Laurie Gheddar
email: l.gheddar@gmail.com

Jean-Sébastien Raul

Pascal Kintz

of SARMs and metabolic modulators screening in hair using UHPLC-MS/MS: Application to a doping case and first identification of S23 in authentic human hair

Keywords: SARMs, modulator metabolic, hair, UHPLC-MS/MS, HRMS

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

Introduction

In recent years, a new family of doping agents, named selective androgen receptor modulators (SARMs), has emerged. The class of SARMs includes a wide variety of chemical structures and can be classified into steroidal (example: MK-0773) and non-steroidal (example: ostarine, andarine) compounds [START_REF] Thevis | Detection of SARMs in doping control analysis[END_REF].

The first SARMs were developed in the 1990s based on a non-steroidal anti-androgen, named bicalutamide, to fight diseases manifested by muscle loss (e.g., cachexia, sarcopenia) or osteoporosis [START_REF] Dalton | Discovery of nonsteroidal androgens[END_REF]. They have similar properties to androgenic anabolic steroids (AAS) but have less marked androgenic properties. They activate the androgen receptors in muscles and bones selectively and avoid the appearance of side-effects (e.g., virilization). On the opposite of AAS, they also have good oral bioavailability. It should be noted that today, all SARMS are still in clinical trials without any therapeutic indication [START_REF] Christiansen | Selective androgen receptor modulators: the future of androgen therapy?[END_REF].

Since 2008, these products have been banned in-and-out of competition by the World Anti-Doping Agency (WADA) and classified as other anabolic agents (class S1.2). They are increasingly used by top athletes, as evidenced by the WADA statistics. In 2015, they accounted for only 32 from 3809 of adverse analytical findings (AAF), compared to 142 from 4180 in 2019 [START_REF]Anti-Doping testing figures report[END_REF].

Like AAS, the misuse of SARMs could extend to the general public as they are easily available on the Internet. Therefore, it seems important for a toxicology laboratory to develop a screening method to identify SARMs. Contrary to urine, in which it is necessary to test for metabolites that are not generally marketed, hair analysis seems to be the gold choice because the target analyte in hair is often the parent molecule. Furthermore, hair specimens allow increasing the window of detection, to document the consumption pattern when segmentation is possible and to trace back a possible source of contamination.

In this study, the authors focused on SARMs available on the Internet. Among these SARMs, two are classified as metabolic modulators (GW501416, SR9009) and categorized as SARMs by users and sellers. Therefore, the authors decided to include them in the screening. GW501516 or cardarine is a peroxisome proliferator-activated receptor delta (PPARδ) agonist. It acts on the receptor to increase fat-burning capacity and muscle production. SR9009, also known as stenabolic, is REV-ERB agonist. It increases fat-burning, muscle production and endurance. These two products have been prohibited by the WADA and are classified as S4. hormone and metabolic modulators. In 2019, GW501516 and SR9009 represented 32 cases and 2 cases on 362 of AAF, respectively [START_REF]Anti-Doping testing figures report[END_REF]. Moreover, Leaney et al. [START_REF] Leaney | Analysis of supplements available to UK consumers purporting to contain selective androgen receptor modulators[END_REF] have detected these two compounds in SARMs tablets purchased on the Internet. Indeed, some products are associated with GW501516 and/or SR9009; this is the case for capsules that include andarine, GW501516 and SR9009. In addition, in this previous study, GW501516 was found in one tablet for which it was not declared on the product label. This capsule was supposed to contain RAD-140, LGD-4033 and ostarine [START_REF] Leaney | Analysis of supplements available to UK consumers purporting to contain selective androgen receptor modulators[END_REF].

In this paper, the authors describe the development and the validation of a screening method for 9 SARMs and 2 metabolic modulators (categorized as SARMs on the Internet) in hair and apply it to a real doping case.

Material and method

Chemicals and reagents

Andarine (S-4), ACP-105, GW501516 (GW1516), Ostarine (S-22 or enobosarm), S23, RAD 140 (testolone), MK 677 (ibutamoren), SR 9009 and bicalutamide-D4 were obtained from Toronto Research Chemicals (Toronto, Canada). Ligandrol (LGD-4033) and AC262536 were obtained from Cayman Chemical (Ann Arbour, USA) and Sigma-Aldrich (Saint-Louis, USA) respectively. MK-0773 (Dynamic Performance) was obtained on the Internet. Ammonium formate 99 % was provided by Alfa Aesar (Schiltigheim, France). Dichloromethane was purchased from Merck (Darmstadt, Germany). Methanol for HPLC LC-MS Grade (MeOH), acetonitrile for HPLC LC-MS Grade (ACN) and formic acid were purchased from VWR Chemicals Prolabo (Fontenay-sous-Bois, France). Diethyl ether was purchased from Honeywell Riedel-de Haën (Seelze, Germany). HPLC-grade acetonitrile (ACN), methanol (MeOH), dichloromethane, hexane and ethyl acetate were obtained from Merck (Darmstadt, Germany). Water for the formate buffer (eluent A for chromatographic separation) was purified using a Milli-Q system (Millipore) (Molsheim, France).

The standard solutions, prepared in MeOH at 100 µg/L, 10 µg/L, 1 µg/L and 100 ng/mL, were stored at -20 °C or at + 4°C according to the supplier recommendations.

Instrumentation

UHPLC-MS/MS analysis

Chromatographic separation was performed with a liquid chromatography Waters Acquity UPLC TM system (Waters corporation, Milford, MA, USA) using an Acquity UPLC HSS C18 column (150 mm x 2.1 mm, i.d. 1.8 µm particle size). The column temperature was set to 50°C, and the flow rate at 0.4 mL/min. A gradient elution was performed using formate buffer adjusted to pH 3 (eluent A) and 0.1% formic acid in acetonitrile LC-MS (eluent B). Analysis time was 6.0 min and the following gradient pattern (eluent A) was used for the separation of SARMs and bicalutamide-D4, used as internal standard (IS): 0 min, 60%; 0.5 min, 60%; 1.50 min, 5 %; 3.0 min, 5%; 4.00 min, 60%; 6.00 min, 60%.

Retention times are presented in table 1.

Detection was performed using a triple quadrupole (XEVO TM TQS micro, Waters Corporation, Milford, MA, USA) mass spectrometer equipped with a Z-spray TM -electrospray ionization source (ESI) and used in the positive and negative mode (ES+/ES-). Nitrogen was used as drying gas as well as nebulizing gas. Parameters were optimized during the development period. The desolvation gas flow was set to 800 L/h and the desolvation temperature was 550°C. The source temperature was 150°C.

The capillary voltage was tested and set at 1.50 kV (ES-) and 3.0 kV (ES+). Cone voltage and collision energy were automatically adjusted to optimize the signal of the 2 most abundant product ions of SARMs and bicalutamide-D4 using Intellistart software. Transitions, cone voltage and collision energy are resumed in table 1. MassLynx 4.1 software was used for identification and quantification.

HRMS analysis

HRMS system enables to confirm the presence of SARMs in samples targeting the exact mass. HRMS is used for this analysis to have a second and independent method as it is achieved in anti-doping laboratories. Additionally, it is in accordance with the Society of Hair Testing consensus for doping agents [START_REF] Sachs | Consensus of the society of hair testing on hair testing for doping agents[END_REF]. Chromatographic separations were performed with identical equipment than UHPLC-MS/MS. The applied parameters (column temperature, mobile phase, gradient pattern) were the same compared to UHPLC-MS/MS system.

For the analysis of the doping case described in 5.2 section, the gradient pattern was different as follow: 0 min, 50%; 0.5 min, 50%; 1.50 min, 5 %; 2.5 min, 5%; 3.00 min, 50%; 4.00 min, 50% to analyse only ostarine (tR=1.91 min), andarine (tR=1.67 min) and S23 (tR=2.24 min). Another gradient was used for HRMS as the aim of this later procedure was to confirm and not to screen for what has been detected by MS/MS. Detection was performed using a high-resolution (XEVO TM G2XS Q-TOF, Waters Corporation, Milford, MA, USA) mass spectrometer operating in positive-and negative-ion mode and in sensitivity mode.

