Margaux Cheval
email: margaux.cheval@gmail.com

Marion Houot

Nathalie Chastan

William Szurhaj

Cécile Marchal

Hélène Catenoix

Luc Valton

Martine Gavaret

Bastien Herlin

Arnaud Biraben

Stanislas Lagarde

Laure Mazzola

Lorella Minotti

Louis Maillard

Sophie Dupont
email: sophie.dupont@aphp.fr

Early identification of seizure freedom with medical treatment in patients with mesial temporal lobe epilepsy and hippocampal sclerosis

Keywords: mesial temporal lobe epilepsy, hippocampal sclerosis, seizure freedom, medically-treated, predictive factors, machine-learning MTLE-HS: mesial temporal lobe epilepsy with hippocampal sclerosis MTLE: mesial temporal lobe epilepsy TLE: temporal lobe epilepsy ASM: antiseizure medications HS: hippocampal sclerosis FTBS: focal to bilateral tonic-clonic seizures SEEG: stereo-electroencephalography

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

1/Introduction

Patients with mesial temporal lobe epilepsy (MTLE) present different clinical profiles, showing poor or good response to antiseizure medication (ASM). [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF][START_REF] Aguglia | Mild non-lesional temporal lobe epilepsy. A common, unrecognized disorder with onset in adulthood[END_REF][START_REF] Coan | Patterns of seizure control in patients with mesial temporal lobe epilepsy with and without hippocampus sclerosis[END_REF] Actually, there is no unifying theory to explain the variety of response to ASM, with several potential contributing factors such as disease duration and severity, structural brain lesions, network disturbances with ongoing neural reorganization, and genetic/ metabolic abnormalities. [START_REF] Rambeck | Comparison of brain extracellular fluid, brain tissue, cerebrospinal fluid, and serum concentrations of antiepileptic drugs measured intraoperatively in patients with intractable epilepsy[END_REF][START_REF] Löscher | The clinical impact of pharmacogenetics on the treatment of epilepsy[END_REF] Among these factors, etiology of seizures seems to play a critical role. [START_REF] Semah | Is the underlying cause of epilepsy a major prognostic factor for recurrence?[END_REF][START_REF] Stephen | Does the cause of localisation-related epilepsy influence the response to antiepileptic drug treatment?[END_REF] Unilateral hippocampal sclerosis (HS), that is the most frequent aetiology of MTLE, is often associated to pharmacoresistance [START_REF] Semah | Is the underlying cause of epilepsy a major prognostic factor for recurrence?[END_REF] but intriguingly, some MTLE-HS patients may achieve a good seizure control with ASM. Kim et al. report 25% of seizure-free MTLE-HS patients with medical treatment (i.e medically-treated seizure-free patients) in their cohort [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF] and Labate et al. found 38.6% of hippocampal sclerosis on brain MRIs of patients with a "mild" form of MTLE. [START_REF] Labate | MRI evidence of mesial temporal sclerosis in sporadic "benign" temporal lobe epilepsy[END_REF] Nevertheless, none of these studies addressed the specificities of medically-treated seizure-free MTLE-HS patients compared to the classical drug-resistant MTLE-HS population.

To gain further insight into the characterization of medically-treated seizure-free MTLE-HS patients and into a better understanding of factors leading to pharmacoresistance in HS, we performed a retrospective multicentric study to determine the clinical factors associated with seizure freedom under medical treatment in MTLE-HS. Our secondary objective was to develop a machine-learning classifier based on clinical data in order to discriminate medically-treated seizure-free patients from drug-resistant patients. The aim was to improve individual patient counselling and determine if presurgical evaluation should be delayed.

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2/ Materials and methods 2.1 Selection criteria of the population and control group

Medically-treated seizure-free MTLE-HS patients were identified from thirteen French tertiary epilepsy centres.

Selection criterion were: 1) adult patients with well-defined MTLE, assessed by experimented epileptologist on medical and EEG data, 2) hippocampal sclerosis on structural brain MRI (characterized by a T1 hippocampal atrophy and a T2 hypersignal, visually assessed by a neuroradiologist, and approved by the epileptologist), 3) MTLE evolving for at least 5 years and 4) seizure freedom under medical treatment (including focal seizures without impaired awareness) for at least 2 years. The clinical and demographic data of these patients were retrospectively reviewed from medical charts.

These selection criterions took into account two possibilities of drug response already described in MTLE-HS patients: 1) a delayed pharmacoresistance during the first years of evolution 9,10 and 2) alternating periods of seizure-freedom and seizure recurrence [START_REF] Coan | Patterns of seizure control in patients with mesial temporal lobe epilepsy with and without hippocampus sclerosis[END_REF] . We thus chose to include only patients with more than 5 years of evolution and with a long and sustained period of at least 2 years of seizure freedom with ASM (versus 12 months to 2 years in another similar studies). [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF][START_REF] Coan | Patterns of seizure control in patients with mesial temporal lobe epilepsy with and without hippocampus sclerosis[END_REF][START_REF] Kwan | Early Identification of Refractory Epilepsy[END_REF][START_REF] Kwan | Definition of drug resistant epilepsy: Consensus proposal by the ad hoc Task Force of the ILAE Commission on Therapeutic Strategies: Definition of Drug Resistant Epilepsy[END_REF] The control group consisted of 200 patients with unilateral surgically-treated drug-resistant MTLE-HS. These patients were selected from a previously published larger cohort of adult drug-resistant MTLE-HS patients who underwent surgery at the Pitié-Salpêtrière Hospital in Paris between November 1990 and December 2015 (diagnosis of HS confirmed by pathological analysis). [START_REF] Mathon | Predictive factors of long-term outcomes of surgery for mesial temporal lobe epilepsy associated with hippocampal sclerosis[END_REF] Among this cohort of 391 patients, 191 were excluded due to missing data on educational level, medical, and psychiatric history. It should be noted that the control group did not include patients with bilateral hippocampal sclerosis, as this was an exclusion criterion for surgery in this centre.

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Standard Protocol Approvals, Registrations, and Patient Consents

The study was approved by the local ethics committee of Sorbonne University (CER-2021-089). The patients signed an informed consent form for the use of medical reports and data.

Recorded variables

Sociodemographic data such as age, sex, level of education, profession, marital status were collected. Education levels were classified by referring to high school graduation (High school graduation or lower, one to 3 years of higher education, more than 3 years of higher education).

Medical history included age at the onset of epilepsy, family history of epilepsy, psychiatric comorbidity (mood disorders, anxiety and psychosis), and the occurrence of an initial precipitating injury (meningitis, encephalitis, head trauma, febrile convulsion [both simple and prolonged]). We collected data on the semiology of seizures. We categorized auras according to their semiological description into seven categories: abdominal, experiential, autonomic, psycho-affective, visual, sensory and non specific. For the medically-treated seizure-free group, we also collected the time to achieve seizure freedom, the response to the first ASM, current medication and the number of seizure-free years at the last evaluation. Semiological data for medically-treated seizure-free MTLE-HS patients were collected mainly from medical charts and in 13 cases from video-EEG recordings. Neuropsychological data were not available for the medically-treated seizure free MTLE-HS patients as they did not undergo a presurgical evaluation. We did not collect the standard 20-min EEG data that are not considered as a performant biomarker of seizure control in the literature. [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF][START_REF] Aguglia | Age at onset predicts good seizure outcome in sporadic non-lesional and mesial temporal sclerosis based temporal lobe epilepsy[END_REF][START_REF] Steinhoff | Is routine electroencephalography (EEG) a useful biomarker for pharmacoresistant epilepsy?[END_REF]

Statistical analyses

To compare demographical characteristics between the two groups (medically-treated seizurefree MTLE-HS patients and drug-resistant MTLE-HS patients), Wilcoxon-Mann-Whitney test ACCEPTED MANUSCRIPT / CLEAN COPY was performed for numerical variables and Fisher's exact test for categorical variables. Pvalues were corrected for multiple comparisons using the Benjamini-Hochberg method. Statistical significance was defined as p < .05.

In a second step, we developed a classifier that aimed to discriminate between medically-treated seizure-free and drug-resistant MTLE-HS patients based on demographics, medical history, and semiological data (List of data included in the model detailed in the Supplementary Materials). Our objective was to better identify MTLE-HS patients with a greater chance of response to ASM. Therefore, we minimized the risk of error when categorizing a patient as "Seizure-free" by selecting a positive predictive value higher than 0.9, to reduce the probability of inappropriately delay referral to pre-surgical assessment in patient misclassified as "seizurefree". With respect to the group balance, dataset was split in a 80%-set used to find the best algorithm, which was then applied on the testing set (20%), i.e. on patients independent from the classifier construction, in order to evaluate its performances. The process to select the best algorithm using repeated 3-fold cross-validation is detailed in the Supplementary Materials.

3/ Results

Comparisons between medically-treated seizure-free and drug-resistant MTLE-HS patients

The proportion of missing data in the overall database was 11.8% in both groups (mostly employment and marital status data). Sixty-four medically-treated seizure-free MTLE-HS patients from 13 French tertiary epilepsy centres and 200 drug-resistant MTLE-HS patients were included. Comparisons between medically-treated seizure-free and drug-resistant MTLE-HS patients are shown in Table 1.

Social and demographic data

The two groups were not different in terms of sex-ratio, employment, or marital status.

The only significant statistical differences concerned the education level. 59.6% of medicallytreated seizure-free MTLE-HS patients stopped their studies before obtaining a high-school degree, while only 39.0% of drug-resistant MTLE-HS patients did. But when focusing on those who performed high-school education, only 0.5% of drug-resistant MTLE-HS patients achieved at least 3 years of higher education, while 11.5% of medically-treated seizure-free MTLE-HS patients did.

Medical history

Medically-treated seizure-free MTLE-HS patients were seizure-free for a median time of 7 years [5;11] at last evaluation. 34.5% of medically-treated seizure-free MTLE-HS patients were seizure free after the first ASM trial. The most frequently used ASM in the seizure-free group were levetiracetam (26.1%), lamotrigine (19.6%), valproic acid (9.8%), carbamazepine (8.7%) and lacosamide (8.7%) (Respective proportions of the other ASM are shown in Supplementary Materials). These patients had tried a median of 2 [2;3] treatments before achieving seizurefreedom. Regarding the findings on interictal EEG recording, 3.1% of the patients had bilateral anomalies, 54.7% had anomalies homolateral to the HS and 42.2% had no anomalies.

