Geoffrey Castillo
email: geoffrey.castillo@univie.ac.at.i

Preference reversals with social distances

Keywords: preference reversals, social distance JEL Codes: C91, D63, D81, D90

Introduction

Preference reversals between choice and valuation are one of the most studied violations of rationality. 1 They show that people reveal different preferences whether they pick an option in a straight binary choice or report a monetary value independently for each option.

1 Preference reversals of this kind are often interpreted as failures of procedure invariance, which says that revealed preferences should not vary with the elicitation method. They have been documented mostly in risky choice and in intertemporal choice, but since this interpretation does not depend on a particular domain they should be observable in other domains.

I extend preference reversals to a new domain, the domain of social distances.

Many important decisions involve social distances, such as the choice between different public policies impacting different communities, so it is important to know whether the preferences driving such decisions are stable to the elicitation method.

I show that preference reversals carry over to social distances and that preference reversals with social distances exist. The experiment also illustrates how one can import and study social distances in the laboratory.

The options used in standard preference reversal experiments can be written as general two-attribute options in which one attribute is an amount of money and the other is a measure of distance. Interpreting the distance as a probability or as a time delay leads to the usual preference reversals encountered in the risk or in the time domain. The preference reversal phenomenon occurs if people choose the distance-rich option but report a higher monetary valuation for the money-rich one. Here I interpret the distance attribute as a social distance and so transpose classical preference reversal experiments to this new domain.

In the experiment, subjects face options that give either a small amount of money to a socially close recipient or a large amount of money to a socially distant one. I study two types of social distances. In the 'Faculty Setting' I study social distances between individuals by inviting subjects from a given faculty and asking them to consider allocations that benefit subjects from other faculties. In the 'Charity Setting' I study social distances between individuals and social groups by making the allocations benefit charities. I rely on a survey and on measures of social distance to find faculties and charities that create small and large social distances. I find a clear pattern of preference reversals in the Faculty Setting, but less so in the Charity Setting. A closer look at social distances shows that the difference in social distance between the options was greater in the Charity Setting than in the Faculty Setting. The scale compatibility hypothesis [START_REF] Tversky | The Causes of Preference Reversal[END_REF] provides a compelling explanation to these findings. According to scale compatibility people's response to an elicitation method depends on the scale used; people thus tend to overweight amounts of money when forming monetary valuations. For the overweighting to create a preference reversal it needs to overturn the preference revealed in choice for the option with the smaller social distance. In the Faculty Setting, where the social distances of the two options are similar, scale compatibility can easily tip people into reversing their preference when reporting their valuation.

On the other hand, in the Charity Setting, social distances are further apart; scale compatibility would need to be stronger to cause people to reverse their choice. As a result we observe more preference reversals in the Faculty Setting than in the Charity Setting.

Preference reversals with social distances have clear implications. They mean that recruiters who judge job candidates on their potential productivity and on how well they would fit in the company, inferred from the social distance between recruiters and candidates, would make different hiring decisions if they directly chose between the candidates or if they first assigned a value to each candidate and then selected the candidate with the highest value. To take another example, contingent valuation studies rely on monetary valuations to compare projects.

If the projects involve social distances-if they impact different communities or countries-the preferences revealed under this method would not be the same as the preferences revealed under straight choice, which might lead one to implement the wrong project. This paper contributes to the large preference reversal literature (see [START_REF] Seidl | Preference reversal[END_REF] for a review) by showing that preference reversals also carry over to choice across social distances. To my knowledge no other paper has studied preference reversals in this domain. In this sense it follows the tradition of testing to what extent preference reversals generalise to new domains; for example, preference reversals have been found under ambiguity [START_REF] Ball | Do preference reversals generalise? Results on ambiguity and loss aversion[END_REF][START_REF] Maafi | Preference Reversals Under Ambiguity[END_REF] or when comparing cash and non-cash incentives [START_REF] Shaffer | Preference reversals in evaluations of cash versus non-cash incentives[END_REF]. The scale compatibility hypothesis is currently the most compelling explanation of preference reversals, supported by decision times and eye-tracking [START_REF] Alós-Ferrer | Preference reversals: Time and again[END_REF][START_REF] Alós-Ferrer | Attentional shifts and preference reversals: An eye-tracking study[END_REF].

In the next Section I translate preference reversals to social distances. Section 3 details the experimental design and Section 4 reports the results. Section 5 closes the paper with a discussion of the results.

Extending preference reversals to social distances

Denote the options by ω = (d, x) where d is a measure of distance and x an amount of money. Preference reversals experiments use options ω d and ω x where ω d is better in d while ω x is better in x. A standard preference reversal happens when people pick ω d in choice and so reveal ω d ω x ; but report a higher monetary valuation V (•) for ω x , V (ω x ) > V (ω d ), and so reveal ω x ω d .

When d is a probability p the options become binary gambles, for example used by [START_REF] Plott | Economic theory of choice and the preference reversal phenomenon[END_REF]: ω p is a gamble that offers a large probability of winning a small amount of money while ω x offers a smaller probability of wining a larger amount. When d is a time delay t the options are delayed payments, used by [START_REF] Tversky | The Causes of Preference Reversal[END_REF]: ω t is a small amount of money to be received soon while ω x is a larger amount to be received later.

Transposing these to social distances, the options become (s, x) where s is the social distance measured between a subject and the recipient of an amount of money x. Then, ω s gives a small amount of money to a recipient socially close to the subject, and ω x , a larger amount of money to a recipient more socially distant. If preference reversal carry over to social distances we should observe subjects revealing ω s ω x in choice but reporting monetary valuations such that V (ω x ) > V (ω s ). The experiment will test this prediction.

3 Experimental design

Social distances in the laboratory

In principle social distances can be of two types: social distance between individuals, and social distance between individuals and groups. The experiment studies both in two separate settings. In the 'Faculty Setting' I invited subjects from a given faculty of the University of Nottingham, and the allocations benefited students from other faculties. In the 'Charity Setting', the allocations benefited charities.

To construct ω s and ω x we need to know what are small and large social distances, therefore we need to measure them. For the distance between individuals I use the Inclusion of Other in the Self scale [START_REF] Aron | Inclusion of Other in the Self Scale and the structure of interpersonal closeness[END_REF]. This measure has proven popular in psychology (see for example Aron and Mashek, 2004;Aron et al., 2004;[START_REF] Cialdini | Reinterpreting the empathy-altruism relationship: When one into one equals oneness[END_REF] and has recently entered the toolbox of economists [START_REF] Gächter | Measuring the Closeness of Relationships: A Comprehensive Evaluation of the 'Inclusion of the Other in the Self' Scale[END_REF][START_REF] Gächter | Revealing the economic consequences of group cohesion[END_REF]. Its counterpart to measure the social distance between individuals and groups is the Inclusion of Ingroup in the Self scale [START_REF] Schubert | Overlap of self, ingroup, and outgroup: Pictorial measures of self-categorization[END_REF][START_REF] Tropp | Ingroup identification as the inclusion of ingroup in the self[END_REF][START_REF] Wright | Including others (and groups) in the self: Self-expansion and intergroup relations[END_REF].

I conducted online surveys using these measures to find small and large social To isolate the effect of social distances, the experiment controlled for selfish motives by having allocations that never benefited the subjects themselves. For example, in the Faculty Setting subjects from the Faculty of Arts chose between a member of the Faculty of Social Sciences receiving a small amount or a Member of the Faculty of Engineering receiving a large amount, but the subjects themselves received the same show-up fee regardless. The experiment also controlled for reputation concerns and second-order beliefs by making the recipients of the allocations unaware of the experiment. For them receiving money was a surprise and appeared to come from the experimenters. Finally I controlled for social image concerns by running the experiment double-blind. The assistants checking the register were the only ones to know the names of the subjects, they stayed outside the laboratory and they were blind to the treatment. Effectively, subjects knew we could never find who made which choice.

Tasks and procedures

In the two settings, the payment in ω s was fixed at £5 and the payment in ω x varied between £6 and £10, resulting in five pairs of allocations: (£5, £6), (£5, £7), (£5, £8), (£5, £9) and (£5, £10). For each pair subjects made a pairwise Choice and reported their Monetary Valuation for each allocation, so in total subjects made 5 Choices and reported 6 Monetary Valuations. Figure 1 gives a sample of these and subsequent tasks.

In addition, subjects completed the Inclusion of Other in the Self (in the Faculty Setting) or the Inclusion of Ingroup in the Self (in the Charity Setting) scales. This way, we can check for each subject whether what we call small and large social distances matches their perception of the social distances. Subjects in the Charity Setting also indicated how familiar they were with Cancer Research UK and The Salvation Army. The order of the tasks was randomised independently for each subject.

The ordinal payoff scheme [START_REF] Cubitt | Testing Explanations of Preference Reversal[END_REF][START_REF] Tversky | The Causes of Preference Reversal[END_REF] made Choice and Monetary Valuation strategically equivalent: At the end of the experiment a pair of allocations was randomly selected. Then, for this pair of allocations Choice or Monetary Valuation was randomly selected. If Choice was selected then the allocation that the subject chose was implemented; if Monetary Valuation was selected then the allocation that received the higher valuation was implemented.

The instructions (see the Supplementary Materials) explained the ordinal payoff scheme in details and featured control questions.

I implemented the allocations as follows. If, in the Faculty Setting, the allocation to implement was, for example, £7 to a member of the Faculty of Engineering, a member of the Faculty of Engineering was invited to participate in the Charity How much money given to you would be just as good as us giving £6 to the Salvation Army ?

Please write the amount here:

We will refer to this amount as your equivalence valuation of giving £6 to the Salvation Army.

Please consider the member of the Faculty of Engineering. Select the pair of circles that best represents how you feel toward the member of the Faculty of Engineering:

You Other Y O Y O Y O Y O Y O Y O
Indicate your answer by drawing a line around the pair of circles you select.

