Elnur Garayev
email: elnurgar@mail.ru

Carole Di Giorgio
email: carole.di-giorgio@univ-amu.fr

Gaëtan Herbette
email: gaetan.herbette@univ-amu.fr

Fathi Mabrouki
email: fathi.mabrouki@univ-amu.fr

David Roux
email: david.roux@univ-avignon.fr

Huguette Sallanon
email: huguette.sallanon@univ-avignon.fr

Evelyne Ollivier
email: evelyne.ollivier@univ-amu.fr

Riad Elias
email: riad.elias@univ-amu.fr

Béatrice Baghdikian
email: beatrice.baghdikian@univ-amu.fr

Bioassay-guided isolation and UHPLC-DAD-ESI-MS/MS quantification of potential antiinflammatory phenolic compounds from flowers of Inula montana L

Keywords: 1, 5-O-Dicaffeoylquinic acid, Chlorogenic acid, Anti-inflammatory activity, UHPLC-DAD, Accuracy profile

Ethnopharmacological relevance: Flowers of Inula montana L. (Asteraceae), commonly known as "Arnica de Provence", are used in the traditional medicine of Provence in France with the same indication as Arnica montana, for the relief of bruises, as an anti-inflammatory agent.

Aims of the study: The aim of our study is to evaluate its anti-inflammatory properties and to justify its traditional uses. Its potential valorization is evaluated in order to propose Inula montana as an alternative to Arnica montana.

Materials and methods: Bio-guided fractionation of ethanolic extract allowed the isolation of compounds, responsible of the inhibition of NO production. The fractionation was realized using chromatographic techniques and structure elucidation was conducted by ESI-MS and NMR spectral data. Anti-inflammatory effect of ethanolic extract, different fractions and isolated pure compounds was studied in vitro on immortalized mouse macrophages RAW 264.7. An analytical UHPLC-DAD-ESI-MS/MS method was developed for the identification of these compounds in the herbal drug. This UHPLC-DAD method was validated and was used to compare the phenolic profile and content in plant material from the two collection sites: Bonnieux and Merindol.

Results: Eleven compounds were identified by UHPLC-MS. Chlorogenic acid (1), Luteolin (2), Nepetin (3), 3,5-O-Dicaffeoylquinic acid (4), 1,5-O-Dicaffeoylquinic acid (5), Nepitrin (6), Hispiduloside (7) and Jaceosid (8) were isolated and identified by NMR. Compounds 9, 10 and 11 were confirmed to be 6-Hydroxykaempferol 3,7-dimethyl ether, Hispidulin and Chrysosplenol C, respectively by comparing retention times and MS/MS data with those of the authentic substances.

Six compounds: 1 and 4-8 are reported for the first time in Inula montana L. Compounds 2-8 showed promising anti-inflammatory activity with the release of NO with IC 50 value <7 µM. The UHPLC-DAD method of quantification of three major bioactive compounds (1, 3 and 5) was validated.

Conclusion:

Flowers extracts and isolated compounds present promising anti-inflammatory activity which provides a scientific basis for the traditional use of Inula montana and may be proposed in the same indications as Arnica montana. The developed and validated simple, accurate and rapid UHPLC method can be used for the quality control of the herbal drug.

Introduction

The genus Inula is mainly distributed in Africa, Asia and Europe, predominantly in the Mediterranean Area. Flavonoids and terpenes are the major compounds of this genus [START_REF] Seca | The genus Inula and their metabolites: From ethnopharmacological to medicinal uses[END_REF][START_REF] Wu | Anti-inflammatory Inositol Derivatives from the Whole Plant of Inula cappa[END_REF][START_REF] Zhao | Chemical constituents of plants from the genus Inula[END_REF].

Inula montana L. is a member of Asteraceae family. Flowers are used in traditional medicine in France, Spain and Morocco. In France, I. montana is called "Arnica of Provence" and is used for the relief of bruises. The flowers are macerated with ethanol at 60-70% (v/v) called "eau de vie" or with olive oil. These preparations are used for cutaneous application with the same indication as Arnica montana [START_REF] Amir | Les cueillettes de confiance[END_REF]. In Spain, decoction, infusion and maceration in alcohol or olive oil of flowers, leaves and fresh aerial parts are used locally for the relief of bruises, joint pains, rheumatism and orally for stomach ache [START_REF] Obón | Árnica: A multivariate analysis of the botany and ethnopharmacology of a medicinal plant complex in the Iberian Peninsula and the Balearic Islands[END_REF]. In Morocco, decoction or infusion of leaves are used to treat orally gastrointestinal, otolaryngological and respiratory diseases [START_REF] Teixidor-Toneu | An ethnomedicinal survey of a Tashelhit-speaking community in the High Atlas, Morocco[END_REF].

Few studies report the phytochemical composition and pharmacological activity of I. montana.

Previous phytochemical studies led to the isolation of flavonoid aglycones, sesquiterpene lactones and inositol derivatives [START_REF] Garayev | New sesquiterpene acid and inositol derivatives from Inula montana L[END_REF][START_REF] Romero | Phytochemistry and pharmacological studies of Inula montana[END_REF][START_REF] Reynaud | Free Flavonoid Aglycones from Inula montana[END_REF]. Crude extracts and some purified constituents of I. montana demonstrated antioxidant, protozoocidal and anti-inflammatory activities [START_REF] Garayev | New sesquiterpene acid and inositol derivatives from Inula montana L[END_REF][START_REF] Martín | Screening for Protozoocidal Activity of Spanish Plants[END_REF][START_REF] Belyagoubi-Benhammou | Phenolic contents and antioxidant activities in vitro of some selected Algerian plants[END_REF].

The aim of our study is to validate the traditional use of Inula montana by evaluating its potential anti-inflammatory properties and to evaluate the possible use of I. montana as an alternative to Arnica montana, a plant species that is endangered and difficult to cultivate [START_REF] Pasquier | L'arnica des montagnes, entre culture et cueillette[END_REF].

Both species are members of Asteraceae family. They contain similar chemical classes of compounds, essentially sesquiterpene lactones, phenolic acids, and flavonoids. Sesquiterpene lactones of I. montana are different from those of A. montana, however we can find two flavonoid aglycones described in both species, luteolin and hispidulin [START_REF] Kriplani | Arnica montana L . -a plant of healing: review[END_REF].

Inula montana is a perennial plant showing a rosette (10-40 cm circumference) of thin lancet-shaped sessile leaves (highly variable number). All aerial organs are hairy, leaves and bracts also display numerous glandular trichomes. From May to June, this rhizomatous plant emits a few ≈20 cm floral stems (rarely branched), with a few smaller alternate leaves, holding at the top a solitary capitulum (5-8 cm diameter) of yellow florets (ray florets with long and thin ligules) displaying a discrete odor. Inula montana grows from 50 to 1300 m altitude and is frequent from eastern Italy to southern Portugal. This calcicolous and xerophilous plant is locally abundant in Southeast France in the calcareous low scrubland ecoregion named Garrigue [START_REF] Girerd | Flore du Vaucluse: troisière inventaire, descriptif, écologique et chronologique[END_REF][START_REF] Romero | Phytochemistry and pharmacological studies of Inula montana[END_REF][START_REF] Botanica | Tela Botanica[END_REF]. The rare and protected Arnica montana is also a rhizomatous herbaceous perennial herb but, in contrast to Inula montana, it grows on acid (siliceous) and poor soils mainly in high altitude (600-2700 m) environment such as the alpine mountain. It displays larger oval leaves which are hairy on the upperside. From June to July, it emits a single ≈50 cm floral stem showing one (frequent) to tree opposite pairs of smaller leaves and a bright yellow to orange 7-8 cm capitulum with a pleasant perfume. Ligules of ray florets are far more developed in comparison to Inula montana [START_REF] Rameau | Flore forestière française: guide écologique illustré[END_REF][START_REF] Sugier | Propagation and Introduction of Arnica montana L. into Cultivation: A Step to Reduce the Pressure on Endangered and High-Valued Medicinal Plant Species[END_REF]. This paper describes the isolation, structure elucidation, quantitative analysis and antiinflammatory effects of the isolated compounds by measuring their capacity to inhibit nitric oxide production. The identification and the structure elucidation were carried out using NMR and UHPLC-DAD-ESI-MS/MS. This UHPLC-DAD method was validated for the quantification of the three major active compounds. To our knowledge, this is the first time that an analytical method for the quality control of I. montana is reported. This validated UHPLC-DAD method was used to compare the phenolic profile and the content of the two collection sites: Bonnieux and Merindol, in order to observe potential modification of the phenolic composition and to select the plant with the highest content.

Materials and methods

Plant material

The plant material of Inula montana was collected in Merindol in 2015 and 2017 and Bonnieux in 2017 located in the Regional Natural Park of Luberon, in the south of France (Table 1).

The voucher specimens were deposited in the Laboratory of Pharmacognosy of Aix Marseille University (Marseille, France). Flowers were air-dried and ground.

Instruments and reagents

1D and 2D NMR spectra were recorded on Bruker AVANCE III 600 spectrometer ( 1 H 600.13MHz) (Bruker, USA) with tetramethylsilane (TMS) as an internal standard at 300K. The chemical shifts (δ) were expressed in ppm with TMS as an internal reference. UHPLC-DAD-ESI-MS/MS analysis were performed using an UHPLC Agilent Infinity 1290 Liquid chromatography system equipped with a binary pump solvent delivery system G4220A and UV photodiode array detector G4212A (Agilent Technologies, Inc., Germany), coupled with Bruker Esquire 2000 (Bruker, USA) which is equipped with an electrospray ionization source (ESI). Preparative HPLC was performed on Gilson PLC2020. Flash chromatography was performed using a CombiFlash Rf 200 (Teledyn Isco, USA). A grinder (Moulinex, France) was used for the homogenization of flowers. The optical density for the evaluation of anti-inflammatory activity was read by fluorescence-luminescence reader Infinite M200 Pro (Tecan, Switzerland). Extraction solvents, methanol and formic acid of LC-MS quality were purchased from Carlo Erba (Italy).

The standards of Chlorogenic acid, 1,5-O-Dicaffeoylquinic acid and Nepetin were purchased from PhytoLab GMBH (Germany).

Reagents for the anti-inflammatory evaluation were purchased from Sigma-Aldrich (USA).

Nitric oxide (NO) production assay

This assay is based on the ability of immortalized mouse macrophages RAW 264.7 to generate a strong inflammatory response leading to NO (Nitric Oxide) release, after the stimulation by antigens of E. coli. Mouse macrophages were seeded into 48-well tissue culture plates at the concentration of 1.10 5 cells/mL (200µL/well) for 24 hours at 37°C (5% CΟ 2 ). At the end of the incubation period, the culture medium was replaced by 200 µL of medium containing appropriate concentrations of evaluated compound and cells were incubated at 37°C (5% CO 2 ) for one hour.

