Catherine Féart

PhD Karine Pérès

PhD Cécilia Samieri

PhD Luc Letenneur

PhD Jean-François Dartigues

PhD Pascale Barberger-Gateau

Catherine Feart
email: catherine.feart@isped.u-bordeaux2.fr

Adherence to a Mediterranean diet

Keywords: Mediterranean diet, disability, activities of daily living

TEXT INTRODUCTION

Absence of disability or activity limitation is a major component of successful aging. Disability is defined as difficulty or dependency in carrying out activities necessary for independent living, including roles, tasks needed for self-care and household chores and other activities important for a person"s quality of life [START_REF] Fried | Untangling the concepts of disability, frailty, and comorbidity: implications for improved targeting and care[END_REF]. The disablement process leads to a poorer quality of life, a rise in morbidity and mortality without predictable pattern [START_REF] Keeler | The impact of functional status on life expectancy in older persons[END_REF][START_REF] Seeman | Disability trends among older Americans: National Health And Nutrition Examination Surveys, 1988-1994 and 1999-2004[END_REF][START_REF] Verbrugge | The disablement process[END_REF]. Despite a trend to lower disability rates in the last years, the prevalence of disability increases again as a consequence of increasing number of older subjects, surviving to severe disease or with long course poorer health status, such as obesity and overweight [START_REF] Seeman | Disability trends among older Americans: National Health And Nutrition Examination Surveys, 1988-1994 and 1999-2004[END_REF][START_REF] Alley | The changing relationship of obesity and disability, 1988-2004[END_REF]. Therefore, strategies to prevent or delay the entry into the disablement process, to slow down its progression and / or to contribute to recover from disability must be developed [START_REF] Guralnik | Underestimation of disability occurrence in epidemiological studies of older people: is research on disability still alive?[END_REF]. Nutrition constitutes an interesting approach against the development and progression of disability, which deserves further investigation [START_REF] Stuck | Risk factors for functional status decline in community-living elderly people: a systematic literature review[END_REF][START_REF] Schroll | Aging, food patterns and disability[END_REF]. Some epidemiological studies have suggested that specific foods or nutrients (i.e. dairy products, fruits, vegetables, calcium, vitamins D, E, B6, B12, folate and selenium) may have potential protective effects against disability [START_REF] Houston | Dairy, fruit, and vegetable intakes and functional limitations and disability in a biracial cohort: the Atherosclerosis Risk in Communities Study[END_REF][START_REF] Bartali | Low micronutrient levels as a predictor of incident disability in older women[END_REF][START_REF] Ortega | Functional and psychic deterioration in elderly people may be aggravated by folate deficiency[END_REF][START_REF] Sharkey | Nutrient intake and BMI as predictors of severity of ADL disability over 1 year in homebound elders[END_REF][START_REF] Tomey | Dietary intake related to prevalent functional limitations in midlife women[END_REF][START_REF] Karlamangla | Light to moderate alcohol consumption and disability: variable benefits by health status[END_REF][START_REF] Vercambre | Longterm association of food and nutrient intakes with cognitive and functional decline: a 13-year follow-up study of elderly French women[END_REF]. While many components of food have been studied in isolation, nutrients are consumed in combination in foods, which are themselves components of various dietary patterns [START_REF] Kant | Dietary patterns and health outcomes[END_REF]. In this context, the Mediterranean diet (MeDi) combines several foods and nutrients already proposed as potential protective factors against several conditions that may contribute to the disablement process [START_REF] Milaneschi | Nutritional determinants of mobility[END_REF][START_REF] Sofi | Adherence to Mediterranean diet and health status: meta-analysis[END_REF][START_REF] Trichopoulou | Anatomy of health effects of Mediterranean diet: Greek EPIC prospective cohort study[END_REF]. Greater MeDi adherence has already been associated with longer survival, reduced risk for cardiovascular and cancer mortality, lower cancer and Alzheimer"s disease incidence and slower cognitive decline [START_REF] Sofi | Adherence to Mediterranean diet and health status: meta-analysis[END_REF][START_REF] Trichopoulou | Anatomy of health effects of Mediterranean diet: Greek EPIC prospective cohort study[END_REF][START_REF] Sofi | Accruing evidence on benefits of adherence to the Mediterranean diet on health: an updated systematic review and meta-analysis[END_REF][START_REF] Scarmeas | Physical activity, diet, and risk of Alzheimer disease[END_REF][START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF][START_REF] Feart | Mediterranean diet and cognitive function in older adults[END_REF]. To our knowledge, the association between MeDi adherence and disability in older persons has never been assessed. The purpose of the current study was to investigate the relationship between MeDi adherence and disability in activities of daily living over 5 years among French older communitydwellers from the Three-City Study.

METHODS

Participants

The Three-City (3C) study is a prospective cohort study of vascular risk factors of dementia whose methodology has been described elsewhere [START_REF] The | Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. The 3C study protocol was approved by the Consultative Committee for the Protection of Persons participating in Biomedical Research at Kremlin-Bicêtre University Hospital (Paris). A sample of 9294 community dwellers aged 65 and over was selected in 1999-2000 from the electoral rolls of three French cities (Bordeaux, Dijon and Montpellier). All participants gave written informed consent. Three follow-up examinations were performed 2 (wave 1, 2001-2002), 4 (wave 2, 2003-2004) and 7 y (wave 3, 2006-2007) after baseline examination. Data collection included sociodemographic information, lifestyle, symptoms and medical complaints, medical history, blood pressure, smoking status, drug use, anthropometric data, neuropsychological testing, and blood sampling. The present study is based on waves 1, 2 and 3 in Bordeaux, the only centre where the standard data collection was completed with a comprehensive dietary survey. The study sample, whose characteristics were detailed previously [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF], comprises 1410 nondemented participants without missing nutritional data and with at least one follow-up reexamination over 5 years. At wave 2, 1340 subjects (95%) were re-examined and 1213 (86% of the initial study sample) were re-examined at wave 3. Among the 114 subjects (53 men and 61 women) with nutritional data at wave 1 but without follow up examination over time, 69.8% of men and 57.4% of women were deceased and 42 subjects refused or were lost-to-follow-up between waves 1 and 3. Wave 3 was closed on December 2007 in the Bordeaux Center.

Assessment of disability

Two domains of disability based on the concepts of basic activities of daily living (ADL) and instrumental ADL (IADL) were investigated as outcomes. Based on the Lawton-Brody scale [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF], disability in IADL was assessed according to the ability of the participants to use a telephone, manage medication, manage money, use public or private transport, and do shopping, for both genders, and additionally, to do the laundry, do housework and prepare meals for women. Disability in ADL was assessed using five items of the Katz scale [START_REF] Katz | Progress in development of the index of ADL[END_REF]: bathing, dressing, toileting, transferring from bed to chair, and eating. Incontinence, which is an impairment rather than a disability [START_REF] Spector | Combining activities of daily living with instrumental activities of daily living to measure functional disability[END_REF], was not considered here. At each wave, a subject was considered as disabled if he/she could not perform at least one activity of the domain without a given level of assistance, as defined in the respective instruments [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF][START_REF] Katz | Progress in development of the index of ADL[END_REF].

