Sofia Tachtalidou

Nicolas Sok

Franck Denat

Laurence Noret

Philippe Schmit- Kopplin

Maria Nikolantonaki
email: maria.nikolantonaki@u-bourgogne.fr

Régis D Gougeon

Direct NMR evidence for the dissociation of sulfur-dioxide-bound acetaldehyde under acidic conditions: Impact on wines oxidative stability

Keywords: antioxidant activity, white wine, oxidation, chardonnay, aldehydes

SO2 reaction with electrophilic species present in wine, including in particular carbonyl compounds, is responsible for the reduction of its protective effect during wine aging. In the present study, direct 1 H NMR profiling used to monitor the reactivity of SO2 with acetaldehyde under wine-like oxidation conditions. The dissociation of acetaldehyde bound SO2 was evidenced suggesting that released free SO2 can further act as an antioxidant. EPR and DPPH assays showed an increasing antioxidant capacity of wine with the increase in the concentration of acetaldehyde sulfonate. The presence of acetaldehyde sulfonate in wines was correlated with the overall antioxidant activity of wines. The first evidence of acetaldehyde bound SO2 dissociation provides a completely new representation of the long-term protection efficiency of SO2 during bottle aging.

Introduction

Sulfite is usually the most common preservative that is widely used in wine production due to its antioxidant, antimicrobial, and anti-enzymatic properties (Li, Guo, et al., 2008). However, due to existing health concerns, there is a current trend to limit its use and to look for alternatives [START_REF] Castro Marín | Chitosan as an antioxidant alternative to sulphites in oenology: EPR investigation of inhibitory mechanisms[END_REF][START_REF] Kontaxakis | Resveratrol: A Fair Race Towards Replacing Sulfites in Wines[END_REF]. Yet, in situations where the microbial stability is achieved through various chemical and/or physical processes, the main issue about shelf-life matter remains the chemical stability against chemical oxidation. This is particularly true for wines, where the high added value relies on their aging potential.

In a context of restrictive regulations about additives in food and wine, a deeper understanding of the chemistry of the widely used sulfites is required, in order to be able to reduce reliably their use and find efficient alternatives.

When added to wine, SO2 reaches an equilibrium involving free forms, including molecular SO2 and the bisulfite anion HSO3 -, and bound forms, which result from the reaction of HSO3 -with a diversity of wine components, including aldehydes, ketonic acids, sugars, sulfur-containing compounds, acids (Arapitsas, Guella, et al., 2018;Jackowetz & Mira de Orduña, 2013;[START_REF] Roullier-Gall | Sulfites and the wine metabolome[END_REF]. Enology had acknowledged for decades that free bisulfite and weakly bound forms are actually assuming protection against oxidation, whereas strongly bound SO2 consider to be lost for wine preservation [START_REF] Ribereau-Gayon | Handbook of Enology Volume 1 The Microbiology of Wine and Vinifications[END_REF]. Among the various binding compounds, acetaldehyde is considered as the strongest binder, through the electrophilic character of its carbonyl group, with a dissociation equilibrium constant for its sulfonate as low as 1.5x10 -6 at pH=3.3 (Burroughs & Sparks, 1973). Acetaldehyde has been extensively studied in wine since it is a major carbonyl compound that is formed upon alcoholic fermentation, but which can further result from chemical oxidation of ethanol promoted by oxygen exposure during aging.

Acetaldehyde concentrations are usually higher in white wines than in red wines, with values of the order of 80 mg/L and 30 mg/L, respectively (Jackowetz & Mira de Orduña, 2013;[START_REF] Liu | An overview of formation and roles of acetaldehyde in winemaking with emphasis on microbiological implications[END_REF]. As 98% of acetaldehyde will bind SO2 within 90 minutes, it has been considered that in the presence of 30mg/L of free SO2, only 0.04% of acetaldehyde will remain in the free form [START_REF] Coetzee | The Use of SO2 to Bind Acetaldehyde in Wine: Sensory Implications[END_REF].

The first step of oxidation involves the oxidation of an Fe (II)-tartrate complex to Fe (III)-O2, which then reacts with o-diphenols. The final product of oxidation are the quinones, form polymerization products (brown pigments) that cause the browning of wine (Li, et al., 2008). [START_REF] Danilewicz | Reaction Mechanisms of Oxygen and Sulfite in Red Wine[END_REF] have proposed a mechanism by which sulfites will react with quinones to restore catechol and to protect the wine from oxidation. Consequently, an accurate evaluation of the antioxidant activity of wines as well as the identification of the wine components that contribute to their oxidative stability, is necessary. Numerous methods have been developed for the evaluation of the antioxidant activity of wines, especially spectrophotometric ones. DPPH (2,2-Diphenyl-1-picrylhydrazyl) is widely used as well, for the rapid and easy determination of the radical scavenging properties of compounds and wines. This colored radical is stable and when it reacts, the purple colour changes to yellow and the absorbance can be measured at 525nm. A recent study from our research group has shown the complementary results that DPPH assay and Electron Paramagnetic Resonance spectroscopy (EPR) provide, for the better understanding of the correlation between the antioxidant capacity of wines and their chemical composition by targeting essentially sulfur containing compounds (R. [START_REF] Romanet | Exploring the chemical space of white wine antioxidant capacity: A combined DPPH, EPR and FT-ICR-MS study[END_REF]. EPR spin trapping approach has been successfully used for the analysis of wines oxidative mechanisms [START_REF] Elias | Identification of Free Radical Intermediates in Oxidized Wine Using Electron Paramagnetic Resonance Spin Trapping[END_REF]. [START_REF] Nikolantonaki | Measurement of white wines resistance against oxidation by Electron Paramagnetic Resonance spectroscopy[END_REF] proposed a more effective way to use EPR spectroscopy for the determination of the antioxidant activity of wines and to classify them according to their resistance to oxidation by modeling the kinetic of 1-hydroxyethyl radical formation after Fenton based initiation of the chemical oxidation.

In this study, the antioxidant capacity associated with acetaldehyde sulfonate in model and real white wines had been investigated by DPPH and EPR measurements, combined with high-resolution LC-MS-Q-ToF molecular profiling. Direct Nuclear Paramagnetic Resonance ( 1 H NMR) analysis has been used for quantifying the rate of production of the sulfonate and for deciphering the possible mechanisms responsible for the antioxidant capacity of acetaldehyde sulfonate.

Materials and methods

Chemicals

The water used in this study was ultrapure water (18.2 MΩ, Millipore, Germany).

Ethanol and methanol were purchased from Honeywell (United states) and were used for the preparation of DPPH solution. Acetaldehyde, ethanol MS grade (>99.8%), tartaric acid, citric acid, sodium phosphate, alpha-(4-pyridyl-1-oxide)-N-tertbutylnitrone (POBN), 2,2-Diphenyl 1-picrylhydrazyl (DPPH), trimethylsilyl propionic acid sodium salt (TSP) and 4-methylcatechol have been purchased from Sigma-Aldrich (St. Louis, MO, USA). Fe (II) sulfate heptahydratewas from Carlo Erba (Milan, Italy) reagents. Na2S2O5 was from Merck (Darmstadt, Germany). Sodium hydroxide and hydrogen peroxide were from ChemLab, methanol (MS grade) and acetonitrile (MS grade) were purchased from Biosolve Chimie (Dieuze, France) and D2O from euriso-top (Saarbrücken, Germany).

Synthesis of acetaldehyde sulfonate

A model wine solution was prepared freshly by adding 12% (v/v) of ethanol and 5g/L of tartaric acid in ultrapure water. The pH has been adjusted to 3.5, with drops of 10% w/v NaOH solution. For the synthesis of acetaldehyde sulfonate, acetaldehyde (1mM) and SO2 have been added in model wine solution in a molar ratio of 1/1 (solution a) and 2/1 (solution a´). The yield of each reaction has been monitored by 1 H NMR spectroscopy.

Wine samples

18 freshly made Chardonnay wines from the region of Burgundy, France, were used for the spike addition experiments with 50 and 100mg/L of SO2. Wines were stored in dark place at room temperature for a week in order to achieve equilibrium before EPR spin trapping and DPPH analysis.

Solutions for simulation of wines chemical oxidation conditions in presence of acetaldehyde sulfonate

4-methylcatechol (4mM), acetaldehyde sulfonate (1 mM) (solution a) and Fe (II) (0.44mM) have been added to a model wine solution in excess of oxygen (7.68mg/L).

The device for measuring the oxygen was NOMASense O2 P6000 and the sensor type was SP-PSt3-NAU-D5-CAF. The yield of the reaction has been monitored by 1 H NMR spectroscopy. There was no control in this experiment, as it is already known that 4-methylcatechol would be oxidized in the absence of a source of SO2.

Acquisition of NMR spectra and apparatus

NMR experiments were performed on a Bruker 600 MHz spectrometer using a cryoprobe at 298 K. Samples were mixed with 100μL of D2O that contained 0.1% of TSP. TSP has been used as an internal standard for quantification. 1 H NMR spectra were obtained by using WET multiple solvent suppression experiment for simultaneous suppression of water and ethanol proton peaks. 2D NMR spectra have been performed for the identification of all forms of acetaldehyde (free, hydrated and bound). The 2D acquisition parameters for the COSY experiment were as follows:

The standard cosygpprqf Bruker pulse sequence was used, with a spectral width of 9014.4 Hz (F2) and 9003.4 Hz (F1), the acquisition time was 0.11 sec, 2K(t2) × 256 (t1) data points, 64 scans and 16 dummy scans. Acquisition parameters for the 1 H-13 C HSQC experiment were as follows: The hsqcedetgpsisp2.3 Bruker pulse sequence, which also incorporates DEPT editing of the signals based on carbon multiplicity (methine, methylene, methyl) was used, with a spectral width of 9615.3 Hz in the F2 dimension and 24904.9 Hz in the F1 dimension. The acquisition time was 0.106 sec, 2K (t2) × 256 (t1) data points, 96 scans and 16 dummy scans. Acquisition parameters for the 1 H-13 C HMBC experiment were as follows: The hmbcgplpndqf Bruker pulse sequence was used, with a spectral width 7812.5Hz in the F2 dimension and 33207.4 Hz in the F1 dimension. Acquisition time was 0.262 sec, 4K (t2) × 128 (t1) data points, 128 scans and 16 dummy scans. All NMR data were processed using Topspin 4.0.6 NMR (Bruker) software.

Optimized DPPH assay

Optimized DPPH assay for sulfur containing compounds was carried out based on the protocol proposed by Romanet et al (Remy Romanet, christian, et al., 2019). 10 mL of DPPH (0.63 mM in methanol) and 90 mL of citrate/phosphate buffer (12.5 mM Na2HPO4 and 14.6 mM citric acid were dissolved in an H2O: Methanol (4:5 v/v) solution) were added in a 100 mL amber volumetric flask. The ultrapure water was degassed for 2 minutes using argon (Ar) gas. In the case of wine samples, they were degassed with CO2 for 15 minutes right before the analysis to ensure the lack of residual free SO2 as Pegram et al have proposed (2013). After the experimental procedure and letting the samples for four hours in a dark place, EC20 (the quantity of the antioxidant needed to scavenge 20% of DPPH radical) has been calculated spectrophotometrically at 525nm using 10 mm QS absorption cells. The control used in model wine was the model wine without addition of any compound (Control a), while the control in the wine experiments was the white wine before the addition of acetaldehyde sulfonate (Control b).