Experiments were performed in duplicate and dexamethasone was used as a positive control at the concentrations of 1, 5, 10, 50 and 100 µM. At the end of the incubation period, pro-inflammatory LPS from E.coli were added to the cell cultures (1 µg/mL). Then the cells were incubated at 37°C (5% CO 2 ) for 18 hours. NO release was measured in the culture supernatant by the Griess reaction: 50 µL of the supernatants were transferred into the 96-well tissue culture plate, and 50 µL of the Griess modified reagent were added to each well. After a 15 min period at room temperature, Optical Density (OD) of each well was read at 540 nm. Results obtained for wells, treated with the evaluated compound, were compared to the untreated control wells (DMSO, 100% viability) and converted into the percentage values.

Inhibition of NO release was expressed as percentages comparing to the negative controls:

Percentage of NO release =

The concentrations of the test material causing a 50% decrease of NO release (IC 50 ) were calculated through non-linear regression analysis using software voltage was set at 3.5 kV, sheath gas (N 2 ) flow rate was set at 50 psi, auxiliary gas (N 2 ) flow rate was set at 12 l/min, capillary exit voltage was set at 102.4 V, tube lens voltage was set at -60 V and capillary temperature was set at 300°C. Fragmentation amplitude was set at 1.0 V.

Extraction and isolation

Powdered flowers (0.2 kg) were macerated with EtOH 95% (v/v) at ambient temperature for 18 h and then extracted by percolation with the same solvent three times (3 x 1.2L

Validation of UHPLC-DAD method

The recommendations of the "Société Française des Sciences et Techniques

Pharmaceutiques" (SFSTP) were respected for the validation of the analytical method in herbal medicinal products (D. [START_REF] Bellenot | Problématiques liées au développement d'une procédure analytique de dosage de constituants dans les médicaments à base de plante(s) IV. Recommandations pour le développement et la validation[END_REF]. Two kinds of samples, calibration and validations standards, were prepared separately. Calibration standards were prepared twice per day over three days. Validation standards were prepared three times per day over three days. and NPT are 60.0 µg/mL, 130.4 µg/mL and 24.0 µg/mL, respectively. Filter 2 mL of this solution through a PVDF filter (0.22 µm, Millipore) for UHPLC analysis.

Preparation of calibration standards

Preparation of validation standard

Put 10 g of powdered flowers into the grinder and homogenize for 5 seconds three times.

Weigh 1.000 g of homogenized flowers and mix with 90 mL of MeOH into a round-bottomed flask of 250 mL. Extract under reflux for 30 minutes. Cool for 15 minutes. Filter into a volumetric flask of 100.0 mL through a paper filter, rinse two times with 5 mL of MeOH and dilute to 100.0 mL with MeOH. Filter 2 mL of this solution through a PVDF filter (0.22 µm, Millipore) for UHPLC analysis.

Statistical analysis

All statistical analyses were based on two-sided p-values, where p < 0.01 indicates statistical significance. They were performed with SAS version 9.4 (SAS Institute, Cary, NC).

Results and discussion

UHPLC-DAD-ESI-MS/MS study of L2E

The ethanol crude extract (L2) showed promising anti-inflammatory effect activity with IC 50 of NO release of 0.77 µg/mL. In a bio-guided fractionation approach, each extract obtained after solvent partitioning was evaluated for the inhibition of NO release its anti-inflammatory activity.

The ethyl acetate extract (L2E) has been found to be the most active with IC 50 of NO release of 0.42 µg/mL and was selected for the next step study. Petroleum ether (L2P) and dichloromethane (L2D) extracts were less active with IC 50 of NO release of 5.06 µg/mL and 0.58 µg/mL, respectively.

The UHPLC-DAD-ESI-MS/MS method was used for the separation and identification by analytical chromatography of eleven compounds from the extract L2E. The spectroscopic data are shown in the Table 2. The compound (1) with Rt 2.8 min and parent ion m/z 355 was suggested to be Chlorogenic The compound (8) with Rt 14.43 min, UV spectra characteristic of flavonoids and parent ion m/z 493 can be also glycosylated.

The chromatogram shows also three compounds (9, 10 and 11) with Rt 19.9, 19.2 and 20.5 min and parent ions m/z 331, 301 and 361 [M+H] + respectively. These compounds were confirmed to be 6-Hydroxykaempferol 3,7-dimethyl ether, Hispidulin and Chrysosplenol C, respectively by comparing their retention time and MS/MS data with those of the authentic substances [START_REF] Garayev | New sesquiterpene acid and inositol derivatives from Inula montana L[END_REF].

Confirmation by NMR of the structures of isolated compounds from L2E

To confirm the structural assumptions, the compounds have been isolated and identified by NMR 1 H, 13 C and 2D NMR data were compared to the spectroscopic data reported in the literature.

The following compounds are present in the flowers of I. montana (Figure 1).

Compound trans-caffeoyl group. The second part of proton at δ H 5.33 (H-5), 3.72 (H-4 ax ), 2.23 (H-6 eq ), 2.18 (H-2 ax ), 2.08 (H-6 ax ), 2.05 (H-2 eq ) were characteristic of the quinic acid. The deshielded proton 5 at δ H 5.33 and the 3 J = HMBC correlation with ester carbon at δ C 168.7 (C-9') showed that transcaffeoyl moiety was attached on C-3 of the quinic acid. Based on further comparison of the NMR data with those in the literature [START_REF] Pauli | Solvent effects in the structure dereplication of caffeoyl quinic acids[END_REF], the structure of 1 was determined to be chlorogenic acid.

1
The molecular formula of compound 4 was deduced to be C Assignments for all proton and carbon resonances were achieved by COSY, HSQC and HMBC experiments, which indicated the presence of β-glucosyl as sugar units. The position of the βglucosyl as sugar units was established by the 3 J crosspeak in HMBC spectrum between the anomeric proton H-1'' at δ H 5.12 and carbon signal at δ C 157.6 (C-3), showed that the β-glucosyl [START_REF] Hong | Phenolic Constituents of Veronica linariifolia[END_REF], the structure of 8 was determined to be Jaceosid.

moiety
Compounds 1 and 4-8 are reported in I.montana for the first time.

Based on literature data, the study of the phenolic composition of A. montana shows that five compounds (1, 2, 4, 5 and 7) isolated from I. montana are also present in A. montana flowers [START_REF] Jaiswal | Determination of the hydroxycinnamate profile of 12 members of the Asteraceae family[END_REF][START_REF] Merfort | Caffeoylquinic acids from flowers of Arnica montana and Arnica chamissonis[END_REF][START_REF] Merfort | Caffeoylquinic acids from flowers of Arnica montana and Arnica chamissonis[END_REF].

R 1 R 2 R 3 R 1 R 2 R 3 1 H H Caf 2 H H OH 4 H Caf Caf 3 OCH 3 H OH 5 Caf H Caf 6 OCH 3 Glucose OH Caf = Caffeoyl 7 OCH 3 Glucose H 8 OCH 3 Glucose OCH 3

Evaluation of the anti-inflammatory potential of isolated compounds

Nitric oxide (NO), which is produced by the NO synthase when induced by pro-inflammatory cytokines, has been shown as an important element of the inflammatory mechanisms. During bacterial infection more particularly, a 1000-fold increase of NO production may be observed in neutrophils and macrophages. NO plays two essential roles in cellular-response to bacterial infection: when excreted in the extracellular environment, it acts as a mediator of the inflammatory response, whereas in the lysosomal environment it contributes to microbe destruction with other oxygen radicals. However, although NO is essential in the cellular defense against infectious diseases, its overproduction may lead to adverse effects such as autoimmune reactions and neurodegenerative syndromes. As a consequence, inhibition of NO synthesis or NO scavenging may be considered as possible strategies in the treatment of chronic inflammatory diseases [START_REF] Sharma | Role of nitric oxide in inflammatory diseases[END_REF].

In the present study, NO-inhibition was assessed in murine macrophages (RAW 267.4 cell line), in which a strong inflammatory response, leading to NO synthesis and secretion in the culture medium, was initiated by LPS of E. coli [START_REF] Gao | Hexahydrobenzophenanthridine alkaloids from Corydalis bungeana Turcz. and their anti-inflammatory activity[END_REF][START_REF] Pansanit | Vermelhotin, an Anti-inflammatory Agent, Suppresses Nitric Oxide Production in RAW 264.7 Cells via p38 Inhibition[END_REF].

Results are presented in Table 3. All compounds showed inhibitory effects of NO production with IC 50 < 32 µM, however compounds 2, 3, 5-8 have been found to be more active than positive control dexamethasone.

Most of the isolated phenolic compounds in from the flowers of I. montana present promising antiinflammatory.

These results allow to validate scientifically the traditional use of flowers for the relief of bruises.

The study provides evidence that Inula montana presents potential anti-inflammatory effect. Further experiments are necessary to confirm these first results in order to propose Inula montana as an alternative to Arnica montana. Moreover, active compounds 2, 5 and 7 are present in the both species.

Analytical validation of UHPLC-DAD method

Three major active compounds Chlorogenic acid (1), Nepetin (3) and 1,5-O-Dicaffeoylquinic acid (5) were retained for their quantification in the plant material, in order to evaluate the quality of Inula montana. This UHPLC-DAD method was validated in term of analysis of specificity, response functions, trueness, precision, accuracy, linearity and limits of detection and quantification. The validation summary can be found in Table 4.

The specificity of the method has been investigated. Calibration and validation standards were prepared and analyzed in terms of specificity. No interference and great resolution were observed for each compound which was assessed by comparing the retention time and the UV spectra with the standard. Furthermore, the purity of the investigated peak was confirmed by DAD purity studies.

The calibration curves were based on the through-origin linear regression model which was Trueness and intermediate fidelity were used for the accuracy profile. The acceptance thresholds were set at ±5% and the results are acceptable with -4.90 and 4.97% for CA, -4.92 and 4.99% for DCQA and -4.88 -4.92% for NPT, as it is shown in Table 4.

The linearity of this method was confirmed by a linear regression function of estimated concentrations to introduced concentrations calculated all the series of validation standards (N = 45). The following equations of the regression line were determined y=0.9889x (R 2 =0.9987), y=0.9979x (R 2 =0.9994) and y=0.9993x (R 2 =0.9997) for CA, DCQA and NPT, respectively, as it is shown in Table 4.

Limit of Detection (LOD) and Limit of Quantification (LOQ) were determined by calculating LOD and LOQ as 3 and 10 times of intensity of background noise, respectively. LOD and LOQ of CA, DCQA and NPT were calculated to be 0.12 and 0.39 µg/mL, 0.17 and 0.56 µg/mL, 0.09 and 0.31 µg/mL, respectively.

The retention times of CA, DCQA and NPT were 2.79, 9.61 and 16.05 min, respectively. The chromatograms of the calibration and validation standards are presented in Figure 2. These results

show that UHPLC-DAD method is validated for the quantification of Chlorogenic acid, 1,5-O-Dicaffeoylquinic acid and Nepetin in flowers of Inula montana L. The validated method was applied for the quantitative analysis of the three major compounds in flowers: Chlorogenic acid (CA), 1,5-O-Dicaffeoylquinic acid (DCQA) and Nepetin (NPT), from the two collection sites of Regional Natural Park of Luberon. One-way ANOVA confirms significant results between the mean content of phenolic compounds at p<0.01 (Table 5). The content of Chlorogenic acid and 1,5-O-Dicaffeoylquinic acid in flowers from Merindol was higher in 2015 with 0.61 and 1.44 %, respectively. In order to compare the influence of altitude on the content of phenolic compounds, two collection sites were compared: Merindol (170 m) and Bonnieux (680 m). Despite the fact that the altitude in Merindol (170 m) is four times lower than in Bonnieux (680 m), no significant differences in the content of each phenolic compound were observed between the two collection sites.

