Ellis Ja

Taylor M Decourtney

Ruehlman Ls

Aiken Ls

Ritter Pl

González Vm

Laurent Dd

Lorig Kr J Rheumatol

Liebman M, Bijur PE Gallagher Ej

Ann

Deisinger Ja

Cassisi Je

Lofland Kr

Shibata T Pain

Wenk R, Baldioceda F, Bernal R, Ovalle A, Torrubia R, Baños Whizar-Lugo Vm

Evaluating chronic pain impact among patients in primary care: further validation of a brief assessment

Background: Pain is one of the most common symptom reported in general practice. For this purpose, one-dimensional and multidimensional pain scales have been developed to assess acute or chronic pain. There is no gold standard in pain assessment in general practice. This study aimed to list all pain scales studied for the general population.

Method: A systematic review questioning Cochrane database, according to the PRISMA protocol, was conducted until February 2018. Only English articles assessing pain scales in general population were included. The bibliography of selected studies and recent systematic reviews had been examined to include additional articles in a second search.

Results: On 1431 results found on Cochrane, 4 relevant articles were included. With the second search, 203 additional articles were detected and 84 were added. Finally, 88 articles were selected and 35 pain scales were found: 6 unidimensional-scales with 13 variants and 12 multidimensional scales with 4 variants.

Discussion: Pain scales were different by explored dimensions and the number of item. Some of them are not studied in this current review by a lack of articles available on the database responding to eligibility criteria. Reliability data of available pain assessment scales were recently collected in a systematic review. It will be necessary to consider external validity data in further studies and to choose which pain scale is the most appropriate for daily practice.

REMERCIEMENTS

Au Professeur Jean-Yves LE RESTE, merci de me faire l'honneur de présider ce jury de thèse, de m'avoir confié ce sujet qui, j'en suis sûr, m'aidera au quotidien dans l'approche globale de ma prise en charge. Merci de tes conseils avisés, de ta réactivité et de ta gentillesse dans tes explications. Sois assuré de toute ma gratitude et de toute mon admiration, tant pour ta pratique universitaire au sein du DUMG que pour ton exercice libéral. Immense merci.

Au Docteur Patrice NABBE, merci de me faire l'honneur de participer à l'évaluation de ce travail au sein de ce jury. Merci pour les formations que vous apportez aux étudiants brestois, qui nous servent au quotidien. Soyez assuré de mon profond respect et de ma sincère gratitude.

Au Docteur Benoit CHIRON, merci de me faire l'honneur de votre présence au sein de ce jury. Je garde un particulièrement bon souvenir de vos interventions en cours lors de mon Internat, avec tout le sérieux et la décontraction nécessaires pour un bon apprentissage. Veuillez croire en mon profond respect et ma sincère estime.

Au Docteur Elouan PERON, merci de me faire l'honneur d'être dans mon jury et de pouvoir évaluer ce travail. Connaissant ton aversion pour tout ce qui touche de près ou de loin à la langue de Shakespeare, je ne pouvais que te le proposer. Mais aussi, sache que tu fais un merveilleux médecin, un ami extraordinaire et un expert dans les problèmes articulaires les plus divers.

A tous mes maîtres et chefs de stage, avec qui j'ai eu l'honneur d'apprendre mon métier. Je ne serais jamais assez reconnaissant pour vos conseils et les bons moments passés. Amitiés sincères pour Dr Bazire, Dr Etienne, Dr Muller, Dr Bourillet et les membres du centre médical Laennec de Saint Pol.

A tous les soignants que j'ai pu croiser. Quel plaisir d'avoir partager ces temps de travail à vos côtés, mêlés de partage de connaissance et de bonne humeur. Pensée nostalgique à tout le service d'hépato-gastro-entérologie du CHU de Saint Pierre à la Réunion, de Dermatologie de Brest et les paramédicaux de Gynéco-obstétrique de Morlaix.

A Myriam et Elise, merci infiniment de m'avoir aidé au début de ce travail. Vos conseils ont été forts précieux. Bonne continuation à vous deux.

A Monsieur Georges Beuzet, merci pour votre aide précieuse dans la recherche bibliographique.

A mes parents, Maman, Papa, merci de tout l'amour que vous m'apportez, d'avoir cru en moi, de votre éducation. Le soutien indéfectible en première année et le rythme « militaire » ont surement était une des clés de l'obtention du précieux sésame. Vous êtes des modèles au quotidien. Je vous aime.

A mes frères. Guillaume, j'aimerai, tout simplement, que tu deviennes le fer de lance de la scène artistique brestoise et que tu fasses bouger cette ville qui en a bien besoin. Ton parcours mérite tout mon respect. Jean-Marie, la ville de Lille possède surement le directeur chargé d'éducation le plus humain et le plus incroyable. Je vous aime les frangins ! A Papy et Elisabeth, merci de votre soutien dans ce long cursus. Merci pour vos discussions si enrichissantes autour d'un bon repas. Papy, merci de m'avoir transmis ta passion pour la cuisine et pour la randonnée. Je vous aime.

Au quarté gagnant, Elou, Ben, Maxime. Messieurs, je ne saurais assez vous remercier pour tout ce que vous m'apportez. On a pu en faire des tonnes de choses intelligentes en votre compagnie ! En espérant que cela continue ! La vie avec vous : c'est bieng.

Aux rennais, Franwad, Tienou et Bastouche, trop heureux de vous retrouver. Le slip de Francois, le batman d'Etienne et le jogging de Bastien me manquaient.