Acquisition of EPR spectra and apparatus

EPR analysis was carried out based on the protocol proposed by [START_REF] Romanet | Exploring the chemical space of white wine antioxidant capacity: A combined DPPH, EPR and FT-ICR-MS study[END_REF]. FeSΟ4•7Η2Ο (0.072 mM) and H2O2 (0.215 mM) were used as a source of hydroxyl radicals and 29.9 mM of POBN solution as a spin trap. Finally, each sample was added to achieve a final volume of 1mL. EPR measurements were performed using an EMXnano EPR spectrometer (Bruker, Germany). The parameters used for the experiments were as follows: modulation frequency 100 kHz, modulation amplitude 1.500 G, time constant 1.28 ms, conversion time 15 ms, attenuation 10 dB, and receiver gain 40. Spectra were recorded at room temperature (298 K). Serial 2min EPR acquisitions were performed by using the function 2D FieldDelay. SpinFit has been used for the quantification of the EPR signal. The SpinFit parameters that have been used were diameter 0.8 mm, center 125 mm, and length 25 mm. Model wines were analyzed daily to ensure repeatability of the experimental conditions and the radical oxidation. To increase repeatability day-to-day, the intensity of POBN-1-HER adducts in model wine and Chardonnay wine samples was normalized by the maximal intensity obtained from model wine. For data analysis, the maximum amount of radical produced (Cmax), the time to reach the maximum amount of radical formed (TMax), and the initial slope of the kinetic curve (1st Slope) were used as parameters to evaluate the antioxidant capacity. The control used in model wine was the model wine without addition of any compound (Control a), while the control in the wine experiments was the white wine before the addition of acetaldehyde sulfonate (Control b).

UPLC-MS-Q-ToF analysis

MS targeted analysis of acetaldehyde sulfonate in white wine samples has been performed using an ultra-high-pressure liquid chromatography (Dionex Ultimate

Data analysis

All experiments were performed at least in triplicate and if not specified, results were expressed as average ± standard deviation for the triplicate. Non parametric t-student rank sum test and spearman test were used for median comparison and correlation estimation, respectively. When acetaldehyde was dissolved in model wine, the hydrated form ( 1 H: (-CH) 5.23 ppm, q / 13 C: 88.80 ppm and (CH3) 1 H: 1.31 ppm, d / 13 C: 23.84 ppm)) was also produced. After the addition of bisulfite, the quadruplet from the -CH group ( 1 H: 9.66 ppm, q / 13 C: 207.19 ppm) and the doublet of the -CH3 ( 1 H: 2.23 ppm, d / 13 C: 30.75 ppm) group of the free acetaldehyde are protected and shifted to a higher field region at 4.54 ppm (q) ( 13 C: 80.99 ppm) and 1.46 ppm (d) ( 13 C: 17.56 ppm) respectively.

Results and Discussion

3

The yield of acetaldehyde sulfonation reaction was calculated for the molar ratios 1/1 and 2/1 (Acetaldehyde / HSO3) (Figure 2). The sulfonation reaction of acetaldehyde was fast and almost complete (yield 95% ± 5%), in concordance with already reported results [START_REF] Cassino | The effects of sulphur dioxide on wine metabolites: New insights from 1H NMR spectroscopy based in-situ screening, detection, identification and quantification[END_REF][START_REF] Nikolantonaki | Direct Analysis of Free and Sulfite-Bound Carbonyl Compounds in Wine by Two-Dimensional Quantitative Proton and Carbon Nuclear Magnetic Resonance Spectroscopy[END_REF]. Since the reaction is almost complete, in the ratio 1/1 we observe no and minimal signal of free and hydrated acetaldehyde respectively.

Global evaluation of acetaldehyde sulfonate importance on model and real white wine antioxidant activity

The antioxidant activity of free and sulfonated acetaldehyde has been evaluated by combined optimized DPPH assay and EPR spin trapping spectroscopy first in model wine and second in white wine matrices. According to Romanet et al (2021) these methods show good complementarity and affinity with sulfur containing compounds.

Based on the same study, the maximum amount of POBN-1-HER radical produced (Cmax); the time to reach the maximum amount of radical formed (Tmax); the initial slope of the kinetic curve (1st Slope) and the concentration of the compound that is required to scavenge 20% of DPPH initial concentration (EC20), derived analytical parameters from EPR spin trapping and optimized DPPH assay, respectively, were used to compare samples. According to our analytical approach, the lower the values of Cmax, Tmax, 1st slope and EC20, the higher the antioxidant activity.

Table 1 shows the antioxidant capacity results of free and sulfonated acetaldehyde compared to catechin which was used as a reference compound presenting high antioxidant capacity (Remy Romanet, et al., 2019). The EC20 value was also estimated for free SO2 in order to use it as a negative control. DPPH results put in evidence for the first time that sulfonated acetaldehyde has an antioxidant capacity compared to free acetaldehyde which showed no response. EC20 values of free SO2 and bound to acetaldehyde were estimated at 0.18 and 0.15, respectively. These results after conversion to the concertation of each substrate reacted with DPPH radical, assume the release of 4.17x10 -4 M of free SO2 which corresponds to 83% of sulfone dissociation after reaction with 6x10 -6 M of DPPH. Our results are contradictory to those published by Danilewich et al. (2016) where no reactivity of SO2 was shown with DPPH. This difference can be explained by the use of DPPH (1.59x10 -4 M) in concentration 300 times higher compared to our experimental conditions, which lead to the no-linear decrease of DPPH during the time of reaction. In comparison with a very good antioxidant compound as catechin, sulfonated acetaldehyde appeared not to be a waste while presenting five times lower antioxidant capacity of that reported for catechin (Remy Romanet, et al., 2019). The use of EPR spin trapping analysis confirmed the overmentioned result and put in light that sulfonated products have greater hydroxyl radical scavenging activity than polyphenols (due to the fact that catechin generates radicals in presence of ROS (reactive oxygen species) and acts both as antioxidant and prooxidant (Caro, Davis, et al., 2019), resulting in lower responses in Cmax, Tmax and slope.

Acetaldehyde-HSO3 was added to a white wine in order to test the dependence of DPPH decrease and 1-hydroxyethyl radical production on its concentration. When acetaldehyde sulfonate was spiked in white wine using increasing concentrations (5x10 -4 M, 10 -3 M, 5x10 -3 M and 10 -2 M), DPPH and EPR spin trapping analyses at room temperature confirmed its antioxidant activity observed in model wine solutions. DPPH results were subjected to a hierarchical linear model in order to detect significant differences among compounds considering repeated measurements.

The student test (t-test) for all the wine samples has shown that there is a difference between samples with different concentrations of acetaldehyde sulfonate (p-values below 0.02), while the two wine samples with concentration of acetaldehyde sulfonate 5x10 -4 M and 10 -3 M have shown that there is no difference between them, since pvalue = 0.28.

In conclusion, the antioxidant effect of acetaldehyde-HSO3 in wines was modulated to a concentration dependent manner. Going from the lowest (5x10 -4 M) to the highest (10 -2 M) concentration, the antioxidant capacity of the wines increases in a significant manner (t-student, p<0.01). Moreover, EPR spin trapping results showed that at high concentration (10 -2 M) acetaldehyde-HSO3 inhibited completely the 1-hydroxyethyl radical formation. The rate of spin adduct degradation may have exceeded speed adduct formation in 10 -2 M treatments, however, which may explain this result. These results lead us to the conclusion that acetaldehyde sulfonate has a major role on wine oxidative stability indirectly by the release of SO2 under wine oxidation conditions as proved in section 3.4. Moreover, in a context of restrictive regulations about sulfite additions in wine, these results could enable to reduce reliably sulfite use for wines oxidative stability.

Identification of acetaldehyde-HSO3 in wine samples and its correlation with wines antioxidant capacity

Complementary to existing data on the antioxidant activity of SO2 released from acetaldehyde sulfonate, the radical scavenging properties as well as the molecular profile of eighteen Chardonnay samples, have been combined in a spearman correlation analysis. DPPH and EPR analyses give information about the antioxidant activity of wines, while LC-MS analysis provides us instant wines molecular fingerprints. Targeted analysis of the data has been acquired and the area of the peak of acetaldehyde sulfonate has been calculated. The spectrum is presented in Figure S1, where the retention time is 0.72 min and the exact m/z = 124.9917. The identification was confirmed after comparison with the standard compound and the isotopic profile.

Figure 3 shows the antioxidant capacity results as well as the content of acetaldehyde-HSO3 of 18 wine samples before (control) and after spiking in 50 and 100 mg/L SO2. These results were subjected to a statistical analysis in order to detect significant differences among the treatments considering repeated measurements and biological duplicates. Acetaldehyde-HSO3 was identified only in 7 out of 18 wine samples and in lowest intensity compared to both 50 and 100 mg/L SO2 treatments. It is important to mention that free SO2 is removed before sampling according to the applied protocols [START_REF] Nikolantonaki | Measurement of white wines resistance against oxidation by Electron Paramagnetic Resonance spectroscopy[END_REF]Remy Romanet, et al., 2019) in order to avoid artefacts. In that respect, considering the complex chemical environment of wine matrixes, the sulfonation reaction of acetaldehyde occurred rapidly after the addition of sulfites for the two considered concentrations. Wine's antioxidant capacity was positively related to sulfonation reactions for both treatments. However, significant differences among treatments (Control, 50 and 100 mg/L SO2) were observed only when EPR spin trapping method was used. Optimised DPPH assay was not significatively discriminant for the low (50 mg/L SO2) concentration of sulfites.

These results agree with those obtained in model wine and put in evidence the importance of using complementary analytical methods in order to study the antioxidant capacity of complex matrices.

In order to investigate the direct impact of the presence of acetaldehyde-HSO3 on wines global antioxidant capacity, the association between each pair of estimated variables (EC20, Cmax, Tmax and slope) with the relative intensity of the detected sulfonate was analysed using Spearman's rank-order correlation coefficients (r). This revealed significant anti-correlation of acetaldehyde-HSO3 with Cmax (r = -0.542, p<0.001) and the slope (r = -0.872, p<0.0001) indicating a great contribution of acetaldehyde sulfonate on wines general antioxidant capacity monitored by EPR spin trapping analysis. On the other hand, in the present study, no significant correlation was established between the acetaldehyde-HSO3 intensity and EC20 and Tmax.

However, Cmax was positively correlated with Tmax (r = 0.874, p<0.001) and slope (r = 0.528, p<0.000), while Tmax was positively correlated with EC20 (r = 0.468, p=0.002). These results agree with the observations made by [START_REF] Romanet | Exploring the chemical space of white wine antioxidant capacity: A combined DPPH, EPR and FT-ICR-MS study[END_REF] who proposed for the first time the modelization of white wines antioxidant capacity by coupling optimized DPPH assay and EPR spin trapping analysis, based on the different antioxidant mechanisms measured in each. Based on the same study, among the 365 identified molecular markers of wines with high antioxidant capacity, 32%

were CHO compounds including phenolics and sugar compounds, 20% were CHOS compounds and 36% were CHONS compounds. This diversity is due to the different antioxidant mechanisms studied with hydrogen atom or electron transfer identified through DPPH and phenolic regeneration, nucleophilic properties and 1-HER trapping properties identified through EPR analysis. We understand so that sulfonation reaction occurring during winemaking and aging (Arapitsas, et al., 2018) is a mechanism related to wines antioxidant capacity, and the present study is an addition to the current knowledge on sulfonation reactions and a direct proof of the dissociation of bound SO2 to acetaldehyde.