The desolvation gas flow was set to 600 L/h at a temperature of 400 °C with the cone gas set to 20 L/h and the source temperature set to 150 °C. The capillary voltage and the cone voltage were set to 2000 (ESI +) / 800 (ESI -) and 20 V, respectively. Leucine-enkephalin was used as the lock mass (m/z 556.2766 -m/z 554.2620) for accurate mass calibration and introduced using the lockspray interface at 10 µL/min and a concentration of 200 pg/µL in 50% aqueous ACN, 0.1 % formic acid. In MS scanning, data were acquired from 50 to 1000 m/z. As for MS/MS fragmentation of target ions, collision energy ranging from 10 to 40 V was applied. Unifi software was used for data, chromatograms and spectra acquisition.

Sample treatment

The hair strands were decontaminated twice with 5 mL of dichloromethane for 1 min at room temperature and dried. For the validation, hair was obtained from the staff of laboratory and verified to be negative. into the laboratory routine, four extraction methods, commonly used in our laboratory, were tested at the concentration of 100 pg/mg in triplicate.

(1) The first consisted of incubation of hair in 1 mL borate buffer pH 9.5 for 15 h at 40 °C, in the presence of 250 pg of bicalutamide-D4. After cooling, 5 mL of the mix solution (ether/dichloromethane/hexane/isoamylic alcohol (50/30/20/0.5)) were added. After agitation for 15 min and centrifugation (15 min, 3000 rpm), the organic phase was collected and evaporated to dryness under a nitrogen flow. The dry residue was dissolved in 30 µL of initial mobile phase.

(2) The second method tested was described by Kintz et al. for anabolic steroid analysis in hair [START_REF] Kintz | Simultaneous testing for anabolic steroids in human hair specimens collected from various anatomic locations has several advantages when compared with the standard head hair analysis[END_REF]. 250 pg of bicalutamide-D4 were used and no derivatization step was applied.

(3) The third method was described by Gheddar et al. for stanozolol analysis [START_REF] Gheddar | Testing for stanozolol, using UPLC-MS/MS and confirmation by UPLCq-TOF-MS, in hair specimens collected from five different anatomical regions[END_REF]. Instead of stanozolol-D3, bicalutamide-D4 was used as IS.

(4) Finally, a fourth method was tested. It consisted of incubation of hair in 1 mL of MeOH, in the presence of 250 pg of bicalutamide-D4, in an ultra sound bath for 2 hours. After cooling, MeOH was evaporated and the dry residue was dissolved in 30 µL in MeOH.

Detection and signal to noise ratios at 100 pg/mg from methods 2, 3 and 4 were compared for each compound to method 1.

Method validation

The validation process was applied to the extraction method chosen by the authors (section 5.1).

Determination and evaluation of linearity, LOD and LOQ

A nine-point calibration curve (n = 3) was obtained by spiking blank hair at 0.1, 1, 5, 10, 50, 100, 200, 500 and 2000 pg/mg. The limit of quantification (LOQ) was defined as the first point of calibration.

The limit of detection (LOD) was the lowest concentration of analyte that could be detected with signal to noise ratio superior to 3.

Intra and inter -day precision

For the intra-day precision, three quality controls (QC) at 10, 100 and 500 pg/mg were evaluated 5 times. And for the inter-day precision, QCs at 3 levels of concentration are evaluated on three days spread over two weeks. The laboratory will accept coefficients of variation (CV) of less than 30% for the 10 pg/mg controls and less than 20% for the 100 and 500 pg/mg controls.

Matrix effect

added at the end of extraction. These specimens were compared to 10 µL of standard solution of mixture of SARMs in MeOH (1 mg/L).

Recovery

It was not possible to establish suitable extraction efficiency in authentic hair due to the lack of reference material available for validation. Nonetheless, authors wanted to approach and control the extraction efficiency. To do that, 5 spiked hair samples at 500 pg/mg of SARMs were extracted and compared to a standard solution of SARMs in MeOH at the same concentration (n=4).

Application to an authentic hair specimen

The method developed at the laboratory was applied to the case of a woman top athlete, a practitioner of a combat sport (MMA), following an adverse analytical finding of ostarine in her urine (around 1 ng/mL). The athlete did not admit the anti-doping rules violation and her lawyer requested our laboratory for additional hair analysis in order to find an explanation. A lock of hair was collected by the lawyer one month after the adverse analytical finding and sent to the laboratory. The hair was received in an envelope and stored at room temperature and protected from UV light until the analysis. Hair was brown in colour and 20 cm in longer. The lock of hair was oriented from root to the tip. Hair was segmented (6 x 1 cm to cover the period of the urine control) and analysed according to the protocol of SARMs screening.

Results and discussion

Method validation

Validation of the screening of SARMs was achieved only with UHPLC-MS/MS because this is the gold method for quantification. The choice of the UHPLC-MS/MS system was obvious because it is necessary to use reverse phase to identify all analytes with different chemical structures. Accurate mass spectrometry was used for qualitative confirmation.

Blank hair specimens from laboratory staff, used for validation method, were chosen after checking the absence of response at the retention times of the various SARMs and the IS. It appears that bicalutamide-D4 is a good internal standard for the analysis of SARMs in hair thanks to its close chemical structure with other SARMs as previously published in urine [START_REF] Gadaj | Enhanced UHPLC-MS/MS screening of selective androgen receptor modulators following urine hydrolysis[END_REF].

The optimization of the mass parameters was not a problem. MK-0773 was purchased on the Internet at the dosage at 10 mg. This tablet was checked using HRMS to verify its composition. HRMS proved the presence of MK-0773 with the mass exact of m/z [M+H] + of 480.2775 with a low mass error (1.1 ppm). No other drug or SARM was detected in MK-0773 tablet. Therefore, theses tablet was used as material reference. Chromatographic separation was tested to obtain the best of resolution and peak separation (resolution superior than 1.5) and the time run faster. All analytes are separated in 6 min.

For each analyte, the area after extraction 2, 3 and 4 were compared with the area from extraction 1.

The results are expressed as a gain or loss (in percentage) relative to the results of method 1 and are summarized in table 2. Methods 1, 2, 3 and 4 produced approximatively the same response (table 2).

However, methods 2 and 3 failed to detect MK-0773 and MK 677, and SR9009 and RAD140 respectively. Method 4 is able to detect all analytes but did not provide to obtain good chromatograms with signal to noise ratios superior than 10 even for concentration at 100 pg/mg. Therefore, method 1 was chosen for the simultaneous screening SARMs. Only the first method was validated.

The linearity ranges, LOD and LOQ were resumed in table 3 for each compound. The method showed a good linearity in the ranges defined in table 3 with correlation coefficients superior to 0.988. The analysis showed good inter-and intra-day precisions for all analytes with CVs < 20% (table 3). Matrix effect (ME) was established for all analytes and was evaluated between 75 and 130 %. However, the ME was similar in the five different tested matrices with a CV < 20% with a IS-normalised ME at 115 % and a CV at 14 %. Finally, recovery extraction was established between 53 and 84% (table 3). It should be noted that the evaluation of the extraction coefficient remains an approximate approach since there is no reference material with certified concentrations.

For all but ACP-105, the limits of quantification were found satisfactory, given the method allows identifying and quantifying simultaneously 11 analytes in hair samples within 6 minutes. The ion chromatograms of 11 analytes and IS in a spiked hair at 100 pg/mg (for ACP-105) and 10 pg/mg (for the other SARMs) are presented in figure 1.

Simultaneous analysis of SARM has been already described in urine and plasma [START_REF] Gadaj | Enhanced UHPLC-MS/MS screening of selective androgen receptor modulators following urine hydrolysis[END_REF][START_REF] Ventura | Development and validation of a semi-quantitative ultra-high performance liquid chromatography-tandem mass spectrometry method for screening of selective androgen receptor modulators in urine[END_REF][START_REF] Polet | Development and validation of an open screening method for doping substances in urine by gas chromatography quadrupole time-of-flight mass spectrometry[END_REF][START_REF] Hansson | Investigation of the selective androgen receptor modulators S1, S4 and S22 and their metabolites in equine plasma using high-resolution mass spectrometry[END_REF]. Contrary to urine and plasma, there are few data in literature on the analysis of SARMs in hair. Some authors have reported data in equine hair. For example, Cutler et al. have identified AC262536 in equine hair with a LOQ at 1 pg/mg [START_REF] Cutler | Equine metabolism of the selective androgen receptor modulator AC-262536 in vitro and in urine, plasma and hair following oral administration[END_REF]. Another method of quantification in equine hair for several AAS and other products namely andarine, GW501516 and ostarine was published in literature with LOD at 0.2 pg/mg, 0.1 pg/mg and 0.1 pg/mg respectively [START_REF] Gray | Detection of prohibited substances in equine hair by ultra-high performance liquid chromatography-triple quadrupole mass spectrometry -application to doping control samples[END_REF]. Only two groups have reported data in human hair. One was the group of Kintz et al. with the identification of ostarine between 3 and 168 pg/mg, andarine between 0.1 and 0.7 pg/mg and LGD-4033 between 14 and 42 pg/mg [START_REF] Kintz | Identification of S22 (ostarine) in human nails and hair using LC-HRMS. Application to two authentic cases[END_REF][START_REF] Kintz | Perspective in evaluating selective androgen receptor modulators in human hair: a short communication[END_REF]. In another paper, Kintz et al. reported GW501516 hair concentrations at 22 and 32 pg/mg after incubation in MeOH [START_REF] Kintz | Testing for GW501516 (cardarine) in human hair using LC/MS-MS and confirmation by LC/HRMS[END_REF]. Finally, Rading et al. identified GSK2881078 in human hair at 1.7 pg/mg [START_REF] Rading | Detection of the selective androgen receptor modulator GSK2881078 and metabolites in urine and hair after single oral administration[END_REF].