Indicate

Implementation

The experiment was conducted in the CeDEx laboratory of the University of Nottingham. Subjects were recruited with ORSEE [START_REF] Greiner | Subject pool recruitment procedures: Organizing experiments with ORSEE[END_REF]. 108 subjects participated in the experiment (56 in the Faculty Setting and 52 in the Charity Setting) over 6 sessions between mid-December 2014 and mid-January 2015. 3 The average payment was £10.9 (SD = £3.5) and the average session lasted 1 hour 15 minutes. receipt corresponding to their donation. For each session I thus pooled the donations, made a single payment, and sent a single receipt to all the participants of the session. 3 Since subjects in the Charity Setting are paid with the allocations selected by the ordinal payoff scheme in the Faculty Setting, there should be the same number of subjects in both Settings. Recruitment issues, however, lead to only 52 subjects participating in the Charity Setting. It was subsequently decided not to run an additional session for the missing subjects. At a glance we see that the preference reversal phenomenon is prevalent in the Faculty Setting but that it is much smaller in the Charity Setting.

Results

Preference reversals

To test for the preference reversal phenomenon I rely on one-sided McNemar tests. 4 Table 2 reports the corresponding χ 2 values and significance levels. We see that in the Faculty Setting the preference reversal phenomenon is significant for all parameter sets. In the Charity Setting, it is significant only for one parameter set and at the aggregate level. This result confirms what we saw in Figure 2: preference reversals are more pronounced in the Faculty Setting. 5

In the Charity Setting, however, some subjects might have been unfamiliar with Cancer Research UK or The Salvation Army. To control for this, 'Known'-in 5 Instead of looking at the net proportion of reversals, the literature sometimes looks at conditional reversal rates-for example, the rate at which a subject commits a standard preference reversal conditional on choosing ω s . Appendix B in the Supplementary Materials reports the analysis using these conditional rates and finds similar results. preference reversals decrease but Table 1 shows that they stay significant at the aggregate level.

We can also control for the perception of social distances using the Inclusion of Other in the Self (in the Faculty Setting) and the Inclusion of Ingroup in the Self (in the Charity Setting) scores reported by the subjects during the experiment. The allocations are correctly constructed when the recipient of ω s is socially closer than the recipient of ω x , s s ≤ s x , otherwise subjects face no trade-off between amounts of money and social distance.

The scores range from 1 to 7, with a larger score corresponding to a smaller social distance. Imposing the requirement that s s ≤ s x -that the recipient of ω s received a weakly higher score than the recipient of ω x -decreases the number of subjects to 45 in the Faculty Setting and to 27 in the Charity Setting. With a strict inequality, s s < s x , the number of subjects drops to 20 and 17. Figure 2 shows that these requirements increase the net proportion of preference reversals in the Faculty Setting but decrease it in the Charity Setting. Despite the small sample size, Table 2 shows that preference reversals remain significant in the Faculty Setting.

In the Charity Setting, however, the significance vanishes.

Therefore, while on the surface we observe preference reversals in both Settings, preference reversals in the Faculty Setting are more robust than those in the Charity Setting.

Social distances

Next we can compare the social distances observed in the two Settings. reported the same scores.

Discussion

That preferences reversals carry over to social distances is perhaps not surprising.

The scale compatibility hypothesis [START_REF] Cubitt | Testing Explanations of Preference Reversal[END_REF][START_REF] Tversky | The Causes of Preference Reversal[END_REF], the leading explanation to preference reversals, is domain-agnostic: it would thus apply equally to options featuring probabilities, delays, or social distances.

There remains the question of why we observe more preference reversals in the It is plausible that reporting Inclusion of Other in the Self scores for more than two faculties allowed subjects to better contextualise their answers and use the full range of the scale.

Further, even if social distances were similar in the Faculty Setting, subjects still cared about them. Table 3 reports the estimates from a conditional logit model looking at the determinants of straight choice in the Choice task, or of choice as implied by the monetary valuations reported in the Monetary Valuation task.

We see that, in the Faculty Setting, the coefficient placed on social distance is positive and significant in Choice: subjects tended to choose more often options with a smaller social distance. In fact, since the coefficient placed on money is not significant, social distance was the sole determinant of choice.

Table 3 also explains why we observe more preference reversals in the Faculty Setting. In Monetary Valuation of the Faculty Setting, only the coefficient placed on money is significant: subjects, when forming their valuations, relied only on the monetary component of the option. The pattern of subjects considering only social distance in Choice and only money in Monetary Valuation is in line with scale compatibility and would automatically generate preference reversals.

On the other hand, in the Charity Setting, the two coefficients are always significant: subjects always took into account the amount of money and the social distance. Scale compatibility still manifests itself: the ratio of the coefficient placed on money over the one placed on social distance is higher in Valuation than in Choice, 0.444/0.652 > 0.292/0.567. But it is not strong enough to make subjects abandon social distances when forming their monetary valuation.

  distances. For the Faculty Setting I invited students from all faculties at the University of Nottingham and administered the Inclusion of Other in the Self scale with targets being students from other faculties. Members of the Faculty of Arts reported the greatest difference between members of the Faculty of Social Sciences and the Faculty of Engineering. Therefore, I decided to invite members of the Faculty of Arts in the Faculty Setting, with members of the Faculty of Social Sciences serving as recipients in ω s and members of the Faculty of Engineering serving as recipients in ω x . For the Charity Setting I administered the Inclusion of Ingroup in the Self scale with one of several charities as the target. Participants reported that Cancer Research UK was their closest charity, and The Salvation Army, their most distant; these charities were thus selected as recipients in ω s and ω x .

  Setting and was paid £7 at the end of this experiment. Participants in the Faculty Setting were provided the date, time and location of the experiments featuring the participants of the Charity Setting and they were actively encouraged to come monitor the payments. In the Charity Setting, Cancer Research UK and The Salvation Army were also paid as a result of the ordinal payoff scheme and the choices of the participants. Participants in the Charity Setting were told that we would send them the receipt of the donations, which we did. 2 Option A: We give £5 to the member of the Faculty of Social Sciences Option B: We give £10 to the member of the Faculty of Engineering Choose A or B:

Figure 1 :

 1 Figure 1: Examples of the tasks.

Figure 2 :

 2 Figure 2: Net proportion of preference reversals, at the aggregate and by Setting.

  The scores obtained with the Inclusion of Other in the Self scale and the Inclusion of Ingroup in the Self scale range between 1 and 7, 7 corresponding to the smallest social distance. Error bars represent one standard deviation.

Figure 3 :

 3 Figure 3: Average of the reported Inclusion of Other in the Self (Faculty Setting)and Inclusion of Ingroup in the Self (Charity Setting) scores.

  Figure 3 reports the averages of the Inclusion of Other in the Self (Faculty Setting) and the Inclusion of Ingroup in the Self (Charity Setting) scores. As predicted, subjects in the Faculty Setting perceived similarly a member of the Faculty of Social Sciences and a member of the Faculty of Engineering: the difference is not significant (two-sided Wilcoxon signed rank test, exact p = 0.1428) and 25 out of 55 subjects (45%) reported the same score for both. On the other hand subjects in the Charity Setting thought Cancer Research UK and The Salvation Army were different: the difference is significant (two-sided Wilcoxon signed rank test, exact p < 0.001) and only 11 out of 52 subjects (21%)

Faculty

  Setting. Only students from the Faculty of Arts participated to the Faculty Setting while students from the Faculty of Social Sciences and from the Faculty of Engineering participated to the Charity Setting; if students from the Faculty of Arts are more prone to preference reversals, preference reversals would naturally be more common in the Faculty Setting. At the University of Nottingham these faculties are very broad: the Faculty of Arts encompasses both the School of Cultures, Languages and Area Studies and the School of Humanities; the Faculty of Social Sciences includes the School of Education and the School of Law alongside the usual School of Economics and School of Sociology and Social Policy; and there are architecture students in the Faculty of Engineering, right next to civil or mechanical engineering students. Unless we make broad generalisations in terms of numeracy or cognitive abilities, there is no particular reason to expect students from one faculty to be more prone to preference reversals. Moreover, by now preference reversals experiments have relied on a variety of subject pools and the existence of a preference reversal phenomenon is a consistent finding. Another potential explanation is that the options in the Faculty Setting were misconstructed. As explained earlier preference reversal experiments use options ω d and ω x where ω d is better in d while ω x is better in x. But we have seen that subjects perceived similarly the two Faculties, thus seemingly violating one of the ingredients of preference reversal experiments.In the online survey conducted to find close and distant faculties, members of the Faculty of Arts reported an average Inclusion of Other in the Self score of 2.71 toward members of the Faculty of Engineering, and an average Inclusion of Other in the Self score of 4.42 toward members of the Faculty of Social Sciences. The difference there was clearly significant (two-sided Wilcoxon signed rank test, exact p < 0.001) and one of the largest reported across all faculties. These Inclusion of Other in the Self scores are in stark contrast to those reported in the experiment (Figure3). Since subjects came from the same pool and were randomly invited to the online survey and to the laboratory experiment, sampling alone is unlikely to explain the difference. Instead, remember that in the online survey subjects had to report an Inclusion of Other in the Self score toward members of all other existing faculties, while in the experiment subjects had to do it only toward members of the Faculty of Social Sciences and toward members of the Faculty of Engineering.

Table 1 :

 1 Frequencies of the different patterns, for each parameter set and for each Setting.Notes. ss ≤ sx: recipient of ωs received a weakly higher Inclusion of Other in the Self (in the Faculty Setting) or Inclusion of Ingroup in the Self (in the Charity Setting) score than recipient of ωx; ss < sx: strictly higher. Known: subject indicated for both charities 'I know the name but I have only a vague idea of what it does' or 'I know the name and I have a good idea of what it does'.Notes. ss ≤ sx: recipient of ωs received a weakly higher Inclusion of Other in the Self (in the Faculty Setting) or Inclusion of Ingroup in the Self (in the Charity Setting) score than recipient of ωx; ss < sx: strictly higher. Known: subject indicated for both charities 'I know the name but I have only a vague idea of what it does' or 'I know the name and I have a good idea of what it does'.