Mechanistic understanding of acetaldehyde sulfonate suggested antioxidant activity

In order to provide a mechanistic approach of the antioxidant activity of acetaldehyde sulfonate, we simulated wine's oxidation conditions by monitoring a Fenton-like reaction (excess of oxygen and Fe (II)), in the presence of 4-methylcatechol and acetaldehyde sulfonate, expecting the observation of the decrease concentration of acetaldehyde sulfonate and the presence of free acetaldehyde. The reaction has been monitored by 1 H NMR spectroscopy. The table of peak assignments is presented in Table S1. As it is observed in Figure 4, the first week of reaction, there is a slight appearance of free acetaldehyde and the concentration of 4-methylcatechol remains stable. Additionally, in the first two weeks, the concentration of 4-methylcatechol remains quite stable, while acetaldehyde sulfonate's concentration has decreased more than a half. These observations lead to the conclusion that at the first step of oxidation, quinone consumes the available free SO2 (yield of sulfonated acetaldehyde: 95% ± 5%, so <10% of SO2 remains in free form) and then, the equilibrium between free and bound acetaldehyde should be re-established by the dissociation of the bound form and the release of SO2. Thus, the most of the initially bound SO2 is used in order to restore catechol from the quinone. As Nikolantonaki and Waterhouse (2012) have shown, 89% of quinone is reduced back to catechol, while 11% forms three quinone-SO2 adducts in ratio 5:1:2. Danilewicz had previously observed that sulfites or other good nucleophiles are necessary to have rapid oxygen consumption (Danilewicz, 2011). This could be explained by the fact that the oxidation of o-diphenols to quinone is thermodynamically unfavorable, and nucleophiles are needed to drive the reaction to completion (Danilewicz, 2013). The current study appears to show that the small amount of free SO2 formed from acetaldehyde sulfonate, facilitates this pathway.

From week 2 until week 4, free and bound acetaldehyde remain stable, due to the competition between acetaldehyde and quinone for the free/available SO2. This is an indication that reinforces the belief that acetaldehyde is a strong binder. The next weeks, SO2 release is slower than at the beginning of the reaction. Thus, the concentration of 4-methylcatechol starts to decrease, due to oxidation, while a part of SO2 will form three different sulfonated adducts of quinone (Nikolantonaki & Waterhouse, 2012). It is concluded that during slow oxidation, acetaldehyde releases the bound SO2 in order to restore the catechol from quinone [START_REF] Danilewicz | Reaction Mechanisms of Oxygen and Sulfite in Red Wine[END_REF]. These results could explain the decrease of the amount of total SO2 bound fraction during wine aging as observed by [START_REF] Sacks | Free, Bound, and Total Sulfur Dioxide (SO2) during Oxidation of Wines[END_REF] and indicates that strongly bound SO2 is not a waste but it contributes on wines resistance against oxidation during bottle aging.

Conclusions

Combined direct 1 H NMR kinetic monitoring, DPPH assay and EPR spectroscopy under wine oxidation conditions supported the more recent studies on the dissociation of acetaldehyde sulfonate and showed that this adduct has antioxidant properties in contrast to previous studies claiming the opposite [START_REF] Ribereau-Gayon | Handbook of Enology Volume 1 The Microbiology of Wine and Vinifications[END_REF]. We have provided a direct proof that acetaldehyde sulfonate can be dissociated under wine oxidation conditions, with the concomitant release of free SO2 which can act as antioxidant by restoring 4-methylcatechol. These results could possibly enhance the knowledge on the compounds contributing to wine's oxidative stability and initiate the reconsideration of SO2 management in wine production. Protons used for quantification are indicated in bold for each compound.

Figure and table captions

  3000, ThermoFischer) coupled to a MaXis plus MQ ESI-Q-TOF mass spectrometer(Bruker, Bremen, Germany). The column used was an Acquity BEH C18 1.7µm, 100x 2.1 mm by Waters (Guyancourt, France) in reverse phase to analyze non-polar compounds. The mobile phase was (A) acidified water with 5% of acetonitrile (0.1% v/v of formic acid) and (B) acidified acetonitrile (0.1% v/v of formic acid). The temperature of elution was 40°C using the gradient: 0-1.10 min 5% (v/v) of B and 95% (v/v) of B at 6.40 min. The flow was 400µL/min. The negative ionization takes place in electrospray. End plate offset (500V) and capillary voltage (4500V) permit the ions transfer. Before each analysis batch, the mass spectrometer was calibrated using Na Formate in enhanced quadratic mode. The mass range was between 100 and 600 m/z in negative ionization mode. Quality controls were analyzed before and throughout each batch, to verify the stability of the LC-MS system. All samples were analyzed randomly. Detection was carried out in negative ionization mode with the following parameters: Nebulizer pressure = 3.0 bar, dry gas flow = 10.0 l/min, dry gas temperature = 200 °C, capillary voltage = 4500 V, end plate off set= 500 V.

  .1 1 H NMR Monitoring of acetaldehyde sulfonation reaction in model wine 1 H NMR methodology was used for directly investigating acetaldehyde interaction with sulfur dioxide under wine like acidic conditions. The presence of 12% of ethanol as well as the excess of water limited the sensitivity of 1D 1 H NMR to low concentrations substances as the intensities of both water and ethanol signals obscured some other signals. In that respect WET experiment was applied, which allowed their suppression. The 1 H NMR method was used to determine the yield of sulfonation reaction of acetaldehyde in different ratios under wine-like acidic conditions at room temperature (298 K). 1 H NMR spectra for both ratios are presented in Figure 1, while 2D NMR spectra of acetaldehyde sulfonate in white wine are presented in supplementary information (Figures S2, S3 and S4).

Figure 1 :

 1 Figure 1: Assignments of 1D 1 H NMR signals for quantification of acetaldehyde

Figure 2 :

 2 Figure 2: Average free, hydrated, and sulfonated acetaldehyde concentrations during

Figure 3 :

 3 Figure 3: Acetaldehyde-HSO3 peak-intensity from LC-MS analysis, EC20, Cmax and

Figure 4 :

 4 Figure 4: (A) 1 H NMR monitoring of acetaldehyde sulfonation reaction products: (i)

Table 1 :

 1 Optimized DPPH assay and EPR spin trapping evaluation (EC20, Cmax, Tmax and slope) of free, sulfite bound acetaldehyde and catechin, in model wine like solution and in a white wine at room temperature (20°C).

  Figure 2

  N.R.: No response; ! MW: model wine; # AS: acetaldehyde-HSO3; § T.H: total inhibition; different letters indicating significant statistical differences among samples from the same column (p<0.01). Control a: model wine, Control b: wine without the addition of acetaldehyde sulfonate.

	Table 1
		EC20	Cmax	Tmax	1 st slope
	Model wine
	conditions
	Control a ( ! MW)	*N.R.	100	69.43 ± 3.4	0.01 ± 0.00
	Acetaldehyde	N.R.	94.4 ± 4.7	86.82 ± 4.3	0.0093 ± 0.00
	Acetald-HSO3	0.15 ± 0.00	64.5 ± 3.2	32.27 ± 1.6	0.0118 ± 0.00
	SO2	0.18 ± 0.00	7.26 ± 0.36 52.4 ± 2.6	1.3525 ± 0.07
	Catechin	0.029 ± 0.00	80.7 ± 4.0	57.42 ± 2.8	1.052 ± 0.05
	White wine
					0.0063 ± 0.00
	Control b	17.87 ± 0.16a 59.34 ± 2.9a 107.42 ± 5.3a
					a
			48.34	±	0.0086 ± 0.00
	# AS 5*10 -4 M	16.32 ± 0.23b		50.28 ± 2.5b
			2.4b		b
					0.0084 ± 0.00
	AS 10 -3 M	16.59 ± 0.05b 44.46 ± 2.2c 32.62 ± 1.6b
					b
	AS 5*10 -3 M	12.31 ± 0.22c § T.H. d	T.H. c	T.H. c
	AS 10 -2 M	9.81 ± 0.14d T.H. d	T.H. c	T.H. c

*

Acknowledgments

This work is part of the project MetaBox ISITE-BFC (2018-2021) supported by the Conseil Régional de Bourgogne Franche-Comté and the European Union through the PO FEDER-FSE Bourgogne 2014/2020 programs. We also thank the OIV for the scholarship of Sofia Tachtalidou.


Sébastien Fueyo

Yacine Chitour

A corona theorem for an algebra of Radon measures with an application to exact controllability for linear controlled delayed difference equations

.

Introduction

Corona problems are relevant in linear infinite-dimensional control theory, especially for delay equations see [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF][START_REF] Yamamoto | Behavioral controllability and coprimeness for a class of infinite-dimensional systems[END_REF]. Exact controllability in finite time is often characterized in terms of a Bézout identity over appropriate functional algebras and hence obtaining an exact controllability criterion is tantamount to the resolution of a corona problem for measures or distributions compactly supported algebras.

Since the resolution of the corona problem in one dimension for holomorphic bounded functions in the unit disk by the celebrated paper [START_REF] Carleson | Interpolations by bounded analytic functions and the corona problem[END_REF], the corona problem received a large attention. Carleson's result has been extended in various way, as for more general domains or algebras, see for instance a matrix version in the polydisk [START_REF] Treil | The matrix-valued hp corona problem in the disk and polydisk[END_REF] or in a multiply connected domains [START_REF] Brudnyi | Matrix-valued corona theorem for multiply connected domains[END_REF], for some functions algebra on planar domains [START_REF] Mortini | Corona-type theorems and division in some function algebras on planar domains[END_REF] or for the algebra of almost periodic function with a Bohr-Fourier negatively supported [START_REF] Frentz | A subalgebra of the hardy algebra relevant in control theory and its algebraic-analytic properties[END_REF]. The most closely corona theorem related to the controllability of difference delay equations is stated in [START_REF] Sasane | Generators for rings of compactly supported distributions[END_REF]Corollary 3.3] for distributions positively compactly supported, but at the current state of the literature, it does not apply directly to the exact controllability of linear controlled delayed difference equations (LCDDE).

In this paper, we establish two results. The first one consists in the resolution of a corona theorem for a subalgebra of M (R -), the commutative Banach algebra made of Radon measures compactly supported in R -. More precisely, for a finite number of f 1 , ..., f N , each of them being a finite sum of Dirac measures supported in R -, we give a necessary and sufficient condition on the Laplace transform of the measures f 1 , ..., f N to obtain the existence of g 1 , ..., g N ∈ M (R -) such that

f 1 * g 1 + ... + f N * g N = δ 0 , (1)
where * denotes the convolution product and δ 0 the Dirac distribution at zero. That result is then used to derive an L 1 exact controllability criterion (in finite time) for LCDDE expressed in the frequency domain, thus solving an open question raised in [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF]. We emphasize that LCDDE can sometimes be used to address some control theoretic questions for 1-D hyperbolic partial differential equations [START_REF] Baratchart | Sufficient stability conditions for time-varying networks of telegrapher's equations or difference-delay equations[END_REF][START_REF] Coron | Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems[END_REF]. The strategy of proof for the corona problem goes as follows : in a first step, we reduce the corona problem (1) to a corona problem in a quotient Banach algebra. The second step goes by contradiction and relies on Gelfand representation theory characterizing maximal ideal as the kernel of homomorphisms, in the spirit of [START_REF] Böttcher | On the corona theorem for almost periodic functions[END_REF][START_REF] Frentz | A subalgebra of the hardy algebra relevant in control theory and its algebraic-analytic properties[END_REF]. It is not immediate how to deduce our corona theorem from these references and we include a proof of it for sake of clarity (yet very similar to that of [START_REF] Frentz | A subalgebra of the hardy algebra relevant in control theory and its algebraic-analytic properties[END_REF]). As for our second main result, it answers a question raised in [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF] where the sufficiency of a frequency domain criterium for L 1 exact controllability of a LCDDE was reduced to establishing the corona theorem established previously.

Prerequisites and definitions

We introduce the notations and the distributional framework needed in this article.

Notations

In this paper, we denote by N and N * the sets of nonnegative and positive integers, respectively. The set {1, . . . , N } is denoted by 1, N for any N ∈ N * . We use R, R + = [0, +∞), R * + , R -= (-∞, 0] and C to denote the sets of real numbers, nonnegative, positive, nonpositive real numbers and complex numbers respectively. For s ∈ C, ℜ(s) and ℑ(s) denote the real and imaginary part of s, respectively.

