PhD Isabelle Jaussent
email: isabelle.jaussent@inserm.fr

MD Jean-Paul Cristol

MD, PhD Benedicte Stengel

PhD Marie-Laure Ancelin

MD, PhD Anne-Marie Dupuy

PhD Alain Besset

MD, PhD Catherine Helmer

PhD Karen Ritchie

MD, PhD Claudine Berr

MD, PhD Yves Dauvilliers

Impact of sleep disturbances on kidney function decline in the elderly

While sleep disturbances are frequent in renal disease patients, no studies have examined prospectively the associations between sleep disturbances and kidney function decline in community dwelling elderly subjects. Glomerular filtration rates (eGFRs) were estimated at baseline and at 11-year follow-up. A glomerular filtration decline over the follow-up was defined as a percentage decline greater than or equal to the cut-off value of the highest tertile of kidney function decline (22%) in 1105 subjects. Excessive daytime sleepiness (EDS), insomnia complaints were self-rated at baseline. Restless legs syndrome (RLS) and its age at onset were assessed at study endpoint. Ambulatory polysomnography recording was performed during the follow-up in 277 subjects. Apnea-hypopnea index (AHI), periodic leg movements during sleep (PLMS) and total sleep time (TST) were analyzed. An increased risk of eGFR decline was associated with EDS (OR=1.67 95%CI=1.18-2.34) and RLS (OR=1.98 95%CI=1.18-3.30) independently of potential confounders including cardiovascular risk factors. Among insomnia complaints, a borderline association with eGFR decline was found for early morning awakening only. High AHI (≥30/hour), short TST (<6 hours) but not PLMS were linked to eGFR decline in crude associations, but only AHI remained significantly associated after multi-adjustments. EDS, RLS and AHI constitute independent risk factors for kidney glomerular function decline.

INTRODUCTION

Sleep disturbances and chronic kidney disease (CKD) are often comorbid and are both common conditions in the elderly. Their prevalence increases with age [START_REF] Coresh | Prevalence of chronic kidney disease in the United States[END_REF][START_REF] Ohayon | Meta-analysis of quantitative sleep parameters from childhood to old age in healthy individuals: developing normative sleep values across the human lifespan[END_REF]. Sleep disturbances such as insomnia, sleep apnea syndrome (SAS), restless legs syndrome (RLS) and excessive daytime sleepiness (EDS) are very common in patients with early or late endstage renal disease [START_REF] Plantinga | Association of sleep-related problems with CKD in the United States, 2005-2008[END_REF][START_REF] Turek | Sleep disturbances as nontraditional risk factors for development and progression of CKD: review of the evidence[END_REF]. However, it remains unclear as to whether the associations between sleep disorders and CKD are unidirectional or bidirectional.

Few longitudinal studies on sleep disorders and renal function have been conducted in clinical populations only. One study reported that patients with various degrees of chronic renal failure had progressively worse sleep quality over a three-year follow-up [START_REF] Sabbatini | Sleep quality in patients with chronic renal failure: a 3-year longitudinal study[END_REF]. Conversely, another study reported that patients newly diagnosed with CKD improved their sleep quality over a four-year period [START_REF] Santo | A longitudinal study of sleep disorders in early-stage chronic kidney disease[END_REF]. These inconstancies could be attributed to differences in design, sample size, clinical setting, and heterogeneity in CKD stages. The impact of sleep disorders on the development and progression of cardiovascular disease, hypertension, diabetes, and obesity has been previously observed [START_REF] Wolk | Sleep and cardiovascular disease[END_REF][START_REF] Pepin | Hypertension and sleep: overview of a tight relationship[END_REF], all factors being associated with decline in glomerular filtration rate [START_REF] Yu | Progression of chronic renal failure[END_REF]. It has been suggested that sleep fragmentation due to periodic limb movements (PLMS), respiratory events, and pain may induce a renin-angiotensinaldosterone system hyperactivation, an autonomic nervous system dysregulation, and consequently increases in blood pressure in turn increasing risk for CKD progression. High rates of insomnia and EDS complaints in the CKD population may also contribute to the burden of cardiovascular disease in this at-risk population, as already reported in the general population [START_REF] Blachier | Excessive daytime sleepiness and vascular events: the Three City Study[END_REF][START_REF] Jaussent | Insomnia, daytime sleepiness and cardiocerebrovascular diseases in the elderly: a 6-year prospective study[END_REF].

In view of these conflicting results, further prospective studies are required to better understand the directionality of these associations, notably to determine whether sleep disturbances aggravate CKD or if CKD increases the severity of sleep problems per se. So far, no studies have assessed prospectively whether sleep disturbances alter kidney function in the general elderly population. We thus proposed to assess, over an 11-year follow-up, the relationships between insomnia complaints, EDS, SAS, RLS and glomerular filtration decline in community dwelling elderly people, taking into account confounding factors comorbid with renal alteration.

METHODS

Study population

Participants were recruited as part of the Three-City Study, an ongoing multi-site longitudinal study involving three French cities: Bordeaux, Dijon and Montpellier. Briefly, noninstitutionalized participants aged ≥65 years were randomly selected from electoral rolls between 1999 and 2001. The study protocol was approved by the ethical committee of the University Hospital of Kremlin-Bicêtre and CPP Sud Méditérannée III, and written informed consent was obtained from each participant. The participants were administered standardized questionnaires and underwent clinical examinations at baseline and at two, four, eight and 11year follow-up. Serum creatinine measurement was performed at baseline and at 11-year follow-up in both Montpellier and Bordeaux centers only.

