Giovanni Oriolo

Lison Huet

Sandra Dexpert

Cédric Beau

Damien Forestier

Patrick Ledaguenel

Eric Magne

Rocío Martin-Santos

Lucile Capuron
email: lucile.capuron@inra.fr

History of major depression is associated with neuropsychiatric symptoms but not systemic inflammation in a cross-sectional study in obese patients

Keywords: Obesity, Inflammation, C-reactive Protein, Neuropsychiatric symptoms, Vulnerability, Major Depressive Disorder (MDD)

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

Introduction

In the last decades, obesity has reached pandemic proportions worldwide [START_REF] Arroyo-Johnson | Obesity epidemiology trends by race/ethnicity, gender, and education[END_REF][START_REF] Bhupathiraju | Epidemiology of obesity and diabetes and their cardiovascular complications[END_REF]. Based on recent estimates, the prevalence of obesity has nearly tripled since 1975 (World Health Organization, 2017) and up to 30% of the European population is now considered as being obese [START_REF] Abdelaal | Morbidity and mortality associated with obesity[END_REF].

Accordingly, the pandemic of obesity represents a major public health burden, notably due to its substantial impact on morbidity, mortality, and healthcare cost [START_REF] Bray | Obesity: a chronic relapsing progressive disease process. A position statement of the World Obesity Federation[END_REF][START_REF] Spieker | Economic Impact of Obesity[END_REF]. Among the morbidities that are frequently associated with obesity, neuropsychiatric alterations, including depression, are of specific concern as they strongly contribute to social and occupational impairment, compromise treatment adherence and weight-management strategies, and heighten the risk of subsequent medical complications [START_REF] Abdelaal | Morbidity and mortality associated with obesity[END_REF].

Obese individuals are frequently afflicted with mood alterations and present an increased risk of depression during lifetime [START_REF] Castanon | Neuropsychiatric comorbidity in obesity: Role of inflammatory processes[END_REF][START_REF] Dawes | Mental Health Conditions Among Patients Seeking and Undergoing Bariatric Surgery: A Meta-analysis[END_REF][START_REF] De Wit | Depression and obesity: A meta-analysis of community-based studies[END_REF]. Similarly, cognitive disturbances, including alterations in memory, executive function and flexibility, have been documented in multiple cross-sectional and longitudinal studies of obese subjects [START_REF] Dahl | Body mass index across midlife and cognitive change in late life[END_REF][START_REF] Lokken | Cognitive performance of morbidly obese patients seeking bariatric surgery[END_REF]. While the directionality of the relationship between obesity and neuropsychiatric symptoms is hard to determine, clinical and preclinical findings comfort the hypothesis that obesity, notably through its inflammatory component, contributes to the pathophysiology of neuropsychiatric symptoms [START_REF] Capuron | Role of Adiposity-Driven Inflammation in Depressive Morbidity[END_REF]. Obesity is considered as being both a metabolic disorder and an inflammatory condition, which is associated with a chronic low-grade inflammatory state and an enhanced susceptibility to immune-mediated diseases [START_REF] Castanon | Neuropsychiatric comorbidity in obesity: Role of inflammatory processes[END_REF][START_REF] Kanneganti | Immunological complications of obesity[END_REF]. Several mechanisms are believed to contribute to the generation and maintenance of systemic low-grade inflammation in obesity, including the infiltration of immune cells in the adipose tissue, the adipose secretion of inflammatory factors, and alterations in the gut microbiota and gut permeability [START_REF] Gregor | Inflammatory mechanisms in obesity[END_REF][START_REF] Kanneganti | Immunological complications of obesity[END_REF]. The chronic activation of inflammatory processes in obesity may easily contribute to the development of obesity-related neuropsychiatric comorbidities, consistent with findings documenting significant associations between systemic inflammation and depressive symptoms in obese patients [START_REF] Delgado | Depressive symptoms in obesity: Relative contribution of low-grade inflammation and metabolic health[END_REF] and with a large literature substantiating the role of inflammation in the pathogenesis of neuropsychiatric symptoms in clinical settings [START_REF] Capuron | Immune System to Brain Signaling: Neuropsychopharmachological Implications[END_REF][START_REF] Liu | Evidence for Inflammation-Associated Depression[END_REF][START_REF] Raison | Cytokines sing the blues: inflammation and pathogenesis of depression[END_REF].

Despite mounting evidence about the close association between inflammation and psychopathology, it is well admitted that neuropsychiatric symptoms do not develop in all patients afflicted with chronic inflammation [START_REF] Lotrich | Inflammatory cytokine-associated depression[END_REF], and that not all depressed patients exhibit immune activation [START_REF] Rosenblat | Inflamed moods: A review of the interactions between inflammation and mood disorders[END_REF]. These findings suggest that inflammation may trigger neuropsychiatric symptoms only in vulnerable individuals or in conjunction with other risk factors. Supporting this notion, factors such as psychological trauma, repeated sub-chronic stressors or early exposure to trauma were found to robustly increase the vulnerability to neuropsychiatric symptoms while impacting neuroimmune circuits [START_REF] Berk | So depression is an inflammatory disease, but where does the inflammation come from?[END_REF]. For instance, it was shown that exposure to childhood trauma increases the risk of developing mood symptoms while inducing persistent increases in inflammatory markers at adulthood [START_REF] Coelho | Childhood maltreatment and inflammatory markers: a systematic review[END_REF][START_REF] Danese | Elevated inflammation levels in depressed adults with a history of childhood maltreatment[END_REF][START_REF] Rasmussen | Cumulative childhood risk is associated with a new measure of chronic inflammation in adulthood[END_REF]. Moreover, a personal history of major depressive disorder (MDD), notably when characterized by multiple episodes, was found to be associated with a greater risk of relapse [START_REF] Kessing | Evidence for clinical progression of unipolar and bipolar disorders[END_REF] and with higher levels of inflammatory markers [START_REF] Celik | The association between serum levels of neopterin and number of depressive episodes of major depression[END_REF][START_REF] Maes | Increased autoimmune activity against 5-HT: A key component of depression that is associated with inflammation and activation of cell-mediated immunity, and with severity and staging of depression[END_REF]. Nevertheless, the impact of a personal history of depression on inflammation and related neuropsychiatric comorbidities in vulnerable subjects free of current depression has not been addressed yet. Accordingly, the aim of the present study was to assess the influence of depressive history on the relationship between inflammation and neuropsychiatric symptoms in obese subjects, characterized by both a systemic low-grade inflammatory state and an increased vulnerability to mood disorders. We hypothesized that obese individuals with a personal history of depression will present greater neuropsychiatric symptoms, as well as increased levels of inflammatory markers, than obese subjects with no history of depression, and that history of depression will modulate the relationship between obesity-related inflammation and neuropsychiatric symptoms.

Patients and Methods

Participants

Obese subjects: Sixty-six obese patients eligible for bariatric intervention were recruited from the services of digestive and bariatric surgery of two private clinics (Tivoli and Jean-Villar) from Bordeaux area, France. Participants were drawn from a larger study on vulnerability factors and pathophysiological mechanisms of neuropsychiatric comorbidities in obesity (OBEPSY Study, ANR ANR-11-JSV1-0006, LC). All participants had a body mass index (BMI) ≥ 35 kg/m 2 and were between 18 and 65 years old. Subjects were excluded if they had current neurologic and/or psychiatric disease, including MDD, as well as a score >20 on the Montgomery-Asberg Depression Rating Scale (MADRS) [START_REF] Montgomery | A New Depression Scale Designed to be Sensitive to Change[END_REF] to warrant the absence of undiagnosed MDD, or if they were currently on antidepressant treatment. Individuals were also excluded if they presented active infection or uncontrolled medical disease, with the exception type-2 diabetes, dysthyroidism, hypertension (HT), dyslipemia and obstructive sleep apnea (OSA), all of which being medical conditions strongly related with obesity.

