Julien Coelho
email: julien.coelho@u-bordeaux.fr

Jacques Taillard

Adèle Bernard

Régis Lopez

Guillaume Fond

Laurent Boyer

Guillaume Lucas

François Alla

Daniel J Buysse

Meredith L Wallace

Catherine Verdun-Esquer

Pierre-Alexis Geoffroy

Emmanuel D'incau

Pierre Philip

Jean-Arthur Micoulaud-Franchi

Emotional Exhaustion, a Proxy for Burnout, Is Associated with Sleep Health in French Healthcare Workers without Anxiety or Depressive Symptoms: A Cross-Sectional Study

Keywords: sleep health, burnout, healthcare workers, cross-sectional

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

Introduction

Burnout, first described by Maslach et al. [START_REF] Maslach | Burnout Inventory Manual[END_REF], is a state of psychological, emotional, and physical stress in response to prolonged exposure to an occupational trigger. It encompasses feelings of emotional exhaustion (EE, depletion of emotional resources), depersonalization (developing cynical attitudes about others, such as patients), and reduced professional accomplishment (negative evaluation of oneself). Burnout is related to adverse health outcomes, including mortality, cardiometabolic morbidity, depression and suicide, cognitive impairment, and reduced work performance [START_REF] Bayes | The Biology of Burnout: Causes and Consequences[END_REF]. The frequency of burnout among healthcare workers (HCWs) is high [START_REF] De Hert | Burnout in Healthcare Workers: Prevalence, Impact and Preventative Strategies[END_REF]. Recent meta-analyses yielded 6.0% and 11.2% burnout prevalence rates among physicians and nurses, respectively, worldwide [START_REF] Woo | Global Prevalence of Burnout Symptoms among Nurses: A Systematic Review and Meta-Analysis[END_REF][START_REF] Karuna | Prevalence of Burnout among GPs: A Systematic Review and Meta-Analysis[END_REF]. Indeed, HCWs are consistently subjected to emotionally draining stressors in the provision of complex patient care and treatment [START_REF] Gómez-Urquiza | Risk Factors, and Levels of Burnout Among Oncology Nurses: A Systematic Review[END_REF], including heavy workloads, time pressure, and conflicts of value associated with low rewards, contradictory demands, and the lack of resources [START_REF] Lucas | Work Environment and Mental Health in Nurse Assistants, Nurses and Health Executives: Results from the AMADEUS Study[END_REF]. For instance, during the period of work overload due to the 2019 coronavirus (COVID-19) pandemic, HCWs reported increased exhaustion, anxiety, and depressive symptoms, as well as insomnia [START_REF] Geoffroy | Psychological Support System for Hospital Workers During the Covid-19 Outbreak: Rapid Design and Implementation of the Covid-Psy Hotline[END_REF]. Sleep alterations due to the disturbance of sleep-wake rhythms by stress and work conditions (i.e., atypical work schedule with frequent night shifts and on-call duties) are also implicated in the development of burnout in HCWs [START_REF] Kancherla | What Is the Role of Sleep in Physician Burnout?[END_REF][START_REF] Stewart | Sleep Disturbances in Frontline Health Care Workers During the COVID-19 Pandemic: Social Media Survey Study[END_REF].

The COVID-19 health crisis aggravated HCWs' critical situation by massively disrupting their sleep [START_REF] Jahrami | Sleep Problems during the COVID-19 Pandemic by Population: A Systematic Review and Meta-Analysis[END_REF] and mental health [START_REF] Geoffroy | Psychological Support System for Hospital Workers During the Covid-19 Outbreak: Rapid Design and Implementation of the Covid-Psy Hotline[END_REF][START_REF] Pappa | Prevalence of Depression, Anxiety, and Insomnia among Healthcare Workers during the COVID-19 Pandemic: A Systematic Review and Meta-Analysis[END_REF], worsening their work conditions, and increasing the prevalence of burnout [START_REF] Ghahramani | A Systematic Review and Meta-Analysis of Burnout Among Healthcare Workers During COVID-19[END_REF]. Beyond facilitating the study of sleep alterations and their causes and consequences, the sleep health framework provides a new approach to the promotion of sleep as a positive dimension for the benefit of health [START_REF] Buysse | Sleep Health: Can We Define It? Does It Matter?[END_REF]. It constitutes a privileged dimension for interventions that attempt to improve the sleep and workplace well-being of HCWs. A short, practical self-reported scale (RU-SATED) was recently developed and validated in a variety of languages [START_REF] Ravyts | Sleep Health as Measured by RU SATED: A Psychometric Evaluation[END_REF][START_REF] Becker | Sleep Health Assessment: A Scale Validation[END_REF][START_REF] Benítez | Validation of the Satisfaction Alertness Timing Efficiency and Duration (SATED) Questionnaire for Sleep Health Measurement[END_REF][START_REF] Coelho | Toward a Multi-Lingual Diagnostic Tool for the Worldwide Problem of Sleep Health: The French RU-SATED Validation[END_REF][START_REF] Furihata | The Validity and Reliability of the Japanese Version of RU-SATED[END_REF], and it is a reliable and valid tool that has shown promise for the rapid evaluation of sleep health. It is a six-item self-reported questionnaire exploring the six sleep health dimensions ("Regularity", "Satisfaction", "Alertness", "Timing", "Efficiency", and "Duration"). Good sleep health has been associated with several positive health outcomes, such as good cardiometabolic markers [START_REF] Buxton | Sleep Health and Predicted Cardiometabolic Risk Scores in Employed Adults From Two Industries[END_REF], a low risk of motor vehicle accidents [START_REF] Coelho | Toward a Multi-Lingual Diagnostic Tool for the Worldwide Problem of Sleep Health: The French RU-SATED Validation[END_REF], good physical [START_REF] Kubala | The Association between Physical Activity and a Composite Measure of Sleep Health. Sleep Breath[END_REF] and mental health [START_REF] Furihata | Study of Osteoporotic Fractures (SOF) Research Group An Aggregate Measure of Sleep Health Is Associated with Prevalent and Incident Clinically Significant Depression Symptoms among Community-Dwelling Older Women[END_REF], and good self-perceived health status [START_REF] Dalmases | Impact of Sleep Health on Self-Perceived Health Status[END_REF]. To date, good sleep health and its relationship with the absence of burnout have not been studied among HCWs. Burnout [START_REF] Koutsimani | The Relationship Between Burnout, Depression, and Anxiety: A Systematic Review and Meta-Analysis[END_REF], anxiety, and depressive symptoms are closely associated with sleep [START_REF] Alvaro | A Systematic Review Assessing Bidirectionality between Sleep Disturbances, Anxiety, and Depression[END_REF]; therefore, they deserve particular attention in studies of the relationship between sleep and burnout. A study of 6307 firefighters revealed no interaction between insomnia and self-reported diagnoses of anxiety and depression in relation to EE [START_REF] Wolkow | Associations between Sleep Disturbances, Mental Health Outcomes and Burnout in Firefighters, and the Mediating Role of Sleep during Overnight Work: A Cross-Sectional Study[END_REF], whereas a study conducted with 84 adults revealed an interaction between sleep quality and mental health in association with heart-rate variability, an objective indicator of chronic stress [START_REF] Thielmann | Subjective and Objective Consequences of Stress in Subjects with Subjectively Different Sleep Quality-A Cross-Sectional Study[END_REF].

Thus, we used the RU-SATED questionnaire to (1) assess sleep health in a large population of HCWs (physicians and nurses) in a French public hospital at the end of the first COVID-19 lockdown (17 March-11 May 2020) and (2) investigate the relationship between the RU-SATED score and burnout. We hypothesized that good sleep health would be associated with a lesser likelihood of burnout. We also determined the consistency of the results among the six sleep health dimensions and evaluated the roles of anxiety and depressive symptoms in any such association.

Materials and Methods

This observational cross-sectional study was conducted from June to October 2020. The 8211 HCWs in Bordeaux University Hospital were asked to complete an Internetbased questionnaire. They were informed of the research objective (i.e., to evaluate sleep health). The STROBE (Strengthening the Reporting of Observational Studies in Epidemiology) statement was used to report this study [START_REF] Von Elm | STROBE Initiative The Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) Statement: Guidelines for Reporting Observational Studies[END_REF].

The following sociodemographic data were collected: age, sex (male, female), job category (physician, nurse), and perceived socioeconomic status (comfortable, adequate, difficult).

The following data on work conditions, related to sleep-wake rhythms, were collected: work schedule (fixed or shift), night work (defined as >3 h work between 9 pm and 6 am two times per week or more) [START_REF] Leger | Le travail posté et de nuit et ses conséquences sur la santé: État des lieux et recommandations[END_REF], and telecommuting (yes, no). The study was conducted during the COVID-19 pandemic; therefore, we collected data concerning COVID-19 exposure (work in a COVID-19 unit; yes, no).

Self-perceived health was assessed using a single item from the Minimum European Health Module [START_REF] Cox | The Reliability of the Minimum European Health Module[END_REF]: "How is your health in general?" rated on a 4-point Likert scale ranging from 0 ("very good") to 3 ("very bad"). We considered that participants who responded "bad" or "very bad" had poor self-perceived health.

Anxiety and depressive symptoms were measured using the Patient Health Questionnaire-4, a reliable and valid four-item tool to which responses regarding the previous 2 weeks are provided on a 3-point Likert scale [START_REF] Kroenke | An Ultra-Brief Screening Scale for Anxiety and Depression: The PHQ-4[END_REF]. Responses to the first two items (feeling nervous, anxious, or on edge and not being able to stop or control worrying) are summed to obtain the anxiety score and those to the last two items (little interest or pleasure in doing things and feeling down, depressed, or hopeless) are summed to obtain the depressive symptom score. Cronbach's alpha values for the anxiety and depressive symptom subscales were 0.82 and 0.81, respectively. Scores ≥ 3 were considered to indicate significant anxiety or depressive symptoms.