Dietary assessment and MeDi score

At wave 1, participants were visited at home by a specifically trained dietician who administered a food frequency questionnaire (FFQ, not semi-quantitative) and a 24H dietary recall [START_REF] Feart | Energy, macronutrient and fatty acid intake of French elderly community dwellers and association with socio-demographic characteristics: data from the Bordeaux sample of the Three-City Study[END_REF][START_REF] Samieri | Dietary patterns derived by hybrid clustering method in older people: association with cognition, mood, and self-rated health[END_REF]. Data from the 24H recall were only used to estimate total energy intake in kcal/d and to compute the MUFA-to-SFA ratio (ratio of the nutrient intake in g/d). Based on the FFQ, the frequency of consumption of 40 categories of foods and beverages for each of the 3 main meals and 3 between-meals snacks was recorded in 11 classes. The food items were converted into number of servings per week and then aggregated into 20 food and beverage groups as described earlier [START_REF] Samieri | Dietary patterns derived by hybrid clustering method in older people: association with cognition, mood, and self-rated health[END_REF]. We then identified the food groups considered to be part of the MeDi: vegetables, fruits, legumes, cereals (including bread, pasta and rice, without distinction between whole and refined grains), fish, meat and dairy products. The number of servings/week for each food group was determined and the MeDi score was computed as follows: a value of 0 or 1 was assigned to each food group using sex-specific medians of the population as cut-offs, as previously described [START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF]. Briefly, subjects received one point if their intake was higher than the sex-specific median for a presumed protective component (vegetables, fruits, legumes, cereals, fish and MUFA-to-SFA ratio) and lower than the sexspecific median for a presumed deleterious component (meat and dairy products). For alcohol, one point was given to mild-to-moderate consumers. Cut-offs, chosen to be close to the second quartile of distribution of total alcohol consumption, were defined in men and women separately. One point was attributed for men if consumption was within 7-to-14 glasses per week (10-to-20g/d) and for women if consumption was within 1-to-4 glasses per week (1.4to-5.7g/d). The MeDi score was generated by adding the scores for each food category for each participant. Thus, the MeDi score could range from 0 to 9, with higher scores indicating greater MeDi adherence [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF]. Three MeDi categories (Low MeDi adherence, score 0-3; Middle MeDi adherence, score 4-5; or High MeDi adherence, score 6-9) were defined so as to be nutritionally relevant, close to tertiles of the distribution of the MeDi score in our sample and similar to those used in previous studies [START_REF] Scarmeas | Physical activity, diet, and risk of Alzheimer disease[END_REF][START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF][START_REF] Trichopoulou | Adherence to a Mediterranean diet and survival in a Greek population[END_REF].

Covariates

Socio-demographic information included age, sex, education (in four classes: no education or primary school only, secondary (middle) school, high school, vocational school or university), income (less than 750 euros, 750 to 1500 euros, 1500 to 2250 euros, more than 2250 euros per month and refuse to answer) and marital status (in four classes: married, divorced or separated, widowed, single). Global cognitive functioning was assessed using the Mini-Mental State Examination (MMSE), used as continuous variable [START_REF] Folstein | Mini Mental State : A practical method for grading the cognitive state of patients for the clinician[END_REF]. Vascular risk factors included Body Mass Index (BMI in kg/m 2 , used as continuous variable), smoking status (in three classes: never, ex-and current smokers), self-reported history of cerebrovascular disease, diabetes, hypercholesterolemia and measured hypertension (blood pressure ≥ 140/90 mmHg or treated), all used as dichotomous variable. Moreover, taking five medications/d (median of total medication consumption/d) or more was considered an indicator of comorbidity and used as dichotomous variable [START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF][START_REF] The | Vascular factors and risk of dementia: design of the Three-City Study and baseline characteristics of the study population[END_REF]. Depressive symptomatology was assessed on the Center for Epidemiological Studies-Depression scale (CES-D), used as continuous variable [START_REF] Berkman | Depressive symptoms in relation to physical health and functioning in the elderly[END_REF][START_REF] Radloff | The CES-D scale : a self-report depression scale for research in the general population[END_REF]. Practice and intensity of physical exercise was assessed by two questions: "Do you practice sports?" and "Do you perspire when you practice sport?". A threelevel variable (no, moderate and intensive) was computed to describe intensity of physical exercise as earlier described [START_REF] Feart | Energy, macronutrient and fatty acid intake of French elderly community dwellers and association with socio-demographic characteristics: data from the Bordeaux sample of the Three-City Study[END_REF].

Statistical analyses

All statistical analyses were performed with SAS Statistical package (Version 9.1 SAS Institute). Participants were classified according to their disability status at baseline in each domain separately (IADL or ADL) and also according to categories of MeDi adherence (low, middle or high). Baseline disability status and dietary characteristics were compared between men and women using Chi-Square statistics or Student"s t test. Baseline demographic, dietary and clinical characteristics were compared, in men and women separately, between subjects with and without disability in IADL or ADL using Chi-Square statistics or Student"s t test. We then explored associations between the presence of disability in IADL or ADL and MeDi adherence at baseline (wave 1) by multi-linear logistic regression. Associations between MeDi adherence and incidence of disability were investigated in subjects without baseline disability by Cox proportional hazards models with delayed entry and taking age as a time scale [START_REF] Commenges | Modelling age-specific risk: application to dementia[END_REF]. Odds ratios (OR), Hazard ratios (HR) and 95% confidence intervals (95% CI) were estimated for the MeDi score considered as a continuous variable (MeDi score from 0 to 9, 1point increase) or as a categorical variable (low MeDi category, score 0-3 as reference vs middle, score 4-5 and high, score 6-9, categories respectively). For logistic regressions and Cox analyses, two models were performed: model 1 was adjusted for (age) education (the first class (no education or primary school only) was chosen as reference), income (reference: less than 750 euros), marital status (reference: married) and total energy intake, used as continuous variable. Then, cardiovascular risk factors (smoking (reference: never), BMI, hypertension (reference: no hypertension) and diabetes (reference: no diabetes), stroke (reference: no history of stroke), taking ≥ 5 medications/d (reference: less than 5 medications/d), MMSE and CES-D scores were entered as additional adjustment variables in model 2. Additional adjustment for physical activity (reference: no practice) was performed in secondary analyses. Separate analyses were performed for each gender since gender differences in disability have been widely documented [START_REF] Newman | Gender gap in longevity and disability in older persons[END_REF][START_REF] Murtagh | Gender differences in physical disability among an elderly cohort[END_REF], three specific items were additionally used when assessing women as in the original IADL scale [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF] and the MeDi score was computed according to the sex-specific medians of consumption of food groups. In sensitivity analyses, we explored the association between MeDi adherence and disability in IADL focusing only on the five items administered for both genders (to use a telephone, manage medication, manage money, use public or private transport, and do shopping) in order to perform similar analyses in women and men.