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The two groups did not exhibit significant differences in terms of early events history such as prematurity, meningitis/encephalitis or head trauma, or in terms of psychiatric history. There was a trend for more frequent familial history of epilepsy in the drug-resistant MTLE-HS group (23.6% vs 10.0%, p=0.062). Two significant differences were noted: 1) the frequency of childhood febrile seizures was significantly lower in medically-treated seizure-free MTLE-HS patients (39.0% vs 57.5%, p=0.035) and 2) medically-treated seizure-free MTLE-HS patients were significantly older at the onset of epilepsy (22.5 [13.0;41.5] vs 8.0 [4.0;14.0], p<0.001).

In addition, there was a lower proportion of ASM polytherapy in the medically-treated seizurefree group (number of ASM at inclusion, 1.0 [1.0; 2.0] vs 2.00 [2.0; 3.0], p<0.001).

Seizure semiology

Video EEG recording of seizure was available for only 13 (20.3%) medically-treated seizurefree MTLE-HS patients, and for all drug-resistant patients.

The occurrence of auras was significantly lower in the medically-treated seizure-free group (56.7% vs 90.0%, p<0.001). Among MTLE-HS patients who exhibited auras, medically-treated seizure-free MTLE-HS patients were less likely to have an autonomic aura (17.6% vs 59.4%, p<0.001) whereas no differences were noted regarding the other auras subtypes. Regarding focal seizures with impaired awareness, medically-treated seizure-free MTLE-HS patients had less gestural (27.4% vs 94.0%, p<0.001), oro-alimentary (32.3% vs 75.5%, p<0.001) or verbal automatisms (12.9% vs 36.0%, p=0.001). They also showed less dystonic posturing of the limbs (9.8% vs 47.0%, p <0.001).

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4/ Discussion

In this multicentric retrospective study, we present a substantial cohort of MTLE-HS patients who have achieved long term seizure freedom with ASM. We showed that medically-treated seizure-free MTLE-HS patients differed from the classically described population of drugresistant MTLE-HS patients in terms of medical history and seizure semiology. Using a machine-learning approach, we were able to develop a reliable classifier to identify MTLE-HS patients with a good response to ASM. The distinctive clinical features that we used for this machine-learning approach are easily assessable by a non-expert physician and do not require the facilities of an expert centre. This algorithm could be an easily accessible and manageable tool to counsel adequately MTLE-HS patients during the first consultation and delay the referral to a pre-surgical evaluation if a good response to ASM could be predicted.

Distinctive features between medically-treated seizure free and drug-resistant MTLE-HS patients

While the presence of hippocampal sclerosis is strongly associated with drug resistance [START_REF] Semah | Is the underlying cause of epilepsy a major prognostic factor for recurrence?[END_REF] , the first data published in 1999 by Kim et al. [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF] reported a 25% rate of seizure-freedom with medication in a MTLE-HS cohort. Studies that have looked specifically at medically-treated seizure-free patients with hippocampal sclerosis are rare (26 cases found in the literature). [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF] Most studies of MTLE with a good prognosis named "mild" or "benign" MTLE include both lesioned (including HS) and non-lesioned patients, 2,3,8,14,16-21 some of whom still having rare seizures that are not considered disabling, making comparisons with studies in the literature complex.

To our knowledge, our medically-treated seizure-free MTLE-HS highly selected cohort is the largest one published to date.

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Demographic and anamnestic data

Our results concerning educational background show that the distribution of educational levels is different between the two populations. In an intriguing way, there was a higher proportion of medically-treated seizure-free MTLE-HS patients who stopped their studies before obtaining a high-school degree. But logically, patients from both groups who stopped within the first three years their high school education had an earlier onset of epilepsy than those who did more than 3 years of higher education and a higher proportion of medically seizure-free MTLE-HS patients achieved at least 3 years of higher education. This outlines the impact of age at onset of epilepsy on patients' education as already demonstrated by Asadi-Pooya et al. [START_REF] Asadi-Pooya | Seizure and social outcomes in patients with non-surgically treated temporal lobe epilepsy[END_REF] In contrast to previous studies, [START_REF] Asadi-Pooya | Seizure and social outcomes in patients with non-surgically treated temporal lobe epilepsy[END_REF][START_REF] Hitiris | Predictors of pharmacoresistant epilepsy[END_REF][START_REF] Brodie | Road to refractory epilepsy: The Glasgow story[END_REF] we found no difference between the two groups in terms of employment rate, probably due to the relatively high employment rate in both groups, nor in psychiatric history.

Interestingly, we found a tendency to have less family history of epilepsy in the medicallytreated seizure-free group. Inconsistent results have been reported in the literature. Some teams found that patients with "mild" TLE are closely associated with familial forms of TLE or that MTLE patients with familial history of epilepsy had a better prognosis, [START_REF] Aguglia | Mild non-lesional temporal lobe epilepsy. A common, unrecognized disorder with onset in adulthood[END_REF][START_REF] Labate | Benign mesial temporal lobe epilepsy[END_REF][START_REF] Josephson | The "Natural" History of Medically Treated Temporal Lobe Epilepsy: What Can an Evidence-Based Approach Tell Us?[END_REF] findings supported by the existence of hippocampal sclerosis on MRI of asymptomatic relatives of patients with a familial form of MTLE. [START_REF] Kobayashi | Magnetic Resonance Imaging Evidence of Hippocampal Sclerosis in Asymptomatic, First-Degree Relatives of Patients With Familial Mesial Temporal Lobe Epilepsy[END_REF] At the opposite, as in our study, other studies have reported similar [START_REF] Andrade-Valença | Clinical and neuroimaging features of good and poor seizure control patients with mesial temporal lobe epilepsy and hippocampal atrophy[END_REF] or lower proportions 1 of family history of epilepsy in well-controlled MTLE patients.

Two major significant differences were noted here: 1) the frequency of childhood febrile seizures was significantly lower in medically-treated seizure-free MTLE-HS patients and 2) medically-treated seizure-free MTLE-HS patients were significantly older at the onset of epilepsy.

A lower incidence of childhood febrile seizures in well-controlled MTLE patients was already described by Aguglia. [START_REF] Aguglia | Age at onset predicts good seizure outcome in sporadic non-lesional and mesial temporal sclerosis based temporal lobe epilepsy[END_REF][START_REF] Labate | Benign mesial temporal lobe epilepsy[END_REF] In these studies, the proportion of patients with febrile seizures was ACCEPTED MANUSCRIPT / CLEAN COPY lower than in our study (11% versus 38.9%), related to the fact that they included patients with and without hippocampal sclerosis. Usually, hippocampal sclerosis as in our drug-resistant group, is highly associated with a history of febrile seizures. [START_REF] No | Medial temporal lobe epilepsy associated with hippocampal sclerosis is a distinctive syndrome[END_REF][START_REF] Heuser | Is Temporal Lobe Epilepsy with childhood febrile seizures a distinctive entity? A comparative study[END_REF] As in our study, most studies found that patients with good seizure control are older at the onset of epilepsy. [START_REF] Kim | The prognosis for control of seizures with medications in patients with MRI evidence for mesial temporal sclerosis[END_REF][START_REF] Aguglia | Mild non-lesional temporal lobe epilepsy. A common, unrecognized disorder with onset in adulthood[END_REF][START_REF] Kwan | Early Identification of Refractory Epilepsy[END_REF][START_REF] Aguglia | Age at onset predicts good seizure outcome in sporadic non-lesional and mesial temporal sclerosis based temporal lobe epilepsy[END_REF][START_REF] Labate | Benign mesial temporal lobe epilepsy[END_REF][START_REF] Hitiris | Predictors of pharmacoresistant epilepsy[END_REF][START_REF] Brodie | Road to refractory epilepsy: The Glasgow story[END_REF] A late age at onset of epilepsy has been identified as a positive prognostic factor in all forms of epilepsy including MTLE with or without HS. [START_REF] Aguglia | Age at onset predicts good seizure outcome in sporadic non-lesional and mesial temporal sclerosis based temporal lobe epilepsy[END_REF] At the opposite, MTLE patients with hippocampal sclerosis have usually an earlier onset than other MTLE patients. [START_REF] No | Medial temporal lobe epilepsy associated with hippocampal sclerosis is a distinctive syndrome[END_REF]

Clinical data

Medically-treated seizure-free MTLE-HS patients differed from drug-resistant MTLE-HS patients by a lower proportion of 1) auras, 2) autonomic auras in presence of aura, 3) gestural, oro-alimentary or verbal automatisms and 4) dystonia of the limbs. There are many arguments that MTLE-HS is a network disease rather than one centered on a single lesion.