Application to an authentic hair specimen

The hair test results of athlete are described in figure 2. Ostarine was detected in all 6 segments, with concentrations increasing from 0-1 cm to 1-2 cm then decreasing from 1-2 cm to the tip, ranging from 1 to 9 pg/mg. Ostarine was the identified agent during the urine control.

The screening revealed the presence of two other SARMs in hair, namely andarine and S23. They were also detected in 6 segments, at concentrations between 120 and 1644 pg/mg for andarine, and 0.6 and 16 pg/mg for S23. The concentration patterns, presented in figure 2, are similar to those observed for ostarine, with increasing concentrations from 0-1 cm to 1-2cm segment then decreasing from 1-2 cm to the tip. The presence of 3 SARMs was confirmed using UPLC-q-TOF-MS system (figure 3) with mass error from -2.5 to -0.4 ppm. The laboratory accepted mass error < 5.0 ppm and a time retention ± 0.2 min compared to the reference material. Ostarine concentrations are in favour of repeated exposure, but in view of the concentrations described in the literature (12 -168 pg/mg) [START_REF] Kintz | Identification of S22 (ostarine) in human nails and hair using LC-HRMS. Application to two authentic cases[END_REF][START_REF] Kintz | Perspective in evaluating selective androgen receptor modulators in human hair: a short communication[END_REF], the concentrations could also be characteristic of environmental or dietary supplement contamination. High concentrations of andarine can be interpreted as a doping practice. For S23, there is no data available in the literature about its identification in hair.

In view of the high concentrations of andarine measured and the information available in the literature, the presence of ostarine and S23 in the hair can be interpreted as a consequence of contamination of andarine by these two products, in an andarine tablet. According to Leaney et al., a single tablet or capsule can contain up to 5 different SARMs in the same item [START_REF] Leaney | Analysis of supplements available to UK consumers purporting to contain selective androgen receptor modulators[END_REF]. Switching a SARM to another SARM has also been described for ostarine [START_REF] Kintz | LGD-4033, S-4 and MK-2866 -Testing for SARMs in hair: about 2 doping cases[END_REF]. It is probably due to the same production line that has also been demonstrated many times for contamination occurring with AAS [START_REF] Geyer | Nutritional supplements cross-contaminated and faked with doping sustances[END_REF].

Andarine and ostarine drugs are regularly identified as AAF in contrast to S23 that was found in urine for the first time in 2019 by anti-doping laboratories [START_REF]Anti-Doping testing figures report[END_REF]. Nonetheless, the presence of these three molecules in the hair of this top athlete is in favour of doping attitude.

Conclusion

In view of the major emergence of SARMs in the field of sport and the interest that this class of drugs has on the general public, the development of an analytical method dedicated to their identification has become necessary. The authors presented a specific and rapid analytic method to 11 "true" and categorized SARMs. This validated method allows in some cases to document the simultaneous presence of several SARMs in the hair of a subject and to trace a possible source of contamination. This is the first time that S23 was reported in human hair.

Figure 1 :Figure 3 :

 13 Figure 1: Ion quantification chromatogram of SARMs in spiked hair at a concentration of 100 pg/mg for ACP-105 and 10 pg/mg for all other SARMs

Table 1 : UPLC-MS/MS conditions for SARMs testing in hair samples

 1

	Analyte	Type of compound	TR (min) Transition (m/z) Cone (V)	CE (eV)	ESI polarity
	AC262536	SARM		2.75	279.1 > 195.0*	14	22	+
					279.1 > 235.0		22
	Andarine	SARM		2.43	440.2 > 149.9*	2	36	-
					440.2 > 261.0	36	20
	ACP-105	SARM		2.92	291.2 > 233.0*	30	21	+
					279.1 > 79.9	44	28
	LGD-4033	SARM		2.80	337.1 > 170.0*	40	30	-
					337.1 > 239.0		20
	MK-0773	SARM		1.62	480.3 > 132.0*	88	40	+
					480.3 > 149.1		30
	MK677	SARM		1.80	529.4 > 91.0*	4	52	+
					529.4 > 267.2		18
	Ostarine	SARM		2.63	388.1 > 117.9*	8	18	-
					388.1 > 269.1		14
	RAD-140	SARM		2.54	394.2 > 223.1*	36	10	+
					394.2 > 169.9		30
	S23	SARM		2.91	415.1 > 144.9*	8	18	-
					415.1 > 185.0	10	36
	GW501516	Metabolic	modulator	3.16	454.0 > 257.0*	58	30	+
		(PPARδ receptor agonist)		452.1 > 137.9	18	26	-
	SR9009	Metabolic	modulator	3.32	438.2 > 124.9*	50	24	+
		(Rev-ErbA agonist)		438.2 > 89.2		78
	Bicalutamide-D4	SARM		2.55	433.1 > 255.1	14	14	-

Table 3 : Linearity range, LOD, LOQ, intra-inter day precision (%), mean matrix effect (%) and mean extraction coefficient (%)

 3

	Analyte	Linearity range (pg/mg) LOD (pg/mg) LOQ (pg/mg)	QC concentration (pg/mg)	Intra-inter CV (%) Matrix effect (%) Recovery (%)
	AC262536	10-1000	5	10	10 100	14.2 -17.2 14.7 -17.3	85	56
	Andarine	1-1000	0.1	1	10 100	7.4 -5.2 13.1 -9.3	130	57
	ACP-105	50-1000	20	50	100 500	8.5 -16 12.3 -14.6	94	59
	LGD-4033	5-1000	3	5	10 100	19 -18 6.5 -2.2	125	54
	MK-0773	1-1000	0.5	1	10 100	17.5 -14.9 10.8 -19.2	112	75
	MK677	10-1000	5	10	10 100	19.8 -11.7 8.2 -14.6	96	53
	Ostarine	0.5-1000	0.1	0.5	10 100	5.9 -13.5 14.8 -10.5	117	83
	RAD-140	5-1000	3	5	10 100	7.3 -6.9 5.8 -4.3	107	68
	S23	1-1000	0.5	1	10 100	9.4 -12.1 15 -14.9	115	84
	GW501516	5-1000	3	5	10 100	5.6 -3.6 14.8 -13.0	72	67
	SR9009	20-1000	10	20	10 100	18.2 -17.8 14.6 -14.1	123	79

Conflict of interest and source of funding

The authors declare no conflict of interest and no funding.


Bexultan Sabyrbay

Dorian Davarzani

Christophe Dicharry

Sagyn Omirbekov

Amir Alamooti

Mélanie Lorthioy

Mohamed Krimissa

Stéfan Colombano

Hossein Davarzani

Mobilization of LNAPL by using polymer-based alcohol emulsion in contaminated aquifers

Keywords: soil remediation, porous media, biopolymer, alcohol, LNAPL

Abstract

Environmental pollution, notably soil pollution, is one of the urgent problems at present. Light refined petroleum products (gasoline, diesel, engine oil, etc.) which represent light non-aqueous phase liquids (LNAPL) are among the most dangerous environmental pollutants. Penetration of LNAPL into the soil or groundwater makes it unsuitable for plant, animal, and microorganism life. Therefore, the search for effective ways to dispose of hydrocarbon pollution is of great importance. One of the most widely known LNAPL remediation methods is Pump-and-treat (PT), which is considered inefficient due to long operation, and low recovery efficiency (around 60%). Injecting non-Newtonian fluids such as polymers presents a significant industrial application interest for in situ remediation of contaminated soil, especially for aquifers with significantly high permeability and strong heterogeneity. Nevertheless, there is some limitation to using high-viscosity fluids. For instance, the injection of a high-viscosity polymer solution during soil remediation can harm the environment by increasing the injection pressure, leading to aquifer erosion or soil uplift. Hence, developing eco-friendly, sufficiently viscous, and biodegradable fluid is highly in demand.