Conclusion

Bio-guided fractionation of 95% ethanolic crude extract allowed the identification of eleven compounds from Inula montana. Chlorogenic acid (1), 3,5-O-Dicaffeoylquinic acid (4), 1,5-O-Dicaffeoylquinic acid (5), Nepitrin (6), Hispiduloside (7) and Jaceosid (8) are reported in this species for the first time. Among them, 1, 4, 5 and 7 are also present in Arnica montana.

A simple, accurate and rapid UHPLC-DAD method was developed as the first report of a simultaneous determination of three main phenolic compounds, chosen as analytical markers of this species, Chlorogenic acid, 1,5-O-Dicaffeoylquinic acid and Nepetin in Inula montana. This validated method allows the quality control of the herbal drug and was applied for the comparison of the phenolic content of the two collection sites: Bonnieux and Merindol. We can conclude that the content of phenolic compounds was close for these collection sites. ACCURACY PROFI L E AT 95% (RI SK = 5% ) A CCUR ACY PROFI L E AT 95% (RI SK = 5% ) A CCUR ACY PROFI L E AT 95% (RI SK = 5% )

Min Biais Max +5% limit -5% limit

H- 2

 2 ), 2.14 (1H, dd, J = 14.0, 5.7 Hz, H-6 ax ); 13 C NMR (150.9 MHz, CD 3 OD) δ: 176.4 5-O-Dicaffeoylquinic acid) was obtained as a an amorphous solid.C 25 H 24 O 12; ESI-MS m/z: 517 [M+H] + . 1 H NMR (600.13 MHz, CD 3 OD) δ: 7.60(1H, d, J = 16.0 Hz, H-7'), 7.57 (1H, d, J = 16.0 Hz, H-7''), 7.06 (2H, d, J = 2.2 Hz, H-2', H-2''), 6.96 (2H, dd, J = 8.2, 2.2 Hz, H-6', H-6''), 6.78 (2H, d, J = 8.2 Hz, H-5', H-5''), 6.30 (1H, d, J = 16.0 Hz, H-8'), 6.27 (1H, d, J = 16.0 Hz, H-8''), 5.42 (1H, brtd, J = 8.9, 4.1 Hz, H-5 ax ), 4.31 (1H, brq, J = 4.0, H-3 eq ), 3.81 (1H, dd, J = 8.3, 3.2, H-4 ax ), 2.61 (1H, dd, J = 13.6, 2.7 Hz, H-6 eq ); 2.52 (1H, dd, J = 15.1, 5.0, H-2 ax ), 2.44 (1H, dd, J = 15.1, 2.6, H-2 eq ), 2.08 (1H, dd, J = 13.6, 9.1 Hz, H-6 ax ); 13 C NMR (150.9 MHz, CD 3 OD) δ: 175.0 (C-7), 170.5 (C-9''), 168.5 (C-9'), 149.8 (C-4'), 149.8 (C-4''), 147.1 (C-7'), 147.1 (C-7''), 146.5 (C-3'), 146.5 (C-3''), 127.5 (C-1''), 127.5 (C-1'), 122.8 (C-6'), 122.8 (C-6'), 116.2 (C-5'), 116.2 (C-5''), 115.1 (C-2''), 115.1 (C-2''), 115.1 (C-8'), 114.9 (C-8''), 80.8 (C-1), 72.6 (C-4), 71.2 (C-5), 69.3 (C-3), 36.7 (C-6), 35.5 (C-2).Compound 6 (Nepitrin) was obtained as an amorphous yellowish solid. C 22 H 22 O 12; ESI-MS m/z: 479 [M+H] + . 1 H NMR (600.13 MHz, CD 3 OD) δ: 7.43 (1H, dd, J = 8.3, 2.0 Hz, H-6'), 7.41 (1H, d, J = 2.0 Hz, H-2'), 6.96 (1H, s, H-8), 6.90 (1H, d, J = 8.3 Hz, H-5'), 6.61 (1H, s, H-3), 5.12 (1H, d, J = 7.6 Hz, H-1''), 3.95 (1H, dd, J = 12.2, 2.0 Hz, H-6''a), 3.73 (1H, dd, J = 12.2, 6.1 Hz, H-6''b), 3.57 (1H, dd, J = 9.1, 7.6 Hz, H-2'' ax ), 3.57 (1H, m, H-5'' ax ), 3.52 (1H, brt, J = 9.1 Hz, H-3'' ax ), 3.42 (1H, brt, J = 9.1 Hz, H-4'' ax ), 3.89 (3H, s, CH 3 O); 13 C NMR (150.9 MHz, CD 3 OD) δ: 184.0 (C-4), 166.7 (C-2), 157.6 (C-7), 154.5 (C-9), 153.8 (C-5), 150.8 (C-4'), 146.7 (C-3'), 133.7 (C-6), 123.2 (C-1'), 120.1 (C-6'), 116.5 (C-5'), 113.9 (C-2'), 107.2 (C-10), 103.4 (C-3), 101.7 (C-1''), 95.5 (C-8), 78.2 (C-5''), 77.6 (C-3''), 74.4 (C-2''), 71.0 (C-4''), 62.2 (C-6''), 61.2 (CH 3 O). Compound 7 (Hispiduloside) was obtained as an amorphous yellowish solid. C 22 H 22 O 11; ESI-MS: m/z 463 [M+H] + . 1 H NMR (600.13 MHz, CD 3 OD) δ: 7.88 (2H, d, J = 8.7, H-2', H-6'), 6.95 (1H, s, H-8), 6.92 (2H, d, J = 8.7 Hz, H-3', H-5'), 6.64 (1H, s, H-3), 5.12 (1H, d, J = 7.6 Hz, H-1''), 3.96 (1H, dd, J = 12.1, 2.1 Hz, H-6''a), 3.73 (1H, dd, J = 12.1, 6.1 Hz, H-6''b), 3.58 (1H, dd, J = 9.1, 7.6 Hz, H-2'' ax ), 3.58 (1H, m, H-5'' ax ), 3.53 (1H, brt, J = 9.1 Hz, H-3'' ax ), 3.42 (1H, brt, J = 9.2 Hz, H-4'' ax ), 3.89 (3H, s, CH 3 O); 13 C NMR (150.9 MHz, CD 3 OD) δ: 184.0 (C-4), 166.6 (C-2), 162.6 (C-4'), 157.6 (C-7), 154.1 (C-9), 153.8 (C-5), 133.8 (C-6), 129.3 (C-2', C-6'), 122.8 (C-1'), 116.7 (C-3', C-5'), 107.2 (C-10), 103.3 (C-3), 101.7 (C-1''), 95.5 (C-8), 78.3 (C-5''), 77.6 (C-3''), 74.5 (C-2''), 71.0 (C-4''), 62.2 (C-6''), 61.2 (CH 3 O).Compound 8 (Jaceosid) was obtained as amorphous yellowish solid. C 23 H 24 O 12; ESI-MS m/z : 493 [M+H] + . 1 H NMR (600.13 MHz, CD 3 OD) δ: 7.56 (1H, dd, J = 8.3, 1.8 Hz, H-6'), 7.52 (1H, d, J = 1.8 Hz, H-2'), 7.02 (1H, s, H-8), 6.94 (1H, d, J = 8.3 Hz, H-5'), 6.71 (1H, s, H-3), 5.13 (1H, d, J = 7.7 Hz, H-1''), 3.97 (3H, s, CH 3 O-3'), 3.95 (1H, dd, J = 12.1, 2.0 Hz, H-6''a), 3.90 (3H, s, CH 3 O-6), 3.71 (1H, dd, J = 12.1, 6.2 Hz, H-6''b), 3.58 (1H, m, H-5'' ax ), 3.57 (1H, ov, H-2'' ax ), 3.52 (1H, brt, J = 9.1 Hz, H-3'' ax ), 3.41 (1H, brt, J = 9.3 Hz, H-4'' ax ); 13 C NMR (150.9 MHz, CD 3 OD) δ: 184.2 (C-4), 166.6 (C-2), 157.7 (C-7),154.7 (C-5), 154.2 (C-9), 152.3 (C-4'), 149.6

  Weigh 15.0 mg of Chlorogenic acid (CA) into a volumetric flask of 10.0 mL and dilute to 10.0 mL with MeOH. Weigh 16.3 mg of 1,5-O-Dicaffeoylquinic acid (DCQA) into a volumetric flask of 5.0 mL and dilute to 5.0 mL with MeOH. Weigh 12.0 mg of Nepetin (NPT) acid into a volumetric flask of 20.0 mL and dilute to 20.0 mL with MeOH. Place 2.0 mL of each solution into a volumetric flask of 50.0 mL and dilute to 50.0 mL with MeOH. Final concentrations for CA, DCQA

  acid. Two compounds (2 and 3) with the retention times (Rt) 15.0 and 16.1 min, UV spectra characteristic of flavonoids and the parent ions m/z 287 and 317 [M + H] + were supposed to correspond to Luteolin and Nepetin, respectively. Two compounds (4 and 5) with Rt 8.2 and 9.6 min and UV spectra characteristic of phenolic acids show the same parent ions m/z 517 [M + H] + and suggest that they possess the same caffeoyl chromophore. Two compounds (6 and 7) with Rt 10.6 and 13.6 min, UV spectra characteristic of flavonoids and the parent ions m/z 479 and 463 [M + H] + , could be glucosides of Nepetin and Hispidulin because the fragment ions 317 and 301 can correspond to their aglycones, respectively. The difference between the parent and fragment ion of 162 Da, for both compounds, can suggest the loss of a glucose residue.