Assessment of change in glomerular filtration rate (GFR)

Serum creatinine was measured in a single laboratory using the colometric Jaffé method at both times. To standardize creatinine values, 1720 frozen serum baseline samples from all three centers [START_REF] Stengel | Epidemiology and prognostic significance of chronic kidney disease in the elderly--the Three-City prospective cohort study[END_REF] and 301 from the Montpellier center at endpoint were remeasured using an Isotope Dilution Mass Spectrometry (IDMS) traceable enzymatic assay (Roche assay) [START_REF] Pieroni | A multicentric evaluation of IDMS-traceable creatinine enzymatic assays[END_REF].

Equations were developed at both times to standardize the results obtained from the Jaffé assay to enzymatic measurements. GFR expressed as mL/min/1.73m 2 was estimated (eGFR) using the CKD-EPI formula without correction for ethnicity (not available in the 3C study and not recommended in France by Haute Autorité de Santé) [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF]. At baseline, eGFR categories were defined according to the National Kidney Foundation guidelines [START_REF]National Kidney Foundation DIGO 2012 Clinical Practice Guideline for the Evaluation and Management of Chronic Kidney Diseasee[END_REF] as 1) stages 1-2: normal to mild decrease, eGFR≥60 2) stage 3: moderate alteration, eGFR between 30 and 59 and 3) stage 4 or higher: severe alteration, eGFR<30mL/min/1.73m 2 . Percentage change in eGFR over time was calculated as: (baseline eGFR-11-year eGFR)/(baseline eGFR)x100. A significant eGFR decline ≥22% (highest tertile of eGFR change) was used to study kidney function decline in our cohort.

Sleep assessment

Sleep complaints were assessed at baseline as part of a clinical interview carried out by a psychologist or nurse, followed by the self-completion of a sleep questionnaire. The participants were invited to answer "never, rarely, frequently, or often" to the following questions: "Do you feel very sleepy during the day?" (EDS), "Do you have any difficulty in falling asleep?" (difficulty in initiating sleep: DIS), "Do you wake up during the night?" (difficulty in maintaining sleep: DMS), "Do you often wake up early in the morning without being able to go back to sleep?" (early morning awakening: EMA), "Do you snore loudly?"

In the analyses, EDS was defined as reporting frequently/often being excessively sleepy. Insomnia complaints based on DIS, DMS, and EMA were also dichotomized as frequently/often versus never/rarely and summed to obtain the number of insomnia complaints (range 0-3). The risk of a "potential" SAS was defined as being obese (body mass index (BMI)>30 kg/m 2 ) with frequent/often loud snoring and EDS.

RLS was assessed at study endpoint using the four questions designed to address the minimal diagnostic criteria of the International RLS Group [START_REF] Allen | Restless legs syndrome: diagnostic criteria, special considerations, and epidemiology. A report from the restless legs syndrome diagnosis and epidemiology workshop at the National Institutes of Health[END_REF]. Diagnosis of RLS was based on the four positive answers, with a further question asked on age at RLS onset. We excluded for this analysis 29 participants with Parkinson's disease or treated with dopaminergic therapy.

Among the participants with creatinine measured at both evaluations, 277 (all from the Montpellier center) underwent, on a voluntary basis, an ambulatory polysomnography (PSG) recording during the follow-up [median at 9.70 years (range, 4.80-12.20)]. No selection was done on sleep complaints and renal function. PSG recordings took place in the subjects' home environment using the Deltamed (Natus) coherence system that include five electroencephalography leads, right and left electro-oculograms, electromyography of chin and tibialis anterior muscles, electrocardiogram, nasal cannula/pressure transducer, mouth thermistor, chest and abdominal bands, body position and pulse oximeter. Sleep stage, microarousals, periodic limb movements (PLM) and respiratory events were scored manually according to standard criteria [START_REF] Iber | The AASM manual for the scoring of sleep and associated events: rules, terminology, and technical specifications[END_REF]. Obstructive apnea was defined as a complete airflow cessation for more than 10 seconds associated with thoracoabdominal movements, and hypopnea as airflow reduction of more than 50% associated with a drop in SpO2 (blood oxygen saturation measured using a pulse oximeter) of more than 3% or a micro-arousal. The average number of apneas/hypopneas per hour of sleep (apnea-hypopnea index (AHI)) was calculated. In the present study, only AHI and its related O 2 saturation parameters (i.e. mean and minimum O 2 saturation, and percentage of time<90%), PLMS and total sleep time (TST) were taken into account with TST<6 hours, AHI≥30/hour and PLMS≥15/hour being considered as pathological conditions.

Assessment of covariates

The standardized interview at baseline included questions on socio-demographic characteristics, alcohol consumption, caffeine intake and smoking status, health status and medication use. Case-level depressive symptoms were defined as a score≥16 on the Center for Epidemiological Studies-Depression Scale [START_REF] Radloff | The CES-D Scale: a self-report depression scale for research in the general population[END_REF], or taking current antidepressant treatment. Cognitive impairment was defined as a score<26 on the Mini-Mental State Examination [START_REF] Folstein | Mini-mental state". A practical method for grading the cognitive state of patients for the clinician[END_REF]. Disability was assessed by the Instrumental Activities of Daily Living scale (IADL) [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF]. Anthropometric measurements including height and weight were performed during the clinical examination to calculate BMI defined as weight (kg) divided by height squared (m 2 ). Hypertension was defined by measured systolic blood pressure≥160 mmHg or diastolic blood pressure≥95 mmHg or current antihypertensive treatment. Diabetes was defined as fasting glucose level≥7.0 mmol/L or treatment for diabetes, and hypercholesterolemia as total cholesterol level≥6.2 mmol/L or treatment with lipid-lowering agents. History of respiratory and cardio-cerebrovascular diseases (angina pectoris, myocardial infarction, cardiovascular surgery, arteritis, and stroke) was investigated. Drugs were coded according to the World Health Organization's Anatomical Therapeutic Chemical Classification [START_REF]Guidelines for ATC Classification and DDD Assignment[END_REF]. Hypnotics were classified as benzodiazepine (BZD), BZD-like compounds (zolpidem, zopiclone), and miscellaneous medications (including barbiturates, antihistamines, and other pharmacological categories such as neuroleptics) [START_REF] Jaussent | Hypnotics and mortality in an elderly general population: a 12-year prospective study[END_REF].