Non-obese subjects: A sample of 22 healthy, non-obese participants (BMI<30 kg/m 2 ), with no clinical sign of systemic inflammation, as attested by circulating levels of highsensitive C reactive protein (hs-CRP) lower than 5 mg/L, was included as control group. All subjects were free of any current depressive disorder and/or antidepressant treatment, diabetes, HT, OSA, dysthyroidism or inflammatory disease.

All participants provided written informed consent before enrollment. The study was approved by the local Committee for the Protection of Persons (CPP) of Bordeaux, France.

Methods

Group Stratification

The group of obese subjects was divided in two subgroups based on history of at least one MDD episode. Accordingly, the first subgroup included 32 obese patients with a personal history of MDD (HMDD+) and the second subgroup included 34 obese patients with no personal history of MDD or other mental disorders (HMDD-). In the group of obese subjects with a personal history of depression, MDD age of onset and number of MDD episodes were also assessed. Early onset was considered when diagnosis of MDD was achieved before the age of 20.

Similar to obese subjects, the group of non-obese subjects was divided in two subgroups based on history of depression. The two subgroups included respectively 13 nonobese subjects with a personal history of MDD and 9 non-obese subjects with no past history of MDD.

Sociodemographic and Clinical Assessments

Sociodemographic and clinical variables, including medical and neuropsychiatric history, current pathologies and treatments, tobacco consumption, BMI and obesity onset, were gathered in all participants. Early obesity onset was considered when the diagnosis of obesity was made before the age of 18 years old.

Behavioral Assessments

All subjects were assessed through the MINI international neuropsychiatric interview [START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): The Development and Validation of a Structured Diagnostic Psychiatric Interview for DSM-IV and ICD-10[END_REF] to rule out current MDD episode according to DSM criteria. The Neurotoxicity Rating Scale (NRS), a self-reported scale used to assess general and behavioral symptoms associated with inflammatory conditions [START_REF] Valentine | Treatment of neurotoxic side effects of interferon-alpha with naltrexone[END_REF] was administered.

This scale includes 39 items scored from 0 ("not present") to 4 ("extremely severe") measuring symptom severity of a wide range of symptoms. To perform dimensional analyses, symptoms were further grouped in five separate dimensions, corresponding respectively to the dimensions of mood, cognitive, sleep, sickness and motor symptoms, as previously described (Capuron et al., 2011[START_REF] Capuron | Interferon-alpha-induced changes in tryptophan metabolism: Relationship to depression and paroxetine treatment[END_REF][START_REF] Capuron | Neurobehavioral effects of interferon-alpha in cancer patients: phenomenology and paroxetine responsiveness of symptom dimensions[END_REF]. The dimension of mood included depressive symptoms [depressed mood, mood swing, anhedonia (i.e., loss of interest in people/work/activities), loss of motivation, decreased emotions] and anxiety symptoms (anxiety, tension, irritability, worries about health and agitation). The cognitive dimension included symptoms of impaired memory, distractibility, indecisiveness, word-finding problems and episodes of confusion.

The dimension of sleep included symptoms of difficulty getting to sleep or staying asleep and sleeping too much. The dimension of sickness included pain symptoms (headache, body aches, joint pain), fatigue, fever and general malaise. Finally, the dimensions of motor symptoms included motor slowing, walking problems and tremor/shakiness. To homogenize scores across dimensions, an average score was calculated for each symptom dimension.

Measurement of Systemic Inflammation

Fasting blood samples were obtained the day of behavioral assessments for the measurement of serum concentration of the inflammatory marker, hs-CRP. Collected samples were centrifuged (10 min, 1000g, 4°C) after clotting and sera were stored at -80°C until the assays. Hs-CRP was assayed by enzyme-linked immunosorbent assay (ELISA), according to the manufacturer's specifications (CYT298, Millipore, Billerica, Massachussets). Intra-and inter-assay variability was ±4.6% and ±6.0%, and sensitivity was 0.20 ng/mL.

Data Analyses and Statistics

Hs-CRP was log-transformed due to non-normality of the distribution. Sociodemographic data, behavioral scores and serum concentrations of biological markers were compared between obese patients and non-obese participants regardless of MDD history using t-test for continuous variables and chi-square test for categorical variables. Analyses were repeated in the form of analyses of covariance (ANCOVAs) controlling for age, gender, and variables showing a significant difference between the two groups to control for the potential effect of these variables on the outcome measures. Similar analyses, followed by post-hoc comparisons (Tukey test) when appropriate, were performed stratifying groups on the basis of MDD history.

The relationship of log hs-CRP with BMI and neuropsychiatric symptoms (NRS total score) was estimated using Bravais-Pearson correlations in the whole sample under study and/or multiple regression analyses. Multiple regression analyses adjusting for age, gender and relevant variables, including comorbidities (diabetes, dysthyroidism, HT, OSA), educational level and tobacco consumption, were also carried out to assess the respective contribution of obesity and history of MDD to neuropsychiatric symptoms (NRS total scores) and systemic inflammation (log hs-CRP) in study participants. Obesity and history of MDD were first introduced in separate models and then together in order to test their possible interaction. Of note, 4 subjects (18%) in the non-obese group had a BMI between 25 kg/m 2 and 29.9 kg/m 2 , indicative of overweight. Albeit none of these subjects had clinical signs of systemic inflammation, as attested by circulating levels of hs-CRP lower than 5 mg/L, all analyses were still systemically redone excluding those 4 subjects to confirm that it did not impact results. Exclusion of those subjects did not alter the results and conclusions of the study (results not shown).

Statistical analyses were performed with XLStat version 18.06. Test of significance were two tailed with the degree of significance set at p<0.05.

Results

Sociodemographic and Clinical Characteristics of Obese and Non-Obese Subjects

Sociodemographic and clinical characteristics of study participants are shown in Table 1. No significant differences were observed between obese and non-obese subjects with respect to age, gender, marital status and history of depression. As expected, obese subjects exhibited higher BMI (t = 19.8, p < .0001), lower education level (χ 2 = 6.4, p = .04) and higher rate of obesity-related comorbidities (χ 2 = 10.3, p < .01), notably OSA (χ 2 = 15.2, p < .001) compared to non-obese subjects. Finally, tobacco consumption was higher in non-obese participants compared to obese subjects (χ 2 = 5.5, p < .05).

When obese and non-obese subjects were stratified based on personal history of MDD, no significant difference was found between subgroups in terms of age, gender and marital status (Supplementary Table,S1). Overall, history of MDD did not affect BMI, education level, obesity onset, comorbidites and tobacco consumption in obese patients. Similar results were obtained in the group of non-obese subjects. Finally, no significant relationship was found between obesity and characteristics of depressive history; the rate of subjects with more than one MDD episode and/or with early MDD onset being comparable between obese patients and non-obese participants (χ 2 = 0.2, p = .67 and χ 2 = 0.1, p = .79).

Inflammation and Neuropsychiatric Symptoms in Obese Patients and Non-Obese Subjects

Circulating levels of hs-CRP (raw and log-transformed) were significantly higher in obese subjects compared to non-obese participants to (t = 9.3, p < .0001) (Table 2).