RESULTS

The study sample consisted of 1410 non-demented individuals (527 men and 883 women), aged 75.9y on average (range 67.7-94.9), with at least one follow-up re-examination over 5 years. The median follow-up was 4.1y (range 1.6-6.1y). Participants with missing follow-up (n=114) had a non-significant lower mean MeDi score (4.1 vs. 4.4 P=0.09), were significantly more disabled in IADL (29.5% vs. 13.6%, P<0.0001) but not in ADL (2.6% vs. 1.1%, P=0.16) at baseline than those with complete follow-up. Disability in IADL and ADL was present in 190 (13.6%) and 16 (1.1%) participants at baseline respectively (Table 1). Women were significantly more often disabled in IADL than men at baseline. Because of the small number of ADL disabled subjects at baseline, participants with disability in ADL or IADL were brought together at each wave and termed as subjects with disability in basic or instrumental ADL (B-IADL). The MeDi score ranged from 0 to 8. The mean (SD) MeDi score was 4.4 (1.7) and was slightly but significantly higher in men (Table 1). As expected, greater MeDi adherence was characterized by higher intake of vegetables, fruits, legumes, cereals and fish and lower intake of meat and dairy products. High MeDi adherence was significantly more frequent in men. Men with baseline disability in B-IADL were significantly older, had lower income, higher mean CES-D scores and took significantly more medications/d than the control men (Table 2). The mean MeDi score did not differ between men with and without baseline disability in B-IADL (Table 2). Women with baseline disability in B-IADL were significantly older, had lower educational level and income, were more often widowed and had a lower practice of physical exercise than control women. The mean MeDi score was also significantly lower in women with baseline disability in B-IADL than in control women, with a borderline significant trend to a higher frequency of women with low MeDi adherence (Table 2). Women with baseline disability had also a worse health status than control women (Table 2). The association between baseline disability in B-IADL and MeDi adherence was assessed in multivariate logistic regression models in 513 men and 839 women without missing data on any covariate (Table 3). In men as in women, MeDi score, either as a continuous or categorical variable, was not associated with baseline disability in B-IADL. Among the 1212 participants without disability in B-IADL at baseline, incident disability in B-IADL occurred for 284 subjects. The incidence of disability for B-IADL was higher in women than in men (6.7 for 100 persons-years vs. 4.8 respectively, P=0.02). Among 470 men without missing data on adjustment variables, there was no association between MeDi score, either as a continuous or categorical variable, and the risk of disability in B-IADL over time (Table 4). In women (n=709 without missing data), each additional unit of MeDi score was associated with a 10 % (95% CI 2% to 18 %) reduced risk of incident disability in B-IADL over time (Table 4). When considering the MeDi score as a categorical variable, women in the highest MeDi category (score 6-8) compared with those in the lowest MeDi category (score 0-3) had a 50% (95% CI 22% to 68%) relative risk reduction of incident disability in B-IADL over time (Table 4). Additional adjustment for physical activity did not modify these associations (data available on request).

In sensitivity analyses, the association between MeDi adherence and incident disability in B-IADL was assessed in women excluding the three women-specific items of the IADL scale. We identified 720 women without baseline disability in the 5-common IADL items and without missing data on adjustment variables. Among them, incident disability in the same items occurred for 185 women. As for the whole study sample of women, each additional unit of MeDi score was associated with a reduced risk of incident disability in the 5-common IADL items over time (HR=0.90, 95% CI 0.82 to 0.98, P=0.02). Moreover, a 50% relative risk reduction for incidence of disability in the 5-common IADL items was observed for women in the highest MeDi category (score 6-8) compared with those in the lowest MeDi category (score 0-3) (HR=0.50, 95% CI 0.32 to 0.78, P=0.002, P for trend=0.01).

DISCUSSION

This large population-based prospective study in older persons living in South-western France shows no association between MeDi adherence and the presence of disability in B-IADL in men as in women in fully adjusted models. By contrast, women with higher MeDi adherence were less likely to become disabled in B-IADL over time. Moreover, there was no association between MeDi adherence and incidence of disability in B-IADL in men. These associations were independent of energy intake, depressive symptomatology, cardiovascular risk factors and baseline global cognitive performances.

Few studies have linked dietary factors with disability in B-IADL among older persons and to our knowledge, none concerned the MeDi [START_REF] Schroll | Aging, food patterns and disability[END_REF][START_REF] Willett | Mediterranean diet pyramid: a cultural model for healthy eating[END_REF]. Among prospective studies, intake of dairy products, fruits and vegetables were inversely associated with risk of disability in IADL and ADL over time, mainly in women [START_REF] Houston | Dairy, fruit, and vegetable intakes and functional limitations and disability in a biracial cohort: the Atherosclerosis Risk in Communities Study[END_REF][START_REF] Vercambre | Longterm association of food and nutrient intakes with cognitive and functional decline: a 13-year follow-up study of elderly French women[END_REF]. No association with other food groups was reported in these analyses. Despite the lack of assessment of disability in IADL and ADL at baseline in these studies, which constituted a major drawback, results are in agreement with a potential benefit role of particular food groups in the disablement process. Concerning dietary micronutrient intake, women of the E3N study with IADL impairment reported past lower intakes of vitamins B6 and B12 [START_REF] Vercambre | Longterm association of food and nutrient intakes with cognitive and functional decline: a 13-year follow-up study of elderly French women[END_REF]. Biological data confirm these findings since low serum concentration of vitamins B6 and B12, were significant and independent predictors of ADL disability 3 years later in women enrolled in the Women"s Health and Aging Study while carotenoids, zinc and folate levels, which are also yet common in the MeDi, were not significant predictors [START_REF] Bartali | Low micronutrient levels as a predictor of incident disability in older women[END_REF].

The biological basis for the apparent health benefits of the MeDi involves a decrease in oxidative stress and inflammation which also participates in the dysregulation of cellular function, muscle damages and the exacerbation of degenerative diseases, such as cognitive decline, sarcopenia, cardiovascular diseases, atherosclerosis, cancer and frailty, leading to B-IADL disability [9-10, 14, 38-41]. Individuals with higher MeDi adherence have been shown to have higher reduced-to-oxidized glutathione ratio, and lower insulin resistance, oxidized low-density lipoprotein, C-reactive protein and interleukin-18 levels [START_REF] Bach-Faig | Evaluating associations between Mediterranean diet adherence indexes and biomarkers of diet and disease[END_REF][START_REF] Dai | Association between adherence to the Mediterranean diet and oxidative stress[END_REF][START_REF] Giugliano | Mediterranean diet and metabolic diseases[END_REF].

The striking differences of the association between MeDi adherence and incident disability in B-IADL according to gender may be explained by several factors. First of all, some chronic and acute conditions impacting on the disablement process may differ in men and women [START_REF] Verbrugge | The disablement process[END_REF][START_REF] Newman | Gender gap in longevity and disability in older persons[END_REF][START_REF] Murtagh | Gender differences in physical disability among an elderly cohort[END_REF][START_REF] Wray | Explaining the role of sex on disability: a population-based study[END_REF]. For instance, the incidence of cardiovascular disease is higher in men, leading to sudden death without prior disability. The inverse association between adherence to a MeDi and lower risk of cardiovascular mortality has been widely documented [START_REF] Scarmeas | Physical activity, diet, and risk of Alzheimer disease[END_REF][START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF]. Hence, in men a protective effect of the MeDi may be more easily evidenced on mortality than on disability. However, the mean MeDi score was not significantly different between men with follow-up compared to those who deceased between 2001-2002 and 2006-2007 (data available on request). A second set of explanations for these gender differences may lie in methodological aspects related to the measurement of dietary intake. Firstly, computation of the MeDi score is based on sex-specific medians. In our sample, the medians consumption of most of food groups were very similar in men and women, except for that of dairy products and meat. Second, the FFQ used in the present study assessed number of servings but not portion size. Although our analyses were adjusted for total energy intake, we cannot exclude that the quantity of presumed protective or deleterious nutrients consumed by men and women with a similar MeDi score may differ. This misclassification in the levels of exposure is probably not related to the incidence of disability in B-IADL. Therefore, our results may have been an underestimation of the true associations. This could in part explain the lack of association between MeDi adherence and incidence of disability in B-IADL in men. Finally, the lack of an association between MeDi adherence and disability in B-IADL at baseline may be due to a lack of statistical power, since few men are disabled at baseline.