Radon measures framework

We give the Radon measures spaces that we use in this paper and for further details see for instance [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF]Section 2]. Denote C 0 (R) and C 0 (R + ) the Fréchet spaces of continuous functions with the topology induced by the uniform convergence on compact sets on R and R + respectively. The (topological) support of a function ϕ ∈ C 0 (R) is the closure of the set {x ∈ R| f (x) ̸ = 0}. We note by M (R -) and M + (R) the spaces of Radon measures defined on R with compact support included in R -and bounded on the left respectively. The support of a Radon measure α ∈ M + (R), denoted supp(α), is the complement of the largest open set on which α is zero. We note δ λ ∈ M (R -) the Dirac distribution at λ ∈ R -. Endowed with the convolution * , the two spaces M + (R) and M (R -) become commutative unital algebras where the unit is δ 0 .

For T > 0, we denote by Ω T -the subspace of M (R -) made of the elements h ∈ M (R -) of the form

h = N j=0 h j δ -λ j , λ j ∈ [0, T ], h j ∈ R, N ∈ N, (2)
where we assumed with no loss of generality that λ i ̸ = λ j when i ̸ = j. We introduce the subalgebra Ω bd -:= ∪

T ∈R + Ω T -of M (R -).
We define the (bilateral) Laplace transform in the complex plane C for µ ∈ M + (R) as

µ(s) = +∞ -∞ dµ(t)e -st , s ∈ C, (3)
provided that the integral exists. We have µ * ρ(s) = µ(s) ρ(s), for all µ, ρ ∈ M + (R) and s ∈ C. For all λ ∈ R, e sλ is the Laplace transform of the element δ -λ in s ∈ C. For an element µ ∈ M (R -), the Laplace transform reads:

µ(s) = 0 -∞ dµ(t)e -st , s ∈ C, (4)
where the previous integral is understood as a Lebesgue integral on (-∞, 0].

The truncation operator

We now define the truncation to positive times of a measurable function f defined on R as the following mapping π satisfying the equation

(πf ) (t) = f (t), if t ≥ 0, 0, if t < 0.
Let us introduce the space C 0,+ (R) the space of continuous functions with support bounded on the left. The following properties of the truncation operator can be easily proved.

Lemma 2.1. The following assertions hold true:

i) For α ∈ C 0,+ (R), we have π(α) = 0 if and only if supp(α) ⊂ (-∞, 0]. ii) π(α * β) = π(α * πβ) for every α ∈ M (R -) and β ∈ C 0,+ (R).
3 Topological properties of the quotient algebra M (R -)/(p)

The aim of this section is to study the topological structure of the quotient algebra M (R -)/(p) where (p) is the principal ideal generated by any p ∈ Ω T -⊂ M (R -), for some T > 0, with the assumption that p ̸ = 0 and the support of p is not reduced to the singleton {0}. In another words, p is given by

p = N j=0 p j δ -λ j , λ j ∈ [0, T ], p j ∈ R, N ∈ N, (5)
and there exists j ∈ {0, ..., N } such that p j λ j ̸ = 0. We next recall the framework developed by Y. Yamamoto in [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. Consider the bilinear form (

•, •) on M (R -) × C 0 (R + ) defined by (w, γ) := (w * πγ)(0) = 0 -∞ dw(τ )γ(-τ ), w ∈ M (R -), γ ∈ C 0 (R + ).
The space of Radon measures M (R -) is a normed algebra with the total variation norm

∥w∥ TV := sup ∥γ∥∞≤1, γ∈C 0 (R + ) |(w, γ)|, w ∈ M (R -),
where ∥γ∥ ∞ := sup

t∈R + |γ(t)| for γ ∈ C 0 (R + ). We define X p := {γ ∈ C 0 (R + ), π(p * πγ) = 0} .
and we introduce the orthogonal complement of X p

(X p ) ⊥ := {w ∈ M (R -), (w, γ) = 0, ∀γ ∈ X p } .
One can see from the definition of the orthogonal complement that (X p ) ⊥ is a closed subspace of M (R -). Thus we can define the normed quotient space M (R -)/ (X p ) ⊥ , see for instance [19, Proposition 3.1, (ii), Chap. 3], endowed with the norm

∥[w]∥ := inf γ∈(X p ) ⊥ ∥w + γ∥ TV , [w] ∈ M (R -)/ (X p ) ⊥ , (6)
where [w] ∈ M (R -)/ (X p ) ⊥ denotes any class of equivalence of M (R -)/ (X p ) ⊥ . We denote by (p) := {p * ψ| ψ ∈ M (R -)} the two-sided ideal generated by p over the commutative algebra M (R -). It turns out that the orthogonal complement of X p is in fact (p) and we give a proof of that similar in the spirit of [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]Lemma 2.18].

Lemma 3.1. The following equation holds,

(X p ) ⊥ = (p). Proof. Pick p * ψ ∈ (p) with ψ ∈ M (R -). For all γ ∈ X p , we have (p * ψ, γ) = (ψ * p * πγ)(0) = 0, (7)
because γ ∈ X p implies that p * πγ(t) = 0 for t ≥ 0. Thus (p) ⊆ (X p ) ⊥ . Conversely, let w ∈ (X p ) ⊥ . Take any ϕ ∈ D(R -), the space of smooth functions defined on R with compact support included in R -. A Neumann series argument proves that p is invertible in M + (R) with respect to the convolution, and we denote p -1 ∈ M + (R) its inverse. From Item ii) in Lemma 2.1, we have that the function

t ∈ R + → γ(t) := π(p -1 * ϕ)(t) belongs to X p and π(w * p -1 * ϕ)(0) = π(w * π(p -1 * ϕ))(0) = (w, γ) = 0, (8)
because w ∈ (X p ) ⊥ . If we take, for all t ∈ R + , δ -t * ϕ instead of ϕ in Equation ( 8), we get that π(w * p

-1 * ϕ)(t) = π(w * p -1 * δ -t * ϕ)(0) = 0. (9)
From ( 9), we have that π(w * p -1 * ϕ) is zero so that Item i) in Lemma 2.1 implies that the support of w * p -1 * ϕ ∈ C 0,+ (R) is included in (-∞, 0]. Since it holds for any ϕ ∈ D(R -), we have that w * p -1 lies in M (R -). In particular, there exists ψ ∈ M (R -) such that w = p * ψ.

We deduce that (X p ) ⊥ ⊆ (p), achieving the proof of the lemma.

Thanks to Lemma 3.1, we have that the quotient normed space M (R -)/ (X p ) ⊥ is in fact the normed quotient algebra equal to M (R -)/(p) with unit [δ 0 ], see for instance [13, Sect 1.4, Lemma 1.4.4] for a reference on normed quotient algebras. In particular, we have that

[w 1 * w 2 ] = [w 1 ] * [w 2 ] and [w 1 + w 2 ] = [w 1 ] + [w 2 ] for all w 1 , w 2 ∈ M (R -)/(p)
. Our next step is to derive the following properties for the quotient algebra M (R -)/(p), which are a specification of [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]Lemma 2.21] in the framework of our article.

Theorem 3.2. The quotient algebra M (R -)/(p) is a commutative unital Banach algebra with [δ 0 ] as unit. Furthermore, we have

∥[w]∥ = sup ∥γ∥ [0,T ] ≤1, γ∈X p |(w, γ)|, [w] ∈ M (R -)/(p). (10)
Proof. We already know that M (R -)/(p) is a commutative unital algebra with unit [δ 0 ]. It remains to prove that it is a Banach algebra. We have that X p ⊂ C 0 (R + ) with the topology induced by the uniform convergence on compact sets. By the definition of p ̸ = 0, we have that γ ∈ X p if and only if γ ∈ C 0 (R + ) and it satisfies the difference delay equation

N j=0 p j γ(t + λ j ) = 0, t ≥ 0, (11)
where p j λ j ̸ = 0 for some j ∈ {0, ..., N }. Thanks to Equation [START_REF] Hale | Introduction to Functional Differential Equations[END_REF], we have that the values on R of the function γ are entirely constrained by the value of γ on the interval [0, T ]. Thus, the topology on X p is equivalent to the topology induced by the uniform convergence on the interval [0, T ]. Therefore X p is a Banach space endowed with the norm ∥ϕ∥ [0,T ] = sup

t∈[0,T ] |ϕ(t)|
with ϕ ∈ X p . We denote by (X p ) ′ the topological dual of X p , i.e. the space of continuous linear forms on X p with respect to the topology induced by the norm ∥ • ∥ [0,T ] . We have that the space (X p ) ′ is a Banach space endowed with the norm

∥x∥ (X p ) ′ := sup ∥ϕ∥ [0,T ] ≤1, ϕ∈X p |⟨x, ϕ⟩ X p |, x ∈ (X p ) ′ ,
where ⟨•, •⟩ X p denotes the duality product on X p . We define the linear map

h : M (R -)/ (X p ) ⊥ → (X p ) ′ [w] → (ϕ ∈ X p → (w, ϕ)) . (12)
We claim that the linear map h is well-defined and is an isometric isomorphism between M (R -)/ (X p ) ⊥ and (X p ) ′ , which is the conclusion of our theorem because, thanks to Lemma 3.1, we have (X p ) ⊥ = (p).

For every [w] ∈ M (R -)/ (X p ) ⊥ , we have that y ∈ [w] if and only if y = w + ψ with ψ ∈ (X p ) ⊥ . Thus by definition of the orthogonal complement, the map h is well defined because it does not depend on the choice of the represent w ∈ M (R -).

The linear map h is injective: reasoning by contradiction, there exists [w] ̸ = 0 ∈ M (R -)/ (X p ) ⊥ such that h([w]) = 0, i.e., w ∈ (X p ) ⊥ , which is a contradiction.

We finally show now that the map h is onto and it is an isometry. An element f ∈ (X p ) ′ is a continuous linear functional for the topology induced by the convergence on compact sets. By the Hahn-Banach extension theorem, we can extend f on a continuous linear functional f belonging to (C 0 (R + )) ′ , the dual space of C 0 (R + ) with the duality product ⟨

•, •⟩ C 0 (R + ) , such that ⟨ f , x⟩ C 0 (R + ) ≤ ∥f ∥ (X p ) ′ max t∈[0,T ] |x(t)|, x ∈ C 0 (R + ). ( 13
)
By the Riesz representation theorem, there exists

ψ ∈ M (R -), with compact support in- cluded in [-T, 0] such that ⟨ f , x⟩ C 0 (R + ) = (ψ, x) for all x ∈ C 0 (R + ) and ∥ψ∥ TV = ∥f ∥ (Xp) ′ .
Furthermore, for all ϕ ∈ (X p ) ⊥ , we have

∥ψ + ϕ∥ TV = sup ∥x∥∞≤1, x∈C 0 (R + ) |(ψ + ϕ, x)| ≥ sup ∥x∥∞≤1, x∈X p |(ψ + ϕ, x)| = sup ∥x∥ [0,T ] ≤1, x∈X p |(ψ, x)| = ∥ψ∥ TV . (14)
Thus we have

∥[ψ]∥ = ∥ψ∥ TV . We deduce that h([ψ]) = f and ∥h([ψ])∥ (X p ) ′ = ∥f ∥ (X p ) ′ = ∥[ψ]∥.
To sum up, we proved that the map h is an isometric isomorphism between M (R -)/ (X p ) ⊥ and (X p ) ′ , achieving the proof of our theorem.