Consistent with this, BMI strongly correlated with log hs-CRP in the whole population under study (R = .680, p < .0001). No significant association was found between log hs-CRP and obesity onset in the group of obese patients (F(1,63) = 1.3, p = .27). Similar to inflammation, NRS total scores were higher in obese patients compared to non-obese subjects, notably in the dimensions of mood, sickness and motor symptoms (Table 2). Differences remained significant in multivariate analyses controlling for age, gender, education level, tobacco consumption and comorbidities. In line with the increased NRS scores in obese patients, significant associations were found between BMI and total NRS scores (R = .265, p < .05), sickness (R = .372, p < .001) and motor symptoms (R = .332, p < .01) in the whole population under study.

Relationship between History of Depression and Systemic Inflammation

When groups were stratified on the basis of MDD history, no significant difference was found in log hs-CRP between obese patients with history of MDD and obese patients with no personal history of MDD. Similarly, history of MDD did not affect log hs-CRP values in non-obese subjects (Figure 1). Significant differences were only observed between obese versus non-obese subjects, regardless of depressive history. To further assess the contribution of MDD history to hs-CRP concentrations in the population under study, multiple regression analyses, adjusting for age, gender, education level, tobacco consumption and comorbidities were performed, taking into account obesity status. Consistent with results from cross-sectional analyses, when history of MDD and obesity were entered separately in regression models, only obesity significantly predicted log hs-CRP in the whole sample (models 1 & 2). This relationship was not impacted by history of MDD when the two variables were simultaneously entered in the model (model 3). Similarly, the combination of obesity and MDD history did not significantly increase the β value, stretching further the primary contribution of obesity to systemic inflammation. In both obese and non-obese subjects (models 4 & 5), a personal history of MDD did not significantly impact log hs-CRP (Table 3).

Relationship between History of Depression and Neuropsychiatric Symptoms

Overall, obese patients with a personal history of depression exhibited higher NRS scores than obese subjects with no history of depression and the two subgroups of non-obese participants (Figure 2A), notably in the dimensions of mood, sickness, and motor symptoms.

(Figure 2B-F). Interestingly, this effect was modulated by characteristics of depressive history (i.e., number of past depressive episodes) since obese subjects with a history of multiple MDD episodes (N=13) exhibited significantly higher NRS scores in the dimensions of cognitive and sickness symptoms (6.1 vs. 2.6, t = 3.5, p < .01 and 12.8 vs. 9.9, t = 2.3, p < .05 respectively) than obese subject with a history of one depressive episode only (N=21).

Regression analyses were carried out to further assess the contribution of MDD history to NRS scores, with respect to obesity status, in the whole population. When history of MDD and obesity were entered separately in regression models (models 1 & 2), both variables significantly predicted the magnitude of NRS total scores. When the same variables were entered concomitantly in regression models (model 3), their relationship with NRS total scores remained significant. Finally, when the combination of obesity and MDD history was considered in the model (model 4), the β value increased somewhat, suggesting synergetic effect of both factors on NRS scores. When obese and non-obese subjects were considered separately (models 5 & 6), history of MDD significantly predicted NRS scores in the two groups of subjects (Table 4).

Effects of History of Depression on the Relationship between Inflammation and NRS Scores

In the whole population under study, increased log hs-CRP values, as seen in obese subjects, correlated with increased NRS total scores (β = .262, p < .05), and notably with the dimensions of sleep, sickness and motor symptoms (β = .254, p < .05; β = .330, p < .01 and β = .280, p < .01 respectively). While the relationship of log hs-CRP with NRS total score and the NRS dimension of sleep symptoms became more marginal when the analysis controlled for history of MDD (β = .173, p = .08 and β = .203, p = .07 respectively), the associations with the dimensions of sickness and motor symptoms remained significant (β = .246, p < .05 and β = .215, p < .05 respectively), albeit the β coefficients were somewhat reduced. In the same analyses, history of MDD remained associated with all NRS symptom dimensions when controlling for log hs-CRP.

Discussion

Results from the present study comfort the notion that obesity represents a chronic inflammatory condition that is associated with greater neuropsychiatric symptoms, as attested by increased scores on the neurotoxicity rating scale in obese subjects. Findings also indicate that, while history of MDD contributes to increased neuropsychiatric scores, it does not significantly impact obesity-related systemic inflammation. Consistent with this finding, history of MDD was not found to significantly modulate, or only to a minimal extent, the relationship between obesity-related inflammation and neuropsychiatric symptoms. To our knowledge, no study has been previously published focusing on the influence of past MDD on a cohort of patients afflicted with a low-grade inflammatory condition such as obesity and vulnerable to depression but currently free of depressive episode.

Overall, results from this study are consistent with multiple reports indicating higher levels of CRP and neuropsychiatric symptoms in obese subjects [START_REF] Castanon | Neuropsychiatric comorbidity in obesity: Role of inflammatory processes[END_REF][START_REF] Daly | The relationship of C-reactive protein to obesity-related depressive symptoms: a longitudinal study[END_REF][START_REF] Dixon | Raised CRP levels in obese patients: symptoms of depression have an independent positive association[END_REF]. In line with a large literature documenting the role of inflammation in the development of neuropsychiatric symptoms, increased circulating levels of log hs-CRP, as measured in obese subjects, were associated with higher NRS scores in the whole population under study. Peripheral inflammation can lead to neuropsychiatric symptoms through different pathways (humor, neural and cellular) and mechanisms (effects on neurotransmitter metabolism and function, neuroendocrine function, neural plasticity and neurocircuitry) that have been largely reviewed elsewhere [START_REF] Capuron | Immune System to Brain Signaling: Neuropsychopharmachological Implications[END_REF]. Those pathways and mechanisms are likely to be involved in the development of neuropsychiatric comorbidities in obese subjects given the chronic activation of inflammatory processes in obesity. Supporting this scenario, increased log hs-CRP values were associated with greater NRS scores in the population under study. These results are in line with previous findings indicating strong relationship between CRP and mood disorders [START_REF] Azar | Mild depressive symptoms are associated with elevated Creactive protein and proinflammatory cytokine levels during early to midgestation: a prospective pilot study[END_REF][START_REF] Dowlati | A Meta-Analysis of Cytokines in Major Depression[END_REF][START_REF] Felger | Inflammation is associated with decreased functional connectivity within corticostriatal reward circuitry in depression[END_REF][START_REF] Haapakoski | Cumulative meta-analysis of interleukins 6 and 1β, tumour necrosis factor α and C-reactive protein in patients with major depressive disorder[END_REF]. Moreover, the associations found between log hs-CRP and specific symptom dimensions, including sleep, sickness and motor symptoms, corroborate reports documenting greatest correlations with the neurovegetative features of depression [START_REF] Jokela | Inflammation and Specific Symptoms of Depression[END_REF].

In the present study, lifetime diagnosis of MDD was associated with more intense neuropsychiatric symptoms in the whole population under study, regardless of obesity.

