Marianne Jover
email: marianne.jover@univ-amu.fr

Sylvie Ray-Kaeser

Anne Martine Bertrand

Noémie Memoli

Jean Michel Albaret

Tanya Rihtman

Psychometric properties of the French European Little Developmental Coordination Disorder Questionnaire (LDCDQ-FE): a pilot study

Keywords: Developmental coordination disorder, screening, psychometric properties, questionnaire, preschool, Little Developmental Coordination Disorder Questionnaire (LDCDQ)

Aims. The Little Developmental Coordination Disorder Questionnaire (LDCDQ) is a parental questionnaire designed to identify preschool children at risk of Developmental Coordination Disorder (DCD). This study aimed to translate and cross-culturally adapt the LDCDQ for French European informants (Little Developmental Coordination Disorder Questionnaire -French European [LDCDQ-FE]) and to undertake a pilot examination of its psychometric properties on a French sample. Methods. A thorough process of cultural adaptation was completed. The psychometric properties were examined with a sample of 154 French children aged to 5y11m (control=121; clinically referred=33). A sub-group of 34 children was assessed using the MABC-2 to measure convergent validity. Results. Principal component analysis demonstrated a fourcomponent structure, accounting for 67.5% of the variance. Internal consistency was acceptable to good (α=0.74-0.89). Significant correlation between the LDCDQ-FE and the MABC-2 total scores showed convergent validity. Discriminant validity was supported by significant score differences between the clinically referred and a matched control sub-group. Using ROC curves, a cut-off of 67 was proposed for a sensitivity of 81.3% and a specificity of 77.8%. Conclusions. Results show initial evidence of the psychometric properties of the LDCDQ-FE and are encouraging of its use to identify young preschoolers at risk for DCD. In future studies, the testretest reliability should be investigated, and study sample sizes expanded.

Developmental coordination disorder (DCD) refers to a condition characterized by difficulties in performing age-appropriate motor skills. According to the DSM-5 (American Psychiatric Association, 2013), DCD manifests in motor performance below age-expected levels, with significant impacts upon activities of daily living or academic achievement. Even though DCD is not commonly diagnosed before the age of 5 years [START_REF] Blank | International clinical practice recommendations on the definition, diagnosis, assessment, intervention, and psychosocial aspects of developmental coordination disorder[END_REF], the onset of symptoms is in the early developmental period (American Psychiatric Association, 2013), and many parents of children who gain a later diagnosis of DCD report having noticed atypicalities in their child's motor abilities before the age of 4 [START_REF] Missiuna | A trajectory of troubles: Parents' impressions of the impact of developmental coordination disorder[END_REF]. Increasingly, evidence suggests that DCD has long-term impacts on children's social participation [START_REF] Chen | Social participation for children with developmental coordination disorder: Conceptual, evaluation and intervention considerations[END_REF], mental health [START_REF] Pratt | Anxiety profiles in children with and without developmental coordination disorder[END_REF] and physical health [START_REF] Hendrix | Developmental coordination disorder and overweight and obesity in children: A systematic review[END_REF]. Early identification of children at risk of DCD is therefore crucial to reduce these negative impacts and facilitate effective intervention [START_REF] Missiuna | Early identification and risk management of children with developmental coordination disorder[END_REF]. The identification and monitoring of younger children who may be at risk of a later DCD diagnosis should start before the age of 5 to prevent longer-term complications through the provision of early developmental support [START_REF] Camden | Best practice principles for management of children with developmental coordination disorder (DCD): Results of a scoping review[END_REF][START_REF] Chambers | The identification and assessment of young children with movement difficulties[END_REF][START_REF] Wall | The developmental skill-learning gap hypothesis: Implications for children with movement difficulties[END_REF].

A central diagnostic criterion for DCD is evidence of motor performance substantially below expected levels, given the person's chronologic age and previous opportunities for skill acquisition (American Psychiatric Association, 2013). For children over the age of 5, the best approach for verifying this criterion is by the use of standardised motor assessments such as the MABC-2 [START_REF] Henderson | Movement assessment battery for children -Second Edition[END_REF] or the Bruininks-Oseretsky Test of Motor Proficiency, 2nd ed (BOT) [START_REF] Bruininks | Oseretsky Test of motor proficiency-Second Edition manual[END_REF]. However, due to cost-limitations, full-scale motor testing is not feasible for all children, and questionnaires have been developed to facilitate the screening and detection of children who may be at risk for DCD. Questionnaires contribute to the early detection of children with suspected or probable DCD, are cost-effective (compared to standardised physical assessment), practical (in terms of time and level of skill of respondent and administrator), useful in clinical settings, provide relatively fast results, and relatively easy analysis (Gabbard & Tamplain, 2021, p.3). Parent-report questionnaires also yield valuable information about motor skills within the child's daily environment [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF]. In addition to addressing the broader challenges of access to lengthier standardised assessment, questionnaires that support the identification of motor coordination difficulties before the age of 5 have the potential to identify children who would benefit from monitoring and the provision of early intervention support.

Despite these benefits, [START_REF] Gabbard | The Strengths and Limitations of DCD-Related Screening Questionnaires[END_REF] emphasise the fact that screening questionnaires should not be used without consideration of their potential limitations that are linked to their psychometric properties. Effective appraisal of the clinical utility of screening questionnaires depends on context-specific, careful consideration of validity, reliability, specificity and sensitivity, as well as reported predictive values cut-off scores. However, in many contexts, the lack of valid, reliable and culturally appropriate tools makes it challenging to screen for early markers of DCD. To be clinically usefuleven if not intended to be used diagnostically screening questionnaires require evidence of sound, context-specific psychometric properties.

The Developmental Coordination Disorder Questionnaire (DCDQ) is a parental questionnaire designed to screen for DCD among children between 5 and 15 years [START_REF] Wilson | Psychometric properties of the revised developmental coordination disorder questionnaire[END_REF]. It has been translated and cross-culturally adapted in many countries, and displays satisfactory psychometrics properties [START_REF] Cancer | Identifying developmental motor difficulties: A review of tests to assess motor coordination in children[END_REF]. In light of its wide-spread usefulness, and since it is accepted that markers for DCD are evident in early childhood (American Psychiatric Association, 2013), [START_REF] Parmar | Psychometric properties of the DCD-Q-07 in children ages to 4-6[END_REF] explored the psychometric properties of the DCDQ if used with children under the age of 5. In their study, the usefulness of the DCDQ with 4-6 year olds was called into question, reinforcing the need for questionnaires exploring motor tasks that are more appropriate for younger children.

Based on the DCDQ, [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF] developed The Little Developmental Coordination Disorder Questionnaire (LDCDQ), which was designed to identify 3-and 4-yearold children at risk of DCD with items reflecting more age-appropriate motor tasks. Since its original development in the Hebrew language [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF], the LDCDQ has been adapted and validated for cross-cultural use in multiple countries with numerous validation projects on-going [START_REF] Rihtman | Can a Little instrument make a big noise? A cross-cultural collaboration for identifying motor delay in young preschoolers[END_REF]. The different language versions of the LDCDQ show good psychometric properties [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF][START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF][START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF][START_REF] Venter | Suitability of the 'Little DCDQ' for the identification of DCD in a selected group of 3-5-year-old South African children[END_REF][START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF]. For example, internal consistency has been shown to be good to excellent for the total LDCDQ score [Cronbach's alpha coefficients ranging from 0.85 [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF] to 0.95 [START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF]], with fair to excellent test-retest reliability (Intra Class Correlation coefficients (ICC) =0.80-0.98 ; [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF]. Across versions (e.g. Cantell et al. [2018]; [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF]; [START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF]), both construct and concurrent validity are supported by differences in scores of children who were typically developing and those who were with DCD or at risk for DCD. In some versions of the LDCDQ [START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF], the inter-rater reliability between teachers and parents was poor for the questionnaire total score (ICC=0.47), but the total score of the LDCDQ and the total score of the MABC-2 test have been found to be moderately correlated (r=0.29 [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] to 0.52 [START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF]). However, the factorial structure of the questionnaire has differed across versions, and the sensitivity and the specificity varies according to the cross-cultural adaptation: 86% and 63% in [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF]), 80% and 40% in Cantell et al. (2018), 96% and 68% in Fu et al. (2022).