Sleep health was measured using the French version of the RU-SATED v2.0 scale [START_REF] Coelho | Toward a Multi-Lingual Diagnostic Tool for the Worldwide Problem of Sleep Health: The French RU-SATED Validation[END_REF], a self-administered questionnaire evaluating the six dimensions of sleep health: sleep "Regularity" (bedtimes and wake times occurring within a 1-h period across days), sleep "Satisfaction" (subjective assessment of "good" or "poor" sleep), "Alertness" during waking hours (awake all day without dozing), appropriate "Timing" (asleep or trying to sleep between 2:00 and 4:00 a.m.), "Efficiency" (<30 min wake time during the night), and adequate "Duration" (6-8 h/day). Items are rated on a 3-point Likert frequency scale (0, "rarely/never"; 1, "sometimes"; 2, "usually/always"). The total score, obtained by summing the item scores, ranges from 0 to 12; higher scores indicate better sleep health. In the absence of a validated cut-off, the total score was dichotomized based on the median, with scores > 8 indicating good sleep health, as in previous studies [START_REF] Dalmases | Impact of Sleep Health on Self-Perceived Health Status[END_REF][START_REF] Yuksel | Sleeping When the World Locks down: Correlates of Sleep Health during the COVID-19 Pandemic across 59 Countries[END_REF].

Emotional exhaustion, measured by the single-item version of the Maslach Burnout Inventory, was used as a proxy for overall burnout [START_REF] West | Single Item Measures of Emotional Exhaustion and Depersonalization Are Useful for Assessing Burnout in Medical Professionals[END_REF]. It consists of a single statement ("I feel burned out from my work") rated on a 7-point Likert-scale ranging from "never" to "daily." This item correlates strongly with the global score for EE (r ≥ 0.76), has good external validity in association with workplace well-being outcomes (suicidality, serious thoughts of dropping out, perceived major medical error) [START_REF] West | Concurrent Validity of Single-Item Measures of Emotional Exhaustion and Depersonalization in Burnout Assessment[END_REF], and has greater predictive power for burnout than does depersonalization [START_REF] Flickinger | Single-Item Burnout Measure Correlates Well with Emotional Exhaustion Domain of Burnout but Not Depersonalization Among Medical Students[END_REF]. Responses of "once a week" or more frequently were considered to indicate the presence of EE as suggested in the original validation study [START_REF] West | Single Item Measures of Emotional Exhaustion and Depersonalization Are Useful for Assessing Burnout in Medical Professionals[END_REF]. Conversely, responses of "a few times per month" or less frequently were considered to indicate the absence of EE.

Descriptive statistics (frequencies for categorical variables, means, and standard deviations for continuous variables) were calculated. Univariate associations with EE were assessed using the chi-squared test for categorical variables and Student's t test for continuous variable. Multivariable logistic regression analysis was performed to estimate adjusted odds ratios (ORs) with 95% confidence intervals (CIs) for the association between good sleep health (the main explanatory variable) and the absence of EE (the dependent variable). Odds ratios were displayed as >1 for ease of interpretation. We selected confounding factors according to a review of the literature for factors influencing both sleep and burnout (age, sex, job category, work schedule, night work, telecommuting, work in a COVID-19 unit, and self-perceived health) [START_REF] Ghahramani | A Systematic Review and Meta-Analysis of Burnout Among Healthcare Workers During COVID-19[END_REF][START_REF] Lim | Individual and Work-Related Factors Influencing Burnout of Mental Health Professionals: A Meta-Analysis[END_REF][START_REF] Demir Zencirci | Morning-Evening Type and Burnout Level as Factors Influencing Sleep Quality of Shift Nurses: A Questionnaire Study[END_REF][START_REF] Demir | Investigation of Factors Influencing Burnout Levels in the Professional and Private Lives of Nurses[END_REF]. The perceived socioeconomic status was also included, as it was associated with sleep health and EE at the 0.20 threshold in univariate analyses. The linearity hypothesis for quantitative variables was verified using fractional polynomials [START_REF] Royston | The Use of Fractional Polynomials to Model Continuous Risk Variables in Epidemiology[END_REF], and analysis of the residuals was conducted to confirm the suitability of the model. To explore the dose-response relationship, a sensitivity analysis was performed with sleep health represented as a continuous variable in a linear regression model. Covariate interactions with sleep health were examined, and no result was significant at the 0.20 threshold [START_REF] Selvin | Statistical Analysis of Epidemiologic Data[END_REF]. Sleep health interacted significantly with anxiety (p = 0.152) and depressive symptoms (p = 0.037) in relation to EE. Thus, stratified analyses were conducted to explore whether the relationship between good sleep health and the absence of EE was persistent and similar in HCWs according to the presence or absence of anxiety and depressive symptoms.

To determine the consistency of associations among the six sleep health dimensions, we performed a secondary analysis of the association between each dimension and the absence of EE. For all tests, the accepted significance level was 5%. The data analyses were conducted using R v. 4.1.2.

Results

Characteristics of the Study Population

The sample comprised 1069 HCWs, or 13.0% of the 8211 HCWs at Bordeaux University Hospital. The mean age was 39.2 ± 11.3 years, 900 (84.2%) participants were female, and 866 (81.0%) participants were nurses. A total of 514 (48.1%) HCWs had shift schedules and 199 (18.6%) were engaged in night work. Anxiety and depressive symptoms were present in 384 (35.9%) and 167 (15.6%) HCWs, respectively. Further descriptive data are presented in Table 1. The mean RU-SATED score was 7.9 ± 2.4, with 474 (44.3%) HCWs reporting good sleep health. Good sleep health was more common among physicians, those with comfortable socioeconomic status, and HCWs on fixed schedules. Conversely, poor sleep health was more common among night workers and those who worked in a COVID-19 unit and/or had poor self-perceived health, anxiety, and/or depressive symptoms.

A total of 143 (13.4%) HCWs reported EE. This condition was more common among females, those in difficult socioeconomic status, and those with poor self-perceived health, anxiety, and/or depressive symptoms.

Association between EE and Sleep Health

HCWs reporting good sleep health had a two-fold lower likelihood of EE (n = 38 [8.0%]) than those reporting poor sleep health (n = 105 [17.6%]; OR = 2.46 [1.66-3.64], p < 0.001). In the analysis adjusted for age, sex, job category, perceived socioeconomic status, work schedule, night work, telecommuting, work in a COVID-19 unit, and self-perceived health, good sleep health remained associated with a lesser likelihood of EE (OR = 2.50 [1.61-3.88], p < 0.001). Males, nurses, and those with a comfortable socioeconomic status had lesser likelihoods of EE than females, physicians, and those with difficult socioeconomic status, respectively. Shift work and night work were not associated with EE in the multivariate models (Figure 1). Sensitivity analyses showed similar results using sleep health and EE as continuous variables (ß = -0.12 [-0.16; -0.08], p < 0.001; Figure S1).

Association between EE and Sleep Health according to ANXIETY and Depressive Symptoms

In adjusted analyses, EE was less common in HCWs with good sleep health among those without anxiety (n = 685; OR = 2.66 [1.20-5.89], p = 0.014) and without depressive symptoms (n = 902; OR = 2.56 [1.47-4.46], p < 0.001). Among HCWs with anxiety (n = 384) and those with depressive symptoms (n = 167), associations were not significant, and estimates were closer to 1 (OR = 1.52 [0.83-2.77], p = 0.166 and OR = 1.24 [0.50-3.12], p = 0.641, respectively; Figure 2). Sensitivity analyses showed similar results when including anxiety and depressive symptoms and their interaction term as covariates in the multivariate model (OR = 2.35 [1.10-5.04], p = 0.028; Figure S2).

Associations between EE and Sleep Health Dimensions

In adjusted analyses, HCWs reporting that they were sometimes and usually/always satisfied with their sleep had two-and five-fold lesser likelihoods of EE, respectively, than those reporting that they were rarely/never satisfied. Similarly, HCWs reporting that they were usually/always alert had a two-fold lesser likelihood of EE than those reporting that they were rarely/never alert. The other sleep health dimensions were not associated with EE (Figure 3).

Discussion

Key Results

Good sleep health was associated with a two-fold lesser likelihood of EE, a proxy for burnout, during the COVID-19 health crisis in a sample of 1069 French HCWs. Males, nurses, and HCWs in comfortable socioeconomic status had lower likelihood of EE. None of these covariates explained the association between sleep health and EE in an adjusted analysis; neither did age, work schedule, night work, telecommuting, work in a COVID-19 unit, or self-perceived health. The sleep health dimensions of "Satisfaction" and "Alertness" were associated with low likelihoods of EE, as indicated by the at least five-fold lesser likelihood of EE among HCWs reporting that they were usually/always satisfied with their sleep than among those reporting that they were rarely/never satisfied. These associations were also significant among HCWs without anxiety or depressive symptoms. The overlap between burnout and depression [START_REF] Bianchi | Burnout-Depression Overlap: A Review[END_REF], associated with decreased power in subgroup analyses, might explain the lack of a significant association among HCWs with anxiety or depressive symptoms.

Interpretation

Our findings are consistent with previous findings on relationships between sleep alterations and burnout [START_REF] Kancherla | What Is the Role of Sleep in Physician Burnout?[END_REF][START_REF] Stewart | Sleep Disturbances in Frontline Health Care Workers During the COVID-19 Pandemic: Social Media Survey Study[END_REF]. The confirmation of this relationship using the RU-SATED scale was important from an interventional perspective. In the sleep health framework, sleep is a positive dimension including measurable sleep quality and modifiable sleep behaviors [START_REF] Buysse | Sleep Health: Can We Define It? Does It Matter?[END_REF]. Thus, improvement of the sleep health of individuals, independently of the presence or absence of sleep disorders, may benefit their workplace well-being [START_REF] Buysse | Sleep Health: Can We Define It? Does It Matter?[END_REF]. Work is an important contributor to the societal and social constraints that limit sleep opportunities for individuals [START_REF] Grandner | Sleep and Health[END_REF][START_REF] Taillard | Sleep Timing, Chronotype and Social Jetlag: Impact on Cognitive Abilities and Psychiatric Disorders[END_REF]; thus, employers and HCWs should be aware of this framework and use it to prevent burnout. Nevertheless, further studies should consider other work characteristics, such as those measured by the Job Content Questionnaire (i.e., psychological demands, emotional demands, job control, social support) [START_REF] Karasek | The Job Content Questionnaire (JCQ): An Instrument for Internationally Comparative Assessments of Psychosocial Job Characteristics[END_REF], that are important contributors to burnout [START_REF] Aronsson | A Systematic Review Including Meta-Analysis of Work Environment and Burnout Symptoms[END_REF]. Taking these numerous and highly correlated factors into account will likely require structural equation modeling to explore interactions with sleep health, burnout, and mental health [START_REF] Stein | Structural Equation Modeling[END_REF], as for sleep duration [START_REF] Lucas | Work Environment and Mental Health in Nurse Assistants, Nurses and Health Executives: Results from the AMADEUS Study[END_REF]. Such studies could lead to the identification of organizational characteristics that are detrimental or beneficial to sleep and workplace well-being, enabling recommendations for organizational changes. At the same time, individual factors such as mental health and socioeconomic status are also crucial contributors to burnout. The implementation of support systems, including psychological evaluation, could facilitate early diagnosis and thus improve the prognosis of sleep alterations and their consequences [START_REF] Geoffroy | Psychological Support System for Hospital Workers During the Covid-19 Outbreak: Rapid Design and Implementation of the Covid-Psy Hotline[END_REF].