				Faculty				Charity
			All s s ≤ s x s s < s x All Known	Known + s s ≤ s x	Known + s s < s x
	(£5,£6)	        	Consistent for ω s Consistent for ω x Standard reversal 18 11 22 Counter reversal 4	10 17 16 2	6 4 9 1	23 22 5 2	13 20 3 0	12 13 2 0
	(£5,£7)	        	Consistent for ω s Consistent for ω x Standard reversal 19 9 23 Counter reversal 4	9 17 18 1	4 6 9 1	16 23 10 3	8 21 6 1	8 13 5 1
	(£5,£8)	        	Consistent for ω s Consistent for ω x Standard reversal 20 4 26 Counter reversal 5	2 19 19 5	1 7 10 2	15 29 4 4	7 26 1 2	7 17 1 2
	(£5,£9)	        	Consistent for ω s Consistent for ω x Standard reversal 25 3 25 Counter reversal 2	3 17 23 2	2 7 10 1	13 30 5 4	5 27 2 2	5 19 1 2
	(£5,£10)	        	Consistent for ω s Consistent for ω x Standard reversal 16 1 34 Counter reversal 4	1 27 14 3	0 10 10 0	11 29 8 4	5 27 3 1	5 18 3 1
	Aggregate	        	Consistent for ω s Consistent for ω x 130 28 Standard reversal 98 Counter reversal 19	25 97 90 13	13 34 48 5	78 133 32 17	38 121 15 6	37 80 12 6

Table 1

 1 [START_REF] Cubitt | Testing Explanations of Preference Reversal[END_REF] observe the smallest preference reversal phenomenon in parameter set 5 (condition MV), with 21.92% of Standard Reversals and 1.75% of Counter Reversals. Using α = 0.05 and 1 -β = 0.8, the required sample size to detect such an effect with a one-sided McNemar test is N = 34 (Stata command power pairedproportions), well below the sample size in the current experiment.

and Figure 2-only looks at subjects who indicated for both charities 'I know the name but I have only a vague idea of what it does' or 'I know the name and I have a good idea of what it does'. This requirement decreases the number of subjects available for analysis in the Charity Setting to 36. In Figure 2 we see that 4

Table 2 :

 2 Tests of the preference reversal phenomenon. Notes. One-sided McNemar tests. A symbol indicates significance at α = 0.05. ss ≤ sx: recipient of ωs received a weakly higher Inclusion of Other in the Self (in the Faculty Setting) or Inclusion of Ingroup in the Self (in the Charity Setting) score than recipient of ωx; ss < sx: strictly higher. Known: subject indicated for both charities 'I know the name but I have only a vague idea of what it does' or 'I know the name and I have a good idea of what it does'.

			Faculty			Charity
		All	s s ≤ s x s s < s x All	Known	Known + s s ≤ s x	Known + s s < s x
	(£5,£6)	8.91 * 10.89 *	6.40 *	1.29	3.00 *	2.00	1.00
	(£5,£7)	9.78 * 15.21 *	6.40 *	3.77 *	3.57 *	2.67	1.00
	(£5,£8)	9.00 *	8.70 *	5.33 *	0.00	0.33	0.33	2.00
	(£5,£9) 19.59 * 17.64 *	7.36 *	0.11	0.00	0.33	0.33
	(£5,£10) 7.20 *	7.12 * 10 *	1.33	1.00	1.00	0.33
	Aggregate 53.34 * 57.56 * 34.89 *	4.59 *	3.86 *	2.00	0.08

Table 3 :

 3 Estimates from a conditional logit model.Notes. Coefficients, standard errors (clustered at the subject's level) in parentheses. Money in £; social distance measured with the Inclusion of Other in the Self (Faculty Setting) or the Inclusion of Ingroup in the Self (Charity Setting). p < 0.05, * * p < 0.01, * * * p < 0.001.

		Faculty	Charity
		Choice	Valuation	Choice	Valuation
	Money	0.104	0.619 * * *	0.292 * * *	0.444 * * *
		(0.059)	(0.122)	(0.081)	(0.120)
	Social distance	0.264 *	0.002	0.567 * *	0.652 *
		(0.121)	(0.128)	(0.197)	(0.270)
	Observations	550	510	520	444
	Log-likelihood -181.766	-101.934	-152.048	-114.049

*

Because of the double-blindness of the experiment it was not possible to send each subject the


Grégoire Justeau
email: gregoire.justeau@chu-angers.fr

MD Chloé Gervès-Pinquié

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Between Nocturnal Hypoxemia and Cancer Incidence in Patients Investigated for OSA

Keywords: obstructive sleep apnea, intermittent hypoxia, cancer AHI: apnea-hypopnea index BMI: body mass index CI: confidence interval COPD: chronic obstructive pulmonary disease Excessive daytime sleepiness: EDS ESS: Epworth Sleepiness Scale HR: hazard ratios INCA: French National Cancer Institute IH: intermittent hypoxemia IQR: interquartile range OSA: obstructive sleep apnea PAP: positive airway pressure PMSI: French Hospital Discharge database PSG: polysomnography RP: respiratory polygraphy SD: standard deviation SNDS: French administrative health care database T90: percent nighttime with oxygen saturation <90%

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

Introduction

Recent data from experimental and clinical studies have suggested that obstructive sleep apnea (OSA) might contribute to the development and progression of cancer. Experimental studies in animal models have strongly suggested that intermittent hypoxemia (IH), the hallmark feature of OSA, might promote tumor growth and metastasis. Several putative intermediate mechanisms linking OSA with cancer incidence and prognosis have been proposed including sympathetic overactivity, systemic inflammation, angiogenesis, and immunological alterations (see references [START_REF] Gozal | Obstructive sleep apnea and cancer: Epidemiologic links and theoretical biological constructs[END_REF] for comprehensive review). However, population-and clinic-based studies have yielded inconsistent findings regarding the association between OSA severity and cancer in humans. The epidemiologic study from Nieto et al. was the first to demonstrate a dose-response association between OSA and cancer mortality [START_REF] Nieto | Sleep-disordered Breathing and Cancer Mortality[END_REF] . Most [START_REF] Marshall | Sleep Apnea and 20-Year Follow-Up for All-Cause Mortality, Stroke, and Cancer Incidence and Mortality in the Busselton Health Study Cohort[END_REF][START_REF] Christensen | Symptoms of Sleep Disordered Breathing and Risk of Cancer: A Prospective Cohort Study[END_REF][START_REF] Fang | Risk of Cancer in Patients with Insomnia, Parasomnia, and Obstructive Sleep Apnea: A Nationwide Nested Case-Control Study[END_REF] but not all [START_REF] Gozal | Sleep Apnea and Cancer: Analysis of a Nationwide Population Sample[END_REF] subsequent population-based studies reported elevated cancer incidence and/or mortality in subjects with OSA. The Spanish clinic-based cohort study was the first to demonstrate an association between sleep-disordered breathing severity and cancer incidence among patients investigated for clinical suspicion of OSA [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] . The percentage of nighttime spent with oxygen saturation lower than 90% (T90) was the only marker of OSA severity associated with cancer incidence in the entire cohort. Among three subsequent clinic-based studies [START_REF] Brenner | Increased Risk for Cancer in Young Patients with Severe Obstructive Sleep Apnea[END_REF][START_REF] Sillah | Obstructive sleep apnea severity and subsequent risk for cancer incidence[END_REF][START_REF] Kendzerska | Obstructive sleep apnea and the prevalence and incidence of cancer[END_REF] , two found no association between OSA severity and cancer incidence [START_REF] Sillah | Obstructive sleep apnea severity and subsequent risk for cancer incidence[END_REF][START_REF] Kendzerska | Obstructive sleep apnea and the prevalence and incidence of cancer[END_REF] and one concluded that severe OSA was associated with a higher all-type cancer incidence only in patients under 45 years of age [START_REF] Brenner | Increased Risk for Cancer in Young Patients with Severe Obstructive Sleep Apnea[END_REF] . The abovementioned studies had some limitations. In several studies the diagnosis of OSA was based on questionnaires or administrative data [START_REF] Gozal | Obstructive sleep apnea and cancer: Epidemiologic links and theoretical biological constructs[END_REF][START_REF] Christensen | Symptoms of Sleep Disordered Breathing and Risk of Cancer: A Prospective Cohort Study[END_REF][START_REF] Fang | Risk of Cancer in Patients with Insomnia, Parasomnia, and Obstructive Sleep Apnea: A Nationwide Nested Case-Control Study[END_REF] . Most clinic-based studies were monocentric [START_REF] Brenner | Increased Risk for Cancer in Young Patients with Severe Obstructive Sleep Apnea[END_REF][START_REF] Sillah | Obstructive sleep apnea severity and subsequent risk for cancer incidence[END_REF][START_REF] Kendzerska | Obstructive sleep apnea and the prevalence and incidence of cancer[END_REF] . Several studies included a limited proportion of severe OSA, had a relatively small number of events and/or did not consider competing risk of death from other causes [START_REF] Nieto | Sleep-disordered Breathing and Cancer Mortality[END_REF][START_REF] Marshall | Sleep Apnea and 20-Year Follow-Up for All-Cause Mortality, Stroke, and Cancer Incidence and Mortality in the Busselton Health Study Cohort[END_REF][START_REF] Brenner | Increased Risk for Cancer in Young Patients with Severe Obstructive Sleep Apnea[END_REF][START_REF] Sillah | Obstructive sleep apnea severity and subsequent risk for cancer incidence[END_REF] . In most studies, adjustment was not made for important confounders such as socioeconomic status, alcohol intake or diabetes [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF][START_REF] Brenner | Increased Risk for Cancer in Young Patients with Severe Obstructive Sleep Apnea[END_REF][START_REF] Sillah | Obstructive sleep apnea severity and subsequent risk for cancer incidence[END_REF] . Furthermore, in most studies sleep-related oxygen desaturations were assessed with the T90, a surrogate marker of global nocturnal hypoxemia, but not with the oxygen desaturation index (ODI) [START_REF] Nieto | Sleep-disordered Breathing and Cancer Mortality[END_REF][START_REF] Marshall | Sleep Apnea and 20-Year Follow-Up for All-Cause Mortality, Stroke, and Cancer Incidence and Mortality in the Busselton Health Study Cohort[END_REF][START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF][START_REF] Brenner | Increased Risk for Cancer in Young Patients with Severe Obstructive Sleep Apnea[END_REF][START_REF] Kendzerska | Obstructive sleep apnea and the prevalence and incidence of cancer[END_REF] .