Interestingly, in obese subjects, history of depression was associated with even worse neuropsychiatric outcomes, as indicated by higher NRS scores in comparison to obese subjects with no history of depressive episode. Moreover, obese subjects with history of multiple episodes exhibited more severe cognitive and sickness symptoms than obese individuals with history of just one depressive episode. Multiple reports have identified history of depression as one of the main risk factor for the development of neuropsychiatric alterations, including depression and cognitive impairment [START_REF] Byers | Depression and risk of developing dementia[END_REF][START_REF] Kessing | The effect of episodes on recurrence in affective disorder: A case register study[END_REF][START_REF] Pereira-Morales | Network analysis of multiple risk factors for mental health in young Colombian adults[END_REF]. In a recent longitudinal study, history of depression was associated with a 21% prevalence rate of depressive disorders at 12 months against 4.4% in persons with no history of MDD [START_REF] Markkula | Predictors of new-onset depressive disorders -Results from the longitudinal Finnish Health 2011 Study[END_REF]. Interestingly, this prevalence increased up to 27% in persons with history of dysthymia. While the present study, because of its cross-sectional design, did not allow performing predictive longitudinal analyses, the relationship between history of depression and current neuropsychiatric symptoms in a susceptible group of individuals (i.e., obese subjects) can be interpreted as supporting evidence for an increased neuropsychiatric vulnerability. This hypothesis is supported by a large database documenting a greater risk of depression in obese individuals (de [START_REF] De Wit | Depression and obesity: A meta-analysis of community-based studies[END_REF] and by reports indicating that individuals with subclinical depressive symptoms are at higher risk for developing later depression [START_REF] Markkula | Predictors of new-onset depressive disorders -Results from the longitudinal Finnish Health 2011 Study[END_REF] and that history of MDD increases the risk of recurrence [START_REF] Bulloch | Recurrence of major depressive episodes is strongly dependent on the number of previous episodes[END_REF]. Alternatively, the possibility of residual depressive symptoms after remission can be considered given that a substantial proportion of patients who recovered from depression do not achieve a complete functional remission due to the persistence of sub-threshold symptoms [START_REF] Stotland | Recovery from depression[END_REF]. Low mood, insomnia and somatic complaints have been described as common residual symptoms of depression [START_REF] Israel | The impact of residual symptoms in major depression[END_REF]. Interestingly, residual and subthreshold symptoms after remission have been shown to be predictive of recurrence and relapses [START_REF] Hardeveld | Prevalence and predictors of recurrence of major depressive disorder in the adult population[END_REF][START_REF] Richards | Prevalence and clinical course of depression: A review[END_REF], stretching further the notion of increased vulnerability.

Whether they reflect or not an increased neuropsychiatric vulnerability, the higher NRS scores measured in obese patients with MDD history compared to obese patients with no MDD history and non-obese subjects, suggest the role of underpinning factors and/or potential synergetic effects between those factors and history of depression. The strong associations found between log hs-CRP and NRS scores, notably with the symptom dimensions that are particularly increased in obese subjects with MDD history, and the finding that those associations remain significant when controlling for history of depression, highly support the role of inflammation as one potential synergetic factor. It is thus highly probable that chronic systemic inflammation in obese subjects contributes to exacerbate vulnerability to neuropsychiatric disorders, in particular in subjects with a depressive history. This is in line with a large literature indicating that inflammation interacts with vulnerability factors to promote the development of clinically relevant neuropsychiatric disorders, including depression [START_REF] Capuron | Role of Adiposity-Driven Inflammation in Depressive Morbidity[END_REF].

Albeit the present data are supporting the notion that history of depression interacts with inflammation to influence neuropsychiatric status and vulnerability, they failed to show any predictive association of MDD history with obesity-related inflammation. In multivariate regression analyses, only obesity was associated with log hs-CRP, and this regardless of lifetime history of MDD. Moreover, history of MDD did not modulate the relationship between obesity and systemic inflammation. These findings indicate that the likely synergetic interaction between MDD history and obesity-driven inflammation, resulting in increased neuropsychiatric symptoms, does not rely on a mediatory effect of MDD history on systemic inflammation. A possible explanation is that history of MDD, which may be potentially associated with a subtle and mild immune activation, is not sufficient enough to influence the chronic, robust and well-installed inflammatory state that characterizes obesity. This scenario is supported by the results showing that recurrent depressive episodes did not influence hs-CRP concentrations in obese individuals.

Taken together, these results add valuable information on the risk factors (in particular related to depressive history) that may help identifying patients at increased risk for neuropsychiatric disorders in conditions of greater susceptibility, such as obesity.

Nevertheless, several limitations of the current study must be pointed out and need to be considered for future investigations. First, the cross-sectional design of this study reduces the possibilities of inference. The absence of longitudinal assessments does not allow performing predictive analyses. Complementary studies using longitudinal designs in larger cohorts of obese subjects are needed to address this issue. Second, restricted inclusion criteria were used as only obese subjects with BMI > 35 kg/m 2 and free of current depression were selected. This limitation warrants further investigations including more heterogeneous profiles of obese subjects so as to extend the present findings. Third, only one inflammatory marker, i.e. CRP, was assessed in the study. Thought CRP is considered one of the most sensitive indicators of inflammation related to neuropsychiatric alterations [START_REF] Kohler | Peripheral cytokine and chemokine alterations in depression:a meta-analysis of 82 studies[END_REF][START_REF] Raison | A randomized controlled trial of the Tumor Necrosis Factor-alpha antagonist infliximab in treatment resistant depression: role of baseline inflammatory biomarkers[END_REF], the use of other markers such as interleukin-6 or tumor necrosis factor-α should be considered in future studies. It could be also interesting to assess other biomarkers that contribute to maintain of inflammation, such as oxidative and nitrosative stress markers [START_REF] Moylan | The neuroprogressive nature of major depressive disorder: pathways to disease evolution and resistance, and therapeutic implications[END_REF][START_REF] Oriolo | Pathways Driving Neuroprogression in Depression: The Role of Immune Activation[END_REF]. Fourth, the sample size was relatively small, a limitation that may have contributed to reduce the statistical power of this study. Finally, the high prevalence of women in our sample may represent another limitation. Albeit the results were systematically controlled for gender, the fact that women were more represented in the present study may also reduce the generalizability of the results, given recent evidence on sexual differences in stress-related neuroimmune effects or behavioral responses to inflammatory challenges [START_REF] Bekhbat | Sex differences in the neuro-immune consequences of stress: Focus on depression and anxiety[END_REF][START_REF] Moieni | Sex differences in depressive and socioemotional responses to an inflammatory challenge: Implications for sex differences in depression[END_REF].

Conclusions

In conclusion, results from this study comfort the notion that chronic systemic inflammation represents a major contributor of neuropsychiatric symptoms and neuropsychiatric vulnerability in obesity. In obese subjects exempt of current diagnosis of MDD, history of depression worsened neuropsychiatric outcomes, but had no effect on hs-CRP concentrations. Similarly, the relationship of systemic inflammation with neuropsychiatric scores was not influenced, or only minimally, by history of MDD. These findings provide relevant information on the risk factors and mechanisms that may help identifying obese subjects who are at increased risk of neuropsychiatric comorbitity, in order to improve preventive strategies. Further research is needed, using large and longitudinal designs and assessing complementary inflammatory markers, to confirm causal relationships between vulnerability markers and disease development. Abbreviations: MDD = Major Depressive Disorder; Log hs-CRP = high-sensitivity C-reactive protein (log-transformed)

Figure 1 .Figure 2 .

 12 Figure 1. Log Hs-CRP in obese and non-obese subjects stratified on the basis of personal

Table 1 : Characteristics of study participants

 1 Analyses were performed using Student T-tests for continuous variables and Chi-square test for categorical variables. Any comorbities correspond to the presence of at least one comorbidity among diabetes, hypertension, dysthyroidism and obstructive sleep apnea.

	Abbreviations: BMI = Body Mass Index; SD = Standard Deviation.

Table 2 : Hs-CRP concentrations and NRS scores in obese and non-obese subjects

 2 Analyses were performed using Student T-tests for continuous variables.

	Obese patients	Non-obese	p
		participants

Abbreviations: Hs-CRP = High-sensitivity C-reactive protein; NRS = Neurotoxicity Rating Scale.