Previous studies have involved the exploration of relationships between separate sleep health dimensions and mental health. The RU-SATED scale dimensions of "Regularity" and "Satisfaction" had the strongest estimated associations with depression, and the "Timing" and "Efficiency" dimensions had the weakest associations [START_REF] Barham | Sleep Health Mediates the Relationship between Physical Activity and Depression Symptoms[END_REF]. The inclusion of night schedule data in our models may explain the lack of association between "Regularity" and burnout. A post hoc analysis showed that "Regularity" was associated with burnout among night workers (n = 199). The "Satisfaction", "Alertness", and "Efficiency" dimensions of the Spanish version of the SATED scale (the RU-SATED instrument without "Regularity") correlated strongly with anxiety and depression [START_REF] Benítez | Validation of the Satisfaction Alertness Timing Efficiency and Duration (SATED) Questionnaire for Sleep Health Measurement[END_REF]. In another assessment of the same five sleep health dimensions, "Satisfaction" and "Alertness" had the strongest estimated longitudinal associations with incident depression and "Duration" had the weakest association [START_REF] Furihata | Study of Osteoporotic Fractures (SOF) Research Group An Aggregate Measure of Sleep Health Is Associated with Prevalent and Incident Clinically Significant Depression Symptoms among Community-Dwelling Older Women[END_REF]. Overall, the "Satisfaction" and "Alertness" dimensions appear to be important for mental health, as suggested in a study conducted to validate the French version of the RU-SATED scale [START_REF] Coelho | Toward a Multi-Lingual Diagnostic Tool for the Worldwide Problem of Sleep Health: The French RU-SATED Validation[END_REF]. Thus, they could be used to assess sleep health in relation to burnout. The discrepancy in association strength among the six sleep health dimensions suggests the need for a better understanding of the concept of sleep health and the distinction between sleep quality dimensions ("Alertness" and "Satisfaction") and sleep behavior dimensions ("Regularity", "Timing", and "Duration") [START_REF] Coelho | Toward a Multi-Lingual Diagnostic Tool for the Worldwide Problem of Sleep Health: The French RU-SATED Validation[END_REF]. Each dimension (except for "Satisfaction") can also be measured objectively [START_REF] Wallace | Multidimensional Sleep Health Domains in Older Men and Women: An Actigraphy Factor Analysis[END_REF] and each is related to physiological processes that may have different implications for health [START_REF] Buysse | Sleep Health: Can We Define It? Does It Matter?[END_REF]. Indeed, "Regularity", "Timing", "Efficiency", and "Duration" showed the strongest associations with mortality, whereas "Satisfaction" was borderline significant and "Alertness" was not significant in a study of 2887 men followed during 11 years [START_REF] Wallace | Osteoporotic Fractures in Men (MrOS) Study Research Group Which Sleep Health Characteristics Predict All-Cause Mortality in Older Men? An Application of Flexible Multivariable Approaches[END_REF]. This point is important because behavioral sleep interventions have different efficacies, depending on the dimension targeted [START_REF] Albakri | Sleep Health Promotion Interventions and Their Effectiveness: An Umbrella Review[END_REF]. Thus, the relationships of sleep health dimensions to other burnout dimensions need to be investigated. Indeed, as described by the job demand-control model, personal accomplishment could compensate for situations with high levels of EE and depersonalization [START_REF] Karasek | Job Demands, Job Decision Latitude, and Mental Strain: Implications for Job Redesign[END_REF]. Future studies should assess the relationship of sleep health to the other two dimensions of burnout (depersonalization and personal accomplishment) using a suitable questionnaire.

Limitations

This study has several limitations. First, participants were included on a voluntary basis, leading to a low response rate (13.0%) and a probable over-representation of persons interested in sleep. However, the sample was representative of Bordeaux University Hospital HCWs in terms of age (mean, 42.5 years in the Bordeaux University Hospital population and 39.2 years in our sample), sex (83.3% and 84.2% female), and job category (87.7% and 81.0% nurses). Moreover, the mean RU-SATED score (7.9) was comparable to that from an international study conducted during the COVID-19 lockdown (8.1) [START_REF] Yuksel | Sleeping When the World Locks down: Correlates of Sleep Health during the COVID-19 Pandemic across 59 Countries[END_REF]. The rates of anxiety and depressive symptoms were high (35.9% and 15.6%, respectively), but comparable to those reported for the post-COVID-19 lockdown period (27.8% and 26.9%, respectively) [START_REF] De Sousa | Mental Health in COVID-19 Pandemic: A Meta-Review of Prevalence Meta-Analyses[END_REF]. Except for mental health, there were no effect-modifying factors for the association between sleep health and burnout in the literature and no interactions were found in our data. Therefore, a selection bias that could alter the association between sleep health and burnout was unlikely. The lack of representativeness of this sample should not hinder the interpretation of associations between variables of interest [START_REF] Rothman | Why Representativeness Should Be Avoided[END_REF]. Second, objective sleep measurements were not taken in this study. Thus, the presence or absence of sleep disorders (e.g., obstructive sleep apnea syndrome, chronic insomnia disorder, and restless legs syndrome) were not assessed, although they constitute risk factors for poor sleep heath and burnout. Further studies should explore the relationship between sleep health and burnout while considering sleep disorders in order to distinguish general and clinical population. Third, we used a single-item scale to assess EE, a proxy for burnout, which may seem to be insufficient to identify burnout with precision. However, the scale has been validated [START_REF] West | Single Item Measures of Emotional Exhaustion and Depersonalization Are Useful for Assessing Burnout in Medical Professionals[END_REF][START_REF] West | Concurrent Validity of Single-Item Measures of Emotional Exhaustion and Depersonalization in Burnout Assessment[END_REF], and the burnout rate among nurses in our sample (13.4%) is consistent with that obtained in an international meta-analysis (11.2%) [START_REF] Woo | Global Prevalence of Burnout Symptoms among Nurses: A Systematic Review and Meta-Analysis[END_REF]. Moreover, mental health assessment did not rely on a clinical diagnosis of anxiety and depression disorders, but used a short self-reported scale (PHQ-4) that is a valid and reliable tool to evaluate the severity of the anxiety and depressive symptoms. Fourth, the study was conducted during the COVID-19 health crisis with a population of HCWs with increased psychological distress. Indeed, sleep health and workplace well-being were disturbed during this period [START_REF] Jahrami | Sleep Problems during the COVID-19 Pandemic by Population: A Systematic Review and Meta-Analysis[END_REF][START_REF] Ghahramani | A Systematic Review and Meta-Analysis of Burnout Among Healthcare Workers During COVID-19[END_REF]. However, working in a COVID-19 unit had no impact on the results of this study, probably because it was conducted at the end of the first lockdown in a region relatively spared by the outbreak and in a period with few health restrictions and no saturation of the healthcare system. Replication of this study should confirm that the findings can be extrapolated to other health contexts. Fifth, the study was limited by its cross-sectional design. No causal inference can be made between burnout and sleep health. Due to the entanglement of many associated factors in this relationship, longitudinal approaches using structural equation modeling based on documented mechanistic assumptions are needed.

Conclusions

Although the cross-sectional design of this study hampered the drawing of causal inferences, our findings suggest: (1) that sleep health, and in particular the "Satisfaction" and "Alertness" RU-SATED dimension scores, are associated with HCW burnout; (2) that sleep health alteration could constitute a risk factor of HCW burnout; and (3) that sleep health promotion should be investigated as a strategy to reduce HCW burnout and could be used to complement work organization-centered strategies.

Supplementary Materials:

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/jcm12051895/s1, Figure S1: Adjusted linear associations between emotional exhaustion and sleep health; Figure S2: Adjusted associations between emotional exhaustion and sleep health with additional adjustment for anxiety and depressive symptoms.

Citation:

  Coelho, J.; Taillard, J.; Bernard, A.; Lopez, R.; Fond, G.; Boyer, L.; Lucas, G.; Alla, F.; Buysse, D.J.; Wallace, M.L.; et al. Emotional Exhaustion, a Proxy for Burnout, Is Associated with Sleep Health in French Healthcare Workers without Anxiety or Depressive Symptoms: A Cross-Sectional Study.

Figure 1 .

 1 Figure 1. Adjusted associations between the absence of emotional exhaustion and socio-demographic data, working condition, and sleep health. Forest plot of the associations with the absence of emotional exhaustion after adjustment for age, sex, job category, perceived socioeconomic status, work schedule, night work, telecommuting, work in a COVID-19 unit, self-perceived health, and sleep health.

Figure 2 .

 2 Figure 2. Adjusted associations between sleep health and the absence of emotional exhaustion (EE) according to anxiety and depressive symptoms. Forest plot of the associations with the absence of emotional exhaustion (EE) after adjustment for age, sex, job category, perceived socioeconomic status, work schedule, night work, telecommuting, work in a COVID-19 unit, self-perceived health, and sleep health, stratified for anxiety and depressive symptoms.

Figure 3 .

 3 Figure 3. Adjusted associations between each sleep health dimension and the absence of emotional exhaustion (EE). Forest plot of the associations with the absence of emotional exhaus- tion (EE) after adjustment for age, sex, job category, perceived socioeconomic status, work schedule, night work, telecommuting, work in a COVID-19 unit, self-perceived health, and sleep health.