Statistical analyses

Logistic regression models were used to compare the characteristics of participants according to the eGFR categories (<60 vs ≥60 mL/min/1.73/m 2 ) at baseline after adjustment for study center, age and gender. To analyze the associations of eGFR decline (≥22% vs <22%) with sleep disturbances, logistic regression models were used to estimate odds ratios (OR) and their 95% confidence intervals (CI). Multivariate models included study center and covariates associated with eGFR decline at p<0.15. Where appropriate, the interaction terms were tested using the Wald- 2 test. Secondary analyses were implemented in a subsample of subjects free of CKD (i.e. eGFR≥60 mL/min/1.73m 2 ) at baseline, and in those with ambulatory PSG, to assess the impact of AHI, PLMs and TST on eGFR decline. Significance level was set at p<0.05. Analyses were performed using SAS-version 9.4 (SAS Inc, Cary, NC, USA).

RESULTS

Subject characteristics

The final sample included 1105 participants from Bordeaux (n=438) and Montpellier (n=667) with a median baseline age of 70.9 years (range, 65.0-87.8) of whom 57.9% were women. As detailed in the flow-chart diagram (Figure 1), these subjects were free of dementia, had fully completed the sleep questionnaire, with eGFRs evaluated at baseline and 11-year follow-up, and without missing data in baseline adjustment covariates. Participants excluded from the study had significantly a lower education level, were older, living alone with activity limitations, more cardio-vascular risk factors, chronic disease and sleep disturbances, taking more hypnotics and with a lower baseline eGFR.

At baseline, 19.2% of the participants had frequent/often EDS, and 74.8% insomnia complaints of whom 30.2% had three insomnia complaints. Furthermore, 17.1% of the participants used regularly sleep medication (67.4% BZD, 32.1% BZD-like compounds, 11.6% miscellaneous medication). Among those subjects, 45.8% had three insomnia complaints and 13.7% none. RLS was reported by 22.2% (n=196) of subjects (62.9% with DMS symptoms, 41.4% with DIS and 35.7% with EMA, and 27.1% without any insomnia complaints), of whom 35.7% declared having RLS before study inclusion.

Median baseline eGFR was 81 mL/min/1.73m 2 (interquartile range, 72-88). Only 9.0% (n=74) had a baseline eGFR<60 including one participant<30. The participants with eGFR<60 differed significantly from those with an eGFR≥60 mL/min/1.73m 2 in being frequently with hypercholesterolemia and hypertension after adjustment for center, age and gender. No significant associations were found between eGFR levels and sleep disturbances (Supplementary table).

Association between sleep disturbances and eGFR decline over 11-year follow-up

The median delay between the collections of both biological samples was 11 years (range, 10.0-12.5). Over this period, 32.1% (n=355) had a moderate to severe eGFR decline including 18 participants with a severe eGFR. The median percentage of eGFR decline was 14% (interquartile range, 7-26) which corresponds to 1.32% per year (interquartile range, 0.59-2.32).

Baseline sociodemographic and clinical characteristics of participants according to kidney function decline are given in Table 1. Participants with a decline ≥ 22% (i.e. highest tertile of kidney function decline) were more frequently men, older, overweight or obese, had diabetes mellitus and hypertension. They also tended to be past or current smoker and had a history of cardiovascular disease (p<0.10). Subsequent analyses were thus adjusted for these factors.

Table 2 shows the associations of baseline sleep disturbances with eGFR decline during the follow-up. Whereas DIS, DMS and the number of insomnia complaints were not significantly associated with eGFR decline, a borderline association was observed for EMA after adjustment for potential confounders (model 2, OR=1.32 95%CI=0.99-1.75). The risk of eGFR decline increased with loud snoring (model 2, OR=1.37 95%CI=1.04-1.81), and clinically-defined apnea (model 2, OR=2.59 95%CI=1.16-5.78). A positive association was also observed between EDS and eGFR decline even after multiple adjustments (model 2, OR=1.67 95%CI=1.18-2.34) and persists even after further adjustment for loud snoring and EMA (OR=1.58 95%CI=1.12-2.23 p=0.009). No significant interactions were found for eGFR decline between EDS and gender or age.

RLS also increased the risk of eGFR decline when its age at onset preceded study inclusion (model 2, OR=1.98 95%CI=1.18-3.30) even after adjustment for EDS (OR=1.90 95%CI=1.13-3.18 p=0.02). In contrast, no significant increased risk of eGFR decline was found when RLS was reported at endpoint only.

Sensitivity analyses

Supplementary analyses were performed 1) excluding the 74 participants with moderate to severe eGFR impairment (<60 mL/min/1.73m 2 ) at baseline 2) taking into account the new cerebro-cardiovascular events reported over the follow-up (n=71) and the results remained unchanged.