The aim of this large multicenter longitudinal cohort study was to determine whether cancer incidence was independently associated with OSA severity and indices of nocturnal hypoxemia after appropriate adjustment for confounders.

Methods

The study was conducted on the Pays de la Loire Sleep Cohort [START_REF] Gagnadoux | Influence of marital status and employment status on long-term adherence with continuous positive airway pressure in sleep apnea patients[END_REF] , linked with data from the French administrative health care database (SNDS), which contains individualized, anonymous, and comprehensive data on health spending reimbursements [START_REF] Fagot | Does insulin glargine increase the risk of cancer compared with other basal insulins?: A French nationwide cohort study based on national administrative databases[END_REF]

Baseline evaluation and questionnaires

Each patient enrolled in the Pays de la Loire Sleep Cohort completed surveys including anthropometric data, smoking habits, alcohol consumption, medical history, and medication use (see e-Table 1 for the definition of covariates). Excessive daytime sleepiness and depressive symptoms were assessed by the Epworth score and the QD2A depression score respectively [START_REF] Gagnadoux | Depressive symptoms before and after long-term CPAP therapy in patients with sleep apnea[END_REF] . As previously described, socioeconomic status included the marital status and the occupational category, determined according to the Institut National de la Statistique et des Etudes Economique (INSEE) nomenclature [START_REF] Gagnadoux | Influence of marital status and employment status on long-term adherence with continuous positive airway pressure in sleep apnea patients[END_REF] .

Sleep studies and positive airway pressure treatment

According to French guidelines, patients with a high clinical probability of OSA were investigated by respiratory polygraphy (RP) (CID102L TM ; CIDELEC, Sainte-Gemmes-sur-Loire, France) [START_REF] Meurice | Recommendations for clinical practice in the management of obstructive sleep apnea syndrome in adults. Preface][END_REF] . Patients with a low likelihood of OSA and/or coexisting sleep disorders received a polysomnography (PSG) (CID102L8D TM , CIDELEC, Sainte-Gemmes-sur-Loire, France). Recorded data included nasal-oral airflow (thermistor and nasal pressure cannula), chest and abdominal wall motion (respiratory inductance plethysmograph belts), arterial oxygen saturation (pulse oximetry), body position and electrophysiological signals for sleep evaluation (PSG). Respiratory events were scored manually using recommended criteria [START_REF] Berry | Rules for scoring respiratory events in sleep: update of the 2007 AASM Manual for the Scoring of Sleep and Associated Events. Deliberations of the Sleep Apnea Definitions Task Force of the American Academy of Sleep Medicine[END_REF] . Apnea was defined as an at least 90% decrease in the oronasal thermal sensor signal and hypopnea was defined as an at least 30% decrease in the nasal pressure signal combined with either ≥ 3% arterial oxygen desaturation or an arousal (PSG), both lasting at least 10 s. According to the criteria defined by the French national health insurance, positive airway pressure (PAP) therapy was prescribed in patients with severe OSA and in those with moderate OSA and cardiovascular comorbidities or severe daytime sleepiness. As previously described [START_REF] Gagnadoux | Influence of marital status and employment status on long-term adherence with continuous positive airway pressure in sleep apnea patients[END_REF] , a single home respiratory care company (ALISEO, Beaucouzé, France) was involved for PAP device delivery and follow-up support program. Patients were reviewed in consultation by the sleep specialist at 5 months, 12 months then at least annually. Objective daily PAP use was monitored at each follow-up visit and recorded in the database. The average PAP adherence during the follow-up period was calculated to determine treatment status.

Endpoint

The endpoint of the study was all-cancer incidence as defined as the first occurrence of a malignant neoplasm at any time between the sleep study and the final follow-up date of December 31, 2017.

The occurrence of cancer was identified on the French Hospital Discharge database (Programme de Medicalisation des Systemes d'Information [PMSI]), which is linked to the SNDS database [START_REF] Fagot | Does insulin glargine increase the risk of cancer compared with other basal insulins?: A French nationwide cohort study based on national administrative databases[END_REF] . Data from SNDS were available from May 2006. The PMSI database provides detailed medical information on all admissions in public and private hospitals, including discharge diagnoses ICD-10 codes and medical procedures performed during the hospital stay. The procedure that we used to identify incident cancers from the PMSI database was adapted from the algorithm developed by the French National Cancer Institute (INCA) [START_REF]Algorithme de sélection des hospitalisations liées au cancer en MCO -Étude de validation -Ref : DONALGOHOSK18[END_REF] . The sensitivity, specificity, positive and negative predictive values of the INCA algorithm has been found to be respectively 85.0% (95%CI [84.1-85.9]), 98.5% [98. 4-98.6], 90.5% [89.8-91.3], 97.5% [97. 3-97.6]. The algorithm was mainly based on primary, related or associated PMSI diagnostic codes of malignancies (ICD-10: C00-C97, D00-D09, D37-D463). When several admissions occurred for the same patient, only the first hospital stay was considered as reflecting an incident case. The accurateness of PMSI database to identify cancer incidence has been pointed out by previous reports [START_REF] Remontet | Is it possible to estimate the incidence of breast cancer from medico-administrative databases?[END_REF][START_REF] Mitton | A Suitable Approach to Estimate Cancer Incidence in Area without Cancer Registry[END_REF][START_REF] Quantin | Estimation of National Colorectal-Cancer Incidence Using Claims Databases[END_REF] who demonstrated a good concordance with data from cancer registries. Patients who did not develop cancer were censored at the date of death or at the final follow-up date.

Statistical analysis

The primary dependent variable of interest was all-cancer incidence. The primary independent variables were OSA severity as assessed by the apnea-hypopnea index (AHI), and two different indices of nocturnal hypoxemia, ODI and T90. In line with recent findings [START_REF] Azarbarzin | The hypoxic burden of sleep apnoea predicts cardiovascular disease-related mortality: the Osteoporotic Fractures in Men Study and the Sleep Heart Health Study[END_REF] , the hypoxic burden has also been calculated for a subgroup of our population. Missing values were imputed using a multiple imputation method (MI procedure from SAS) when the proportion of missing values was < 20%. Five datasets were generated using logistic regression for qualitative variables or linear regression for quantitative variables, considering that missing values were random. The baseline differences between OSA categories were compared using the nonparametric Kruskal-Wallis test for quantitative and the chi-square test for qualitative characteristics (see Online Content and e-Table 2 for details on statistical analyses).

Kaplan-Meier survival analysis with log-rank test and Cox proportional hazard models were used to evaluate the association of all-cancer incidence with categories of OSA severity and indices of nocturnal hypoxemia. To adjust for potential confounders, study site and variables for which there was a significant difference between patients with and without incident cancer were entered in the multivariate analysis. The results were expressed as hazard ratios (HR) and 95% confidence intervals (CI). P values for linear trends across categories of OSA severity and indices of nocturnal hypoxemia were calculated by the Cochrane-Armitage trend test. The association of continuous forms of AHI, ODI and T90 with cancer incidence was also examined.

To evaluate a potential treatment effect, a stratified analysis was conducted according to treatment status. Patients with no therapy or mean objectively measured PAP adherence <4 h per night during the follow-up period were considered as untreated. Patients with mean objective PAP adherence ≥4 h per night during the follow-up period were considered as treated. Based on previous reports 2,7 a priori-defined interactions between exposures and factors that may influence the risk of cancer including age, gender and obesity were also tested. Subgroup analyses were conducted to evaluate the association between of OSA and nocturnal hypoxemia with the most frequent cancer types. Fine and Gray competing-risk regressions were used to account for the competing risk of death [START_REF] Kendzerska | Sleep Apnea Increases the Risk of New Hospitalized Atrial Fibrillation: A Historical Cohort Study[END_REF] .

All statistical analyses were performed with SAS 9.4 software (SAS Institute, Cary, NC). Associations were considered statistically significant for a p value <0.05. 3 for baseline characteristics of the entire cohort, the study population, and patients not included).

Results

Out

As shown in Table 1, the study population consisted of typical OSA patients, predominantly male, obese or overweight, frequently presenting cardio-metabolic comorbidities. Among the 8,748 patients included, 1,161 (13.0%) had no OSA, 1,878 (21.5%) had mild OSA, 1,826 (21.0%) had moderate OSA, and 3,883 (44.5%) had severe OSA. As expected, patients with increasing OSA severity were of significantly higher age and BMI, and were more frequently male with medical history of diabetes, hypertension, cardiac disease and alcohol consumption. There were also significant differences between OSA categories for smoking habits, socio-professional status and the type of sleep recording. The comparison of patients with and without incident cancer showed significant differences for age, gender, body mass index, smoking status, alcohol intake, diabetes, hypertension, medical history of cardiac and chronic obstructive pulmonary disease, indices of OSA severity, Epworth score, marital status, and type of sleep study (e-Table 4).