Author Contributions:

  Conceptualization, J.C., J.-A.M.-F., P.P., F.A., and C.V.-E.; methodology, J.-A.M.-F., P.P., and J.T.; software, J.-A.M.-F. and P.P.; validation, J.-A.M.-F., P.P., and J.T.; formal analysis, J.C. and J.-A.M.-F.; investigation, J.C. and J.-A.M.-F.; resources, J.-A.M.-F. and P.P.; data curation, J.-A.M.-F. and P.P.; writing-original draft preparation, J.C. and J.-A.M.-F.; writing-review and editing, J.C., J.T., A.B., R.L., G.F., L.B., G.L., F.A., D.J.B., M.L.W., C.V.-E., E.d., P.-A.G., P.P., and J.-A.M.-F.; visualization, J.C. and J.-A.M.-F.; supervision, J.-A.M.-F. and P.P.; project administration,

Table 1 .

 1 Characteristics of the study population and their distributions according to sleep health and emotional exhaustion. Measured by the RU-SATED with good sleep health defined as a score >8; b measured by the single-item version of the Maslach Burnout Inventory with emotional exhaustion defined as a frequency of once a week or more; the student t-test was used for continuous variables, while the chi-square test was used for categorical variables to test the univariate associations of each characteristic with good sleep health and the absence of emotional exhaustion.

					Sleep Health a			Emotional Exhaustion b
			Total
			n = 1069	Good n = 474	Poor n = 595	Statistic	p-Value	Yes n = 143	No n = 926	Statistic	p-Value
	Age (years)	39.2 ± 11.3	39.8 ± 10.8	39.8 ± 11.7	t = 1.41	0.159	40.4 ± 10.8	39.1 ± 11.4	t = 1.39	0.167
	Sex --	Male Female	169 (15.8%) 900 (84.2%)	80 (16.9%) 394 (83.1%)	89 (15.0%) 506 (85.0%)	X 2 = 0.593	0.393	12 (8.4%) 131 (91.6%)	(17.0%) (83.0%)	X 2 = 6.20	0.009
	Job category -Nurses -Physicians	866 (81.0%) 203 (19.0%)	332 (70.0%) 142 (30.0%)	534 (89.8%) 61 (10.2%)	X 2 = 65.3	<0.001	116 (81.1%) 27 (18.9%)	(81.0%) (19.0%)	X 2 < 0.001	0.972
	Perceived socioeconomic
	status	501 (46.9%)	287 (60.6%)	214 (36.0%)			56 (39.2%)	(48.1%)
	-	Comfortable	501 (46.9%)	172 (36.3%)	329 (55.3%)	X 2 = 67.4	<0.001	67 (46.9%)	(46.9%)	X 2 = 17.8	<0.001
	-	Adequate	67 (6.3%)	15 (3.2%)	52 (8.7%)			20 (14.0%)	(5.1%)
	-	Difficult
	Work schedule: -Shift -Fixed	514 (48.1%) 555 (51.9%)	192 (40.5%) 282 (59.5%)	322 (54.1%) 273 (45.9%)	X 2 = 19.0	<0.001	67 (46.9%) 76 (53.2%)	(48.3%) (51.7%)	X 2 = 0.051	0.821
	Night work: -Yes -No	199 (18.6%) 870 (81.4%)	50 (10.6%) 424 (89.4%)	149 (25.0%) 446 (75.0%)	X 2 = 35.6	<0.001	22 (15.4%) 121 (84.6%)	(19.1%) (80.9%)	X 2 = 0.905	0.342
	Telecommuting: Yes	168 (15.7%)	81 (17.1%)	87 (14.6%)	X 2 = 1.03	0.271	24 (16.8%)	(15.6%)	X 2 = 0.064	0.706
	Work in a COVID-19 unit: Yes	234 (21.9%)	90 (19.0%)	144 (24.2%)	X 2 = 3.90	0.041	30 (21.0%)	(22.0%)	X 2 = 0.030	0.777

a

Acknowledgments:

The RU-SATED scale is copyrighted by the University of Pittsburgh. Permission to use it should be requested from Daniel J Buysse. The authors express their sincere acknowledgement to all the professionals involved in the questionnaire design, including Rachel Debs. They also thank Christophe Gauld and Kevin Ouazzani for their expertise in clarifying the different dimensions of sleep and sleep health. Finally, the authors acknowledge Bordeaux University Hospital for its help in recruiting the participants.

Funding: This research received no external funding.

Institutional Review Board Statement:

The project was conducted in accordance with the Declaration of Helsinki. The protocol was in accordance with General Data Protection Regulation (GDPR) and the Commission Nationale de l'informatique et des libertés (CNIL) requirements, which is the national authority that ensures that research data collection does not violate freedoms, rights, and human privacy. Due to the data collection method, which was completely anonymous, this research was outside the scope of the provisions governing biomedical research.

Informed Consent Statement: Informed consent was obtained from all participants in the study.

Data Availability Statement: Not applicable.

J.-A.M.-F. and P.P.; funding acquisition, P.P. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.


Fréderic Thomas
email: frederic.thomas2@ird.fr

Eric Elguero

Jacques Brodeur

Jérôme Le Goff

Dorothée Missé

Frédéric Thomas

Jérôme Le Goff

Herpes simplex virus type 2 and cancer: A medical geography approach

Introduction

Since the mid-1970s, it has been increasingly recognized that infectious agents can cause cancers (see De [START_REF] Martel | Infections and cancer: established associations and new hypotheses[END_REF] for a recent review). For instance, worldwide incidences of cancers of the liver, stomach, and cervix uteri are largely attributable to hepatitis B and C viruses, to the bacteria Helicobacter pylori, and to human papilloma virus, respectively. However, the complete list of oncogenic pathogens is far from being definitively established (e.g., [START_REF] Cochran | Infectious causation of disease: an evolutionary perspective[END_REF][START_REF] Ewald | An evolutionary perspective on parasitism as a cause of cancer?[END_REF]. Identifying infectious agents that directly or indirectly contribute to oncogenesis remains a priority in the war on cancer for an obvious reason: since infectious diseases are often preventable or treatable, cancers associated with these infections could potentially be preventable as well [START_REF] Martel | Infections and cancer: established associations and new hypotheses[END_REF].

Herpes viruses (approximately 100 members, with eight found to infect humans) possess numerous attributes of oncogenic viruses [START_REF] Ewald | An evolutionary perspective on parasitism as a cause of cancer?[END_REF]: they persist in the host by establishing a latent infection, they are mutagenic and inhibit apoptosis, and they stimulate host DNA synthesis. Such alterations can result in the derailment of the normal cell cycle and ultimately cause cancer. Not surprisingly, herpes viruses have been implicated as aetiological factors in several human malignancies [START_REF] Shillitoe | Oral cancer and herpes simplex virus. A review[END_REF][START_REF] Eglin | Detection of RNA complementary to herpes simplex virus in human oral squamous cell carcinoma[END_REF][START_REF] Young | Epstein-Barr virus: 40 years on[END_REF][START_REF] Parker | Head and neck cancer associated with herpes simplex virus 1 and 2 and other risk factors[END_REF][START_REF] Horikawa | Twist and epithelial-mesenchymal transition are induced by the EBV oncoprotein latent membrane protein 1 and are associated with metastatic nasopharyngeal carcinoma[END_REF][START_REF] Filippakis | Herpesviruses: hijacking the Ras signalling pathway[END_REF][START_REF] Mesri | Kaposi's sarcoma and its associated herpesvirus[END_REF][START_REF] Petrova | Epstein-Barr virus: silent companion or causative agent of chronic liver disease[END_REF]. Among them, herpes simplex virus type 2 (HSV-2), which is the most common cause of recurrent genital ulcers in the world, has been linked to prostate cancer [START_REF] Boldogh | Human cytomegalovirus and herpes simplex type 2 virus in normal and adenocarcinomatus prostate glands[END_REF][START_REF] Haid | Immunofluorescent evidence of prior herpes simplex virus type-2 infection in prostate carcinoma[END_REF][START_REF] Dennis | Sexually transmitted infections and prostate cancer among men in the US military[END_REF] and to invasive cervical cancer when occurring in conjunction with human papillomavirus (Smith et al., 2002). However, the exact role of HSV-2 in causing or promoting cancers remains controversial because HSV-2 is not systematically detected in tumours (see for instance [START_REF] Herbert | Herpes simplex virus type 2 and cancer of the prostate[END_REF][START_REF] Baker | The relationship of herpesvirus to carcinoma of the prostate[END_REF][START_REF] Serfling | Human papillomavirus and herpes virus DNA are not detected in benign and malignant prostatic tissue using the polymerase chain reaction[END_REF][START_REF] Taylor | Prostate cancer and sexually transmitted diseases: a meta-analysis[END_REF][START_REF] Sutcliffe | Sexually transmitted infections and risk of prostate cancer: review of historical and emerging hypotheses[END_REF]. In addition, it is difficult to determine whether HSV-2 is a true causal factor or whether it is correlated to other oncogenic sexually transmitted infections. Finally, it has also been demonstrated that cells may acquire increased susceptibility to HSV-2 during tumour progression [START_REF] Jensen | Human herpes simplex viruses in benign and malignant thyroid tumours[END_REF].

Here, we explore the links between infection by HSV-2 and a large variety of cancers in humans using a medical geography approach. We conducted a comparative study using data from populations throughout the world since marked variations exist between countries in both the incidence of cancers and the seroprevalence of HSV-2.

Material and methods

Data sources

International statistics on 46 cancers in men and women (lip oral cavity; nasopharynx; other pharynx; oesophagus; stomach; Herpes simplex virus type-2 (HSV-2) has been identified as a possible aetiological agent of cancer in humans, especially prostate cancer, but results remain controversial. Here, we have addressed this question using a medical geography approach based on the national incidence of various cancers and seroprevalence of HSV-2 in 64 countries worldwide. We corrected reports of cancer incidence for national gross domestic product (GDP) because living in a wealthy nation likely increases the probability of having a cancer detected. Data were also corrected for latitude and diet. Our analysis not only confirms that prostate cancer and HSV-2 seroprevalence are positively associated, but it also reveals the existence of a positive relationship between HSV-2 and melanoma incidence in both men and women. These results, though correlational, suggest that HSV-2 should continue to be investigated as a possible oncogenic pathogen of humans. 1 colorectum; liver; gallbladder; pancreas; larynx; lung; melanoma of skin; breast; cervix uteri; corpus uteri; ovary; kidney; bladder; brain, nervous system; thyroid; Hodgkin lymphoma; non-Hodgkin lymphoma; multiple myeloma; leukaemia; prostate; testis) were obtained from the International Agency for Research on Cancer (IARC GLOBOCAN project, 2008, http://globocan.iarc.fr/). We did not include mortality data in our analysis since this variable is influenced by the access to therapies-a parameter that strongly varies between countries. Instead, we used incidence data (agestandardised rate) that derive from population-based cancer registries. While data from most of the developing countries might not be of the highest quality, this information is still of unique importance as it often remains the only relatively unbiased source of information available on the profile of cancer (see http:// globocan.iarc.fr/). Data on dietary energy consumption, in kcal/ person/day, were from FAO (http://www.fao.org/economic/ess/ food-security-statistics/en/).