The main goal of this study is to improve the removal of LNAPL (here diesel fuel) with a new polymerbased formulation that promotes its dissolution or mobilization. The use of viscous polymer solutions can contribute to stable oil displacement in heterogeneous porous media and simultaneously carry active matters (surfactants, co-solvents) to hard-to-reach low-permeability media.

To embody the goal, the remediation of diesel was tested in one-dimensional 30 cm long silica-based sand-pack columns. Biopolymer "xanthan gum" was used due to its rheological and biodegradability properties. Primary flushing by xanthan gum with a concentration of 0.1% w/w, promoted stable displacement in saturated porous media and allowed the recovery of up to 80% of diesel fuel. Further, the polymer solution was formulated with the addition of alcohol solvents to improve sweeping performance. Therefore, fifteen types of biodegradable homologous alcohols (from ethanol to n-octanol) were applied for batch experiments to formulate polymer-based alcohol mixtures. These homologous alcohols are known to exhibit different partitioning behavior upon contact with diesel fuel. Environmentally friendly surfactant sodium dodecyl sulfate (SDS) was used to stabilize a mixture of alcohol and polymer solution. To obtain a stable formulation, batch experiments were carried out at various surfactant concentrations and various volume fractions of polymer solution, alcohol, and diesel fuel.

Batch experiments were carried out in small polypropylene vials to test the mechanisms of alcohol partitioning in the presence of diesel fuel. According to the results of batch experiments, ethanol, 1propanol, and 2-propanol exhibit a mechanism of diesel dissolution into an aqueous phase, which leads to a decrease in the volume of diesel fuel. Homologous alcohols from n-butanol to n-octanol promoted partitioning into the organic phase. As a result, swelling of the volume of diesel fuel was observed, which is typical for the mobilization mechanism. However, only 1-pentanol, 1-hexanol, 1-heptanol, 1octanol, and 2-octanol among mobilization alcohols showed stable emulsions when mixed with a xanthan-SDS solution.

Further, the rheological behaviors of xanthan-SDS-alcohol mixtures were characterized through a rheometer. The preliminary results showed that they exhibit non-Newtonian shear-thinning behavior. Rheological experiments are still ongoing to select the best formulation regarding viscosity properties. Thus, the final polymer-based alcohol emulsion will be used for the secondary flushing (after polymer flushing) in a 1D column to remove residual diesel. References:

[1] S.  Rheological experiments:

 1-D column experiments:

Alcohol partitioning type № 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15  The new-developed XG-SDS-1-pentanol emulsion called "COCKTAIL 50%" exhibited an improved recovery of 99.4% upon direct treatment in 1D columns.

Alcohol Ethanol 1-propanol 2-propanol 1-butanol 2-butanol Isobutanol Tert-butanol 2-methyl-1- butanol 1-pentanol 2-pentanol 3-pentanol 1-hexanol 1-heptanol 1-octanol 2-ocatnol

Materials & Methods 2

Results & discussions 3

Main Conclusions 4

 Post-injection of "COCKTAIL 50%" after XG-SDS flushing improved diesel removal to 11.3% (total 93.8%).

 COCKTAIL 50% emulsion preserves non-Newtonian shear-thinning behavior despite the addition of alcohol.

 The main mechanism of 1-pentanol in "Cocktail 50%" is the swelling of the diesel phase in a porous media and thereby its further mobilization.

Direct treatment: LNAPL saturated Porous Media

 All mixtures  non-Newtonian shear-thinning behavior.  SDS decreases the viscosity of XG solution due to repulsive electrostatic forces.  Water-insoluble "mobilization" alcohols (50% vol.) increase the viscosity  Water-soluble "solubilization" alcohols (50% vol.) decrease the viscosity when mixing with XG-SDS solution  "COCKTAIL 50%"  XG-SDS-1-Pentanol emulsion at 50% alcohol content.

 1-pentanol  lower toxicity and higher biodegradability. b.sabyrbay@brgm.fr

Fig. 2 :

 2 Fig. 2: Schematic of the experimental setup

Fig. 3 :

 3 Fig. 3: Alcohol partitioning mechanism in the presence of water and diesel

Fig. 4 :

 4 Fig. 4: Development and selection of a new single-phase formulation

Fig. 1 :

 1 Fig. 1: Methodology of batch experiments

Fig. 7 :Fig. 6 :

 76 Fig. 7: LNAPL displacement in 1D column by direct-treatment Fig. 8: Pressure drop as a function of injected PV for different formulations


Phuc Minh-Tuan Truong

Do Van

Laurent Mendizabal

Benoît Iung

Phuc Do
email: phuc.do@univ-lorraine.fr

Benoit Iung

An improved accelerated degradation model for LED reliability assessment with self-heating impacts

Keywords: LED, reliability, degradation modelling, accelerated degradation testing, stochastic process

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

Introduction

Since the 2000s, Gallium nitrogen (GaN) light-emitting diodes (LEDs) have been widely used in multiple fields of application (e.g., display, communications, medical services, etc.) [START_REF] Cho | White light-emitting diodes: History, progress, and future[END_REF]. However, the growing complexity of LED technologies and the miniaturization of assembly technologies make harder for the industry and academic research to evaluate the reliability of the new LED products [START_REF] Chang | Light emitting diodes reliability review[END_REF]. The direct consequence is that the reliability of these systems is more and more challenging to estimate. Lumen degradation is one of the main indicators of LED failure and has been studied in various works, see for instance [START_REF] Fukuda | Reliability and degradation of semiconductor lasers and leds[END_REF][START_REF] Tuttle | Projecting long term lumen maintenance of led light source[END_REF][START_REF] Tsai | Inference from lumen degradation data under wiener diffusion process[END_REF][START_REF] Huang | Degradation modeling of mid-power white-light leds by using wiener process[END_REF][START_REF] Park | Lifetime estimation of led lamp using gamma process model[END_REF][START_REF] Sun | A review of prognostic techniques for high-power white leds[END_REF][START_REF] Ibrahim | Lumen degradation lifetime prediction for high-power white leds based on the gamma process model[END_REF][START_REF] Ibrahim | Machine learning and digital twin driven diagnostics and prognostics of light-emitting diodes[END_REF]. According to TM-21 standard [START_REF] Tuttle | Projecting long term lumen maintenance of led light source[END_REF], the LED's lifetime is evaluated, using an exponential law for modelling lumen degradation, higher than 50.000h.

Besides, stochastic processes, such as Wiener [START_REF] Tsai | Inference from lumen degradation data under wiener diffusion process[END_REF][START_REF] Huang | Degradation modeling of mid-power white-light leds by using wiener process[END_REF] or gamma process [START_REF] Park | Lifetime estimation of led lamp using gamma process model[END_REF][START_REF] Ibrahim | Lumen degradation lifetime prediction for high-power white leds based on the gamma process model[END_REF], at several accelerated conditions, are also used to model the LEDs' lumen degradation. In both methods, thermal stress is one of the most commonly used methods to accelerate the failure/degradation of LED using the Arrhenius model, which describes the degradation rate of the luminous intensity of LED following the operational temperature [START_REF] Nelson | Accelerated testing: statistical models, test plans, and data analysis[END_REF]. In many studies, the junction temperature used in the Arrhenius model is considered to be constant and equal to the ambient temperature because it can be controlled with a temperature chamber during the acceleration test. However, it has been shown that the component's lifetime prediction, which used the ambient temperature to model the degradation rate, may lead to significant prediction errors and uncertainties with the actual application life [START_REF] Ke | Analysis of junction temperature and modification of luminous flux degradation for white leds in a thermal accelerated reliability test[END_REF]. Indeed, according to the literature, the junction temperature, which directly affects the LEDs' failures (e.g., semiconductor failure, interconnect failure), significantly influences on the degradation [START_REF] Chang | Light emitting diodes reliability review[END_REF][START_REF] Fukuda | Reliability and degradation of semiconductor lasers and leds[END_REF][START_REF] Meneghini | A review on the physical mechanisms that limit the reliability of gan-based leds[END_REF]. Thus, for the modelling of the LEDs' degradation rate, the junction temperature as thermal stress is more relevant than the ambient temperature in conventional ADT approaches. However, direct measurement of the junction temperature is complex and may lead to an important error [START_REF] Chang | Light emitting diodes reliability review[END_REF][START_REF] Trevisanello | Thermal stability analysis of high brightness led during high temperature and electrical aging[END_REF][START_REF] Della Corte | Temperature sensing characteristics and long term stability of power leds used for voltage vs. junction temperature measurements and related procedure[END_REF]. Besides, based on LEDs' failure analyses (e.g., optical and infrared thermal microscopy, scanning electron microscope, etc.), many studies have shown that LED's junction temperature is not constant and may increase during the acceleration test due to the LED degradation impact [START_REF] Buffolo | Long-term degradation mechanisms of mid-power leds for lighting applications[END_REF][START_REF] Ke | Analysis of junction temperature and modification of luminous flux degradation for white leds in a thermal accelerated reliability test[END_REF]. In addition, the increment of junction temperature may lead to further accelerate the LEDs' degradation rate [START_REF] Chang | Light emitting diodes reliability review[END_REF][START_REF] Meneghini | A review on the physical mechanisms that limit the reliability of gan-based leds[END_REF]. To sum up, due to the degradation impact, there is a loop of thermal stress, subjected to a self-heating on the LED component, which affects the LED degradation process over time. In that way, the self-heating phenomenon, shown through the increment of junction temperature, is an important issue and needs to be considered in the LEDs' degradation/failure modelling and reliability assessment. However, very few studies on this research topic are found in the literature [START_REF] Ke | Analysis of junction temperature and modification of luminous flux degradation for white leds in a thermal accelerated reliability test[END_REF].