A corona theorem for a subalgebra of Radon measures negatively and compactly supported

For the Banach algebra M (R -)/(p), we call homomorphism a continuous linear mapping ϕ :

M (R -)/(p) → C satisfying ϕ(F G) = ϕ(F )ϕ(G) for all F, G ∈ M (R -)/(p).
Recall that a character χ is application from R + to C such that |χ(t)| = 1 and χ(t + τ ) = χ(t)χ(τ ) for all t, τ ∈ R + . We first give in Proposition 4.1 a description of the nonzero homomorphisms on M (R -)/(p). (1) for every h ∈ Ω bd -given by (5):

ϕ([h]) = h j , if there is λ j = 0 and h j ̸ = 0, 0, otherwise.

(2) or there exist σ ∈ R and a character χ such that, for every h ∈ Ω bd -given by (5),

ϕ([h]) = N j=0
h j e σλ j χ (λ j ) ,

Proof. Let ϕ be a nonzero homomorphism ϕ ̸ = 0 on M (R -)/(p). In particular, by the continuity property, there exists C > 0 (in fact C can be taken equal to one because we are in a unital Banach algebra) such that:

|ϕ([h])| ≤ C∥[h]∥, ∀[h] ∈ M (R -)/(p). ( 17
)
For t ≥ 0, set L(t) = |ϕ([δ -t ])|, yielding a well-defined map from R + to R + . We deduce from the equations ( 10) and ( 17) that L is bounded over the interval [0, T ]. Furthermore, from the property of homomorphisms, we deduce that L is a multiplicative map, that is,

L(t 1 + t 2 ) = L(t 1 )L(t 2 ), t 1 , t 2 ∈ R + . (18)
Equation ( 18) is a Cauchy equation of exponential type, see for instance [START_REF] Kannappan | Functional equations and inequalities with applications[END_REF]. Since ϕ is a nonzero homomorphism, there exists t 0 ≥ 0 such that L(t 0 ) = c > 0 for some t 0 ∈ R + . Thus we have c = L(t 0 ) = L(t 0 )L(0) = cL(0) and we deduce that L(0) = 1. Following the discussion in [14, Paragraph 1.5.1], if there exists t * > 0 such that L(t * ) = 0 then L(t) = 0 for every t > 0. In that case, L is called the trivial solution to the Cauchy equation of exponential type. Otherwise, the application of [14, Theorem 1.37] gives the existence of σ ∈ R such that L(t) = e σt for every t ≥ 0. In summary, L(0) = 1 and we have the following alternative:

(a) either L(t) = 0 for t > 0 and then ϕ([δ -t ]) = 0 for t > 0 and ϕ([δ 0 ]) = 1, i.e., this corresponds to Item (1) in the theorem with the help of Equation ( 5);

(b) or there exists σ ∈ R such that L(t) = e σt for t ≥ 0, and then ϕ([δ -t ]) = e σt χ(t) with χ(t) equal to ϕ([δ -t ])e -σt which verifies |χ(t)| = 1 for t ≥ 0, i.e., χ is a character. According to Equation ( 5), one gets Item (2) in the theorem.

We can now state and prove the corona theorem of this paper.

Theorem 4.2. Let K be a positive integer and T be a strictly positive real number. Consider

f i ∈ Ω T -for i = 1, . . . , K. If there exists α > 0 such that K i=1 fi (s) ≥ α, ∀s ∈ C, (19)
then there exist

g i ∈ M (R -) for i = 1, . . . , K satisfying K i=1 f i * g i = δ 0 . ( 20
)
Remark 4.3. Condition ( 19) is the same as that the condition of the corona theorem for H ∞ by Carleson [START_REF] Carleson | Interpolations by bounded analytic functions and the corona problem[END_REF]. However, our corona theorem is much simpler because we worked in the algebra M (R -) and we stated an interpolation result just for the elements belonging to Ω T -, for some T > 0. More precisely, the properties of the homomorphisms given in Proposition 4.1 are harder to obtain for the algebra H ∞ . Furthermore, contrary to the corona theorem in H ∞ , we did not provide an estimate on the Laplace transform of the g i depending on K and α.

Proof. Notice first that if K = 1 then the conclusion holds trivially (since in that case f 1 = h 1 δ -λ 1 with h 1 ̸ = 0) and we will assume then that K ≥ 2 in the sequel. Moreover, one deduces from Condition [START_REF] Swartz | Elementary Functional Analysis[END_REF] that either every f i is zero or a nonzero multiple of δ 0 (and the result is again immediate or at least one of the f i 's (let say f K ) has a nonempty support with a non zero element in its support. We will assume the latter in the sequel.

The first step of the proof consists in reducing the corona problem as stated in M (R -) into a corona problem in the commutative unital Banach quotient algebra A = M (R -)/ (f K ), where (f K ), the two-sided ideal generated by f K over the commutative normed algebra R -, is defined as {f K * h| h ∈ M (R -)}. We note by [•] a class of equivalence of the quotient algebra A. Hence, we can interpret Equation (20) as

K-1 i=1 [f i ] * [g i ] = [δ 0 ]. (21)
Proving the theorem amounts to prove the existence of [g i ] ∈ A, i = 1, ..., K -1 satisfying [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. Thanks to Theorem 3.2, A is a commutative unital Banach algebra, and so we can use the Gelfand theory [8, Chap VII, §8]. Equation ( 21) is equivalent to the fact that [δ 0 ] belongs to the two-sided ideal ([

f 1 ], • • • , [f K-1 ]) generated by [f 1 ], • • • , [f K-1
] over the commutative algebra A and defined as

{[f 1 ] * [h 1 ] + . . . + [f K-1 ] * [h K-1 ]| [h 1 ], . . . , [h K-1 ] ∈ A}.
In other words, Equation ( 21) is equivalent to the fact that ([

f 1 ], • • • , [f K-1 ]) is equal to A.
Reasoning by contradiction, let us assume that ([

f 1 ], • • • , [f K-1 ]
) is not equal to A and hence it is a proper ideal of A which is, according to Krull's theorem (see for instance [START_REF] Rudin | Functional Analysis. International series in pure and applied mathematics[END_REF]Theorem 11.3]), included into a maximal ideal of A. In particular, [f 1 ], ..., [f K ] belong to a maximal ideal of A. The Gelfand representation theory states that the maximal ideals are in bijection with the nonzero complex homomorphisms of A so that a maximal ideal is included into the kernel of a unique nonzero homomorphism, see for instance [8, Proposition 8.2, Chap. VII]. Hence, there exists a nonzero homomorphism ϕ of A for which

ϕ([f 1 ]) = ϕ([f 2 ]) = • • • = ϕ([f K ]) = 0. ( 22
)
If ϕ is given by Item (1) of Proposition 4.1, then the limit of the left-hand side of (19) tends to zero as ℜ(s) tends to -∞, which contradicts Equation [START_REF] Swartz | Elementary Functional Analysis[END_REF]. Assume now that ϕ is given Item (2) of Proposition 4.1. For every k ∈ 1, K , the function fk can be written as

fk (s) = n k l=0 f k,l e sλ k,l , s ∈ C, (23)
where n k is an integer, f k,l a real number and λ k,l ∈ [0, T ]. We deduce from ( 22), ( 23) and ( 16) in Item (2) of Proposition 4.1 that there exist σ ∈ R and a character χ such that

n k l=0 f k,l e σλ k,l χ(λ k,l ) = 0, k ∈ 1, K . (24)
We remark that, thanks to [6, Proposition 3.9], there exist a positive integer q, a rationally independent family (r 1 , . . . , r q ) of positive real numbers, and nonnegative integers m k,l,j for l ∈ 1, n k , k ∈ 1, K and j ∈ 1, q such that λ k,l = q j=1 m k,l,j r j .

(25)

Since |χ(t)| = 1 for all t ∈ R, we have χ(r j ) = e 2πiγ j for some γ j ∈ R and for j = 1, ..., q. It follows that

χ(λ k,l ) = e 2πi q j=1 m k,l,j γ j , l ∈ 1, n k , k ∈ 1, K . (26)
By the Kronecker approximation theorem (see e.g. [12, Theorem 2, Chapter 2]), for every ϵ > 0, there exist a real number β and integers p 1 , ..., p q such that |βr j -γ j -p j | ≤ ϵ, for j = 1, ..., q.

(27)

From Equations ( 25)-( 26 m k,l,j (γ j +p j -βr j )

Using (27) in the above equation, one gets that there exists C > 0 such that, for all k = 1, ..., K and l = 1, ..., n k , we have:

χ(λ k, l ) -e 2πiβλ k, l ≤ Cϵ. (29)
Let us define:

s ϵ = σ + iβ ∈ C and C = sup k∈ 1,K n k l=0 |f k,l |. (30)
Hence, from equations ( 24)-( 29)-(30), we get for all k = 1, ..., K:

|f k (s ϵ )| = f k (s ϵ ) - n k l=0 f k,l e σλ k,l χ(λ k,l ) , = n k l=0 f k,l e σλ k,l χ(λ k,l ) -f k,l e σλ k,l e 2πiβλ k,l , ≤ n k l=0 |f k,l | χ(λ k,l ) -e 2πiβλ k,l , ≤ C Cϵ. (31)
Letting ϵ tend to zero and using (31), we build a sequence of complex numbers (s

n ) n∈N such that lim n→+∞ f1 (s n ) = • • • = lim n→+∞ fK (s n ) = 0, (32)
which contradicts Equation [START_REF] Swartz | Elementary Functional Analysis[END_REF]. That completes the proof of Theorem 4.2.

Remark 4.4. Two questions remain open. Is it possible to find g 1 , ..., g K ∈ M (R -) (resp. Ω bd -), in the case where f 1 , ..., f K ∈ M (R -) (resp. Ω bd -), satisfying [START_REF] Treil | The matrix-valued hp corona problem in the disk and polydisk[END_REF] if [START_REF] Swartz | Elementary Functional Analysis[END_REF] holds? Corona questions for measures can fail to have positive answers hold true as proved by the Wiener-Pitt phenomenon, see for instance [START_REF] Nikolski | In search of the invisible spectrum[END_REF]. For M (R -), the characterization of the nonzero homomorphisms of M (R -) does not seem to be stated in the literature and therefore a corona theorem for this algebra is an open question.

As application, we use Theorem 4.2 to establish a L 1 -exact controllability of linear controlled delayed difference equations.

L 1 exact controllability of linear difference delay control systems

The motivation to prove Theorem 4.2 arises from the study of the exact controllability problem of LCDDE. More precisely, let us consider a linear difference delay control system of the form

x(t) = N j=1 A j x(t -Λ j ) + Bu(t), t ≥ 0, (33)
where, d and m are two integers, the state x and the control u belong to R d and R m respectively, and A 1 , . . . , A N and B are constant matrices with real entries of appropriate size. Without loss of generality, the delays Λ 1 , . . . , Λ N are positive real numbers so that Λ

1 < • • • < Λ N .
Since an LCDDE defines a infinite-dimensional dynamical system, we must introduce the functional spaces defining the state space and the control space of System (33). If I is a bounded interval of R and n ∈ N * , we note L 1 (I, R n ) the space of integrable functions on I with values in R n .