International statistics on HSV-2 infection prevalence were obtained from the few review papers available on this topic [START_REF] Smith | Age-specific prevalence of infection with herpes simplex virus types 2 and 1: a global review[END_REF][START_REF] Malkin | Epidemiology of genital herpes simplex virus infection in developed countries[END_REF][START_REF] Pebody | The seroepidemiology of herpes simplex virus type 1 and 2 in Europe[END_REF][START_REF] Patnaik | Type-specific seroprevalence of herpes simplex virus type 2 and associated risk factors in middle-aged women from 6 countries: the IARC multicentric study[END_REF] and completed by an extensive search of the literature mainly using ISI Web of Knowledge and PubMed. Search keywords included HSV-2, genital herpes prevalence, seroprevalence, epidemiology, and seroepidemiology. The only data used were from peer-reviewed articles that provided a clear description of the type-specific serologic methodology for detection of HSV-2 antibodies as well as the age (median or mean) of the participants (except for Barbados, for which the age information was obtained by contacting the author). We considered data only on non-high risk populations. When herpes data were available for both males and females, we calculated the mean value.

Analyses

The influence of HSV-2 prevalence on the incidence of several cancers was examined using linear regression; data on cancer incidence were log-transformed for the analysis. Since herpes prevalence often increases with age (e.g., [START_REF] Smith | Age-specific prevalence of infection with herpes simplex virus types 2 and 1: a global review[END_REF], we included the mean or median age of the populations used to estimate HSV-2 prevalence in the models. In addition, cancer detection might vary among nations: living in a wealthy country probably increases the likelihood of cancer detection. We controlled for national wealth by including the logarithm of per capita GDP (gross domestic product) in the models. Two other potentially confounding variables were included: the average daily caloric intake and the absolute value of latitude as a means to summarize environmental factors. Male and female cancers were analyzed independently, standard Bonferroni corrections being applied separately for the 22 male cancers and the 24 female cancers.

Results

Data on the seroprevalence of HSV-2 were obtained for 64 countries (Table 1).

The results of the linear regression models for the three cancers associated with HSV-2 prevalence-prostate cancer and melanoma in both men and women are summarized in Table 2, where oddsratios are given for a 10% increase in HSV-2 prevalence. Prostate cancer was significantly associated with HSV-2 after Bonferroni correction, whereas the association between male and female melanoma and HSV-2 prevalence was only marginally significant. Among the confounding variables, per capita GDP and caloric intake were positively associated with those cancer incidences, as expected

Discussion

Despite considerable research, the causes of most cancers remain a mystery. Because our approach is correlative, it is not possible to prove causation or to exclude the possibility of spurious results (i.e., a non-causal correlation between cancer and ecological variables that influence HSV-2 transmission). Despite these limitations, the present work suggests that HSV-2 should continue to be investigated as a possible oncogenic pathogen of humans.

HSV-2 seroprevalence was significantly associated with the incidence of prostate cancer. The existence of a positive association between HSV-2 and prostate cancer is consistent with previously published studies (e.g., [START_REF] Herbert | Herpes simplex virus type 2 and cancer of the prostate[END_REF][START_REF] Boldogh | Human cytomegalovirus and herpes simplex type 2 virus in normal and adenocarcinomatus prostate glands[END_REF][START_REF] Haid | Immunofluorescent evidence of prior herpes simplex virus type-2 infection in prostate carcinoma[END_REF][START_REF] Dennis | Sexually transmitted infections and prostate cancer among men in the US military[END_REF], but to our knowledge this is the first time that such an association is detected at this geographical scale. Our results could also support the hypothesis that prostate cancer is associated to sexually transmitted diseases in general, not necessarily HSV-2, and/or that HSV-2 could just act as a cofactor [START_REF] Korodi | No serological evidence of association between prostate cancer and infection with herpes simplex virus type 2 or human herpesvirus type 8: a nested case-control study[END_REF][START_REF] Huang | Sexually transmissible infections and prostate cancer risk[END_REF]. In addition, we cannot totally exclude a surveillance bias: for instance, if men with HSV-2 are followed more closely than uninfected men and are therefore screened for prostate cancer more accurately, we would expect a positive association between HSV-2 and prostate cancer even if no causal link exists. Nevertheless, if a causal relationship exists, it can only be in the direction of STIs to prostate cancer.

The statistical association between HSV-2 and melanoma in both men and women, although only marginally significant, deserves to be considered, since first, the Bonferroni correction is known to be overly conservative, and second, it is unlikely that the male and female incidences of a same disease were picked up by hazard, especially since latitude was controlled for. This association is intriguing and invites further exploration before we can consider this virus as a possible causative agent for melanoma genesis. At present, very few studies have linked herpes viruses and human melanoma. However, metastatic spread of melanomas mimicking herpes zoster has been reported in three studies [START_REF] North | Recurrent malignant melanoma presenting with zosteriform metastases[END_REF][START_REF] Evans | Zosteriform metastasis from melanoma[END_REF][START_REF] Zalaudek | Cutaneous zosteriform melanoma metastases arising after herpes zoster infection: a case report and review of the literature[END_REF]. Here, it is relevant to first note that the association is detected despite the fact that melanomas are usually more common in people with lightly pigmented skin while HSV-2 seroprevalence is highest in southern countries. [START_REF] Lundberg | Human herpes virus DNA is rarely detected in non-UV light-associated primary malignant melanomas of mucous membranes[END_REF] pointed out that melanomas can also emerge on sun-sheltered body surfaces, i.e., nasal cavity, anus, rectum, vulva, and penis. However, none of the eight herpes viruses infecting human seems to play a major role in the development of these extracutaneous melanomas [START_REF] Lundberg | Human herpes virus DNA is rarely detected in non-UV light-associated primary malignant melanomas of mucous membranes[END_REF]. In addition, extracutaneous melanomas are few in absolute numbers compared with cutaneous ones. Skin cancer has been linked to herpes viruses, but it concerns non-melanoma skin cancer and cytomegalovirus [START_REF] Zafiropoulos | Human herpes viruses in non-melanoma skin cancers[END_REF]. Thus, the positive link between HSV-2 seroprevalence and the incidence of melanoma detected here remains enigmatic.

Further research is necessary to determine the potential mechanism(s) for HSV-2 initiating or promoting prostate cancer and melanomas. Similarly, given the high number of people who are infected with HSV-2 worldwide compared with those who have cancers, it appears clear that this virus alone does not cause cancer. Determining the other underlying causes of cancer initiation in people infected with HSV-2 is therefore essential. In addition, because HSV-2 transmission is potentially correlated to that of other sexually transmitted pathogens, our study draws attention to the need to assess whether other sexually transmitted pathogens are responsible for prostate cancer and melanoma.

We would like to underline some potential limitations of our study. Sources of environmental heterogeneity at the largest spatial scale are extraordinarily various and undoubtedly cannot be all considered in our analysis when attempting to understand variation in cancer incidence. As frequently argued, comparative analyses at the largest scale have to be interpreted with caution because data are collected using different methods or they come from different sources. Although this argument holds true when no significant result is detected (i.e., data are not strong enough to detect a potentially significant result), it is usually unlikely to be relevant when significant trends are found, since a biological trend has a priori no reason to be correlated to background noise in the data set [START_REF] Brown | Macroecology[END_REF][START_REF] Lawton | Are there general laws in ecology?[END_REF]. However, we must be aware that despite our efforts to control for such effects, we cannot exclude the possibility that other parameters, at a different scale, may confound our conclusions.

