oxygen atoms. The datasets are further described in the Suppl. Info. 5 file. It collects, in a balanced way, alkanes, alkenes, ethers, alcohols, aldehydes and ketones, carboxylic acids, esters, amines, amides, nitriles and imines, which were also chosen so that they do not exhibit conformational complexity. The ΔH∘ CBS-QB3 dissoc values span a large range (about 60 kJ mol -1 , 16% of the average value), between 346.9 and 407.5 kJ mol -1 with a mean value equal to 379.6 kJ mol -1 , proving evidence that the C-H bond strength in CH 2 group can be considerably influenced by its substituents.

Similarly, a set of 15 molecules featuring a tertiary carbon atom was considered. The range for ΔH∘ CBS-QB3 dissoc values is close to that for secondary cases (from 351.7 to 417.3 kJ mol -1 , with an average value equal to 383.5 kJ mol -1 ). This suggests that secondary and tertiary C-H can be both prone to dissociation, and that one cannot determine the preferential oxidation site only based on the intrinsic carbon class.

These results also give evidence that the ML model systematically underestimates ΔH∘ dissoc , the mean signed deviation (MSD) with respect to CBS-QB3 being equal to -6.1 kJ mol -1 for secondary carbons and -9.4 kJ mol -1 for tertiary ones. This hints that the ML model, which was not specifically trained for such bonds, can be however considered a suitable departure point for the studied issue. Noteworthy, the ML and CBS-QB3 enthalpy series are strongly linearly correlated (R 2 = 0.98 for secondary group and 0.99 for tertiary subset, see Fig. 7), according to:

ΔH∘ model dissoc = -45.7 + 1.14ΔH∘ ML dissoc ⃒ ⃒ kJ mol -1 (secondary) (1a) ΔH∘ model dissoc = -13.6 + 1.06ΔH∘ ML dissoc ⃒ ⃒ kJ mol -1 (tertiary) (1b)
These models afford a significant improvement since the corresponding MAD value is equal to 1.7 (secondary) and 1.3 kJ mol -1 (tertiary). In conclusion, these corrected ML models provide a reliable and very fast prediction of ΔH∘ dissoc , equivalent to CBS-QB3, but at negligible computational cost.

On the oxidizability of networks under study

The models were then applied to the molecules of interest, as depicted later in Fig. 9. These molecules were chosen as they replicate the constitutive repetitive units of the network except for methyl groups on the outer branching to the next unit. It should be noted that these molecules can also be obtained following the reaction between 1,2epoxy-3-phenoxypropane EPP and the studied hardeners, and will be further called EPP-EDA, EPP-DETA, and EPP-TETA.

Their molecular structure present all types of chemical reactive groups that may go through oxidation. For conciseness, only the CRU of the DGEBA-EDA network is presented in Fig. 8 but all three systems share similar structure. Each CRU is constituted of diethylene amine hardener moieties branched with epoxy moieties, in different proportions for each network depending on the number of reactive N-H bonds of the initial aliphatic amine hardeners. The concentration of each oxidizable site is reported in Table 1.

One can observe that the three networks present a rising total concentration of amide-yielding sites (16.2 mol L -1 for DGEBA-EDA, 18.9 for DGEBA-DETA, and 20.6 mol L -1 for DGEBA-TETA), with similar concentrations of amide-yielding sites on the epoxy moiety, but very different concentrations for the hardener moiety (almost twice as much for DGEBA-TETA than for DGEBA-EDA for instance). Finally, the selectivity of oxidation is predicted by the ratio k p hardener [>N-CH 2 ] hardener /k p epoxy [>N-CH 2 -] epoxy which is always higher than 1 here so that the rate of degradation depends in first approach of the concentration in the utmost reactive sites ("-N-CH 2 -hardener") given in Table 2 (NB: assessment of propagation rate constant will be detailed below). Oxidation is more selective at low temperature whereas oxidation kinetics appear similar when increasing the ageing temperature (see Suppl. Info. 5). A better understanding of the reactivity of those sites (expressed by the rate constant of kinetic model and their activation) remain to be determined and discussed in a future work.

Given that BDEs for the EPP-hardener molecules are representative of the polymer molecular structure, one can observe that for all three networks, O-CH 2 -sites present the highest value of BDE at 413.6 kJ mol -1 making it the least likely to be abstracted, whether it be on a side or mid epoxy branch. -CH-OH-sites present values of 392.4 kJ mol -1 , making them more likely to be abstracted than O-CH 2 -sites.

As for the amide-yielding groups within the structure, one can observe that "hardener" sites present BDE values between 377.9 and 380.2 kJ mol -1 , while "epoxy" sites present higher BDE values close between 396.0 and 397.4 kJ mol -1 . This significant difference of BDEs - Deconvoluted peaks of the corresponding graphs. Baseline was subtracted and the deconvolution method is described in the Suppl. Info.file.

Table 1

Assignment of FTIR peaks in the 2750-3000 cm -1 region [START_REF] Celina | A perspective on the inherent oxidation sensitivity of epoxy materials[END_REF]. close to 20 kJ mol -1 -indicates that "hardener" sites are more prone to abstraction by peroxy radical. Those data are in accordance with previously presented data showing that "hardener" C-H groups are consumed faster than any other sites in epoxy amine networks. Korcek et al. [START_REF] Korcek | Absolute rate constants for hydrocarbon autoxidation. XXI. activation energies for propagation and the correlation of propagation rate constants with carbon-hydrogen bond strengths[END_REF] have shown that the reaction rate, k p , for the abstraction reaction of hydrogen atoms from unreacted sites by previously formed peroxy radicals (POO • + PH → POOH + P • ) is directly related to the BDE of the P-H bond, according to:

logk p (30 • C) = 16.2 -0.2 BDE (2)
E P = 0.55(BDE -62.5)

Where BDE is expressed in kcal mol -1 .

Here, the use of those equations using for instance BDE = 380 kJ mol - 1 for >N-CH 2 hardener, and 396 for >N-CH 2 epoxy suggests that at room temperature, the hardener methylene react almost 10 times faster than epoxy ones, but the ratio decreases at higher temperature (almost 3 at 120 • C) which seems in line with the relative decrease of absorbances at 2933 and 2830 cm -1 (Fig. 5) BDE estimations and the use of Eqs. ( 2) and (3) also justify in part that >CH-OH are stronger than others, so that their oxidation is slow and the contribution to 1740 cm -1 absorbance is low (Fig. 3).

Let us now adress the effect of oxidation on the reactivity of epoxies. Some representative units of oxidized DGEBA-EDA are depicted in 2. -The oxidation of a hardener methylene with no chain scission (EE=O_Nh) significantly weakens (-8 kJ mol -1 ) the vicinal hardener methylene. -The oxidation of an epoxy methylene with no chain scission (EE=O_Ne) weakens (to a less extent) the closest hardener methylene (-4 kJ mol -1 ), while it has almost no effect (+ 1.4 kJ mol -1 ) if there is a chain scission (EE=O*_Ne).