These findings from the various LDCDQ validation studies demonstrate that it has the potential to be a highly appropriate screening tool in identifying preschoolers at risk for DCD [START_REF] Lee | Early identification of children with/at risk of developmental coordination disorder: a scoping review[END_REF], but reinforce the fact that validated screening tests or questionnaires cannot simply be exported for use in other locations, as an instrument developed in one country may not be psychometrically sound when implemented in a different cultural context. For instance, educational contexts are informed by factors such as socio-political structures, school organisation, and local schoolyard games, all of which should be taken into consideration if ecological validity is to be achieved. Rigorous processes of cross-cultural adaptation of screening and assessment tools should be adhered to, to ensure equivalence between the original and translated versions [START_REF] Beaton | Guidelines for the process of cross-cultural adaptation of self-report measures[END_REF].

In European French speaking countries, no validated tools are available for early screening of young children at risk of a later DCD diagnosis. The availability of a reliable and valid first-step DCD screening questionnaire in French would represent a major advancement in identifying and supporting children. Since the LDCDQ is designed for use with children aged 3 to 4 years, with a range of studies exploring its cultural adaptation, the development of a Little Developmental Coordination Disorder Questionnaire -French European (LDCDQ-FE) version seems timely. Although the original LDCDQ was validated for use with Israeli children between the ages of 3 years and 4 years 11 months [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF], children living in European-French speaking countries attend kindergarten until the age of 6, therefore psychometric testing of the LDCDQ-FE included children aged between 3 years and 5 years 11 months, similar to other validation studies [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF][START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF]. This age-range extension was agreed after discussion and consultation with the original authors.

This paper aims to describe the translation and cross-cultural adaptation of the LDCDQ into the European French language (LDCDQ-FE), as well report on pilot investigation of the instrument's psychometric properties when used with French preschoolers. We expected the LDCDQ-FE to present sound psychometric properties and to allow the detection of preschool children who may be at risk of a later diagnosis of DCD.

Methods

Phase 1: Instrument Translation and Cultural Adaptation

The LDCDQ [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF] contains 15 statements describing motor-based tasks that are commonly performed by young children. The instructions emphasise that the parent should compare the child's task performance with that of other children of the same age and sex. Parents are asked to rate the ability of their child to perform each task, on a 5-point Likert scale ranging from Not at all to Strongly -like my child. Each item is scored from 1 to 5 points, giving a total score of 15 to 75 points, with higher scores reflecting better reported performance. The questionnaire yields three sub-scores (each ranging between 5-25 points): control during movement (CDM), fine motor (FM), and general coordination (GC). Sound psychometric properties of the original LDCDQ have been reported [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF].

Translation and cross-cultural adaptation of the English published version of the LDCDQ into

European French was conducted according to international recommendations [START_REF] Beaton | Guidelines for the process of cross-cultural adaptation of self-report measures[END_REF]. Translation was performed by the first two authors, who are French and Swiss, yielding a draft version of the LDCDQ-FE. Back-translation was performed by a professional translator. Eight native English speakers were then asked to compare between the two English language versions on a 6-point scale ranging from 0 (The two versions have exactly the same meaning) to 5 (The two versions have different meanings). The mean score across items was 1.1 (SD=1) suggesting satisfactory linguistic equivalence of the initial translation.

At the next stage, a panel of experts (n=6 occupational therapists) were asked to rate the suitability of the translated items to European French speaking children of preschool age. A 5point scale ranging from 1 (very appropriate) to 5 (not appropriate at all) was used, with mean expert scores varying from 1 (SD=0.3) to 1.6 (SD=0.6). During this process, written feedback was also requested from respondents, a process which resulted in additional linguistic adjustments before confirming the final version of the LDCDQ-FE with the original authors. For instance, cup (tasse) was replaced with glass (verre) as French-speaking children in Europe do not usually use the word cup. In addition, a more commonly used term (crayon) was identified to replace writing instrument (instrument d'écriture), which is not used in everyday language in European French [START_REF] Jover | European French adaptation of the Little DCD-Q questionnaire (Little DCDQ E-French)[END_REF]. This version of the LDCD-FE was then carried forward for psychometric testing as described in Phase 2.

Phase 2: Psychometric Testing

Participants

Participants were parents of preschool children living in France. Children in the clinically referred group were recruited via occupational therapists, after being referred due to atypical motor development concerns (n=33). Children with coordination difficulties related to a medical condition or disease (e.g. cerebral palsy, muscular dystrophy) or a suspicion of intellectual impairment were excluded from both groups. Children in the control group were recruited via convenience sampling, in liaison with local kindergartens, and did not have any known developmental concerns based on parents and teachers report (n=121). In addition to the larger control group, a matched control sub-group (n=33) was formed by matching each child in the clinically referred group to a child from the control group of the same sex and similar age (± 3 months). One questionnaire from the control group was excluded due to incomplete completion; the final sample was n=154.

For convergent validity and to determine the cut-off score, all parents were asked whether they would agree to bring their children for standardised assessment with the MABC-2 or to provide the results of a recent MABC-2 assessment; a sub-group of n=34 children (n=18 control; n=16 clinically referred) was recruited in this manner and composed the convergence study group (Table 1). Note : F = Female, M = Male, T = Total, SS = standard score, n = number, SD = standard deviation

Measure

Little Developmental Coordination Disorder Questionnaire -French European (LDCDQ-FE).

As per the original LDCDQ, each of the 15 items of the LDCDQ-FE is scored from 1 to 5 points, giving a total score of 15 to 75 points. Lower scores suggest a higher risk for DCD.

Movement Assessment Battery for Children -2nd edition (MABC-2) [START_REF] Henderson | Movement assessment battery for children -Second Edition[END_REF], French version Marquet-Doléac et al., 2016). The MABC-2 is a standardized assessment of motor performance. It evaluates manual dexterity, ball skills, and static and dynamic balance in children between the ages of 3-16. Children scoring between the 16 th and the 5 th percentile are considered to be at-risk for DCD and children scoring below the 5 th to have DCD if the other DMS-5 criterion are fulfilled. The MABC-2 has demonstrated sound psychometric properties: test-retest reliability (ICC=0.83-0.96) and inter-rater reliability (ICC=0.96-0.99, [START_REF] Griffiths | Psychometric properties of gross motor assessment tools for children: a systematic review[END_REF].

Procedure

The study received ethical approval from Aix-Marseille University (approval #32.110213). Parents were informed about the purpose of the study and provided their written consent to participate prior to completing the LDCDQ-FE. Children of the convergence study group were assessed either by their regular occupational therapist or by an experienced research assistant within the 2 months following LDCDQ completion (M=25.1 days, SD=24.3).

Data Analysis

Data were analyzed using Statistical Package for the Social Sciences, Version 22 (SPSS Statistics; IBM Corp. 2017). The structure of the questionnaire was analyzed on the complete sample using a Principal Component Analysis with Varimax rotation. We opted for this analysis to be able to compare the results to those obtained by [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] and Cantell et al. (2018) with the same questionnaire. Internal consistency was assessed using Cronbach's alpha and item-to-total correlations. We used [START_REF] Gliem | Calculating, interpreting, and reporting Cronbach's alpha reliability coefficient for Likert-type scales[END_REF] rule of thumb which considers Cronbach alphas >.9 as Excellent, >.8 as Good, >.7 as Acceptable, >.6 as Questionable and >.5 as Poor. Concerning the Corrected item total correlation, the value should be at least .40 [START_REF] Gliem | Calculating, interpreting, and reporting Cronbach's alpha reliability coefficient for Likert-type scales[END_REF]. A chi-square test of independence was performed to compare the clinically referred and the matched control sub-group concerning sex and two sample t-tests were used to compare the group's age and MABC-2 standard score.

The effects of the children's age and sex on the LDCDQ-FE scores were explored in the large control group and in the clinically referred group separately. Considering the ordinal level of the LDCDQ response scale, nonparametric tests were used for correlations and comparisons. Correlations between age and LDCDQ-FE scores were analyzed using Spearman's rho correlations. Mann-Whitney U tests were carried out to compare LDCDQ-FE scores between sexes.

Although formal DCD diagnosis does not usually occur prior to the age of 5 such that adherence to formal DCD criteria was not feasible [START_REF] Blank | International clinical practice recommendations on the definition, diagnosis, assessment, intervention, and psychosocial aspects of developmental coordination disorder[END_REF], children were initially grouped according to the recruitment from clinical settings (clinically referred group) or not (matched control sub-group). The known-group method was used to test the discriminant validity. We compared the clinically referred group and matched control sub-group using Mann-Whitney U tests for the LDCDQ-FE total score and for the LDCDQ-FE subscores obtained from the principal component analysis. Effect sizes were expressed using epsylon squared (ε²), the rule of thumb for its interpretation is the same as for an adjusted R2 [START_REF] Vogt | Dictionary of Statistics & Methodology (Vols. 1-0)[END_REF].