Table 3 : Respective contribution of history of MDD and obesity to circulating levels of hs-CRP Log hs-CRP

 3

	Predictors	β	F	P
	Model 1 (all subjects)
	History of MDD	.183	3.1	.08
	Model 2 (all subjects)
	Obesity	.786	85.5	< .0001
	Model 3 (all subjects)
	Obesity (controlling for history of MDD)	.772	81.2	< .0001
	History of MDD (controlling for obesity)	.090	1.4	.23
	Model 4 (all subjects)
	Obesity + MDD history	.385	15.1	< .001
	Model 5 (obese subjects only)
	History of MDD	.108	0.9	.36
	Model 6 (non-obese subjects only)
	History of MDD	.222	0.8	.39
	Analyses were performed using multivariate regression analysis. All models were
	systematically adjusted for age, gender, educational level, tobacco consumption and any
	comorbidities.

Acknowledgments:

The authors thank the patients and volunteers for their participation in the study and Dr J.J. Peroua, Dr M. Salzmann-Lovato, and the medical assistants for their help with recruitment.

Funding: This work was supported by funds from the French National Institute for Agricultural Research (INRA), French National Research Agency, ANR (ANR-11-JSV1-0006, Dr Capuron), and Fondation pour la Recherche Medicale (DPP20151033943, Dr Capuron).

Conflict of interest: None


L Iglesias

M T Boudjiet

I Dufour

Discrimination and Concentration Measurement of Different Binary Gas Mixtures with a Simple Resonator through Viscosity and Mass Density Measurements

Keywords: Gas Sensing, Gas Discrimination, MEMS, Viscosity, Density, Uncoated Cantilever

Compared to physical microsensors based on the measurement of physical properties of gases, chemical microsensors are attractive for detection of gas due to their selectivity and relatively high sensitivity. However, most of the time they present a considerable drift in their output which either requires recalibration or makes them unsuitable for several applications. Physical gas microsensors used for chemical detection on the other hand lack of selectivity which makes them unsuitable for applications where more than one gas concentration is susceptible to change. In this paper, a new method is exposed, that allows physical microsensors to discriminate different binary gas mixtures in applications such as industrial gas monitoring, where the viscosity and mass density of the gases prone to manifest are known. The method is based on the measurement of both the resonant frequency and the quality factor in microresonators. The small variations of such signals are interpreted in terms of viscosity and mass density variations with respect to a reference gas such as nitrogen (N 2 ). This method allows the discrimination of the gases mixtures as well as the determination of the gas concentration.

Introduction

Stability and long time reliability are two well known limitations regarding chemical gas micro sensors [START_REF] Korotcenkov | Instability of metal oxide-based conductometric gas sensors and approaches to stability improvement[END_REF][START_REF] Romain | Three years experiment with the same tin oxide sensor arrays for the identification of malodorous sources in the environment[END_REF][START_REF] Romain | Long term stability of metal oxide-based gas sensors for e-nose environmental applications[END_REF]. Although metal oxide based (MOX) sensors, according to Romain et al. [START_REF] Romain | Three years experiment with the same tin oxide sensor arrays for the identification of malodorous sources in the environment[END_REF][START_REF] Romain | Long term stability of metal oxide-based gas sensors for e-nose environmental applications[END_REF], have relatively good stability, they typically require either a recalibration, or to be replaced after a few weeks or a couple of months due for instance to aging [START_REF] Sharma | Investigation of stability and reliability of tin oxide thin-film for integrated micro-machined gas sensor devices[END_REF] of the sensitive film. A lot of effort has been put into making chemical 5 sensors more stable over time such as improving the materials used [START_REF] Göpel | New materials and transducers for chemical sensors[END_REF][START_REF] Yamazoe | New approaches for improving semiconductor gas sensors[END_REF]. Some researchers even proposed to apply statistical models [START_REF] Padilla | Drift compensation of gas sensor array data by orthogonal signal correction[END_REF] or even machine learning algorithms like support vector machines [START_REF] Vergara | Chemical gas sensor drift compensation using classifier ensembles[END_REF] and artificial neural networks [START_REF] Hossein-Babaei | Compensation for the drift-like terms caused by environmental fluctuations in the responses of chemoresistive gas[END_REF] to compensate for the drift. This, naturally, requires a fairly important amount of computation and processing. Another approach is to simply not use a sensitive film at all which has resulted in an increasing interest towards using uncoated resonators for chemical detection [START_REF] Rosario | Piezoelectric excited millimeter sized cantilever sensors for measuring gas density[END_REF][START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical[END_REF][START_REF] Boudjiet | Preliminary results of the feasibility of hydrogen detection by the use of uncoated silicon microcantilever-based sensors[END_REF][START_REF] Boudjiet | Geometry optimization of uncoated silicon microcantilever-based gas density[END_REF]. [START_REF] Rosario | Piezoelectric excited millimeter sized cantilever sensors for measuring gas density[END_REF] Tétin et al. [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical[END_REF] showed that the shift in the resonant frequency of an uncoated cantilever can be associated to both the density and the viscosity of its surrounding fluid. This allowed for measurements of molar fractions for several gases such as carbon dioxide (CO 2 ) [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical[END_REF] and hydrogen (H 2 ) [START_REF] Boudjiet | Preliminary results of the feasibility of hydrogen detection by the use of uncoated silicon microcantilever-based sensors[END_REF][START_REF] Boudjiet | Geometry optimization of uncoated silicon microcantilever-based gas density[END_REF].

Nevertheless, one of the difficulties of this approach is the lack of selectivity resulting from removing the sensor's sensitive film which strongly limits its applications. Indeed, every chemical specie is susceptible of 15 changing the physical properties of the sensor environment which makes it difficult to differentiate between different species.

Nonetheless, as presented in Table 1, measuring particular properties of the surrounding gas such as viscosity (η) or mass density (ρ) is most certainly a first step towards discriminating different chemical entities. The physical properties of some common industrial gases can vary considerably from one gas to another. Although in the following parts we will focus on measuring both the viscosity and the mass density; the thermal conductivity and the speed of sound could potentially improve the uniqueness of a gas signature to identify it. There exist several ways to measure a fluid's viscosity and mass density simultaneously [START_REF] Bircher | Real-time viscosity and mass density sensors requiring microliter sample volume based on nanomechanical resonators[END_REF][START_REF] Igarashi | Development of simultaneous measurement system for instantaneous density[END_REF][START_REF] Lötters | Integrated multi-parameter flow measurement system[END_REF][START_REF] Seibt | Simultaneous measurements on helium and nitrogen with a newly designed viscometerdensimeter over a wide range of temperature and pressure[END_REF] although very often it consists of a system of two sensors such as a flow and a pressure sensor. Integrated versions are rare but they can be found in the literature [START_REF] Lötters | Integrated multi-parameter flow measurement system[END_REF]. Resonators also have been used to measure both mass density and viscosity [START_REF] Bircher | Real-time viscosity and mass density sensors requiring microliter sample volume based on nanomechanical resonators[END_REF] but due to the small viscosity of gases at atmospheric pressure some techniques are useful for liquids only. Despite this challenge Lötters et al. managed to measure simultaneously density, viscosity, heat capacity and mass flow of several gases included H 2 by integrating a pressure, a thermal and a Coriolis microsensor [START_REF] Lötters | Integrated multi-parameter flow measurement system[END_REF]. In this article we present a method that not only can measure both quantities from a single microresonator for a given gas mixture, but it can also discriminate from different binary gas mixtures and estimate the gas concentration.

In this paper the idea of using both the viscosity and the mass density of a binary gas mixture to discriminate different gases and to estimate their concentration is presented (Section 2). Then the method to determine both the viscosity and the mass density of a gas from the measurement of both the resonant frequency and the quality factor of an uncoated microresonator is exposed (Section 3). The silicon microcantilever and the measurement equipments used to test the proposed method are presented in Section 4 and the obtained results are presented in Section 5. Finally, because the method used a calibration step, the impact of the choice of gases and concentrations for the calibration is studied (Section 6).