A tous les copains brestois de la fac,

INTRODUCTION

Pain is one of the most common symptom reported in the general population (1). It is recognized as "a major global healthcare problem"(2) by its repercussions on the quality of life, its cost for the society and its societal representation. The overall prevalence of pain complaints in primary care was 45,2% in a Malaysian study, [START_REF] Zailinawati | Pain morbidity in primary carepreliminary observations from two different primary care settings[END_REF], 40% in a Finnish study [START_REF] Mäntyselkä | Pain as a reason to visit the doctor: a study in Finnish primary health care[END_REF], 30% in Swedish study [START_REF] Hasselström | Prevalence of pain in general practice[END_REF] and 43 % in a French study [START_REF] Tajfel | La douleur en médecine générale[END_REF]. Furthermore moderate-severe chronic pain occurs in 19% of European adults, seriously affecting their daily, social and working lives [START_REF] Breivik | Survey of chronic pain in Europe: prevalence, impact on daily life, and treatment[END_REF].

Whether acute or chronic, pain has a warning role. Some describe it as the « Fifth vital sign »(8), as blood pressure, pulse, breathing or temperature. It holds a major place in any clinical examination and shows the importance of using the best assessment tool as possible.

Pain is a subjective, complex and multidimensional symptom. The measurement of pain intensity is complicated by its nature as a "personal, subjective experience influenced by cultural learning, the meaning of the situation, attention, and other psychological variables" (9).

Its evaluation is the basis of pain management. There is no gold standard for pain assessment.

Because of their ease of use, unidimensional scales have become popular tools to quantitate pain relief and pain intensity but tend to focus only on pain intensity with increased risk of oversimplification of the experience. So unidimensional pain scales are more appropriate in the setting of acute pain than chronic. The assessment of chronic pain often requires multidimensional scales or more complex evaluation tools [START_REF] Ho | Review of pain-measurement tools[END_REF] which explored disabilities, belief, quality of life, etc. This study aimed to list all pain scales studies in the general population in a literature review using Cochrane database.

General population included subjects who have no particular disease, no particular pain and no age limit. This population underwent pain assessment to test pain scales. As there is no gold standard, there is no pain scale comparator required. The study of pain scales in general population is the main outcome sought in this review.

A recent study examined the psychometric values of the pain scales selected in a first study from a different database. [START_REF] Belhadj | What pain scales assessment are usable in general population ? A systematic literature review of Medline according to the PRISMA statement[END_REF] [START_REF] Payelle | Reliability of the pain assessment scales available in the general population: a systematic review[END_REF] Then the best pain scales to use in a GP activity will be chosen. For the present review, no particular study design is needed to be included. This study is meant to be an update of the 2017 study which used Medline database.

METHOD GENERAL FRAMEWORK

This study was a systematic review of literature according to the PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses ) protocol [START_REF] Liberati | The PRISMA statement for reporting systematic reviews and meta-analyses of studies that evaluate health care interventions: explanation and elaboration[END_REF]. The purpose of this method was to obtain a reproducible systematic review from a 27 items list called "PRISMA 2009 Checklist".

INCLUSION AND EXCLUSION CRITERIA

Eligibility criteria were the same as the literature review based on Medline database. They were chosen during a study group composed by training doctors, students and directed by senior researchers. Decisions were undertaken using everyone's knowledge and were validated during this study group.

a. Inclusion Criteria

-About pain Pain had to be non-focused on a site and non-specific to a particular disease. It could be an acute or a chronic pain. Musculoskeletal pain had been included because of this non-specific site.

-About scales Only scales assessing pain were included, whether one-dimensional and multidimensional scales. Were included pain self-assessment scales and observational rating scales.

-About population To be included, studies had to evaluate general population. General population meant that the studied population did not have a particular disease (were accepted studies which included 3 or more different diseases). Studied populations had not to be selected according to their age.

-About studies Studies had to be written in English, in an IMRAD format, to have an abstract available, to deal with human. All types of studies were accepted but they had to study pain scales.

b. Exclusion criteria -About pain

Pain caused by a particular disease or focused on a site were excluded.

-About scales

Were excluded all scales which the principal purpose was not pain evaluation (quality of life scales, behavior scales, etc.) as well as scale for diagnosis (Ex: neuropathic pain scales).

-About population

Studies with specific population were excluded (for example: children, old people, pregnant women etc.). Concerning multi-dimensional scales, parameters were a little different: number of time quote in an article, number of time used as a reference, number of item, did it derived from another scale, the explored dimension and the type of evaluation used.

RISK OF BIAS IN INDIVIDUAL STUDIES

A quality assessment was applied on included articles to reduce the risks of bias. It followed the "Exercer" quality grid (Appendix 4) which is based on 18 points. The studies were included if the score of 9 was reached. Less, they were excluded.

RESULTS

STUDY SELECTION

This systematic review was conducted between December 2017 and February 2018. The search returned 1431 results. Then, articles were screened on basis of titles and abstracts: 1332 studies were excluded. Among them, 1316 were off topics (with 38 duplicates), 5 were not written in English, 7 had not an available abstract and 4 were not written in an IMRAD format.

Next, the application of inclusion/exclusion criteria allowed to exclude 95 articles: 6 were duplicates and 89 met exclusion criteria. 4 studies were therefore included.

A quality assessment with the "Exercer" grid was applied (APPENDIX 4). This grid allows to limit the number of bias. Finally, the 4 articles were included with 6 different available scales.

To use the studies cited by but unavailable in Cochrane database, a review of the bibliography of each included article was added. Another screening of the first study's bibliography (based on Medline) was also realized. 203 studies were found. 21 were excluded because there were 2 duplicates and 19 not available on database. After application of inclusion criteria, 98 articles were excluded with 15 off topics, 2 which were not written in an IMRAD format, 5 because full articles were not available, 33 because of a specific population and 43 with a specific pain. After the same quality assessment as first part of the search, 84 articles were added to the 4 first. A total of 88 studies were included and 35 pain scales were found.

See Figure 1.

Psychometric properties and validation of the Italian version of the

SYNTHESIS OF RESULTS

a. One-dimensional pain scales

One-dimensional pain scales are mostly used to assess acute pain. They evaluate only one parameter, which is commonly pain intensity.