The distinct clinical symptomatology in medically-treated seizure-free MTLE-HS can certainly be explained by differential spreading networks from an identical focus: the sclerotic hippocampus. In MTLE with HS, studies indicate that the amygdala and hippocampus are the hub regions and that secondary activation occurs preferentially to insular and medial frontal regions [START_REF] Russo | The localizing value of epileptic auras: pitfalls in semiology and involved networks[END_REF] or to the basal ganglia. [START_REF] Vercueil | Seizures and the basal ganglia: a review of the clinical data[END_REF] Interestingly, our data suggest that in medically-treated seizure-free MTLE-HS patients, secondary propagation could not occur preferentially in these regions. Signs that are supposed to result mainly from: i) an insulo-opercular cortex involvement, i.e autonomic aura and oroalimentary automatisms, ii) a frontal involvement, i.e gestural or verbal automatisms or iii) a basal ganglia involvement, i.e dystonic posturing of the limbs, are less frequent in the medically-treated seizure-free MTLE-HS patients subgroup. Indeed, autonomic symptomatology is supposed to be produced by activation of the insular cortex, anterior cingulum, or supplementary sensorimotor area, all areas that have been found to share the same subcortical relay station as the hypothalamus. [START_REF] Calleja | Orgasmic epilepsy[END_REF][START_REF] Fish | Clinical responses to electrical brain stimulation of the temporal and frontal lobes in patients with epilepsy. Pathophysiological implications[END_REF][START_REF] Bouilleret | Insular cortex involvement in mesiotemporal lobe epilepsy: a positron emission tomography study[END_REF] Similarly, stereo-EEG ACCEPTED MANUSCRIPT / CLEAN COPY (SEEG) data have indicated that the occurrence of oro-alimentary automatisms in seizures originating in anterior and medial temporal lobe depended on their propagation to insuloopercular areas, especially to their anterior part. [START_REF] Li | Oral and manual automatisms elicited by electrical stimulation of the pars opercularis cortex in a patient with frontal lobe epilepsy[END_REF][START_REF] Aupy | Insulo-opercular cortex generates oroalimentary automatisms in temporal seizures[END_REF][START_REF] Maillard | Semiologic and Electrophysiologic Correlations in Temporal Lobe Seizure Subtypes[END_REF] SEEG exploration have also shown that, in MTLE, gestural automatisms only occurred when the discharges spread to extratemporal cerebral structures, especially the opercular and cingular cortices. [START_REF] Li | Oral and manual automatisms elicited by electrical stimulation of the pars opercularis cortex in a patient with frontal lobe epilepsy[END_REF][START_REF] Maillard | Semiologic and Electrophysiologic Correlations in Temporal Lobe Seizure Subtypes[END_REF][START_REF] Munari | Electro-clinical semiology of subintrant temporal lobe seizures[END_REF][START_REF] Talairach | The cingulate gyrus and human behaviour[END_REF] Brain mechanisms underlying dystonic posturing obviously reflect a specific primary spread of the ictal discharge to the basal ganglia. [START_REF] Vercueil | Seizures and the basal ganglia: a review of the clinical data[END_REF] Imaging studies that ictal dystonic posturing was significantly associated to an increase in perfusion or a decrease in metabolism of the basal ganglia contralateral to the dystonic limb. [START_REF] Dupont | The underlying pathophysiology of ictal dystonia in temporal lobe epilepsy: An FDG-PET study[END_REF][START_REF] Newton | Dystonia, clinical lateralization, and regional blood flow changes in temporal lobe seizures[END_REF] The hypothesis is that connection patterns in a given pathology, i.e MTLE-HS, are malleable and may reflect individual plastic changes that have occurred in the course of a patient's lifetime, both in response to physiological events and to seizures. These distinct epileptogenic networks could partly explain the existence or absence of pharmacoresistance. [START_REF] Tang | Drug-Resistant Epilepsy: Multiple Hypotheses, Few Answers[END_REF] In line to this hypothesis, Fang et al. [START_REF] Fang | A new hypothesis of drug refractory epilepsy: neural network hypothesis[END_REF] recently proposed the neural network hypothesis, which states that seizure-induced degeneration and remodeling of the neural network suppress the endogenous antiseizure system and lead to ASM failure. An association between the presence of auras and automatisms and the drug-resistant status of MTLE patients has also been addressed by Stefanatou et al. who examined 83 consecutive MTLE patients (with (n=28) and without HS). [START_REF] Stefanatou | Drugresponsive versus drug-refractory mesial temporal lobe epilepsy: a single-center prospective outcome study[END_REF] As in our study, they found a lower frequency of auras and automatisms in drug responsive MTLE patients. These results are not in accordance with those of Agulia et al. [START_REF] Aguglia | Mild non-lesional temporal lobe epilepsy. A common, unrecognized disorder with onset in adulthood[END_REF][START_REF] Aguglia | Age at onset predicts good seizure outcome in sporadic non-lesional and mesial temporal sclerosis based temporal lobe epilepsy[END_REF][START_REF] Labate | Benign mesial temporal lobe epilepsy[END_REF] who described a higher frequency of autonomic, experiential and abdominal symptoms in "mild" MTLE as compared to drug-resistant MTLE. The explanation may again come from the non-homogeneous population of MTLE patients of these studies.

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Finally, the proportion of focal to bilateral tonic-clonic seizures (FTBS) was higher in both groups (medically-treated seizure-free MTLE-HS: 66.1% vs drug-resistant MTLE-HS: 71.9%), than previously reported in literature. [START_REF] Maillard | Semiologic and Electrophysiologic Correlations in Temporal Lobe Seizure Subtypes[END_REF] This can be explained by the fact that we took into account here the existence of at least one FTBS during the evolution of the epilepsy and not their frequency, a parameter difficult to estimate from a retrospective study. Similarly, Bone et al. [START_REF] Bone | Secondarily generalized seizures in temporal lobe epilepsy: Secondarily Generalized Seizures[END_REF] have shown an association between the presence of HS and a higher frequency of FTBS (80% in MTLE-HS versus 67% in other TLE patients), and a high rate of FTBS was also found in Aguglia's study (58 to 63.3%). 14

Prediction of seizure outcome with medical treatment in MTLE-HS patients

As a result, we were able to develop a reliable and accurate classifier. Another team had tried to develop an algorithm to predict the ASM response in MTLE [START_REF] Silva-Alves | A Prediction Algorithm for Drug Response in Patients with Mesial Temporal Lobe Epilepsy Based on Clinical and Genetic Information[END_REF] but its diagnostic performances were significantly lower than those of our classifier. Our classifier is simple to use as the variables included are easy to collect for the clinician. It has a good diagnostic performance, including a very high positive predictive value. The objective is twofold for newly diagnosed MTLE patients with hippocampal sclerosis: 1) to provide them with appropriate advice covering their future life and 2) to delay or avoid the referral to a presurgical assessment. It is clear that life choices, especially professional ones, are very much influenced by the drugsensitivity of the epilepsy. In MTLE-HS patients, the usual pharmacoresistance encourage practitioners to address the issue of a presurgical assessment at a very early stage. Our classifier could help to identify the few patients who have a high chance to achieve seizure-freedom with ASM and do not require early referral to presurgical evaluation. The classifier needs to be tested on a larger population and in a prospective approach.

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Limitations and strength of the study

Our analyses were retrospective, based on medical and seizure charts and we acknowledge the limitations of a retrospective study: 1) the collection of objective signs such as automatisms or dystonia may have been biased in the medically-treated seizure-free group since only 20% were objectively assessed by video-EEG recordings, and 2) we were unable to establish the time to drug resistance in the drug-resistant group.

Another limitation of our study is its limited recruitment to tertiary centres, which may have lowered the number of included patients, as seizure-free patients are less often referred to these centres.

The strengths of our study are the inclusion of well-characterized medically-treated seizure-free MTLE-HS patients with stringent and well-defined inclusion criterion. Indeed, our patients had been seizure-free for a median of 7 years, which can be considered as long-term seizurefreedom. We also decided to include only strictly seizure-free MTLE-HS patients in opposition to previous studies, [START_REF] Aguglia | Mild non-lesional temporal lobe epilepsy. A common, unrecognized disorder with onset in adulthood[END_REF][START_REF] Labate | MRI evidence of mesial temporal sclerosis in sporadic "benign" temporal lobe epilepsy[END_REF][START_REF] Andrade-Valença | Clinical and neuroimaging features of good and poor seizure control patients with mesial temporal lobe epilepsy and hippocampal atrophy[END_REF][START_REF] Labate | Benign mesial temporal lobe epilepsy[END_REF][START_REF] Labate | Hippocampal and thalamic atrophy in mild temporal lobe epilepsy A VBM study[END_REF][START_REF] Labate | Late drugresistance in mild MTLE: Can it be influenced by preexisting white matter alterations?[END_REF] where patients gathered as "mild", "benign" or "drugresponsive" TLE formed a heterogeneous group that ranged from complete seizure freedom to less to three seizures per year (focal with impaired awareness or focal to bilateral tonic-clonic).

It would be interesting to perform afterwards a dedicated MRI study of hippocampal volumetry and connectivity to assess if there are distinctive patterns between these two populations and thus define if these patients suffer from a different form of epilepsy than the drug-resistant MTLE-HS patients.

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5/ Conclusions

Our multicentric study presents data from the larger cohort of medically-treated seizure-free MTLE-HS, population that has been overlooked in the literature. We have shown that these patients present atypical clinical features compared to the commonly described drug-resistant MTLE-HS population: absence of childhood febrile seizures, later age at onset of epilepsy, absence of auras, lack of ictal dystonic posturing or automatisms. These features constitute "green flags" that can enable to hope for a sustained seizure-freedom with ASM. Combined in an efficient predictive model with other easily accessible anamnestic data, these "green flags" may help in a very early stage to reliably identify medically-treated seizure free MTLE-HS from drug-resistant MTLE-HS patients in order to improve patient counselling and the referral to an early pre-surgical evaluation. These findings will need validation from larger prospective studies. The study was approved by the local ethics committee of Sorbonne University (n° CER-2021-089). The patients signed an informed consent form for the use of medical reports and data. The best algorithm was defined as the one that maximized the median Youden index among those with a median positive predictive value greater than 0.9. Performance indicators of testing set was calculated using the best algorithm trained on the 80%-set.

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Authors contributions

Figures legends

-Tutorial: How to download and run our classifier?

-Get to https://github.com/mhouot/MTLE-HS_algorithm -You can download the all data and classifier or just run the app to test a patient.

-Testing a patient without R download on: https://mhouot.shinyapps.io/mtle-hs_algorithm/ -To get the algorithm on your computer:

-Download the free software R and R studio and install it on your computer from https://cran.r-project.org .

-Then, you need to install the shiny package in R: Copy and paste in the R console if (!require('shiny')) install.packages("shiny") -To run the application, you have to load the library and use the function runGitHub().

Copy and paste in the R console: library(shiny) shiny::runGitHub("MTLE-HS_algorithm", "mhouot") -Choose between the two tabs: ACCEPTED MANUSCRIPT / CLEAN COPY

  Statistical analyses were performed using R 4.1.2. (R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.) and with packages randomForest version 4.7-1 ; e1071 version 1.7-9; and xgboost version 1.5.2.1.

  M.C and S.D had the idea for the study, included patients and wrote the initial protocol and analysis plan. M.C collected the data. M.C, M.H and S.D did the data analysis. M.H did the statistical analysis and developed the machine-learning classifier. M.C. produced the table and wrote the first draft of the manuscript. M.H and S.D further reviewed the manuscript and contributed to the final version of the manuscript. All other authors (N.C, W.S, C.M, H.C, L.V, M.G, B.H, A.B, S.L, L.Maz, L.Min, L.Mai) included in the study patients from their epilepsy centres and contributed to editing and commenting on the final version.

1 .

 1 Select one patient's data: to obtain the probability of being medically seizure-free by entering patient data one by one 2. Upload a file (.xlsx) of the data patients: to obtain the probability of being medically seizure-free of several patient at a time. See the data_test.xlsx file in the datas folder for example.

Statements Conflict of interest

  None of the authors has any conflict of interest to disclose.