In the 277 participants having ambulatory PSG recording, only 9.3% had an index of AHI≥30/h and this was associated with loud snoring (p=0.007) but not with EDS (p=0.90) or BMI (p=0.22). A high AHI was also associated with an eGFR decline (OR=2.80 95%CI=1.21-6.44 p=0.02) and the results remained significant after adjustments for age, gender, BMI, smoking status, diabetes mellitus, hypertension, history of cardiovascular disease and EDS (OR=2.50 95%CI=1.01-6.20 p=0.04) (data not shown). No significant interaction was found between AHI and EDS for eGFR decline (p=0.99). Furthermore, no significant associations were found either for mean (p=0.34) and minimum O 2 saturation (p=0.16) or for percentage of time<90% (p=0.28). A short TST (<6 hours) was found in 50.4%, without any association with insomnia complaints. A short TST was associated with an eGFR decline (OR=1.75 95%CI=1.05-2.93 p=0.03), however the results became nonsignificant after multi-adjustments (p=0.08). An index of PLMS≥15/h found in 62.4% was associated with RLS at endpoint only (p=0.04) but not when already reported at study inclusion (p=0.10). Finally, no significant associations were found between PLMS and eGFR decline neither in crude (p=0.20) nor in adjusted association (p=0.40).

DISCUSSION

We examined associations between a large range of sleep disturbances and eGFR decline at 11-year follow-up in a general elderly population. Subjects with EDS had a 1.7-fold greater risk of having eGFR decline after adjustment for numerous potential confounders including baseline cardiovascular risk factors. RLS was also found associated with a twice higher risk of eGFR decline independently of EDS. At the exception of EMA which showed a borderline association, insomnia complaints were not related to eGFR decline. In a subsample, high AHI and shorter TST but not PLMS were linked to eGFR decline in crude associations but only AHI remained significant after adjustments.

Although associations between renal functioning and sleep have been previously observed [START_REF] Turek | Sleep disturbances as nontraditional risk factors for development and progression of CKD: review of the evidence[END_REF], to our knowledge no study has investigated the impact over time of sleep disturbances on eGFR decline in a general population sample. The few clinical studies which have examined the relationship between sleep quality and renal failure in CKD patients with early or endstage have produced conflicting results [START_REF] Sabbatini | Sleep quality in patients with chronic renal failure: a 3-year longitudinal study[END_REF][START_REF] Santo | A longitudinal study of sleep disorders in early-stage chronic kidney disease[END_REF]. The association between EDS and eGFR decline reported in our study was independent of cardiovascular diseases which failed to support the hypothesis mediating effect of cardiovascular diseases in the relationship between EDS and eGFR in the elderly. Several studies have suggested that cardiovascular and kidney diseases are closely linked through multiple interact effects including neuroendocrine perturbations, fluid dysregulation, inflammation and immune disturbances resulting in amplification loops defined as cardio-renal syndrome. In addition to cardiovascular diseases, diabetes, hypertension and obesity can participate also in the progression of CKD. We report here that EDS is associated with eGFR decline independently of vascular and metabolic risk factors. On the other hand, in previous analyses, we reported that EDS increased the risk of cardiovascular events both fatal and nonfatal, whether first events or recurrent over 6-year follow-up [START_REF] Jaussent | Insomnia, daytime sleepiness and cardiocerebrovascular diseases in the elderly: a 6-year prospective study[END_REF]. The underlying pathophysiological mechanism by which EDS interacts with eGFR decline thus appears to be complex and multifactorial.

Sleep disturbances were often associated with activations of hypothalamic-pituitary-adrenal axis and sympathetic nervous system, as well as chronic inflammation that may promote a non-dipping pattern, hypertension and subsequently alter renal function [START_REF] Pepin | Hypertension and sleep: overview of a tight relationship[END_REF]. In our sample, the relationship between EDS and eGFR decline was neither modified by BMI nor by the presence of snoring; however the latter parameters were weaker predictors of SAS in the elderly than in middle aged adults [START_REF] Young | Predictors of sleep-disordered breathing in community-dwelling adults: the Sleep Heart Health Study[END_REF]. In the subsample, no associations were found between AHI and EDS or BMI; however we reported an association between AHI and eGFR decline independently of EDS and cardiovascular risk factors without any impact of nocturnal hypoxia (i.e. no between-group difference for mean and minimum O 2 saturations). This notably differs from the direct nocturnal hypoxia effect already reported on the progression of CKD through a renin-angiotensin-aldosterone system hyperactivation [START_REF] Ahmed | Nocturnal hypoxia and loss of kidney function[END_REF][START_REF] Nangaku | Chronic hypoxia and tubulointerstitial injury: a final common pathway to end-stage renal failure[END_REF]. The link between EDS and eGFR decline may involve a shorter nocturnal sleep duration, a condition frequently associated with increased systemic low-grade inflammation that may trigger cardiovascular disease and subsequent CKD [START_REF] Cappuccio | Sleep duration predicts cardiovascular outcomes: a systematic review and meta-analysis of prospective studies[END_REF]. However the association loses its significant in multivariate model.