  177.2 (C-7), one was an ester carbon at δ C 168.7(C-9'), two were an oxygenated aromatic carbon according to its chemical shift at δ C 149.6 (C-4'), 146.8 (C-3'). Among the eight methine groups, five corresponded to aromatic carbons according to their chemical shifts at δ C 147.1 (C-7'), 123.0 (C-6'), 116.5 (C-5'), 115.2 (C-8'), 115.1 (C-2') and tree corresponded to oxymethine carbons according to their chemical shifts at δ C 73.5 (C-4), 72.0 (C-5), 71.3(C-3). All methylene groups are alkane to their chemical shifts at δ C 38.2(C-2) and 36.8 (C-6). The 1 H NMR spectrum of 1 displayed an ABX system at δ H 7.05 (1H, d, J = 2.0 Hz, H-2'), 6.96 (1H, dd, J = 8.1, 2.0 Hz, H-6'), 6.79 (1H, d, J = 8.1 Hz, H-5') and two trans olefinic protons as doublets (J = 16.0 Hz) at δ H 7.56 (H-7'), 6.26 (H-6') giving a crosspeak with an ester function at δ C 168.7 (C-9') in HMBC spectrum and a crosspeak with the aromatic carbon at δ C 127.8 (C-1'), 123.0 (C-6'), 115.1(C-2') corresponded to a

  25 H 24 O 12 by positive mode electrospray ionization mass spectrometry (ESI-MS) data at m/z 517 [M+H] + . Compared to compound 1, the 13 Cand 1 H NMR spectra of 4, indicated the presence of two trans-caffeoyl groups and a quinic acid.The position of the two trans-caffeoyl groups on quinic acid was established by the downfield shift of the proton signals at δ H 5.42 (H-3 eq ) and δ H 5.42 (H-5 ax ) and carbon signals at δ C 73.0 (C-5) and δ C 72.2 (C-3) and by the 3 J HMBC correlation of H-3 eq at δ H 5.42 with ester carbon at δ C 168.5 (C-9') and H-5 ax at δ H 5.42 with ester carbon at δ C 169.0 (C-9'') showed that two trans-caffeoyl moiety were attached on C-3 and on C-5 respectively of the quinic acid. Based on further comparison of the NMR data with those in the literature[START_REF] Kodoma | 3,5-Di-O-caffeoylquinic acid, an infection-inhibiting factor from Pyrus pyrifolia induced by infection with Alternaria alternata[END_REF], the structure of 4 was determined to be 3,5-O-Dicaffeoylquinic acid.Compound 5 presents a molecular formula to be C 25 H 24 O 12 by positive mode electrospray ionization mass spectrometry (ESI-MS) data at m/z 517 [M+H] + . The 13 C and 1 H NMR spectra of 5 indicated the presence of two trans-caffeoyl groups and a quinic acid and were also similar to those of compound 4. The major differences were the chemical shift of C-1, C-3 and H-3 eq at the quinic acid moiety in compound 5. The C-1 of 5 was deshielded at δ C 80.8 and C-3, H-3 eq were shielded at δ C 69.3 and at δ H 4.31 respectively, showed that the second trans-caffeoyl group was attached on C-1 of the quinic acid. Based on further comparison of the NMR data with those in the literature[START_REF] Merfort | Caffeoylquinic acids from flowers of Arnica montana and Arnica chamissonis[END_REF], the structure of 5 was determined to be 1,5-O-Dicaffeoylquinic acid.The molecular formula of compound 2 was deduced to be C 15 H 10 O 5 by positive mode electrospray ionization mass spectrometry (ESI-MS) data at m/z 287 [M+H] + . The 13 C NMR and edited-HSQC NMR spectra of 2 displayed 15 carbon signals included nine quaternary (δ C184.3, 166.2, 166.0, 163.2, 159.4, 151.0, 147.1, 123.8, 105.4) and six oxymethine carbons (δ C103.5, 120.4, 116.9, 114.3, 100.2, 95.3). Among the nine quaternary carbons, one was an aromatic ketone carbon at δ C 184.3 (C-4), six were oxygenated aromatic carbon according to their chemical shifts at δ C 166.2 (C-2), 166.0 (C-7), 163.2 (C-5), 159.4 (C-9), 151.0 (C-4'), 147.1 (C-3') and two are aromatic carbon according to their chemical shifts at δ C 123.8 (C-1'), 105.4 (C-10). All oxymethine carbons were aromatic carbons according to their chemical shifts at δ C 120.4 (C-6'), 116.9 (C-5'), 114.3 (C-2'), 100.2 (C-6), 95.3 (C-8). The 1 H NMR spectrum of 2 displayed an ABX system at δ H 7.35 (1H, m, H-2'), 7.35 (1H, m, H-6'), 6.89 (1H, d, J = 8.8 Hz, H-5') due to a 3',4'-disubstitution pattern of ring B. Moreover, two meta-coupled doublets at δ H 6.41 (1H, J = 2:1 Hz, H-8) and 6.19 (1H, J = 2:1 Hz, H-6) were consistent with 5,7-dioxygenated ring A. These resonances together with the singlet at δ H 6.52 (H-3) and the corresponding carbon signals revealed the presence of a luteolin. Based on further comparison of the NMR data with those in the literature (Azimova and Vinogradova, 2013), the structure of 2 was determined to be luteolin. Compound 3 shows the molecular formula C 16 H 12 O 7 deduced by positive mode electrospray ionization mass spectrometry (ESI-MS) data at m/z 317 [M+H] + . Compared to compound 2, the 13 C and 1 H NMR spectra of 3 are similar. The major differences were the presence of the chemical shift at δ C 60.9 and at δ H 3.88 (3H, s) corresponding of methoxy group, and a disappeared of H-6 signal. The position of the methoxy group was established by the downfield shift of the carbon signals at δ C 132.8 (C-3) and the 3 J crosspeak in HMBC spectrum between methoxy group at δ H 3.88 and carbon signals at δ C 132.8 (C-3), showed that the methoxy group was attached on C-3 of the flavone. Based on further comparison of the NMR data with those in the literature (Azimova and Vinogradova, 2013), the structure of 3 was determined to be Nepetin. Compound 6 presents a molecular formula C 22 H 22 O 12 deduced by positive mode electrospray ionization mass spectrometry (ESI-MS) data at m/z 479 [M+H] + . Compared to compound 3, the 13 C and 1 H NMR spectra of 3 are similar. Additionally, the resonances of on anomeric proton, indicative of the presence of one βlinked sugar unit δ H 5.12 (1H, d, J = 7:6 Hz, H-1'').

Figure 1 .

 1 Figure 1. Isolated phenolic acids and flavonoids from I.montana

Figure 2 .

 2 Figure 2. Chromatograms of Calibration and Validation Standards by UHPLC-DAD at 335 nm Calibration standard

Table 5

 5 Quantification of Chlorogenic acid (CA), 1,5-O-Dicaffeoylquinic acid (DCQA) and Nepetin (NPT) in flowers of Inula montana L. from different collection sites

Fig. 1

 1 Fig.1 Response function of Calibration standard (CS). Two points per day for each

Fig. 4 Fig. 5

 45 Fig.4 Response function of Calibration standard (CS). Two points per day for each

Fig. 7 Fig. 8

 78 Fig.7 Response function of Calibration standard (CS). Two points per day for each

Table 1

 1 Flowers of Inula montana L. from different collection sites

	Place of collect	Collect date	Altitude (m)	Voucher specimen
	Merindol	03/06/2015	170	IMFLF151
	Bonnieux	31/05/2017	680	IMFLFB171
	Merindol	31/05/2017	170	IMFLFMe171

Table curve

 curve

	2.4 UHPLC-DAD-ESI-MS/MS analysis
	Chromatographic separation was achieved on Agilent 1290 liquid chromatograph using
	Zorbax SB-Phenyl (100 × 2.1 mm, 1.8 µm, Agilent, Germany) column, operated at 43 °C. Water
	acidified with 0.1% of formic acid (v/v) as Solvent A and Methanol as Solvent B were used for the
	gradient elution at 0.5 mL/min. The gradient program was: 18% B (from 0 to 3.5 min), 18 to 30% B
	(from 3.5 min to 3.6 min), 30% B (from 3.6 min to 12.0 min), 30 to 40% B (from 12.0 min to 12.1
	min), 40% B (from 12.1 min to 16.2 min), 40 to 42% B (from 16.2 min to 16.3 min), 42% B (from
	16.3 min to 22.0 min), 42 to 80% B (from 22.0 min to 22.1 min), 80% B (from 22.1 min to 26.0
	min) with post time of 5.0 min and injection volume of 0.3 µL. The UV detection wavelength was
	100 x (OD of test well -OD of blank) OD of DMSO control -OD of blank
	Version 2.0.

set at 335 nm. The following MS parameters were employed: on positive ionization mode spray

Table 2

 2 Spectroscopic data of compounds of the extract L2E analyzed by UHPLC-DAD-ESI-MS/MS

	Cmpd n o Rt (min)	λ max (nm)	[M+H] + (m/z)	MS/MS (m/z)	Identification
	1	2.8	240-326	355	185-215	Chlorogenic acid
	4	8.2	242-328	517	-	3,5-O-Dicaffeoylquinic acid a
	5	9.6	242-328	517	499	1,5-O-Dicaffeoylquinic acid
	6	10.6	272-348	479	317	Nepitrin
	7	13.6	232-272-334	463	301	Hispiduloside
	8	14.4	224-260-346	493	331	Jaceosid
	2	15.0	254-348	287	-	Luteolin a,c
	3	16.1	254-348	317	302	Nepetin c
	10	19.2	280-338	331	298-316	6-Hydroxykaempferol 3,7-dimethyl ether b
	9	19.9	222-272-336	301	286	Hispidulin b,c
	11	20.5	268-344	361	346	Chrysosplenol C b

a No MS/MS fragments were detected; b

Garayev et al., 2017; c Reynaud and Lussignol, 1999

  shows a molecular formula to be C 16 H 18 O 9 by positive mode electrospray ionization mass spectrometry (ESI-MS) data at m/z 355 [M+H] + . The 13 C NMR and edited-HSQC NMR

spectra of 1 displayed 16 carbon signals included six quaternary carbons (δ C

177.2, 168.7, 149.6, 146.8, 127.8, 76.

2), eight methine (δ C .147.

1, 123.0, 116.5, 115.2, 115.1, 73.5, 72.0, 71.3

) and two methylene groups (δ C .38.2, 36.8). Among the four quaternary carbons, one was an acid carbon at δ C

Table 3 .

 3 Anti-inflammatory potential of compounds 1-8 Each of three calibration points was prepared in duplicate and each of five validation points was prepared in triplicate over three consecutive days. The coefficients of determination of each-day equation of the regression line were all greater than R 2 >0.999 for the calibration and validation standards.The precision of the method was determined by computing the Relative Standard Deviation (RSDs) for repeatability and intermediate precision at each concentration level of the validation standards. The precision of the validation standards did not exceed 5% for each compound, as shown in Table4. The comparable RSDs between repeatability and intermediate fidelity were mainly due to non-significant intergroup variances, validating the precision of the developed method.

	Compounds	NO release. IC 50 (µM)
	1	31.46 ± 5.20
	2	0.52 ±0.01
	3	3.04 ± 0.27
	4	6.94 ± 0.98
	5	2.54 ± 0.34
	6	1.23 ± 0.09
	7	0.48 ± 0.00
	8	0.34 ± 0.01
	Dexamethasone a	3.89 + 0.94
	a Positive control

Table 4

 4 Results of validation of UHPLC-DAD method for the quantitative analysis of Chlorogenic acid (CA), 1,5-O-Dicaffeoylquinic acid (DCQA) and Nepetin (NPT).