The convergent validity was explored in children assessed with the MABC-2 (convergence study group). We estimated the association between the LDCDQ-FE total score and subscores and the MABC-2 standard scores using a Spearman's rho correlation.

To compute preliminary cut-off scores, two new study groups were reconstituted based on MABC-2 outcomes regardless of initial group membership. The not-at-risk for DCD group (scores above or at the 16 th percentile) was composed of 18 children (15 from the control group; 3 from the clinically referred group). The at risk for DCD group (scores below the 16 th percentile) was composed of 16 children (3 from the control group; 13 from the clinically referred group). Receiver operating characteristic (ROC) curves were used to determine preliminary optimum cut-off values, using the MABC-2 score as the state variable (at-risk and not-at-risk groups). The point on the ROC curve that allowed for a sensitivity of around 80% was selected as the cut-off (Schoemaker & Wilson, 2015). Sensitivity and specificity values, as well positive and negative predictive values according to the cut-off, were calculated. For all analyses, the statistical significance level was set at 0.05.

Results

Sample characteristics (Table1

). The participants ranged in age from 36 to 71 months in the large control group and from 40 to 71 months in the clinically referred group/matched control sub-group. As expected, the age and the sex distribution did not differ between the clinically referred group and the matched control sub-group. The MABC-2 standard scores were lower in the clinically referred group than in the large control (t(32) = 4.45, p < 0.001) and in the matched control sub-group (t(26) = -3.64, p = 0.001).

Principal Components Analysis and internal consistency. The Principal Components Analysis with Varimax rotation resulted in four factors (components) with an eigenvalue higher than one (Table 2). The factors accounted for 67,5% of the variance. The factor loadings were between 0.46 and 0.82 with one item loading on two factors. The four factors of the LDCD-FE were labelled Daily activities (4 items), Fine motor activities and games (5 items), Ball skills (3 items) and Posture and global coordination (3 items). Internal consistency of the LDCDQ-FE was acceptable to good as the Cronbach's alpha coefficient was 0.89 for the total score, 0.82 for F1, 0.80 for F2, 0.74 for F3 and 0.74 for F4. Corrected item-total correlations demonstrated a moderate contribution of each item to the overall questionnaire (range=0.48-0.69), which was slightly increased when considering each component separately (range=0.48-0.72). Cronbach's alpha coefficient for the total score remained good if items were deleted for the total score (0.88-0.89), and it was poor to good for the components considered separately (0.58-0.83, Table 3). As the instructions of the LDCDQ-FE guide parents to consider their child's performance in relation to other children of the same age and sex, we expected the LDCDQ-FE scores (Total score, F1, F2, F3 and F4) to not depend on age or sex. As predicted, none of the correlations between age and LDCDQ-FE scores were significant (Total score: control group r = 0.08, p > 0.4; clinically referred group r = 0.06, p > 0.7). Likewise, no significant difference between girls and boys we found across the LDCDQ-FE scores.

Discriminant validity. The known-group method was used. LDCDQ-FE total score and subscores were computed across clinically referred group and matched control sub-group (Table 4). The difference between the clinically referred group and matched control sub-group was tested with Mann-Whitney U tests and was significant for all scores. The children who had been referred to occupational therapy due to motor difficulties had lower scores than the children of the matched control sub-group for the total score and each subscore of the LDCDQ-FE. The magnitude of the effect size was small for F3 Ball skills, medium for F1 Daily activities and large for F2 Fine motor activities and games and F4 Posture and general coordination. Convergent validity. The LDCDQ-FE total score was correlated to the MABC-2 total standard score (Table 5). Correlations between the LDCDQ-FE and the MABC-2 subscores were also considered. F1 and F2 were the only factors correlated to the MABC-2 total score and were also the factors which correlated the most with the MABC-2 subscales. The F1 Daily activities was correlated to the Manual dexterity and the Balance subscales of the MABC-2. The F2 Fine motor activities and games was correlated to each MABC-2 subscore (Manual dexterity, Aiming and catching, Blance). On the contrary, F3 and F4 were correlated to only one subscale of the MABC-2. Consistently, F3 Ball skills was correlated to Aiming and catching, but F4 Posture and general coordination was correlated to the Manual dexterity subscale of the MABC-2. Cut-off scores, sensitivity and specificity. The optimal cut-off in the sample was explored through the ROC method. The ROC curve is presented Figure 1, the x-axis represents the false positive rate (1-specificity) and the y-axis the true positive rate (sensitivity). The estimated area under the ROC curve was 0.82 (CI95=0.66-0.97) and significantly different than 0.50 (p=0.002). Considering the LDCD-FE total score, a cut-off of ≤67 led to an associated sensitivity of 0.81, a specificity of 0.78, a predictive positive value of 0.76 and a predictive negative value of 0.82.

Discussion

The aim of the present study was to cross-culturally adapt and conduct a preliminary assessment of the psychometric properties of the LDCDQ for use with 3-to 5-year-old children living in French-speaking European countries. The translation of the questionnaire and its cultural adaptation were conducted and resulted in a pilot version, which was then psychometrically tested with a French sample. Results provided sound initial evidence of the internal consistency and the validity of the LDCDQ-FE.

The finding of very good internal consistency (substantially above the level of .70; Bland, Altman,1997) implies that the LDCDQ items were homogeneous in relation to the construct of motor coordination and warranted included in the French European version of the questionnaire. This finding, which aligns with that of other validated versions of the LDCDQ [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF][START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF][START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF], supports the assumption of a central construct related to early motor development and reinforces the importance of understanding the manner in which young children perform day to day motor tasks as an indicator of their early motor development.

Examination of the structural organization of the LDCDQ-FE revealed four components specific to this version of the questionnaire (Daily Activities: items related to eating, moving from place to place; use of playground equipment; Fine Motor Activities and Games: puzzles, building a block tower, sticking stickers; Simon Says; Ball Skills: all ball-related items; Posture and Global Coordination: sitting, running). This factorial structure explained a large part of the variance, however, differs from the factor structures of other LDCDQ versions. For instance, the original study comprised 3 subscales relying on experts' categorization of the items between three categories and in an attempt to propose alignment with the general structure of the DCDQ [START_REF] Wilson | Psychometric properties of the revised developmental coordination disorder questionnaire[END_REF]: control during movement, fine motor, general coordination [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF]. [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] and [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF] both used principal component analysis but obtained 2 factors (gross motor and fine motor) and 3 factors (fine motor skills, locomotor skills, ball skills), respectively. The structure that is emerging in different versions may be describing differing associations between items made by the parents within specific cultural contexts as well as reflect culturally-specific differences in opportunities for motor development. For example, the Simon Says game (or equivalent) was associated with fine motor skills in France and the Netherlands [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF], but with gross motor skills in Canada [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] and general coordination in Israel [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF]. More research is needed to identify how cultural differences potentially shape parental assessment, and how the parental assessment differs from the traditionally accepted views of professionals in relation to components of motor function.