Table 2 Estimated [C-H] concentrations (in mol kg

As hardener methylenes feature the lowest BDE, they are more likely to generate amides, and in turn, the vicinal hardener methylene becomes weaker and more susceptible to hydrogen abstraction by a peroxy radical. These results suggest that any kinetic model built at low conversion rate (when only a small part of hardener methylene has been consumed) should be completed by including the self-sensibilisation induced by the first oxidation reactions.

On the link between CH 2 consumption and carbonyl build up

Let us now look back at the amide build-up. Several sorts of amides (Fig. 11) can be considered:

Their population depend on the relative reactivity of CH 2 on the hardener and epoxy sides, and on the relative proportion of amide over chain scission produced, which are expected to impact the macromolecular architecture and later macroscopic properties.

To discuss on the temperature effect on the relative ratio between mechanisms leading to species depicted in Fig. 11, the depletion of hardener CH 2 was plotted against amide build-up (Fig. 12). As the molar attenuation coefficient for the FTIR signals at 2830 cm -1 and 1660 cm -1 are unknown for the systems under study, and as there is no reason that amides depicted in Fig. 11 even have the same, maxima of absorbances were plotted.

A strong correlation between the rising absorbance of amide products at 1660 cm -1 and the decreasing absorbance of hardener CH 2 sites is depicted. Such observations may confirm that irrespectively of ageing temperature, hardener methylene are the most affected by oxidation.

Conclusion

Experimental and theoretical approaches were used in this paper to study the thermal oxidation of epoxy-amine networks and its selectivity. Ageing was monitored by FTIR with an innovative discussion of the consumption of each possible oxidizable groups. It was supported by theoretical calculations with a QC corrected machine-learning approach to determine the BDE of oxidizable groups of interest in the structure. The higher sensitivity to oxidation of methylene groups located on hardener moieties was thus explained from their lower BDE. To describe the co-oxidation of several methylene groups in α-position of nitrogen atoms simultaneously, a more complex kinetic model would have to be implemented. However, the existence of a clear correlation between methylene groups of hardener moieties consumption and amide groups formation suggests that the mechanism is representative in the Fig. 10, where each structure represent the possible structure resulting from radical attack of each sort of >N-CH 2 -. By comparison with Fig. 9, it can mainly be seen that:

Data will be made available on request. experimental conditions under investigations, and that it can therefore be extrapolated towards lower temperatures for predicting the structural changes occurring during epoxies oxidation.

CRediT authorship contribution statement

Romain Delannoy: Formal analysis, Investigation, Resources, Data curation, Writingoriginal draft. Vincent Tognetti: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writingreview & editing, Supervision. Emmanuel Richaud: Conceptualization, Methodology, Software, Validation, Formal analysis, Writingoriginal draft, Writingreview & editing, Visualization, Supervision, Project administration, Funding acquisition.

Declaration of Competing Interest

We hereby confirm that we have no conflict of interest with the paper "Experimental and theoretical insights on the thermal oxidation of epoxy-amine networks" submitted to Polymer Degradation and Stability.

Fig. 1 .

 1 Fig. 1. Chemical formulas of monomers used for the formation of epoxyamine networks.

Fig. 2 .

 2 Fig. 2. Curing process analysis. (left) DSC ramps at 10 • C min -1 after each step show the disappearance of an exothermic curing peak -(right) FTIR spectra of unaged samples.

Fig. 3 .

 3 Fig. 3. FTIR spectra of the carbonyl region during the thermal oxidation of epoxy-amine samples at 120 • C. Thickness were normalized at 30 μm.

Fig. 4 .

 4 Fig. 4. Changes in amide groups concentrations at 120 • C under air. Open symbols represent carbonyls with FTIR signal around 1730 cm -1 (using ε = 300 l mol -1 cm -1 ) and closed symbols amides at 1660 cm -1 .

Fig. 5 .

 5 Fig. 5. (top) C-H bonds depletion during the thermal oxidation at 120 • C under air. Thicknesses were normalized at 30 μm for ease of comparison -(bottom)

  Isopropyl -CH 3 methyl symmetric stretching 2933 (peak) N-C-H 2 asymmetric stretching (DGEBA) 2910 (shoulder) N-C-H 2 asymmetric stretching (hardener) O-C-H 2 asymmetric stretching 2890 (shoulder) Isopropyl -CH 3 methyl asymmetric stretching 2872 (peak) N-C-H 2 symmetric stretching (DGEBA) O-C-H 2 symmetric stretching 2830 (peak) N-C-H 2 symmetric stretching (hardener)

Fig. 6 .

 6 Fig. 6. Tentative assignment of the CH 2 and CH 3 groups within the molecular structure of DGEBA-EDA aged at 120 • C after deconvolution.

Fig. 7 .

 7 Fig. 7. C-H standard dissociation enthalpies (in kJ mol -1 ) for C-H bonds in 40 molecules (see Supp Info.), predicted by machine learning (x-axis) and CBS-QB3 theory (y-axis).

Fig. 8 .

 8 Fig. 8. Molecular structure of the constitutive repetitive unit (CRU) of DGEBA-EDA. Symbols here are further used for a better understanding of Table2.

  -1 ) in ideal structures of the three epoxy-amine networks. N is the number of considered group, f the number of C-H reactive group and [C-H] = N.f/M CRU .

Fig. 9 .

 9 Fig. 9. BDE estimations for C-H bonds under study. BDE are expressed here in kJ mol -1 .

Fig. 10 .

 10 Fig. 10. Structure of oligomers representative of partially oxidized EPP-EDA and effect on selected Bond Dissociation Energies (in kJ mol -1 ).

Fig. 11 .

 11 Fig. 11. Possible structures of oxidation products.

Fig. 12 .

 12 Fig. 12. Absorbance correlation between build-up of amide C=O groups and disappearance of C-H bonds. Blue squares represent DGEBA-EDA, red diamonds represent DGEBA-DETA, and yellow triangles represent DGEBA-TETA.

R. Delannoy et al.

Acknowledgments

The Agence Nationale de la Recherche is gratefully acknowledged for funding the project ANR DUREVE (18-CE-0028, 2019-2022).

Data Availability


Genane Youness

Ahmad Ramadan

Use of PLS path modeling to study customer satisfaction of two Lebanese mobile operators

Keywords: PLS path modeling, ECSI, Latent variable, satisfaction, bootstrap

Introduction

The purpose of this study is to measure customer satisfaction of two mobile operators in Lebanon, Alfa and MTC Touch, from a sample of the telephone customers for these two operators; We draw 500 phone active numbers randomly selected from their system database. After telephone contacts, we were able to record and completed 600 questionnaires referred to the EPSI (European Performance Satisfaction Index) questionnaire made on 23 questions and distributed equally between the two operators; the data collection was carried out between May 2012 and June 2012.