	Funding
	No funding to disclose.
	Ethical Publication Statement
	-We confirm that we have read the Journal's position on issues involved in ethical publication
	and affirm that this report is consistent with those guidelines.
	-We used the STROBE reporting guidelines.
	(von Elm E, Altman DG, Egger M, Pocock SJ, Gøtzsche PC, Vandenbroucke JP, et al. The
	Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) Statement:
	guidelines for reporting observational studies. Int J Surg. déc 2014;12(12):1495-9.)
	-IRB Statement:

Table 1 : Comparisons between medically-treated seizure-free and drug-resistant MTLE-HS patients

 1 Percentages of specific aura were calculated on patients who experienced aura. ‡ Wilcoxon-Mann-Whitney test was used to compare groups for numerical variables and Fisher's exact test for categorical variables.∫ pvalues correctd for multiple testing using Benjamini-Hochberg method. § at evaluation for Medically-treated seizure-free group, at surgery for Drug-resistant group Abbreviations: MTLE-HS = medial temporal lobe epilepsy with hippocampal sclerosis, ASM = antiseizure medication, SF = seizure-free, PR = drug-resistant, y = years, HS =hippocampal sclerosis MRI= magnetic resonance imaging ACCEPTED MANUSCRIPT / CLEAN COPY Boosting). Each machine learning technique was fitted several times with different combination of parameters : for regression logistic, no parameter needed to be fixed ; for random forests (RF), 8 combinations were tested (mtry ∈ {2;√p}, ntree ∈ {500;1000}, maxnodes ∈ {4;10}) ; for support vector machine (SVM), 9 combinations were tested (gamma ∈ {0.01;1;10}, cost ∈ {0.01;1;10}) ; for eXtreme Gradient Boosting (XGBoost), 8 combinations were tested (eta ∈ {0.1;0.3}, gamma ∈ {0;0.5}, max.depth ∈ {3;6}). Parameters names were those from R packages.

	ACCEPTED MANUSCRIPT / CLEAN COPY

¥

Acknowledgments:

The authors thank Drs. J.P.Vignal, A.Gales, V.Frazzini, C.Adam and Pr P.Thomas for their contribution to the recruitment of this study

Data Availability

The anonymized data that support the findings of this study are available on request from the corresponding author. The web-application of the algorithm is available at https://github.com/mhouot/MTLE-HS_algorithm .

Data availability statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

Conflict of interest

None of the authors has any conflict of interest to disclose.

Prediction of seizure-free status with medical treatment in MTLE-HS patients

The best performances were obtained with support-vector machine (SVM) method. The median

-Process to select the best algorithm

Performances were estimated using 500 samples of 3-fold cross-validation, leading to estimate median indicators and their 95% credibility intervals. Indeed, for each bootstrap sample, predicted probabilities of the validation set (as an independent cohort) (i.e. 1/3 of the sample size) were obtained using the algorithm fitted on the training set (i.e. 2/3 of the sample size).

The threshold of fitted probabilities on the training set, obtained by maximizing the Youden index among those with positive predictive value higher than 0.9, was then used to classify subjects of the validation test as true positive, false positive, true negative, or false negative.


Emilian Tichtchenko

Benoit Bedat

Olivier Simonin

Ludovic Glavier

David Gauchard

Alain Esteve

Carole Rossi

Comprehending the influence of the particle size and stoichiometry on Al/CuO thermite combustion in close bomb: a theoretical study

The paper is a theoretical exploration of complex Al/CuO thermite combustion processes, using a zero-dimensional (0D) model which integrates both condensed phase and gas phase reactions, and considers all thermodynamic stable molecular or atomic species identified during the Al+CuO reaction. We found that the particle size mainly influences the reaction kinetics and pressure development. Thermite with nano-sized particles (nanothermites) burns ~10 times faster than the same thermite with micron-sized particles (microthermites). This is due to the fact that the thermite reaction occurs mainly in condensed phase, i.e. in the melted Al phase, as all gaseous oxygens released by the CuO decomposition are spontaneously absorbed on the huge specific surface area of metallic Al. As a consequence, the pressure development in nanothermites follows the thermite chemical reaction, the gas phase is mostly composed of a metal vapor (mostly Cu and Al), Al suboxides, but is free of molecular oxygen. In contrast, when dealing with microthermites, an oxygen pressure peak occurs prior to the thermite reaction due to the gaseous O2 released by the early CuO decomposition, that cannot be absorbed on the Al particles surface in real time. The powder stoichiometry greatly impacts the final pressure. Al lean thermites generate a higher final pressure (× 3) than stoichiometric and

Al rich mixtures, due to unreacted gaseous oxygen which remains in the gas phase after the full consumption of the metallic Al.

Introduction

Thermites composed of metallic reactant (mostly Al) coupled to an inorganic oxidizer (CuO, Fe2O3, Bi2O3, MoO3 …) are non-explosive reactive materials which stay inert and stable until subjected to a sufficiently strong thermal stimulus, after which they undergo fast burning with release of high amount of chemical energy (up to 16 kJ.cm -3 ). They represent an interesting class of energetics because of their high adiabatic flame temperature (> 2600 °C) [START_REF] Fischer | Theoretical energy release of Thermites, Intermetallics, and Combustible Metals[END_REF][START_REF] Kabra | Recent trends in nanothermites: Fabrication, characteristics and applications[END_REF] and tunable combustion properties. Indeed, a number of factors, in addition to the reactant (metal and metal oxide) nature and size, influence the combustion properties of these reactive materials. These additional factors include powder density, the thickness of the natural oxide layer that chemically passivates the particles, the stoichiometry (metal oxide over metal ratio) [START_REF] Polis | Quo Vadis, Nanothermite? A Review of Recent Progress[END_REF][START_REF] Wu | Unexpected enhanced reactivity of aluminized nanothermites by accelerated aging[END_REF][START_REF] Chen | Reactivity adjustment from the contact extent between CuO and Al phases in nanothermites[END_REF][START_REF] Ramachandran | Influence of Stoichiometry on the Thrust and Heat Deposition of On-Chip Nanothermites[END_REF][START_REF] Zapata | Self-propagating combustion of sputter-deposited Al/CuO nanolaminates[END_REF][START_REF] Palussiere | The role of alkylamine in the stabilization of CuO nanoparticles as a determinant of the Al/CuO redox reaction[END_REF]. This tunability makes thermites promising in replacement of CHNO energetics for a large variety of applications such as propulsion [START_REF] Staley | Fast-Impulse Nanothermite Solid-Propellant Miniaturized Thrusters[END_REF], in-situ welding [START_REF] Dombroski | Joining and welding with a nanothermite and exothermic bonding using reactive multi-nanolayers -A review[END_REF][START_REF] Swiston | Thermal and microstructural effects of welding metallic glasses by self-propagating reactions in multilayer foils[END_REF], material synthesis [START_REF] Groven | Combustion synthesis and characterization of nickel aluminide-carbon nanotube composites[END_REF][START_REF] Farley | Self-propagating high-temperature synthesis of nanostructured titanium aluminide alloys with varying porosity[END_REF], generation of biocidal-agents [START_REF] Xu | Tuning the reactivity and energy release rate of I2O5 based ternary thermite systems[END_REF][START_REF] Liu | Preparation, ignition, and combustion of magnesium-calcium iodate reactive nano-composite powders[END_REF], initiations [START_REF] Rossi | Engineering of Al/CuO Reactive Multilayer Thin Films for Tunable Initiation and Actuation[END_REF][START_REF] Fu | Metal-interlayer-metal structured initiator containing Al/CuO reactive multilayer films that exhibits improved ignition properties[END_REF][START_REF] Pouchairet | PyroMEMS as Future Technological Building Blocks for Advanced Microenergetic Systems[END_REF], actuations in microsystems [START_REF] Salvagnac | Layered Al/CuO Thin Films for Tunable Ignition and Actuations[END_REF][START_REF] Suhard | When Energetic Materials, PDMS-Based Elastomers, and Microelectronic Processes Work Together: Fabrication of a Disposable Microactuator[END_REF][START_REF] Rodriguez | A microactuator based on the decomposition of an energetic material for disposable lab-on-chip applications: fabrication and test[END_REF][START_REF] De Koninck | Foil-Level Inkjet-Printed pyroMEMS Balloon Actuators: Fabrication, Modeling, and Validation[END_REF] and chips self-destruction [START_REF] Pandey | Self-Destructing Secured Microchips by On-Chip Triggered Energetic and Corrosive Attacks for Transient Electronics[END_REF][START_REF] Ma | Additive-Free Energetic Film Based on Graphene Oxide and Nanoscale Energetic Coordination Polymer for Transient Microchip[END_REF][START_REF] Sevely | Developing a highly responsive miniaturized security device based on a printed copper ammine energetic composite[END_REF]. Currently only general correlations between microscopic (particles size, stoichiometry…) and mesoscopic properties (powder compaction) of the thermite and its combustion behavior do exist, mostly derived from three decades of experimentations on thermite systems in research laboratories [START_REF] Glavier | Nanoenergetics as pressure generator for nontoxic impact primers: Comparison of Al/Bi2O3, Al/CuO, Al/MoO3 nanothermites and Al/PTFE[END_REF][START_REF] Williams | On the Pressure Generated by Thermite Reactions Using Stress-Altered Aluminum Particles[END_REF][START_REF] Martirosyan | Nanoenergetic Gas-Generators: principles and applications[END_REF]. This study investigates theoretically the influence of two important microscopic parameters (particle size and stoichiometry) on the thermite combustion behavior in terms of combustion temperature, pressure development, gas phase composition and combustion residues. As a case study, an Al/CuO powdered thermite (mixture of Al and CuO particles) is considered because it is welldocumented from both theoretical and experimental perspectives. Three Al/CuO systems are considered: nanothermites (nT, with particles diameter equal to 100 nm), submicrothermites (sµT, with particles diameter equal to 0.7 µm), and microthermite (µT with particles diameter equal to 2 µm). The combustion of each Al/CuO system is simulated considering three stoichiometries: Al lean, stoichiometric and Al rich. To this end, we use an original multi-phasic zero-dimensional (0D) model that explicitly treats the energy and mass exchanges between the condense and gas phases. In addition, the model integrates an ensemble of chemical reactions, and physical transformations that are known to occur during the thermite reaction [START_REF] Baijot | Modeling the Pressure Generation in Aluminum-Based Thermites[END_REF]. The detailed description of the model developed for thermite combustion prediction is outside the scope of this paper as it was already published [START_REF] Tichtchenko | Combustion model for thermite materials integrating explicit and coupled treatment of condensed and gas phase kinetics[END_REF]. Results not only confirm the experimental findings, e.g. the particles size mainly influences the reaction kinetics and pressure development, but importantly give an understanding of their underlying mechanisms.