The LDCDQ-FE scores did not depend on sex or age, which is in line with the questionnaire instruction to respondents to consider the child as compared to other children of the same sex and age, and an important aspect of verifying construct validity of the instrument. Concerning sex, this result aligns with findings from other versions of the questionnaire [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF] and that of the French European adaptation of the DCDQ, designed for 5-to 15-year-old children [START_REF] Ray-Kaeser | Psychometric assessment of the French European Developmental Coordination Disorder Questionnaire (DCDQ-FE)[END_REF]. However, some adaptations of the LDCDQ appear to elicit lower scores from boys than from girls despite this instruction [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF][START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF][START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF]. Again, the cultural context may explain this divergence. Parental expectation might be relatively comparable for boys and girls in France, whereas they may be higher for boys in Canada, Netherland, or Taiwan, leading to lower scores in this group. Alternatively, culturally-based, sex-related expectations for different motor activities may underlie these different findings. These comparative findings suggest the need for further exploration of the underlying reasons that may explain these differences (see [START_REF] Rihtman | Can a Little instrument make a big noise? A cross-cultural collaboration for identifying motor delay in young preschoolers[END_REF]. Concerning the age, the present adaptation focuses on a group of 3-, 4-and 5-year-old children, whereas the original LDCDQ was designed to assess 3-and 4-year-olds. The correlation between age and LDCD-FE score was very low, suggesting that the inclusion of older children did not affect the parent's responses. The inclusion of 5-year-old children was supported by other studies [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF][START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF] and the instructions for the questionnaire are compliant as they require parents to rate their children 'in relation to other children of the same age' [START_REF] Wilson | Psychometric properties of the revised developmental coordination disorder questionnaire[END_REF][START_REF] Wilson | Reliability and validity of a parent questionnaire on childhood motor skills[END_REF]. Except in [START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF]'s study which showed a decrease of the LDCDQ-C score with age, no age effect has ever been obtained with the LDCDQ between 3 years and the end of the 4 th year [START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF][START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] or between 3 years and the end of the 5 th year [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF]. Thus, further validation should continue to include children up to the end of their 5 th year, because children attend kindergarten across these age ranges in many European French speaking countries. As the DCD-FE comprises children between 5 and 15 years [START_REF] Ray-Kaeser | Psychometric assessment of the French European Developmental Coordination Disorder Questionnaire (DCDQ-FE)[END_REF], the addition of the LDCDQ-FE means that there are now two questionnaires available to support screening of European French speaking 5-year-old children. We recommend choosing the questionnaire according to the child scholarship: LDCDQ-FE for children attending kindergarten and DCDQ-FE for children already going to school.

To test the construct validity of the LDCDQ-FE, the scores of children in the clinically referred group and a matched control sub-group were compared. The significant difference suggests that the LDCDQ-FE captures parental perception of their child's early motor difficulties (Missiuana et al., 2007), in each domain explored: daily activities, fine motor activities and games, ball skills, posture and general coordination. Moreover, the correlation between the MABC-2 and the LDCDQ-FE attested to a satisfactory convergent validity of the questionnaire: it was slightly higher than those obtained by [START_REF] Fu | Psychometric properties of the Little Developmental Coordination Disorder Questionnaire-Taiwan[END_REF] or [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF] on children aged from 3 to 5 years (0.52 and 0.36 respectively), and also higher than those obtained by [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] and [START_REF] Venter | Suitability of the 'Little DCDQ' for the identification of DCD in a selected group of 3-5-year-old South African children[END_REF] on children aged from 3 to 4 years (0.30 and 0.29). The first two factors of the questionnaire (F1 Daily activities and F2 Fine motor activities and games), in particular, strongly correlated with the MABC-2 physical assessment and need careful consideration in case of suspected DCD. These two factors may be particularly good examples of the day-to-day effect of motor coordination difficulties in children before 6 years (e.g. [START_REF] Wang | Functional performance of children with developmental coordination disorder at home and at school[END_REF]. The lack of correlation between F3 Ball skills and F4 Posture and general coordination and the MABC-2 Total score potentially rely on the fact that these components count less items (3) and a reduced variability of the scores. Each of these components however correlated with one subscore of the MABC-2. On the whole, the correlation between the MABC-2 and the LDCDQ-FE confirms the usefulness of the questionnaire to screen for children below six years at-risk for DCD as a reflection of diagnostic criterion B (APA, 2013), although it is not intended to replace a formal motor assessment (which addresses diagnostic criterion A [APA, 2013]). The questionnaire constitutes a means to encourage parents to monitor their child's motor development and obtain a standardized motor assessment if he/she keeps falling behind peers at a later age (Cantell et al., 2019, p.34), and the addition of a validated European French version is a welcome, culturally-specific addition to support quality clinical practice.

The cut-off scores of the LDCDQ-FE were calculated using ROC analysis, setting the standard of sensitivity to 80%, as recommended by [START_REF] Schoemaker | Screening for developmental coordination disorder in children[END_REF]. These authors assumed that sensitivity should be higher than specificity when a screening instrument is used to detect DCD, where early diagnosis is beneficial, diagnosis is easy to confirm using a noninvasive motor test, and a false positive has few psychological consequences [START_REF] Schoemaker | Screening for developmental coordination disorder in children[END_REF][START_REF] Schoemaker | Psychometric properties of the Movement Assessment Battery for Children-Checklist as a screening instrument for children with a developmental co-ordination disorder[END_REF]. A cut-off score of 67 was the best way of distinguishing children at-risk from children not-at-risk for DCD. In other words, children scoring at or below this cut-off should be considered for further motor assessment, as they may be considered as being at risk for having DCD. This value is comparable to the one proposed by [START_REF] Wilson | Psychometric properties of the Canadian Little Developmental Coordination Disorder Questionnaire for Preschool Children[END_REF] (67-68 -children aged 3 and 4 years) yet lower than the one suggested by [START_REF] Cantell | Age-related validity and reliability of the Dutch Little Developmental Coordination Disorder Questionnaire (LDCDQ-NL)[END_REF] (70 -children aged 3, 4 and 5 years). The findings reported here reinforce the necessity to ensure psychometric testing of tools when adapted for use in other cultures [START_REF] Jover | European French adaptation of the Little DCD-Q questionnaire (Little DCDQ E-French)[END_REF]. The questionnaire's sensitivity (81%) and specificity (78%) attested its ability to correctly classify a child as being at-risk for DCD (true positive) or not-at-risk for DCD (true negative). Likewise, the questionnaire can help to determine if a formal motor assessment or motor support may be needed.

The present study has limitations. First, it is a pilot study, and the limited sample size mitigates the strength and generalizability of its results. Second, the participants come from only one country and the educational level of the mother was not measured as a potential confounding variable. Further studies should be conducted before using the questionnaire in other Frenchspeaking European countries. Finally, due to the limited number of participants who performed the MABC-2, it was not possible to calculate the cut-off score by age category. Therefore, future research investigating the LDCDQ-FE should aim to establish age-band specific cut-off scores. Future research should also aim to estimate the test-retest reliability.

Conclusions

In conclusion, the results of the current study suggest that the LDCDQ-FE has preliminary evidence of satisfactory psychometric qualities as a screening instrument (internal consistency, discriminant and convergent validity, sensitivity and specificity). As the LDCDQ-FE focuses on functional skills in a range of contexts, it has noteworthy potential to detect those preschool children who would benefit from early support1 .

Figure 1 :

 1 Figure 1: LDCDQ-FE Receiver Operating Characteristic (ROC) curve (diagonal segments are produced by ties) for a cut off ≤67. The cut off point is represented by the circle.

Table 1 . Group characteristics: number of children and age according to sex, and MABC-2 standard score

 1

			Clinically referred group	Matched control sub-group		Control Group		Convergence study Group
			n	mean (SD)	n	mean (SD)	n	mean	(SD)	n	mean	(SD)
	Age	F	11	54.2	(9.9)	11	54.2	(10.1)	48	50.6	(9.0)	14	51.3	(9.3)
	(month)	M	22	57.1	(8.4)	22	57	(8.4)	73	52.2	(9.6)	20	53.9	(9.7)
		T	33	56.2	(8.9)	33	56.1	(9.0)	121	51.5	(9.4)	34	52.8	(9.5)
	MABC-2		16	4.19	(3.1)	12	9.1	(4.0)	18	9.8	(3.9)	34	7.1	(4.5)
	(SS)

Table 2 . Principal Components Analysis and factor loading by items (n=154)

 2

		F2		F4
		Fine		Posture and
	F1	motor		general
	Daily	activities	F3	coordinatio
	activities	and games	Ball skills	n

Note: I = Item; Letters in parentheses refer to the original subcategories in

[START_REF] Rihtman | Development of the Little Developmental Coordination Disorder Questionnaire for preschoolers and preliminary evidence of its psychometric properties in Israel[END_REF]

: CDM control during movement; FM fine motor; GC general coordination.

Table 3 . Internal consistency of the LDCDQ-FE items (n=154)

 3

	LDCDQ-FE items	Corrected item-total correlation (total score)	Alpha if item deleted (total score)	Corrected item-total correlation (F1, F2, F3, F4)	Alpha if item deleted / component (F1, F2, F3, F4)
	I5 Moves from place to place	0.635	0.883	F1 0.677	0.763
	I6 Drinks from cup	0.613	0.883	F1 0.697	0.753
	I7 Uses cutlery	0.505	0.886	F1 0.545	0.834

Table 4 . LDCDQ-FE total score and subscores in the clinically referred (n=33) and matched control (n=33) sub-groups

 4

	Clinically	Matched control
	referred Group	Sub-Group	Mann-Whitney U test
	mean (SD)	mean (SD)

Note: SD: standard deviation

Table 5 . Correlation between LDCDQ-FE total score and subscores and the MABC-2 standard scores (n=34)

 5

	MABC-2

The questionnaire is available through request to the authors.