Table 1

 1 HSV-2 prevalence estimates from non-high-risk population by country.

	Country	Seroprevalence	Mean or	Reference
		of HSV-2	median age
	Australia	14	32	Smith and Robinson (2002)
	Bangladesh	15.4	27.9	Bogaerts et al. (2001)
	Barbados	34.2	35	Levett (2005)
	Belgium	11.1	32	Pebody et al. (2004)
	Benin	26.5	32	Smith and Robinson (2002)
	Brazil	34.4	36	Cowan et al. (2003)
	Bulgaria	23.9	32	Pebody et al. (2004)
	Burkina Faso	17.9	24	Kirakoya-Samadoulougou
				et al. (2008)
	Cameroon	55	32	Smith and Robinson (2002)
	Canada	19	32	Smith and Robinson (2002)
	Central African	82	27	Smith and Robinson (2002)
	Republic
	China	29	52	Smith and Robinson (2002)
	Colombia	56.9	47	Patnaik et al. (2007)
	Costa Rica	39	34.5	Smith and Robinson (2002)
	Croatia	8.7	39	Rode et al. (2008)
	Czech republic	6	27	Pebody et al. (2004)
	Denmark	40	32	Smith and Robinson (2002)
	Erythrea	23	28	Smith and Robinson (2002)
	Estonia	15.9	30.5	Cowan et al. (2003)
	Ethiopia	35.1	26.5	Mihret et al. (2002)
	Finland	16	33	Malkin (2004)
	France	15.7	37.5	Malkin (2004)
	Gabon	66	37	Ozouaki et al. (2006)
	Gambia	32.5	32	Smith and Robinson (2002)
	Germany	13.9	32	Pebody et al. (2004)
	Groe ¨nland	74	32	Smith and Robinson (2002)
	(Denm)
	Haiti	54	25	Smith and Robinson (2002)
	Hungary	4	34.5	Hettmann et al. (2008)
	India	12.4	31.1	Cowan et al. (2003)
	Indonesia	18.6	31	Davies et al. (2007)
	Israe ¨l	6.4	40.5	Smith and Robinson (2002)
	Italia	4.8	30	Malkin, 2004
	Japan	1	38	Malkin (2004)
	Ke ´nya	71	32	Smith and Robinson (2002)
	Malawi	64.3	29.5	Sutcliffe et al. (2002)
	Mali	43.3	45	Patnaik et al. (2007)
	Mexico	24	44.5	Smith and Robinson (2002)
	Morocco	26	40	Patnaik et al. (2007)
	Netherlands	8.8	27	Pebody et al. (2004)
	New Zealand	11.5	26	Smith and Robinson (2002)
	Norway	34	35	Malkin (2004)
	Papoua New	27.4	32	Suligoi et al. (2005)
	Guinea
	Peru	35,7	48	Patnaik et al. (2007)
	Philippines	9.2	47	Smith and Robinson (2002)
	Poland	9.15	34	Smith et al. (2002)
	Porto Rico (US)	15.5	34.5	Pe ´rez et al. (2010)
	Russia	20.3	46.4	Khryanin and Reshetnikov
				(2007)
	Senegal	22	25.2	Diawara et al. (2008)
	South Africa	66.5	23	Smith and Robinson (2002)
	Spain	9.4	52	Patnaik et al. (2007)
	Sri Lanka	17.65	37.6	Cowan et al. (2003)
	Sweden	35	33	Malkin (2004)
	Switzerland	11.3	34.5	Malkin (2004)
	Syria	0	25.5	Smith and Robinson (2002)
	Tanzania	57	29.5	Smith and Robinson (2002)
	Thaı ¨land	34.6	51	Patnaik et al. (2007)
	Turkey	37	35.5	Smith and Robinson (2002)
	Uganda	82	34.5	Smith and Robinson (2002)
	England	4.2	37	Pebody et al. (2004)
	USA	28	34.5	Malkin (2004)
	Vanuatu	30	25.8	Haddow et al. (2007)
	Vietnam	21.9	39.5	Le et al. (2009)
	Zambia	63.5	32	Smith and Robinson (2002)
	Zimbabwe	33	33	Smith and Robinson (2002)

Table 2

 2 Association of HSV-2 prevalence and cancer incidences. Odds-ratios and 95% confidence intervals are given for a 10% increase in HSV-2 prevalence.

	Cancer	Odds-ratio	P-value	Bonferroni-corrected
		(95% C.I.)		significance threshold
	Prostate	1.26 (1.12-1.41)	0.00024	0.0022
	Melanoma (male)	1.25 (1.09-1.44)	0.0026	0.0022
	Melanoma (female)	1.27 (1.09-1.49)	0.0031	0.0021

Acknowledgement

We thank Paul Ewald and an anonymous referee for the helpful comments they made to improve the paper.

Conflict of interest statement

We declare no conflict of interest.


Yacine Oussar

Cedric Margo

Stéphane Jérôme Lucas

Fast circular shapes detection in cylindrical ECT sensors by design selection and nonlinear black-box modeling

teaching and research institutions in France or abroad, or from public or private research centers.

Fast circular shapes detection in cylindrical ECT sensors by design selection and nonlinear black-box modeling

Introduction

Electrical Capacitance Tomography (ECT) is a well-known and a widely used sensing technique [START_REF] Yang | Design of electrical capacitance tomography sensors[END_REF]. Generally, an ECT sensor is formed by a set of electrodes arranged circularly. The sensing technique consists in measuring the electrical capacitances between all possible couples of electrodes. Typically, an ECT sensor can yield information on the spatial configuration (size, position) of a non-conducting material whose permittivity is different from that of the environment. Such situation is common when gas bubbles appear in a fluid conveyed in a pipe.

The cross-sectional image of the sensed area of interest can be obtained using capacitance measurements. Usually, the pixels correspond to nodes from a spatial discretization scheme that involves a linear knowledge-based model. This model describes the relationship between the measurements and the permittivity at each node. The image resolution increases with the space discretization. Constructing the image leads to invert an underdetermined set of linear equations [START_REF] Yang | Image reconstruction algorithms for electrical capacitance tomography[END_REF]. 1 imaging in fluids, flow measurement. While both the number and the size of electrodes have great importance regarding the sensor performances, they are usually set arbitrarily and seldom discussed [START_REF] Chen | Image reconstruction for an electrical capacitance tomography system based on a least-squares support vector machine and a self-adaptive particle swarm optimization algorithm[END_REF]. In addition, in most of the studies, all the physically independent measurements are used [START_REF] Wang | Identification of two-phase flow regimes based on support vector machine and electrical capacitance tomography[END_REF]. One can wonder if they are all relevant or just a few of them are sufficient to achieve the best image reconstruction given the sensor structure. Moreover, one might ask: what should be a good structure of the sensor for a given application?

The present study focuses on structure selection (number and size of electrodes) in cylindrical ECT sensors that permits a fast detection of circular shapes with sufficient accuracy. In this work, it is assumed that tuning the number and the size of electrodes allows to optimize the sensor performances for a given application. More precisely, the goal consists in determining the sensor architecture that best predicts the size and the position of a single circular shape in the sensor cross-section. Two different situations are considered. In the first one, the environment permittivity is fixed. In the second one, the environment permittivity can take any value in a known domain. To achieve this goal, non linear black box models were implemented using Least-Square Support Vector Machines (LS-SVM) to establish the relationship between the measured capacitances and the circular shape in terms of size and position.

The paper is organized as follows: Section 2 recalls briefly the ECT sensors background and states the problem. Section 3 describes the LS-SVM modeling technique and the validation method implemented in this work. Section 4 is dedicated to data generation. Sections 5 and 7 present the selection of the most efficient design of ECT sensors, among those considered, in respectively, fixed permittivity and variable permittivity environments. Section 6 illustrates the impact of noisy measurements on the sensor structure that achieves the best prediction performance.

ECT sensor and problem statement

Brief recall of ECT sensor background

Figure 1 illustrates the structure of a typical ECT sensor considered in this study. It is formed by a set of guarded electrodes arranged circularly. The electrodes consist for instance of two conducting layers separated by an insulator. The internal layer is the measurement electrode on which the charges are measured.

The external layer is always driven with the same potential as the measurement electrode but the charges it holds are not measured. As long as the external electrode and the insulating layer are sufficiently thin, the field inside the sensor is little modified. Since the the potentials are the same on both the guard and the electrode, the field between them is null. Consequently, the insulting layer does not contribute to the signal.

Due to the circular symmetry of the sensor, the guard electrodes insure that the electrical field originating from the measurement electrodes is restricted to the closed region circumscribed by the sensor electrodes.

Finally, the designed ECT sensor is only sensitive to its inside region.

The environment delimited by the sensor is characterized by a permittivity ε 1 . A non conducting single circular shape with permittivity ε 2 lies inside the sensor . This circular shape has unknown size and position respectively pointed by a radius R and two coordinates: x and y. According to this formulation, the image reconstruction consists in estimating as accurately as possible the radius and the coordinates of the circular shape.

The measurement of the capacitances appearing between the sensor electrodes are performed by polarizing each electrode independently. When a potential V j is applied to electrode j while all the other electrodes are grounded, the measurement of the charge Q i appearing on electrode i allows the capacitive

ε 1 ε 2 R x y V 4 V 3 V 2 V 1 V 5 V 6 V 7 V 8
Figure 1: An 8-electrode ECT sensor tensor c ij to be determined according to the relation:

Q i = c ij V j (1)
For a given P-electrode sensor, the complete capacitive sensor c ij can be expressed using the capacitances C ij connecting electrode i to electrode j as follows:

c ij = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∑ j C 1j -C 12 -C 13 • • • -C 1P -C 12 ∑ j C 2j -C 23 • • • -C 2P -C 13 -C 23 ∑ j C 3j • • • -C 3P . . . . . . . . . . . . . . . -C 1P -C 2P -C 3P • • • ∑ j C Pj ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (2)
Since the capacitive sensor is symmetric and the diagonal components are a linear combination of other capacitances, there are at most 1 2 P (P -1) independent capacitances where P is the number of electrodes. Usually, the image reconstruction consists in determining the permittivity distribution by solving an underdetermined set of linear equations using an iterative algorithm [START_REF] Yang | Image reconstruction algorithms for electrical capacitance tomography[END_REF][START_REF] Jang | Modified iterative landweber method in electrical capacitance tomography[END_REF]. The algorithm minimizes a cost function given by:

J = m ij -S ijk V εk 2 (3)
where m ij is the measurement at electrode i when electrode j is under the voltage V while all the other electrodes are grounded. Here m ij is homogeneous to a charge, but it could be a current or a potential. Indeed it was proven that they are all equivalent [START_REF] Lucas | Analytical capacitive sensor sensitivity distribution and applications[END_REF][START_REF] Lucas | Simple and direct calculation of capacitive sensor sensitivity map[END_REF]. εk is the permittivity of a small element k resulting from the spatial discretization of the cross-section of interest. S ijk is the sensor sensitivity matrix. It is usually defined as [START_REF] Xie | Electrical Capacitance Tomography for Flow Imaging: System Model for Developement of Image Reconstruction Algorithms and Design of Primary Sensors[END_REF]:

S ijk = δv max δv k × C ij (ε k = ε max ) -C ij ε k = ε min ε max -ε min (4)
where δv k is the "volume" of kth element resulting from the spatial discretization of the cross-section and δv max is the "volume" of the largest element. C ij (ε k = ε max ) and C ij ε k = ε min are respectively the capacitances when the permittivity takes its maximum value ε max and its minimum value ε min in the kth element of volume.

Relation 3 assumes that an additive noise acts on the measurement process. It is generally the case for capacitive systems. Thus, if the predictive model is relevant and its parameters properly estimated, the cost function of relation 3 will be equal to the noise variance.

Problem statement

In the present study, an ECT sensor is fully defined by providing the number and the size of the electrodes.

Indeed, this couple along with the circular geometry set the complete sensor structure. While it is assumed that a single circular pattern lies in the sensor, the basic motivation for this study is twofold: (i) lead to the most accurate and computationally cost effective prediction of its size and position, (ii) selecting an efficient sensor architecture regarding the assigned objective.