After a median (interquartile range [IQR]

) follow-up of 5.8 [3.8-7.8] years, 718 patients (8.2%) had received a diagnosis of malignancy. Colorectal (n=129), lung (n=69), prostate (n=66) and breast (n=62) were the most common incident cancer sites. As shown in table 1, the incidence of cancer increased from 5.7 to 9.1% (p=0.0005) between the non OSA group and the severe OSA group (incidence density rates: 10.0 to 16.6 per 1000 person-years). Unadjusted Kaplan-Meier survival curves indicating cancer-free survival according to categories of OSA severity, quartiles of ODI and T90 are shown in Figure 2. Log-rank tests comparing curves showed that cancer-free survival was negatively associated with increasing severity of OSA (p<0.0005) and nocturnal hypoxemia (p<0.0001 for both ODI and T90).

Cox proportional hazard models demonstrated a significant association between all-cancer incidence and increasing nocturnal hypoxemia assessed by T90. Compared with the lowest severity category, the adjusted HR, 95% [95%CI] for incident cancer was 1.33 [1.05-1.68] for T90 ≥13% versus <0.01%; p=0.02). Cancer incidence was not associated with OSA severity and ODI after adjustment for confounders (Table 2). As shown in e-Table 5, all-cancer incidence was associated with continuous forms of AHI, ODI, and T90 but the association was no longer significant after adjustment for confounders.

Hypoxic burden data were available in 5,683 patients. Slight but significant differences were observed between the entire cohort and the subgroup of patients with hypoxic burden measurement (e-Table 6). As shown in Table 3, increasing hypoxic burden was associated with a higher incidence of cancer but the association was no longer significant after adjustment for confounders.

Cox proportional hazard models assessing the association between all-cancer incidence and T90 were then conducted separately in 5,724 untreated patients with no active OSA therapy or PAP adherence <4 h per day (median [IQR]: 2.4 [1.0-3.4] h per day), and 3,024 treated patients with objective PAP adherence ≥4 h per day (median [IQR]: 6.5 [5.5-7.5] h per day). As expected, patients receiving adequate PAP therapy had more severe disease compared to untreated patients (e-Table 7). As shown in Figure 3, there was a dose-response relationship between T90 and all-cancer incidence in the entire study population (p=0.04 for linear trend) and in untreated patients (p=0.004 for linear trend), but not in patients with adequate PAP therapy (p value for interaction = 0.39).

There was no association of all-cancer incidence with OSA severity (p=0.54 for linear trend) and quartiles of ODI (p=0.88 for linear trend) in untreated patients (data not shown).

Further stratified analyses evaluating the association between cancer incidence and severe nocturnal hypoxemia are presented online in e-Figure 1. The association was stronger among patients aged > 60 years compared with those aged ≤ 60 years (p< 0.0001 for interaction). The association between severe nocturnal hypoxemia and cancer incidence appeared also stronger among patients with obesity but the interaction was not statistically significant (p=0.89).

Subgroup analyses demonstrated a dose-response association between nocturnal hypoxemia and incident lung cancer (p=0.03 for linear trend). Breast cancer was associated with the second (adjusted HR 2.04 [1.04-3.98]; p=0.04), but not with third or fourth quartiles of T90 as compared with the first quartile. Colorectal and prostate malignancies were not associated with nocturnal hypoxemia (Table 4).

The magnitude of the association between all-cancer incidence and T90 was unchanged after controlling for a competing risk of death (e-Table 8).

Discussion

The main finding of our study is that T90, a marker of overnight hypoxemia predicts all-cancer incidence in patients investigated for suspected OSA, independently of major confounding factors.

Previous cohort studies have demonstrated an independent association between T90 and cancer mortality 2 or incidence [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] . Conversely, recent data from a large cohort of 10,149 patients with suspected OSA, have shown no independent association between cancer incidence and AHI or T90 [START_REF] Kendzerska | Obstructive sleep apnea and the prevalence and incidence of cancer[END_REF] . However, several characteristics of the study population may have contributed to these negative findings including younger age and lower OSA severity. Altogether our findings and those from previous studies [START_REF] Nieto | Sleep-disordered Breathing and Cancer Mortality[END_REF][START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] suggest that hypoxia may be the main intermediate mechanism linking OSA to cancer. In the present study, cancer incidence was associated with T90 but not with ODI, which is considered the best marker of IH in clinical setting. Whether IH can directly or indirectly induce tumorigenesis has not been clearly established, but current evidence strongly suggests that IH enhances cancer progression in animal models 1 . However, using FiO2 of about 5% or less, animal models of IH reproduces severe forms of the human OSA. In mice exposed to chronic IH, the degree of metabolic dysregulation is dependent on the severity of the hypoxic stimulus [START_REF] Li | Hyperlipidemia and lipid peroxidation are dependent on the severity of chronic intermittent hypoxia[END_REF] . Our findings and those from the Spanish clinic-based cohort study [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] suggest that T90 would be a better predictor of cancer risk than AHI and ODI, as it has also been demonstrated for atrial fibrillation, cardiovascular events and all-cause mortality [START_REF] Kendzerska | Sleep Apnea Increases the Risk of New Hospitalized Atrial Fibrillation: A Historical Cohort Study[END_REF][START_REF] Kendzerska | Obstructive sleep apnea and risk of cardiovascular events and all-cause mortality: a decade-long historical cohort study[END_REF] . As previously discussed [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] , these findings might be explained by the fact that a significant proportion of respiratory events taken into account in the calculation of AHI and ODI are accompanied by small drops in oxygen saturation resulting in low T90. The T90 characterizes not only IH resulting from obstructive events but also persistent hypoxemia, such as from COPD, obesity hypoventilation or cardiac failure. In the present study, the association between T90 and cancer incidence remained statistically significant after adjustment for BMI, cardiac diseases and COPD. However, as pulmonary function tests and arterial blood gases were not available in our database, the contribution of comorbid cardiac and respiratory disorders cannot be excluded.

A recent study identified the hypoxic burden as an independent predictor of cardiovascular mortality [START_REF] Azarbarzin | The hypoxic burden of sleep apnoea predicts cardiovascular disease-related mortality: the Osteoporotic Fractures in Men Study and the Sleep Heart Health Study[END_REF] . While a strong association was found between hypoxic burden and cardiovascular mortality, there was a weaker association with all-cause mortality suggesting that there may be a distinct biological pathway linking OSA-related hypoxemia to cardiovascular death. In the present study, we found no independent association between hypoxic burden and cancer incidence. It is important to note that the hypoxic burden does not distinguish short and deep desaturations from and long and shallow ones. The hypoxic burden was found to be poorly correlated with T90 and its association with conventional polysomnographic indices weakened for patients with moderate to severe OSA [START_REF] Azarbarzin | The hypoxic burden of sleep apnoea predicts cardiovascular disease-related mortality: the Osteoporotic Fractures in Men Study and the Sleep Heart Health Study[END_REF] .

On stratified analysis, Cox proportional hazard models assessing the association between T90 and cancer incidence showed a significant dose-response relationship in untreated patients, but no association in patients receiving adequate PAP therapy of OSA. Furthermore, the association between T90 and cancer risk was stronger in untreated patients than in the entire cohort. However, taking into account unbalanced sample sizes, the observational design of our study, the healthy adherer effect [START_REF] Simpson | A meta-analysis of the association between adherence to drug therapy and mortality[END_REF][START_REF] Vestbo | Adherence to inhaled therapy, mortality and hospital admission in COPD[END_REF] , and the fact that formal tests for interaction did not reach statistical significance, our finding should not be interpreted as the demonstration of a preventive effect of PAP therapy on cancer incidence. Further studies are required to evaluate whether adequate OSA therapy might reduce the risk of cancer in patients with marked nocturnal hypoxemia.

Our finding that severe nocturnal hypoxemia was significantly associated with cancer risk in patients with obesity but not in non-obese subjects is consistent with previous data underlying the interaction between obesity, nocturnal oxygen desaturations and cardiovascular risk [START_REF] Kendzerska | The Interaction of Obesity and Nocturnal Hypoxemia on Cardiovascular Consequences in Adults with Suspected Obstructive Sleep Apnea. A Historical Observational Study[END_REF] . It has been demonstrated that body weight is a significant predictor of the amount of oxygen desaturation during apnea's and hypopnea's, potentially exacerbating the adverse health impact of OSA in obese individuals [START_REF] Peppard | The Impact of Obesity on Oxygen Desaturation during Sleep-disordered Breathing[END_REF] . We observed that the association between T90 and cancer incidence was stronger in older patients contrary to the Spanish cohort study, which however included a relatively small sample of older subjects (n=1,203) [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] . The result from these stratified analyses need to be interpreted with caution due to limited and/or unbalanced sample sizes.

In line with previous reports [START_REF] Gozal | Sleep Apnea and Cancer: Analysis of a Nationwide Population Sample[END_REF][START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF][START_REF] Seijo | Obstructive sleep apnea and nocturnal hypoxemia are associated with an increased risk of lung cancer[END_REF][START_REF] Chang | Sleep apnea and the subsequent risk of breast cancer in women: a nationwide population-based cohort study[END_REF][START_REF] Li | Target of obstructive sleep apnea syndrome merge lung cancer: based on big data platform[END_REF] , we found that among the most frequent cancer sites, nocturnal hypoxemia was associated with the incidence lung and breast cancer but not with colorectal and prostate malignancies. A significant association between oxygen desaturation and the incidence of smoking-related cancers have been found in previous clinic-based cohort studies [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF][START_REF] Kendzerska | Obstructive sleep apnea and the prevalence and incidence of cancer[END_REF] . Sleep apnea and nocturnal hypoxemia were associated with an increased prevalence of lung cancer in a recent crosssectional study [START_REF] Seijo | Obstructive sleep apnea and nocturnal hypoxemia are associated with an increased risk of lung cancer[END_REF] . Li et al. recently reported that OSA severity was a risk factor that contributed to short overall survival in patients with lung cancer [START_REF] Li | Target of obstructive sleep apnea syndrome merge lung cancer: based on big data platform[END_REF] . The authors identified molecular convergence between hypoxia and lung cancer that reflected their clinical profiles and revealed molecular pathways involved in hypoxic-induced lung cancer progression. The reasons why breast cancer risk was increased in patients with mild nocturnal hypoxemia but not in those with more severe oxygen desaturation remain unclear and the interpretation may be limited by sample size. Further studies are required to determine whether the presence OSA and nocturnal desaturation may affect all types of oncological conditions or only selective types of tumors.