Nanothermites burns ~10 time faster than the microthermites because the oxidation of Al occurs mainly in the condensed phase, e.g. in the melted Al, as all gaseous oxygens released by the CuO decomposition are spontaneously absorbed by the melted Al particles. The temporal pressure evolution thus follows the thermite reaction and the gas phase is mostly composed of a metallic (Al and Cu) vapor. There is no gaseous Al suboxides and no gaseous oxygen. In contrast, in microthermites, an oxygen pressure peak occurs prior to the Al oxidation reaction as the oxygens released by the CuO decomposition cannot be all absorbed on the limited Al particles surface. The powder stoichiometry greatly impacts the final pressure. Al lean mixtures generate a higher final pressure (× 3) than stoichiometric and Al rich ones, due to unreacted gaseous oxygens which remain in the gas phase after full consumption of the metallic Al.

Overall, this study, as the first theoretical exploration of complex thermite combustion processes, offers valuable guidelines for experimental investigations.

Computational details

The thermite system consists of a powder made of a homogeneous mixture of Al and CuO particles in a close chamber initially filled with ambient air. The Al particle purity (depending on the Al2O3 shell thickness), the powder density, the particles diameters (dAl and dCuO), and stoichiometric ratio (Al over CuO ratio, ) define the main parameters of the system. To initiate the Al/CuO reaction, an input energy of 600 kJ.kg -1 is injected uniformly (Figure 1). The first key mechanism before the initiation of the thermite reaction is the CuO decomposition [START_REF] Yi | High heating rate decomposition dynamics of copper oxide by nanocalorimetry-coupled time-of-flight mass spectrometry[END_REF][START_REF] Abdallah | Structure and Chemical Characterization at the Atomic Level of Reactions in Al/CuO Multilayers[END_REF][START_REF] Monk | Combustion Characteristics of Stoichiometric Al-CuO Nanocomposite Thermites Prepared by Different Methods[END_REF][START_REF] Sullivan | In situ microscopy of rapidly heated nano-Al and nano-Al/WO3 thermites[END_REF][START_REF] Umbrajkar | Exothermic reactions in Al-CuO nanocomposites[END_REF] releasing gaseous O2. Then, oxygen species (either in atomic or molecular state) diffuse across the alumina shell to oxidize spontaneously metallic Al at the Al core/Al2O3 shell interface. This strong exothermic reaction relays the external supplied energy to complete thermite initiation and reach a steady state at the time defined as tsteady. Steady-state is defined as the moment when the temperature variation of each phase decreases below 10 K.µs -1 .

After the initiation, in addition to oxygen transport though Al2O3 growing layer followed by the oxidation of the melted Al as previously evoked, a wide range of mechanisms contribute to the self-sustained reaction. The chemical reaction and mass transfer mechanisms are listed in Table 1 with corresponding mathematical model and parameters. Energy transfers are summarized in Table 2. In addition to the heterogeneous reactions occurring in both gaseous and condensed phase, the model considers homogeneous reactions that take place only in the gaseous phase.

Although no species transfer is considered between each particle phase, exchanged heat is modelled using a unit cell approach for the conduction and an enclosed assumption for the radiation. The model obeys the conventional mass and energy conservation equations (Equations 1-4), which are solved in the two distinct condensed phases, namely Al and CuO, and in the gas phase composed of Al, Cu, O, O2, N2, Al2O, Al2O2, AlO, AlO2.

The particles being considered uniformly distributed in the closed volume (Figure 1), a 0D

approach is used to calculate the system temporal state evolution considering the mass and energy exchanges between Al, CuO particles and the gas as briefly summarized in subsections 2.1 and 2.2. All model details can be found in [START_REF] Tichtchenko | Combustion model for thermite materials integrating explicit and coupled treatment of condensed and gas phase kinetics[END_REF].

Mass and energy conservation in the condensed phases

𝑁 𝑘 𝑑𝑚 𝑖,𝑘 𝑑𝑡 = ∑ 𝜔 ˙𝑖,𝑘 𝑟 𝑟∈𝑅 (1) 𝑁 𝑘 𝑑𝐻 𝑘 𝑑𝑡 = ∑ 𝑄 𝑘′→𝑘 + 𝑄 𝑖𝑛𝑖𝑡 𝑘′≠𝑘 (2)
where 𝑁 𝑘 is the number of particles in a given phase k (k = Al, CuO); 𝑚 𝑖,𝑘 is the mass of the species i in the particle k and 𝜔 ˙𝑖,𝑘 𝑟 is the reaction rate of the species i from reaction r relative to particle k; 𝐻 𝑘 is the enthalpy of one particle of the particle phase k, assuming that all the particles within a given phase k are at the same temperature, and, integrating liquid to solid phase transformation; 𝑄 𝑘′→𝑘 is the heat flux received by a given particle phase k from either the other particle phase or the gas, noted k'. 𝑄 𝑖𝑛𝑖𝑡 is the input energy per unit of time used to ignite the self-sustained reaction. Heat fluxes are symmetric, hence for any condensed or gaseous phase k' and k, 𝑄 𝑘'→𝑘 = -𝑄 𝑘→𝑘'

Mass and energy conservation in the gas phase

𝑑𝑚 𝑔 𝑑𝑡 = ∑ ∑ 𝜔 ˙𝑖,𝑔 𝑟 = 𝛤 𝑔 𝑖 𝑟∈𝑅 (3) 𝑑 𝑑𝑡 (𝑚 𝑔 𝑌 𝑖,𝑔 ) = ∑ 𝜔 ˙𝑖,𝑔 𝑟 𝑟∈𝑅 (4) 𝑑 𝑑𝑡 (𝑚 𝑔 𝑢 𝑔 ) = ∑ 𝑄 𝑘→𝑔 𝑘=𝑝,𝑞 (5)
where 𝜔 ˙𝑖,𝑔 𝑟 and 𝛤 𝑔 are the gas production rate of the species i and the total mass flux going from all the particles to the gas, respectively; 𝑌 𝑖,𝑔 is the mass fraction of the species i; 𝑢 𝑔 is the specific total internal energy (summing all species contribution) of the gas at the overall gas temperature; 𝑄 𝑘→𝑔 is the energy fluxes received by the gas from the particle phase k. 𝑚 𝑔 is the mass of the gas phase.

Gas phase kinetics

We consider the gas phase kinetics based on the scheme by Catoire et al. [START_REF] Catoire | Kinetic model for aluminum-sensitized ram accelerator combustion[END_REF] from which only the reactions involving Al and O species are considered. Table 3 lists the gas phase reactions along with their kinetic parameters. The open-source tool Cantera dedicated to rate theory calculations was used to solve the set of differential equations associated with the gas phase reactions [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics[END_REF]. Finally, the gas transport properties were computed based on the Lennard-Jones potential using Cantera considering parameters found in the literature.

Physical and kinetic parameters

All thermodynamic data, for condensed and gaseous species, are taken from JANAF tables [START_REF] Chase | NIST-JANAF thermochemical tables[END_REF] and NASA7 polynomials [START_REF] Mcbride | NASA Glenn coefficients for calculating thermodynamic properties of individual species[END_REF]. The kinetics of CuO, Al2O3 decomposition and oxygen species diffusion follows a single Arrhenius dependence on temperature. The Arrhenius parameters, activation energy, and pre-exponential factor are critical to correctly predict the combustion kinetics, despite the fact that the decomposition mechanisms for both materials are still not well characterized nor fully quantified. For CuO, the prefactor 𝑘 0 and the activation energy 𝐸 𝑎 are fitted such as the decomposition of copper oxide happens between 800 K and 1200 K [START_REF] Zhou | Time-Resolved Mass Spectrometry of the Exothermic Reaction between Nanoaluminum and Metal Oxides: The Role of Oxygen Release[END_REF][START_REF] Kwon | Interfacial Chemistry in Al/CuO Reactive Nanomaterial and Its Role in Exothermic Reaction[END_REF].

The oxygen diffusion is also thermally activated following an Arrhenius law, with prefactor 𝑘 0 and activation energy 𝐸 𝑎 fitted using experimental data [START_REF] Glavier | Nanoenergetics as pressure generator for nontoxic impact primers: Comparison of Al/Bi2O3, Al/CuO, Al/MoO3 nanothermites and Al/PTFE[END_REF]. Finally, vaporization/condensation of Al and Cu are treated based on the gas/liquid equilibrium and Spalding formulation. Their vaporization points evolve with respect to the pressure as a function of a Clapeyron law.

Results and discussion

Three different sizes of Al and CuO particles were investigated: dCuO = dAl = 100 nm (noted as nT), dCuO = dAl = 700 nm (noted as sµT), and, dCuO = dAl = 4 µm (noted as µT). A constant alumina shell thickness of 4 nm coats the Al particles, whatever their particles' size. This gives an Al purity of 0.7, 0.95, 0.99 for nT, sµT and µT, respectively. For each thermite (nT, sµT, µT), three equivalence ratios (, Equation 6) are considered: 0.8 (fuel lean), 1

(stoichiometric), 1.5 (fuel rich).

 = 81 × 𝑚 𝑝𝑢𝑟𝑒𝐴𝑙 160 × 𝑚 𝐶𝑢𝑂 (6)
Finally, the powder density is set at 50 % of the theoretical density of Al/CuO thermite. The theoretical density of the nT system is 2686, 2598, 2431 kg.m 3 for  = 0.8, 1 1.5 respectively.

For each powder, the total (Stot) and aluminum (SAl) surface area per volume of powder is reported in Table 4. Temperature evolution. In the temperature temporal curve (orange, blue and green curves in obtained for nT, sµT and µT, respectively. The differences are simply due to the differences in Al particle purity from one system to another, as the final pressure with heat losses is only function of the thermodynamic properties of the mixture, and independent of the particle size.