For every t ≥ 0, u ∈ L 1 [0, t], R m , and

x 0 ∈ L 1 [-Λ N , 0], R d , there exists a unique solution x ∈ L 1 [-Λ N , t], R d such that x(θ) = x 0 (θ) for almost all θ ∈ [-Λ N , 0] and x(•) satisfies Equation (33) for almost all t ∈ [0, t], cf. [7, Proposition 2.2].
We aim at reaching elements of L 1 [-Λ N , 0], R d with an integrable control in a finite time along trajectories of (33). For that purpose, we introduce the following definition of exact controllability. Definition 5.1. System (33) is L 1 exactly controllable in time T > 0 if for every x 0 , ϕ ∈ L 1 [-Λ N , 0], R d , there exists u ∈ L 1 ([0, T ], R m ) such that the solution x(•) of System (33) starting at x 0 and associated with the control u verifies

x(T + θ) = ϕ(θ), for almost all θ ∈ [-Λ N , 0]. (34)
In [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF], it is proved that the L 1 exact controllability of System (33) is equivalent to the resolution of a Bézout identity over the algebra of Radon measures compactly supported in R -, see [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF]Theorem 5.13]. This characterization allows one to give a necessary condition for the L 1 exact controllability but the remaining question whether this condition is also sufficient or not was left open in that reference. Using Theorem 4.2, we bring a positive answer to this question. Since the L 1 -controllability criterion is expressed in the frequency domain, we introduce the matrix-valued holomorphic map

H(s) := I d - N j=1 e -sΛ j A j , s ∈ C, (35)
where I d is the identity operator on R d . The matrix H(•) relates the control frequency with the state space frequency. More precisely, assuming that u ∈ L 1 (R, R m ) and u(t) = x(t) = 0 for t < 0, we take the one-sided Laplace transform in (33) and we obtain that there exists α > 0 such that: H(s)X(s) = BU (s), s ∈ C, ℜ(s) > α, We state a sufficient and necessary criterion for the L 1 exact controllability for (33) in the frequency domain. Proof. Theorem 4.2 solves [5, Conjecture 5.18] in the particular case where the q i belongs to Ω T -instead of M (R -) for all i = 1, ..., N . Hence, Remark 5.19 and the discussion just below in the paper [START_REF] Chitour | Hautus-yamamoto criteria for approximate and exact controllability of linear difference delay equations[END_REF] allow us to conclude that [5, Conjecture 5.18] is true for q = 1, which is the result that we wanted.

Proposition 4 . 1 .

 41 If ϕ ̸ = 0 is a homomorphism in M (R -)/(p) then either:

  ), we obtain for all k = 1, ..., K and l = 1, ..., n k χ(λ k, l ) -e 2πiβλ k, l = e

  )e -st dt and U (s) = +∞ -∞ u(t)e -st dt. (37) The existence of α > 0 such that Equation (36) is satisfied follows from classical exponential estimates for difference delay equations, see [11, Chapter 9]. We note H(C) the closure of the holomorphic matrix H(•) in the complex plane C. The d × (d + m) matrix [M, B] denotes the concatenation of a d × d matrix M and the matrix B. Furthermore, rank [M, B] denotes the dimension of the range of the matrix [M, B].

Theorem 5 . 2 .

 52 System (33) is L 1 exactly controllable in time dΛ N if and only if the two following conditions hold:i) rank [M, B] = d for every M ∈ H(C), ii) rank[A N , B] = d.

Acknowledgement:

The authors would like to thank N. Nikolski for discussions on corona problems for measures and Brett D. Wick for literature references on corona theorems for almost periodic functions.


Carole Vitellius
email: carole.vitellius@chu-angers.fr

Anita Paisant
email: anita.paisant@chu-angers.fr

Adrien Lannes
email: adrien.lannes@chu-angers.fr

Julien Chaigneau
email: julien.chaigneau@univ-angers.fr

Frédéric Oberti
email: froberti@chu-angers.fr

Jérôme Lebigot
email: jelebigot@chu-angers.fr

Isabelle Fouchard
email: isabelle.fouchard@chu-angers.fr

Jérôme Boursier
email: jeboursier@chu-angers.fr

Pascal David
email: p.david3@wanadoo.fr

Christophe Aubé
email: chaube@chu-angers.fr

Paul Calès
email: paul.cales@univ-angers.fr

Liver fibrosis staging by computed tomography: prospective randomized multicentric evaluation of image analyses

Keywords: ICCC: intra-class correlation coefficient NAFLD: non-alcoholic fatty liver disease computed tomography, liver fibrosis, cirrhosis, staging, non-invasive diagnosis

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

INTRODUCTION

The evaluation of liver fibrosis is essential in the management of patients with chronic liver disease (CLD). However, due to multiple biases, liver biopsy cannot be considered a gold standard, but rather only a best standard [START_REF] Bedossa | Liver biopsy: the best, not the gold standard[END_REF]. Many teams have developed non-invasive assessments of liver fibrosis, particularly blood fibrosis tests and transient elastography [START_REF] The | EASL Clinical Practice Guidelines On Non-Invasive Tests For Evaluation Of Liver Disease Severity And Prognosis-2021 Update[END_REF].

Radiological imaging could provide a double advantage of enabling both a fibrosis assessment based on dysmorphia and an exploration of the entire liver in a single examination [START_REF] Laghi | Multiphase multislice spiral CT for liver assessment: optimization in cirrhotic patients[END_REF]. Computed tomography (CT) is a widely available and frequently employed imaging technology, but its potential in this setting has received little attention and, despite such recent techniques as texture analysis [START_REF] Budai | Three-dimensional CT texture analysis of anatomic liver segments can differentiate between low-grade and high-grade fibrosis[END_REF], no classification of liver fibrosis stages is available for it.

Our objective was to evaluate the contribution of descriptors for fibrosis staging while avoiding classical biases: observer expertise and liver biopsy as reference. Our ensuing working hypothesis was two-fold. First, a way to avoid expertise bias would be to employ a naive population of investigators. Second, the bias due to liver biopsy might by avoided by using Metavir F concordant over several independent non-invasive tests (NITs) as a reference, as done in diagnostic studies without gold standards [START_REF] Rutjes | Evaluation of diagnostic tests when there is no gold standard. A review of methods[END_REF]. Thus, our primary objective with the present proof of concept study was to validate radiological staging for liver fibrosis. Our secondary objectives were to compare image analysis fibrosis stagings done by radiologist to those done by computer, and stagings done by clinicians to those done by radiologists. The present study, called comprehensive dynamic medical imaging report (CDMIR), was a prospective diagnostic study with stratified randomization of observers and patients.

METHODS

Overall presentation

Images from 10 patients were compared under three formats: native raw images as internal controls; images enriched by expert annotations; and images enriched by computerized morphometry. The main evaluators were naive hepato-gastroenterology (HGE) residents.

Patients

Patients were selected from the MALAH2 CLD population (details in Supplemental Material). A patient flowchart is provided in Figure S1 in Supplemental Material. In a preliminary study, liver CT images were evaluated by three senior radiologists in 106 patients [START_REF] Cales | Morphometry on CT improves non-invasive diagnosis of hepatic fibrosis[END_REF]. Ten of those 106 patients were included in the present CDMIR study, based on fibrosis staging concordance over four independent diagnostic tools for liver fibrosis (Metavir F, blood test, elastometry and CT morphometry, Table S1 in Supplemental Material) to optimize the histological Metavir reference for liver fibrosis [START_REF] Bedossa | Liver biopsy: the best, not the gold standard[END_REF][START_REF] Rutjes | Evaluation of diagnostic tests when there is no gold standard. A review of methods[END_REF]. Patients were stratified for the five Metavir F stages measured on liver specimens, i.e., two in each Metavir F stage. The main characteristics of the 10 patients included in the present study were: age: 56.8±15.7 years, male sex: 40.0%, body mass index: 30.4±5.5 kg/m², non-alcoholic fatty liver disease (NAFLD): 80.0%. Their individual liver fibrosis descriptors are provided in Tables S1 andS2.

Investigators

Observers -Seventy HGE residents ("observers" hereafter) working in five university hospital centres ("centre" hereafter) participated in the study. Their inclusion into three groups was randomly stratified according to two criteria: centre (n=5) and internship year (n=4). There were no significant differences in observer characteristics between the three groups (Table S3).

Radiologists -Four senior radiologists participated in the study. First, an HGE expert radiologist (9 years of experience; "descriptor expert" hereafter) selected and annotated the images used for observer evaluation. Three radiologists evaluated images: two specialists in HGE (22 and 28 years of experience) and one non-specialist in HGE (29 years in general radiology). These three radiologists ("evaluators" hereafter) independently evaluated the same raw (i.e. those without annotations) CT images as the HGE observers did. Unlike HGE observers, the radiologists had no time constraints.

Liver imaging

Classical semiology -Based on findings reported by expert HGE radiologists in previous studies [START_REF]Imaging modalities for the diagnosis of hepatic fibrosis and cirrhosis[END_REF][START_REF] Indiran | Right posterior hepatic notch sign[END_REF][START_REF] Aube | Ultrasonographic diagnosis of hepatic fibrosis or cirrhosis[END_REF][START_REF] Kudo | Diagnostic accuracy of imaging for liver cirrhosis compared to histologically proven liver cirrhosis. A multicenter collaborative study[END_REF], seven conventional, simple descriptors of liver fibrosis and cirrhosis were selected [START_REF] Cales | Morphometry on CT improves non-invasive diagnosis of hepatic fibrosis[END_REF]: 1/ liver edge (normal, moderate or marked irregularity), 2/ right posterior notch (present/absent) [START_REF] Indiran | Right posterior hepatic notch sign[END_REF], 3/ segment I hypertrophy (absent, moderate or marked), 4/ segment IV hypotrophy (absent, moderate or marked), 5/ splenomegaly (absent or present according to spleen size measurement along its major axis), 6/ coarse appearance of liver parenchyma (absent, moderate or marked) and 7/ portosystemic collaterals (absent, minimal or marked) (Figure 1A).

CT -CT images without or with contrast medium injection at portal time were used. The technique is detailed in the Supplemental Material.

New CT classification of liver fibrosis

The CT classification of liver fibrosis stages was developed during a consensus meeting between the most experienced radiologist and the principal investigator. Histological Metavir F staging was the reference [START_REF]Intraobserver and interobserver variations in liver biopsy interpretation in patients with chronic hepatitis C. The French METAVIR Cooperative Study Group[END_REF] and the seven previous radiological descriptors were considered. The extreme radiological liver fibrosis stages (FR) were defined first as normal (FR0) and cirrhosis (FR4) according to the absence or presence of classical signs. Thereafter, two intermediate stages were defined, i.e., moderate or marked abnormalities without a firm cirrhosis diagnosis (FR3) and moderate abnormalities with no doubt as to the absence of cirrhosis (FR2). A theoretical FR1 stage was grouped with FR0 as the experts estimated that their distinction was not possible in principle. Thus, the four FR stages were: stage FR0/1 for a (near) normal liver, FR2 for significant fibrosis, FR3 for marked fibrosis or possible cirrhosis and FR4 for definitive cirrhosis.

Morphometric analysis -This technique included computerized image analysis as described elsewhere [START_REF] Cales | Automated morphometry provides accurate and reproducible virtual staging of liver fibrosis in chronic hepatitis C[END_REF]. From the preliminary study, we selected images of six native descriptors among 43 morphometric descriptors [START_REF] Cales | Morphometry on CT improves non-invasive diagnosis of hepatic fibrosis[END_REF]. Coloured masks were possible (Figure 1B). The six descriptors were the independent predictors of fibrosis stages or cirrhosis included by multivariate analysis into two diagnostic scores (for Metavir F or cirrhosis). The ensuing eight measurements (the six descriptors plus the two scores) are detailed in Table S2.

Study design

Study conduct

We conducted the observer evaluation during a workshop session performed in Angers on 16 October 2018. All participants were blinded to the biological/clinical background of patients until the end of the evaluation session.

Definitions

Objectives -The primary objective was to develop radiological staging for liver fibrosis and evaluate its performance (mainly discrimination). The secondary objectives were to a) compare diagnostic accuracies for fibrosis staging between images enriched by radiologist annotations and images enriched by morphometric analysis, and between HGE observers and radiologists; b) measure fibrosis staging agreement between observers and liver pathology (i.e. calibration) or observers themselves (i.e. reproducibility); c) compare agreement of fibrosis stagings obtained by the observers on CT and those obtained by different pathologists on liver biopsy.