To overcome this issue, this paper proposes a novel ADT model considering the self-heating phenomenon impacting the LED's degradation process. In that way, the self-heating impact is first analyzed and modelled mathematically. In fact, the junction temperature of a LED component depends not only on the heat generation (e.g., forward current, dispersing heat) but also on the current state (degradation level) of the component. Next, a modified stochastic difference equation (SDE) is developed for modelling and formulating the degradation process by considering the self-heating impact. The LED reliability formulation is then derived. To estimate the model's parameters from experimental data, an estimation method based on the maximum likelihood is also developed. This paper develops an extended version of the work presented in [START_REF] Truong | A novel degradation model for led reliability assessment with accelerated stress and self-heating consideration[END_REF].

To describe the added value of our contributions, the rest of the paper is organized as follows:

we first introduce, in Section 2, the LED device used for our acceleration test. The failure and degradation behavior of the LED is then analyzed. In addition, the experimental platform is also presented. Next, the self-heating impact is mathematically modeled in Section 3. The degradation model and the reliability assessment considering the self-heating impact are formulated in Section 4.

Furthermore, a maximum likelihood-based method for estimating the proposed model's parameters is also presented herein. Section 5 analyses some obtained results when applying the proposed model to our experimental dataset. Finally, the conclusions drawn from this work are presented in the last section.

Notations

Experiment description

In this section, we present the detail of an experiment for a LED in order to investigate the self-heating phenomenon, i.e., the increment of junction temperature due to the power dissipation and the LED degradation impact, and evaluate the LEDs' reliability. In that way, the LED studied, which was chosen for the acceleration test, is firstly presented. Next, we present an experimental platform and the test condition. The failure and degradation behaviors are then analyzed based on the experimental dataset. In addition, the existence of an increment in junction temperature due to the degradation of the LED is also highlighted.

Device description and degradation characteristic

Device description In this study, gallium nitrogen (GaN) LED was chosen as the research object.

The elementary structure consists of a GaN chip on the interconnected metal layer and the ceramic substrate soldered on an Aluminium (Al) support. Figure 1 presents an example of LED structure and its packaging on the printed circuit board (PCB). This is a LED Cree EZ1000, and it has a vertical structure with two topside 150 × 150µm square bond pad cathode (-) terminals and an 80:20

AuSn terminated anode (+) on the bottom of the silicon substrate [START_REF] Sheet | Cree's ezbright leds[END_REF]. The LED emitting layer is metallically bonded to the silicon substrate, and the periphery of the emitting mesa is passivated.

These vertically structured, low forward voltage LED chips are approximately 170 µm in height. the open and short failure rate is very small. While the gradual degradation mechanisms, which could come from the deterioration of the Ohmic contacts or/and semiconductor chips [START_REF] Chang | Light emitting diodes reliability review[END_REF][START_REF] Fukuda | Reliability and degradation of semiconductor lasers and leds[END_REF][START_REF] Sun | A review of prognostic techniques for high-power white leds[END_REF][START_REF] Ibrahim | Machine learning and digital twin driven diagnostics and prognostics of light-emitting diodes[END_REF][START_REF] Meneghini | A review on the physical mechanisms that limit the reliability of gan-based leds[END_REF], affects significantly the LED failure. The previous studies revealed that gradual degradation of LED could suffer from gradual performance degradation, such as lumen degradation [START_REF] Chang | Light emitting diodes reliability review[END_REF][START_REF] Fukuda | Reliability and degradation of semiconductor lasers and leds[END_REF]. It is shown that heat and current are two main causes of lumen degradation, and they occur gradually over a long period of operating time [START_REF] Chang | Light emitting diodes reliability review[END_REF][START_REF] Fukuda | Reliability and degradation of semiconductor lasers and leds[END_REF][START_REF] Sun | A review of prognostic techniques for high-power white leds[END_REF][START_REF] Ibrahim | Machine learning and digital twin driven diagnostics and prognostics of light-emitting diodes[END_REF].

Failure criteria

The luminous intensity of a LED decreases when the LED degrades. A LED is considered as failed when its luminous intensity reaches a critical threshold. According to the Illuminating Engineering Society of North America (IESNA), the critical threshold for a LED is about 70% of the initial value of the luminous intensity of the LED, called L 70 [START_REF] Tuttle | Projecting long term lumen maintenance of led light source[END_REF].

Experimental platform and test condition

Experimental platform: Figure 2 presents a global view of the experimental platform to characterize the LEDs' performance during the acceleration test. This system consists of a DC source meter, temperature chamber, and a spectrometer for optical measurement. In which:

• A KEITHLEY 2600 source meter generator is connected by an IEEE bus to the computer unit that controls the bench. This device is used as a bias current source and allows measuring the LED voltage.

• A spectrometer (USB2000+) is used to measure the spectral and intensity of LEDs.

• A climatic chamber (CLIMATS Excal) is used for thermal regulation during measurements and aging tests.

• An infrared camera (FLIR SC7000) is used to evaluate the LED temperature (LED surface temperature).

The infrared camera, which takes the image of LED surface, is a detector of indium antimony (InSb). However, these elements (e.g., GaN, substrate, wire bonds) have different emissivities, and each of them has its particular response to infrared. Thus, at a given temperature, all materials will be seen differently by the camera. Consequently, the camera must be calibrated before being used to estimate the LED temperature value accurately. Indeed, the calibrated camera is made as Figure 3.

More details, we place the LED (non-powered) at three temperatures T 1 , T 2 and

T 3 (T 1 < T 3 < T 2 ),
then, the images are measured in units of digital levels (DLs) at these three temperature levels (i.e., Im 1 (i, j), Im 2 (i, j) and Im 3 (i, j) are the values provided by the pixel (i,j) at temperature T 1 , T 2 and T 3 ). The image given by the detector placed in front of a LED is therefore not uniform for all pixels due to technological dispersion and an optic presence that induces aberrations. Thus, it is necessary to bring the slope and offset all the pixels back to the average values. And, this three-point correction makes it possible to standardize the response of the pixels. After correction over the range of 23 o C -90 o C, all pixels provide the same output level. After calibrating the camera, LED's temperature is calculated and analyzed from the thermal image by using a proprietary software tool (Flir System Inc.). When observing a scene, an image (Im(i, j)) in units (DL) is recorded. The temperature of the scene observed by the pixel (i, j) of the matrix is given by [START_REF]FLIR Systems | Imagerie thermique, vision nocturne et systèmes de caméras infrarouges[END_REF]:

T f rame (i, j) = T 1 + (T 2 -T 1 ) Im(i, j) -Im 1 (i, j) Im 2 (i, j) -Im 1 (i, j) (1)
Test condition: The choice of temperature condition, T k with k = 1, 2, was determined based on the LED Cree EZ1000 data sheet [START_REF] Sheet | Cree's ezbright leds[END_REF]. The test condition of data used in this paper is shown in Table 1. In this case, the forward current is fixed at 350 mA for all tests.