Importantly, we observe two major differences in the pressure development between µT and the two other thermites. First, the pressurization rate obtained for nT and sµT systems is 8 times superior to those of poorly-reactive µT, which agrees with the experimental findings pointing to higher reactivity of nanothermites [START_REF] Pantoya | Combustion behavior of highly energetic thermites: Nano versus micron composites[END_REF][START_REF] Bahrami | Magnetron Sputtered Al-CuO Nanolaminates: Effect of Stoichiometry and Layers Thickness on Energy Release and Burning Rate[END_REF] due to the enhanced interfacial contact between condensed and gas phases compared to that of the microscale mixtures. Second, the µT features a pressure peak at ~150 MPa before the thermite combustion. Interestingly, the oxygen released by the CuO decomposition between 600 -1200 K is not consumed readily in time on the Al particles to further react with metallic Al core, as the Al particle surface area is limited to 0.26 km 2 .m -3 only. This is also characterized by the time evolution of the O2 pressure in the gas phase in Figure 3, considering only stoichiometric systems (nT, sµT and µT taken at =1). The surface area not only allows for faster thermalization of the different phases as discussed in the previous paragraph, but also greatly impacts the kinetic of absorption of reactive species such as O2 from the gas phase to the Al particles through condensation mechanisms. In nT, the oxygen released by the CuO decomposition, starting ~ 600 K, is consumed almost spontaneously by the high Al particle surface (SAl=12.13 km 2 .m -3 , Table 4) and redox reaction occurs in the Al particles (Figure 3, dotted curve). In sµT, the oxygen released by the CuO decomposition is also consumed to react with Al in the Al phase (SAl=1.5 km 2 .m -3 , Table 4)

but with a slight delay (oxygen peak  in Figure 2b,e,h, i curves). But, in µT, the gaseous oxygens cannot be consumed and thus reacts in the gas phase with vaporized Al. Gas phase composition. The gas phase composition during the different steps of the combustion is given in Figure 4 for each thermite, nT, sµT and µT taken at =1. Initially, only oxygen and nitrogen are present in the gas phase, according to a standard dry atmospheric condition. When temperature approaches the boiling temperature of aluminum, evaporation increases and reacts with gaseous oxygens producing AlOx intermediates: a majority of AlO, but also Al2O. It has to be noted that Al2O2 is in anecdotic proportion. We observe first the formation of AlO, followed by O and Al2O which might originate from the progressive gas phase enrichment with atomic Al during its vaporization. AlxOy sub-oxides also condensate on Al and CuO particles as the reaction goes to completion. Gaseous Cu appears in the gas phase when temperature tends to its boiling point depending on pressure conditions. The gas phase composition greatly varies between nT and the two other systems: the gas phase of the nT contains almost no aluminum suboxide. As the oxygen is instantaneously absorbed on the huge aluminum particle surface (12.13 km².m -3 ), there is no oxygen in the gas phase to react with Al vapor when this latter starts to vaporize. In addition, the aluminum suboxides formed by the oxidation reaction directly condense onto both particles, again due to the high specific surface of the particles (30 km².m -3 ). By contrast, for µT and sµT systems, the total gaseous mass fraction of AlxOy reach up to 20% and 26% respectively, and are present during at least one microsecond.

Final products analysis. Finally, the chemical composition and the particle sizes of the final products are analyzed for each thermite system. Similar trends are observed for the three systems (Table 5), e.g. Al particles increase in size (by 5% for nT and 8% for µT) and get fully oxidized in Al2O3. CuO particles decrease in size by roughly 10 % leaving a majority of pure copper on which a non-negligible amount of aluminum oxide is formed through the condensation of Al suboxides.

Key role of thermite stoichiometry (equivalence ratio) on the final pressure

For each 9 Al/CuO systems, e.g. stoichiometric nT, sµT, µT, Al rich nT, sµT, µT and Al poor nT, sµT, µT, Table 6 summarizes the simulation results in terms of pressurization rate, pressure pre-peak, final pressure, temperature and time to reach the steady-state combustion regime. The pressure and temperature time evolution curves for each phases and thermite systems are plotted in Figure 5.

Table 6. Summary of simulated combustion characteristics (pressurization rate, pressure, temperature and time delay to reach the steady state) as a function of equivalence ratio ()

and for the three thermite systems: nT, sµT and µT. Clearly, the equivalence ration affects the combustion behavior and characteristics (Table 6 and Figure 2): in particular, the lack of Al in the mixtures leads to an excess of oxygens released in the gas phase, explaining why the final pressure of all Al lean mixtures is × 5 those of others.

The final pressure for  = 0.8 is in the 150 -185 MPa range whatever the thermite systems, while it is < 40 MPa for  = 1 and 1.5. µT systematically produces the pressure peak prior the steady state due to oxygens release as discussed in the preceding section. The intensity of these oxygen pre-peaks increases with decreasing the stoichiometry (from 152 MPa to 125 MPa for =1 to 1.5) as Al surface (SAl) on which O2 is absorbed increases with  as summarized in Table 6. µT systems reach their steady state within ~10 µs: 8 µs for Al rich and 15 µs for Al lean. Considering nT and sµT systems, a much faster initiation and combustion regime is observed, as the time needed to reach the steady state falls within 2 -2.3 µs. Interestingly, the steady state is reached faster when reducing the Al content ( = 0.8). In contrast, µT system behaves oppositely as it features a higher reactivity for higher Al content ( = 1.5), because of the increase of aluminum specific surface area.

It is also interesting to note that for both nT and sµT, there is an anecdotic oxygen pressure prepeak (30 and 39 MPa) at  = 1, which is not present for Al lean and Al rich mixtures. This pressure overshot is explained by the fact that aluminum which is vaporized and reacts in the gas to form aluminum suboxide, prior the condensation of these latter on particles. In the Al lean and Al rich mixtures, the energy is not sufficient to permit enough aluminum to vaporize to observe this phenomena. Regarding final combustion temperature, stoichiometric mixtures are systematically ~600 K higher than those of Al rich and lean ones due to a higher heat of reaction of the mixture. Typically, the heat of reaction of is of 3381 kJ.kg -1 , 4071 kJ.kg -1 , 3728 kJ.kg -1 for  = 0.8, 1 and 1.5, respectively.

The temperature profiles of each phase (Al, CuO and gas) during the combustion are particularly interesting to analyze from Figure 2 curves. In µT thermite system, whatever , each phase behaves thermally differently: the temperature of Al particles rises first as the thermite reaction occur into them, and noteworthy the highest Al phase temperature is obtained for  = 1 (5220 K). Then, when all the core particle aluminum is fully oxidized (at t = 9 μs), the temperature of the Al phase decreases as the heat is transferred to the gas and copper oxide particles. This is the reason why CuO particle temperature rises after the Al ones for all μT systems. Gas phase is the third to heat up but, because of the excess in oxygen in the gas phase (pressure pre-peak), the temperature of the gas increases suddenly once Al is vaporized, giving birth to gas phase exothermic reactions, which occurs only for  = 1 and 1.5, as Al lean system has no enough Al to induce gas phase reactions.

In stoichiometric and Al rich sµT systems, both the CuO and Al condensed phases are almost thermalized whereas the gas heats up faster ( in Figure 2 e-f)). However, in Al lean sµT, the Al and gas phases heat up faster than the CuO phase as indicated by the overtemperature peak ( in Figure 2 g).

Influence of gas phase treatment: kinetic theory vs thermodynamic equilibrium

For calculations presented in section 3.1 and 3.2, we used kinetic theory of gases based on the molecular interactions, which leads to the macroscopic relationships detailed in the computational section. Considering that the molecular chemical processes in the gas phase should be fast compared to condensed phases processes limited by the atomic mass transport, it is legitimate to wonder if it is necessary to solve the differential rate equations of the molecular reactions or replacing it by a gas phase equilibrium determination, as already used in [START_REF] Lahiner | A diffusion-reaction scheme for modeling ignition and self-propagating reactions in Al/CuO multilayered thin films[END_REF]. This section aims at discussing this point. For that purpose, we simulated the combustion of the 3 stoichiometric Al/CuO systems, nT, sµT and µT, considering the gas phase at the equilibrium and we compared the combustion temperature, pressurization rate, maximum pressure and gaseous composition with those obtained in section 3.1 for which a kinetic treatment was considered for the gas phase. No difference is observed for µT and sµT (Figure 5) so that assumption of gaseous equilibrium can be done for particle size smaller than 0.7 nm.

For µT and sµT, the global reaction characteristic time can be considered longer than gaseous molecular reaction time. However, in nT, the gaseous composition temporal evolution differs both in the amount and temporal distribution of gaseous species that are in minority (mass ratio < 10 -2 ). Assuming the gas at the equilibrium, AlO appears first and and is the dominant species among other Al suboxide species in the gas phase which can be considered as inexistent. AlO is similarly present using the kinetic algorithm, but is followed by Al2O in almost the same quantity (3×10 -2 mass ratio in Figure 5d). Therefore, the equilibrium assumption is not fully valid for nT system as the characteristic time of the different interphase mechanisms decreases to fall in the order of magnitude of the gaseous kinetics characteristic time scale. However, the impact of this assumption on the global behavior of the total system, even in nT systems, is negligible. and, µT (c,f) using the equilibrium approximation (top) and the gas phase kinetic approach (bottom).

Concluding remarks

We theoretically investigated the influence of two microscopic parameters, the particle size and mixture stoichiometry, on Al/CuO powdered thermite combustion. We detailed and analyzed the temperature, pressure, gas phase composition temporal evolution, and, combustion residues.

Three Al/CuO systems were considered: nanothermites (nT, with particles diameter equal to 100 nm), submicrothermites (sµT, with particle diameter equal to 0.7 µm), and microthermite

Figure 1 .

 1 Figure 1. Schematic of the simulated thermite system: (a) initial system, (b) system during the

Figure 2 .

 2 Figure 2. Pressure (dotted lines) and temperature (solid lines) temporal evolution of each phase

Figure 2 )Figure 2 )

 22 Figure2), we can distinguish the preheating stage during which the temperature increases

Figure 3 .

 3 Figure 3. Total pressure (solid blue), copper pressure (dotted red), aluminum pressure (dotted

Figure 4 .

 4 Figure 4. Gas phase species temporal evolution for stoichiometric nT (a), sµT (b), and, µT (c).

Figure 5 .

 5 Figure 5. Time evolution of the gas phase species (𝑌 𝑖,𝑔 ) for stoichiometric nT (a,d), sµT (b,e),

(

  µT with particle diameter equal to 4 µm). For each, three stoichiometries were simulated (Al lean, stoichiometric and Al rich) using a multi-phasic 0D combustion model integrating chemical reactions in both condensed and gas phases, and typical physical transformations occuring during the thermite reaction. Nanothermites burns ~10 time faster than the microthermites because the oxidation of Al occurs mainly in the melted Al, as all gaseous oxygens released by the CuO decomposition are spontaneously absorbed onto the high surface area of melted Al particles. The temporal pressure evolution follows the thermite reaction and the gas phase is mostly composed of a metallic (Al and Cu) vapor. No Al suboxides and oxygen are present. In contrast, in microthermites, an oxygen pressure peak occurs prior to the Al oxidation reaction as the oxygens released by the CuO decomposition cannot be all absorbed on the Al particles surface. The powder stoichiometry greatly impacts the final pressure. Indeed, Al lean mixtures generate a higher final pressure (× 5) than stoichiometric and Al rich ones, due to unreacted gaseous oxygens which remain in the gas phase after the full consumption of the metallic Al.

Table 1 .