The goal is to construct the structural equations modeling and to estimate the measure of customer satisfaction using the ECSI [START_REF] Eklöf | European Customer Satisfaction Index Pan-European Telecommunication Sector Report Based on the Pilot Studies[END_REF] model by PLS approach and build a satisfaction index for each operator. In our work, we discuss the ECSI model (European Customer Satisfaction Index). Indeed, this model solves the main difficulties in measuring customer satisfaction. It provides best reliable results and do better understand and monitor the quality of customer relationship business. The hypothesis are tested and discussed.

Empirical study.

It should be noted that all manifest variables to the corresponding latent variables have a reflective measurement model. After the convergence of the algorithm of PLS, the structural path significance testing of both inner and outer model is verified by a bootstrap procedure with 300 subsamples . The results showed that the factor "Customer Complaints" has to be removed from both models; From Alfa model, the relation of Perceived Quality and Image to Satisfaction are not significant and have to be removed ; From the Mtc model, the relation of Perceived Quality"," Image" and "customer expectation" to "Satisfaction" have to be removed. After applying these modifications and using the algorithm PLSPM by SmartPLS software [START_REF] Wong | Partial Least Squares Structural Equation Modeling (PLS-SEM) Techniques Using SmartPLS[END_REF], we obtain the final models for Alfa and Mtc Touch as follows: For Alfa model, the coefficient of determination R 2 for Satisfaction is 0.55 this means that Expectation and Perceived Value are moderately explain 55% of the variance in customer Satisfaction. For Loyalty R 2 is 0.482 this means that Satisfaction and Image explain 48.2% of the variance of Loyalty, the important factor in mobile phone industry . In the other side, the inner model suggests that Expectation has the strongest effect on Satisfaction (0.501) followed by Perceived Value (0.309). Satisfaction has the strongest effect on Loyalty (0.427) followed by Image (0.348). For Mtc model, Perceived Value is moderately explain 54.6% of the variance of Satisfaction. Satisfaction and Image explain 57.6% of the variance of Loyalty. In the other side , the inner model suggests that the only important impact on Satisfaction is Perceived Value (0.739). Satisfaction has the strongest effect on Loyalty (0.45) then Image (0.39). Image has no direct impact on Satisfaction. For Alfa: Satisfaction = 0.5014Expectation + 0.3085PerceivedValue. Loyalty=0.427 Satisfaction+0.34 Image. For MTC Touch: Satisfaction = 0.7392PerceivedValue. Loyalty=0.45 Satisfaction+0.39 Image. So Perceived Value turned out to be such an important factor in the two models but the surprise that for Mtc operator it is the only strong predictor of Satisfaction. For Alfa, the most important impact is Expectations.

________________________________________________ * Corresponding author. E-mail: genane.youness@cnam.fr, Beirut, Lebanon, BP 11-4661 Beirut, Lebanon.

Comparison of the two models

For both mobile operators, the quality indices considered by the communality, the redundancy and the Goodness Of Fit found have good predictive and discriminant validity. Results are shown as follows:

Conclusion

In this paper customer satisfaction of both Alfa and Mtc mobile operators were compared by PLS approach using ECSI format. Perceived value was found to be the only driver of customer satisfaction for Mtc while expectation was found the most important one for Alfa;

Figure 1 .

 1 Figure 1. Final Path diagram of Alfa and MTC touch operators.

Table 1 .

 1 Comparison of results found for Alfa and MTC touch models. This led to the development of an overall composite index of customer satisfaction Alfa and MTC Touch. It represents the average score in terms of overall satisfaction given to Alfa or MTC Touch for all customers which is around 7.75. It also identifies the weights (external and internal) criteria satisfaction in overall satisfaction. The Alfa and Mtc models are not significantly different in term of satisfaction indices (for H0: dif=0 versus dif≠ 0, tvalue=0.2077 and Pvalue=0.8356).

		Alfa		Mtc Touch
		Initial Model	Final Model	Initial Model	Final Model
	Reliability	0.8630	0.8630	0.8515	0.8515
	A.V.E	0.6780	0.6778	0.6570	0.6539
	R 2	0.5644	0.5502	0.5920	0.5464
	Redundancy	0.2738	0.3059	0.1065	0.3203
	GOF	0.5283	0.5562	0.5208	0.5454
	Satisfaction Index	7.7259	7.7256	7.7499	7.7480


Michael Defoin Platel

Sébastien Vérel

Manuel Clergue

Malik Chami

Density Estimation with Genetic Programming for Inverse Problem Solving

This paper addresses the resolution, by Genetic Programming (GP) methods, of ambiguous inverse problems, where for a single input, many outputs can be expected. We propose two approaches to tackle this kind of many-to-one inversion problems, each of them based on the estimation, by a team of predictors, of a probability density of the expected outputs. In the first one, Stochastic Realisation GP, the predictors outputs are considered as the realisations of an unknown random variable which distribution should approach the expected one. The second one, Mixture Density GP, directly models the expected distribution by the mean of a Gaussian mixture model, for which genetic programming has to find the parameters. Encouraging results are obtained on four test problems of different difficulty, exhibiting the interests of such methods.

1 In fact, there are three sufficient conditions for ill-posedness which are the existence, the continuity and the redundancy of the solutions.

Introduction

One of the main application of Genetic Programming (GP) is for the approximation of unknown functions, a task known as Symbolic Regression [START_REF] Koza | Genetic Programming: On the Programming of Computers by Means of Natural Selection[END_REF]. To construct a such approximator, a GP system is trained on a given dataset that consists of pairs of inputs and desired outputs, representative of an unknown function. This is a direct problem.

While a direct problem describes a Cause-Effect relationship, an Inverse Problem (IP) consists in retrieving the causes responsible of some observed effects. For example, inferring gene regulatory networks from gene activity data or deriving some water constituants from the ocean colour are actually IP. GP has already been introduced as a method for solving IP, such as in [START_REF] Collet | Polar IFS+parisian genetic programming=efficient IFS inverse problem solving[END_REF][START_REF] Chami | Inversion of oceanic constituents in case i and case ii waters with genetic programming algorithms[END_REF][START_REF] Streichert | Comparing genetic programming and evolution strategies on inferring gene regulatory networks[END_REF][START_REF] Platel | Teams of genetic predictors for inverse problem solving[END_REF]. However, IP are often far more difficult to solve than direct problems, since different causes may produce the same effect, i.e. the solution of an IP may be not unique. In that case, the IP is said to be redundant or ambiguous or, in a more formal way, ill-posed 1 . In the context of learning from datasets, a redundant IP corresponds to a Many-To-One mapping inversion, i.e. to a given input y i , a set of outputs X i is expected. The purpose of this study is to enhance the inversion of Many-To-One mappings with GP. Here, the set of expected outputs X i is seen as a set of realisations of a random variable with an unknown probability density which has to be estimated. We propose two different ways to estimate the probability density of these answers and both are based on the possibility of producing multiple outputs with GP.

The section 2 highlights the limits of the classical Symbolic Regression approach for ambiguous IP and reviews the possibilities of producing multiple outputs with GP. Then, two original methods to tackle the redundancy problem are proposed in section 3 and tested on four benchmark problems in section 4. Finally, the application of this work and the possible further developments are discussed in the conclusion.