Input current 350mA

Temperature

T 1 85 o C(358 o K) T 2 100 o C(373 o K)
Sample size 9 units under T 1 and 10 units under T 2

Preliminary analysis from experimental dataset

Evolution of the luminous intensity

Our experiment has been realized for 19 LEDs during 1500 hours with two test stress levels (9 LEDs with T 1 = 85 o C and 10 LEDs with T 2 = 100 o C). The luminous intensity, denoted P (t), was measured every 48 hours. To better show the evolution of the luminous intensity over time, we used the normalized luminous intensity (P out (t)) that is defined as the ratio of the light output (P (t)) and its initial value P (0). The obtained results are shown in Figure 4.

P out (t) = P (t) P (0) (2)
Figure 4: Lumen degradation at two stress conditions ; a)

T 1 = 85 o C; b) T 2 = 100 o C
It is clear that under a stress temperature, the luminous intensity decreases with time, and the evolution of luminous intensity is not deterministic. Therefore, it is important to model this evolution in order to predict the failure time of the LED under a given stress temperature. In that way, a degradation model for the luminous intensity process is developed and presented in Section 4.

Impact of the forward current on the junction temperature

We analyze herein the variance of the LED's junction temperature using an infrared camera. Figure 5 shows the thermal image in units (DLs) taken before (Figure 5 Therefore, based on the obtained results, it is shown that an increment of junction temperature with an average of 10.14 degrees and a standard deviation of 1.14 degrees. Thus, with a 350 mA of forward current, the junction temperature becomes 95.14 o C and 110.14 o C when the stress condition of 85 o C and 100 o C, respectively, are applied. The impact of the forward current on its junction temperature will be modeled and presented in the next section.

Impact of the lumen degradation on the junction temperature

Figure 6 shows the different images in units (DLs) provided by the infrared camera FLIR SC7000 at different time points t (before the acceleration test t = 0, t = 576h, t = 720h and t = 1500h). It is shown that the junction temperature of LED increases during the acceleration test. Indeed, from these images, the increment on the LED junction temperature is evaluated by using the protocol presented in Section 2.2. The obtained results are reported in Table 3 and Table 4. It should be noticed that the increasing of the junction temperature may also affect the LED degradation process. Thus, there is herein a loop of thermal stress. The impact of the LED degradation on its junction temperature will be modeled and presented in the next section. 3 Mathematical modelling of self-heating impact

In this section, we try to model the impact of self-heating phenomenon in the LED's accelerated degradation process.

As the normalized luminous intensity is decreasing overtime under a given stress temperature, the degradation indicator of the LED is built as follows

X(t) = 1 -P out (t) (3)
In that way, X(t) is an increasing process, see Figure 7.

Conventional degradation rate model

According to the Arrhenius reaction rate theory [START_REF] Nelson | Accelerated testing: statistical models, test plans, and data analysis[END_REF], the degradation rate µ(T k ) under stress temperature T k can be formulated by:

µ(T k ) = α × exp - β T j (4)
Where

• T j is the junction temperature and it is considered as the stress temperature, i.e., T j = T k .

• α and β represent the characteristics of the LED failure mechanism and test conditions. β can be expressed as

β = E a k B ,
with E a is the activation energy for the reaction and k B is the Boltzmann constant.

According to this model, the degradation rate depends only on the failure mechanism and the magnitude of the junction temperature T j , which is considered the constant stress temperature. This means that the degradation rate is constant.

Modified degradation rate model with self-heating impact

As shown in section 4.3, the LED's junction temperature might increase during the acceleration tests due to the appearance of defects [START_REF] Buffolo | Long-term degradation mechanisms of mid-power leds for lighting applications[END_REF][START_REF] Ke | Analysis of junction temperature and modification of luminous flux degradation for white leds in a thermal accelerated reliability test[END_REF]. In fact, as mentioned above, the increment in the junction temperature can be divided into two parts:

• Power dissipation effect: when a forward current is applied to a LED, the LED's junction generates heat, which in turn increases the temperature of the LED's junction. This heat depends not only on the forward current but also on the LED's design (related to thermal resistance), see Figure 5.

• The impact of degradation: this means that when the degradation occurs, it increases the LED junction temperature, see Figure 6.

However, this variation in junction temperature is not considered in the conventional degradation rate model. We propose herein a modified degradation rate allowing considering the self-heating impact, i.e., the increasing of the junction temperature. In this way, the increasing of the junction temperature, with respect to the stress temperature, can be expressed as follows:

T (t) = a + f (X(t)) (5)
where,

• a is a non-negative real number and depends on the forward current and the characteristics of structure and material of the LED, see Appendix A. In this study, we assume that a is constant if the forward current is constant; it can be estimated in the analysis step.

• f (X(t)) indicates the effect of the degradation state, X(t), on the LED degradation process.

Moreover, we suppose that the effect of the degradation state, f (X(t)), can be described as Eq. ( 6)

for a simplified model. A detailed explanation is shown in Appendix A.

f (X(t)) = b.X(t) (6)
where b is a non-negative real number that qualifies the influence of the LED's degradation state on its accelerated degradation rate. According to Eq. ( 5) and Eq. ( 6), the junction temperature can be calculated by

T j = T k + ∆T = T k + a + b.X(t).
So, the formulation of the LED's degradation rate can be extended by considering the self-heating as follows:

µ(T k , t) = α × exp - β T k + a + b.X(t) (7)
This model somehow shows the state-dependence effect, i.e., the degradation evolution depends on the current degradation level [START_REF] Bian | Stochastic framework for partially degradation systems with continuous component degradation-rate-interactions[END_REF][START_REF] Assaf | Wear rate-state interactions within a multicomponent system: a study of a gearbox-accelerated life testing platform[END_REF].

Degradation modelling and reliability assessment

From the degradation rate, the evolution of the degradation process considering the self-heating impact can be derived using modified stochastic difference equation (SDE) [START_REF] Protter | Stochastic differential equations[END_REF][START_REF] Bian | Stochastic framework for partially degradation systems with continuous component degradation-rate-interactions[END_REF].

dX(t) = µ(T k , t).dt + σ.dB(t) (8)
where B(t) is a standard Brownian motion with the following properties:

• B(0) = 0, where B(t) ∈ [0, ∞),

• Station and independent increments, namely B(t + t) -B(t) ∼ N (0, t); where σ is the diffusion parameter, σ is a constant and σ > 0.

Degradation model

Following the accelerated degradation rate model considering the self-heating, we formulate a degradation model of a LED and its reliability calculation. Note however that determining the analytically formulation of X(t) based on Eq. ( 8) and Eq. ( 7) is very challenging [START_REF] Protter | Stochastic differential equations[END_REF]. Thus, we propose to use an approximation by the Taylor series, and the degradation rate of the component can be expressed by :

µ(T k , t) = α.   e -β T k + b × β T 2 k .e -β T k × X(t)    (9)
where, T k = T k + a.

By using Eq. ( 9), Eq. ( 8) can be written as follows:

dX(t) = λ. [κ + X(t)] dt + σ.dB(t) (10)
where:

κ = T 2 k bβ (11) λ = b × αβ T 2 k .exp - β T k (12)
Finally, the LED' degradation model considering the self-heating is found as the explicit solution of Eq.( 10), [START_REF] Bian | Stochastic framework for partially degradation systems with continuous component degradation-rate-interactions[END_REF]:

X(t) = X(0).e λ.t + κ e λ.t -1 + t 0
σ.e (λ(t-s)) dB(s).ds

A detailed mathematical development of Equation ( 10) is shown in Appendix B.

Reliability assessment

With a failure threshold, denoted D, the failure time is defined as the first hitting time of the stochastic degradation process exceeding the failure threshold value D. Then the cumulative failure distribution function can be expressed as:

F (t) = P (X(t) > D) = Φ E (X(t)) -D V ar (X(t)) (14)
where E (X(t)), V ar (X(t)) are the mathematical expectation and the variance of X(t) respectively.