The proportion of subjects with RLS (8.3% at baseline) was in the range of previous reported studies in the elderly [START_REF] Allen | Restless legs syndrome prevalence and impact: REST general population study[END_REF], and it is known to increase with end-stage renal disease [START_REF] Unruh | Restless legs symptoms among incident dialysis patients: association with lower quality of life and shorter survival[END_REF]. We found that eGFR decline was predicted by RLS at inclusion but not when present at study endpoint only, independently of cardio-vascular risk factors and EDS. These findings suggested that RLS could be an early risk factor for eGFR decline through the bias of an association with sleep fragmentation and periodic-related nighttime increased blood pressure [START_REF] Pennestri | Nocturnal blood pressure changes in patients with restless legs syndrome[END_REF], conditions at risk for renal dysfunctioning. However, no associations were found between eGFR decline and PLMS. Except for a borderline association with EMA, other subtypes of insomnia complaints, and depressive symptoms were not significantly associated eGFR decline. Altogether these results supported our previous findings that insomnia complaints were not risk factors for cardiovascular diseases [START_REF] Jaussent | Insomnia, daytime sleepiness and cardiocerebrovascular diseases in the elderly: a 6-year prospective study[END_REF], insomnia being more likely a consequence of these chronic diseases.

The present study benefits from several strengths including standardized creatinine measurements which reduced the bias in the estimate eGFR decline, a large sample size, an 11-year follow-up period of elderly subjects not pre-selected for kidney disease, and a large range of subjective and objective sleep measures.

The present study has some limitations. Selection bias concerned the recruitment of volunteers from electoral rolls and the exclusion of institutionalized elderly people, which limits the extent to which these findings can be generalized to all older adults. Selection bias concerned also the exclusion of subjects due to missing data, loss to follow-up and death. The excluded subjects were older, had more often chronic disorders, a lower baseline eGFR and were more likely to report sleep complaints and to take hypnotics. This progressive selection of the cohort may cause biases which lead to 1) an underreporting of sleep disorders prevalence and kidney failure incidence rate and 2) a modification (to a lesser extent) of the associations between sleeps complaints and eGFR decline.

Sleep complaints were self-reported using a short questionnaire already used [START_REF] Blachier | Excessive daytime sleepiness and vascular events: the Three City Study[END_REF][START_REF] Jaussent | Insomnia, daytime sleepiness and cardiocerebrovascular diseases in the elderly: a 6-year prospective study[END_REF][START_REF] Jaussent | Hypnotics and mortality in an elderly general population: a 12-year prospective study[END_REF], which while lacking external confirmation, remains the most common method for initial diagnosis of sleep pathologies in the primary healthcare setting. Unfortunately, validated sleep measures such as the Epworth Sleepiness Scale which provides a measurement of the subject's general level of daytime sleepiness were not available for this study.

Due to the study design and population (large multicentric elderly cohort), it was not possible to perform PSG in laboratory for the whole sample and this was only possible in the center of Montpellier (one quarter of the sample) in an ambulatory way based on a voluntary basis without selection made on sleep complaints and renal function. PSGs were performed under naturalistic conditions, carried out by skilled sleep specialized technologists with only 2% of recordings being unusable. PSGs were recorded during the follow-up precluding examination of the relationships with baseline sleep complaints. However, the natural course of sleep disturbances is slow in the elderly [START_REF] Sforza | A 3-year longitudinal study of sleep disordered breathing in the elderly[END_REF] thus our small sub-sample with ambulatory PSG recording helped to further understand the underlying mechanisms involved in the relationships between sleep disturbances and eGFR decline.

Finally, despite extensive adjustments, the possibility remains that unmeasured factors such as inflammatory biomarkers may also be involved and confound associations.

CONCLUSION

The results of this large 11-year prospective study in the elderly showed that EDS, RLS and AHI constitute risk factors of renal function alteration at a very early stage of decline.

ACKNOWLEDGMENTS

The 3C Study is conducted under a partnership agreement between Inserm, the Victor Segalen -Bordeaux II University and Sanofi-Synthélabo. The Fondation pour la Recherche Médicale funded the preparation and first phase of the study. The 3C-Study is also supported by the Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, MGEN, Institut de la Longévité, Agence Française de Sécurité Sanitaire des Produits de Santé, the Regional Governments of Aquitaine, Bourgogne and Languedoc-Roussillon and, the Fondation de France, the Ministry of Research-Inserm Programme "Cohorts and collection of biological material".The Lille Génopôle received an unconditional grant from Eisai. The Fondation Plan Alzheimer funded the follow-up of the cohort. Part of this project is financed by grants from the Agence Nationale de la Recherche (projects ANR 2007-LVIE-004 and 06-PNRA-005).

Figure 1 .

 1 Figure 1. Flow chart of participant inclusion

Table 1 :

 1 Baseline sociodemographic and clinical characteristics of participants according to the estimated GFR (eGFR) decline over 11-year follow-up.

	eGFR decline over 11-year
	follow-up
	<22%	≥22%
	N=736	N=369

Table 2 :

 2 Baseline sleep complaints of participants according to the percentage decline in estimated GFR decline over 11-year follow-up.

eGFR decline over [START_REF] Jaussent | Insomnia, daytime sleepiness and cardiocerebrovascular diseases in the elderly: a 6-year prospective study[END_REF] (1) Model 1 (2) Model 2 (3) (1) Model 1 (2) Model 2 (3)

Variable n % N % OR [95% CI] p OR [95% CI] p OR [95% CI] p

(1) Model 0: Crude associations (2) Model 1: Adjusted for center, age, gender (3) Model 2: adjusted for all the covariates on model 1 plus BMI, smoking status, diabetes mellitus, hypertension and history of cardiovascular disease. (4) For apnea, the adjustment for BMI was not applied because of the colinearity.