We found an inverse association between MeDi adherence and risk of disability in B-IADL over time but not at baseline in fully adjusted models. First, it is important to consider that women included in cross-sectional and longitudinal analyses partly differed. Indeed, women with baseline disability in B-IADL are excluded from longitudinal analyses, leading to a sub-sample of women with high baseline MeDi score and better health status for longitudinal analyses. Second, most late-life disability is believed to be the result of an insidious process in which risk factors for disablement accumulate over many years and slowly exert deleterious effects that impair the ability to live independently [START_REF] Fried | Disability in older adults: evidence regarding significance, etiology, and risk[END_REF]. The protective association between the MeDi and risk of B-IADL disability is in accordance with previous findings of slower cognitive decline in older persons adhering more closely to a MeDi [START_REF] Feart | Mediterranean diet and cognitive function in older adults[END_REF]. Indeed, B-IADL, and more specifically the so-called "Instrumental" activities [START_REF] Lawton | Assessment of older people: self-maintaining and instrumental activities of daily living[END_REF], include complex tasks such as managing finances, medication or means or transportation that require integrity of cognitive functioning [START_REF] Barberger-Gateau | Neuropsychological correlates of self-reported performance in instrumental activities of daily living and prediction of dementia[END_REF]. Limitation on these activities is strongly associated with cognitive decline and risk of dementia [START_REF] Peres | Natural history of decline in instrumental activities of daily living performance over the 10 years preceding the clinical diagnosis of dementia: a prospective population-based study[END_REF]. Hence, the MeDi could contribute to postpone disability in B-IADL by slowing down cognitive decline in older persons. Indeed, the MeDi provides dietary antioxidants, from fruits and vegetables, B vitamins, from cereals and green leafy vegetables and omega3 fatty acids, from fish, whose protective role against brain aging has been suggested by many observational studies [START_REF] Cunnane | Fish, docosahexaenoic acid and Alzheimer's disease[END_REF][START_REF] Guyonnet | IANA Task Force on Nutrition and Cognitive Decline with Aging[END_REF][START_REF] Feart | Adherence to a Mediterranean diet and plasma fatty acids: data from the Bordeaux sample of the Three-City study[END_REF]. This effect was independent of baseline cognitive performance and cardiovascular risk factors in our study. Nevertheless, this protective association was observed only in women and not explained by the three women-specific items of the IADL scale. Some studies observed a higher incidence of dementia in women [START_REF] Letenneur | Are sex and educational level independent predictors of dementia and Alzheimer's disease? Incidence data from the PAQUID project[END_REF] which may have increased the power of the study to detect a protective effect against functional decline associated with neurodegenerative disease in women.

Our results should be interpreted with caution because of some methodological limitations. First, the overall good health of participants is highlighted by the low prevalence of baseline disability in subjects enrolled in the current study compared with those who were not included in these analyses. This may in part explain the low incidence of disability in IADL and ADL observed and may have decreased our chance to evidence additional or stronger associations. In a sub-sample of the French PAQUID study, we already observed similar incidence of disability in IADL and ADL [START_REF] Sauvel | Factors associated with a 1-year development in the functional autonomy of elderly persons living at home[END_REF][START_REF] Deschamps | Nutritional status of healthy elderly persons living in Dordogne, France, and relation with mortality and cognitive or functional decline[END_REF]. Usually, women experience longer disability before death than men [START_REF] Newman | Gender gap in longevity and disability in older persons[END_REF]. Hence, more new "cases" of disability may have been missed in men who may have died in a disabled state between two waves of follow-up, and thus never detected. Second, the MeDi score is based on a traditional Mediterranean reference pattern defined a priori which does not consider the overall correlation between all foods [START_REF] Kant | Dietary patterns and health outcomes[END_REF]. The use of sex-specific cut-off points does not measure adherence to a universal traditional MeDi pattern but rather to a specific pattern, which precludes generalising results of the present study to other populations who have on the whole different food intake and MeDi adherence [START_REF] Bach | The use of indexes evaluating the adherence to the Mediterranean diet in epidemiological studies: a review[END_REF]. However, advantage of the MeDi score is the potential to consider additional, synergistic or opposite effects of 9 food groups and therefore to minimize confounding and modification effects among nutritional variables. In the same way, we adjusted our analyses for energy intake, to achieve of an isocaloric diet and to reduce measurement error in the score [START_REF] Willett | Adjustment for total energy intake in epidemiologic studies[END_REF][START_REF] Balzi | Risk factors for disability in older persons over 3-year follow-up[END_REF]. Finally, we cannot rule out the possibility of residual confounding by other unknown risk or protective factors such as a general healthier lifestyle of MeDi adherents. Despite these limitations, the strengths of the present study are its size, the population-based design, and control for several potential confounders. In particular, we controlled for depressive symptomatology since links between poor performance in IADL and depression are well documented [START_REF] Covinsky | Depressive symptoms in middle age and the development of later-life functional limitations: the long-term effect of depressive symptoms[END_REF][59]. We also controlled our analysis for BMI, cardiovascular risk factors and cognitive performances, which reinforced our results [START_REF] Alley | The changing relationship of obesity and disability, 1988-2004[END_REF][START_REF] Scarmeas | Physical activity, diet, and risk of Alzheimer disease[END_REF][START_REF] Feart | Adherence to a Mediterranean diet, cognitive decline, and risk of dementia[END_REF]. Although physical activity would most likely impact the association between dietary intake and physical functioning and disability, additional adjustment for this variable did not modify the result of the analyses. We did not adjust our main analyses for physical activity because we hypothesized that it might lie in the causal pathway, and thus was not a confounder [START_REF] Paterson | Physical activity and functional limitations in older adults: a systematic review related to Canada's Physical Activity Guidelines[END_REF]. In conclusion, this cohort study conducted in French elderly community dwellers showed an inverse association between adherence to a Mediterranean type diet and risk of B-IADL disability in women. No association was evidenced in men. The Mediterranean-style diet pattern probably does not fully explain the better health of persons who adhere to it but it likely contributes directly. The possible beneficial effect of the MeDi identified in the current study on the onset of disability, at least in older women, could be added to its widely acknowledged health benefits.

Table 2 .

 2 Demographic, dietary and clinical characteristics according to baseline disability in basic or instrumental ADL among older persons living in Bordeaux, The Three-City study, 2001-2002 : ADL activities of daily living; MeDi Mediterranean diet; BMI, Body Mass Index; CES-D Center for Epidemiological Studies-Depression scale; MMSE Mini-Mental State Examination P-value for the Chi-Square test between subjects with baseline disability in B-IADL and control subjects, except for age, Mediterranean diet score, total energy intake, BMI, CES-D score and MMSE score whose means were compared by Student"s t test.