E (X(t)) =E X(0).e λ.t + κ e λ.t -1

+ E t 0
σ.e λ(t-s) dB(s).ds [START_REF] Della Corte | Temperature sensing characteristics and long term stability of power leds used for voltage vs. junction temperature measurements and related procedure[END_REF] Note that E X(0).e λ.t + κ e λ.t -1 are deterministic values for any given t, whereas E t 0 σ.e λ(t-s) dB(s).ds = 0. Thus,

E (X(t)) = X(0).e λ.t + κ e λ.t -1 ( 16
)
The variance of X(t) can be expressed as:

V ar (X(t)) = V ar t 0 σ.exp (λ(t -s)) dB(s).ds = σ 2 × t 0 exp (2.λ(t -s)) .ds = σ 2 2.λ (exp(2.λ.t) -1) (17)
Finally, from Eq. ( 14) the reliability of the LED can be evaluated by:

R(t) = 1 -F (t) = 1 -Φ E (X(t)) -D V ar (X(t)) (18)

Parameters estimation

The proposed model is characterized by a set of parameters θ = {α, β, a, b, σ} which need to be estimated from the experimental data. Noting that a can be directly estimated with the analytical step [START_REF] Meneghini | Analysis of the role of current, temperature, and optical power in the degradation of ingan-based laser diodes[END_REF] thanks to an infrared camera. To estimate the remaining parameters (α, β, b, σ) from the experimental data, we develop herein an estimation approach based on the discrete maximum likelihood (DML) [START_REF] Florens-Zmirou | Approximate discrete-time schemes for statistics of diffusion processes[END_REF][START_REF] Bian | Stochastic framework for partially degradation systems with continuous component degradation-rate-interactions[END_REF]. Indeed, the degradation level at time t can be expressed by:

X(t j ) = X(t j-1 ) + κ e λ. t) -1 + e λ. t .X(t j-1 ) + σ * .dB(t j ) (19)
where,

(σ * ) 2 = σ 2 2.λ e 2.λ. t -1 (20)
Suppose that degradation signals are monitored at separated times, and t = t j -t (j-1) is the measurement frequency and t is small enough. According to Ito's stochastic integral [START_REF] Steele | Stochastic calculus and financial applications[END_REF], the conditional probability density of an observation X(t j ) given the previous observation X(t j-1 ), with t being time step, is given by

f (X(t j )|X(t j-1 ), κ, λ, σ * ) = 1 2π (σ * ) 2 * exp - X(t j ) -κ e λ. t -1 -e λ. t .X(t j-1 ) 2 2 (σ * ) 2 (21)
Then, the log-likelihood function of an observation data can be as follows:

l(α, β, b, σ) = K k=1 n k i=1 m k j=1 ln (f (X(t j )|X(t j-1 ), α, β, b, σ) (22)
The partial derivations for maximum likelihood estimation are required and set equal to zero, i.e.,

∂l(α, β, b, σ) ∂α = 0 ∂l(α, β, b, σ) ∂β = 0 ∂l(α, β, b, σ) ∂b = 0 ∂l(α, β, b, σ) ∂σ = 0 (23)
By using a numerical method for nonlinear algebraic equations, the parameters (α, β, b, σ) can be obtained by integrally solving the partial derivative equations using a MATLAB procedure of the Nelder-Mead simplex method [START_REF] Lagarias | Convergence properties of the nelder-mead simplex method in low dimensions[END_REF]. A detailed mathematical development of ( 23) is shown in Appendix C.

Result analyses

In this section, we analyse some obtained results when applying the proposed degradation model to our experimental dataset.

Estimation of degradation parameters and reliability

First, a can be easily estimated by the fact that the forward current at 350ma leads to an increasing of 10 degrees in the junction temperature (see again section 2), i.e., â = 10. The other parameters α, β, b, σ are estimated by the estimation approach presented in section 4.3. The obtained results are reported in Table 5. Especially, the decay rate following the convention ADT under two stresses of temperature 85 o C and 100 o C is found at 3.5E -4, 5.7E -4, respectively. Note that the conventional ADT and the estimation method are presented in Appendix D. Next, an additional analysis on LED shows some small darkenings in the junction mesa; see Figure 8. These defects are directly related to LEDs degradation and failure. Besides, we did not observe any defects in the electrode metal contact region after the acceleration test. This failure mechanism of bulk defects coincides with the energy activation, about E a = 0.31eV , which is found based on the model parameter's estimated value. Therefore, it can be concluded that the performance of the estimates seems to be adequate.

Comparison analysis

According to Eq. ( 14), the cumulative distribution function (CDF) based on the proposed degradation model is calculated using the estimated parameters reported in table 5. The obtained results are then compared with the experiment dataset in tables 6 and 7, see figures 9 and 10. It is clearly found that the CDF based on the proposed degradation model, which considers self-heating, is well-fitting with the experiment values. In contrast, without self-heating consideration, the conventional ADT model (see Appendix D) leads to a significant error in the calculation of CDF.

Evolution of the junction temperature

As an illustration, figure 13 shows simulated paths of the junction temperature over time using the proposed model with estimated parameters which are reported table 5. It is clear that the simulated results are very closed to the experimental ones (see tables 3 and 4).

Sensitivity analysis to data samples:

In this section, we examine the impact of the data samples on the reliability assessment or CDF of the LED. In that way, two data samples are used. The first is the data samples from 0 to 576h, and the second is from 0 to 720h. Figures 14 and15 present the CDF of the proposed degradation model with two different data samples. The results show that the CDF built with the samples of 720h is well-fitting with the experiment values. It is also very closed to the results obtained with the full dataset (t = 1500h). Besides, a significant error is obtained when using the samples of 576h. This can be explained that 576h data samples are not sufficient to provide good estimators while an accelerated dataset until t = 720h is quite enough to get a good estimator that allows obtaining a good accuracy for the LED's reliability assessment. This is an interesting result that allows us to stop the accelerated test from t = 720h while ensuring the accuracy of the reliability assessment of the LED component. sessing more precisely the reliability of LED components under a stress condition. In that way, the self-heating impact is firstly analysed from experimental data and then mathematically formulated.

To consider the self-heating impact on the LED's degradation process, a modified accelerated degradation model is developed based on a modified stochastic difference equation. Then the reliability of LED component is formulated. To estimate the proposed model's parameters, an estimation approach is also proposed. Finally, the proposed degradation model is applied for a real dataset of 19 identical LED components. The obtained results show that the proposed degradation model fits well to the experimental data. Also, it is shown that without considering the self-heating may lead to a significant error in the reliability assessment of LED components. Our future research works will focus on the development of the proposed model for a more complex system with self-heating and thermal interactions between components.

d (exp(-λ.s) × X(s)) = -λ.exp(-λ.s).X(s).ds + exp(-λ.s) × dX(s) + 0 (28)

• Then, we have d (exp(-λ.s) × X(s)) = -λ.exp(-λ.s).X(s).ds

+ exp(-λ.s) [λ. [κ + X(s)] ds + σ.dB(s)]
= exp(-λ.s).λ.κ.ds + σ.exp(-λ.s).dB(s)

• Integrating both side from 0 to t, we have:

exp(-λ.t) × X(t) -X(0) = t 0 exp(-λ.s).λ.κ.ds + t 0 σ.exp(-λ.s).dB(s) (30)
• So • t : operating time in hours;

X(t) = exp(λ.t).X(0) + t 0 exp [λ.(t -s)] .λ.κ.ds + t 0 σ.exp [λ.(t -s)] .dB(s) (31) ∂l(α, β, b, σ) ∂α = 2αβ 2 e - β T k f b 2 e 2αβe - β T k f b T k 2 - β T k                       X(t j ) -e αβe - β T k f b T k 2 X(t j-1 ) - T k 2            e αβe - β T k f b T k 2 -1            βb                       2 T k 4 σ 2          e 2αβe - β T k f b T k 2 -1          2 - βe - β T k b                       X(t j ) -e αβe - β T k f b T k 2 X(t j-1 ) - T k 2            e αβe - β T k f b T k 2 -1            βb                       2 T k 2 σ 2          e 2αβe - β T k f b T k 2 -1          -- αβe - β S b            2e β S σ 2 f e 2αβe - β S f b S 2 - β S π α - S 2 e β S σ 2      e 2αβe - β S f b S 2 -1      π α 2 βb            2S 2 σ 2     e 2αβe - β S f b S 2 -1     π - 2αβe - β T k b            - βf be αβe - β T k f b T k 2 - β T k X(t j-1 ) T k 2 -f e αβe - β T k f b T k 2 - β T k                                  X(t j ) -e αβe - β T k f b T k 2 X(t j-1 ) - T k 2            e αβe - β T k f b T k 2 -1            βb                       T k 2 σ 2          e 2αβe - β T k f b T k 2 -1          (32) ∂l(α, β, b, σ) ∂β = αβe - β T k b     2αe - β T k f b T k 2 -2αβe - β T k f b T k 3     e 2αβe - β T k f b T k 2                       X(t j ) -e αβe - β T k f b T k 2 X(t j-1 ) - T k 2            e αβe - β T k f b T k 2 -1            βb                       2 T k 2 σ 2          e 2αβe - β T k f b T k 2 -1               -     αe - β T k f b T k 2 -αβe - β T k f b T k 3     e αβe - β T k f b T k 2 X(t j-1 ) - T k 2       αe - β T k f b T k 2 - αβe - β T k f b T k 3       e αβe - β T k f b T k 2 βb + T k 2            e αβe - β       T k 2 σ 2       2αe - β T k f b T k 2 - 2αβe - β T k f b T k 3      
• φ(t) : averaged normalized of luminous intensity at time t;