2 Inverse Problem and GP

Learning with Redundancy

With few hypothesis on the instructions set used to build programs, GP is an universal approximator [START_REF] Yao | Universal approximation by genetic programming[END_REF] that can learn an arbitrary target function t : X → Y from a dataset D t = {(x 1 , y 1 ) . . . (x N , y N )}. Here, we consider the pairs (x i , y i ) as the realisations of two random variables x and y and we note f the function implemented by a GP program. When training a GP system to approximate an unknown mapping defined by D t , an error function, such as for example the classical Root-Mean-Square Error, is minimized. It is known that in the theoretical case, this process leads to find the optimal answer f * (x) which is the conditional mean E(y|x), see [START_REF] Bishop | Mixture density networks[END_REF] for details.

When solving an IP, the dataset D t used can always be seen as the set of the N reversed pairs (y i , x i ). If the direct function t is not injective, the learning of D t corresponds to a Many-To-One mapping inversion. Hence, for each y i a set of of outputs X i is expected, and the single answer f (y i ) given by GP can be very poorly adapted. Indeed, during the training phase, f tends to converge towards the theoretical optimal answer f * (x) = E(x|y), which is, in the better case, only one of the expected answer.

To illustrate this, we have created a dataset D d from the target function d : → such that y = sin(x2 ) + , with a random variable with normal distribution N (0, 0.2). An (nearly good) approximation of the corresponding theoretical f * (x) is given by a standard GP system from D d with N = 500 learning examples in the range [-2, 2], see Figure 1. In a same way, in Figure 2, the output of GP corresponding to the inversion of d is plotted 2 . We can see that GP has produced a very unstable function that highly overfits the dataset D d .

Multiple Outputs

In the GP field, several studies are related to IP solving (see for example [START_REF] Collet | Polar IFS+parisian genetic programming=efficient IFS inverse problem solving[END_REF][START_REF] Chami | Inversion of oceanic constituents in case i and case ii waters with genetic programming algorithms[END_REF][START_REF] Streichert | Comparing genetic programming and evolution strategies on inferring gene regulatory networks[END_REF][START_REF] Platel | Teams of genetic predictors for inverse problem solving[END_REF]) but very few of them have investigated the question of the redundancy. However a noticeable exception can be found in [START_REF] Paris | Genetic programming with boosting for ambiguities in regression problems[END_REF] and will be discussed further. To overcome the non-uniqueness of the solution, instead of predicting the conditional mean, one way is to ensure that the output of an inverse model is at least one of the expected solutions [START_REF] Jordan | Forward models: Supervised learning with a distal teacher[END_REF]. A second idea is to produce a reduced list of illustrative examples [START_REF] Morishita | Finding multiple solutions based on an evolutionary algorithm for inference of genetic networks by s-system[END_REF] and a third possibility consists in approximating the distribution of the plausible solutions, that is the conditional probability density p(x|y) as explained in [START_REF] Bishop | Mixture density networks[END_REF].

In this study, we are interested in the latter two possibilities, namely, those that require for GP to produce multiple outputs.

A lot of previous work in GP deals with multiple outputs. Probably the most straightforward way of doing this is by implementing sophisticated systems evolving programs able to manipulate a complex data structure such as a vector or a matrix [START_REF] Spector | Autoconstructive evolution: Push, pushGP, and pushpop[END_REF]. Thus, one single GP program is responsible of producing multiple outputs. In [START_REF] Paris | Genetic programming with boosting for ambiguities in regression problems[END_REF], a technique based on boosting, that usually creates a posteriori mixtures of potential solutions, was extended to handle the ambiguity problem. Similarly, with the Parisian approach, used also to tackle an IP in [START_REF] Collet | Polar IFS+parisian genetic programming=efficient IFS inverse problem solving[END_REF], a subset of the population is selected to build the final answer. In these two previous examples, some GP programs are optionally associated to form a set of multiple outputs. Conversely, it is also possible to definitely link together several programs as being the co-operative members of a team and then, to make the whole team evolve [START_REF] Brameier | Evolving teams of predictors with linear genetic programming[END_REF][START_REF] Platel | Teams of genetic predictors for inverse problem solving[END_REF]. In what follows, we will see how the multiple outputs of a team of programs can be used to estimate a probability density.

Density Estimation with GP

From Team to Probability Distribution

Practically, to solve an IP, a dataset D consisting of pairs (y i , x i ) is used and for a given input y i , the goal is to approximate the distribution of the associated outputs X i . In the statistical inversion theory (see [START_REF] Tarantola | Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation[END_REF]), the pairs (y i , x i ) are considered as the joint realisations of two random variables y and x and the solution of an IP has the form of a conditional probability density p(x|y). We propose two different ways to approximate the subsequent distribution.

In the first approach, called Stochastic Realisation GP (SR-GP), for each input y i , we consider the n outputs of a GP team T as the n realisations of an unknown random variable. We note f j , the function implemented by the j th member of T . With SR-GP, the evolutionary system try to find teams which outputs f j (y i ) report distributions similar to the distributions p(x|y = y i ) for all the y i of the dataset.

The second approach is called Mixture Density GP (MD-GP). Here, a parametric model, namely the finite Gaussian Mixture Model (GMM), is used as explained in [START_REF] Bishop | Mixture density networks[END_REF]. An unknown density p(x|y) can always be represented as a finite sum of G Gaussian densities such as :

p(x|y) = G g=1 w g (y) φ g (x|y)
with φ g (x|y) a normal density N (µ g (y), σ g (y)). In MD-GP, each of the n = 3× G members of a team T approximates one of the functions φ g , µ g and σ g that actually tune the GMM. It is worth noticing that, except for the means µ g , the parameters of a GMM are constrained for a given y i , since the deviations σ g (y i ) are positive real numbers, and since the weights w g (y i ) are also positive numbers but with w g (y i ) = 1. We note W and S, two functions that transform the team outputs into respectively valid w g (y i ) and σ g (y i ) parameters for GMM 3 . So, for a given input y i , the answer of a team T is :

T (y i ) =    w 1 (y i ) = W ( f 1 (y i ) ) . . . w G (y i ) = W ( f n-2 (y i ) ) µ 1 (y i ) = f 2 (y i ) . . . µ G (y i ) = f n-1 (y i ) σ 1 (y i ) = S( f 3 (y i ) ) . . . σ G (y i ) = S( f n (y i ) )   
Hence T (y i ) is directly used to tune the GMM from which a set of r realisations can be produced (with usually r >> n). With MD-GP, the evolutionary system tries to find teams tuning GMM, so that the GMM realisations according to y i report distributions similar to the distributions p(x|y = y i ).

Intuitively, we understand that in SR-GP, a huge number of parameters have to be retrieved, since the size of the teams have to be big enough to produce a sufficient number of realisations (probably more than 10 3 ) so that the subsequent distributions can be significantly tested. However, one can presume that with this representation, the search space is "smooth" since the modification of one team member only affects one realisation and so slightly modify the distribution. At the contrary, with MD-GP, fewer parameters have to be retrieved to properly tune the GMM (less than 30 in this paper) and so to produce significant results but it is clear that the modification of only one team member can induce important consequences on the fitness of the whole team.