In Figure 1, the gap between electrodes is relatively small. This may not be always the case since the size and the number of electrodes have a significant impact on the sensor performances. In an ECT sensor such as on Figure 1, the size of electrodes can be deduced from their number and from the ratio of the sensor perimeter they cover. Therefore, for a given covering ratio, studying the influence of the number of electrodes allows to study simultaneously the influence of the size of the electrodes. Thus, the study was performed considering two parameters: the number of electrodes and the covering ratio. However, see for • Wire electrodes: they cover a small ratio of the sensor perimeter, less than 10% typically. They were successfully used by the authors for the design of ECT sensors dedicated to the measurement of water content in wood-chips [START_REF] Cuvigny | Measuring the retrievable energy from wood-chips using an electrostatic method[END_REF][START_REF] Margo | Wood-chip water content sensor with capacitive tomography[END_REF]. Figure 2 shows an ECT sensor whose electrodes cover 6% of the perimeter. It is connected to an electronic device that carries out charge measurements.

• Intermediate electrodes: they cover any other ratio of the sensor perimeter. Three values are considered in this study: 25%, 50% and 75%.

• Contiguous electrodes: they cover the major sensor perimeter, more than 90%.

Figure 3 illustrates these three types of ECT sensors.

Instead of dealing with an image reconstruction founded on inverting an underdetermined set of equations by minimizing the cost function given by relation 3, a black-box inverse model that predicts the size and the position of single circular pattern is preferred and can be designed using data generated by numerical experiments. The black-box models are implemented using the LS-SVM technique. Since the output of a LS-SVM model is computed using a set of basic arithmetic operations, this approach allows an almost instantaneous prediction of the pattern configuration which makes it highly attractive for a real-time implementation. 3 Training and validating LS-SVM models

ε 1 ε 1 ε 1 ε 2 ε 2 ε 2

LS-SVM model description

In spite of several and efficient techniques for non linear static modeling, such as neural networks, Least Square Support Vector Machines (LS-SVM) are attractive candidates thanks to various interesting properties: they are linear-in-their-parameters models, their training algorithm consists in a quadratic minimization and they have a built-in regularization mechanism [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]. As a result, these properties confer to them the ability to build models with high generalization capabilities by avoiding overfitting and controlling model complexity. Given a set D formed by N examples of S variables and a scalar z:

D = (t i , z i ) ∈ R S × R, i = 1, ...N , nonlinear models of the form: ẑ = w T ϕ (t) + b (5)
are implemented, where t is the vector of input variables, ẑ is the model output, w and b are unknown parameters and function ϕ, usually introduced by a kernel function, is a nonlinear mapping from the original feature space to a higher dimensional feature space [START_REF] Cristianini | Support Vector Machines and other Kernel-based Learning Methods[END_REF]. The LS-SVM training procedure consists in minimizing the cost function defined by:

J (w, e) = 1 2 ∥w∥ 2 + C N ∑ i=1 e 2 i ( 6
)
subject to the equality constraints e i = z iẑi with e i the prediction error taking into account the target output z i and the predicted output ẑi . N is the size of the training set and C is the regularization parameter.

This optimization problem can be cast into a dual form with unknown parameters α and b, α being the vector of the Lagrange multipliers [START_REF] Bertsekas | Nonlinear Programming[END_REF]. The parameters can be computed by resolving the following set of linear equations:

K + 1 2C I N 1 N 1 T N 0 matrix M α b = z 0 (7) with 1 N = [1, 1, ..., 1] T , α = [α 1 , α 2 , ..., α N ]
T and I N is the identity matrix. K is the kernel matrix: its component k ij is defined as the dot product between the ϕ(t i ) and the ϕ(t j ) mappings. In our study, the Gaussian kernel is used. It is a common choice when practicing non linear modeling with SVM. It introduces an additional parameter σ which is its standard deviation. The expression of k ij is given by:

k ij = K t i , t j = exp - t i -t j 2 σ 2 (8)
Parameters σ and C are called the hyperparameters of the LS-SVM model. Their values can be opti-mized using a validation procedure. Hence, the expression of the model becomes:

ẑ = N ∑ i=1 α i K (t i , t) + b (9)
where α and b are the solutions of the system given by (7).

LS-SVM model selection procedure

Since LS-SVM models are linear in their parameters models, the solution of the training phase is unique and can be computed straightforwardly using the set of linear equations given by relation ( 7). This holds when the hyperparameters C and σ are known with fixed values. Usually, these hyperparameters are unknown and must be computed prior to the training phase. A suitable way to proceed consists in selecting the couple (C, σ) that best validates the LS-SVM model. In practice, the generalization capabilities of such black box model are estimated by computing the validation error. Various validation techniques exist in the literature [START_REF] Hastie | The Elements of Statistical Learning[END_REF]. The most popular technique is probably the cross validation method and the Leave-One-Out (LOO) technique. Since a fine search is desirable to best optimize the model performance, the computational burden can rapidly become untractable when using either methods. In order to reduce substantially the computational time of the selection procedure without compromising its efficiency, the validation error is estimated using the Virtual Leave-One-Out (VLOO) method. This method, first proposed for linear models [START_REF] Belsley | Regression Diagnostics: Identifying Influential Data And Sources Of Collinearity[END_REF] and later extended to nonlinear models [START_REF] Laurent | Leverage, local influence and curvature in nonlinear regression[END_REF], allows an estimation of the validation error to be computed by performing only one training involving the whole available data. This estimation is exact when dealing with linear-in-their-parameters models, such as LS-SVM models, while it remains an approximation for models which are non-linear with respect to their parameters. More recently, a framework was described [START_REF] Cawley | Preventing over-fitting during model selection via bayesian regularization of the hyper-parameters[END_REF] to implement the VLOO method for LS-SVM models. For a given LS-SVM model, the VLOO error is computed as:

VLOO error = 1 N N ∑ i=1 α i (M -1 ) ii 2 (10)
where N is the size of the training set, and (M -1 ) ii is the i-th diagonal element of the inverse of matrix M that appears in relation ( 7). Thus, the VLOO permits a fast and exact estimation of the validation error which consists in a great benefit when optimizing the values of the hyperparameters according to a grid search.

Data generation

As discussed in section 2.2 and presented above, the configuration of a circular shape is predicted using LS-SVM models. These black box models must be trained using data. The data can be generated using either a real experimental setup or numerical experiments. Since a large number of real experiments is complicated to carry out, the study uses data generated by simulations. Indeed, the simulation of electric fields is considered as very efficient. Moreover, it makes it possible to finely study parasitic effects such as noise. Since electrodes are considered of sufficient length compared to the sensor and bubble radii, invariance by translation can be assured. As a consequence, most of the simulations were carried out in 2D.

that the sensor sensitivity is sufficient. Consequently, the generated data can be considered informative and relevant to be used to design black box models with machine learning techniques. Thus, data was generated using a knowledge-based model given by a set of differential equations. The spatial discretization of the sensor is implemented using the finite element modeling approach with Gmsh software [START_REF] Dular | [END_REF]). The differential equations are solved using GetDP software (Dular & Geuzaine (accessed January 2012)). Figure 4 illustrates the electrical field lines computed by GetDP when a circular form is present in the sensor. The three electrodes are alternately polarized. When an electrode is polarized to 1 V, the 2 others are grounded. Note that guard electrodes considered in our simulations do not appear on Figure 4 since they are held at the same potential as their corresponding measurement electrodes and they can not be distinguished from the measurement ones at the figure scale. This work considers circular forms of air with relative permittivity ε 2 = 1 in an oil flow having a relative permittivity ε 1 = 3. Thus, the field lines appear almost straight in the circular shape since its permittivity is smaller than the environment permittivity. One can also observe that the field lines are exploring the whole area bounded by the sensor. This does not happen consistently. In particular, when the number of electrodes is large and/or the electrodes are contiguous, the field lines stand in the vicinity of the gaps between the electrodes and miss a large part of the domain circumscribed by the sensor.

1V 0V 0V 0V 0V 1V 0V 1V 0V ε 1 ε 1 ε 1 ε 2 ε 2 ε 2
Figure 4: Typical field lines distribution in a 3-electrode sensor in presence of a single circular form. Each time a different electrode is polarized to 1 V while the two others are grounded. For the sake of clarity, only internal field lines are represented.

For a given sensor structure defined by the number of electrodes and their size, 500 to 1000 circular forms with random sizes and positions are generated. Each time a triplet (x, y, R) is drawn from a uniform distribution, it is necessary to check if the shape of the circular form is limited to the closed region circumscribed by the sensor electrodes. If so, the item is selected and the corresponding capacitances calculated.

Otherwise, the form does not correspond to a realistic configuration. Thereby, it is rejected. This procedure is iterated until the required number of numerical experiments is reached.

Selecting an efficient sensor architecture for a fixed permittivity environment

In this section,the design of ECT sensors with a fixed value of the environment permittivity is considered.

Basically, the design of the inverse model is implemented using three independent models: each of them predicts a component of the triplet (x, y, R) as given by equation ( 11).

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x = ϕ 1 (independent measurements) ŷ = ϕ 2 (independent measurements) R = ϕ 3 (independent measurements) (11)
The search of an efficient sensor architecture leads to a combinatorial optimization. A grid search is applied by considering all the sensors with a number of electrodes in {2, 3, 4, 5, 6, 8, 12, 16} and electrode sizes in {1%, 25%, 50%, 75%, 99%}.

A first attempt to build inverse models to predict the configuration of circular shapes with a linear LS-SVM leads to poor generalization capabilities. This result concerns the modeling technique and not the sensor selection. It means that linear modeling is not sufficient to capture the relationship between the measurements and the output variables to be predicted. Therefore, nonlinear models were implemented using the Gaussian kernel.

Figure 5 shows the validation error for the prediction of the radius R (top graph), the coordinates x (middle graph) and y (bottom graph). Obviously, a 2-electrode sensor is not satisfactory to predict the circular shape configuration. With 3 or more electrodes, all the sensors provide enough information to estimate accurately the circular shape size and position with some differences depending on electrodes size.

Regarding coordinates x and y, a more thorough analysis of the results shows that the most accurate model is obtained with 3 electrodes of intermediate size 50%. The VLOO error computed using relation ( 10) is less than 3% of the sensor radius. Increasing the number of electrodes leads systematically to less efficient models. For intermediate size electrodes, the error increases almost monotonically. However, the behavior of sensors with contiguous electrodes is quite different. At least 5 electrodes are required to achieve a satisfactory prediction accuracy. This can be explained physically by the penetration of the electric field in the sensor. Contrarily to intermediate electrodes, the electric field with contiguous electrodes is indeed concentrated at the junctions between electrodes. Therefore, with contiguous electrodes, increasing their number better distributes the electrical field in the sensor.