The authors gratefully thank Yohann Kraut for his help in collecting the data. They are also thankful to the parents and children who participated to the study.

Funding: This work was supported by a grant from the Foundation for Research in Psychomotricity and Diseases of Civilization (FRPMC) to MJ. The study was also supported through funding from the HES-SO University of Applied Sciences and Arts of Western Switzerland, health domain (26-015) to SRK and AMB. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.


P Bokov

P Flaud

J.-M Fullana
email: jose.fullana@upmc.fr

M Rossi

Implementing Boundary Conditions in Simulations of Arterial Flows

Computational hemodynamic models of the cardiovascular system are often limited to finite segments of the system and therefore need well-controlled inlet and outlet boundary conditions. Classical boundary conditions are measured total pressure or flow rate imposed at inlet and impedances of RLR, RLC or LR filters at outlet. We present a new approach based on a unidirectional propagative hypothesis (UPH) to model the inlet/outlet boundary conditions on the axisymmetric Navier-Stokes equations. This condition is equivalent to a non-reflecting boundary condition in a fluid-structure interaction model of an axisymmetric artery. First we compare the UPH to the best impedance filter (RLC). Second, we apply this approach to a physiological situation i.e. the presence of a stented segment into a coronary artery. In that case a reflection index is defined which quantifies the amount of pressure waves reflected upon the singularity.

Introduction

As widely documented [ [START_REF] Nerem | The role of fluid mechanics in atherogenesis[END_REF][START_REF] Zarins | Carotid bifurcation atherosclerosis. quantitative correlation of plaque localization with flow velocity profiles and wall shear stress[END_REF][START_REF] Ku | Pulsatile flow and atherosclerosis in the human carotid bifurcation. positive correlation between plaque location and low oscillating shear stress. arteriosclerosis[END_REF][START_REF] Giddens | The role of fluid mechanics in the localization and detection of atherosclerosis[END_REF][START_REF] Tropea | Reduction of aortic wall motion inhibits hypertension-mediated experimental atherosclerosis[END_REF][START_REF] Meyerson | The effects of extremely low shear stress on cellular proliferation and neointimal thickening in the failing bypass graft[END_REF][START_REF] Cheng | Atherosclerotic lesion size and vulnerability are determined by patterns of fluid shear stress[END_REF][START_REF] Coskun | Computational fluid dynamics tools can be used to predict the progression of coronary artery disease[END_REF]], hemodynamical conditions at the arterial wall play an important role in atherosclerosis: a low mean shear stress, an increase of the shear oscillatory component, or an abnormal stagnation time are known to be correlated to the location of atheromateous plaques. A comprehensive study of such a phenomenon requires the accurate knowledge of blood flow, i.e. velocity profiles and pressure distributions inside the artery. Unfortunately, these data remain difficult to measure experimentally, particularly when studying atheromateous plaques: blood velocity measurements using Doppler devices are difficult to assess, in part because of the size and location of the region of interest and, more importantly, because of the presence of distal walls that deteriorates the signal to noise ratio due to wave reflections. Another important point concerns the evaluation of in vivo pressures, which cannot be directly obtained in an atraumatic way. Because of these experimental limitations, computational hemodynamics has been used as an alternative tool for studying arterial flow. As large numerical resources are required to simulate a complex 3D arterial tree, numerical works are often limited to the investigation of a unique vascular segment [ [START_REF] Botnar | Hemodynamics in the carotid artery bifurcation: a comparison between numerical simulations and in vitro mri measurements[END_REF][START_REF] Tang | Simulating cyclic artery compression using a 3d unsteady model with fluid-structure interactions[END_REF][START_REF] Cebral | Efficient pipeline for image-based patients specific analysis of cerebral aneurysm hemodynamics: technique and sensitivity[END_REF]]. In all cases the main problem remains the choice of inlet, outlet and arterial wall conditions. Different approaches have been proposed to determine an optimal choice for inlet and outlet conditions: for instance, [START_REF] Li | Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses[END_REF] imposed inlet and outlet pressure to be the experimentally measured quantities. Resulting velocity profiles are nevertheless sensitive to errors resulting from imprecise estimations of pressure. Other authors used a RLC system (like Winkessel models) or more complex systems with inductive elements [ [START_REF] Vignon | Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure wave in arteries[END_REF][START_REF] Vignon-Clementel | Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries[END_REF][START_REF] Alastruey | Modelling the circle of williw to assess the effects of anatomical variations and occlusions on cerebral flows[END_REF][START_REF] Kim | Patient-specific modeling of blood flow and pressure in human coronary arteries[END_REF]] . These methods however failed to completely eliminate inlet and outlet boundary reflections [ [START_REF] Willemet | Inlet boundary conditions for blood flow simulations in truncated arterial networks[END_REF]]. This latter point is crucial since spurious reflections generated by outlet or inlet boundaries may flaw the investigation of true reflections due to arterial stenosis or stents. In the following, we present a numerical model of an axisymmetric flow in a flexible tube and an approach based on a unidirectional propagative hypothesis (UPH). This models the inlet/outlet boundary conditions for the Navier-Stokes equations. We compare the UPH with the condition obtained when assuming a semi-infinite tube or a classical RCL system. Since spurious reflections disappear in that approach we apply the UPH to investigate the hydrodynamic role of a singularity; more precisely we define a reflection index that quantifies the amount of pressure waves reflected upon a stent model.

Material and methods

We have tested the different boundary conditions on a Newtonian flow in a model of artery which is a flexible cylinder of length L and radius R(t). The temporal dependence on radius R(t) is due to the coupling between flow and wall. Thereafter, we present the numerical description of the model and its validation.

Description of the model

The numerical simulations were performed using the COMSOL software (COMSOL Inc.). This tool based on a finite element approach, numerically solves partial differential equations (PDE) with the corresponding boundary conditions. We use two different solvers one for the fluid domain and one for the solid one. COMSOL also handles fluidstructure problems flow by coupling both fluid and solid solvers. For the fluid part, COMSOL solves the Navier-Stokes equation and mass conservation over the computational domain. The fluid is assumed incompressible of constant density ρ = 1060kg/m3 and Newtonian of dynamic viscosity µ = 5.10 -3 Pa.s. The blood viscosity for a high shear rate is around 4 10 -3 and can rise up to 40 10 -3 Pa.s for low shear rates. We decided to use a moderate high viscosity value to model large or middle size arteries. The flow is assumed to remain axisymmetric; as a consequence, only radial v r (r, z,t) and longitudinal v z (r, z,t) velocities and pressure p(z,t) appear into the mathematical model. The fluid equations thus read :

∂v r ∂t + v r ∂v r ∂r + v z ∂v z ∂z = - 1 ρ ∂p ∂r + ν ∂ 2 v r ∂ 2 r + 1 r ∂v r ∂r + ∂ 2 v z ∂ 2 z - v r r 2 ∂v z ∂t + v r ∂v z ∂r + v z ∂v z ∂z = (1)
- 1 ρ ∂p ∂z + ν ∂ 2 v z ∂ 2 r + 1 r ∂v z ∂r + ∂ 2 v z ∂ 2 z ∂v r ∂r + v r r + ∂v z ∂r = 0
where ν = µ/ρ denotes the kinematic viscosity and the longitudinal variable z runs between 0 and L, and the variable r between 0 and R(t). Furthermore, by symmetry, radial velocity and radial derivative of axial velocity are zero at r = 0 and a no slip condition is imposed on the arterial wall (r = R(t)).

In COMSOL, the inlet and outlet boundary conditions (z = 0 and z = L) are expressed as follows: the pressure is imposed at both ends of the computational domain. Depending on models (see below), it can be given directly as a given input (P(z = 0/L,t)), or can be obtained through the solution of an ordinary differential equation relating pressure and flow rate.