Motivation

Figure 1 shows the theoretical mass density difference (∆ρ) against the theoretical viscosity difference (∆η), both with respect to nitrogen, for different molar fractions in percent and for different gases.

The mixture mass density was computed using Equation 1 which comes from the definition of the mass density:

ρ G-N2 = ρ G x G + (1 -x G )ρ N2 , (1)
where x G , ρ G , ρ N2 and ρ G-N2 are the molar fraction of the gas, G, present in a small amount, its mass density, the mass density of nitrogen and the one of the mixture, respectively. The mixture viscosity was computed using the approximated equation from the kinetic gas theory [START_REF] Sutherland | [END_REF]19,[START_REF] Wilke | A viscosity equation for gas mixtures[END_REF]:

η G-N2 = x G η G x G + (1 -x G )Φ G-N2 + (1 -x G )η N2 1 + (Φ N2-G -1)x G , (2)
where η G , η N2 and η G-N2 refer to the viscosity of the gas present in a small amount, nitrogen and the mixture, respectively. Φ G-N2 and Φ N2-G are given in Equations 3 and 4, respectively.

Φ G-N2 = [1 + ( η G η N 2 ) 1 2 ( M N 2 M G ) 1 4 ] 2 [8(1 + M G M N 2 )] 1 2 , (3)
Φ N2-G = [1 + ( η N 2 η G ) 1 2 ( M G M N 2 ) 1 4 ] 2 [8(1 + M N 2 M G )] 1 2 , (4)
where M N2 and M G are the molar weight of nitrogen and of the gas present in a small amount, respectively. It can be noticed that in Figure 1 some gases with similar mass density variations for a given molar fraction such as hydrogen and helium can differ considerably in their viscosity variation. Similarly, gases with close viscosity variations for a given molar fraction such as oxygen and helium can be differentiated by their mass density variations. There is, however, an additional advantage of measuring both variables ∆ρ and ∆η. As illustrated in Figure 1 and Table 2, the slope of the curves of Figure 1 (

η N 2 ρ N 2 ∆ρ ∆η
) is a good parameter to discriminate between different gases. Finally, once the gas mixture is determined, its molar fraction is given either by ∆ρ or ∆η depending on which signal is easier to detect. For instance, as shown in Table 2, the sensitivity to mass density ( ∆ρ

x G ρ N 2
) for H 2 is much higher than the one to viscosity ( ∆η

x G η N 2
).

Model

In Sader's theory [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF][START_REF] Chon | Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids[END_REF], in the case of a beam respecting Euler-Bernoulli conditions (L >> b >> h where L, b, and h are the length, width and thickness of the cantilever, respectively), it has been shown [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical[END_REF] that for small variations ∆ρ and ∆η of mass density and viscosity respectively, these quantities are related to the shift in resonant frequency ∆f r as described by Equation 5.

∆f r f r = - π 8 ρ ρ beam b h (a 1 ∆ρ ρ + a 2 8Re(f r ) ∆η η ), (5)
where ρ beam is the mass density of the beam's material, a 1 = 1.0553, a 2 = 3.7997 are Maali's [START_REF] Maali | [END_REF] parameters of the real part of the hydrodynamic function and Re(f r ) is the Reynolds' number at the resonant frequency defined in Equation 6:

Re(f r ) = πb 2 ρf r 4η . (6)
Under the same assumptions we get a similar expression for the quality factor Q shown in Equation 7:

∆Q Q = - 1 2 ( ∆ρ ρ + ∆η η ). (7)
Despite the fact that Equations 5 and 7 are valid for a very specific case (Euler-Bernoulli conditions), they are useful to give an insight in the parameters that play a role in the relative weight between ∆ρ and ∆η in both, the resonant frequency and the quality factor of a resonator.

In order to extract ∆ρ and ∆η in a more general case in terms of geometry (even if the Euler-Bernouilli conditions are not valid), we will assume that a first order approximation remains valid for small variations. Thus, ∆f r and ∆Q can be written as in Equations 8 and 9 where α 1 , α 2 , β 1 and β 2 can be considered constant since they depend on the geometry and technology of the resonator and on the predominant gas (N 2 in our case) but not on the gas present in a small amount:

∆f r f r = α 1 ∆ρ ρ + β 1 ∆η η , (8)
∆Q Q = α 2 ∆ρ ρ + β 2 ∆η η . (9)
Therefore if we are able to obtain such constants through calibration, then ∆ρ and ∆η are simply given by Equations 10 and 11, respectively:

∆ρ ρ = 1 α 1 β 2 -α 2 β 1 (β 2 ∆f r f r -β 1 ∆Q Q ), ( 10
) ∆η η = 1 α 1 β 2 -α 2 β 1 (α 1 ∆Q Q -α 2 ∆f r f r ). (11)
In order to get the four parameters (α 1 , α 2 , β 1 and β 2 ) it is needed to measure ∆f r and ∆Q, for two different binary mixtures (nitrogen and either gas A or B) at a given concentration for each gas mixture (the corresponding ∆ρ and ∆η being determined using Equations 1 and 2). The system described by Equations 12 to 15 is then solved where the subscripts 'A' and 'B' designate the quantities of gases A and B respectively:

∆f r,A f r = α 1 ∆ρ A ρ + β 1 ∆η A η , ( 12
) ∆f r,B f r = α 1 ∆ρ B ρ + β 1 ∆η B η , (13)
∆Q A Q = α 2 ∆ρ A ρ + β 2 ∆η A η , (14)
∆Q B Q = α 2 ∆ρ B ρ + β 2 ∆η B η . ( 15
)
Which finally leads to the following equations to determine the four needed parameters:

α 1 = ρ f r ∆f r,A ∆η B -∆f r,B ∆η A ∆ρ A ∆η B -∆ρ B ∆η A , (16)
β 1 = η f r ∆f r,B ∆ρ A -∆f r,A ∆ρ B ∆ρ A ∆η B -∆ρ B ∆η A , (17)
α 2 = ρ Q ∆Q A ∆η B -∆Q B ∆η A ∆ρ A ∆η B -∆ρ B ∆η A , (18)
β 2 = η Q ∆Q B ∆ρ A -∆Q A ∆ρ B ∆ρ A ∆η B -∆ρ B ∆η A . (19)
This model fails when the influence of either the viscosity or the mass density is negligible in both the quality factor and the resonant frequency since the system becomes unsolvable. Typically one would only be able to measure the mass density as the sensors become smaller since the influence of the viscosity tends to decrease due to high resonant frequency and low quality factor.

Sensor Description

In order to validate the proposed method, a sensor not respecting the Euler Bernoulli condition b << L has been chosen. This choice has been motivated in order to prove that even if the Equations 5 and 7 are not valid, the Equations 8 and 9 can be used (in the general case the expression of the parameters α 1 , α 2 , β 1 and β 2 as functions of the geometrical parameters and material properties of the resonant MEMS are different). For this study the sensor used had the following geometry: h = 10 µm, b = L = 1 mm. Figure 2a shows a picture of it. The details of the sensor manufacturing process, proof of its reversibility along with other details can be found in reference [START_REF] Boudjiet | Geometry optimization of uncoated silicon microcantilever-based gas density[END_REF].