Predicting the circular shape size is clearly less critical than the coordinates x and y. A 2-electrode sensor of intermediate size 50% allows to predict radius R with an error of less than 1% of the sensor radius. Similarly to the coordinates, the radius prediction with a contiguous electrodes sensor requires at least 5 electrodes.

The sensors formed by 8, 12 or even 16 electrodes are often preferred in the literature devoted to ECT sensors and show satisfactory performances. However, it appears clearly that such an amount of electrodes is not necessary to estimate efficiently the size and the position of a single circular shape when using the approach proposed in this study.

These promising results were obtained with data obtained with 2D simulations. One can wonder if they are still sound when dealing with data that come from 3D simulations. In order to have a better insight of the effectiveness of the proposed methodology in the 3D case, a set of nonlinear LS-SVM models were built and selected using data that take into account the physical properties of ECT sensors in 3D.

ECT sensors formed by 2 to 8 electrodes of intermediate size 50% were considered. Figure 6 shows the best validation errors obtained for the three coordinates. These errors are compared with those obtained with 2D simulations and the same sensor configurations. Predicting the circular shapes coordinates in a 3D configuration comes with larger errors. They are roughly: (i) 2 times larger for coordinates x and y, (ii) 3 times larger for predicting the radius R. However, these errors remain satisfactory. Coordinates

x and y are best predicted by a 3-electrode sensor with errors representing 6.8% and 5.2% of the sensor radius respectively. The estimation of radius R is achieved with an error less then 1% of the sensor radius regardless of the number of electrodes. Thus, it is reasonable to consider that validating the proposed method with 2D simulations is representative of what would be obtained with 3D simulations. Using 2D simulations has also the advantage to be computationally faster.

Noise robustness

In the previous experiments, data were noise free. If it were real measurements, data would be impacted by noise. This may lead to a decrease in the prediction performance. Since our aim is to select an efficient design of ECT sensors, one can wonder how does the presence of noise in the data influence the sensor structure that achieves an efficient coordinates estimation.

To study the effect of noise in the measurements, several numerical experiments were conducted. A time varying noise was introduced in a test set as realizations of a uniformly distributed random variable in the interval [-a, a]. Six different distributions were considered corresponding to different values of a: 0.1%, 0.3%, 1%, 3%, 10% and 30% of the average signal amplitude.

Figure 7 illustrates the test error for the prediction of radius R (top graph), coordinate x (middle graph)

and coordinate y (bottom graph) for all the sensors formed by a number of electrodes in {3, 4, 5, 6, 8, 12, 16} and covering 50% of the sensor perimeter. For the sake of clarity and in order to limit the figure number, this electrode size was preferred since it showed better performances than other sizes as presented in section 5.

The test error for radius R never exceeds 4% of the sensor radius whatever the sensor architecture between 3 and 6 electrodes. Sensors with larger number of electrodes have worse performances. Thus, the prediction of radius R is not much affected by the presence of noise: a 4-electrode sensor provides the best performance whatever the noise amplitude. However, the best sensor architecture for predicting coordinates x and y evolves differently when the noise amplitude increases. Indeed, for a noise amplitude under 3% of the average signal amplitude, a 3-electrode sensor remains the most efficient to predict the circular shape position. With more electrodes, the test error increases. On the contrary, for a noise amplitude of 10% or more, the performance of sensors formed by 3 to 6 electrodes deteriorates drastically. The 8-electrode sensor offers the best compromise. Since beyond 3 electrodes, the information brought by the measurements does not improve the prediction performance, the latest observation indicates that the information redundancy helps to overcome the presence of noise.

Selecting an efficient sensor architecture with a variable permittivity environment

In this section, the design of an ECT sensor that can be used with different values of the environment permittivity is studied. Typically, such sensor is helpful when several fluids with different permittivities can be conveyed in the same pipe. Indeed, the performance of the sensors designed according to the method described above decreases drastically when the environment permittivity changes. In our experiments the environment permittivity value ε 1 is known and can take any value in the range from 2 to 4 (Olive Oil, Gasoline, Phenol). The values of ε 1 are uniformly distributed between these two values. The goal remains the detection of both size and position of circular shapes made of air. Thereby, the circular shapes permittivity value is constant:

ε 2 = 1.
When the environment permittivity is fixed, it is not a relevant information to design the models given by equation ( 11). However, when the environment permittivity is variable, it influences the measurements and then should be a priori taken into account as a relevant input in the design of the nonlinear inverse models as shown by relation ( 12). Similarly to the previous sections, models of equation ( 11) are built using the LS-SVM modeling technique. Sets of data were generated for each sensor structure formed by 2 up to 7 electrodes. Indeed, the results presented above show that sensors formed by more than 6 electrodes are oversized to predict the configuration of a single circular form using noise free measurements.

Sets formed by 500 samples failed to train efficiently models that achieve a satisfactory performance in the prediction of the triplet ( x, ŷ, R). This means that the relationship between the independent measurements and the variables x, y and R is more complex to model when ε 1 is a variable parameter. As a result, training and validation were conducted using larger sets formed by 1000 examples. The size of the training set was set to 1000 samples since this value ensures good results. Smaller sets formed by 500 to 1000 samples might be sufficient. While the size of the training set can be an important issue when modeling from data, it is out of our concern in this study.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x = ψ 1 (independent measurements, ε 1 ) ŷ = ψ 2 (independent measurements, ε 1 ) R = ψ 3 (independent measurements, ε 1 ) (12)
Figure 8 illustrates the validation error for the prediction of radius R (top graph), coordinate x (middle graph) and coordinate y (bottom graph) for all sensors formed by 2 to 7 electrodes. Dark bars correspond to models given by relation ( 12): ε 1 is an input variable, while white bars correspond to models given by relation ( 11): ε 1 is not provided to the model as an input variable. For radius R, the prediction error is less than 1% of the sensor radius regardless of the number of electrodes when ε 1 is as an input variable.

However, if ε 1 is excluded from the set of input variables, an efficient prediction of R is obtained with sensors formed by 4 electrodes or more. Thus, the redundancy in the information compensates for the lack of input ε 1 . The prediction of both coordinates x and y is almost independent of variable ε 1 as an input. However, the redundancy also plays a role here. Indeed, the best validation error is obtained with a 5-electrode sensor while for a fixed permittivity environment, the best performance was obtained with a 3-electrode sensor.

Although coordinates x and y are predicted with an error of 7% against less than 3% when the permittivity environment is fixed, theses results show the ability of the nonlinear modeling to take into account the variation of the permittivity of the environment solely from the measurements of the capacitances.

Conclusion

This study is dedicated to the selection of an efficient design in ECT sensors for detecting the size and the position of a single circular shape in both fixed and variable permittivity environment. The influence of both the number and the size of the electrodes were investigated and discussed. To this aim, non linear inverse models involving the LS-SVM technique were implemented. Theses models were validated by an efficient validation procedure. The nonlinear inverse models predict the size and the position of a single circular shape using independent measurements.

Regarding the number of electrodes, our results show that increasing this parameter does not lead systematically to an improvement of the sensor performance. For a fixed permittivity environment, the best accuracy is achieved with a sensor formed by 3 electrodes. For a variable permittivity environment, the best performance is obtained with a 5-electrode sensor. This result is counterintuitive since most of the other studies involve ECT sensors with much more electrodes. In the presence of moderate noise, this result holds. However, when the measurements are very noisy, more electrodes are required to achieve a good performance. An 8-electrode sensor shows the best performance. Albeit to a lesser extent, the electrode size also has an impact on the sensor performance. In a fixed permittivity environment, the most accurate sensor was obtained using intermediate size electrodes that cover 50% of the sensor perimeter. The overall results show that the sensor performances are roughly stable with wire or intermediate size electrodes (between 25% and 75% of the sensor perimeter). However, the use of contiguous electrodes requires more electrodes to achieve satisfactory performances. Besides, this technique is efficient with circular shapes. Whenever non circular shapes are presented to the predictive models, the results will be irrelevant. In such situation, new models have to be designed with relevant data drawn from either numerical experiments or physical measurements.

The main result of our study: the use of a 3-electrode sensor in a fixed permittivity environment and 5electrode sensor in a variable permittivity environment, associated to an image reconstruction via a direct calculation for a single circular shape localization and sizing, leads to a computational cost effective sensing approach. This makes it highly attractive for real-time implementations.

Yacine

  Oussar (LPEM, ESPCI Paris, PSL Research University, Sorbonne Universités, UPMC Univ Paris 06, CNRS, Paris, France) Cedric Margo (LPEM, ESPCI Paris, PSL Research University, Sorbonne Universités, UPMC Univ Paris 06, CNRS, Paris, France) Jérôme Lucas (LPEM, ESPCI Paris, PSL Research University, Sorbonne Universités, UPMC Univ Paris 06, CNRS, Paris, France) Stéphane Holé (LPEM, ESPCI Paris, PSL Research University, Sorbonne Universités, UPMC Univ Paris 06, CNRS, Paris, France)

  instance[START_REF] Lucas | Physical limitations on spatial resolution in electrical capacitance resolution[END_REF] section 3.4, it is relevant to consider three types of electrode size : wire, intermediate and contiguous electrodes.

Figure 2 :

 2 Figure 2: ECT sensor formed by a set of wire electrodes and dedicated to the measurement of water content in wood-chips.

Figure 3 :

 3 Figure 3: ECT sensors with wire electrodes (left), intermediate electrodes (middle) and contiguous electrodes (right)

Figure 5 :Figure 6 :

 56 Figure 5: VLOO error for predicting radius R (top), coordinate x (center) and coordinate y (bottom) in a fixed permittivity environment of ε 2 = 1.

  the average signal)

Figure 7 :

 7 Figure 7: Test error for radius R (top), coordinate x (center) and coordinate y (bottom) with noisy measurements.

Figure 8 :

 8 Figure 8: VLOO error for predicting radius R (top), coordinate x (center) and coordinate y (bottom) in a variable permittivity environment.