Tomer Libal
email: tomer.libal@inria.fr

Towards Deciding Second-order Unification Problems Using Regular Tree Automata

The second-order unification problem is undecidable [START_REF] Goldfarb | The undecidability of the second-order unification problem[END_REF]. While unification procedures, like Huet's pre-unification, terminate with success on unifiable problems, they might not terminate on non-unifiable ones. There are several decidability results for unification problems with infinitely-many pre-unifiers, such as for monadic second-order problems [START_REF] William | A unification algorithm for second-order monadic terms[END_REF]. These results are based on the regular structure of the solutions of these problems and by computing minimal unifiers.

Beyond the importance of the knowledge that searching for unifiers of decidable problems always terminates, one can also use this information in order to optimize unification algorithms, such as in the case for pattern unification [START_REF] Miller | Unification of simply typed lambda-terms as logic programming[END_REF].

Nevertheless, being able to prove that the unification problem of a certain class of unification constraints is decidable is far from easy. Some results were obtained for certain syntactic restrictions on the problems (see Levy [START_REF] Levy | Decidable and undecidable second-order unification problems[END_REF] for some results and references) or on the unifiers (see Schmidt-Schauß [START_REF] Schmidt-Schauß | A decision algorithm for stratified context unification[END_REF], Schmidt-Schauß and Schulz [START_REF] Schmidt-Schauß | Solvability of context equations with two context variables is decidable[END_REF][START_REF] Schmidt-Schauß | Decidability of bounded higher-order unification[END_REF] and Jeż [START_REF] Jez | Context unification is in PSPACE[END_REF] for some results).

Infinitary unification problems, like the ones we are considering, might suggest that known tools for dealing with the infinite might be useful. One such tool is the regular tree automaton. The drawback of using regular automata for unification is, of course, their inability to deal with variables. In this paper we try to overcome this obstacle and describe an on-going work about using regular tree automata [START_REF] Comon | Tree automata techniques and applications[END_REF] in order to decide more general second-order unification problems.

The second-order unification problems we will consider are of the form λz n .x 0 t . = λz n .C(x 0 s) where C is a non-empty context [START_REF] Comon | Completion of rewrite systems with membership constraints. part i: Deduction rules[END_REF] and x 0 does not occur in t or s. We will call such problems cyclic problems.

An important result in second-order unification was obtained by Ganzinger et al. [START_REF] Ganzinger | Rigid reachability[END_REF] and stated that second-order unification is undecidable already when there is only one second-order variable occurring twice. The unification problem they used for proving the undecidability result was an instance of the following cyclic problem. Note that we chose to use in the definition only unary second-order variables but that this restriction should not be essential.

x 0 (w 1 , g(y 1 , a)) = g(y 2 , x 0 (w 2 , a)) (1)
Our decidability result is obtained by posing one further restriction over cyclic problems which is based on the existence and location of variables other than the cyclic one.

A sufficient condition for the decidability of second-order unification problems was given by Levy [START_REF] Levy | Decidable and undecidable second-order unification problems[END_REF]. This condition states that if we can never encounter, when applying Huet's preunification procedure [START_REF] Gérard | A unification algorithm for typed lambda-calculus[END_REF] to a problem, a cyclic equation, then the procedure terminates.

It follows from this result that deciding second-order unification problems depends on the ability to decide cyclic problems. The rules of Huet's procedure (PUA) are given in Fig. 1. Imitation partial bindings and projection partial bindings are defined in [START_REF] Snyder | Higher-order unification revisited: Complete sets of transformations[END_REF] and are denoted, respectively, by PB(f, α) and PB(i, α) where α is a type, Σ a signature f ∈ Σ and i > 0.

Sσ ∪ {x . = λz k .t} x ∈ FV(t) ∧ σ = [λz k .t/x] S ∪ {λz k .x(z k ) . = λz k .t} (Bind) S ∪ {x . = u, λz k .x α (sn) . = λz k .f (tm)} u ∈ PB(f, α) S ∪ {λz k .x α (sn) . = λz k .f (tm)} (Imitate) 1 S ∪ {x . = u, λz k .x α (sn) . = λz k .a(tm)} i > 0 ≤ k, u = PB(i, α) S ∪ {λz k .x α (sn) . = λz k .a(tm)} (Project) 2
1. where f ∈ Σ.

2. where either a ∈ Σ or a = zi for some 0 < j ≤ k. In order to describe the algorithm, we give next several technical definitions. These definitions are taken from our previous work on extending PUA to deal with some non-termination [START_REF] Libal | Regular patterns in second-order unification[END_REF]. The discussions there can help to supplement the description of the definitions given here.

The main technique behind the results of this paper is to analyze the application of PUA on cyclic equations. A crucial component for obtaining non-termination is the generation of new cyclic equations in each step. The following definition of progressive contexts describes the form of these new cyclic equations along the execution of PUA.

Let e be a cyclic equation as above where

C = C 1 . . . C m such that for all 0 < i ≤ m, C i = f i (r 1 i , . . . , [.], . . . , r ni i ) where n i = arity(f i ) -1. Define also, for all m < i, C i = f k (y 1 i-m s, . . . , [.], . . . , y n k i-m s)
where k = ((i -1) mod m) + 1 and y j i-m for 0 < j ≤ n k are new variables. We define the progressive context D e i for all 0 ≤ i as D e i = C i+1 . . . C i+m . In the rest of this paper, e will refer to equations of this form and t, s, C, m, k, n i , r j i and y j i will refer to the corresponding values in e.

In order to clarify the definitions, we will use the following (non-unifiable) cyclic equation as an example:

x 0 f (a, a) . = f (x 0 a, f (f (a, a), b)
). Note that PUA does not terminate on this problem.