Visual Analog Pain Scale

Visual Analog scale (VAS) is the most commonly used scale because of its ease of use and its simple construction. It was introduced in the first part of the 20th century. It's a graduated 100-mm line with one-hand marked as "No pain" and the other hand marked as "worst possible pain". Patients move the cursor themselves where it better corresponds to their current pain.

Finally, the distance from "no pain" to the cursor is measured in millimeters. See figure 2.

This scale was studied in 16 articles in this current review. [START_REF] Mohan | The end of the line? The Visual Analogue Scale and Verbal Numerical Rating Scale as pain assessment tools in the emergency department[END_REF][START_REF] Lund | Lack of interchangeability between visual analogue and verbal rating pain scales: a cross sectional description of pain etiology groups[END_REF][START_REF] Kane | Visual Analog Scale pain reporting was standardized[END_REF][START_REF] Williamson | Pain: a review of three commonly used pain rating scales[END_REF][START_REF] Yamamoto | Minimum clinically significant VAS differences for simultaneous (paired) interval serial pain assessments[END_REF][START_REF] Holdgate | Comparison of a verbal numeric rating scale with the visual analogue scale for the measurement of acute pain[END_REF][START_REF] Bijur | Reliability of the visual analog scale for measurement of acute pain[END_REF][START_REF] Gallagher | Prospective validation of clinically important changes in pain severity measured on a visual analog scale[END_REF][START_REF] Svensson | Concordance between ratings using different scales for the same variable[END_REF][START_REF] De C Williams | Simple pain rating scales hide complex idiosyncratic meanings[END_REF][START_REF] Berthier | Comparative study of methods of measuring acute pain intensity in an ED[END_REF][START_REF] Kelly | Does the clinically significant difference in visual analog scale pain scores vary with gender, age, or cause of pain?[END_REF][START_REF] Todd | The minimum clinically important difference in physician-assigned visual analog pain scores[END_REF][START_REF] Jensen | The measurement of clinical pain intensity: a comparison of six methods[END_REF][START_REF] Dixon | Reproducibility along a 10 cm vertical visual analogue scale[END_REF][START_REF] Scott | Graphic representation of pain[END_REF] It was used 14 times as a reference [START_REF] Hayashi | Associations between pain drawing and psychological characteristics of different body region pains[END_REF][START_REF] Arimura | Pain questionnaire development focusing on cross-cultural equivalence to the original questionnaire: the Japanese version of the Short-Form McGill Pain Questionnaire[END_REF][START_REF] Adelmanesh | Reliability, validity, and sensitivity measures of expanded and revised version of the short-form McGill Pain Questionnaire (SF-MPQ-2) in Iranian patients with neuropathic and non-neuropathic pain[END_REF][START_REF] Erdinc | Turkish version of the Cornell Musculoskeletal Discomfort Questionnaire: cross-cultural adaptation and validation[END_REF][START_REF] Zinke | Examining the cross-cultural validity of the english short-form McGill Pain Questionnaire using the matched moderated regression methodology[END_REF][START_REF] Ergün | Trial of a new pain assessment tool in patients with low education: the full cup test[END_REF][START_REF] Oksuz | Nonmalignant chronic pain evaluation in the Turkish population as measured by the McGill Pain Questionnaire[END_REF][START_REF] Ritter | Measurement of pain using the visual numeric scale[END_REF][START_REF] Kitisomprayoonkul | Thai Short-form McGill Pain Questionnaire[END_REF][START_REF] Perkins | Development and validation of a brief, descriptive Danish pain questionnaire (BDDPQ)[END_REF][START_REF] Bijur | Validation of a verbally administered numerical rating scale of acute pain for use in the emergency department[END_REF][START_REF] Lázaro | Psychometric properties of a Spanish version of the McGill Pain Questionnaire in several Spanish-speaking countries[END_REF][START_REF] Hasegawa | The McGill Pain Questionnaire, Japanese version, reconsidered: confirming the theoretical structure[END_REF][START_REF] Ferrari | Internal structure and validity of the multidimensional pain inventory, Italian language version[END_REF]. The Numerical Pain Rating Scale (or Verbal Numerical Rating Scale)

Numerical Pain Rating Scale (NPRS) is a unidimensional measure of acute or chronic pain intensity. The most commonly used is the 11-item NPRS. From 0 ("No pain") to 10 ("Worst possible pain"), patients select the number that best reflects the intensity of their pain. The NPRS can be administered verbally or graphically. See figure 3.

This scale was studied in 14 articles [START_REF] Lorenz | How reliable is pain as the fifth vital sign?[END_REF][START_REF] Mohan | The end of the line? The Visual Analogue Scale and Verbal Numerical Rating Scale as pain assessment tools in the emergency department[END_REF][START_REF] Lorenz | Exploring alternative approaches to routine outpatient pain screening[END_REF][START_REF] Angst | Responsiveness of five condition-specific and generic outcome assessment instruments for chronic pain[END_REF][START_REF] Nicholas | What do the numbers mean? Normative data in chronic pain measures[END_REF][START_REF] Williamson | Pain: a review of three commonly used pain rating scales[END_REF][START_REF] Holdgate | Comparison of a verbal numeric rating scale with the visual analogue scale for the measurement of acute pain[END_REF][START_REF] Bijur | Validation of a verbally administered numerical rating scale of acute pain for use in the emergency department[END_REF][START_REF] Farrar | Clinical importance of changes in chronic pain intensity measured on an 11-point numerical pain rating scale[END_REF][START_REF] De C Williams | Simple pain rating scales hide complex idiosyncratic meanings[END_REF][START_REF] Jensen | Comparative reliability and validity of chronic pain intensity measures[END_REF][START_REF] Berthier | Comparative study of methods of measuring acute pain intensity in an ED[END_REF][START_REF] Jensen | Increasing the reliability and validity of pain intensity measurement in chronic pain patients[END_REF][START_REF] Jensen | The measurement of clinical pain intensity: a comparison of six methods[END_REF].