	Validation criteria	CA	DCQA	NPT
	Response function (m=3; d=3; n=2;) a Y=ax
	Day 1	R 2 =0.9999	R 2 =0.9999	R 2 =0.9998
	Day 2	R 2 =0.9999	R 2 =0.9998	R 2 =0.9997
	Day 3	R 2 =0.9999	R 2 =0.9998	R 2 =0.9997
	Linear Range	29.1-94.0 µg/mL	70.5-221.0 µg/mL	10.4-32.1 µg/mL
	Trueness (m=5; d=3; n=3) a	Concentration (µg/mL)	Relative biais (%)	Concentration (µg/mL)	Relative biais (%)	Concentration (µg/mL)	Relative biais (%)
	Level 1	30.4	0.75	71.9	-0.05	10.5	-0.16
	Level 2	45.7	2.72	108.0	0.39	15.8	0.25
	Level 3	60.8	0.68	143.9	0.49	21.0	-0.02
	Level 4	75.9	-1.70	179.6	-0.26	26.3	0.39
	Level 5	91.2	-2.46	215.6	-0.57	31.5	-0.46
	Precision (d=3; n=3) a	Repeatability RSD (%)	Intermediate precision RSD (%)	Repeatability RSD (%)	Intermediate Precision RSD (%)	Repeatability RSD (%)	Intermediate Precision RSD (%)
	Level 1	0.90	0.90	1.96	1.96	1.41	1.58
	Level 2	0.90	0.90	1.29	1.38	1.66	1.66
	Level 3	1.64	1.64	1.81	1.81	1.95	1.95
	Level 4	1.08	1.08	1.74	1.74	1.49	1.71
	Level 5	0.94	0.94	1.32	1.32	1.03	1.03
	Accuracy (d=3; n=3) a	β-expectation lower and upper tolerance limits of the relative errors (%)	β-expectation lower and upper tolerance limits of the relative errors (%)	β-expectation lower and upper tolerance limits of the relative errors (%)
	Level 1	-1.49; 2.98	-4.92; 4.82	-4.31; 3.99
	Level 2	0.48; 4.97	-3.21; 3.99	-3.90; 4.40
	Level 3	-3.40; 4.77	-4.01; 4.99	-4.88; 4.85
	Level 4	-4.40; 1.01	-4.60; 4.07	-4.14; 4.92
	Level 5	-4.90; -0.02	-3.86; 2.72	-3.04; 2.12
	Linearity slope	0.9889	0.9979	0.9993
	R 2	0.9987	0.9994	0.9997
	LOD b	0.12 µg/mL	0.17 µg/mL	0.09 µg/mL
	LOQ b	0.39 µg/mL	0.56 µg/mL	0.31 µg/mL

a m: number of calibration points; d: days; n: replicates. b LOD = limit of detection, LOQ = limit of quantification.

DAD analysis method of Chlorogenic acid (CA) Tab. 1 Raw data of Calibration standard

  The potential anti-inflammatory effect, based on NO-inhibition, of different extracts and compounds from Inula montana provides the scientific basis for the traditional use of this plant for the relief of bruises and underlines the ethnopharmacological relevance. Nevertheless this interesting biological activity should be investigated by complementary experiments concerning other mediators of the inflammatory response (TNFα, IL-1) in order to confirm the antiinflammatory activity and to propose Inula montana as an alternative to Arnica montana.

	Place of collect Merindol Bonnieux Merindol % = g per 100 g of dry plant material. a The mean of three independent analysis ± Standard Deviation. The results of one-Collect date Altitude (m) CA content a (%) DCQA content a (%) NPT content a (%) 03/06/2015 170 0.61±0.02 1.44±0.03 0.21±0.01 31/05/2017 680 0.39±0.01 1.02±0.11 0.25±0.01 31/05/2017 170 0.39±0.02 0.92±0.05 0.22±0.02 way ANOVA test are significant at p < 0.01; Fig.8 Linearity of Validation standard (VS). Three points per day for each concentration over three days Fig.9 Accuracy profile of VS at 95% (Risk = 5%) over 5 validation points (50, 75, 100, 125 and 150%) Tab.2 Raw data of Validation standard Validation of UHPLC-Level / sample / Day Level Theoretical concentration (mg/mL) Introduced concentration (mg/mL) CA area Results 1.1.1 50% 0.0030 0.0296 Level / sample / Day Level Introduced Results Results Introduced concentration equivalent content [1] (mg/mL) CA area Inverse prediction Calculated concentration X = Y/a Level / sample / Day Level concentration Inverse prediction equivalent CA area Calculated concentration content [1] (mg/mL) X = Y/a 232 1.2.1 0.0030 0.0296 1.1.1 0.0312 254 0.0304 238 1.1.2 0.0030 0.0296 234 1.2.2 0.0030 0.0296 241 1.1.3 0.0030 0.0296 1.2.1 0.0312 256 0.0307 1.3.1 0.0314 256 4.1.1 0.0781 635 0.0757 0.0305 1.1.2 0.0307 258 0.0312 4.2.1 0.0781 615 0.0733 4.3.1 0.0782 626 0.0746 243 1.2.3 0.0030 0.0296 50% 1.2.2 0.0308 255 0.0308 4.1.2 0.0768 626 0.0758 244 2.1.1 100% 0.0060 0.0592 482 2.2.1 0.0060 0.0592 1.3.2 0.0309 252 0.0303 1.1.3 0.0293 245 0.0305 4.2.2 0.0768 615 0.0744 125% 4.3.2 0.0767 616 0.0746 488 2.1.2 0.0060 0.0592 1.2.3 0.0292 246 0.0308 4.1.3 0.0728 594 0.0744 484 2.2.2 0.0060 0.0592 1.3.3 0.0291 245 0.0306 4.2.3 0.0729 602 0.0752 497 2.1.3 0.0060 0.0592 2.1.1 0.0473 397 0.0471 4.3.3 0.0731 592 0.0739 494 2.2.3 0.0060 0.0592 497 3.1.1 150% 0.0090 0.0888 2.2.1 0.0470 396 0.0473 2.3.1 0.0469 393 0.0940 749 0.0891 5.1.1 0.0470 5.2.1 0.0936 747 0.0893 724 3.2.1 0.0090 0.0888 2.1.2 0.0461 389 0.0472 5.3.1 0.0939 733 0.0873 732 3.1.2 0.0090 0.0888 75% 2.2.2 0.0463 388 0.0468 5.1.2 0.0921 740 0.0896 733 3.2.2 0.0090 0.0888 739 3.1.3 0.0090 0.0888 2.3.2 0.0461 384 0.0467 2.1.3 0.0437 380 5.2.2 0.0922 735 0.0890 150% 0.0475 5.3.2 0.0923 732 0.0885 755 3.2.3 0.0090 0.0888 2.2.3 0.0438 370 0.0463 5.1.3 0.0874 721 0.0902 745 2.3.3 0.0438 370 0.0463 5.2.3 0.0875 704 0.0880
	3.1.1 5.3.3	0.0626 0.0875	517 713	0.0617 0.0892
	100% 1 : Constituent Equivalent (mg/ml) : calculated using the mean of Chlorogenic acid content calculated on the 3.2.1 0.0625 515 0.0616 3.3.1 0.0625 513 corresponding day : 0.0614 3.1.2 0.0616 513 Day 1: IMFLF151 contains: 0.624 % Chlorogenic acid, 0.0621 3.2.2 0.0614 493 Day 2: IMFLF151 contains: 0.614 % Chlorogenic acid, 0.0597 Day 3: IMFLF151 contains: 0.582 % Chlorogenic acid.
	3.3.2	0.0616	497	0.0601
	3.1.3	0.0584	505	0.0630
	3.2.3	0.0583	489	0.0611
	3.3.3	0.0586	486	0.0606

Validation of UHPLC-DAD analysis method of 1,5-O-Dicaffeoylquinic acid (DCQA) Tab.3 Raw data of Calibration standard

	Tab.4 Raw data of Validation standard			Results
	Level / sample / Day Level / sample / Day Level Level Introduced Theoretical concentration (mg/mL) concentration equivalent content [2] (mg/mL) Introduced Level / sample / Day Level concentration equivalent content [2] (mg/mL)	Introduced concentration (mg/mL) DCQA area DCQA area	DCQA area Results Results Inverse prediction Calculated concentration X = Y/a Inverse prediction Calculated concentration X = Y/a
	1.1.1 1.2.1 1.1.2 1.2.2 1.1.3 1.2.3 2.1.1 2.2.1 2.1.2 2.2.2 2.1.3 2.2.3 3.1.1 3.2.1 3.1.2 3.2.2 3.1.3 3.2.3 1.1.1 4.1.1 1.2.1 4.2.1 1.3.1 1.1.2 4.3.1 1.2.2 4.1.2 1.3.2 4.2.2 1.1.3 4.3.2 1.2.3 4.1.3 1.3.3 4.2.3 2.1.1 4.3.3 2.2.1 5.1.1 2.3.1 5.2.1 2.1.2 5.3.1 2.2.2 5.1.2 2.3.2 5.2.2 2.1.3 5.3.2 2.2.3 5.1.3 2.3.3 5.2.3 3.1.1 5.3.3	50% 100% 150% 50% 125% 75% 150%	0.0652 0.0652 0.0652 0.0652 0.0652 0.0652 0.1304 0.1304 0.1304 0.1304 0.1304 0.1304 0.1956 0.1956 0.1956 0.1956 0.1956 0.1956 0.0734 0.0732 0.1837 0.0737 0.1837 0.0715 0.1838 0.0717 0.1785 0.0718 0.1787 0.0709 0.1784 0.0707 0.1763 0.0705 0.1766 0.1112 0.1771 0.1105 0.2210 0.1102 0.2201 0.1072 0.2207 0.1077 0.2142 0.1071 0.2143 0.1059 0.2146 0.1060 0.2118 0.1061 0.2119 0.1472 0.2119	Min	Biais 0.0652 Max 0.0652 0.0652 0.0652 0.0652 0.0652 0.1304 0.1304 0.1304 0.1304 0.1304 0.1304 0.1956 0.1956 0.1956 0.1956 0.1956 0.1956 691 694 1717 669 1712 671 1666 646 1670 648 1641 652 1638 658 1570 632 1648 1051 1592 1059 2056 1015 2035 989 2004 981 1995 971 1924 973 1934 975 1945 974 1925 1376 1954	+5% limit	-5% limit	600 593 569 568 597 567 1212 1231 1196 1164 1195 1170 1838 1815 1784 1792 1799 1756 0.0729 0.0734 0.1807 0.0702 0.1802 0.1751 0.0737 0.1831 0.0706 0.1798 0.0707 0.1798 0.0722 0.1747 0.0731 0.1829 0.0704 0.1763 0.1099 0.2158 0.1114 0.2145 0.1070 0.2106 0.1085 0.2188 0.1073 0.2110 0.1068 0.2119 0.1083 0.2162 0.1083 0.2139 0.1082 0.1447 0.2169
	3.2.1 2 : Constituent Equivalent (mg/ml) : calculated using the mean of 1,5-O-Dicaffeoylquinic acid content calculated on 0.1470 1366 0.1439 3.3.1 0.1469 1351 the corresponding day : 0.1424 3.1.2 0.1432 1362 Day 1: IMFLF151 contains: 1.468 % 1,5-O-Dicaffeoylquinic acid, 0.1491 Day 2: IMFLF151 contains: 1.427 % 1,5-O-Dicaffeoylquinic acid,
	3.2.2	100% Day 3: IMFLF151 contains: 1.411 % 1,5-O-Dicaffeoylquinic acid. 0.1429 1305	0.1433
	3.3.2		0.1431		1307			0.1432
	3.1.3		0.1416		1325			0.1470
	3.2.3		0.1413		1322			0.1469
	3.3.3		0.1419		1271			0.1408