• φ o : projected initial constant derived by the least squares curve-fit;

• η : decay rate constant derived by the least squares curve-fit; η = α.exp -β T k ; where α is the pre-exponential factor, β = Ea k B , E a is the activation energy, k B is the Boltzmann's constant;

• B(t) is a standard Brownian motion As a special case without self-heating consideration, Si et al [START_REF] Si | Remaining useful life estimation based on a nonlinear diffusion degradation process[END_REF] gave a probability distribution function (PDF) of the first hitting time of a threshold value D as follows: The parameter estimation of this model, θ = {α, β, σ} was performed according to the properties of the Wiener process. We let x ikj = φ i (t kj ) -φ i (t k(j-1) ) be a signal increment correspondence to the component which is monitored at separated times, and t = t j -t j-1 . So,

f ξ (t) = 1 √ 2πt S 2
x ikj ∼ N t j t j-1
γ(s, θ)ds, σ 2 t j -t (j-1) t 0 γ(s, θ)ds = φ o .exp (-η.t)

η = α.exp - β T k (38)
According to Eq. (38), the likelihood function was given:

L(θ) = K k=1 n k i=1 m k j=1 1 2πσ 2 (t j -t j-1 ) e   -
x ikj -t j t j-1 γ(s,θ)ds 2 2σ 2 ( t j -t j-1)

   (39)
We would not try to provide mathematical proof of the feasibility of this likelihood function. The models' parameters can be obtained by integrally solving the partial derivative equations using the numerical solutions for nonlinear algebraic equation groups thanks to a MATLAB procedure.

Figure 1 :

 1 Figure 1: Details of LED Cree EZ10000; a) LEDs' structure diagram; b) LED packages on the PCB

Figure 2 :Figure 3 :

 23 Figure 2: Global view of the experimental platform

  .a) and after applying the forward current of 350 mA (Figure 5.b). It is apparent that the forward current leads to a significant change (the area in the center of the image is brighter) in the thermal image due to the increment of the junction temperature of the LED. Indeed, this change is converted into the temperature increment by using Equation (1).

Figure 5 :

 5 Figure 5: Thermal analysis of a LED: a) without forward current and b) with forward current at 350mA

Figure 6 :

 6 Figure 6: Thermal analysis of a LED with forward current at 350mA: a) before the test; b) at 576h; c) at 720h and d) at 1500h

Figure 7 :

 7 Figure 7: Degradation indicator at two stress conditions ; a) T 1 = 85 o C; b) T 2 = 100 o C

Figure 8 :

 8 Figure 8: A result of the optical microscopy analysis: a) before the test; b) after the test

Figure 9 :Figure 10 :

 910 Figure 9: Cumulative failure function of LED according to Eq. 14 and the value experimental under stress at 85 o C

Figure 11 :Figure 12 :

 1112 Figure 11: Reliability of LED under stress at 85 o C

Figure 13 :

 13 Figure 13: Simulated paths of the junction temperature with a) T 1 = 85 o C and b) T 2 = 100 o C

Figure 14 :Figure 15 :

 1415 Figure 14: Cumulative failure function estimated with different data samples under stress at 85 o C

Table 1 :

 1 Test conditions

	Terms	Test condition
	Test duration	1.500 h
	Data collection interval	Minimum of every 48h

Table 2

 2

reports the results for 19 LEDs components in our experimentation.

Table 2 :

 2 Increment of junction temperature due to the forward current

	Component	T [ o C] Component	T [ o C]
	1	12.0	11	8.9
	2	10.9	12	11.3
	3	8.3	13	8.4
	4	11.0	14	8.5
	5	12.3	15	8.8
	6	10.7	16	7.6
	7	11.5	17	12.0
	8	11.0	18	10.5
	9	9.6	19	9.0
	10	10.4

Table 3 :

 3 Increment of junction temperature due to the LEDs' degradation under stress of T = 85 o C T (t) [ o C]

	Component t = 0h t = 576h t = 720h t = 1500h
	1	0	2.4	3.4	5.7
	2	0	2.5	3.6	7.0
	3	0	2.7	3.8	7.2
	4	0	1.6	2.6	5.5
	5	0	2.0	3.0	6.2
	6	0	2.4	3.4	6.0
	7	0	2.5	3.3	5.7
	8	0	2.2	3.9	7.0
	9	0	2.3	3.7	5.3
	Average	0	2.28	3.41	5.5

Table 4 :

 4 Increment of junction temperature due to the LEDs' degradation under stress of T = 100 o C T (t) [ o C]

	Component t = 0h t = 576h t = 720h t = 1500h
	1	0	2.6	3.8	6.7
	2	0	3.8	5.3	9.5
	3	0	3.7	4.7	9.0
	4	0	3.5	4.5	9.2
	5	0	3.0	4.3	7.8
	6	0	4.2	5.2	9.2
	7	0	2.9	4.0	8.3
	8	0	2.5	3.1	5.8
	9	0	3.7	4.7	8.7
	10	0	4.8	6.0	10.5
	Average	0	3.47	4.56	8.47

Table 5 :

 5 Parameters estimation

		Parameter
	Terms
		Mean std
	α	4.5	0.1
	β(×10 3 )	3.6	0.2
	â	10	2
	b	16.78 2.72
	σ	0.002	-

Table 6 and

 6 Table 7 respectively.

Table 6 :

 6 Result of acceleration test under stress T 1 = 85 o C

	Component t L70 [h] Uncertainty [h]
	1	867	[-38, +10]
	2	916	[-14, +34]
	3	939	[-28, +20]
	4	997	[-37, +11]
	5	1030	[-24, +24]
	6 and 7	1135	[-22, +26]
	8	1218	[-30, +18]
	9	1569	[-17, +31]

Table 7 :

 7 Result of acceleration test under stress T 2 = 100 o C

	Component t L70 [h]) Uncertainty[h]
	1	605	[-30, +18]
	2 and 3	681	[-20, +28]
	4	712	[-23, +25]
	5	754	[-25, +23]
	6	769	[-18, +30]
	7	840	[-25, +23]
	8	912	[-37, +11]
	9 and 10	1230	[-18, +30]

Table 8 :

 8 Increment of the junction temperature obtained from the simulated paths

				T (t) [ o C]
	Stress t = 0 t = 576h t = 720h t = 1500h
	85 o C	0	2.28	3.29	6.23
	100 o C	0	3.43	4.53	8.37

Appendix

Appendix A. Increasing of the junction temperature caused by the self-heating impact As a semiconductor device which emits the light, the electrical power consumed by the LED principally converts into light and heat dissipation.

Then, [START_REF] Jayasinghe | Is the thermal resistance coefficient of high-power leds constant?[END_REF][START_REF] Chu | Thermal conductivity measurements of thin-film resist[END_REF] show the relation between LED junction temperature and the luminous intensity, forward voltage following Eq.( 25).

where, T k is the ambient temperature, R th is the thermal resistance, I.V f is the input electrical power, and P is luminous intensity; A and B represent the constant coefficient.

• When t = 0, due to heat dissipation, the junction temperature increases about a.

• Then, the relation between the luminous intensity and the junction temperature over time according to the following equation:

Thus, the increasing of the junction temperature affecting the LED degradation can be modeled as Eq. ( 6)

Appendix B. Resolution of Equation (10)

We try to find the explicit solution of [START_REF] Ibrahim | Machine learning and digital twin driven diagnostics and prognostics of light-emitting diodes[END_REF]. In order to solve the SDE equation, we introduce then some transformations as follows.

• The expression exp(-λ.s) satisfies the following equation

Appendix D. Conventional ADT model for LED reliability assessment

According to the TM-21, [START_REF] Tsai | Inference from lumen degradation data under wiener diffusion process[END_REF] proposes an exponential degradation model for lumen degradation, as follows:

where,