			Men (n=527)		Women (n=883)
		Control	Disabled	P	Control	Disabled	P
		(n=477)	(n=50)		(n=735)	(n=148)
	Demographic
	characteristics
	Age (y) mean (SD)	75.1 (4.6) 77.8 (4.6) <0.0001	75.4 (4.6)	79.3 (5.0) <0.0001
	Education (%)			0.17			0.0002
	no or primary school	23.7	38.0		33.1	51.4
	secondary		22.0		29.4	23.6
	high school	22.4	18.0		24.1	12.5
	university	28.1	22.0		13.4	12.5
	Monthly income (€) (%)			0.003			0.0005
	< 750	1.5	8.0		9.5	16.4
	750-1500	22.7	34.0		38.0	50.7
	1500-2250	28.4	26.0		23.5	13.7
	> 2250	40.8	32.0		21.9	13.7
	refused to answer	6.7	0.0		7.1	5.5
	Marital Status (%)			0.98			0.008
	married	79.6	82.0		45.0	34.9
	divorced/separated	4.8	4.0		10.0	7.5
	widowed	13.5	12.0		35.6	50.7
	single	2.1	2.0		9.4	6.9
	Practice of physical			0.16			0.0001
	exercise (%)
	No	51.7	62.0		52.7	64.4
	Moderate	16.8	14.0		20.8	6.8
	Intensive	14.7	4.0		4.5	1.4
	No answer	16.8	20.0		22.0	27.4
	Dietary characteristics
	MeDi score, mean (SD)	4.5 (1.7)	4.6 (1.6) 0.79	4.3 (1.6)	3.9 (1.7) 0.003
	MeDi categories (%)			0.81			0.06
	Low MeDi category	26.1	22.0		31.0	41.1
	(score 0-3)
	Middle MeDi	44.3	48.0		43.1	38.4
	category (score 4-5)
	High MeDi category	29.6	30.0		25.1	20.5
	(score 6-8)
	Total energy intake	1876	1810	0.39	1473	1485 (395) 0.77
	(kcal/d), mean (SD)	(513)	(543)		(443)
	Clinical characteristics
	BMI (kg/m²), mean	27.0 (3.5) 27.1 (3.5) 0.88	26.1 (4.3)	26.9 (5.0) 0.06
	(SD)
	CES-D score, mean	5.2 (5.3)	9.2 (9.1) 0.004	8.2 (7.4)	12.4 (8.8) <0.0001
	(SD)
	MMSE score, mean	27.8 (1.7) 27.4 (2.2) 0.16	27.7 (2.0)	27.1 (2.1) 0.001
	(SD)
	Number of drug/d >5	41.0	62.0 0.004	48.5	75.3 <0.0001
	(%)
	Hypertension (%)	80.2	72.0 0.17	71.6	82.9 0.005
	Diabetes (%)	13.1	20.0 0.17	5.6	14.4 0.0001
	Smoking status (%)			0.21			0.14

Abbreviations

ACKNOWLEDGMENTS

Funding / Support: The Three-City Study is conducted under a partnership agreement between the Institut National de la Santé et de la Recherche Médicale (INSERM), the Institut de Santé Publique et Développement of the Victor Segalen Bordeaux 2 University, and Sanofi-Aventis. The Fondation pour la Recherche Médicale funded the preparation and initiation of the study. The 3C Study is also supported by the Caisse Nationale Maladie des Travailleurs Salariés, Direction Générale de la Santé, Mutuelle Générale de l"Education Nationale, Institut de la Longévité, Regional Governments of Aquitaine and Bourgogne, Fondation de France, and Ministry of Research -INSERM Programme "Cohortes et collections de données biologiques." Financial support for 3C-COGINUT project was provided by the French National Research Agency (ANR-06-PNRA-005). C. Féart was funded by the French National Research Agency (ANR-06-PNRA-005). C. Samieri was funded by the Institut Carnot Lisa (Lipides pour l"Industrie et la santé, Lipids for Industry, Safety and Health). Role of the sponsors: Study sponsors played no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; and preparation, review, or approval of the manuscript. Author Contributions: Design and conduct of the study: Barberger-Gateau, Dartigues Collection: Barberger-Gateau, Letenneur, Dartigues Management: Barberger-Gateau Analysis and interpretation of data: Féart; Pérès, Samieri, Letenneur, Dartigues, Barberger-Gateau Preparation of the manuscript: Féart, Pérès, Barberger-Gateau Review and approval of the manuscript: Féart, Pérès, Samieri, Letenneur, Dartigues, Barberger-Gateau

Conflict of interest: P. Barberger-Gateau received fees for conferences from Danone, Lesieur, Bauch&Lomb and Aprifel and benefits from research grants from Danone and Lesieur. The other authors declared no support from any institution for the submitted work. For alcohol intake, we attributed a value of 1 for people whose consumption was mild-to moderate, close to the second quartile of distribution of total alcohol intake. One point was given to men if their consumption was within 7-to-14 glasses per week (10-to-20g/d) and to women if consumption was within 1-to-4 (1.4-to-5.7g/d) glasses per week

TABLES


Validation of basal ganglia segmentation on a 3T MRI template

Introduction Subthalamic nucleus (STN) deep brain stimulation (DBS) has been demonstrated as an efficient surgical treatment in patients with Parkinson's disease suffering from severe disabilities in their motor symptoms (tremor, akinesia, rigidity) [START_REF] Avants | Geodesic estimation of large deformation anatomical shape averaging and interpolation[END_REF]. Sometimes, patients with STN DBS have secondary neuropsychological and/or psychiatric effects because the STN has a small size and a functional subdivision into motor, associative and limbic parts [START_REF] Haegelen | The subthalamic nucleus is a key-structure of limbic basal ganglia functions[END_REF]. Therefore, targeting is an important step of the neurosurgical procedure as the DBS could induce undesirable side-effects. To improve targeting, many authors created either anatomical atlases [START_REF] Bardinet | A three-dimensional histological atlas of the human basal ganglia. II. Atlas deformation strategy and evaluation in deep brain stimulation for Parkinson disease[END_REF][START_REF] Chakravarty | The creation of a brain atlas for image guided neurosurgery using serial histological data[END_REF] or multimodal databases [START_REF] D'haese | Computer-aided placement of deep brain stimulators: from planning to intra-operative guidance[END_REF][START_REF] Guo | Visualization and navigation system development and application for stereotactic deep-brain neurosurgeries[END_REF][START_REF] Guo | Surgical targeting accuracy analysis of six methods for subthalamic nucleus deep brain stimulation[END_REF]. They demonstrated that multimodal databases improved targeting accuracy over anatomical atlases. We presented a validation automatic segmentation of the basal ganglia based on a manual segmentation of these structures completed on a 3T MRI-template. This is the first step to increase accuracy of the targeting of DBS in order to improve the clinical outcome of DBS procedures.

Methods First, we developed a three-dimensional mono-subject template of the human brain by averaging 15 tridimensional (3D) T1 and 7 T2 MRI volumes acquired on a 3T scanner (Philips Medical Systems, Best, The Netherlands). The method to build the template was described in Lalys et al. [START_REF] Lalys | Construction and assessment of a 3T brain template[END_REF]. We obtained high signal-to-noise average T1, T2 and T1-T2 mixing templates (Fig. 1) to improve the visualisation of small deep nuclei. Second, we manually painted slice-by-slice on the T1-T2 template 24 bilateral cerebral structures which included the caudate nucleus, putamen, lateral and medial pallidum, thalamus, hippocampus, amygdala, lateral and medial geniculate nuclei, red nucleus, substantia nigra and STN (Fig. 2). To validate the automatic template-based segmentation procedure, we painted 14 bilateral structures on the cerebral MRI of 5 patients with Parkinson's disease. The patients had 3D T1 without and with gadolinium injection and a 3D coronal T2 MRI on the same 3T machine. Each sequence was corrected for image intensity inhomogeneity [START_REF] Sled | A nonparametric method for automatic correction of intensity nonuniformity in MRI data[END_REF]. The painted structures in the patients were bilaterally the putamen, thalamus, hippocampus, amygdala, substantia nigra and STN (Fig. 3). We registered the template onto the T1 sequence of each patient by linear and non-linear registration [START_REF] Avants | Geodesic estimation of large deformation anatomical shape averaging and interpolation[END_REF], and then registered the patient's T1 sequence to their T1 gadolinum-enhanced sequence with a linear registration. The labels, defined on the template, were mapped through concatenated transformations onto the subject's T1 gadolinium-enhanced volume to achieve automatic structure segmentation. Finnally, the automatic labels were then computed to the manual labels using a Dice Kappa metric.