For the solid dynamics part, the arterial wall is assumed to move only radially which is generally well admitted in hemodynamics. The entrance and the outlet portion of the solid is free to move radially under the pressure forces. The flexible wall is assumed to be nearly incompressible and made of an isotropic linearly elastic solid. It is thus only defined by its Young Modulus E and its Poisson's ratio ν p . The Euler equation of the solid structure is then:

∂σ i j ∂x = ρ d 2 b i dt 2 (2)
where σ i j denotes the component (r, θ, z) of the stress tensor and b i the component of deformation along the i-axis . For a linear elastic, isotropic material the stress tensor components are related to the strain tensor components ε i j by the relation:

σ i j = E 1 + ν p ε i j + ν p 1 -2.ν p ε kk δ i j , (3)
ε ik = 1 2 (∂ k b i + ∂ i b k + ∂ i b l ∂ k b l )
where δ i j is the Kronecker tensor. Using equations ( 2) and (3) COMSOL obtain a global equation for the displacement tensor. Equation (3) defines the two wall models used in our numerical computations. In the first one, the Young modulus E is constant and, in the second one, E = E(z) varies along the arterial axis (z variable) to mimic the effect of the stent. This approach is valid : the Young modulus of the stent is known to be higher than the Young modulus of the arterial wall and, as shown in a previous work [ [18]], the stent behaves as if it only induces a mechanical stiffening of the arterial wall. Practically, Young's modulus is increased for a normal value (artery) to a higher value within the length of the stent (20mm). In order to avoid discontinuous functions in numerical resolution the Young modulus smoothly changes near of the extremities of the stent.

To couple both solvers, COMSOL uses an ALE (Arbitrary Lagrangian-Eulerian) approach: the fluid-structure interaction is taken into account by imposing that radial forces computed through the fluid stress tensor are continuous across the wall at r = R(t).

Computational domain

The computational mesh for the fluid domain is axisymmetric as stated above, therefore (r, z = [0, R(t)] × [0, L]) is composed of rectangular elements: 500 elements in the axial direction and 12 elements in the radial one. We performed several sensitivity tests by simulating a pulsed flow in a long tube (L = 0.5 m). For the axial direction, we studied the dependence of velocity field and arterial radius with the mesh density. The computation of the maximum radius over the simulation Max t [R(L/5,t)] at z = L/5 only varies of 0.04 % when using 1000 instead of 500 axial elements. At the same position, the axial velocity varies by 1.7% and the wall shear stress (WSS) by 1.9%. On the contrary, 250 mesh elements seem to be insufficient. Therefore for all numerical simulations presented here the mesh density in the axial direction was set to 500/50 = 10 elements per cm. This radial mesh is not uniform: step size is growing exponentially as exp(0.03r) from the wall to the center of the tube. This allows meshing correctly the oscillating boundary layer; its typical size is proportional to the inverse of the square of the Womersley number α defined as R(ω/ν) 1/2 . Here ω denotes the typical flow pulsation.

We have checked that 12 elements in the radial direction were sufficient enough for a good evaluation of the WSS. For the elastic wall, only 3 mesh elements were used in the radial direction. We tested the sensitivity of the solution and determined that the relative variation of the solution between 2 elements and 5 elements in the tube thickness was less than 0.3% for the maximal fluid velocity at z = L/5. The axial mesh in the solid is identical to the fluid domain one. Consequently, the sensitivity with respect to the axial mesh in the solid has been performed together with the fluid domain (see above).

Validation of the model

In order to validate our numerical model, we compare the numerical solution with an analytical solution in a semiinfinite medium. Note that this analytical solution is only pertinent in a linear regime where deformations amplitudes are small compared to physiological ones.

Analytical solution in a "semi-infinite" tube

It is possible to obtain an analytical solution for a pressure wave generated by an imposed periodic pressure signal p(z = 0,t) which propagates in a viscous fluid contained in a semi-infinite elastic tube z > 0. Indeed let us Fourier decompose the pressure at z = 0

P(t, z = 0) = N ∑ n=0
p n e iω n t .

The general solution then reads

P(t, z) = N ∑ n=0 p n e i(ω n t-k n z)
where the complex wave numbers k n (ω n ) is obtained by a dispersion relation for a particular propagative solutions of the form e i(ωt-k(ω)z) . Within some approximations (wave velocity larger than the longitudinal fluid velocity, wavelength larger than the mean tube radius R), the existence of this particular solution implies that complex wave number k and the real pulsation ω are related via the dispersion equation

c 2 (ω) = ω 2 k 2 = c 2 0 1 -F(i 3/2 α) , α = R[ ω ν ] 1/2 (4)
where

F(z) = 2J 1 (z)/(zJ 0 (z)) is the Womersley function [ [ 19
]], J 0 and J 1 are Bessel functions of the 1st kind, and c 0 is the phase velocity of the radial mode called Young's mode equal to A ρ dP dA . For thin walled tube hypothesis and linear elasticity, its value is known via the Moens and Korteweg relation [ [START_REF] Mcdonald | Blood Flow in Arteries[END_REF]]

c 2 0 = A 0 ρC p (5)
where A 0 stands for the undeformed tube cross sectional area and C p for the elastic tube compliance. This latter quantity is assumed constant (Hookean model) which is the correct approximation as long as the pressure-induced deformations are negligible compared to the tube radius.

Numerical simulation in a "semi-infinite" tube

The above analytical expression is used to check the numerical code for a long tube with a known inlet pressure. The pressure imposed at inlet P imp (t) mimics a systolic pulse of duration T 1 = 0.4 s and of period T = 1 s, then

P imp (t) = ∆P (1 -cos(2πt/T 1 )) t ∈ [0, T 1 ] 0.0 t ∈ [T 1 , T ] (6)
where ∆P is the maximal pressure pulse. Mechanical and geometrical properties as well as numerical inputs for COMSOL are given in Table 1 where h denotes wall thickness, D ext the external diameter.

These values are chosen close to physiological values of human coronary arteries since our interest is focused towards studying pressure pulse propagation in coronary arteries and the effect of stent placement in such vessels. Physiological h/D ext for coronary arteries have been extensively studied: for instance, [START_REF] Ozolanta | Changes in the mechanical properties, biochemical contents and wall structure of the human coronary arteries with age and sex[END_REF] reported values between 0.08 and 0.44 depending on the age of the subject and the artery (left coronary artery vs right coronary artery).

The simulation is called here "semi infinite" because the tube is long enough (L = 4.92m) to avoid wave reflections at the outlet over the simulation time. In Figure 1, comparisons are performed for z ∈ [0, 1.6 m] and reflected waves have no time to come-back to measurement points and only one wave is observed, the incident wave.

Comparisons (Figure 1) confirms the validity of our numerical code. The pulse attenuation coefficient was estimated by fitting the decrease of peak-to-peak pressure by an exponential function of the distance. The axial velocity of the fluid is of the order of 3.10 -2 m/s and the wave group velocity (peak to peak) computed from the numerical simulation was 5.7 ± 0.1 m/s (the Moens and Korteweg velocity (equation ( 5)) is 6.14 m/s which corresponds to the high frequency phase velocity). In addition, linear hypotheses are fulfilled since non-linear terms are around 0.5% of the linear term ∂u ∂t .

Boundary Conditions

In this section, we propose a new boundary condition to allow unidirectional waves cross boundaries without reflections, the approach is named "unidirectional propagation hypothesis" (UPH). We also build another boundary condition based on a RCL model. Both approaches will be compared in Section 4.

Outlet boundary condition: unidirectional propagation hypothesis (UPH)

Let us consider the axial-component of the linearized Navier-Stokes equation in which one assumes axial derivatives to be negligible with respect to radial derivatives (such hypotheses are justified for blood flow since artery radius is smaller enough than characteristic wavelength). This leads to the simplified equation

∂v z ∂t = - 1 ρ ∂P ∂z + µ 1 r ∂ ∂r (r ∂v z ∂r ) ( 7
)
where P is the transmural pressure. When integrated over a section at constant z, this equation becomes :

dQ dt = -πR 2 1 ρ ∂P ∂z + 2πRµ ∂v z ∂r r=R (8)
Let us consider a monochromatic wave of pulsation ω propagating in the z > 0 direction. Within Prandtl approximation, the associated transmural pressure is independent of the radial coordinate r and tis a plane progressive wave damped in the propagation direction:

P(z,t) = P 0 e γ z e iω(t-z/c) (9)
where c = c(ω) is the phase velocity and γ = γ(ω) a damping coefficient. For this particular solution propagating towards positive z, it is possible to transform the z derivative into a time derivative

∂P ∂z = -γP - 1 c ∂P ∂t (10)
Let us now assume that this relation can be used for any unidirectional pressure wave as long as c and γ only a slightly depend on frequency ω. Using equations ( 8) and ( 10), a relationship is obtained for outlet pressure P out = P(z out ,t) :

-γP out + 1 c dP out dt - ρ πR 2 dQ out dt + 2µ R ∂v z ∂r | r=R,z=z out = 0(11)
In our computations, this equation is implemented as the outlet condition. Indeed pressure P out (t) can be obtained by solving this equation, Q out being numerically computed. The following procedure is called the unidirectional propagation hypothesis (UPH).