Outcome measurements -The first primary outcome measurement was the rate of correctlyclassified patients (accuracy in percentages hereafter). This was expressed with four diagnostic descriptors: pre-and post-analysis accuracies and the accuracy course expressed either as a qualitative descriptor for observers (stable, regression, progression) or as a quantitative descriptor, i.e., post-analysis accuracy -baseline accuracy, called net change hereafter. The classification reference was Metavir F stages by liver biopsy. The main diagnostic targets were fibrosis staging and cirrhosis. The second primary outcome measurement was the FR discrimination between Metavir F either for binary fibrosis targets (≥F1: fibrosis, ≥F2: significant fibrosis, ≥F3: marked fibrosis, F4: cirrhosis) by AUROC or all F by Obuchowski index or between two adjacent F.

The secondary outcome measurements were observer agreement, either interobserver (reproducibility) or with Metavir F staging (i.e. calibration of radiological staging), the comparison of accuracies between HGE observers and radiologists, and the comparison of agreement between observers and pathologists. This last outcome is reported in the Supplementary Material.

Study -This work was a prospective (data collection organized upstream of the performance of index tests and the reference standard with prospective aim) [START_REF] Bossuyt | an updated list of essential items for reporting diagnostic accuracy studies[END_REF] multicentre study with double stratified randomization (patients and observers). It was a type 1a predictive study according to TRIPOD classification [START_REF] Collins | Transparent reporting of a multivariable prediction model for individual prognosis or diagnosis (TRIPOD): the TRIPOD statement[END_REF]. Data were reported according to STARD [START_REF] Bossuyt | an updated list of essential items for reporting diagnostic accuracy studies[END_REF] and Liver FibroSTARD [START_REF] Boursier | An extension of STARD statements for reporting diagnostic accuracy studies on liver fibrosis tests: The Liver-FibroSTARD standards[END_REF] statements.

Evaluation session

The design of the observer evaluation session is summarized in Table 1. Details are reported in Supplemental Material with examples in Figure 1. Briefly, the observers were randomized into three groups according to the images evaluated. Images were evaluated twice. First, the same raw images in all groups. Second, the same images with different formats according to groups: control group 1C with raw images, group 2A with annotated images, and group 3B with images analysed by bio-morphometry.

Regulatory aspects

Patients -The CDMIR study was conducted in the setting of the SNIFF cohort, which included all patients with CLD managed in our centre. Written, informed consent was obtained from all patients.

Observers -the study, termed an 'educational research study', was not considered as 'Research Involving the Human Person' as defined in article R1121-1 of the French Public Health Code, and thus no Ethics Committee approval was required for the observers.

Statistical analysis

Quantitative variables were expressed as mean ± standard deviation. Comparisons between groups were done using an ANOVA parametric test or Chi² test. The 2 to 2 comparisons were carried out using an unpaired t-test for continuous variables and a Fischer test for qualitative variables. For intra-group comparisons (baseline and post-analysis data), a paired t-test was used. AUROCs were compared by the paired Delong test or unpaired reduced gap test.

Discrimination between the five Metavir F stages was described by the Obuchowski index [START_REF] Lambert | How to measure the diagnostic accuracy of noninvasive liver fibrosis indices: the area under the ROC curve revisited[END_REF] (details in Supplemental Material). The comparisons were also performed using nonparametric tests. These latter provided similar results and are thus not reported here. No weighting was introduced for multiple comparisons [START_REF] Katz | Multivariate analysis. A practical guide for clinicians[END_REF]. Multivariate analyses were performed by backward binary logistic regression with or without interaction terms, excluding colinearity >0.8. Inter-observer agreement was assessed by the kappa index and the intra-class correlation coefficient (ICCC). The sample size is discussed in the Supplemental Material.

Statistical analyses were performed mainly with SPSS software version 18.0 (IBM, Armonk, NY, USA).

RESULTS

Accuracy

Fibrosis staging

Individual observer description -Individual baseline and post-analysis accuracies are detailed in Figure S2. In group 1C (n=22, with the same images in the two steps), mean accuracy (in 10 patients) was stable in 4 observers, progressed in 11 and regressed in 7. In group 2A (n=21), mean accuracy progressed in all observers. In group 3B (n=27), mean accuracy was stable in 7 observers, progressed in 15 and regressed in 5. So, the main trend was 100% progression in group 2A vs around 50% in other groups (p<0.001, Table S4). 2A and Figure 2A) -The overall (all groups merged) baseline FR accuracy for Metavir fibrosis stages was 43% without significant differences between the three groups (p=0.186). The overall post-analysis accuracy was 60% with significant differences between the three groups (p<0.001). Specifically, post-analysis accuracy was higher in group 2A (81%) compared to group 1C (51%, p<0.001) and to group 3B (52%, p<0.001), but it did not significantly differ between groups 1C and 3B (p=0.786). Therefore, the net change (absolute difference) was significantly higher in group 2A than in the others.

Univariate analysis (Table

Multivariate analysis -Metavir F had a positive influence (p<0.001) and centre 5 had a negative influence (p=0.017) on baseline accuracy for fibrosis FR staging but not the group (Table S5A). For post-analysis accuracy, group 2A (p<0.001) had a positive influence in addition to the two previous predictors.

Cirrhosis diagnosis

Univariate analysis (Table 2B and Figure 2A) -The overall baseline FR accuracy for cirrhosis was 84% without significant differences between the three groups (p=0.163). The overall post-analysis accuracy was 89% with significant differences between the three groups (p<0.001). Specifically, the post-analysis accuracy was higher in group 2A (96%) compared to group 1C (82%, p<0.001) or 3B (90%, p=0.012) and it did differ significantly between groups 1C and 3B (p=0.012). Therefore, the net change was significantly higher in groups 2A and 3B than it was in controls.

Multivariate analysis -Metavir F (p<0.001) and centre 3 (p=0.023) had a negative influence on baseline accuracy for cirrhosis but not the group (Table S5B). For post-analysis accuracy, Metavir F (p<0.001) had a negative influence while groups 2A (p<0.001) and 3B (p=0.008) had positive influences.

Fibrosis staging vs cirrhosis diagnosis

As expected, the FR accuracy was significantly higher for cirrhosis than for fibrosis staging in the baseline (p<0.001) and post-analysis (p<0.001) steps (Table 2C). In contrast, the net change was significantly lower for cirrhosis than it was for fibrosis staging (p<0.001).

Sensitivity analysis

Centre

Accuracies as a function of centres and groups are shown in Figures 2B-D.

Fibrosis staging -There was no significant difference in baseline accuracy between centres (p=0.258) (Table S6A). There was a significant difference in post-analysis accuracy (p<0.001) or net change (p=0.025) between centres with a significantly lower progression in centre 5 vs other centres due to a negative net change in group 3B (Table S7).

Cirrhosis diagnosis -There was no significant difference in baseline accuracy (p=0.134) (Table S6B), post-analysis accuracy (p=0.655), and net change (p=0.554) between the five centres.

Internship year

Regarding the four accuracy descriptors, there were no significant differences for fibrosis staging and cirrhosis diagnosis according to the internship years (Table S8).

Metavir F

Fibrosis staging -Baseline and post-analysis accuracies for fibrosis staging were significantly different as a function of Metavir F (p<0.001) (Table S9). Both accuracies were the highest in Metavir F4 and the lowest in Metavir F1 whatever the group (Table S10). The net change was significant, with progression in all Metavir F (borderline significance in F4).

Cirrhosis diagnosis -Baseline and post-analysis accuracies for cirrhosis diagnosis were

significantly different as a function of Metavir F with the highest in Metavir F≤2 and the lowest in Metavir F≥3 (Table S9). Therefore, the accuracies for cirrhosis were significantly higher in patients without than in patients with cirrhosis in all groups (Table S11). However, there was a notable exception in group 2A: post-analysis accuracy was 96% in F≤3 vs 98% in F4 (p=1). The net change was significant (progression) only in F3 (Table S9).

AUROCs

The AUROCs of observer fibrosis stagings were measured for four diagnostic binary targets according to Metavir F cut-offs (Table 3): F≥1 (see discussion for this unexpected target), F≥2, F≥3 and F=4. In baseline evaluation, AUROCs were not significantly different between groups (except a borderline significant decrease in group 3B for F≥1). The post-analysis AUROCs of group 2A were significantly increased vs the baseline AUROCs of that group and vs the post-analysis AUROCs of the other groups. The post-analysis AUROC for cirrhosis of group 3B was significantly higher than the post-analysis AUROCs of group 1C.

Finally, baseline overall (all groups merged) AUROCs were satisfactory (≥0.83) for marked fibrosis (F≥3 or cirrhosis) whereas post-analysis AUROCS were excellent (≥0.89) in group 2A for all diagnostic targets (especially ≥0.98 for F≥3 and cirrhosis) and in group 3B for cirrhosis (0.93). The Obuchowski index, reflecting discrimination between Metavir F stages, significantly (p<0.001) progressed in groups 2A and 3B with the greater progression in group 2A, from 0.66 to 0.87 (Table 3). This provided excellent negative (NPV: 98%) and positive (PPV: 95%) predictive values for marked fibrosis (F≥3) in post-analysis group 2A (Table S12).

Observer agreement

Inter-observer agreement -Inter-observer agreement for fibrosis FR staging was assessed by ICCC in the baseline and post-analysis evaluations or both steps merged in each group (Table S13). In the baseline evaluation, agreement was fair in groups 1C and 3C and slightly higher in group 2A. Agreement was not significantly different between evaluation steps in group 1C, as expected. Agreement significantly improved between the two steps in groups 2A and 3B, especially in the former, where it became excellent, reflecting the accuracy results. Indeed, accuracy and ICCC were highly correlated (rp=0.927, p=0.008) (Figure S3).

Agreement between observer FR and Metavir F -Overall agreement (i.e. calibration of radiological staging) was significantly improved in the post-analysis step vs the baseline step in groups 2A and 3B (Table S14). Figure 3 shows that calibration was satisfactory for all Metavir F stages only in the post-analysis step of group 2A, with rs=0.914 and ICCC=0.951. Of note, calibration improved for all F in post-analysis group 2A (Table S15). Thus, in group 2A, post-analysis calibration became excellent in Metavir F2 to F4 while calibration was good in Metavir F1 and fair in Metavir F0. In addition, the discrimination between adjacent Metavir F was very significant, especially between F1 and F0 or F2 (Table S15) as also previously suggested by AUROCs. In contrast, NITs were unable to discriminate Metavir F0 from F1 (Table S16) and their Obuchowski index was significantly lower than that of observer FR in post-analysis group 2A (details in Supplemental Material).

Comparison with senior radiologists

The accuracies of the single evaluations (on raw images of 10 patients) by the three senior evaluator radiologists were compared to those of the two evaluations of observers in group 2A (Table 4, Figure S4). Baseline HGE observer accuracies for fibrosis staging and cirrhosis diagnosis were not significantly different from radiologist accuracies except for fibrosis staging, which was significantly (p<0.001) higher in expert 1 (full-time in HGE for clinical practice). However, post-analysis accuracies for HGE observers became higher to those of non-expert for fibrosis staging (p=0.014) and cirrhosis (p=0.069) and not significantly different from that of expert 1 for fibrosis staging (p=0.215).

DISCUSSION

Main results -Using a simple and original classification of liver fibrosis by CT, the accuracy for cirrhosis (84%) and discrimination for marked fibrosis (AUROC=0.83) by observers were already very satisfactory at baseline. Image analysis, especially on annotated images with simple descriptors, significantly improved accuracy for fibrosis staging or cirrhosis, as well as discrimination between fibrosis stages or for all binary diagnostic targets, and calibration of radiological classification. For fibrosis staging and cirrhosis diagnosis, the best progression was observed in group 2A (annotated abnormalities) (Figure 2A). Thus, simple annotations provided the greatest improvement for fibrosis staging by CT, which consequently became accurate. Thus, NPV and PPV for marked fibrosis (F≥3) were 98% and 95%, respectively.