The working principle of both the actuation and the readout is illustrated in Figure 2b. An alternative current generated by an Agilent E5061B network analyzer passes through a conductive path located close to the edge of the top surface of the cantilever. In presence of a magnetic field #» B, it induces a Lorentz force #» F that makes the beam vibrate orthogonally to its plane (out-of-plane vibration). This was chosen, as opposed to in-plane vibration, because it strengthens the fluid/structure interaction [START_REF] Dufour | Effect of hydrodynamic force on microcantilever vibrations: Applications to liquid-phase chemical sensing[END_REF] which is the basis of the principle of uncoated sensors. A first piezo-resistor then senses the strain close to the clamped part of the beam and another one is used as a reference to compensate for common mode variations such as a deformation of the piezo-resistor due to temperature. The readout is the signal at the middle point of the voltage divider bridge formed by the two piezo-resistors. This signal is then fed to the network analyzer to extract the measured phase (φ) spectrum (phase difference between the actuation signal and the readout signal). Finally a linear fit (φ f it ) of the phase close to the resonant frequency was applied to the phase curves as shown in Equation 20:

φ f it = c 0 + c 1 f, (20)
where φ f it is in radians, c 0 is its bias and c 1 its slope. This was done in order to extract both the resonant frequency and the quality factor as a function of time. Indeed, as explained by Boudjiet et al. [START_REF] Boudjiet | Geometry optimization of uncoated silicon microcantilever-based gas density[END_REF], the equations to extract f r and Q from such fit (φ f it ) are Equations 21 and 22, respectively:

f r = - π 2 + c 0 c 1 , (21)
Q = - c 1 2 f r . (22)
5. Experiment

Measurements

With the previously described setup four gases (H 2 , He, CO 2 and CH 4 ) were used to validate the method.

The microcantilevers are placed in a small gas chamber (total volume: 500 µL) under a controlled gas flow (100 cm 3 • min -1 ). Gas streams containing binary mixtures of the desired species (hydrogen or helium or carbon dioxide or methane) and nitrogen are fed to the chamber using bottles of gas (5% H 2 /95% N 2 or 100% He or 100% CO 2 or 10% CH 4 /90% N 2 ) and a set of mass-flow controllers (Brooks 5850S). They were introduced in three different concentrations 5%, 4% and 3% for 400 seconds each. N 2 was alternated in between each gas mixture for 400 seconds in order to compensate, if needed, for any considerable drift and to show the reversibility of the sensor. Figures 3a and3b show respectively the resonant frequency and quality factor shifts for this experiment where the resonant frequency (f r,N2 ) and the quality factor in nitrogen (Q N2 ) are respectively 23.268 kHz and 500.

It can be noticed that for all of the four gases there is an associated shift in resonant frequency and quality factor (of roughly 100 ppm for ∆f r and 4000 ppm for ∆Q). Although studying this sensor's limit of detection is out of the scope of this article, it can be noticed that measuring smaller concentrations could be done with ease. In addition these curves are consistent with the full reversibility and fast response time (less than 1 min for 5% of H 2 -N 2 ) of uncoated cantilevers in gas sensing with respect to typical sensors with sensitive film which is usually between 1 min and 10 min. The limiting factor in the response time in our experiment is the time that the gas takes to reach the sensor but one could potentially go below 1 min with a faster gas delivery. Moreover, there is nearly no drift (0.15Hz for ∆f r and 0.1 for ∆Q during 3 hours).

The reproducibility of the measurements of both the resonant frequency and quality factor is not shown in the present paper but has already been shown in [START_REF] Boudjiet | Geometry optimization of uncoated silicon microcantilever-based gas density[END_REF]. Finally, we notice that the trends in ∆f r and ∆Q are very similar for H 2 , He and CH 4 which are all lighter gases than N 2 . This could make it potentially difficult to differentiate each one of them by looking only at ∆f r and ∆Q. In such case, transforming the data to obtain the density and viscosity of each mixture can be quite useful, as presented in the next section.

Results

In order to test the proposed method, the model calibration was first done by using the mean value of the steps corresponding to CH 4 and He both at 5%. This allowed us to obtain the four parameters

α 1 = -6.1x10 -3 , β 1 = -1.28x10 -4 , α 2 = -4.4x10 -1
, and β 2 = -1.6x10 -1 . The linear transform from Equations 10 and 11 was then applied to the measured ∆f r and ∆Q presented in Figures 3a and3b. The resulting mass density and viscosity shifts are shown in Figures 4a and4b.

We notice that, in one hand, the computed mass density varies roughly in the opposite way as the resonant frequency which is explained by the fact that |α 1 | |β 1 | and α 1 < 0. On the other hand, the computed viscosity varies in a completely different way than the resonant frequency or the quality factor. Moreover it can be seen that, despite the relatively low drift and noise in ∆f r and ∆Q with respect to the concentration jumps, in the case of the viscosity the noise and drift are much more noticeable. Finally, the data points surrounded by a dashed box in Figures 4a and4b were plot against each other along with the theory from Figure 1 for comparison in Figure 5. In this case, the small drifts were linearly compensated both in the density and the viscosity. For demonstration purposes, the angle bisector between two consecutive theoretical curves was used as frontier between the domains to assign one data point to a gas mixture but this choice might need refining for other applications depending on their context. The slope of such domains frontiers are shown in Table 3.

It can be noticed that, although no information about H 2 or CO 2 was given to the model (they were not used for the calibration) other than their corresponding ∆f r and ∆Q, the experimental points are remarkably close to their theoretical values which validates the model from Section 3. The transform is, naturally, more accurate with the gases used for calibration (CH 4 and He in this case). It can also be seen that the different gases are relatively well distinguishable from one another apart from a few extreme points corresponding to He that end up closer to the theoretical values for H 2 . If we apply the method suggested in Section 2 to the mean values of each concentration step (dark red pentagons in Figure 5), we obtain the results shown in Table 4 where the first Columns 1 and 2 describe the binary mixture injected in the chamber, Column 7 describe the prediction of the binary gas mixture based on the value of the slope (Column 5) and the sign of ∆ρ (Column 6), The column 8 is the estimation of the gas concentration using the variation of the mass density (Column 3).

It can be seen that this method allows a very good discrimination of the different binary mixtures. In addition, it can be noticed that we are able to predict the concentration of the mixture as well (in this case less than 3% of error for three of the mixtures and a maximum of 12% of error for CO 2 ). Nevertheless it seems that how accurate the results are depends on how similar a gas-mixture is from the gases used in the calibration. Indeed, since H 2 is more similar to He than CO 2 is to either CH 4 or He, a better prediction of the concentration of H 2 than of CO 2 is obtained.

Optimizing the Calibration Procedure

In order to understand better the robustness of the proposed method we developed a simulator that, from the theoretical values of both the viscosity and the mass density for the different gases as well as from the theoretical parameters α 1 , α 2 , β 1 and β 2 from Section 3 (assuming a cantilever respecting Euler-Bernouilli conditions), generates the associated shifts pair ∆f r and ∆Q as a function of time. A white Gaussian noise is then added to the model. Since the goal in this section is to optimize the calibration procedure, the mean distance between a simulated data point and its theoretical value (Equation 23) is used as error metric:

= 1 N N i=1 ∆ρ exp i -∆ρ th i ρ N2 2 + ∆η exp i -∆η th i η N2 2 , ( 23
)
where N is the total number of simulated data points. To limit biasing the error computation and covering more parts of the density viscosity plane, two more gases, CO and O 2 , were added to the simulation.

Calibration Concentration

The objective of this part is to determine weather it is better to calibrate with a high concentration or a low concentration (keeping in mind that the concentrations must remain small for the model to hold). To test this, several simulations were ran calibrating with He and CH 4 at different concentration pairs. The resulting error values are shown in Figure 6.

It can be noticed that the mean error decreases as higher concentrations to calibrate are used. Therefore, the calibration should be done with concentrations as big as possible as long as the small variation assumption holds.