 1 List of chemical reaction and mass transfer mechanisms implemented to model the

	Al/CuO thermite initiation/combustion .c, g subscripts stand for condensed and gas,
	respectively.
	Reaction	Mechanism	Mathematical implementation
	2Al(c) + 3/2O2(g) → Al2O3(c)	Diffusion-reaction	Spalding quasi-steady state
	2Al(c) + 3O(g) → Al2O3(c)		formulation/spontaneous reaction
	Al2O3(c) → 2Al(g) + 3/2O2(g)	Decomposition	Arrhenius dependence on
			temperature
	CuO(c) → Cu(c) + 1/2O2(g)	Decomposition	Arrhenius dependence on
			temperature
	Cu(c) ↔ Cu(g)	Evaporation/Condensation	Spalding quasi-steady state
			formulation/Clapeyron
	Al(c) ↔ Al(g)	Evaporation/Condensation	Spalding quasi-steady state
			formulation/Clapeyron
	AlOx(c) reactions on condensed	Detailed in [30]	Spalding quasi-steady state
	phases		formulation/spontaneous reaction
	Al, O, O2, Al2O, AlO, Al2O2,	Reaction	Gas phase kinetics
	AlO2

Table 2 .

 2 List of heat transfer mechanisms implemented to model the Al/CuO thermite initiation/combustion.

		Mechanism	Mathematical implementation
	Particles -particles	Conduction	Unit cell approach taking into
			account the interstitial gas
			conductivity [36, 37]
		Radiation	Enclosed system approximation
			[38]

Table 3 .

 3 Gas phase chemical reactions and kinetic parameters, defined as 𝐴𝑇 𝑛 𝑒 -𝐵 𝑇

	Chemical reaction	A (cm -3 × mol -1 × s -1 )	𝑩 (K)	𝑛
	Al + O 2 ↔ AlO + O	9.7×10 13	80.5	0
	AlO + O 2 ↔ AlO 2 + O	4.6×10 14	10008	0
	Al 2 O 2 ↔ AlO + AlO	1.0×10 15	59335	0
	Al 2 O 2 ↔ Al + AlO 2	1.0×10 15	74937	0
	Al 2 O 2 ↔ Al 2 O + O	1.0×10 15	52466	0
	AlO 2 ↔ AlO + O	1.0×10 15	44564	0
	Al 2 O ↔ AlO + Al	1.0×10 15	67035	0
	Al + O + X ↔ AlO + X	3.0×10 17	0	-1
	O 2 + X ↔ O + O + X	1.2×10 14	54244	0

⁄

[START_REF] Glorian | On the role of heterogeneous reactions in aluminum combustion[END_REF]

. X corresponds to the gas phase species: Al, Cu, O, O2, Al2O, AlO, Al2O2, AlO2, N2.

Table 4 .

 4 Total (Stot) and aluminum (SAl) surface area per volume of powder, and pressurization rate, for nT, sµT, and µT.

	Thermite system	nT	sµT	µT
	Stot (km 2 .m -3 )	30	4	0.75
	SAl (km 2 .m -3 )	12.13	1.5	0.26
	Pressurization rate (MPa.μs -1 ) 65	88	34
	3.1. Dominant role of Al particle surface area on the combustion regime and pressure
	development

Table 5 .

 5 Physico-chemical characteristics of the combustion residues of stoichiometric nT (a), sµT (b), and, µT (c).

	Thermite system	nT	sµT	µT
	Mass percentage of pure Cu in CuO	96	95	99.3
	particle (in %)
	Mass percentage of Al2O3 in final alumina	99	99	100
	particle
	Mass percentage of Al2O3 on final Cu	4	5	0.7
	particle
	Final Al2O3 particles mean diameter (nm)	104	740	4340
	Final Cu particles mean diameter (nm)	84	600	3300
	Initial dAl and dCuO (nm)	100	700	4000

Acknowledgements

The authors grateful acknowledge support from the European Research Council (H2020 Excellent Science) Researcher Award (grant 832889 -PyroSafe). The authors also thank Elliott Pollice for his help in implementing the gas phase kinetic in the model.


Pietro Alfonsi

Olivier Fouché

Emmanuel Bourgeois

A method of statistical reconstruction of the fractures network for numerical studies of rock foundations

INTRODUCTION AND PROBLEMATIC

The purpose of an artificial structural element referred to as a foundation is twofold: (1) to receive and transmit structural and external loads to the underlying rock at a given depth below the ground surface, safely without causing any distress to the superstructure; and (2) to distribute pressure at the base of the footing of the foundation to an intensity allowable on the rock. Concerning the first point, this paper will not address the effect of external loads as wind or seismic shocks. A foundation must be designed so that the resultant load exerted on it does not cause any lateral sliding of the foundation base on rock, ground collapse underneath the structure, tilting or intolerable settlement of the structure. Deep foundations are laid at considerable depth below the lowest part of the superstructure, to rest on, and to transmit structural loads to a firm geological stratum of adequate bearing capacity.

To determine the limit loads for rock foundations, estimates resulting from empirical studies and on the nonlinear criterion of failure from Hoek and Brown [START_REF] Hoek | Empirical strength criterion for rock masses[END_REF] are often proposed. This method, supplemented by theoretical assumptions, allowed the development of more specific procedures of application for various types of rocks depending on their state of fracturing (Serrano and Ollala, 1996).

To identify the class of qualification of the rock on the level of the foundations, the internal structure of a rock mass is hardly accessible. Tools of structural geology allow identifying the main discontinuity sets with the help of field data. Further information on the nature and fractures of the rock mass is given by boreholes, but these data are very local and the real fracture size and shape remain unknown. To deal with this difficulty, we use stochastic algorithms to simulate many realizations of a fracture network conditioned by measured information [START_REF] Fouché | Geological and geometrical characterization of natural fractures in a granitic formation from cored boreholes[END_REF]. The computing tool used in this study was designed by the French Institute for Petroleum (IFP). The fracture network is constrained by an a priori model where the geological medium is stratified and heterogeneous. From the spatial coordinates of the intersections between fractures, we reconstitute a 3D system of rock blocks. Then, we intersect this system with a cross section giving a 2D network of fracture traces.

Comparing estimates made from a specific standard for viaduct foundations on rock slopes with those made with the help of numerical models for continuous media (3D or 2D) would be a first task which is not be addressed here. Then, the comparison with the results of discontinuous medium modeling is not direct [START_REF] Alfonsi | Fondations au rocher: influence de la fracturation sur la répartition des tassements au pied des piles d'un viaduc[END_REF]. In this paper, we present the case of the viaduct of Saulières (France) to compare the rock mass deformation under the foundations predicted by a continuous numerical modeling (finite elements or finite differences) and discontinuous modeling (distinct elements).

FOUNDATION ON A ROCK SLOPE

The different scales of fractures

The description of a rock mass involves multiple scales: figure 1 sums up the complexity of the structure of a rock slope [START_REF] Rachez | Les fondations au rocher de grands viaducs: l'apport de la méthode des éléments distincts[END_REF]. The anisotropy of both the fracture network and the rock constitu-A method of statistical reconstitution of the fracture network for numerical studies of rock foundations ABSTRACT: For the design of foundations on rock, the part of numerical modeling becomes essential with the increase in complexity of construction works, especially when the structure of the rock mass is involved. The lack of information on rock discontinuities makes it difficult to reach enough precision in modeling. Thus, we have to resort to a stochastic approach of fractures from borehole data, here gathered in a viaduct project. To illustrate the differences between two approaches, we present a numerical study of the deformation of the rock around a foundation with a continuous modeling and a discontinuous modeling.

tive materials, as well as the filling of joints, is likely to influence the mechanical behavior of the rock mass. In a volume large enough where the multiple order discontinuities do not favor any potential failure surface, we can represent the rock mass as an equivalent continuum (cf. section 3). Properties for materials should be chosen by taking into account the fracture density [START_REF] Fouché | Assessment of free ground water hazard in tunneling: use of stochastic fracture network simulation and homogenization[END_REF]. At a specified scale characterized by a sufficiently well known system of discontinuities [START_REF] Fouché | Des forages à la simulation mécanique des massifs rocheux, l'apport d'une approche probabiliste 3D des réseaux de discontinuités[END_REF], another method is to represent explicitly the rock mass as a system of homogeneous blocks (section 4). The viaduct of Saulières will be 30 meters high and 450 meters long. It is designed with an ENE / WSW trend to cross a valley and a local interest road. It will be constructed to be a part of a bypass around the regional city of Brive (fig. 2).

The viaduct of Saulières, a rock-founded bridge

The piles of this bridge will be constructed within the sandstone formation of the Permian basin of Brive, away from the principal NE-SW faults. The dip of the two opposed slopes is respectively 30 to 40 degrees (West side) and 50 to 60 degrees (East side). At the surface, the bottom of the valley is made of alluviums whose thickness exceeds 4 meters, and the soil on the slopes is made of colluviums, whose thickness varies from 0.3 to 5 meters. These deposits are lying on sandstones which are often altered at the top and affected by intense fracturing in various directions.

Design of the foundations of the viaduct

For design and construction purposes of foundations on rock, the current methods consist to estimate the bearing capacity on the basis of an empiric classification of rocks mainly according to the rock matrix properties [START_REF] Bieniawski | Rock mass classification in rock engineering[END_REF]. Still, some recommendations of building codes for this application mention the existence of fractures in the rock mass, but generally the estimation is obtained from continuum mechanics. For instance, the approach used for the viaduct of Millau (France) is based on a classification of rocks from field data, laboratory tests and drill core description [START_REF] Rat | Le viaduc de Millau[END_REF]. This approach is global and allows for estimating the mechanical characteristics as deformability and resistance at the scale of the rock volume involved. In the recent years, because of an increase in applied loads in engineering practice, and taking into account seismic alea and other major natural hazards, the designers of road infrastructures in rock masses pay more and more attention to the role of fractures in the stability computations of construction works [START_REF] Maurin | Viaduc des Mercureaux. Etude de la fondation d'un ouvrage d'art exceptionnel au droit d'une faille majeure du faisceau Bisontin[END_REF].

For the viaduct of Saulières, the bearing capacity, Q pu , and the ultimate shaft resistance, Q su , are calculated on the basis of pressuremeter or dilatometer tests. The piled foundations drilled will be anchored in the compact sandstones. The load on each foundation, is evaluated according to the number of piles under consideration for viaduct, about 1.9.10 5 N by pile, and 0.5.10 5 N for the two abutments. These values are tested in the numerical study following.

A CONTINUOUS REPRESENTATION 3.1 A continuous 3D modeling (finite elements)

A model was realized in 3D using the finite element code CESAR-LCPC. The topography is roughly drawn and it gives a simplified image of the load repartition at the bottom of the viaduct piles. The mesh resolution is enhanced in the neighborhood of the piles (fig. 3). But the piles are not explicitly represented with this software.