Construction of Target Distributions

The conditional probability density p(x|y) is unknown and only pairs (y i , x i ) are available in a dataset D. However, illustrative training distributions are required to properly educate the GP system. So for each input y i , we have to build the probability distribution p(x|y = y i ). Our idea is that even if a given value y i is more likely present at most once in the dataset, many comparable values can be found. Thus here, for each value y i , a set of k-nearest neighbours {y i . . . y j } is computed. Then, for each value y i of the dataset, the set {x i . . . x j } is turn into binned data, by grouping the events into C specific ranges and so binned distributions are computed. The underlying assumption being that the distribution of the {x i . . . x j } is similar to p(x|y = y i ).

The dataset D d presented section 2.1 is extended to hold N = 10 5 pairs and in Figure 3, we have plotted three sets of the k = 1000 pairs (y i , x i ) . . (y j , x j ) corresponding to y i = -0.93, -0.19 and 0.49. In Figure 3, three binned distributions are drawn for C = 50. Each of them represents an approximation of the theoretical conditional probability p(x|y = y i ) for y i =-0.93, -0.19 and 0.49.

Fitness Function

In this paper, only the fitness of the teams are computed and no effort have been made to estimate the fitness of the team members. Either with SR-GP or MD-GP, a team will be considered as a good team, if the set of the realisations produced for each y i tested report a binned distribution comparable to the binned distribution obtained from a dataset D, as explained above. To measure the distance between distributions, φ-divergences can be used, as for example the chi-square distance. Let be U (y i ) the expected binned distribution computed from D for a given y i and V (y i ), the distribution produced by a team of programs for the same y i .

We note U c (y i ) and V c (y i ), the number of events in the bin c of the two distributions. The total number of events #U (y i ) is actually independent of y i in this study and corresponds to the value k defined in section 3.2. Here, the partial fitness of a team T for one sample y i is

E T (y i ) = 1 #U (y i ) C c=1 (U c (y i ) -V c (y i )) 2 U c (y i ) + V c (y i )
and the total fitness is simply the average of the E T (y i ) for all the y i of a dataset D.

Experiments

In this section, we aim to verify the ability of the SR-GP and MD-GP systems to approximate conditional probabilities and so to solve an IP. The linear stack-based GP implementation described in [START_REF] Platel | Teams of genetic predictors for inverse problem solving[END_REF] is used to run the experiments but any other GP implementation suitable for Symbolic Regression can be used instead. The number of members in the teams is static and fixed a priori, see [START_REF] Brameier | Evolving teams of predictors with linear genetic programming[END_REF] for details. Different settings for the fitness function and the genetic operators but also various options for the representation and the conversion of teams are investigated. For each experiment, we perform 30 independent runs and a statistical unpaired, two-tailed t-test with 95% confidence determines if results are significantly different. Populations of teams are randomly created and the maximum creation size of the teams members is 50. The instruction set contains: the four arithmetic instructions ADD, SUB, MUL, DIV, the input variable Y and one stack-based GP specific instruction DUP which duplicates the top of the operand stack. We add also into the instructions set, two Ephemeral Random Constants noted ERC1 and ERC2, as described in [START_REF] Koza | Genetic Programming: On the Programming of Computers by Means of Natural Selection[END_REF], respectively in the ranges [-100, 100] and [-1, 1]. The evolution is achieved with elitism, 4-tournament selection and steady-state replacement. Recombination is performed by the standard crossover operator with a rate of 0.7 and mutation with a rate of 1.0, meaning that each program involved in reproduction will undergo, on average, either one insertion or one deletion or one substitution.

Four problems P a , P b , P c , P d are tested. The corresponding datasets consist of N = 10 5 samples from which 100 binned distributions are computed as explained in section 3.2 and 80 are dedicated to the training phase. In fact, the two first test problems P a and P b do not correspond to the inversion of a direct function, but for P a , the GP system has to approximate a bimodal distribution

4 1 2 N 1 10 y i 3 -y i 2 + y i + 9, 0.1 + 1 2 N 5(y i 3 -3y i 2
) * e -yi -3, 0.1 and for P b , for each input y i we have N y i , 2.3y i 2 -1.7y i -5.4 . With an increasing difficulty, P c and P d are true IP that correspond respectively to the inversion of the function c(x) = x + 0.3sin(2πx) + with , a random variable with uniform distribution in the range [-0.1, 0.1] and to the inversion the function d described We note from Table 1 that SR-GP is the best approach for tackling P a even if we allow more generations to SR-GP. Actually, solving P a consists more in retrieving two different functions than in a density estimation, and we think this is exactly the reason why SR-GP is more adapted. Conversely, MD-GP performs better on the three others problems. To figure out the quality of density estimation given by MD-GP, we have plotted for P d , three examples of expected and retrieved distributions corresponding to y i = 0.49, -0.93 and -0.19, (see fig. 6), and we think that a good agreement has been obtained. The retrieved distributions are produced by three Gaussian mixtures with G = 5. The outputs of the corresponding GP team, consisting of 15 programs, is plotted in Figure 7. We can see (middle part), the expected 'Σ' shape realized by the 5 functions µ g controlling the means of the Gaussian. Actually, the unwanted null-function has no influence in the model outputs since the corresponding weight Φ g in the mixture is also null in the top part of Figure 7. We note that there are two different ways of almost "switching-off" a Gaussian since fixing a deviation σ g to zero is another possible option for the system. This possibility should also be investigated for the approximation of non-continuous function with teams.

Conclusion and Perspectives

In many scientific domains, solving IP is now a major concern and the question of their redundancy requires particular answers. A wide range of methods can be used. Often based on a Bayesian approach, they actually report very good performances compared the two GP variants presented here, so that no intercomparisons were made. However, this paper is a promising proof of concept. Indeed, we note that Symbolic Regression has been introduced probably more as a benchmark problem than as a true potential application for GP, but after more than a decade of improvements, the best programs found by GP are now competitive with others methods, which is very encouraging for SR-GP and MD-GP.

One of the difficulty in comparing with published work is that, for the two alternatives presented here, the fitness function used is a φ-divergence, very useful for computing distance between two distributions but unusual in GP. As far as we know, the only previous GP work addressing explicitly the redundancy problem [START_REF] Paris | Genetic programming with boosting for ambiguities in regression problems[END_REF], is much more appropriated for functions decomposition, as the problem P a , than for more complex IP. Our SR-GP system can also easily solve P a , while MD-GP is much more suitable to tackle IP where complex densities have to be estimated. So a broad variety of redundant IP can be addressed with SR-GP and MD-GP and we think that, for a given IP, preliminary statistical analysis of the dataset, will help to decide which method is adapted but also to a priori tune the systems parameters. Moreover, a lot of improvements can be made. We think that further work should address the possibility of : assigning a partial fitness to the team members, producing a variable number of realisations according to the inputs and designing genetic operators appropriate to teams, in particular to teams which members have different semantics in the system as the weights, the means and the deviations of the Gaussian Mixture Model.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Example of direct problem solving with Symbolic Regression

Fig. 3 .Fig. 4 .