Validation of UHPLC-DAD analysis method of Nepetin (NPT) Tab.5 Raw data of Calibration standard

	Tab.6 Raw data of Validation standard
	Level / sample / Day 1.1.1 Level / sample / Day Level Level Introduced Theoretical concentration (mg/mL) 0.0120 Introduced concentration equivalent content [3] (mg/mL) 1.1.1 0.0106 Level / sample / Day Level concentration equivalent content [3] (mg/mL)	Introduced concentration (mg/mL) 0.0136 NPT area 100 NPT area	NPT area Results 126 Results Results Inverse prediction Calculated concentration X = Y/a 0.0105 Inverse prediction Calculated concentration X = Y/a
	Min 3 : Constituent Equivalent (mg/ml) : calculated using the mean of Nepetin content calculated on the corresponding Biais Max +5% limit -5% limit 50% 1.2.1 0.0120 0.0136 128 1.1.2 0.0120 0.0136 125 1.2.2 0.0120 0.0136 130 1.1.3 0.0120 0.0136 127 1.2.3 0.0120 0.0136 126 2.1.1 100% 0.0240 0.0273 257 2.2.1 0.0240 0.0273 265 2.1.2 0.0240 0.0273 261 2.2.2 0.0240 0.0273 264 2.1.3 0.0240 0.0273 260 2.2.3 0.0240 0.0273 265 3.1.1 150% 0.0360 0.0409 395 3.2.1 0.0360 0.0409 399 3.1.2 0.0360 0.0409 400 3.2.2 0.0360 0.0409 404 3.1.3 0.0360 0.0409 405 3.2.3 0.0360 0.0409 398 50% 1.2.1 0.0106 101 0.0106 1.3.1 0.0107 99 4.1.1 0.0266 253 0.0264 0.0103 1.1.2 0.0104 102 4.2.1 0.0266 255 0.0267 0.0108 1.2.2 0.0104 99 4.3.1 0.0267 250 0.0261 0.0105 1.3.2 0.0104 101 4.1.2 0.0259 247 0.0262 0.0106 1.1.3 0.0106 98 125% 4.2.2 0.0259 245 0.0260 0.0103 1.2.3 0.0105 100 4.3.2 0.0258 244 0.0259 0.0105 1.3.3 0.0105 100 4.1.3 0.0262 248 0.0260 0.0105 2.1.1 75% 0.0161 155 4.2.3 0.0263 258 0.0270 0.0161 2.2.1 0.0160 152 4.3.3 0.0263 259 0.0270 0.0158 2.3.1 0.0160 149 5.1.1 0.0321 304 0.0317 0.0156 2.1.2 0.0155 151 5.2.1 0.0319 303 0.0317 0.0160 2.2.2 0.0156 148 5.3.1 0.0320 299 0.0311 0.0156 2.3.2 0.0155 149 5.1.2 0.0310 300 0.0319 0.0158 2.1.3 0.0158 148 150% 5.2.2 0.0310 294 0.0312 0.0155 2.2.3 0.0158 150 5.3.2 0.0311 292 0.0310 0.0157 2.3.3 0.0158 156 5.1.3 0.0315 300 0.0314 0.0163 3.1.1 100% 0.0214 207 5.2.3 0.0315 296 0.0310 0.0216 0.0213 196 0.0205 5.3.3 0.0315 300 0.0314 3.2.1 3.3.1 0.0213 200 0.0209 3.1.2 0.0207 198 day : 0.0210 3.2.2 0.0207 197 Day 1: IMFLF151 contains: 0.213 % Nepetin, Day 2: IMFLF151 contains: 0.207 % Nepetin, 0.0210 3.3.2 0.0207 200 0.0212 Day 3: IMFLF151 contains: 0.210 % Nepetin.
	3.1.3	0.0211	195	0.0204
	3.2.3	0.0210	205	0.0215
	3.3.3	0.0211	204	0.0212

Acknowledgement

We are grateful to Mr. Vincent Lorenzini from the Embassy of France in Azerbaijan for his contribution to the funding of this research project work and to Dr. Didier Morisot and Mr. Laurent Michel for their botanical identification.

The work was supported by the region PACA (France) (grant number 2013-08504), the Embassy of France in Azerbaijan (grant numbers 836060K and 862819K) and the Luberon Natural Regional Park in France (grant number 887312G).

Conflict of interest statement

The authors declare no competing financial interest.

SUPPORTING INFORMATION Validation of UHPLC-DAD analysis method of


Mathieu Aucejo

Olivier De Smet

Bayesian structural source identification using local generalized Gaussian priors

Keywords: Source identification, Vibration, Regularization

The reconstruction of mechanical sources from vibration measurements is known to be an ill-posed inverse problem. A classical solution to overcome this difficulty consists in including prior information on the spatial distribution of the sources to constrain the space of solutions. Among all the methods developed to this end, the Tikhonov regularization is certainly the most popular. However, it assumes a global a priori on the spatial distribution of sources. Incidentally, poor results can be obtained if a structure is subjected to localized and distributed sources. This paper aims at providing an identification methodology able to take advantage of prior local information on both the nature and location of excitation sources. For this purpose, the Bayesian framework is well adapted, since it offers a rigorous probabilistic approach to exploit our a priori knowledge on the sources to identify. The proposed Bayesian formulation is based on the use of generalized Gaussian priors, which provide a flexible way to introduce local a priori information. Practically, the resulting optimization problem is solved from a Generalized Iteratively Reweighted Least-Squares algorithm. The validity of the proposed methodology is illustrated numerically. It is especially shown that local information improves drastically the quality of the source identification.

INTRODUCTION

The source identification of mechanical sources from vibration measurements is an ill-posed inverse problem, meaning that the existence of a unique stable solution is not guaranteed. A classical approach to bypass this difficulty consists in including in the formulation of the inverse problem some prior information on the measurement noise and the spatial distribution of sources to constrain the space of solutions. This idea is at core of Tikhonov-like regularization methods [START_REF] Thite | The quantification of structure-borne transmission paths byinverse methods. Part 2: Use of regularization techniques[END_REF]. In such a regularization procedure, however, the a priori on the spatial distribution of sources is global. Incidentally, this can lead to poor identification if actual sources combine both localized and distributed sources, since the a priori has to reflect a compromise between two contradictory distributions.

The present paper aims at remedying this problem using available local information on the distribution and the nature of sources to identify. The proposed approach relies on the Bayesian inference, which offers a rigorous mathematical framework allowing combining both probabilistic and mechanical data [START_REF] Zhang | Bayesian force reconstruction with an uncertain model[END_REF]. To exploit our a priori knowledge of excitation sources, the structure is divided into different zones in which it is assumed that local priors on the sources to identify follow generalized Gaussian distributions. In doing so, prior information can be accurately tuned in each zone of the structure. From a mathematical point of view, the solution of the problem is defined as the maximum a posteriori estimate. Practically, one seeks the solution of the dual minimization problem, which is solved from a Generalized Iteratively Reweighted Least-Squares algorithm [START_REF] Rodriguez | An Iteratively Weighted Norm Algorithm for Total Variation Regularization[END_REF]. To assess the validity of such an approach, a numerical validation is proposed. Obtained results clearly show that using prior local information significantly improves the quality of the identification.

BAYESIAN SOURCE IDENTIFICATION

Problem description

Let consider the practical situation where the vibration field X, measured over the surface of a structure, is caused by an unknown excitation field F. If the structure is linear, it dynamic behavior is completely determined by the transfer functions matrix H, relating the vibration field X to the unknown field F. Now, let suppose that the measured vibration field is corrupted by a measurement noise N. In such a situation, the measured vibration field X is obtained from the following direct formulation:

N HF X + = . (1)
The structural source identification problem consists in estimating the unknown excitation field F acting on a structure from the vibration field X measured on its surface. To this end, the Bayesian framework is adopted. The Bayesian paradigm consists in considering all the parameters of the problem as random variables. Consequently, the uncertainty on each parameter is modeled by a probability distribution, describing the state of knowledge or the prior on this parameter. The Bayesian source identification formulation relies on the Bayes' rule:

( ) ( ) ( ) F F X X F p p p ∝ , (2) where:
-( )

X F p
is the posterior probability distribution, representing the probability of observing F given a vibration field X. In other words, it defines what it is known about the excitation field F after making vibration measurements; -( )

F X p
is the likelihood function, representing the probability of measuring X given an excitation field F. It reflects the uncertainty related to the measurement of the vibration field X; -( )

F p
is the prior probability distribution, representing our knowledge on the unknown excitation field F before measuring the vibration field X.

In this paper, one will seek the most probable excitation field F ˆ given a measured vibration field X. From the Bayesian point of view, it consists in finding a point estimate of F ˆ corresponding to a mode of the posterior probability distribution. Mathematically speaking, the solution of the identification problem is sought as the maximum a posteriori estimate, that is:

( ) ( ) ( ) F F X X F F F F p p p argmax argmax ˆ= = . (3)
Practically, it is generally easier to find a solution of the following dual minimization problem:

( ) ( ) ( ) F F X X F F F F p p p log log min arg log min arg ˆ-- = - = . (4)

Choice of the likelihood function and the prior probability distribution

Choice of the likelihood function

The likelihood function reflects the uncertainty related to the measurement of the vibration field X. By definition, this uncertainty is mainly related to the measurement noise N. Consequently, the likelihood function ( )

F X p
can be rewritten under the following form:

( ) ( ) N HF X F X - = p p , (5)
representing the probability of obtaining 0 HF X = given the measurement noise N.

If the noise is supposed to be due to multiple independent causes, then the likelihood function ( )

F X p
can be represented by a normal distribution with mean HF and variance 2 α :

( )

        - - ∝ 2 2 2 2 1 exp α HF X F X p , (6) where 2
• is the L 2 -norm. It can be noticed that the form of the probability distribution implicitly assumes that all the elements of the vector HF Xare independent identically distributed variables with same standard deviation α .

Choice of the prior probability distribution

The prior probability distribution reflects the uncertainty related to the unknown excitation field F. It can be seen as a measure of the a priori knowledge of the experimenter on the sources to identify. If one supposes that the structure is excited in different regions by uncorrelated excitations of various types (localized or distributed), then local excitation fields F i can be considered as independent identically distributed variables. As a result, the prior probability distribution is written:

( ) ( ) ∏ = = N i p p 1 i F F , ( 7
)
where N is the number of selected zones and ( )

i F p
is the local prior probability distribution reflecting the prior knowledge of the expert on the nature of the sources in the zone i.