It was used as a reference in 5 articles [START_REF] Pickering | Acute pain evaluation with Algoplus scale in Cambodian patients[END_REF][START_REF] Arimura | Pain questionnaire development focusing on cross-cultural equivalence to the original questionnaire: the Japanese version of the Short-Form McGill Pain Questionnaire[END_REF][START_REF] Ellis | Establishing the psychometric properties and preferences for the Northern Pain Scale[END_REF][START_REF] Spadoni | The evaluation of change in pain intensity: a comparison of the P4 and single-item numeric pain rating scales[END_REF][START_REF] Masedo | Some empirical evidence regarding the validity of the Spanish version of the McGill Pain Questionnaire (MPQ-SV)[END_REF].

Four variant scales were found: the composite scores [START_REF] Jensen | Comparative reliability and validity of chronic pain intensity measures[END_REF], the electronic diary version [START_REF] Heapy | Using multiple daily pain ratings to improve reliability and assay sensitivity: how many is enough?[END_REF] and the 4-Item Pain Intensity Measurement (P4) [START_REF] Spadoni | The evaluation of change in pain intensity: a comparison of the P4 and single-item numeric pain rating scales[END_REF] and the NRS seven point version [START_REF] Svensson | Concordance between ratings using different scales for the same variable[END_REF].

Concerning the composite scores, authors thought the combination of individual pain ratings could be a more valid and reliable measure of pain intensity than individual ratings. First, patients had to assess, with a 11-item NRS, their worst, least and average pain over the past 2 weeks. Then, 3 composite pain intensity scores were created by calculating the averages of different parameters over the past 2 weeks (a 2-item scale: least and average pain ratings, a 3item scale: worst, current and average pain ratings and a 4-item scale: worst, current, least and average pain ratings of the last 2 weeks).

The P4 is a pain intensity tool consisting of 4 items that evaluate pain intensity in the morning, in the afternoon, in the evening, and with activity over the past 2 weeks. Each item is scored on a 0-10 NPRS. P4 scores vary from 0 (no pain) to 40 (highest possible pain level) Visual Rating Scale (VRS)

The Verbal rating Scale is a 5-item one-dimensional pain scale with eligible alternatives -no pain, mild, moderate, severe, worst possible pain. Patients must choose the best adjective that correspond to their current pain. See Figure 4.

This scale was studied in 4 articles in this current study [START_REF] Lund | Lack of interchangeability between visual analogue and verbal rating pain scales: a cross sectional description of pain etiology groups[END_REF][START_REF] Williamson | Pain: a review of three commonly used pain rating scales[END_REF][START_REF] Berthier | Comparative study of methods of measuring acute pain intensity in an ED[END_REF][START_REF] Jensen | The measurement of clinical pain intensity: a comparison of six methods[END_REF].

It

Algoplus

This 5-item hetero-evaluation pain scale had been initially developed in 2011 for detecting acute pain in elderly patients with the inability to communicate. But this behavior scale can also be used for strangers who don't share a common language in order to assess pain severity. The 5 items studied are facial expression (frowning, grimacing etc.), complaints ("ouch-ow", "that hurts" etc.), the look (inattentive, distant etc.), body position (refuses to move, withdrawn etc.)

and atypical behavior (agitation, aggressivity etc.)

This scale was studied in one article in this current review [START_REF] Pickering | Acute pain evaluation with Algoplus scale in Cambodian patients[END_REF]. See Table 3.

Wong and Baker Faces Pain Scale

This scale was created in 1988(102). This auto evaluation pain scale represents 6 various facial expressions progressing from a smiling face at one end, correlating with no pain, to an extremely unhappy face at the other end, correlating with the worst pain. The patient chooses the one that best fits his current pain. See Figure 5.

This scale was studied in 1 article [START_REF] Ellis | Establishing the psychometric properties and preferences for the Northern Pain Scale[END_REF]. It was not used as a reference. the localization of their pain. Different versions of pain drawing exist but they are not consensual (including pain intensity, organic pain or not etc.). Therefore, this scale is classified as a one-dimensional pain scale. See Figure 6.

This scale was studied in four articles in this current review [START_REF] Hayashi | Associations between pain drawing and psychological characteristics of different body region pains[END_REF][START_REF] Southerst | The reliability of body pain diagrams in the quantitative measurement of pain distribution and location in patients with musculoskeletal pain: a systematic review[END_REF][START_REF] Persson | Computer-aided surface estimation of pain drawingsintra-and inter-rater reliability[END_REF][START_REF] Margolis | Test-retest reliability of the pain drawing instrument[END_REF]. It was not used as a reference.

One variant was found: Dynamic Pain Drawings (computerized version) (57) 4.

Number of time quote in an article

Number of time used as a reference

How many derived scales were found?

The explored dimension

Is a graphic support needed?