 34 Fig. 3. Examples of pairs (yi, xi) corresponding to the k-nearest neighbours of yi=-0.93, -0.19 and 0.49

Fig. 5 .

 5 Fig. 5. Density estimation : four test problems

Fig. 6 .Fig. 7 .

 67 Fig. 6. Examples of density p(x|y = yi) estimation with MD-GP for the problem P d with, from top to bottom, yi = 0.49, -0.93 and -0.19

Table 1 .

 1 Average fitness of the best programs found on the four test problems GP Type Pop. Size Team Size Max Size No Gen. k Real. Train Fit. Test Fit.

				Problem Pa
	MD-GP SR-GP	9.10 2	2 × 3 2	2.10 2	10 2	10	0.43σ=0.16 0.49σ=0.24 0.22σ=0.09 0.25σ=0.09
	MD-GP SR-GP	9.10 2 10 4	4 × 3 10 3	Problem P b 10 2 10 2	10 3 0.21σ=0.11 0.23σ=0.14 1.24σ=0.21 1.43σ=0.34
				Problem Pc
	MD-GP SR-GP	9.10 2 10 4	5 × 3 10 3	10 2	2.10 2	10 3 0.55σ=0.11 0.58σ=0.14 1.02σ=0.18 1.03σ=0.24
	MD-GP SR-GP	9.10 2 10 4	6 × 3 10 3	Problem P d 10 2 10 2 10 4	10 3 0.28σ=0.06 0.35σ=0.07 0.42σ=0.15 0.51σ=0.14

The evolutionary parameters settings of the direct case were kept here.

In this paper, S(σg(yi)) is simply the absolute value |σg(yi)| and similarly W (wg(yi)) = |wg(yi)|/ |wg(yi)|.

This problem is strongly inspired by the work of Paris et al.[START_REF] Paris | Genetic programming with boosting for ambiguities in regression problems[END_REF]


Morgan Roupret

Alice Brouquet

Florian Colrat
email: florian.colrat@bms.com

Pauline Diez-Andreu

Alexis Prudent

Melanie Chartier

Anne-Francoise Gaudin

Francoise Bugnard

Mélanie Chartier

Anne-Françoise Gaudin

Françoise Bugnard

Louis Chillotti

Stève Bénard

Sébastien Branchoux

Carine Bellera

Sylvie Negrier

Population characteristics, management, and survival outcomes in muscle-invasive urothelial carcinoma undergoing radical resection: the MINOTAUR study

Keywords: Urothelial carcinoma, Insurance claim review, (neo)Adjuvant chemotherapy, Survival outcomes, Bladder cancer, Renal pelvis, Ureter

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

Introduction

Urothelial carcinomas arise in the bladder (90%) and upper urinary tract (10%). They represent 3% of cancers worldwide and are among the most frequent cancers in France [1]. French Public Health group (Santé Publique France) reported in 2018 over 14,000 new cases, with men being mostly affected [1,[START_REF] Defossez | Estimations nationales de l'incidence et de la mortalité par cancer en France métropolitaine entre 1990 et[END_REF]. Muscle-invasive urothelial carcinomas (MIUCs) encompass both muscle-invasive bladder cancer (MIBC) and upper tract urothelial carcinoma (UTUC). MIUC represents 30% to 50% of urothelial cancers. According to current French and European guidelines, radical surgery (RS)-cystectomy (for MIBC) and/or 1 3 nephroureterectomy (for UTUC)-is the standard treatment for non-metastatic MIUC. This can be done alone, or with neoadjuvant chemotherapy (NAC) and/or adjuvant chemotherapy (AC) [START_REF] Rouprêt | Recommandations françaises du Comité de cancérologie de l'AFU -actualisation 2020-2022 : tumeurs de la voie excrétrice urinaire supérieure[END_REF][START_REF] Rouprêt | Recommandations françaises du Comité de cancérologie de l'AFU -actualisation 2020-2022 : tumeurs de la vessie[END_REF][START_REF] Léon | Surgical landscape of radical cystectomy for bladder cancer in France over the last 10 years[END_REF][START_REF] Rouprêt | European Association of urology guidelines on upper urinary tract urothelial carcinoma: 2020 update[END_REF]. Despite these therapeutics options, MIUCs remain at risk of recurrence and metastatic evolution given notably their locoregional invasion [START_REF] Miyazaki | Epidemiology of urothelial carcinoma[END_REF][START_REF] Thorstenson | Gender-related differences in urothelial carcinoma of the bladder: a population-based study from the Swedish National Registry of Urinary Bladder Cancer[END_REF].

Apart from a previous study from 2015 focusing on MIBC patients, there is no recent real-world data on the epidemiology, management, and survival of patients with resectable MIUC in France [START_REF] Léon | Surgical landscape of radical cystectomy for bladder cancer in France over the last 10 years[END_REF]. The MINOTAUR study therefore aimed to describe the incidence of French patients with MIUC who had undergone RS and their management.

Methods

Design

This is a non-interventional, national, real-world, retrospective study using secondary data from the French National Hospitalization Database (PMSI). The PMSI is a hospital claims database containing patient demographic, clinical, and therapeutic data, including diagnosis codes, procedures performed, expensive treatments, and in-hospital deaths for all hospitalizations in public or private hospitals (~ 65 million individuals) [START_REF] Boudemaghe | Data Resource Profile: The French National Uniform Hospital Discharge Data Set Database (PMSI)[END_REF].

Study population

Adults with resectable MIUC who underwent a first RS (index date) between January 1, 2015, and December 31, 2020, were included. Patients were identified using validated algorithms integrating diagnoses and procedures according to cancer site (MIBC/UTUC) (see Supplementary material). In case of inconsistency between diagnosis code and performed procedure (e.g., UTUC code with cystectomy), the procedure was favored over diagnosis to determine cancer site. Patients were excluded if they developed metastases within the 3 years before and within 3 months after RS, had neuromuscular dysfunction of the bladder, or any history of urologic surgery before RS.

After exclusion of patients with both MIBC and UTUC at index date, two subpopulations were defined according to the year RS was performed: patients with RS performed in 2015 or in 2019. Patients in the 2015 subpopulation were followed up until December 31, 2020, or until in-hospital death (whichever occurred first), allowing up to 6 years of followup for survival analysis. The 2019 subpopulation allowed for detection of potential changes in MIUC management after guideline updates in 2018. The 2020 subpopulation was not used for assessment of changes in management due to the potential effects of the COVID-19 pandemic on healthcare organization and availability, leading to 2019 being the latest and most representative year for management evaluation.