For the example,

t = f (a, a), s = a, C = f ([.], f (f (a, a), b)), m = 1, k = 1, n i = 1, and r 1 i = f (f (a, a), b
) for all 0 < i. The progressive contexts for this example are:

D 0 = C, D 1 = f ([.], y 1 a), D 2 = f ([], y 2 a), etc.
Since the only "don't know" non-determinism in PUA is due to the choices in the search between the rules (Imitate) and (Project) [START_REF] Snyder | Higher-order unification revisited: Complete sets of transformations[END_REF], we can follow the execution of PUA on the cyclic equation e using the search tree in Figure 2 and using the following definitions. Given a cyclic equation e, for all 0 ≤ i, we define I(i), I * (i) and P(i) inductively as follows:

• P(0) = I(0) = I * (0) = ∅. • if 0 < i ≤ m then I * (i) = I * (i -1) ∪ {λz n .y j i t . = λz n .r j i | 1 ≤ j ≤ n i }. • if m < i then I * (i) = I * (i -1) ∪ {λz n .y j i t . = λz n .y j i-m s | 1 ≤ j ≤ n i }.
• for all i > 0,

I(i) = I * (i) ∪ {λz n .x i t . = λz n .D e i (x i s)}.
• for all i > 0,

P(i) = I * (i -1) ∪ {λz n .t . = λz n .D e i-1 (s)}. e I (1)
. . . P(2) (Project),(Bind) (Imitate),(Bind),(Decomp)

P(1)
(Project),(Bind) (Imitate),(Bind),(Decomp)

Figure 2:
The "don't-know" non-determinism in PUA As was noted above, the progressive context is being used in order to describe the form of the new cyclic equations in the sets P(i) and I(i).

From the fact that PUA is complete for higher-order unification [START_REF] Gérard | A unification algorithm for typed lambda-calculus[END_REF] and from the fact that e is a cyclic problem, it follows that e is unifiable iff there is i > 0 such that P(i) is unifiable [START_REF] Libal | Regular patterns in second-order unification[END_REF].

For the example, we have

P(1) = {f (a, a) . = f (f (a, a), b)}, P(2) = {y 1 f (a, a) . = f (f (a, a), b), f (a, a) . = f (a, y 1 a)}, P(3) = {y 1 f (a, a) . = f (f (a, a), b), y 2 f (a, a) = y 1 a, f (a, a) . = f (a, y 2 a)}, etc.
Zaionc [START_REF] Zaionc | The regular expression descriptions of unifier sets in the typed lambda calculus[END_REF] gave a finite description of complete sets of unifiers for some monadic secondorder problems by proving that a structure based on the matching tree generated by Huet's procedure [START_REF] Gérard | A unification algorithm for typed lambda-calculus[END_REF] is regular. We would like to pursue a similar approach for our cyclic problems but unlike the monadic case, where each node in the matching tree contains the same number of equations, for non-monadic signatures there is no bound on the number of equations in the nodes.

In the following definition we remove equations from the P sets in order to obtain similar regularity.

Let P -(i) ⊆ P(i) be the following set of equations.

• if 0 < i ≤ m + 1 then P -(i) = P(i). = f (a, y 2 a)} and P * (3) = {y 2 f (a, a) = y 1 a}. Clearly, P(i) is unifiable only if both P -(i) and P * (i) are (using the same substitution). In [START_REF] Libal | Regular patterns in second-order unification[END_REF] we proved that P -(i) is unifiable only if there is 0 < j ≤ 3m ≤ i such that P -(j) is unifiable. I.e. that we can always decide if there is an i > 0 such that P -(i) is unifiable.

• if m + 1 < i then P -(i) = I * (m) ∪ {λz n .t . = λz n .D e i-1 (
In the present paper, we will describe how to decide the unification problem of these cycles under a very strong restriction. We require that the generated sets P -have at most finitelymany ground unifiers. This is a very strong requirement since even simple first-order problems like f (x) = y have infinitely-many ground unifiers.. Nevertheless, we hope that this preliminary work can be extended to stronger classes. A class subsumed by the above and describable syntactically is the class of second-order problems containing one variable (second or firstorder) occurring at most twice. At the same time, the class described by the cyclic problems in Equation 1 and which was shown to be undecidable [START_REF] Ganzinger | Rigid reachability[END_REF] may generate P -sets having infinitelymany most general unifiers.

In order to decide if such cyclic equations are unifiable, we will first investigate the equations in P -(i) and P * (i) and the relation between them. Let us consider P(i) for some i > 0 and let us pick an arbitrary equation λz n1 .y j k t . = λz n1 .r j k ∈ P -(i) where 0 < k ≤ m and j is some index depending on the arity of the enclosing function symbol in C. This equation is connected to the following set of equations in P * (i): {λz n2 .y j k+m t . = λz n2 .y j k s, λz n3 .y j k+2m t . = λz n3 .y j k+m s, . . . , λz n l .y

j i-1 m t . = λz n l .y j i-1
m -m s}. The last occurrence of a variable in this chain is the occurrence of y j i-1 m s in the equation λz n l+1 .t . = λz n l+1 .D e i-1 (s) ∈ P -(i). Call the equations from P -(i) base equations and the ones from P * (i) inductive equations. In the following algorithm we will consider first the finitely-many P(j) problems for 0 < j ≤ 3m and will then consider the problems P(i) such that P -(i) is isomorphic to P -(j) (this is determined according to the indices i and j, see [START_REF] Libal | Regular patterns in second-order unification[END_REF] for more information). We will call the infinitely-many such problems P(i) the extensions of P(j).