The The MOS 36-Item Short-Form Health Survey (SF-36) was created in 1993 as a physical, mental and psychosocial health assessment scale (103). This scale has been declined in eight different subscales. [START_REF] Cook | Establishing a common metric for selfreported pain: linking BPI Pain Interference and SF-36 Bodily Pain Subscale scores to the PROMIS Pain Interference metric[END_REF][START_REF] Krebs | Comparative responsiveness of pain outcome measures among primary care patients with musculoskeletal pain[END_REF][START_REF] Angst | Responsiveness of five condition-specific and generic outcome assessment instruments for chronic pain[END_REF] The 2-item Bodily pain subscale (SF-36 BPS) is studied in 4 articles in this current review [START_REF] Kean | Comparative responsiveness of the PROMIS Pain Interference short forms, Brief Pain Inventory, PEG, and SF-36 Bodily Pain subscale[END_REF][START_REF] Cook | Establishing a common metric for selfreported pain: linking BPI Pain Interference and SF-36 Bodily Pain Subscale scores to the PROMIS Pain Interference metric[END_REF][START_REF] Krebs | Comparative responsiveness of pain outcome measures among primary care patients with musculoskeletal pain[END_REF][START_REF] Angst | Responsiveness of five condition-specific and generic outcome assessment instruments for chronic pain[END_REF]. It was used as a reference in 3 articles [START_REF] Adachi | Validation of the Japanese version of the pain self-efficacy questionnaire in Japanese patients with chronic pain[END_REF][START_REF] Bracher | Cross-cultural adaptation and validation of a Brazilian Portuguese version of the chronic pain grade[END_REF][START_REF] Krebs | Development and initial validation of the PEG, a three-item scale assessing pain intensity and interference[END_REF]. This subscale assesses the pain during the 4 past weeks. One item evaluates pain interference on a 1-5 verbal rating scale. The other item assesses pain severity on a 1-6 verbal rating scale.

See Table 5 on APPENDIX 5

Brief Pain Inventory (BPI) BPI was created in 1993 initially to estimate pain in a cancer population(104). This 11-item pain scale; using a 0-10 NRS, a pain drawing and questions about treatment; measures two domains of pain: a seven-item about interference dimension and a four-item about severity dimension.

This scale was studied in 5 articles in this current review [START_REF] Kean | Comparative responsiveness of the PROMIS Pain Interference short forms, Brief Pain Inventory, PEG, and SF-36 Bodily Pain subscale[END_REF][START_REF] Lapane | One, two, or three? Constructs of the brief pain inventory among patients with non-cancer pain in the outpatient setting[END_REF][START_REF] Rockett | Characteristics of pain in hospitalized medical patients, surgical patients, and outpatients attending a pain management centre[END_REF][START_REF] Krebs | Comparative responsiveness of pain outcome measures among primary care patients with musculoskeletal pain[END_REF][START_REF] Tan | Validation of the Brief Pain Inventory for chronic nonmalignant pain[END_REF]. It was used as a reference in 6 articles [START_REF] Lorenz | How reliable is pain as the fifth vital sign?[END_REF][START_REF] Dekel | Psychometric properties and validation of the Italian version of the Mainz pain staging system as a tool for pain-patients referral selection[END_REF][START_REF] Wilson | Depression and pain interference among patients with chronic pain after ED encounters[END_REF][START_REF] Lorenz | Exploring alternative approaches to routine outpatient pain screening[END_REF][START_REF] Krebs | Development and initial validation of the PEG, a three-item scale assessing pain intensity and interference[END_REF][START_REF] Broderick | Signaling does not adequately improve diary compliance[END_REF]. See table 6 on APPENDIX 5.

Ultra-Brief 3-item PEG (UB-3-PEG)

This scale, derived from the Brief Pain Inventory, was created in 2009. The PEG ("P" for pain intensity, "E" for enjoyment of life, "G" for interference with general activity) comprises 1 intensity item and 2 interference items using a 0-10 NRS.

This scale was studied in 3 articles in this current review [START_REF] Kean | Comparative responsiveness of the PROMIS Pain Interference short forms, Brief Pain Inventory, PEG, and SF-36 Bodily Pain subscale[END_REF][START_REF] Krebs | Comparative responsiveness of pain outcome measures among primary care patients with musculoskeletal pain[END_REF][START_REF] Krebs | Development and initial validation of the PEG, a three-item scale assessing pain intensity and interference[END_REF]. It was not used as reference. See table 7 on APPENDIX 5 Chronic Pain Grade (CPG) This pain scale was created in 1992. This is a seven-item instrument which provide a pain severity score distributed into 4 hierarchical classes: low disability--low intensity (Grade I); low disability--high intensity (Grade II); high disability--moderately limiting (Grade III); and high disability--severely limiting (Grade IV). Using NRS, sub-scale scores evaluate pain intensity, frequency of pain and limitation of activities due to pain. patients' quality of sleep, comfort, medication consumption etc.) and the degree to which the pain interferes with activities of daily living. These 4 subscales provide a single number between 0 and 100 to describe overall pain and its effects. This scale was studied in 1 article in this current review [START_REF] Gentile | Reliability and validity of the Global Pain Scale with chronic pain sufferers[END_REF]. It was not used as a reference.

See Table 11 on APPENDIX This scale was studied in 1 article in this current review [START_REF] Erdinc | Turkish version of the Cornell Musculoskeletal Discomfort Questionnaire: cross-cultural adaptation and validation[END_REF]. It was not used as a reference.

See Table 12 on APPENDIX 5.

Nordic Musculoskeletal Questionnaire (NMQ)

This scale, created in 1987, assesses 4 pain dimensions using an illustration of the human body seen from behind: annual and weekly pain prevalence, functional incapacity and search for medical assistance in the last 12 months. Pictorial body is divided in nine parts linked to 4 questions, corresponding to pain dimensions. See Table 13 on APPENDIX 5. This scale was studied twice in this current review [START_REF] De Barros | Cross-cultural adaptation of the Nordic musculoskeletal questionnaire[END_REF][START_REF] Dickinson | Questionnaire development: an examination of the Nordic Musculoskeletal questionnaire[END_REF]. It was not used as a reference.

One variant scale was found: the Extended Nordic Musculoskeletal Questionnaire (Ex NMQ) [START_REF] Dawson | Development and test-retest reliability of an extended version of the Nordic Musculoskeletal Questionnaire (NMQ-E): a screening instrument for musculoskeletal pain[END_REF]. Patients have to answered to 11 questions about their pain. This scale is more precise on pain prevalence and consequences of pain in the previous year.