Outcomes and variables

The primary outcome was the annual incidence of patients that underwent resection of MIUC between 2015 and 2020 according to cancer site (MIBC, UTUC, or both MIBC and UTUC). The secondary outcomes were: patient characteristics, initial treatment of MIUC, in-hospital overall survival (OS), and disease-free survival (DFS), according to cancer site (MIBC or UTUC).

Demographic and clinical characteristics (age, sex, comorbidities, cancer site) were described at index date. Medical history of interest (detailed list available in Supplementary material) and treatment history were assessed for a 3-year period before index date. Initial treatment of MIUC included surgery alone or with NAC, AC, or both. NAC was defined as initiation of chemotherapy within 6 months before RS, and AC as initiation of chemotherapy within 3 months after index date. OS was defined as the time between index date and in-hospital death; no outpatient data are available in the PMSI. DFS was a composite endpoint defined as time from index date to either the date of recurrence or in-hospital death (depending on which event occurred first). Recurrence was defined by a subsequent surgery, lymph node involvement, diagnosis of a new MIUC or metastasis, or initiation of radiotherapy or immunotherapy after index date.

Codes and algorithms for patient selection and definition of variables were derived from validated algorithms [START_REF]Méthode : algorithme cancer[END_REF] and are available in Supplementary material.

Statistical analyses

Continuous data were reported as means (± standard deviation) and categorical data as counts and proportions. The Kaplan-Meier method was used to estimate survival rates with 95% confidence intervals [CI 95% ] from patients within the 2015 subpopulation with at least 3 months of follow-up after RS. Disease-free and/or surviving patients at last hospital discharge were censored on December 31, 2020. Median follow-up was estimated using the reverse Kaplan-Meier method [START_REF] Schemper | A note on quantifying follow-up in studies of failure time[END_REF]. Analyses were performed using SAS® version 9.4 (SAS Institute Inc. Cary, NC, USA).

Ethics

This study was designed according to the International Society for Pharmacoepidemiology (ISPE) guidelines and applicable regulatory requirements, including the French Data Protection Agency (CNIL) act n°2018-257 on regulatory requirements and authorization for processing PMSI data (MR-006) [START_REF]of P. Guidelines for good pharmacoepidemiology practice (GPP)[END_REF][START_REF]l'article L. 5311-1 du code de la santé publique aux données du PMSI centralisées et mises à disposition par l'ATIH par l'intermédiaire d'une solution sécurisée (MR 006)[END_REF][14].

Results

Cancer site

Between 2015 and 2020, 21,295 patients with MIUC underwent a first RS. Among them, 68.9% (n = 14,673) had MIBC, 28.9% (n = 6143) had UTUC, and 2.2% (n = 479) had both MIBC and UTUC (Fig. 1). During this period, the annual incidence of patients who underwent resection remained stable at approximately 2450 for MIBC and 1,000 for UTUC (Table 1). Median follow-up time was 66.0 months.

MIBC

In 2015, 2,394 patients with MIBC underwent a first RS. The mean age was 72.1 (± 9.5) years, with 78.0% of patients aged ≥ 65 years, and 90.1% of patients were male (Table 2). The most frequent comorbidities were diabetes (n = 524, 21.9%), obesity (n = 397, 16.6%), kidney disease (n = 381, 15.9%), chronic pulmonary disease (n = 349, 14.6%), peripheral vascular disease (n = 287, 12.0%), and myocardial infarction (n = 239, 10.0%). The sociodemographic and clinical characteristics were similar between the 2015 and 2019 subpopulations (Table 2).

Discussion

This is the first study providing real-world data on the perioperative management and the survival outcomes of MIUC patients treated by RS in France. Sociodemographic characteristics were consistent with prior literature. The median age was 73 years and most of patients were male (approximately 75%) [START_REF] Defossez | Estimations nationales de l'incidence et de la mortalité par cancer en France métropolitaine entre 1990 et[END_REF][START_REF] Chu | Delays in radical cystectomy for muscle-invasive bladder cancer[END_REF][START_REF] Macleod | Comparison of Neoadjuvant and Adjuvant Chemotherapy in Muscle-invasive Bladder Cancer[END_REF]. Chronic pulmonary disease and obesity are two known risk factors for urothelial cancers. Both factors were the most frequent comorbidities and concerned more than 12% of patients included. This percentage is likely underestimated, as for other comorbidities, due to PMSI data availability limitations [START_REF] Colombel | Epidemiology, Staging, Grading, and Risk Stratification of Bladder Cancer[END_REF][START_REF] Burger | Epidemiology and Risk Factors of Urothelial Bladder Cancer[END_REF][START_REF] Seisen | Prognostic Impact of pT3 Subclassification in a Multicentre Cohort of Patients with Urothelial Carcinoma of the Renal Pelvicalyceal System Undergoing Radical Nephroureterectomy: A Propensity Score-weighted Analysis After Central Pathology Review[END_REF].

Among patients who underwent surgery, 30% were UTUC patients. This is a higher rate than currently described in the literature indicating approximately 10% [1,[START_REF] Miyazaki | Epidemiology of urothelial carcinoma[END_REF]. This could be explained by our patient selection criteria. Indeed, we excluded patients with no indication of surgery (notably those with metastatic MIUC at time of diagnosis) as well as those with a history of partial surgery for MIUC. This could have led to a higher proportion of patients with UTUC, since patients with this cancer site are more likely to be indicated for surgery based on the staging, compared to patients with MIBC [START_REF] Galsky | Treatment of patients with metastatic urothelial cancer "unfit" for Cisplatin-based chemotherapy[END_REF].

The current French and European guidelines consider RS, with cisplatin-based NAC or AC for eligible patients, as the standard treatment for the management of resectable MIUC [START_REF] Rouprêt | Recommandations françaises du Comité de cancérologie de l'AFU -actualisation 2020-2022 : tumeurs de la voie excrétrice urinaire supérieure[END_REF][START_REF] Rouprêt | Recommandations françaises du Comité de cancérologie de l'AFU -actualisation 2020-2022 : tumeurs de la vessie[END_REF][START_REF] Léon | Surgical landscape of radical cystectomy for bladder cancer in France over the last 10 years[END_REF][START_REF] Rouprêt | European Association of urology guidelines on upper urinary tract urothelial carcinoma: 2020 update[END_REF]. This study highlighted that RS alone was the most frequent initial management for resectable MIUC. Independent of cancer site or year of surgery, over 70% of patients with MIUC underwent RS alone. This incidence even reached 92% of patients with UTUC in 2019. However, an increase in NAC use was observed among patients with MIBC, reaching 22.2% in 2019, lower than the 50% proportion of patients expected to be fit for it (maximum) [START_REF] Witjes | EAU Guidelines on Muscle-invasive and metastatic bladder Cancer[END_REF]. A similar but smaller increase was observed for AC use in patients with UTUC, reaching 6% in 2019. This could depict the progressive implementation in routine practice of successive guidelines [START_REF] Roupret | EAU Guidelines on Upper Urinary Tract Urothelial Carcinoma[END_REF][START_REF] Drakaki | Real-world" outcomes and prognostic indicators among patients with high-risk muscle-invasive urothelial carcinoma[END_REF]. However, it cannot be ruled out that a change in epidemiology of MIUC (proportion of patients with MIBC/UTUC, severity, resectability) could underlie this change [START_REF] Rouprêt | European Association of urology guidelines on upper urinary tract urothelial carcinoma: 2020 update[END_REF].