Semi-automated morphometric analysis tended to improve the diagnostic performance of fibrosis staging but less so than did the annotated images in the study conditions. Finally, we observed high discrimination of all Metavir F by radiological FR classification on annotated images, which must be clarified as it was unexpected.

Originalities -The Metavir F stages 0 and 1 were not distinguished by definition in the FR classification. Metavir F was an independent predictor of FR accuracy. Thus, accuracy was the highest in F4 and the lowest in F1 which was expected due to the preliminary merging of F0 and F1 in radiological staging. Unexpectedly, the overall (i.e. mean) FR observer staging significantly discriminated F1 from F0 or F2 (Figure 3) in post-analysis group 2A despite the merging of FR0 and FR1 in the present study. This discrimination is noteworthy since NITs are usually unable to distinguish F0 from F1 [START_REF]European Association for the Study of the Liver, Asociacion Latinoamericana para el Estudio del Higado. EASL-ALEH Clinical Practice Guidelines: Non-invasive tests for evaluation of liver disease severity and prognosis[END_REF] as in the present study, even with magnetic resonance elastography [START_REF] Hsu | Magnetic Resonance vs Transient Elastography Analysis of Patients With Nonalcoholic Fatty Liver Disease: A Systematic Review and Pooled Analysis of Individual Participants[END_REF]. These data suggest that the radiological classification can be improved. Thus, some minor radiological abnormalities can distinguish Metavir F1 from F0, and early fibrosis may be detectable by CT. The high discrimination of radiological staging finds reflection in the very high Obuchowski index at 0.873-significantly higher than that of NITs-in post-analysis group 2A. In comparison, with the best-performing tests in NAFLD, the Obuchowski index was, liver stiffness: 0.808, FibroMeter VCTE2G : 0.846 [START_REF] Ducancelle | A single test combining blood markers and elastography is more accurate than other fibrosis tests in the main causes of chronic liver diseases[END_REF].

Accuracy for cirrhosis diagnosis was as high in Metavir F≤3 (96%) as it was in F4 (98%) in post-analysis group 2A (Table S11), resulting in a very high NPV (99%) and a high PPV (85%) (Table S12). This is also noteworthy, since VCTE, still considered as the bestperforming commonly-employed NIT for cirrhosis, provides a PPV of only around 60% [START_REF] Cales | Cirrhosis diagnosis and liver fibrosis staging: transient elastometry versus cirrhosis blood test[END_REF].

Indeed, NITs more frequently classify cirrhosis correctly in patients with F≤3 than they do in patients with F4. Because F4 is infrequent (10-15%), overall accuracy remains good despite a high rate of misclassified F4. Remarkably therefore, CT can avoid this usual pitfall. Finally, agreement between Metavir F, as reference, and FR stagings was excellent in post-analysis group 2A, with an ICCC=0.95, in contrast to Metavir F agreement between expert, as reference, and first line pathologists as shown in Figure S5, with an ICCC=0.87, or with an ICCC=0.69 reported in a previous study [START_REF] Rousselet | Sources of variability in histological scoring of chronic viral hepatitis[END_REF].

Interpretation -In our prospective randomized stratified study, the absence of significant change in the post-analysis accuracy of the control group, with a classical presentation of semiology and the same images in the two steps, suggests good overall reproducibility and argues against a learning phenomenon attributable to the study. Thus, the control group validates the methodology used in our study. The excellent correlation between accuracy and observer agreement further validates the evaluation method. Unexpectedly, we observed only a weak improvement in accuracy with morphometry, which could seem paradoxical given the high level of information provided by it. Indeed, morphometry improved diagnostic accuracy in our preliminary study [START_REF] Cales | Morphometry on CT improves non-invasive diagnosis of hepatic fibrosis[END_REF]. We hypothesise that the information presented for this technique may have been too complex for the observers. By contrast, the simple presentation including annotated images showed the best improvement in accuracy. The trivial explanation is that the commented analysis makes it possible to visualize the anomalies easily and quickly and thus stage disease with greater accuracy. This good result was obtained despite the very novel character of the radiological classification in fibrosis stages, furthermore presented to naive observers. In contrast to fibrosis staging, there was a small but significant accuracy improvement for cirrhosis diagnosis with both image analyses. The moderateness of the improvement was attributable to the already very good baseline accuracy (with a limited progression margin), itself attributable to the classical character of this semiology and the presence of more marked anatomical abnormalities such as signs of portal hypertension.

Considering the centre effect, there was mainly a significant effect in centre 5 of group 3B with a worsening in fibrosis staging (Figure 2D). This might be attributable to the combination of the complexity of morphometric data and poorer initial training. Indeed, this centre was the only one including three residents with unusual university training before internship. Finally, image analysis with annotations (group 2A) provided excellent intercentre reproducibility for accuracy in both fibrosis staging and cirrhosis (Figure 2B/C). The present CT method is simpler and more discriminant than texture analysis [START_REF] Budai | Three-dimensional CT texture analysis of anatomic liver segments can differentiate between low-grade and high-grade fibrosis[END_REF]. However, a next step will be to compare the present modality of fibrosis staging with radiomics which are promising techniques [START_REF] Wei | Radiomics in liver diseases: Current progress and future opportunities[END_REF][START_REF] Decharatanachart | Application of artificial intelligence in chronic liver diseases: a systematic review and metaanalysis[END_REF]. Radiomics have been evaluated in liver fibrosis staging with CT used alone [START_REF] Cui | Predicting the stages of liver fibrosis with multiphase CT radiomics based on volumetric features[END_REF] or combined with biomarkers [START_REF] Hu | A Combination Model of Radiomics Features and Clinical Biomarkers as a Nomogram to Differentiate Nonadvanced From Advanced Liver Fibrosis: A Retrospective Study[END_REF], with magnetic resonance imaging [START_REF] Zheng | Imaging-Based Staging of Hepatic Fibrosis in Patients with Hepatitis B: A Dynamic Radiomics Model Based on Gd-EOB-DTPA-Enhanced MRI[END_REF] and with ultrasound combined with elastometry [START_REF] Xue | Transfer learning radiomics based on multimodal ultrasound imaging for staging liver fibrosis[END_REF]. Therefore, radiomics should be compared with our CT technique which might include two advantages: easy applicability and discrimination between F0 and F1. The radiation doses should also be compared.

Limitations -The limits of this study are mainly methodological, concerning observer choice and image presentation with annotations. The present exploratory study might also include an optimism bias. These limits are detailed in the Supplemental material. One limitation of CT is radiation exposure, especially in patients with recurrent CT imaging [START_REF] Kwee | Which patients are prone to undergo disproportionate recurrent CT imaging and should we worry?[END_REF]. However, this limit 20 can be circumvented or attenuated. First, this limit does not concern other initial validated indications of CT than liver fibrosis staging. For example, cirrhosis is often fortuitously discovered. Second, CT indications can be restricted to patients with discordant NITs for liver fibrosis, or when ultrasound imaging is not complete or inconclusive. Finally, one can use a low dose CT technique [START_REF] Takahashi | Can lowdose CT with iterative reconstruction reduce both the radiation dose and the amount of iodine contrast medium in a dynamic CT study of the liver?[END_REF] or new devices reducing the radiation dose by 81% [START_REF] Leyendecker | Prospective evaluation of ultra-low-dose contrast-enhanced 100-kV abdominal computed tomography with tin filter: effect on radiation dose reduction and image quality with a third-generation dual-source CT system[END_REF].

Application -Our study shows that annotated images improve diagnoses by HGE clinicians.

We believe that this improvement could also concern other physicians as suggested by the accuracy difference between expert and non-expert radiologists (Table 4). Thus, the approach presented here can be applied in clinical practice via three means: 1) a diagnostic grid including the elementary descriptors, especially for first-line diagnoses (Figure 1A); 2) annotations provided by radiologists or by 3) semi-automatic morphometric analysis for second-line diagnoses. A reading grid contributes to structuring diagnostic reasoning. Furthermore, semi-automatic morphometry makes it possible to imagine, in the long term, an application to clinical practice not only as a diagnostic aid for the radiologist evaluating images initially, but also as a means for clinician prescribers and patients themselves to better appropriate results. Upstream of these possibilities, it will be necessary to validate this new semiology (especially for fibrosis FR staging into five stages) with comparisons to a robust reference (liver biopsy, NITs) in a large patient population. Further study will determine independent descriptors and an ensuing diagnostic score. Finally, in the future, a single radiological modality could include a comprehensive liver evaluation comprising fibrosis staging, elastometry, steatosis and masses.

Conclusion -A new CT classification based on simple descriptors can accurately stage liver

fibrosis according to the Metavir system. Unexpectedly, this classification, although based on cirrhosis descriptors, was able to discriminate between all Metavir fibrosis stages, including between F0 and F1. The simple identification of these descriptors was superior to 21 morphometric analysis but clinical application needs to be improved. This simple descriptor identification might result in accuracies by HGE clinicians at least as good as those provided by senior radiologists. Finally, imaging could provide comprehensive liver evaluation including highly discriminative liver fibrosis staging. S2. S14. e With data as in Table S2.

Figure legend Figure 1 .

 legend1 Figure legend

Figure 2 .

 2 Figure 2. Observer FR accuracies. Panel A: Accuracies as a function of observer groups with p values between evaluation steps. Panels B/C: Accuracies for fibrosis staging (B) and cirrhosis (C) as a function of centres, observer groups, and evaluation steps. Panel D: Accuracy course (net change) as a function of observer groups and centres. * indicates an interaction between centre and observer group: # indicates a significant centre effect.

Figure 3 .

 3 Figure 3. Agreement of radiological FR classification vs Metavir F. Agreement as a function of evaluation steps and observer groups. At baseline, calibration was poor in F0 and F1. In post-analysis evaluation, the calibration became excellent for all F in group 2A. rs: Spearman's correlation coefficient for individual observer data; comparisons are detailed in TableS14.

Table 1 .a

 1 Study design: the four steps of the evaluation session. Example in Figure 1D. Patients different from those of the next evaluation steps. b Patient order identical in the 3 groups. c Patient order different from step #2. d 35 comments including at least each of the seven categorized descriptors of liver fibrosis/cirrhosis.

Table 4 .

 4 Comparison of accuracies (%) between group 2A observers and radiologists as a function of evaluation steps. Comparison of radiologist vs observer baseline accuracy in the 10 patients of the present study by Fisher test. b Comparison of radiologist vs observer post-analysis accuracy in the 10 patients of the present study by Fisher test.c The radiologist comparison was performed between accuracies measured in the 10 patients of the core population vs the remaining 96 patients evaluated by radiologists8 .

		Observers			Radiologists
		Baseline	Post-		p a vs	p b vs post-		p c vs 10
			analysis		baseline	analysis		patients by
					observer	observer		radiologist
	Patients (n)	10	10	10	-	-	96	-	106
						Expert 1
	FR staging	47	81	100	<0.001	0.215	51	0.002	56
	Cirrhosis	88	96	100	0.608	1	86	0.607	88
						Expert 2
	FR staging	47	81	60	0.523	0.116	41	0.318	42
	Cirrhosis	88	96	100	0.609	1	83	0.353	85
						Non-expert
	FR staging	47	81	20	0.116	0.014	32	0.721	31
	Cirrhosis	88	96	80	0.353	0.069	78	1	78
	Significant differences in bold.

a

Acknowledgments:

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

d Cumulative rate of observer change per patient (expressed in binary qualitative value). This rate is therefore different from the changes in mean accuracy for 10 patients depicted in the main text (Table S4). e The net change is the absolute difference between baseline and post-test analysis and is expressed as a quantitative variable.