West Haven Yale Multidimensional Pain Inventory (WHYMPI)

This 52-item scale was created in 1985. It's composed by 3 sections evaluating pain intensity and pain interference with daily activities, work and social life; using a 7-points NRS for each question. This scale was studied in 9 articles [START_REF] Laliberté | French translation of the Multidimensional Pain Inventory: L'inventaire multidimensionnel de la douleur[END_REF][START_REF] Angst | Responsiveness of five condition-specific and generic outcome assessment instruments for chronic pain[END_REF][START_REF] Nicholas | What do the numbers mean? Normative data in chronic pain measures[END_REF][START_REF] Deisinger | An examination of the psychometric structure of the Multidimensional Pain Inventory[END_REF][START_REF] Ferrari | Internal structure and validity of the multidimensional pain inventory, Italian language version[END_REF][START_REF] Lousberg | Psychometric properties of the Multidimensional Pain Inventory, Dutch language version (MPI-DLV)[END_REF][START_REF] Bergström | Reliability and factor structure of the Multidimensional Pain Inventory--Swedish Language Version (MPI-S)[END_REF][START_REF] Gagné | Confirmatory factor analysis of a 4-factor model of chronic pain evaluation[END_REF][START_REF] Kerns | The West Haven-Yale Multidimensional Pain Inventory (WHYMPI)[END_REF]. It was used as reference in 3 articles [START_REF] Gentile | Reliability and validity of the Global Pain Scale with chronic pain sufferers[END_REF][START_REF] Ruehlman | The development and preliminary validation of a brief measure of chronic pain impact for use in the general population[END_REF][START_REF] Kerns | Self-monitored pain intensity: psychometric properties and clinical utility[END_REF]. See Table 14 on APPENDIX This scale was studied once in this current review [START_REF] Dekel | Psychometric properties and validation of the Italian version of the Mainz pain staging system as a tool for pain-patients referral selection[END_REF]. It was not used as a reference. This scale was studied in 15 articles in this current review [START_REF] Costa | Systematic review of crosscultural adaptations of McGill Pain Questionnaire reveals a paucity of clinimetric testing[END_REF][START_REF] Nicholas | What do the numbers mean? Normative data in chronic pain measures[END_REF][START_REF] Oksuz | Nonmalignant chronic pain evaluation in the Turkish population as measured by the McGill Pain Questionnaire[END_REF][START_REF] Varoli | Adapted version of the McGill Pain Questionnaire to Brazilian Portuguese[END_REF][START_REF] Hasegawa | Confirming the theoretical structure of the Japanese version of the McGill Pain Questionnaire in chronic pain[END_REF][START_REF] Lázaro | Psychometric properties of a Spanish version of the McGill Pain Questionnaire in several Spanish-speaking countries[END_REF][START_REF] Hasegawa | The McGill Pain Questionnaire, Japanese version, reconsidered: confirming the theoretical structure[END_REF][START_REF] Masedo | Some empirical evidence regarding the validity of the Spanish version of the McGill Pain Questionnaire (MPQ-SV)[END_REF][START_REF] Gagné | Confirmatory factor analysis of a 4-factor model of chronic pain evaluation[END_REF][START_REF] Strand | The development of a Norwegian pain questionnaire[END_REF][START_REF] Stein | The German counterpart to McGill Pain Questionnaire[END_REF][START_REF] Vanderiet | The McGill Pain Questionnaire constructed for the Dutch language (MPQ-DV). Preliminary data concerning reliability and validity[END_REF][START_REF] Reading | The McGill Pain Questionnaire: a replication of its construction[END_REF][START_REF] Ketovuori | A pain vocabulary in Finnish-The Finnish pain questionnaire[END_REF]100). It was used as a reference in 5 articles [START_REF] Kitisomprayoonkul | Thai Short-form McGill Pain Questionnaire[END_REF][START_REF] Perkins | Development and validation of a brief, descriptive Danish pain questionnaire (BDDPQ)[END_REF][START_REF] Ferrari | Internal structure and validity of the multidimensional pain inventory, Italian language version[END_REF][START_REF] Kerns | Self-monitored pain intensity: psychometric properties and clinical utility[END_REF][START_REF] Kerns | The West Haven-Yale Multidimensional Pain Inventory (WHYMPI)[END_REF]. This subscale was studied in 6 articles in this current review [START_REF] Maruo | Validity, reliability, and assessment sensitivity of the Japanese version of the short-form McGill pain questionnaire 2 in Japanese patients with neuropathic and non-neuropathic pain[END_REF][START_REF] Erdinc | Turkish version of the Cornell Musculoskeletal Discomfort Questionnaire: cross-cultural adaptation and validation[END_REF][START_REF] Lorenz | Exploring alternative approaches to routine outpatient pain screening[END_REF][START_REF] Jakobsson | Psychometric testing of the brief screening version of Multidimensional Pain Inventory (Swedish version)[END_REF][START_REF] Gallagher | Prospective validation of clinically important changes in pain severity measured on a visual analog scale[END_REF][START_REF] Elliott | Changes in chronic pain severity over time: the Chronic Pain Grade as a valid measure[END_REF]. It was used as a reference in 4 articles [START_REF] Lorenz | How reliable is pain as the fifth vital sign?[END_REF][START_REF] Gentile | Reliability and validity of the Global Pain Scale with chronic pain sufferers[END_REF][START_REF] Varoli | Adapted version of the McGill Pain Questionnaire to Brazilian Portuguese[END_REF][START_REF] Holdgate | Comparison of a verbal numeric rating scale with the visual analogue scale for the measurement of acute pain[END_REF].

The Short-Form McGill Pain Questionnaire 2 (SF McGPQ 2) was created later, with 7 additional items to measure both neuropathic and non-neuropathic pain, so 22 items that patients rate on a 0-10 NRS. This subscale was studied in 2 articles in this current review [START_REF] Maruo | Validity, reliability, and assessment sensitivity of the Japanese version of the short-form McGill pain questionnaire 2 in Japanese patients with neuropathic and non-neuropathic pain[END_REF][START_REF] Ellis | Establishing the psychometric properties and preferences for the Northern Pain Scale[END_REF]. It was not used as a reference.

d. Summary of multidimensional scales

A summary of different parameters of multidimensional scales is available on table 4.