Since in our example m equals 1, there can be only one chain. For P(i) the chain is just the sequence of equations in P(i).

We will require another restriction for the remaining part of the paper and will consider only the case when there is one chain and renumber the indices of the y variables with 1, . . . , p. We believe that both this restriction and our use of unary cyclic variables are not essential to the results obtained. Since there is only one chain and the chain and the problem P(i) are the same, we will consider both the problems P(i) as extensions of P(j) and the (single) chains of P(i) as extensions to the chain in P(j).

In order to define the algorithm, we need first to define how to construct the regular tree automaton based on three terms. The first term will be of the form λz.u where u can contains z but no subterm of the form v, the second term will be v and the third term will be w such that w contains occurrences of z and has no subterm of the form v. Using these three terms, we define the following tree automaton A = (Q, Q f , ∆) where Q = {q w , q u }, Q f = {q u } and ∆ is defined as follows:

• λz.u 1 → q u (λz.u 2 ) where u 1 is obtained from u by replacing each occurrence of z with q w (x l ) where l is a new index for each occurrence. u 2 is obtained in the same way but we replace each occurrence of z with x l .

• w 1 → q w (w 2 ) where w 1 and w 2 are obtained from w in the same way u 1 and u 2 were obtained from u,

• v → q w (v).

• z → q w (z).

We describe this automaton using aut(u, v, w). The idea behind this construction is that the language accepted by this automaton is exactly the one containing all the possible mappings for y 1 in the chain according to the arbitrarilymany constraints in the inductive part of the chain and based on a given unifier σ for the base part of the chain. Given this automaton, we need just to test if σ(y 1 ) is recognized by it. Note that the base equations contain only occurrences of the variables y 1 and y p and therefore, a unifier for the base part poses no constraints on the values for the variables y 2 , . . . , y p-1 in unifiers for the inductive part and we can freely generate all possible substitutions.

We describe next an algorithm for deciding our cyclic problems.

1. given a cyclic problem e.

2. compute the (finite) complete set of unifiers for some P -(j) where 0 < j ≤ 3m.

3. let σ be such a unifier and for the single chain in P -(j) do the following:

(a) let λz.u be obtained from σ(y p ) by replacing all occurrences of sσ with z..

(b) let v = sσ.
(c) let w be obtained from tσ by replacing all occurrences of sσ in it with z.

(d) fail if σ(y 1 ) is not recognized by aut(u, v, w).

The algorithm tries to find one unifier of a P -(j) which can be extended in order to unify some extension P * (i) of P * (j) for an arbitrary i > 0.

The correctness of the above algorithm is based on the following theorem.

Theorem 1. Given a problem e, 0 < j ≤ 3m, a chain in P(j) over variables y 1 , . . . , y p and a unifier σ of P -(j), σ(y 1 ) is recognized by aut(u, v, w) as above iff there is an extension P(i) of P(j) and a substitution θ, such that θ unifies P(i).

Proof. Proof sketch: the automata for the chain and all its extensions are the same.

• if -by induction on the number of the variables y 1 , . . . , y q in the extension P(i) of P(j).

For the step we need to prove that the first equation in the chain (which is determined last since the base of the terms in the language are determined by θ(y q )) is unifiable only if σ(y 1 ) is recognized by the automaton. Since θ extends σ, we have θ(y 1 ) = σ(y 1 ) and by assuming that θ(y 2 ) is recognized, the rest follows from the definition of the automaton.

• only if -for this direction, we need to choose the extension P(i) and build the substitution θ. This is computed by considering the accepting sequence of transitions for σ(y 1 ). The maximal number of nested transitions determines the number of equations in the chain while the transitions themselves determines the values θ(y 2 ), . . . , θ(y q-1 ) in the chain. In addition, θ(y 1 ) = σ(y 1 ) and θ(y q ) = σ(y p ).

We will now demonstrate this idea on the example. A unifier σ for P -(2) is [y 2 → λz.z, y 1 → λz.f (z, b)]. Using the unifier we get that u = λz.z, v = a and w = f (z, z). Therefore, aut(u, v, w) = (Q, Q f , ∆) where Q = {q w , q u }, Q f = {q u } and ∆ is defined as follows:

• λz.q w (x) → q u (λz.x).

• f (q w (x 1 ), q w (x 2 )) → q w (f (x, y)).

• a → q w (a).

• z → q w (z).

Clearly σ(y 1 ) cannot be generated by aut(u, v, w) and therefore we have proved, according to the previous theorem, that there is no possible extension of σ. By doing the same to all unifiers of P -(1), P -(2) and P -(3), we can prove that the example is not unifiable.

To summarize, we have described a decision algorithm for restricted cyclic second-order unification problems. Although the class is strongly restricted, it includes the class of secondorder problems containing only one variable (first or second-order) which occurs at most twice. The main novelty of the method is the use of tree automata in order to decide unification problems. At the same time, the exact syntactic form of cycles in this class and the possibility to treat problems with more than one chain are still under investigation.

Figure 1 :

 1 Figure 1: PUA-Huet's pre-unification procedure

  s)}. and let P * (i) = P(i) \ P -(i). The corresponding values for our example are P -(1) = P(1), P -(2) = P(2), P -(3) = {y 1 f (a, a) . = f (f (a, a), b), f (a, a) .

 [START_REF] Comon | Tree automata techniques and applications[END_REF]Funded by the ERC Advanced Grant ProofCert.