For practical reasons, one assumes that the each excitation fields F i follows a generalized Gaussian distribution with zero mean, that is:

( )           - ∝ i i i q i q q i q p β i i i F L F 1 exp , (8)
where:

i q is the shape parameter controlling the shape of the probability distribution in the zone i. The shape parameter is defined in the interval ] [ +∞ , 0 ;

- i q • is i q L -norm or quasi-norm; -i
β is a scaling factor which defines the dispersion of the distribution around the mean. It is therefore a generalized measure of the variance in the zone i; i L is a differentiation operator that allows controlling the regularity of the solution.

Practical form of the identification problem

In order to derive the practical form of the identification problem, one just has to introduce Eqs. ( 6) and (8) in Eq.( 4). In doing so, one finally obtains a formulation of the identification problem equivalent to a generalized Tikhonov regularization:

∑ = + - = N i q i q q i i i i q 1 2 2 2 1 2 1 min arg ˆβ α i i F F L HF X F , (9)
At this stage, one has 2N+1 parameters ( i q , i β , α ) to mathematically transcribe prior information on the measurement noise and the nature of the sources to identify. However, it can be emphasized that parameters i q and i β play a similar role. Therefore, only one set of parameters ( i q or i β ) can be used. Here, it has been chosen to work with the scale parameters i q , while setting the values of the scale parameters i q i

β to a unique constant value β . Taking this observation into account, one finally gets the practical form of the identification problem:

∑ = + - = N i q q i i i q 1 2 2 1 2 1 min arg ˆi i F F L HF X F λ , (10)
where

β α λ 2 =
is generally known as the regularization parameter. It can be noticed that the proposed formulation is equivalent to the standard Tikhonov regularization if

2 = i q for all [ ] N i , 1 ∈ .

BAYESIAN SOURCE IDENTIFICATION

In this paper, the solution of the minimization problem given by Eq. ( 10) is computed from a general version of the Iteratively Reweighted Least-Squares (IRLS) algorithm [START_REF] Rodriguez | An Iteratively Weighted Norm Algorithm for Total Variation Regularization[END_REF].

General principle

The core idea of the Generalized Iteratively Reweighted Least-Squares (GIRLS) algorithm is to replace, within an iterative scheme, the i q L -norm, that appears in Eq. ( 10), by a weighted L 2 -norm, so that the functional to minimize has an explicit expression at each iteration. For this purpose, one just has to notice that:

( ) 2 2 1 1 1 , , n n n q n n q q n x x w x q x q q x ∑ ∑ = = ∀ ∀ , (11)
where ( )

2 2 - = q n n x q x w
is the weighting coefficient.

As part of an iterative scheme, one tries to find the solution ( )

1 + k x at iteration k+1 from the solution ( ) k x at iteration k by setting ( ) ( ) ( ) k n n x w x w =
, in order to find the equality (11) when the process has converged. Here, the direct application of this idea consists in replacing the minimization problem given by Eq.( 11) by an equivalent iterative process, for which the excitation field ( )

1 ˆ+ k F
at iteration k+1 is the solution of the following minimization problem:

( ) ( ) ( ) ∑ = + + + - = N i k k k 1 2 2 1 2 2 1 2 2 1 min arg ˆi i i F F L W HF X F λ , ( 12
)
where ( )

k i W
is a definite positive matrix of the form:

( ) ( ) ( )       = k i k T q i i i F L W 2 diag ε , ( 13
)
with:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )      ≤ > = - - k i k q k i k i k q k k x x x x T i i ε ε ε ε si si 2 2 , ( 14
)
where ( ) k i ε is a is a small real positive number acting as a damped parameter. It allows avoiding infinite weights when ( ) 0 → k x and 2 < i q .

Choice of the tuning parameters, initial guess and stopping criterion

According to Eq.( 12), the tuning parameters of the problem are the shape parameters i q , the regularization parameter ( )

1 + k λ and the damping parameter ( ) k i ε .
To properly choose the shape parameters i q , one has to keep in mind that these parameters control the general shape of the distribution. So, using 2 ≥ i q will give priority to smooth solutions, while using

1 ≤ i q
will favor sparse solutions.

Regarding the choice of the regularization parameter ( )

1 + k λ
and the damping parameter ( ) k i ε , automatic selection procedures have been implemented. For the regularization parameter, the L-curve principle is used [START_REF] Hansen | Rank-Deficient and Discrete ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF], while for the damping parameter, its value is set so that a fixed percentage i p of the smallest values of ( )

k i i F L ˆ are below ( ) k i ε . In the next of the paper, 5 = i p %.
Since the algorithm is iterative, another critical issue is the choice of the initial solution as well as that of the stopping criterion. Choosing a good initial guess is a key point of the convergence of the algorithm, since the functional to minimize is non-convex when 1 < i q

. The question that arises here is: What is a good initial guess? Actually, it is a coarse solution of the problem, easy to calculate, but sufficiently close to the final solution to ensure the convergence of the iterative process. Such requirements are fulfilled by the solution of the standard Tikhonov regularization. Finally, it remains to define a stopping criterion for the GIRLS algorithm. In this paper, a stopping criterion related to the variation of the functional

( ) ( ) ( ) ( ) ( ) ( ) ∑ = - + - = N i k k k k k J 1 2 2 1 2 2 2 2 1 ˆi i i F L W F H X F λ between two successive
iterations is used. The algorithm is automatically stopped when a prescribed tolerance defined by the user is reached. Here, the tolerance is set to 8 10 -.

NUMERICAL VALIDATION

In the present numerical validation, one seeks to identify a point force of unit amplitude acting on a thin simply supported steel plate with dimensions 2x1.5x0.017 m 3 . The coordinates of the point force, measured from the lower left corner of the plate, are (x,y) = (1.04 m, 0.93 m). Moreover, to simulate the vibration field X, a finite element model of the plate made up with 567 quadrilateral shell elements has been used. It is worth to mention that an additive Gaussian white noise has been added to the data to simulate the measured vibration field. The noise has been computed so as to have a signal-to-noise ratio of 6 dB. Finally, a FE model of the structure with free boundary conditions is used to compute the transfer functions matrix H. This has the advantage of enabling the identification of the point force acting on the structure as well as reacting forces at boundaries. As shown in Figure 1a, in such a configuration, two types of spatial distributions can be distinguished over the structure: a smooth distribution of the reacting forces at boundaries and a singular distribution around the location of the point force. shows that the location of the point force is properly estimated while the amplitude is greatly underestimated. On the contrary, reacting forces at boundaries are well identified. This contrasted result can be explained by the fact that the global a priori used in the standard Tikhonov regularization gives priority to smooth solutions. To improve the quality of the identification, the prior knowledge on the nature of sources acting on the source has to be exploited. Indeed, as presented in Figure 1a, one knows that the excitation field to identify is very sparse, except in the vicinity of the boundaries of the plate. Consequently, the structure can be divided into two regions: (i) a region, containing the point force, in which a sparsity-promoting regularization term is required ( 5 . 0 1 = q

) and (ii) a region, corresponding to the boundaries of the plate, in which a regularization term that promotes the smoothness of the solution has to be employed ( 2 2 = q ) [see Figure 2a].

CONCLUSIONS

In the present study, the motivation was to propose a Bayesian formulation of the structural source identification problem able to fully exploit spatial information available a priori on the nature and the location of the sources. To this end, the structure is divided into several regions, in which it is supposed that local priors follow generalized Gaussian laws. This formulation is very flexible, insofar as the tuning parameters of these laws can be precisely adjusted to mathematically transcribe the a priori knowledge of the experimenter on the studied system. Formally, the solution of the problem is defined as the maximum a posteriori estimate, but practically, one seeks the solution of the dual minimization problem, which is solved from a Generalized Iteratively Reweighted Least-Squares algorithm. Numerical results show that exploiting local information substantially improves the quality of the source identification.

Figure 1 -

 1 Figure 1 -Numerical validation -(a) Excitation field to identify at 450 Hz and (b) Identified excitation field from standard Tikhonov regularization

Figure 2 -

 2 Figure 2 -Numerical validation -(a) Definition of the selected zones: (o) zone 1 (sparse), (x) zone 2 (smooth) and (◊) location of the point force and (b) Excitation field identified using the proposed approach with Identity L i =

mathieu.aucejo@cnam.fr

olivier.de_smet@cnam.fr


Olivier De Smet
email: olivier.desmet@cnam.fr

Mathieu Aucejo
email: mathieu.aucejo@cnam.fr

BAYESIAN STRUCTURAL SOURCE IDENTIFICATION USING GENERALIZED GAUSSIAN PRIORS

Keywords: Bayesian identification, Structural source, Tikhonov regularization

The reconstruction of mechanical sources from vibration measurements is known to be an ill-posed inverse problem. One of the most widespread method to overcome this difficulty is the Tikhonov regularization. Such an approach consists in including in the formulation of the inverse problem some prior information on the nature of the sources to constrain the space of solutions. However, the reconstructed sources are generally too smooth to properly deal with localized sources. This paper aims at providing an identification methodology able to take advantage of prior information on the nature of excitation sources. More particularly, we will focus on the identification of localized sources. For this purpose, the bayesian framework is well adapted, since it offers a rigourous probabilistic approach to exploit our a priori knowledge on the nature of the sources to identify.

The proposed bayesian formulation is based on the use of generalized gaussian priors, which provide a flexible way to introduce a priori information. Usually one use standard gaussian priors for both uncertainties and source distributions. Because noise and sources are usually of different kind (e.g. white noise on measurement and sparsity of sources distribution), we propose to introduce two different priors. The first one tuned to deal with uncertainties and the second one adapted to the nature of the sources exciting the structure. Practically, the inverse problem is solved from a Generalized Iteratively Reweighted Least-Squares algorithm. The proposed methodology is illustrated on various numerical and experimental examples (i.e. for different sources distribution). It is shown that using priors depending on sources distribution nature are needed to improve drastically the quality of the source identification.

INTRODUCTION

The source identification of mechanical sources from vibration measurements is an ill-posed inverse problem, meaning that the existence of a unique stable solution is not guaranteed. A classical approach to bypass this difficulty consists in including in the formulation of the inverse problem some prior information on the measurement noise and the spatial distribution of sources to constrain the space of solutions. This idea is at core of Tikhonov-like regularization methods [START_REF] Thite | The quantification of structure-borne transmission paths by inverse methods. Part 2: Use of regularization techniques[END_REF]. In such a regularization procedure, however, the a priori on the spatial distribution of sources is global. Incidentally, this can lead to poor identification if actual sources combine both localized and distributed sources, since the a priori has to reflect a compromise between two contradictory distributions.

The present paper aims at remedying this problem using available information on the distribution and the nature of sources to identify. The proposed approach relies on the Bayesian inference, which offers a rigorous mathematical framework allowing combining both probabilistic and mechanical data [START_REF] Zhang | Bayesian force reconstruction with an uncertain model[END_REF]. To exploit our a priori knowledge, we separate a priori on the measurement noise and on the excitation sources like the sparsity or the regularity. In doing so, prior information can be accurately tuned for each term of the problem. From a mathematical point of view, the solution of the problem is defined as the maximum a posteriori estimate. Practically, one seeks the solution of the dual minimization problem, which is solved from a Generalized Iteratively Reweighted Least-Squares algorithm [START_REF] Rodriguez | An Iteratively Weighted Norm Algorithm for Total Variation Regularization[END_REF]. To assess the validity of such an approach, a numerical validation is proposed. Obtained results clearly show that using prior information significantly improves the quality of the identification.