DISCUSSION

This study was prompted by the need of establish an inventory of different pain scales studied in general population, for the daily exercise of general practitioners. Indeed, very few literature reviews are available about this topic.

35 scales have been found in this current review. They have been classified in two different groups: one-dimensional scales, with 6 main scales and 13 derived scales, and multidimensional scales with 12 main scales and 4 derived scales. There is also a bias with the quality of articles because they are mostly cross-sectional studies. Only studies written in English were included, limiting the assessment of non-Anglo-Saxon scales.

Number of time quote in an article

Number of time used as a reference

Number of item

Finally, no psychometric values were collected, as it will be the aim of a further study.

CONCLUSION

Unidimensional scales are used to evaluate only one parameter, most of the time pain intensity. But many other symptoms such as depression or anxiety are assessed by these scales, with a risk of over simplification of pain affects. Mood, factor influencing pain, daily activities etc, are primordial dimensions in a holistic approach to pain assessment. On this principle, onedimensional scales seem to be less useful. Moreover, evaluating pain with numbers is it still justified? Can we quote everything with numbers? Developed primarily for clinical research, these scales are often misused by caregivers.

Multidimensional scales seem to be more appropriate to a complete analysis of pain. They have the disadvantage of being long to achieve and they are not much used in daily practice. To have a better efficiency, pain scales shouldn't take too much time, number of items must be adequate and explored dimensions must be relevant.

Thereby, in this current review, the BPI appears to be the most appropriate scale. A recent systematic review [START_REF] Payelle | Reliability of the pain assessment scales available in the general population: a systematic review[END_REF] tried to collect reliability data of pain scales available in general population and seemed to correlate this idea.

Finally, this study is incorporated into a larger project to determine the most appropriate pain rating scale for daily exercise in general practice. Psychometric validation can lead to answers about this. A field application will then be needed to confirm this work. Because it is often forgotten in clinical trials, patients 'point of view will be eventually essential.

ABBREVIATIONS

  Présentée par Monsieur le Professeur Jean-Yves LE RESTE Titre de la thèse: What pain assessment scales are usable in general population ? A systematic literature review of Cochrane library according to the PRISMA statement ACCORD DU PRESIDENT DU JURY DE THESE SUR L'IMPRESSION DE LA

  ABSTRACT ..................................................................................................................... RESUME ......................................................................................................................... INTRODUCTION .............................................................................................................. METHOD ........................................................................................................................ General framework ................................................................................................................... Inclusion and exclusion Criteria ................................................................................................. a. Inclusion Criteria .................................................................................................................... b. Exclusion criteria ................................................................................................................... Information sources .................................................................................................................. Search ........................................................................................................................................ Study selection .......................................................................................................................... Data collection process and data item ...................................................................................... Risk of bias in individual studies ................................................................................................ RESULTS ........................................................................................................................ Study selection .......................................................................................................................... Synthesis of results .................................................................................................................... a. One-dimensional pain scales ................................................................................................. b. Summary of one-dimensionnal scales ................................................................................... c. Multidimensional scales ........................................................................................................ d. Summary of multidimensional scales .................................................................................... RESUME Introduction : La douleur est un des symptômes les plus courants signalé en médecine générale. Dans ce but, des échelles uni et multidimensionnelles de douleurs ont été développées pour évaluer la douleur aigüe ou chronique. Il n'y a pas de gold standard dans l'évaluation de la douleur en médecine générale. Cette étude avait pour but de lister toutes les échelles d'évaluation de la douleur étudiées en population générale. Méthode : Une revue de littérature questionnant la base de données Cochrane a été réalisée jusqu'en février 2018, basée sur le protocole PRISMA. Seuls les articles anglais évaluant les échelles de douleurs en population générale étaient inclus. La bibliographie des articles sélectionnés et de récentes revues de littératures ont été examinées pour inclure des articles supplémentaires dans une seconde recherche. Résultats : Sur 1431 résultats trouvés sur Cochrane, 4 articles pertinents étaient inclus. Avec la deuxième recherche, 203 articles supplémentaires étaient détectés et 84 ont été ajoutés. Finalement, 88 articles ont été sélectionnés et 35 échelles de douleurs étaient trouvées : 6 échelles unidimensionnelles avec 13 variantes et 12 multidimensionnelles avec 4 variantes. Discussion : Les échelles d'évaluation de douleur étaient différentes par les dimensions explorées et le nombre d'items. Certaines d'entre elles n'ont pas été étudiées dans cette revue par un manque d'articles répondant aux critères d'éligibilité, disponibles sur la base de données. Les données de validité des échelles d'évaluation de douleur disponibles ont récemment été collectées dans une revue systématique. Il sera nécessaire de tenir compte des données de validité externe dans d'autres études et de choisir quelle échelle d'évaluation de douleurs est la plus appropriée pour une pratique quotidienne.