A median DFS time of 16.0 months was reported for patients with MIBC and 27.0 months for patients with UTUC after resection. These results are consistent with international reports. After resection, Drakaki et al. and Birtle et al. demonstrated a median DFS of 13.5 and 29.8 months for patients with MIBC or UTUC, respectively [START_REF] Birtle | Adjuvant chemotherapy in upper tract urothelial carcinoma (the POUT trial): a phase 3, open-label, randomised controlled trial[END_REF][START_REF]Cancer de la vessie[END_REF]. The 5-year survival rate after resection for MIBC is < 50% according to the French health authorities and the French National Cancer Institute, which is also in line with current findings [START_REF] Cagiannos | Surveillance strategies after definitive therapy of invasive bladder cancer[END_REF].

This study used the French PMSI, therefore ensuring exhaustive data analysis and accurately reflecting the current epidemiology and management of MIUC in France. The study design combined medical history assessment up to 3 years before index date and up to 6 years of follow-up for the 2015 subpopulation, allowing for a thorough analysis of patients with MIUC after the first RS as well as assessment of DFS and OS.

On top of being retrospective, the main limitation related to PMSI is the lack of clinical data (e.g., TNM stage, histological results). This limitation has been compensated for by the development of algorithms and medical reviews, as confirmed by our results which are consistent with the literature [START_REF] Schemper | A note on quantifying follow-up in studies of failure time[END_REF][START_REF] Drakaki | Real-world" outcomes and prognostic indicators among patients with high-risk muscle-invasive urothelial carcinoma[END_REF]. However, since risk assessment or documentation of previous history of non-MIBC (based on staging) of included patients was not possible, the study included heterogeneous patients with both low and high risk of recurrence or death, and patients with either primary or secondary MIBC [START_REF] Ge | Oncological outcome of primary and secondary muscle-invasive bladder cancer: a systematic review and meta-analysis[END_REF]. These latter could have lower survival outcomes [START_REF] Spf | Où meurt-on en France ? Analyse des certificats de décès[END_REF].

Another limitation is the absence of outpatient data, meaning that OS and DFS rates were estimated considering in-hospital death only, which could have led to their underestimation for the overall French MIUC population. However, results were consistent with the literature. This implies a low rate of outpatient mortality, as already highlighted by a study from Santé Publique France showing that 70-80% of patients with cancer die in hospital [30].

Overall, the French PMSI is a robust and exhaustive data source for cancer epidemiology, management, and survival [START_REF] Boudemaghe | Data Resource Profile: The French National Uniform Hospital Discharge Data Set Database (PMSI)[END_REF]. Despite not describing the effects of using new drugs at different disease stages (e.g., adjuvant, metastatic), our findings are

Conclusions

This study shows that RS alone remains the main treatment for MIUC initial management. Minor changes in practices were observed between 2015 and 2019, with an increase in NAC use among patients with MIBC and a slighter increase in AC use among patients with UTUC.

Survival among patients undergoing MIUC resection remained poor with a median DFS time of approximately 2 years and a median OS time shorter than 5 years among patients with MIBC. This highlights an unmet medical need which could be partly tackled by the emergence of innovative treatments.

Fig. 1

 1 Fig. 1 Patient disposition. MIBC: Muscle Invasive Bladder Cancer; MIUC: Muscle Invasive Urothelial Carcinoma; RS: radical surgery; UTUC: Upper Tract Urothelial Carcinoma

Fig. 2

 2 Fig. 2 DFS and OS after RS in patients with resected MIBC and UTUC-Kaplan-Meier method. DFS: Disease-free survival; MIBC: Muscle Invasive Bladder Carcinoma; OS: Overall Survival; RS: radi-

Table 1

 1 Patients with resected MIUC who underwent a first radical surgery over the 2015-2020 period

	MIBC	UTUC	Both MIBC and	Overall
	patients	patients	UTUC (N = 479)	(N = 21,295)
	(N = 14,673)	(N = 6143)
	2015 2,394	976	109	3,479
	2016 2,393	962	89	3,444
	2017 2,489	1,028	96	3,613
	2018 2,470	1,047	80	3,597
	2019 2,452	1,097	68	3,617
	2020 2,475	1,033	37	3,545
	MIBC: Muscle Invasive Bladder Cancer; UTUC: Upper Tract
	Urothelial Cancer

Table 2

 2

	Sociodemographic
	and clinical characteristics of
	patients with resected MIBC or
	UTUC in 2015 and 2019

Acknowledgements

This study was funded by Bristol Myers Squibb. Access to some confidential data, on which is based this work, has been made possible within a secure environment offered by CASD -Centre d'accès sécurisé aux données (Ref. https:// doi. org/ 10. 34724/ CASD).

Author contributions Bellera, Chartier, Gaudin, Négrier, Prudent and Rouprêt were involved in protocol development, statistical analysis, critical review of the manuscript. Bénard contributed to data management, data analysis, manuscript drafting/editing, critical review of the manuscript. Branchoux was involved in protocol development, data analysis, manuscript drafting/editing, critical review of the manuscript. Brouquet contributed to data management, statistical analysis, critical review of the manuscript. Bugnard and Chillotti were involved in manuscript drafting/editing, critical review of the manuscript. Colrat contributed to protocol development, statistical analysis, manuscript drafting/editing, critical review of the manuscript, supervision. Diez-Andreu was involved in data management, statistical analysis, manuscript drafting/editing, critical review of the manuscript, administrative support.

Declarations

Conflict of interest MR received personal fees from Astra Zeneca, Astellas, Bristol Myers Squibb, Janssen and Roche. FC, AP, MC, AFG and SBr are employees of Bristol Myers Squibb. AB, PDI, FB and LC are employees of stève consultants, which has a research consultancy contract with Bristol Myers Squibb. SBe is the executive director of stève consultants, which has a research consultancy contract with Bristol Myers Squibb. CB received personal fees from Bristol Myers Squibb. SN received consulting fees from Bristol Myers Squibb, Ipsen, Pfizer, Merck Sharpe & Dome and Eisai.

Research involving human participants

As part of the MR-006 submission process, this retrospective claims study using data from the French national hospitalization database (PMSI) does not require approval from an ethics committee in France.

Data availability statement

The source data base (PMSI) contains personal health data containing potentially identifying and sensitive patient information. In accordance with CNIL opinion n°2018-257 of 7th June 2018 on regulatory requirements and governance for processing PMSI data under MR-006 procedure, no raw data can be shared externally. The request to access PMSI data must be submitted to the Health Data Hub, the French public agency which gathers the requests, as per the MR-006 procedure. We are not permitted to publicly share these data.

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