Choice of the Two Gas Mixtures

Having studied the calibration dependency on the concentration, there is left to test which pair of gas mixtures is more appropriate for the calibration. In order to do so, several simulations were ran calibrating this time with each pair of gas mixtures at a fixed distance from the origin. The points used for calibration in this set of simulations are marked by a red star in Figure 7-a. For all the possible pair of points (corresponding to the gas pair for calibration) the corresponding error was plotted against the angle θ between the theoretical curves (Figure 7-b) of the corresponding mixtures. As an example, the angle between the curves of the mixtures CO 2 -N 2 and CH 4 -N 2 is shown in Figure 7-a.

It can be noticed (Figure 7-b) that most pairs of mixtures used to calibrate have an error below 0.4 except for the combinations He/H 2 , He/CO 2 and O 2 /CH 4 . These three combinations, as shown in Figures 7 a andb, correspond to the curves that are almost aligned (θ close to either 0 • or 180 • ). This allows to conclude that the closer to collinearity the curves of the mixtures used for the calibration are, the worse the performance of the method is. This can be explained by the fact that, in order to compute the four parameters α 1 , α 2 , β 1 and β 2 , it is required to divide by ∆ρ A ∆η B -∆ρ B ∆η A (Equations 16 to 19). However, it can be shown that for two mixtures A and B for which the curves are nearly collinear, this value is close to zero which causes that a small variation due to the noise can lead to a significantly different value of the parameters.

Conclusion

A simple method to extract both the gas mass density and the gas viscosity from the measurement of both the quality factor and the resonant frequency of a microresonator without sensitive coating has been presented. In a gas of reference (which can be a mixture of gas and which is nitrogen in the presented measurements) if one gas appears it is possible with the determination of the two physical properties of gas (mass density and viscosity), to know which gas has appeared and the concentration of this gas. The proposed method can be used in a mixture of three or more mixture of gases only if it is known that only one concentration of the gas is modified the proportion of the other ones staying stable. This could be useful in some industries to detect gas leakages for example. The main advantages of such method are: (1) the smaller time response compared to the one of chemical sensors using sensitive layer, (2) the possibility to discriminate between different gases and to estimated the gas concentration with only one sensor, (3) the aging and reversibility of such sensors are very good due to the absence of sensitive coating. The presented method has been validated with four different gas mixtures using a silicon microcantilever. The discrimination of the four gases has been achieved and the estimation of the gas concentration has been obtained with less than 3% of error for three gases and less than 12% for the last one. This has been performed using a calibration step with two gases at a fixed concentration. The impact of this calibration step has been studied and a general way to choose the gases and the concentration for the calibration has been established.

Figures and

Figure 1 :

 1 Figure 1: Relative mass density variation vs. relative viscosity variation with respect to N 2 for different gases and different concentrations at 25 • C and atmospheric pressure

Figure 2 :

 2 Figure 2: Silicon electromagnetically actuated sensor with piezo-resistive readout used in this study: (a) picture (L = 1 mm, b = 1 mm, h = 10 µm), (b) schematic illustrating both the actuation and the read-out

Figure 4 : 10 -

 410 Figure 4: Computed shifts for different gas mixtures at different concentrations (5%, 4% and 3%) where α 1 = 6.1x10 -3 , β 1 = 1.28x10 -4 , α 2 = 4.4x10 -1 , β 2 = 1.6x10 -1

Figure 5 :

 5 Figure 5: Simultaneous measurements of both the mass density and the viscosity using the data presented in Figures 4a and 4b

Figure 6 :

 6 Figure 6: Mean error (equation 23) for different concentration pairs using CH 4 and He to calibrate

Figure 7 :

 7 Figure7: a: Explanation of the calibration at a fixed distance from the origin and definition of θ, the angle between the theoretical density-viscosity curves for two mixtures. b: Mean error of a calibration at a fixed distance from the origin as a function of the angle between the theoretical density-viscosity curves as shown in Figure7-a of the pair of gas mixtures used in the calibration

Table 1 :

 1 Physical properties of common gases[START_REF] Zuckerwar | Handbook of the Speed of Sound in Real Gases[END_REF] 26,[START_REF]Gas encyclopedia air liquide[END_REF]

	Gas Mass Density	Viscosity	Thermal Conductivity Speed of Sound
		(kg • m -3 )	(µPa • s)	(mW • m -1 • K -1 )	m • s -1
		1.015 bar, 15 • C	1.015 bar, 0 • C	1.015 bar, 0 • C	1.015 bar, 20 • C
	CO 2	1.87	13.7	14.7	267
	O 2	1.35	19.1	24.4	326
	Air	1.22	17.2	24.4	343
	CO	1.18	16.5	24.7	336
	N 2	1.18	16.6	24.0	349
	CH 4	0.68	10.1	30.6	446
	He	0.17	18.7	146.2	1007
	H 2	0.09	8.4	176.2	1270

Table 2 :

 2 Theoretical values of the slope of the curves of Figure1and sensitivities of both density and viscosity for different gases mixtures

	Gas Mixture	∆ρ ρN 2 (x1) ηN 2 ∆η	∆ρ xGηN 2 (x10 -3 /%) (x10 -3 /%) ∆η xGηN 2	Sign of ∆ρ
	CO 2 -N 2	-2.94	5.6	1.9	+
	O 2 -N 2	0.872	1.39	1.6	+
	He-N 2	-5.32	8.4	1.6	-
	CO-N 2	-0.0027	0.001	0.38	-
	CH 4 -N 2	0.99	4.2	4.2	-
	H 2 -N 2	80.7	9.1	0.11	-

Table 3 :

 3 Value of the slope of the frontier between domains Frontier H 2 /He CO 2 /CH 4 He/CO 2 CH 4 /H 2

	Slope (x1)	-11.5	-0.236	0.263	2.3560

Table 4 :

 4 Summary of the method presented in Section 2 applied to the mean values of viscosity and density for each concentration (dark red pentagons in Figure5)

	Used Mixture		Measured		Predicted
	Gases	x G (%) (x10 -2 %) (x10 -2 %) ∆ρ ρN 2 ∆η ηN 2	Slope Sign of ∆ρ (x1)	Gases	x G Using ∆ρ ρN 2 (%)
		3	1.54	-0.322	-4.78	+	N 2 -CO 2	2.8
	N 2 -CO 2	4	2.07	-0.480	-4.31	+	N 2 -CO 2	3.7
		5	2.45	-0.545	-4.5	+	N 2 -CO 2	4.4
		3	-1.25	-1.35	0.926	-	N 2 -CH 4	3.0
	N 2 -CH 4	4	-1.66	-1.68	0.988	-	N 2 -CH 4	4.0
		5	-2.10	-2.09	1.00	-	N 2 -CH 4	5.0
		3	-2.64	0.546	-4.83	-	N 2 -He	3.1
	N 2 -He	4	-3.43	0.517	-6.63	-	N 2 -He	4.0
		5	-4.15	0.667	-6.22	-	N 2 -He	5.0
		3	-2.66	-0.152	17.5	-	N 2 -H 2	2.9
	N 2 -H 2	4	-3.57	-0.299	11.9	-	N 2 -H 2	4.0
		5	-4.48	-0.707	6.33	-	N 2 -H 2	5.0

Acknowledgement

This research was done thanks to the contest Programme d'Investissements d'Avenir of the French Goverment under the supervision of the French National Radioactive Waste Management Agency (ANDRA). Special thanks to Thierry Leïchlé, Laurent Mazenq and Liviu Nicu from LAAS/CNRS for fabricating the silicon microsensor used in this study.