The values of c and γ used in equation ( 11) are determined through simulations as follows. First a computation is performed using outlet condition (equation ( 11)) with constant c given by the Moens-Korteweg relation [START_REF] Tropea | Reduction of aortic wall motion inhibits hypertension-mediated experimental atherosclerosis[END_REF] and γ by the attenuation in the case of Womersley number given in Table 1. Moreover, we use for this first simulation a tube long enough (1 m in this case) to ensure the separation of backward and foreward waves. By following the position and value of the pulse peak, velocity c and attenuation γ are measured. Once these factors computed, a second simulation is performed with the new values of c and γ and the true finite tube length. If reflected waves are observed, one repeats the same procedure and computes again the values of c and γ until the reflective waves disappear. Convergence is generally reached in three iterations. The evaluation of parameter c is important since a wrong value of c would lead to a unsatisfactory non-reflective condition. If one wants to modulate a reflected wave by prescribing a given amplitude index a in which runs between 0 (no reflexion) and 1 (total reflexion), it is possible to use the modified equation

a in -γP out + 1 c dP out dt - ρ πR 2 dQ out dt + 2µ R ∂v z ∂r | r=R,z=z out = 0
It is possible to introduce similar boundary conditions at inlet to minimize reflective waves generated at the entrance. It is however necessary to separate the total pressure as a sum of a wave P i (z,t) propagating in the increasing z direction and a wave P r (z,t) propagating in the decreasing z direction:

P(z,t) = P i (z,t) + P r (z,t) (12)
with P i (z = 0,t) = P imp (t). One can then write equation ( 8) for pressure P and flow rate at inlet (z = 0). The reflected wave P r and the imposed pressure function P imp at that point can be added to get the total pressure P in = P(z = 0,t). Assuming that P i (z,t) is given by a spatio-temporal function similar to equation ( 9) and that P r (z,t) is given by P r (z,t) = P r0 e γz e iω(t-z/c) then the z-derivatives of P(z,t) at the entrance read :

∂P in (z = 0,t) ∂z = ∂P i (z=0,t) ∂z + ∂P r (z=0,t) ∂z = -γP imp -1 c dP imp dt + γP r + 1 c dP r dt z=0 (13)
Using equation [START_REF] Li | Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses[END_REF], one can eliminate P r from equation ( 13) and deduce a boundary condition at the entrance:

γP in + 1 c ∂P in ∂t -γP imp - 1 c ∂P imp ∂t + ρ πR 2 dQ out dt + 2µ R ∂v z ∂r | r=R (14)
Equations ( 11) and ( 14) are used as boundary conditions: they allow to model pressure pulses with minimal reflections at both tube extremities. This model has one major assumption: the non-dispersive hypothesis concerning c and γ.

A priori the viscous term in equation ( 14) must be evaluated at z = 0. Nevertheless at the inlet, the flow is not fully developed and the wall shear rate computed at z = 0 would lead to a wrong estimation of this viscous term. Under the long wavelength hypothesis it is more convenient to evaluate this term at a position sufficiently away from the inlet, larger than the inlet length and negligible as compared to the wavelength. The extent along z of this region scales as R, the radius of the tube. The effect of an inaccurate evaluation of the near wall shear rate can be observed by considering a wave propagating towards the inlet as shown in Figure 2a. If one evaluates the shear rate of equation ( 14) in a point immediately near the entrance (point C in Figure 2a) one can observe reflections (Figure 2c). Furthermore when the evaluation is done in a point distant from the inlet (point B located at a distance of 7R from the inlet), no reflective wave appears (Figure 2b).

Another outlet boundary condition : RLC as a model of terminal impedance

We present in this section a boundary condition based on a RCL filter model. If one assumes low amplitude, the outlet pressure is related to the flow rate by the characteristic impedance Z c (ω)

Z c (ω) = ρc 0 A 0 1 -F(i 3/2 α) -1/2 (15)
This relation however cannot be used with a time advancing computation scheme since, it requires to work in frequency spectral space and thus to know, a priori, the whole time signal. In the general nonlinear regime, we replaced this transformation by a differential equation system approximating the function Z c in a given frequency range, in particular the physiological one, between 1 and 20 Hz. We propose a more complex filter than a standard RLC, where the elements are estimated by fitting the filter impedance as closely as possible to the characteristic impedance. When linear hypothesis are fulfilled, such a condition is strictly equivalent to the semi-infinite tube. This particular filter is given by a resistance R 1 and an inductive element L in series with a capacity C in parallel with an element Z 2 . This leads to the following complex impedance

Z = R 1 + iωL + Z 2 1 + iωCZ 2
In order to obtain the best fit of equation ( 15), we assumed that element Z 2 is complex. Once the best values of coefficient are found (C, R 1 , L, Z 2 ) it is possible to connect such an impedance to a system of differential equations calculating the pressure/flow relationship at the outlet boundary. In Fourier frequency-space, the expression for Z is decomposed into two terms Z = Z 1 + 1/Y 2 , so that the total complex pressure P at the outlet can be splitted into to terms such

P = P1 + P2 P1 = Z 1 Q = (R 1 + iωL) Q P2 = 1 Y 2 Q = 1 1 Z 2 + iωC Q
where Q is the real flow rate. The above algebraic system is equivalent to a system of linear differential equations which will be used as an implicit outlet boundary condition:

P1 -R 1 Q -L dQ dt = 0 P2 Z 2 +C d P2 dt -Q = 0.
Finally the pressure P out is obtained by the following :

P out = R e ( P1 + P2 ).
In reference [START_REF] Bokov | Description expérimentale et numérique de l'interaction entre un stent biodégradable et la paroi artérielle[END_REF], we have shown that this approach is better than the classical RLC filter method to define non reflective boundary conditions.

Results and Discussion

We present numerical test on the application of the UPH approach. Let us denote by S INF the value of the numerical solution of a pulse wave propagating in a semi-infinite domain (see section 2) but reduced to the spatial sector [0, 0.5m]. At z = 0.5m, the numerical data hence present no reflexion and no reflected waves. Therefore this numerical data is the target for our comparisons. S UPH denotes the solution obtained numerically within domain [0, L = 0.5m] using the UPH boundary condition at z = 0.5 and S RLC the associated solution obtained numerically in the domain [0, L = 0.5m] using the RLC outlet boundary condition. We simulate these various approximations to identify which boundary condition is preferable. The mechanical characteristics and simulation parameters are given in Table 2 Moens-Korteweg velocity [m.s -1 ] 6.32

Womersley number (α) 6.14

Computed group velocity [m.s -1 ] 5.87 ± 0.05 We use the imposed pressure signal P imp of equation ( 6) with two values of adimensional pulse pressure amplitude ∆P/E ( 5.10 -4 and 5.10 -3 ) which represents approximately 1% and 10% of physiological values. The last value corresponds to the limit of approximations in the analytical model as v max /c becomes close to 6%. For ∆P/E = 5.10 -4 , the two solutions S RLC to S INF agree (Figure 3a), the relative difference being less than 3.5%. For ∆P/E = 5.10 -3 (Figure 3b), the difference at the tube outlet grows to 7 % due to reflections.

For ∆P/E = 5.10 -4 , the solution S UPH and S INF are in good agreement (Figure 4a). The wave velocity and attenuation coefficient are also well assessed. For ∆P/E = 5.10 -3 (Figure 4b), the difference between the two solutions at the outlet boundary is around 0.25% (evaluated at pressure peak). This result has to be compared to the S RLC condition (7%).

Figure 2a illustrates the rôle of the inlet boundary condition [START_REF] Vignon-Clementel | Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries[END_REF] : in that instance, a pressure wave propagates towards the inlet and exists without reflections. However, when both backward and forward waves interact at z = 0, this inlet condition has not as efficient.