Results

The results are summarized in the table presented in Fig. 4. The median of the Dice Kappa rates were greater than 0.8 for the bilateral thalami (left 0.85, right 0.83) and the left putamen (0.81).

The median of the Dice Kappa rates were between 0.7 and 0.8 for the bilateral amygdala, hippocampus and red nucleus, and for the right putamen. The median of the Dice Kappa rates were between 0.55 and 0.65 for the bilateral substantia nigra (left 0.57, right 0.59) and STN (left 0.58, right 0.62).

Conclusion Our method was efficient to validate based-MRI template segmentation. Our results

showed better kappa correlation within the biggest structures except for the red nucleus which is an anatomically constant structure in the cerebral brain. In the future, the volume and position of these structures will be correlated with clinical outcomes of improvement in motor and neuropsychological scores. Our method could be improved in two ways, first, by validation on more patients, and second, by creating Parkinson's disease-specific template from patients which have more cortico-subcortical atrophy than normal subjects. Our goal is further to build on the template segmentation, a database of implanted electrodes in all the patients with movement disorders in our center.

Legends of figures

Figure 1 .

 1 Figure 1. Coronal, sagittal and axial slices of the T1 (above) and the T1-T2 mixed (below) templates.

Figure 2 .

 2 Figure 2. Anterior 3D view of the based-MRI template segmentation. From the midline to the left, andfrom caudal to cranial, we see the red nucleus (light pink), substantia nigra (dark red), the subthalamic nucleus (white), caudate nucleus (pink), thalamus (light blue), amygdala (dark blue), hippocampus (red), and putamen (grey).

Figure 3 .

 3 Figure 3. Sagittal (left above), coronal (right above) and axial (left below) slices of a cerebral MRI of a patient with Parkinson's disease. We see the segmentation of the following bilateral structures: amygdala (red and green), hippocampus (dark and light blue), thalami (pink and white), putamen (light purple and brown), red nucleus (light green and blue), substantia nigra (green and purple), and subthalamic nucleus (brown and dark red).

Figure 4 .

 4 Figure 4. Distribution of the median of kappa rates (in ordinates) according to eight bilateral anatomical structures (in axis).

Figure 1 .

 1 Figure 1. Coronal, sagittal and axial slices of the T1 (above) and the T1-T2 mixed (below) templates.

Figure 2 .

 2 Figure 2. Anterior 3D view of the based-MRI template segmentation. From the midline to the left, and from caudal to cranial, we see the red nucleus (light pink), substantia nigra (dark red), the subthalamic nucleus (white), caudate nucleus (pink), thalamus (light blue), amygdala (dark blue), hippocampus (red), and putamen (grey).

Figure 3 .

 3 Figure 3. Sagittal (left above), coronal (right above) and axial (left below) slices of a cerebral MRI of a patient with Parkinson's disease. We see the segmentation of the following bilateral structures: amygdala (red and green), hippocampus (dark and light blue), thalami (pink and white), putamen (light purple and brown), red nucleus (light green and blue), substantia nigra (green and purple), and subthalamic nucleus (brown and dark red).

Figure 4 .

 4 Figure 4. Distribution of the median of the Dice Kappa rates (in ordinates) according to eight bilateral anatomical structures (in axis).


Asma Zoghlami
email: asma.zoghlami@etudiant.univ-reims.fr

Cyril De Runz
email: cyril.de-runz@univ-reims.fr

Herman Akdag
email: herman.akdag@lip6.fr

Montaceur Zaghdoud
email: montaceur.zaghdoud@ensi.rnu.tn

Henda Ben Ghezala
email: henda.benghezala@ensi.rnu.tn

Handling imperfect spatiotemporal information from the conceptual modeling to database structures

Keywords: Conceptual model, imperfection, pictograms, database, multi-layer, multivalued set

Many real world systems and applications require a management tool that provides support for dealing with imperfect data. The aim of this paper is to handle the imperfection of spatiotemporal data from the conceptual modeling to the database conception. We propose to add new pictograms in PERCEPTORY in order to build imperfect spatiotemporal class diagrams such as those made using Fuzzy UML. Using those models, we organize the database as a three layer organization: data layer, metadata layer, multivalued layer. Those interlinked layers give a more accurate interaction.

Introduction

As said in [START_REF] Goodchild | Foreword[END_REF], "the quality of spatial data, as indeed of any data, is crucial to its effective use". As data imperfection is a part of (spatial) data quality, Geographic Information Systems (GIS) has to deal with, for instance, uncertain, imprecise and/or incomplete knowledge.

This paper considers imperfect information modeling in GIS, either on its descriptive, temporal or spatial levels. Imperfect information that characterizes the knowledge is most often manifested by the vagueness but also sometimes by uncertainty or lack of data. Much work about handling imprecise information in Information Systems and in spatial field has been done [START_REF] Devillers | Fundamentals of Spatial Data Quality[END_REF][START_REF] Jeansoulin | Methods for Handling Imperfect Spatial Information[END_REF]. Therefore, a lot of conceptual data models have been extended to model fuzzy data [START_REF] Ma | A Literature Overview of Fuzzy Conceptual Data Modeling[END_REF]; fuzzy set theory is the best possible tool that represents data imprecision.

The starting point of our conceptual data modeling is an UML type model which is the class diagram. It is just through this model that different levels of fuzziness were introduced into the concept of class, object and relations between classes.

Thus, the concepts of fuzzy class, fuzzy association, fuzzy aggregation and fuzzy generalization were set in the work of [START_REF] Ma | Fuzzy conceptual information modeling in UML data model[END_REF].

PERCEPTORY, presented in [START_REF] Bédard | Modeling Geospatial Databases with Plug-Ins for Visual Languages: A Pragmatic Approach and the Impacts of 16 Years of Research and Experimentations on Perceptory[END_REF], is a modeling tool that is mainly based on UML and extended with spatial and spatiotemporal stereotypes. These stereotypes have been developed through two distinct PVL (plug-in for visual language): the spatial PVL, for the representation of spatial data and the spatiotemporal PVL that is used to model spatial data and/or temporal data [START_REF] Brodeur | Modelling Geospatial Application Database using UML-based Repositories Aligned with International Standards in Geomatics[END_REF]. However, it does not provide a sufficiently explicit way to cope with information imperfection, such as uncertainty and imprecision.

Even though one considers that we can represent imperfect data using the dictionary associated to the model, the automation of the translation into GIS is not easy. This may be possible using an approach that splits data, imperfection and meta-data storage.

This paper aims to highlight the imperfect information in the PERCEPTORY model through the introduction of new visual symbols to manage imperfect spatial, temporal and descriptive data.

Then, we manage these imperfections from the conceptual data model to the database through a multi-layer approach. In this approach, a multi-valued layer is added to the more classical, data layer and meta layer.

This work is organized as follows. Section 2 is devoted to the insertion of visual variables in PERCEPTORY for the management of data and knowledge imperfection. Then, the structure of the built system is exposed (section 3) with a focus on the explanation of the imperfect management layer so called multivalued layer. The last section (section 5) presents the conclusion.

Highlighting imperfect knowledge in PERCEPTORY: new visual variables

There are a lot of ways to highlight the imperfection of knowledge in conceptual models. This section introduces the ones we have to deal in the objective of building an agronomical observatory, called Observox, in the Vesle Basin (the project challenges are described in [START_REF] De Runz | Issues about qualified spatialization of agricultural practices and data for water contamination study in the Vesle basin[END_REF].