BAYESIAN SOURCE IDENTIFICATION

Problem description

First, let consider the situation where the vibration field X, measured over the surface of a structure, is caused by an unknown excitation field F. If the structure is linear, its dynamic behavior is completely determined by the transfer functions matrix H, relating the vibration field X to the unknown field F. Now, let suppose that the measured vibration field is corrupted by a measurement noise N. In such a situation, the measured vibration field X is obtained from the following direct formulation:

X = HF + N (1)
The structural source identification problem consists in estimating the unknown excitation field F acting on a structure from the vibration field X measured on its surface. The Bayesian paradigm consists in considering all the parameters of the problem as random variables. Consequently, the uncertainty on each parameter is modeled by a probability distribution, describing the state of knowledge or the prior on this parameter. The Bayesian source identification formulation relies on the Bayes' rule:

p(F|X) ∝ p(X|F)p(F) (2)
where:

• p(F|X) is the posterior probability distribution, representing the probability of observing F given a vibration field X. In other words, it defines what it is known about the excitation field F after making vibration measurements;

• p(F|X) is the likelihood function, representing the probability of measuring X given an excitation field F. It reflects the uncertainty related to the measurement of the vibration field X;

• p(F) is the prior probability distribution, representing our knowledge on the unknown excitation field F before measuring the vibration field X.

In this paper, one search the most probable excitation field F given a measured vibration field X. From the Bayesian point of view, it consists in finding a point estimate of F corresponding to a mode of the posterior probability distribution. Mathematically speaking, the solution of the identification problem is sought as the maximum a posteriori estimate, that is:

F = arg max F p(F|X) = arg max F p(X|F)p(F) (3)
Practically, it is generally easier to find a solution of the following dual minimization problem:

F = arg min F -log p(F|X) = arg min F -log p(X|F) -log p(F) (4)
2.2 Choice of the likelihood function and the prior probability distribution

Choice of the likelihood function

The likelihood function reflects the uncertainty related to the measurement of the vibration field X. By definition, this uncertainty is mainly related to the measurement noise N. Consequently, the likelihood function p(X|F) can be rewritten under the following form:

p(X|F) = p(X -HF|N) (5)
representing the probability of obtaining X -HF = 0 given the measurement noise N.

If the noise is supposed to be due to multiple independent causes, then the likelihood function p(X|F) can be represented by a normal distribution with mean HF and variance α 2 :

p(X|F) ∝ exp - 1 2 X -HF 2 2 α 2 (6)
where • 2 is the L 2 -norm. It can be noticed that the form of the probability distribution implicitly assumes that all the elements of the vector X -HF are independent identically distributed variables with same standard deviation α .

Choice of the prior probability distribution

The prior probability distribution reflects the uncertainty related to the unknown excitation field F. It can be seen as a measure of the a priori knowledge of the experimenter on the sources to identify. For practical reasons, one assumes that the excitation field F follows a generalized Gaussian distribution with zero mean. The global prior probability distribution is written as:

p(F) ∝ exp - 1 q LF q q β q (7)
where:

• q is the shape parameter controlling the shape of the probability distribution. The shape parameter is defined in the interval ]0, + inf[;

• • q is the L q -norm or quasi-norm;
• β is a scaling factor which defines the dispersion of the distribution around the mean. It is therefore a generalized measure of the variance;

• L is a differentiation operator that allows controlling the regularity of the solution.

Practical form of the identification problem

To get the practical form of the identification problem, introduce Eqs. 6 and 7 in Eq. 4. One obtains the formulation of the identification problem similar to a generalized Tikhonov regularization:

F = arg min F 1 2 X -HF 2 2 α 2 + 1 q LF q q β q (8)
Here one have three parameters to mathematically describe prior information on the measurement noise and the nature of the sources. However, parameters q and β play a similar role. Only one parameter can be used. By choosing to work with the scale parameter q, while setting β q to a constant value β, one finally gets:

F = arg min F 1 2 X -HF 2 2 + λ q LF q q (9)
where λ = α 2 /β is known as the regularization parameter. If q = 2, the proposed formulation is equivalent to the standard Tikhonov regularization.

SOURCE IDENTIFICATION

The solution of the minimization problem given by Eq. 9 is computed from a general version of the Iteratively Reweighted Least-Squares (IRLS) algorithm [START_REF] Rodriguez | An Iteratively Weighted Norm Algorithm for Total Variation Regularization[END_REF].

General principle

The core idea of the Generalized Iteratively Reweighted Least-Squares (GIRLS) algorithm is to replace, within an iterative scheme, the L q -norm, that appears in Eq. 9, by a weighted L 2norm, so that the functional to minimize has an explicit expression at each iteration. For this purpose, one just has to notice that:

∀x n , ∀q, 1 q x q q = 1 q n |x n | q = 1 2 n w(x n ) |x n | 2 (10)
where w(x n ) = 2 q |x n | q-2 is the weighting coefficient. As part of an iterative scheme, one tries to find the solution x (k+1) at iteration k + 1 from the solution x (k) at iteration k by setting w(x n ) = w(x (k) n ), in order to find the equality 10 when the process has converged. Here, the direct application of this idea consists in replacing the minimization problem given by Eq. 9 by an equivalent iterative process, for which the excitation field F(k+1) at iteration k + 1 is the solution of the following minimization problem:

F(k+1) = arg min F 1 2 X -HF 2 2 + λ (k+1) 2 W (k) LF 2 2 (11)
where W (k) is a definite positive matrix of the form:

W (k) = diag 2 q T L F(k) (12)
with:

T (x (k) ) = x (k) q-2 ifx (k) > (k) (k) q-2 ifx (k) ≤ (k) (13)
where (k) is a small real positive number acting as damped parameter. It avoids infinite weights when x (k) → 0 and q < 2.

Choice of the tuning parameters, initial guess and stopping criterion

According to Eq. 11, the tuning parameters of the problem are the shape parameter q , the regularization parameter λ (k+1) and the damping parameter (k) .

To properly choose the shape parameter q , one has to keep in mind that this parameter control the general shape of the distribution. So, using q ≥ 2 will give priority to smooth solutions, while using q ≤ 1 will favor sparse solutions.

Regarding the choice of the regularization parameter λ (k+1) and the damping parameter (k) , automatic selection procedures have been implemented. For the regularization parameter, the L-curve principle is used [START_REF] Hansen | Rank-Deficient and Discrete ill-Posed Problems: Numerical Aspects of Linear Inversion[END_REF], while for the damping parameter, its value is set so that a fixed percentage p i of the smallest values of |L F(k) | are below (k) . In the next of the paper, p = 5 %.

For iterative algorithm, another critical issue is the choice of the initial solution as well as that of the stopping criterion. Choosing a good initial guess is a key point of the convergence of the algorithm, since the functional to minimize is non-convex when q ≤ 1. The question that arises here is: What is a good initial guess? Actually, it is a coarse solution of the problem, easy to calculate, but sufficiently close to the final solution to ensure the convergence of the iterative process. Such requirements are fulfilled by the solution of the standard Tikhonov regularization. Finally, it remains to define a stopping criterion for the GIRLS algorithm. In this paper, a stopping criterion related to the variation of the func-

tional J( F(k) ) = 1 2 X -H F(k) 2 2 + λ (k) 2 W (k-1) LF (k) 2
2 between two successive iterations is used. The algorithm is automatically stopped when a prescribed tolerance defined by the user is reached. Here, the tolerance is set to 10 -8 .

NUMERICAL VALIDATION

In the present numerical validation, one seeks to identify a point force of unit amplitude acting on a thin simply supported steel plate with dimensions 0.21x0.27x0.007 m 3 . The coordinates of the point force, measured from the lower left corner of the plate, are (x,y) = (0.17 m, 0.12 m). Moreover, to simulate the vibration field X, a finite element model of the plate made up with 616 (22 × 28) quadrilateral shell elements has been used. It is worth to mention that an additive Gaussian white noise has been added to the data to simulate the measured vibration field as shown in Figure 1.

Finally, a FE model of the structure with free boundary conditions is used to compute the transfer functions matrix H.

Figure 2a presents the excitation field F from the standard Tikhonov regularization. This figure clearly shows that the location of the point force is properly estimated while the amplitude is greatly underestimated. This contrasted result can be explained by the fact that the global a priori used in the standard Tikhonov regularization gives priority to smooth solutions. To improve the quality of the identification, the prior knowledge on the nature of sources acting on the source has to be exploited. Indeed, as presented in Figure 1a, one knows that the excitation field to identify is very sparse. Consequently, the structure can be characterized with a sparsitypromoting regularization term (q < 2). Figure 2b presents the excitation field identified from the proposed approach using q = 0.5. The comparison of this result with the reference solution clearly shows that the proposed methodology allows identifying properly not only the location and the amplitude of the point force (0.998 instead of 1).

EXPERIMENTAL VALIDATION

In the experimental validation, one seeks to identify a point force of known amplitude acting on a thin simply supported steel plate with dimensions 0.35x0.29x0.005 m 3 . The coordinates of the point force, measured from the lower left corner of the plate, are (x,y) = (0.241 m, 0.192 m). The measurement of the vibration field is made on a 1015 (35 × 29) quadrilateral mesh centered on the plate. To compute the transfer functions matrix H, a FE model of the structure with free boundary conditions is used. Figure 3a presents the excitation field identified with a standard Tikhonov regularization. The location and amplitude of the acting point is barely visible. Figure 3b presents the excitation field identified from the proposed approach with q = 0.5 for a sparse excitation. The comparison of this result with the Tikhonov solution clearly shows that the proposed methodology allows identifying properly the location. One get a normalized force of 0.6 for coordinates (0.245 m, 0.195 m) and a second force of 0.3 for coordinates (0.235 m, 0.185 m). The identified value of the amplitude of the two points force is due to the fact that the excitation point is not just above a measurement point. A refined mesh in this domain could enhance the quality of the identified solution.

CONCLUSION

In the present study, the motivation was to propose a Bayesian formulation of the structural source identification problem able to fully exploit information available a priori on the nature and the location of the sources. To this end, the nature of the sources and of the noise are characterized with different distributions. A classic Gaussian law is taken for the noise, and a generalized Gaussian law is used for the sources. This formulation is very flexible, as the tuning parameters of these laws can be precisely adjusted to mathematically transcribe the a priori knowledge of the experimenter on the studied system. Formally, the solution of the problem is defined as the maximum a posteriori estimate, but practically, one seeks the solution of the dual minimization problem, which is solved from a Generalized Iteratively Reweighted Least-Squares algorithm. Numerical and experimental results show that exploiting a priori information substantially improves the quality of the source identification.

Figure 1 :Figure 2 :

 12 Figure 1: (a) reference solution; (b) direct identification with noisy data

Figure 3 :

 3 Figure 3: (a) standard Tikhonov regularization (b) proposed method with q = 0.5