The UPH boundary conditions at inlet and outlet is able to eliminate spurious reflective waves. As illustrated in Figure 4, it provides for a short tube in a 2D-axisymmetric configuration, a pressure propagative wave which does not differ from the pressure computed in "infinite" domain by more than 0.25%.

Application of UPH on a stent singularity

Diseases of the arterial system (aneurism, stenosis...) and surgical treatments (stent-grafts, vascular substitute, stent) induce modifications of mechanical and geometrical properties of the vascular wall. For instance the consequences of a stent in an artery have been widely studied (struts protrusion [ [START_REF] Seo | Computational study of fluid mechanical disturbance induced by endovascular stents[END_REF]], distribution of mechanical stresses on the artery [ [START_REF] Ladisa | Circumferential vascular deformation after stent implantation alters wall shear stress evaluated with time-dependent 3d computational fluid dynamics models[END_REF][START_REF] Gijsen | Simulation of stent deployment in a realistic human coronary artery[END_REF]]). In particular it is known that a typical metal stent is around 5 times less compliant than the arterial wall [ [START_REF] Nicoud | A numerical assessment of wall shear stress changes after endovascular stenting[END_REF]]. One practical application of the UPH approach is to describe quantitatively the hydrodynamic effect of such an inhomogeneity in a short vessel segment. In this case it is necessary to separate the induced reflected wave from the total pulse wave (a stent could be responsible for a pressure pulse reflection around 6 %).

Numerous separation methods have been presented in the literature [ [START_REF] Stergiopulos | Nonlinear separation of forward and backward running waves in elastic conduits[END_REF][START_REF] Rogova | Propagation d'ondes en hémodynamique artérielle: application à l'évaluation indirecte des paramètres physiopathologiques[END_REF]] : linear methods, which give fairly good results, however fail when considering high amplitude waves [ [START_REF] Bokov | Description expérimentale et numérique de l'interaction entre un stent biodégradable et la paroi artérielle[END_REF]] and nonlinear methods used until now generally required strong hypothesis for the evaluation of viscous term in the Navier-Stokes equation. The proposed approach evaluates the impact of an inhomogeneity by (a) performing two numerical simulations : without and with inhomogeneity. and (b) by subtracting both solutions, upstream of the singularity. Since reflected waves are only issued from the singularity, the difference between the two solution signals (flow rate, pressure, shear stress...) gives the reflected wave.

Figure 7 presents successive wall displacement profile ∆R(z) in the tube measured each 1 m.sec. Without any singularity (fig. 7b), the envelop of the maximum wall displacements (yellow line) shows the expected attenuation of pressure maxima while the wave is propagating. When a stent is present, this curve exhibits an increase just before the singularity, associated to a backward reflected wave. The characteristics of the numerical simulation are given in Table 2 with a pressure signal defined before, and ∆P/E of 5 10 -3 . We have validated this method of separation by comparing its results within the linear domain with another separation method described in [ [START_REF] Rogova | Propagation d'ondes en hémodynamique artérielle: application à l'évaluation indirecte des paramètres physiopathologiques[END_REF]]. A good agreement was observed (results are not shown; differences are less than 0.5%).

A reflection index is introduced to quantify the effect of the singularity : it is based on the ratio between the backward axial velocity amplitude W - The difference observed between a separation based on a linear theory of pulse reflection and the numerical separation method can be explained by the fact that a linear theory does not take into account the hydrodynamic entrance length in the stent. The reflection index is an indirect evaluation of the compliance of the stent. Such an evaluation could be interesting in monitoring the degradation of biodegradable stents. Indeed it is still unknown how long a biodegradable scaffold effectively assumes a supporting role of the vascular wall [ [START_REF] Wong | Bioabsorbable scaffold: animal and human observations: Take-home message[END_REF]].

Conclusion

We introduced a method that defines a non-reflective boundary condition for pressure waves into a flexible vessel. This method takes also into account the fluid-structure interaction and is based on Navier-Stokes equations using inlet/outlet boundary conditions with an unidirectional propagative hypothesis (UPH). This approach allows to eliminate spurious reflections which appear in classical techniques like RLC and LR circuits. Though it is performed here with an axisymmetric geometry, this method can be easily generalized to 3D simulations. A reflection index is given by the implementation of a method of wave separation where the component of the reflected wave is computed from the values of total pressures or wall displacements with and without the obstacle (the stent). We have applied this approach to a physiological situation i.e. the presence of a stent into a coronary artery.

Fig. 1 .

 1 Fig. 1. Non dimensional pressure (P/E) as a function of time. Each curve corresponds to different positions z (0, 0.4m, 0.8m, 1.2m, and 1.6m). Numerical simulations (red dash-dot line) are compared to analytical solutions (black dashed line). The peak Reynolds number evaluated at entry, is equal to 23.

Fig. 2 .

 2 Fig. 2. (a) Pulse wave, propagating towards negative z : axial velocity is color coded (max in red, deep blue is zero) and the contour is the radial displacement for three times. (b) Pressure as a function of time at two different positions, z = 0.4m and z = 0.1m. The shear rate of equation (23) is evaluated at z = 1cm, i.e. approximately 7R, (c). Idem Fig 2 (b) but the shear rate is now evaluated at the tube entrance.

  .

4. 1

 1 Comparison between UPH versus RLC.

Fig. 3 .

 3 Fig. 3. Adimensional pressure (P/E) as a function of time at several axial positions (z=0 m (entry point); 0.1m; 0.3m; 0.5m (exit point)). Red dash-dot lines correspond to S RLC , black dashed lines to S INF . (a) ∆P/E of 5.10 -4 ; (b) ∆P/E of 5.10 -4 .

Fig. 4 .

 4 Fig. 4. Non-dimensional pressure (P/E) as function of the time at several axial positions (z=0 m (entry point); 0.1 m; 0.3 m; 0.5 m (exit point)). Red dash-dot lines correspond to S UPH with UPH outlet conditions (equation 18), black dashed lines to S INF (a) S UPH and S INF for ∆P/E = 5.10 -4 (b) S UPH and S INF for ∆P/E = 5.10 -3

Fig. 5 .

 5 Fig. 5. Pressure pulse using UPH for a high amplitude (10 4 Pa) in a 50 cm length tube with Young modulus E = 106 Pa and viscosity µ = 7 mPa.s. Wall displacements at successive times (0.05, 0.1, 0.15, 0.2, 0.25 and 0.3 sec) are shown : no apparent reflexion occurs. The wave moves from the bottom to the top. The color bar is scaled in 10 -2 mm, and the bottom axis in mm.

Fig. 6 .

 6 Fig. 6. Pressure pulse using UPH for a high amplitude (10 4 Pa) : the position of the pressure maximum as a function of time. Diamonds (S INF ), squares (S UPH ).

Fig. 7 .

 7 Fig. 7. Radial displacement waves measured every 1e -3 s using UPH: Fig. (a) when the tube presents a singularity a hundred times stiffer than the tube wall situated between z = 0.31 m and z = 0.33 m; Fig. (b) without singularity. A yellow continuous line is superimposed, joining the maximum of the displacement wave at each time.

Figure 8

 8 Figure 8 displays the numerical evaluation of this reflection index as a function of the relative Young's modulus of the stent.The difference observed between a separation based on a linear theory of pulse reflection and the numerical separation method can be explained by the fact that a linear theory does not take into account the hydrodynamic entrance length in the stent. The reflection index is an indirect evaluation of the compliance of the stent. Such an evaluation could be interesting in monitoring the degradation of biodegradable stents. Indeed it is still unknown how long a biodegradable scaffold effectively assumes a supporting role of the vascular wall [[START_REF] Wong | Bioabsorbable scaffold: animal and human observations: Take-home message[END_REF]].

Fig. 8 .

 8 Fig. 8. Reflection index as a function of the relative Young's modulus of the stent (E r = E/E 0 ) where E 0 is the Young modulus of a normal artery. The results using our numerical separation method are given by red squares. The results of a linear theory of reflection (see Annex I) are represented by black squares.

Table 1 .

 1 Mechanical properties and numerical inputs used for the "semi-infinite" simulations.

	h/D ext	0.083
	Tube length [m]	4.92
	∆P/E	5.10 -4
	Womersley number α	8.2
	Dynamic viscosity (blood) [Pa.s]	5.10 -3
	Pulse duration T 1 [s]	0.4
	Computed attenuation coefficient [m -1 ]	0.20

Table 2 .

 2 Mechanical properties and numerical input used in UPH simulations.