The strength of the current study includes a multicenter design, a large sample size with a wide range of sleep-disordered breathing severity, an appropriate adjustment for multiple cancer risk factors, a long and complete follow-up with access to comprehensive data from the National Health Insurance Information System, and the assessment of different measures of hypoxia. The French Health Care database now covers 98.8% of the French population, over 66 million persons, from birth (or immigration) to death (or emigration) [START_REF] Bezin | The national healthcare system claims databases in France, SNIIRAM and EGB: Powerful tools for pharmacoepidemiology[END_REF] . The linkage between clinical cohort's data and SNDS data makes easier the long-term follow-up with limited attrition bias contrary to trials [START_REF] Tuppin | Value of a national administrative database to guide public decisions: From the système national d'information interrégimes de l'Assurance Maladie (SNIIRAM) to the système national des données de santé (SNDS) in France[END_REF][START_REF] Scailteux | French administrative health care database (SNDS): The value of its enrichment[END_REF] . Our study also has limitations. Because of its observational design, potential unmeasured confounding factors cannot be excluded. We also acknowledge that sleep recordings were performed either by PSG or RP and manually scored locally, with no assessment of inter-scorer reliability. It cannot be excluded that inter-scorer variability, particularly for hypopneas which did not cause an oxygen desaturation [START_REF] Grigg-Damberger | The AASM Scoring Manual four years later[END_REF] may have contributed to the lack of association between cancer incidence and categories of OSA severity.

However, this potential bias was mitigated by the adjustment for study site. Furthermore, our finding that T90 was a better predictor of cancer risk than AHI is consistent with previous reports [START_REF] Campos-Rodriguez | Association between obstructive sleep apnea and cancer incidence in a large multicenter Spanish cohort[END_REF] . There are some limitations related to the use of health administrative data. We had no data about cancer stage and the time from cancer symptom onset to diagnosis. In French hospitals, all physicians routinely contribute to PMSI data collection, with annual quality control of coding resulting in a decreased risk of misclassification bias [START_REF] Fagot | Does insulin glargine increase the risk of cancer compared with other basal insulins?: A French nationwide cohort study based on national administrative databases[END_REF] . Furthermore, previous studies have demonstrated that the use of PMSI claims database was a suitable approach to estimate cancer incidence [START_REF] Remontet | Is it possible to estimate the incidence of breast cancer from medico-administrative databases?[END_REF][START_REF] Mitton | A Suitable Approach to Estimate Cancer Incidence in Area without Cancer Registry[END_REF][START_REF] Tuppin | Value of a national administrative database to guide public decisions: From the système national d'information interrégimes de l'Assurance Maladie (SNIIRAM) to the système national des données de santé (SNDS) in France[END_REF][START_REF] Quantin | Estimation of National Colorectal-Cancer Incidence Using Claims Databases[END_REF] . We acknowledge that our study did not have sufficient statistical power to examine the association of OSA and nocturnal hypoxemia with the different cancer subtypes. Finally, the fact that hypoxic burden data were not available in the entire cohort may have also decreased the power to detect a significant association with cancer incidence.

Interpretation

Nocturnal hypoxemia was associated with all-cancer incidence in patients investigated for suspected OSA. Whether OSA therapy might reduce the risk of cancer needs to be further evaluated. 2B), and percent night-time with oxygen saturation below 90% (T90) (Figure 2C).

The log-rank test was used to compare the survival distribution across categories. P=0.39 for the interaction of treatment status and T90 for all-cancer incidence.

HR were adjusted for *Adjusted for age, gender, body mass index, smoking status, alcohol intake, diabetes, hypertension, medical history of cardiac and chronic obstructive pulmonary disease, marital status, type of sleep study, and study site.

  (see Online Content for details on the Pays de la Loire Sleep Cohort and the linkage process with the SNDS database). All patients included in the cohort between May 15, 2007 and December 31, 2015 with available SNDS data were eligible for the present study. Patients were excluded if they had received a diagnosis of cancer at any time before the sleep study. Approval was obtained from the University of Angers Ethics Committee and the 'Comité Consultatif sur le Traitement de l'Information en matière de Recherche dans le domaine de la Santé' (CCTIRS; 07.207bis). All patients had given their written informed consent.

  of 11,218 patients investigated for suspected OSA between May 15, 2007 and December 31, 2015, 1,644 were not linked to SNDS dataset, 818 had received a diagnosis of cancer before the sleep study and AHI data were unavailable for 8. The final study sample size comprised 8,748 cancer-free patients at the time of the diagnostic sleep study (see Figure 1 and e-Table

Figure 1 :Figure 2 :

 12 Figure 1: Flow diagram of the subjects during the study

Figure 3 :

 3 Figure 3: Cox proportional hazard models assessing the association between nocturnal hypoxemia

Table 1 :

 1 Characteristics of the study population according to categories of obstructive sleep apnea (OSA) severity.Data are expressed as median [interquartile range], n or percentages Abbreviations: BMI, body mass index; COPD, Chronic obstructive pulmonary disease; AHI, apneahypopnea index; ODI, 3% oxygen desaturation index; T90, percent night-time with oxygen saturation below 90%; PSG, polysomnography. * OSA categories were compared using the nonparametric Kruskal-Wallis test for quantitative and the chi-square test for qualitative characteristics.

		All	No OSA	Mild OSA	Moderate OSA	Severe OSA	p value*
	n	8,748	1,161	1,878	1,826	3,883
	Age, years	61 [51-70] 52 [41-63] 58 [49-63] 61 [52-70]	64 [55-72] <0.0001
	Men, %	64.5	49.4	56.3	66.4	72.0	<0.0001
	BMI, kg/m 2	29 [26-34] 26 [23-30] 28 [25-30] 28 [26-33]	32 [28-36] <0.0001
	Alcohol intake, g/day	0 [0-20)	0 [0-10]	0 [0-10]	0 [0-20]	10 [0-20]	<0.0001
	Smoking status, %
	Never Current	41.6 22.6	42.5 28.2	42.8 24.4	41.6 21.2	40.6 20.7	<0.0001
	Former	35.9	29.3	32.9	37.2	38.7
	Epworth sleepiness score	10 [6-14]	10 [3-14]	9 [6-14]	10 [6-14]	10 [6-14]	0.4009
	QD2A depression score	3 [1-6]	3 [1-6]	3 [1-6]	3 [1-6]	3 [1-6]	0.1876
	Diabetes, %	14.5	6.2	10.3	13.1	19.5	<0.0001
	Cardiac disease, %	10.4	5.4	9.1	10.5	12.4	<0.0001
	Hypertension, %	33.9	15.7	28.0	31.4	43.2	<0.0001
	COPD, %	9.1	8.2	8.9	8.3	9.8	0.1819
	AHI, events/h	24 [9-41]	2 [1-3]	9 [7-3]	21[18-24]	44 [35-59] <0.0001
	ODI, events/h	17 [6-35]	2 [1-5]	7 [4-5]	15 [9-21]	37 [24-55] <0.0001
	T90, %	2 [0-13]	0 [0-0]	0 [0-0]	1 [0-5]	10 [2-28]	<0.0001
	Occupational category, %
	Farmer	4.2	2.7	3.2	3.6	5.4
	Craftsman	9.6	7.3	8.7	9.3	10.8
	Executive	19.2	17.4	19.5	22.1	18.2	<0.0001
	Intermediate	21.3	25.1	21.6	21.9	19.8
	Employees	19.6	24.8	23.0	18.3	17.0
	Workers	26.1	22.8	24.0	28.7	28.7
	Married or living as a couple, %	78.1	75.5	79.7	78.2	78.0	0.0673
	PSG, %	42.5	44.5	47.0	56.3	33.3	<0.0001
	Follow-up time, person-years	47,909	6,583	10,219	9,787	21,322	0.0089
	Incident cancers, n	718	66	136	161	355	<0.0001
	All-cancer incidence, n (%)	8.2	5.7	7.2	8.8	9.1	0.0005

Table 2 :

 2 Cox proportional hazard models assessing the association of all-cancer incidence with categories of obstructive sleep apnea (OSA) severity and indices of nocturnal hypoxemia.

	Abbreviations: CI, confidence interval
	*Adjusted for age, gender, body mass index, smoking status, alcohol intake, diabetes, hypertension,
	medical history of cardiac and chronic obstructive pulmonary disease, marital status, type of sleep
	study, and study site.

Table 3 .

 3 Cox proportional hazard models assessing the association between hypoxic burden and cancer incidence (n= 5,683) Adjusted for age, gender, body mass index, smoking status, alcohol intake, diabetes, hypertension, medical history of cardiac and chronic obstructive pulmonary disease, marital status, type of sleep study, and study site.

		Follow-up	All-cancer
		time, person-	incidence, n (%)	Unadjusted HR [95%CI]	p value	Adjusted* HR [95%CI]	p value
		years
	Hypoxic burden
	<12 %min/h (ref)	7,306	76 (5.5)	1.00	-	1.00	-
	12 to <36 %min/h	7,421	110 (7.9)	1.41 [1.05-1.88]	0.0220	1.10 [0.81-1.48] 0.5364
	36 to <84 %min/h	7,260	119 (8.6)	1.56 [1.17-2.08]	0.0024	1.01 [0.74-1.36] 0.9763
	≥84 %min/h	7,634	145 (10.5) 1.83 [1.38-2.41]	<0.0001	0.92 [0.68-1.25] 0.5991
	Abbreviations: HR, hazard ratio; CI, confidence interval

*

Table 4 .