The load is imposed at a 10 meters depth, i.e. at the top level of the intact sandstone layer which sup-ports the piles. The vertical stress increases according to the empirically estimated load on each pile. Layers 1 and 2 are both discretized into two mesh strata, and layer 3 has seven mesh strata with increasing thickness along depth. The values of the characteristics of intact and altered sandstones have been estimated including a secondary alteration due to fractures, and the values for the characteristics of colluviums are reduced down to a point imposed by the resolution of our finite element computations.

After loading, some deformation appears in the soil at the bottom of the piles (fig. 4). Deformation is centered on the points where the load is applied, and propagates through the layers in a quite symmetrical way. Notice that the displacements are insignificant since they are found to be in the order of 0.02 mm at the surface and 0.20 mm at the loading point. When the plastic state is attained in the superficial layer, the computations are soon interrupted for nonconvergence. For this reason, it was not possible to reduce the mechanical properties of colluviums in order to increase the displacements. On the basis of the geometry of the geological layers, a computation is performed by finite differences with the UDEC code, at the scale of a single pile in a vertical longitudinal cross-section. The piles of the viaduct are explicitly represented in this software. The bottom of the pile is fixed at the top level of the intact sandstone layer.

A continuous 2D modeling (finite differences)

We use a transfer factor µ which is recommended for estimating 2D equivalent stresses from the 3D load applied at the top of a pile with a square base [START_REF] Rachez | Les fondations au rocher de grands viaducs: l'apport de la méthode des éléments distincts[END_REF]:

B H B BH µ + + = 2 ² 4 = 11
where the height of the pile is H = 30 meters, and the edge length of the square base is B = 6 meters.

After we have deduced the settlement at the interfaces between the layers, we can note on figure 5 that the maximum displacement is close to the result of the 3D model with CESAR-LCPC. A close-up on the pile P3 with a better mesh shows locally a greater settlement around the bottom of the pile, progressively decreasing from a layer to the next with increasing depth.

A DISCONTINUOUS REPRESENTATION

The failure mechanisms of foundations on rock

Loading in continuous models gives a global idea of the intensity of deformation within the soil, but does not make it possible to detect the different orientations of the displacements which arise from the natural stratification of the rocks. For different types of the foundation and of the structure of the rock under loading, different mechanisms of potential instability leading to the failure were considered, from geology conditions (i.e. wedge, toppling or plane failure) or shear and deformation of fractured rock mass due to compression (fig. 6) [START_REF] Wyllie | Foundations on Rock[END_REF]. It is thus very important to improve our knowledge of the rock mass in order to define the better type of foundation in the studied case. To make numerical modeling realistic enough, we must face the difficulty to reproduce the fracture network from punctual data.

Characterization of the rock slope

From field prospecting and aerial photograph analysis we have a good knowledge of the main discontinuities, faults, and folds. Interpretation of well logging and borehole data results in a structural model which takes into account the sedimentary layers and the fractures. However, these data are very expensive to obtain and they are available only in a limited number. Also, the interpolation of data between several boreholes is very difficult. In this project, two boreholes have been realized and they give a 30 meters cumulate length of cores. The first was done at the East end of the projected bridge (CE), the other at the West end (CO). These boreholes are supplemented by 22 pressuremeter profiles [START_REF] Chahine | Projet du viaduc de Saulières, déviation de Brive[END_REF].

An example of a stereographical treatment of data is shown on figure 7a. We identify two fracture sets that are characteristic of borehole CO, near the West end. Within 13 meters of cores, 50 discontinuities have been detected in the depth interval from 2 to 10 meters. The mean dip vector (azimuth and plunge angles) is (212-43°) for set S1 and (58-56°) for set S2. Respectively the mean spacing is 0.24 m and 0.30 m. We do not use the data of the drilling carried out at the end CE because of a low quality in numerical imaging and because the discontinuities have very sparse directions. Thus, the statistical analysis of the data from this borehole is difficult to interpret.

In order to extrapolate the fracture network to a volume larger than that recognized by drilling, we use data-processing tools which allow the stochastic generation of discontinuous 3D models statistically equivalent to real borehole data. The FRACA software, Beicip (IFP) Company, was initially conceived for oil research [START_REF] Cacas | Nested geological modeling of naturally fractured reservoirs[END_REF] but can be adapted for civil engineering problems. By introducing an empiric value of 5 meters for the average size of the fractures, we simulate a series of virtual boreholes to explore the variability of the spatial distribution of our virtual network. An example of discontinuities connected to borehole CO is displayed in figure 7b.

Numerical modeling by the distinct element method

From the rectangles generated in the neighborhood of the abutments and foundations of the piles, we extract the 2D co-ordinates of their intersections with a cross-section (fig. 8). Up to now, we do not have any numerical tool, even in 2D, capable of treating such a complex network of fracture traces. So we choose to run UDEC with oversimplified schemes. To remain within the limits of data-processing resolution of the computer code UDEC, the representation of a cross-section of a discontinuous model over the entire length of the viaduct would imply a considerable reduction in precision. We thus will position our cross-section at the scale of the pile n°3 of the viaduct which is supposed to support the most significant effort. Moreover, the fractured part is reduced to a zone under the pile. Lastly, we tune the distribution of the two fracture sets and the value of mean dip angle of dip (for one set) in order to show the influence of the geometrical uncertainty.

Values equivalent to those of preceding calculations were then applied to constitutive materials. A friction angle of 20 degrees and a stiffness of 10 10 N.m for the joints of the fractured zone.

The first case represents two sets of discontinuities with trace persistence 1 meter, respectively with an apparent dip of 58 degrees and 115 degrees (fig. 9a). Under the effect of the load applied at the top of the pile, the deformations are propagated in the rock mass in an asymmetrical way, with maximal amplitude lower than 2 millimeters. Notice that the distri-bution of the iso-values of displacement follows the dips of the two discontinuity sets.

The second case represents the two same sets but their persistence follows a uniform distribution in a close range of mean value 1 meter (fig. 9b). In this case, the most highly deformed zone is tightened, the amplitude at the bottom of the pile increases slightly and the limit between the iso-values of displacement follows the traces of some discontinuities.

The third case represents the two sets with random persistence but with stronger dips: 72 degrees and 115 degrees (fig. 9c). The pile of the viaduct sinks like a punch, while following the dips of the two sets: the maximum value of displacement at the bottom of the pile approaches 2,5 millimeters.

DISCUSSION

The difficulties encountered when modeling rock slopes are well known: the constitutive materials are diverse, the geological survey is not always adapted to the specificity of the project, and the fractures are partially detected [START_REF] Fouché | Describing the Geometry of 3D Fracture Systems by Correcting for Linear Sampling Bias[END_REF]. It appears that the dimensioning of the works rather rises from a know-how informed by databases and practice. This pragmatic approach has proved to be reliable in the construction industry. It is generally supplemented by a numerical approach using the mechanics of the continuous media. In this paper we have shown that, in the case of loading on the piles of a viaduct, this method gives some useful indications on the spatial repartition of the ground defor- mation. It can be improved by resorting to an homogenization method, which includes the discontinuous character of the rock into an equivalent continuous medium [START_REF] Sulem | Modèles continus pour les structures rocheuses discontinues[END_REF]. But the indications obtained by the former methodology are essentially qualitative and do not allow to work without over-design. If one wishes to adapt the dimension of the foundations to the nature of the rock, we need some quantitative estimate of the displacements and the threshold of failure. Then, we have to explicitly represent the geometry and mechanics of the discontinuous media.

The stages of this kind of study are multiple, from in situ data collecting to numerical simulations. Starting from a minimum of geological and geometrical statements, the stochastic reconstitution of the fracture network makes it possible to limit the number of possible cases of simulations. Then it is necessary to carry out a parametric study of the characteristics of constitutive materials of the rocks and joints to qualify the potential state of failure at the bottom of the foundations. This approach is made possible by the distinct element modeling [START_REF] Cundall | Numerical modelling of discontinua[END_REF].

If 2D modeling is rather easy to use, it however requires considering cross-sections perpendicular to the main fracture set of the rock mass to be a reliable representation of reality. The 3D structure is, on the other hand, rather delicate to work out, with a more restricted number of distinct elements, but it opens more prospects for the study of the stability of the fractured rock volume.

CONCLUSION

Based on the geological analysis of the area, the reconstitution of the fracture distribution over the length of a bridge is possible from borehole data by using a specific modeling tool. The extraction of virtual boreholes at the position of the foundations then allows the construction of 3D or 2D models that turn out to be more rigorous and more complex that those resulting from the surface survey.

The numerical study was limited to particular cases of fracture densities which shows that, at the scale of a pile, the homogenization of the media seems satisfactory.

Nevertheless, in spite of the reduced dimensions of our volume of interest and of the dispersion of the fracturing of the rock, numerical computations of 2D discontinuous models show a strong anisotropy in the distribution of the iso values of displacement, with maximum amplitude locally higher than the values given by the continuous models. While approaching the natural configuration of the rocks, numerical modeling can then highlight the preferential orientations of the stresses or the deformations and contribute to detect the potential risks of failure.

We are grateful to the Beicip (IFP) Company and the Regional Laboratory of Bridges & Roads of Clermont-Ferrand for their invaluable collaboration which enabled us to carry out this first testing of our methodology.

Figure 1 .

 1 Figure 1. The rock mass at different scales.

Figure 2 .

 2 Figure 2. Localization of the projected viaduct on the geological map (1/25000): Fz = alluvium, h5 = sandstone.

Figure 3 .

 3 Figure 3. A 3D stratified model of the Saulières valley with the position of the viaduct, piers, and drillings CO (West) and CE (East).

Figure 4 .

 4 Figure 4. Vertical deformations after punctual loading and 2D vertical cross-section (computed with CESAR-LCPC).

Figure 5 .

 5 Figure 5. Vertical deformation after loading at the top of the pile, computations with UDEC.

Figure 6 .

 6 Figure 6. An example of potential mechanism of failure of a fractured rock mass under loading.

  Figure 7. a) Stereographic representation of the normal vectors of fractures, borehole CO (lower hemisphere, ISRM convention) b) intersection of the borehole with simulated rectangles standing for the fractures of the two identified sets (FRACA)

Figure 8 .

 8 Figure 8. Trace intersections between a vertical plane and the rectangles representing the fractures.

Figure 9 .

 9 Figure 9. Vertical displacements due to loading on pile P5-3 for three discontinuous models, computations with UDEC.

  stochastic generation of discontinuities (3D model from FRACA) extraction of intersections with a vertical plane