Imperfect spatial symbol

Spatial pictograms in PERCEPTORY allow us to define the geometry chosen for spatial elements of a class model. The main geometries are: point, line and polygon. If one can't define with precision the boundaries of a spatial object, there is some imprecision on the geometric shape of this object at the class level. Thus in the PERCEPTORY model, we propose to use the three following basic geometries with dashed outline as shown below in figure 1.

Imperfect temporal symbol

The temporal modeling in PERCEPTORY is based on the concepts of existence and evolution. The existence of an object corresponds to its period of life. Objects, having an instant existence, are represented by a pictogram indicating a date while objects that have a sustainable existence are represented by a pictogram indicating a time interval. At this level, there may be some imprecision in the definition of a date or of a time interval. The question is: when was an object present and when did it disappear? The two temporal pictograms are used with dashed outline to express this imprecision.

On modeling of class relationship imperfection

The modeling of relationships between classes with their imperfection, we use the UML object diagram in which we associate to the link between the two classes a membership degree.

The relation between two classes (A and B) may also have different value depending of the class instances. In order to model that, we link each instance of A to each instance of B with a membership degree. The figure 4 shows an example of modeling an uncertain relationship between two classes A and B according to class instances.

Membership degree of an object to a class

An object can belong to a class with a membership degree. This is shown in the object diagram through the introduction of the word "with a membership degree" after the instance name. At the implementation level, a first classical layer is implemented. This layer, called data layer, contains data in a crisp modeling: the geometric data with the shape and the location of an object, the descriptive data referring to all the descriptive attributes of an object, the temporal data... In our database, the spatial data are represented according to the vector mode in which the objects are represented by points, lines and polygons instead of the raster mode because the vector approach has a lower storage cost.

The data layer is followed by a meta layer that represents the metadata. Metadata usually concerns the content, data sources, data identification, data quality, spatial representation, spatial reference and any other useful characteristic that may qualify the data. It can also store specific ontology, database schema, etc.

A third layer will allow us to link the data layer and the meta layer to a modeling tool that takes into consideration their imperfection. This layer allow to represent the imprecision and uncertainty through a multivalent approach [START_REF] De Runz | Study of spatial fusion of geographical entities and quantitative information in accordance with their imprecision: application to agricultural information in Observox[END_REF].

The principle of the multivalent approach lies in the introduction of several truth values that modulate the information in order to focus on the natural language imperfection. Thus, linguistic expressions such as "very little", "a lot", etc can be used [START_REF] Akdag | Uncertainty Operators in a Many-valued Logic[END_REF]. By building a link between data, metadata and imperfection modeling, we try to provide a more accurate view of the processed information by linking together these three layers through putting a link interface between them (figure 6).

Multivalued layer

The multi valued layer deals with all the spatial, temporal and descriptive imperfections that may be present on the two more classic layers (cf. figure 7). According to [START_REF] Fisher | Models of uncertainty in spatial data[END_REF][START_REF] Dubois | Formal representations of uncertainty[END_REF], the modeling of imperfect (spatial) data may be done using a lot of theories (probabilities, possibilities, fuzzy sets, belief functions, etc.). All of those theories use a paradigm of the attribution of weights (between [0;1]) to each element of the studied domain (ℝ 2 or ℝ 3 for space, ℝ for time, ℝ + for quantitative information, etc.).

In order to reduce the cost and the complexity of storage and also in order to maintain the possibilities of exploitation, an approach based on the α-cute principle -the domain of values for which the weight is higher or equal to α -is developed. Then, the modeling data have been putting into a multivalued form. This view is then adaptable to the more frequent uncertain representation. It allows users to choose the mode of uncertainty representation for every data. The interoperability between theories should after be done by the systems (it is one of our future goals).

The imperfection layer has an impact on the data layer through dealing with the imperfect relations between objects, imperfect object class relations, imperfect attributes, etc. To deal with possible uncertain relations between objects belonging to different database classes, one must associate a membership degree to the object identifiers in a new database table that indicates at what degree they may have a relation between them.

Links between layers (example)

A geographical entity [START_REF] De Runz | Study of spatial fusion of geographical entities and quantitative information in accordance with their imprecision: application to agricultural information in Observox[END_REF] is composed by a fuzzy spatial area and a set of fuzzy quantities. To handle the vagueness at the database level, the imprecise information is stored in a specific table connected to the geographical entity table through an intermediate table which references the fuzzy quantity values stored in the fuzzy quantity table (see figure 8.).

Conclusion

In this paper, we started from the PERCEPTORY conceptual data model to handle spatial, temporal and descriptive data imperfections through the introduction of new visual symbols. At the database level, a multilayer structure is implemented. Thus, a communication between a multi valued layer, a data layer and a meta layer is established.

In perspective, an application of our approach will be done in the archeological domain and more specifically archeological digs in the city of Reims (ARCHEOCHAMP project).

Figure 1 .

 1 Figure 1. Spatial vagueness on the form of spatial objects

Figure 2 .

 2 Figure 2. Temporal imprecision

Figure 3 .

 3 Figure 3. Modeling of imperfect attribute

Figure 4 .

 4 Figure 4. Modeling uncertain relationships between two classes through class instances

Figure 5 .

 5 Figure 5. An object belonging to a class with a membership degree

Figure 6 .

 6 Figure 6. Relations between layers

Figure 7 .

 7 Figure 7. Managing imperfection in the multi valued layer

Figure 8 .

 8 Figure 8. Illustration of the storage of fuzzy quantities as multivalent set of values.

Acknowledgments

We would thank the Champagne-Ardenne Region Council, France and European Union (FEDER) for their funding of the AQUAL Project.


Karima Zayrit
email: karima.zayrit@univ-reims.fr

Éric Desjardin

Cyril De Runz
email: cyril.de-runz@univ-reims.fr

Herman Akdag
email: herman.akdag@lip6.fr

Eric Desjardin
email: eric.desjardin@univ-reims.fr

Propagation of spatial imprecision in imprecise quantitative data in agronomy

Keywords: Imprecision, fuzziness, propagation, agriculture

Introduction

In the past 30 years, the use of GIS has grown and today it is the standard for managing spatial -as located on Earth -and spatiotemporal data. Their use goes from archaeology [START_REF] Conolly | Geographical Information Systems in Archaeology[END_REF][START_REF] De Runz | Imperfect spatiotemporal information modeling and its analysis in a generalization process in a GIS: application to archaeological information[END_REF] to agronomy (the context of this work).

The spatial feature of studied entities is often as imprecise (and/or uncertain) as its quantitative and descriptive features. According to literature [START_REF] Klir | Fuzzy sets and fuzzy logic: theory and applications[END_REF][START_REF] Smets | Probability, Possibility, Belief : which for what ?[END_REF][START_REF] Fisher | Approaches to uncertainty in spatial data[END_REF][START_REF] Runz | Toward handling uncertainty of excavation data into a GIS[END_REF], fuzzy set theory and fuzzy logic are a good approach to deal with this kind of data imperfection. Then, one can build entities where both the spatial and the quantitative features are fuzzy.

The fuzzy set theory allows overlap between fuzzy shapes. The question is: what is the value of fuzzy quantitative attributes in a location where two or more fuzzy spatial shapes overlap? The answer to this question is the heart of this article.

Actually, in order to build an observatory on agricultural practice in the Vesle Basin, we have to deal with multiple sources of information that introduce imprecision in the object. From this situation, spatial and quantitative information may thus be imprecise.