 4 Cox proportional hazard models assessing the association of nocturnal hypoxemia (percent night-time with oxygen saturation below 90% [T90]) with the incidence of different cancer sites.Abbreviations: HR, hazard ratio; CI, confidence interval *Adjusted for age, gender, body mass index, smoking status, alcohol intake, diabetes, hypertension, medical history of cardiac and chronic obstructive pulmonary disease, marital status, type of sleep study, and study site. † p<0.05; † † p<0.01; † † † p<0.0001 # Tested by the Cochrane-Armitage trend test.

		Follow-up	All-cancer			p value
		time,	incidence,	Unadjusted HR	Adjusted* HR	for
		person-	n (%)	[95%CI]	[95%CI]	linear
		years				trend #
				Incident lung Cancer (n=72)
	T90, %
	<0.01 (ref)	16 556.5	12 (0.40)	1.00	1.00
	0.01 to <2.1 2.1 to <13	7 040.81 10 321.16 15 (0.83) 9 (0.73)	1.77 [0.75-4.20] 2.02 [0.95-4.32]	1.07 [0.41-2.76] 1.29 [0.58-2.86]	0.0349
	≥13	11 137.86 36 (1.84)	4.47 [2.32-8.59] † † †	2.14 [1.01-4.54] †
				Incident breast cancer (n=67)
	T90, %
	<0.01 (ref)	6 986.35	20 (1.55)	1.00	1.00
	0.01 to <2.1 2.1 to <13	2 653.35 3 239.83	19 (4.02) 16 (2.79)	2.52 [1.35-4.73] † † 1.74 [0.90-3.36]	2.04 [1.05-3.98] † 1.40 [0.69-2.87]	0.7989
	≥13	3 214.7	12 (2.06)	1.31 [0.64-2.67]	1.14 [0.50-2.58]
				Incident colorectal cancer (n=131)
	T90, %
	<0.01 (ref)	16 625.66 32 (1.06)	1.00	1.00
	0.01 to <2.1 2.1 to <13	7 064.47 10 376.26 38 (2.08) 21 (1.68)	1.55 [0.89-2.69] 1.91[1.19-3.05] † †	1.18 [0.67-2.09] 1.25 [0.76-2.06]	0.5105
	≥13	11 160.47 45 (2.29)	2.10 [1.33-3.30] † †	1.19 [0.70-2.01]
				Incident prostate cancer (n=68)
	T90, %
	<0.01 (ref)	9 648.8	13 (0.75)	1.00	1.00
	0.01 to <2.1 2.1 to <13	4 466.37 7 152.52	13 (1.65) 18 (1.45)	2.16[1.00-4.66] † 1.86 [0.91-3.81]	1.49 [0.67-3.30] 1.13 [0.54-2.38]	0.6268
	≥13	7 903.23	24 (1.74)	2.25[1.15-4.43] †	1.33 [0.62-2.84]

Acknowledgments

We thank the National Health Insurance Fund for giving us access to the French administrative health care database. The ERMES study group: Centre Hospitalier Universitaire, Angers: Christine Person, Pascaline Priou; Centre Hospitalier, Le Mans: Olivier Molinier, Audrey Paris; Centre Hospitalier, La Roche sur Yon: Isabelle Caby, Maël Bellier; Pôle santé des Olonnes, Olonne sur Mer: Marie Langelot-Richard; Centre Hospitalier Universitaire, Nantes: Sandrine Jaffre, Frédéric Corne; Nouvelles Cliniques Nantaises: Marc Normand de la Tranchade; Centre Hospitalier, Machecoul: Béatrice Rouault; Institut Recherche

Guarantor's name

Prof Gagnadoux is the guarantor of this work and, as such, had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis

Author Contributions

G.J., C.G.-P., M.L.V., W.T., N.M., F.Gou., T.P., X.-L.N., L.L.V., P.M., A.B.-T., M.-P.H., C.G., M.B., T.U., F.Gag. were substantially involved in the design of the study and critical revision of the paper for important intellectual content. G.J., C.G.-P., M.L.V., T.U., F.Gag. were substantially involved in drafting the article.

All authors were substantially involved in data acquisition, data analysis, or interpretation of data. All authors approved this final version of the article.


Arthur Sarthou

-Stéphane Vincent

Philippe Angot

-Jean- Paul Caltagirone

The Sub-Mesh Penalty Method

Keywords: High-order penalty methods, fictitious domains, Eulerian/Lagrangian grid coupling

The Sub-Mesh Penalty (SMP) method, a new fictitious domain method of high order is presented. The final aim is to develop an efficient coupling between incompressible multiphase flows and fixed or moving obstacles of complex shape. The flow is solved on a fixed Cartesian grid and the solid objects are represented by Lagrangian surface elements. Several validation problems in 2D and 3D are presented to demonstrate the interest and accuracy of the method.

Introduction

Many works have been devoted to the interaction between multiphase flows and solid obstacles such as the Arbitrary Lagrangian Eulerian (ALE) method [Mau 99], the Immersed Boundary Method (IBM) [PES 00] or the Immersed Interface Method (IIM) [LE 07], the Distributed Lagrangian Multiplier (DLM) approach [ GLO 99] or the Ghost Fluid method [ FED 99]. In order to deal with fluid/solid interactions, our objective is to propose a new penalty-based numerical method, spatially of second order, which can be easily implemented in an implicit finite volume CFD code with minor modifications of the standard discretization schemes. The main interest of this new method is to account for complex solid shapes or immersed interfaces on nonconforming structured grids with second-order accuracy, while former penalty methods are generally of first order only since they consider the projected shape of the fluid-solid interface on the Eulerian grid to define the penalty parameters [RAM 07]. Among the existing methods, we can cite the Darcy Penalty Method (DPM) for the Navier-Stokes equations that can be used to treat fixed obstacles by adding a Darcy term [KHA 00] in the momentum equations. Concerning obstacles moving under flow action the derivatives of the velocity are penalized through a new formulation of the viscous stress tensor ([RAN 05] and [VIN 07]) in order to impose no deformation with a velocity resulting from the effect of the surrounding fluid. This method is called the Implicit Tensorial Penalty Method (ITPM).

As a first step, we present the method for a fixed Cartesian grid in finite volumes, even if the method can a priori be applied to finite elements and unstructured grids. The article recalls the principles of the penalty methods. The new high-order method which is based on a sub-mesh penalty approach is then detailed. The last section presents validations dedicated to scalar and vector problems. Perspectives and conclusions are finally drawn.

Penalty methods for immersed interfaces

Low order penalty methods

Let us consider the following model scalar problem with a Dirichlet boundary condition on the interface Σ :

-∇ • (a∇u) = f in Ω 0 u |Σ = u D on Σ [1]
Some boundary conditions are also imposed on the other part of the boundary ∂Ω 0 such that the whole problem is well-posed. The penalty methods consist adding specific terms in the conservation equations to play with the order of magnitude of existing physical contributions so as to obtain at the same time and with the same set of equations two different physical properties. The Volumic Penalty Method (VPM) ([KHA 00] and the references therein) consists the addition of a penalty term b(u-u D ) in the conservation equations, such that:

-∇ • (a∇u) + b(u -u D ) = f in Ω with b |Ω0 = 0, b |Ω1 = 1 ε , for 0 < ε ≪ 1 [2]
where ε denotes the penalty parameter which tends to 0.

Sub Mesh Penalty algorithm with Dirichlet boundary conditions

A simple version of the SMP algorithm for problem [2] was first proposed in [SAR 08]. Let us consider the original domain of interest denoted by Ω 0 , typically the fluid domain, which is embedded inside a simple computational domain Ω ⊂ R d . The auxiliary domain Ω 1 , typically a solid particle or an obstacle, is then such that : Ω = Ω 0 ∪ Σ ∪ Ω 1 where Σ is an immersed interface (see Fig. 1 left). Let n be the unit outward normal vector to Ω 0 on Σ. Our objective is to numerically impose the adequate boundary conditions on the interface Σ. These conditions will be discretized in space with second order schemes on an Eulerian structured mesh covering Ω.

The computational domain Ω is meshed with a set of cell-centered finite volumes (V I ) for I ∈ E, E being the set of index of the Eulerian structured mesh (typically a Cartesian mesh uniform or not). Let x I be the vector coordinates of the center of each volume V I . The local space step of the volume V I defined as the maximum length Figure 1. Definition of the domains-discretization for the SMP method of V I in each direction is denoted by h I , whereas h denotes the Eulerian mesh step: h = sup I∈E h I . This grid is used to discretized the conservation equations. A dual grid is introduced for the management of the penalty method. The grid lines of this dual cell-vertex mesh are defined by the network of the cell centers x I . The volumes of the dual mesh are denoted by (V ′ I ) We note the Eulerian unknowns u I which are the approximated values of u(x I ), i.e. the solution at the cell centers x I . The discrete interface Σ h , hereafter called the Lagrangian mesh, is given by a discretization of the original interface Σ. It is described by a piecewise linear approximation of Σ :

Σ h = {σ l ∈ P d-1 1 , l ∈ L f }, L f
being the set of index of the Lagrangian mesh, and P d n being the space of polynomial of n degrees in d dimensions. Typically, σ l are segments in 2D and triangles in 3D. The vertices of each face σ l are denoted by x l,i for i = 1, d and the set of all vertices is : {x l , l ∈ L v }. The intersection points between the grid lines of the Eulerian dual mesh and the faces σ l of the Lagrangian mesh are denoted by {x i , i ∈ N } (see Fig. 1 right). Our objective is to discretize the boundary conditions at these interface points with a second-order approximation. The cell centers x I are sorted according to their location inside Ω 0 or Ω 1 by defining the color fonction C defined by : C I = C(x I ) = 1 if x I ∈ Ω 1 , and C I = 0 if x I ∈ Ω 0 . We introduce new sets of Eulerian points x I near the interface such that there it exists one neighbor x J verifying C J = C I , i.e. the segment [x I ; x J ] is cut by Σ h . These Eulerian "interface" points are also sorted according to their location inside Ω 0 or Ω 1 . So we have the two sets {x I , I ∈ N 0 } and {x I , I ∈ N 1 }. The previous equation [2] is then discretized with finite volumes on the Eulerian mesh covering Ω. For each control volume V I , we get a discrete equation formally written as:

-{∇ • (a∇u)} I + {b(u -u D )} I = {f } I , ∀I ∈ E [3]
where {} I denotes the spatial approximation of a quantity. The SMP algorithm introduces a specific discretization procedure for the points x I which are near the interface and inside the fictitious domain Ω 1 , i.e. for I ∈ N 1 . We first describe the case when x I has only one neighbor in Ω 0 . Let us consider one point x I of this type, x J its neighbor in Ω 0 and x l the intersection between [x I ; x J ] and Σ h (Fig. 1 right). Then, the interface unknown u l is approximated by the P 1 1 -interpolation between the Eulerian unknowns u I and u J :

u l = λ I u I + λ J u J with 0 < λ I , λ J < 1 and λ I + λ J = 1 [4]
Hence, for this point, the corresponding penalty term {b(uu D )} I is approximated by {b(u lu D (x l ))} I with b = 1/ε and the discrete equation becomes

-{∇ • (a∇u)} I + 1 ε (λ I u I + λ J u J -u D (x l )) I = {f } I , ∀I ∈ N 1 [5]
So, u l tends to u D when ε tends to 0.

Let us consider an additional point x K such as u J is between u I and u K . For the sake of simplicity, h = 1. The discretization of the Laplacian operator at x J using a centered scheme is (u

I -2u J + u K ). The discretization of [2] at x I tends to (λ I u I + λ J u J -u D (x l )) = 0.
We recall that λ I + λ J = 1, and for h = 1, λ I = d(x J , x l ), d being the distance function. Hence, the discretization at x J of the Laplacian operator can be rewritten as (-

(λ J u J + u D (x l ))/λ I -2u J + u K ) = ((u D -u J )/λ I -(u J -u K )).
We can notice that the term (u D -u J )/λ I is the centered spatial derivative between x D and x J , and the method is equivalent to rewriting the discretization of spatial operators using a modified boundary mesh fitting the objects. If x I has more than one neighbor in different directions, we build a penalty constraint using higher dimension P 1 or Q 1 interpolation. If x I has more than one neighbor in a given direction, the Eulerian mesh is considered as too coarse to accurately describe the interface and VPM is used. In any case, by decreasing the Eulerian mesh step h, we also decrease the number of points x I near Σ h where VPM is used. Hence, the present method is suitable to impose a Dirichlet boundary condition on Σ for Ω 0 , when the solution in Ω 1 has no interest. For nodes in this domain, the penalty term has crushed the initial conservation equation and the solution u I (x I ) for I ∈ N 1 is only useful to build the physical solution in Ω 0 . The solution in Ω 1 far from the interface is obtained using VPM. The imposed solution can be analytical, when possible, or a constant value. The uselessness of the calculation of the solution for these nodes can be treated numerically by switching the solving of u(x I ), x I ∈ {Ω 1 }, I ∈ N 1 off, or by totally removing these nodes. Future investigations will be devoted to these topics.

The SMP method can be extended to any scalar or vector equations, such as the Navier-Stokes model.

Management of the Lagrangian points

The generation of the Lagrangian mesh of the obstacles is achieved using a computer graphics software. Specific algorithms have been developed to interpret this Lagrangian grid on the Eulerian physical grid. We use a fast ray-casting method (working row by row) to obtain the binary phase functions C i of objects (strictly 0 or 1). These phase-functions are used to localize the immersed obstacles. These binary C i functions are used to build an Eulerian Level-Set function near the interface by estimating the distance between the Eulerian points and the neighbor Lagrangian points. The Lagrangian points used to couple the Lagrangian surface of the complex object and the Eulerian grid used to solve the conservation equations are the points x l of Σ h . We use two different methods to determine these points. The first method (SMP-LS) uses the Level-Set function which indicates the signed distance of an Eulerian point from the interface. Using this Eulerian function, we can quite accurately determine the intersection between an edge of V ′ i and the interface. For example, let us consider two Eulerian points x I ∈ Ω 0 and x J ∈ Ω 1 . We denote by d I = d(x I , Σ h ) and d J = d(x J , Σ h ) the unsigned distances between Eulerian points and the interface Σ h . Then, we obtain x l = (x I d J + x J d I )/(d I + d J ). The efficiency of this method increases with the regularity of the Lagrangian interface. The second method (SMP-GI) calculates the geometric intersection between an edge of V ′ i crossing the interface and the interface. This last method is very accurate but a lot slower.

Discretization and solvers

Details on the discretizations and the validations have previously been published by Vincent et al. [VIN 00]. All simulations are based on implicit finite volumes on staggered Cartesian grids. For the Navier-Stokes equations, velocity/pressure coupling is obtained using an Augmented Lagrangian (AL) method. In [SAR 08], the use of 1D linear interpolations only was incompatible with the minimization of the divergence provided by the AL operator. This problem has been solved using multidimensional linear interpolations. Concerning the penalty terms, they are added to the motion equations according to the Eulerian description C i of the object (low order approach) or to their Lagrangian position x l (high order method). The penalization parameter ǫ is chosen according to C i independently of any other numerical or physical parameters. Numerically, b = 1/ǫ = 10 40 C i . A particular class of IBM methods, the direct-forcing method [TSE 03] for the Navier-Stokes equations, uses a technique quite similar to SMP to treat the fluid/solid interface. The main difference lies in the integration of the interfacial constraint into the conservation equation. Our technique lies in an implicit penalty method whereas the IBM direct-forcing approach builds a source term. Direct forcing methods uses fractional step methods to ensure pressure/velocity coupling. The AL ensure in only one step both divergence free and penalty constraint, whereas the time splitting approach requires a corrector step to ensure the divergence free constraint which must be adapted to each fictitious domain method. Gibou et al. have proposed a fictitious domain approach for scalar equations [GIB 05]. The method uses 1D interpolations which are taken into account by directly modifying the discretization of the operators. [TSE 03] and [GIB 05] explain that their methods requires special treatment of the interpolation when a Lagrangian point is very close to an Eulerian point. In fact, implicit constraints modify the matrix which can loose its diagonal dominance. Numerical problems occurs with SMP method too if coupled with a basic solver such as a Jacobi method. When coupled with a PAR-DISO direct solver or an iterative BiCG-Stab method and an ILU preconditioning, our method correctly impose the interfacial constraint, no matter the matrix coefficients.

Validations and applications

3.1. Sub Mesh Penalty method for scalar equation We solve the homogenous Laplace equation in a numerical square of side 4 with a Dirichlet condition of u 1 = 10 on a first circular interface (R 1 = 1) and an analytical solution on the boundary of the Eulerian grid. Practically, the analytical solution which accounts for the presence of a second circle with a radius R 2 = 4 and u 2 = 0 is imposed on the boundary conditions. Fig. 2 left shows a second order of convergence in space for the L 2 relative error. Fig. 2 center and right represent the isovalue u = 10, i.e. the Dirichlet value imposed on R 1 , obtained from the simulations with the VPM (center) and the SMP method (right). Analytically, the isovalue is a circle. As can be seen, the SMP method greatly improves the shape of the isovalue. We simulate the cylindrical Couette flow in a numerical square of side 0.30 m. The inner circle has a rotation speed ω 1 = 1 rad/s and a radius R 1 = 0.05 m. We impose the analytical solution on the numerical boundary as the domain is surrounded by a second circle whose rotation speed is ω 2 = 2 rad/s and radius is R 2 = 2 m. In Fig. 3 left, the convergence of the L 2 relative error is presented. A second order is reached for velocity whereas a first order is observed for pressure. For all meshes, the maximum divergence of the flow is about 10 -14 with a velocity correction of projection type. The third curve (Velocity LS) shows the convergence rate of the method using the Level-Set function to define x l (SMP-LS). This method is less accurante than SMP-GI which take the true intersection, and an average order of only 1.55 is obtained. As a second test case, using SMP-GI, we simulate the flow past a circular cylinder of radius 0.05 m immersed in a rectangular domain [ -1.6 ; 1.4 ] × [ -0.75 ; 0.75 ] at Reynolds 40. Fig. 3 right shows the convergence of the recirculation length against the length for a 2000 × 1000 mesh for which L/d = 2.37. A second order is obtained. The last case, using SMP-GI, is the measurement of the recirculation length for a 3D flow past a sphere for Re = 100. The domain is a box Ω = [ -10 ; 10 ] × [ -5 ; 5 ] × [ -5 ; 5 ]

Discussion and conclusion

On fixed staggered Cartesian grids, a new Sub Mesh Penalty method has been proposed for the simulation of free surface flows interacting with complex shape obstacles (see an example of dam break flow over an obstacle in Fig. 4 right). These obstacles are managed using Lagrangian meshes easily generated with 3D computer graphic softwares. Several penalty methods have been implemented and coupled to obtain the interaction between fluid and solid media. The SMP method is second order for both scalar diffusion and the Navier-Stokes equations. The interest of the interpretation and management of triangularized surface of obstacles can be seen in [MKF 07]. This Web site contain several three-dimensional realistic multiphase flow simulations involving the SMP method with Q 0 interpolations. Future works will be devoted to extending the SMP method, coupled to ITPM or VPM, to moving obstacle of arbitrary motion. The formal proof of the second order of the SMP method is under consideration as well as its compatibility with various approaches dealing with incompressibility, in order to reach the second order convergence on pressure. In addition, Adaptive Mesh Refinement (AMR [DEL 06]) techniques will be associated with the tracking of boundary layers in penalized fluid/solid cells.

Figure 2 .

 2 Figure 2. Relative L 2 convergence of the SMP method and comparison between solutions obtained with VPM and SMP method on a 16 × 16 grid

Figure 3 .

 3 Figure 3. Relative L 2 error of the SMP method for a Couette flow-Error on recirculation length for a flow past a cylinder

Figure 4 .

 4 Figure 4. Mesh and streamtraces for the flow past a sphere-dam break flow over an obstacle