Indeed, in the spatial context, there is two main ways for modelling imprecision [START_REF] Bouchon-Meunier | Qualified topological relations between spatial objects with possibly vague shape[END_REF]. In the first hand, the crisp models extend or transform precise spatial concept in order to represent spatial imprecision as for instance the Egg-Yolk model (Cohn and Gotts, 1996). In the second hand, the models are based on uncertain mathematical theories as the ones, such as [START_REF] Navratil | Indeterminacy and Spatiotemporal Data: Basic Definitions and Case Study[END_REF], using fuzzy sets [START_REF] Zadeh | Fuzzy Sets[END_REF], either those, for example [START_REF] Worboys | Imprecision in finite resolution spatial data[END_REF], exploiting rough sets, or those, as for instance (Pfoser et al., 2005), using probabilities.. The fuzzy models give us a unique and soft framework that allows us to represent imprecision and to better conceptualize the reality (see [START_REF] Smets | Probability, Possibility, Belief : which for what ?[END_REF]). As the aim of our system is to give interpretable information in each location of the monitored space, fuzzy data modeling data is used by us.

This paper exposes our opinion and choices in order to answer to these questions in the context of agronomical data exploitation. It introduces an operator for the propagation of spatial imprecision to imprecise quantitative information. It also presents a global structure for the management of fuzzy geo-entities. This structure is based on a fuzzy data storage impact analysis.

Section 2 is devoted to the imprecise geo-entity modelling in the framework of fuzzy set theory. Then, the propagation of imprecision in overlapping areas is studied (section 3). Finally, the conclusion is presented in section 4.

Fuzzy modeling of agronomical entities

In the sustainable development context, the AQUAL project (a State-Region Project in the Champagne-Ardenne, France) highlights the need of a monitoring environment for the study of agricultural practices and their pressure on the water resources in the Vesle basin. It is called Observox and it exploits data coming from heterogeneous sources: satellite images, land registry, statistical data, Corine Land Cover and other European data. The construction of a unique set of entities implies the combination of information coming from all the sources. The built entities thus induce some imprecision in the definition of spatial features and quantitative attributes [START_REF] Shi | Principle of Modeling Uncertainties in Spatial Data and Spatial Analyses[END_REF].

On the other hand, Fisher in (Ficher, 1996) presents a comparative study between crisp sets and fuzzy sets in order to model landscape. The formers simplify the modeling but could amplify errors. The latters make the models and the treatments more complex. In [START_REF] Fisher | Approaches to uncertainty in spatial data[END_REF], the authors present a taxonomy of uncertainty in spatial context where the vagueness is associated to the fuzzy set theory. According to [START_REF] Duckham | A formal approach to imperfection in geographic information[END_REF], vagueness is a special type of imprecision. Vagueness and imprecision could be both represented by fuzzy sets [START_REF] Bouchon-Meunier | Qualified topological relations between spatial objects with possibly vague shape[END_REF][START_REF] Klir | Fuzzy sets and fuzzy logic: theory and applications[END_REF][START_REF] Smets | Probability, Possibility, Belief : which for what ?[END_REF] introduced in [START_REF] Zadeh | Fuzzy Sets[END_REF].

According to this, in agronomical studies as well as in geography, the geographical entities could be modeled as fuzzy geographical entities. Those entities have a label, a fuzzy spatial shape and a set of fuzzy quantities (each quantity corresponds to a specific attribute such as population or a specific chemical). The definition of a geographical entity may be defined as follows.

Let Ω be the set of studied geographical entities {A 1 ,…,A n }. Let be Ϙ the set of monitoring quantitative information (Q 1 ,…,Q m ) if one supervises m different information (P 1 ,…,P m ) as for instance m different molecules or products . Let us define a fuzzy geographical entity A i in Ω as an object described by:

-A label or concept LA i member of an ontology.

-A fuzzy set FSA i describing its spatial representation. The membership function µSA i of FSA i is defined on ℝ 2 . -A fuzzy quantity FQ j A i for each quantity Q j (of P j ) in Ϙ. The membership function µQ j A i of FQ j A i is defined on ℝ + . An example of an A i is shown in Table 1. Quantity of P j (e.g. Isoproturon)

FQ j A i
If Q j is a precise quantity (with a value a), it could be represented by a singleton in the fuzzy set theory as follows: if q=a then µQ j A i (q)=1 else µQ j A i (q)=0 such as q belongs to ℝ + ). This principle is presented in figure 1. In the context of OBSERVOX, Ϙ is the set of studied chemical (or at a microscale, the set of phytosanitary molecules). It could be for example a fuzzy prescribed dose or an estimation of quantity which was actually spread.

The next section is devoted to the sensibility of quantity values in a space location.

Propagation of imprecision

Let us consider x a location. We consider that the confidence in FQ j A i should be put into perspective with the membership degree ( ) in order to define FQ j A i ,x with its membership function µQ j A i ,x as proposed in (1). In this definition, when a fuzzy geographical entity does not participate to the definition of x ( ( ) ), the quantity of product P j diffused at x by is certain and null.

( ) Q A , ( ) ( ( ) A ( )) Q A , ( ) Q A , ( ) (1)
with q in ℝ + and T an aggregation function, usually a t-norm such as the multiplication or the minimum.

The imprecision, conceptualized using a classical fuzzy number for quantities and by fuzzy area for spatial feature, is the propagated in the consideration of fuzzy quantities at a specific location. As our goal is to consider all the quantities of a specific product at each location of the space, an aggregation operator is now needed for obtaining the combined information. Then, we use the Zadeh"s extension 0 q 1 a principle that allows to extend usual operation in the fuzzy set context such as in our context the sum (due to the additive aspect of product diffusion).

Thus if we deal with an additive information P j , using this hypothesis and Zadeh"s extension principle we define FQ j ,x the overall quantity at the position x by following the equation ( 2) for the definition of its membership function Q , .

( ) ( ( Q A , ( ) Q A , ( )))
(2) In order to test the feasibility of our approach, we illustrate it using two overlapped fuzzy geographic entities (A 1 and A 2 ) at a specific location x (figures 2 and 3). The goal is in this example to determine the total quantity of a chemical Pj (corresponding to Bentazone) at x.

This principle allows us to compute the quantity of each monitored molecule in every location of the studied region. The confidence in the computed fuzzy quantity is lower (or equal) than the original confidence in each fuzzy geographical entities.

Conclusion

In this paper, we propose a study of the imprecision propagation from spatial information to quantitative one. We firstly introduced our context and our approach of a fuzzy geographical entity. Next, we proposed a new operator of imprecision propagation.

This paper is a starter for the future construction of an agricultural practice observatory. In our future work, we will use conceptual approach that allows us to automatically obtain a fuzzy spatiotemporal data storage solution [START_REF] Zoghlami | Handling imperfect spatiotemporal information from the conceptual modeling to database structures[END_REF], but we also want to study the propagation of quantitative imprecise information into other topological relations between fuzzy spatial This paper is a preliminary study before building the observatory. It presents our choice at the beginning of the project. In our future work, we will develop our approach by defining new fuzzy agronomical indices in the observatory.

Figure 1 .

 1 Figure 1. Illustration of a precise quantity a represented in the fuzzy set theory.

Figure 2 .

 2 Figure 2. Illustration of the imprecision propagation on quantitative value Q j of P j for a specific location x, with Ω={A 1 ,A 2 },

Figure 3 .

 3 Figure 3. Illustration of the imprecision propagation: a spatial/quantity view.

Table 1 .

 1 A fuzzy geographical entity A i (only one quantitative information is shown)

	Type of information	Attribute	Value example
	Label/Function	LA i	Vineyard
	Spatial definition	FSA i

Acknowledgements

We would thank the Seine-Normandy Water Agency, Champagne-Ardenne Region Council, France and European Union, through the FEDER, for their funding of the project CPER AQUAL.