-

  About studies Studies about laboratory tests; experimental researches; therapeutic evaluations ; tests evaluations ; evaluation of health care providers' knowledge ; studies about practices and pain belief, research strategies, algorithms, cost analyzes, statistics model evaluation, studies about pain risk factors, predictive factors of therapeutic efficacy or of diseases/diagnosis were excluded. INFORMATION SOURCES Studies were identified by questioning the Cochrane library database. A previous study questioned Medline database with the same purpose. Only English studies were included and there was no restriction with the writing date. The systematic review was conducted between December 2017 and February 2018. The bibliography of included studies and of recent systematic reviews had been screened to include additional articles. All the articles found were include in an excel file (Appendix 3). SEARCH The first step was to create a search equation. This study is meant to be an update of the previous study based on the Medline literature review. A librarian from university of Brest in collaboration with a librarian from university of Paris Descartes helped to create the final search equation. The keyword "general population" was not included because of its lack of use at international. First, the « pain measurement » [MeSH] (Medical Subject Headings) term restricted to « Instrumentation, Methods, statistics and numerical data » gave 2243 results. To have a finest selection, an association of psychometrics keywords was added: (MeSH descriptor: [Reproducibility of Results] or MeSH descriptor: [Sensitivity and Specificity] or MeSH descriptor: [Predictive Value of Tests] or MeSH descriptor: [Psychometrics] or Youden or Cronbach or Cohen ). A new keywords combination was obtained using the « Constructeur de Requête Bibliographiques Médicales du Cismef » which corresponds to the whole words included in the [MeSH] term "Pain measurement". By adding these different combination, it gave 1121 results with the EQUATION 1 (APPENDIX 1). After a quick screening, only 4 articles could be included. By simplifying the equation constructed with the « Constructeur de Requête Bibliographiques Médicales du Cismef », 675 articles were found, giving EQUATION N°2. An other quick reading showed that only 4 articles could be included. In front of too many off topics, the search equation (EQUATION N°3) was reworked with a librarian of the university of Paris Descartes. It gave finally 1431 results with only 4 articles which could be included. STUDY SELECTION All articles obtained from the search equation were included in an excel spreadsheet (Appendix 3). Collected informations helped to create this spreadsheet: title; author; year of publication, abstract available or not. For each article, only titles and abstracts were read. Off Topics, thereby selected, were classified into different categories (Appendix 2). Articles which was not written in English, and in IMRAD format, were excluded. From there; if the subject answered the question research; inclusion and exclusion criteria were applied. DATA COLLECTION PROCESS AND DATA ITEM All included articles were fully read. Scales were divided in two categories: one-dimensional scales and multidimensional scales. For each one-dimensional scale, following parameters were exposed: number of time quote in an article, number of time used as a reference, number of derived scales, the explored dimension, is a graphic support needed and the type of evaluation used.

  Mainz pain staging system as a tool for pain-patients referral selection.(19) Samolsky Dekel BG, Gori A, Vasarri A, Adversi M, Di Nino G, Melotti RM. J Eval Clin Pract Validation of the Japanese version of the pain self-efficacy questionnaire in Japanese patients with chronic pain.(20) Adachi T, Nakae A, Maruo T, Shi K, Shibata M, Maeda L, Saitoh Y, Sasaki J. Pain Med One, two, or three? Constructs of the brief pain inventory among patients with non-cancer pain in the outpatient setting.Acute pain evaluation with Algoplus scale in Cambodian patients.(23) Pickering ME, Bunna P, Rat P, Madeline G, Lebost C, Serrie A, Pereira B. Pain Med The reliability of body pain diagrams in the quantitative measurement of pain distribution and location in patients with musculoskeletal pain: a systematic review.P, Sehn F, Vidor LP, Dall-Ágnol L, Chassot M, Torres IL. Pain Med Prevalence of chronic pain among Libyan adults in Derna City: a pilot study to assess the reliability, linguistic validity, and feasibility of using an Arabic version of the structured telephone interviews questionnaire on chronic pain.

Five

  variants (considered as variants of the Visual Analog Scale because they need graphic support and evaluate only pain intensity) were found in this current review: the full cup test[START_REF] Ergün | Trial of a new pain assessment tool in patients with low education: the full cup test[END_REF] (developed to patients with low education), the Visual Numeric scale (49) (which combines strong visual cues with an 11-point numeric rating scale), the Horizontal Numerical Graphic Rating Scale (54), the Dynamic VAS (computerized version of VAS)(57) and the 11

Figure 2 :

 2 Figure 2: Visual Analog Pain Scale

Figure 3 :

 3 Figure 3: Numeric Pain Rating Scale

  was used as a reference in 3 articles (25,71,72) Four variants were found: the Dynamic Verbal Ratings (computerized version)(57), the Self-Monitored Pain Intensity(90), the 4-points Verbal Rating Scale(94) and the 5-points Verbal Rating Scale(94). The Self-Monitored Pain Intensity is a tool which allowed to patients to record their perceived pain intensity on an hourly basis for 2 weeks, with a 6-points verbal scale like Present Pain Intensity part of the McGill Pain Questionnaire (0 = No pain, 1 = mild, 2 = discomforting, 3 = distressing, 4 = horrible, 5 = excruciating).

Figure 4 :

 4 Figure 4: Verbal Rating Scale

Figure 5 :

 5 Figure 5: Wong and Baker Faces Pain Scale (102)

Figure 6 :

 6 Figure 6: Pain Drawing

5 .

 5 Cornell Musculoskeletal Discomfort Questionnaire for standing worker (CMDQ fsw) This 4-item pain scale evaluates 4 pain dimensions with a VRS (the frequency in the past work week, the severity, work interferences and the localization) linked to a pictorial back body, different for male and female, composed by 20 body parts.

5 .

 5 One variant of the WHYMPI was found in this current review: The Brief screening version of the Multidimensional Pain Inventory (BSv MPI)[START_REF] Jakobsson | Psychometric testing of the brief screening version of Multidimensional Pain Inventory (Swedish version)[END_REF]. It includes 34 of the 52 items and explores the same pain dimensions.The Mainz Pain Staging System (MPSS)Initially developed in the 2000s in Germany, the Mainz Pain Staging System is a 10-item multidimensional tool evaluating 4 axes: pain persistence (items 1-3), spatial aspects (item 4), drug intake and withdrawal treatments (items 5 and 6) and health care system utilization (items 7-10). The Resulting score classifies patients at three chronicity stages: I: Mild; II: Moderate; III: Severe Chronic Pain. See Table15on APPENDIX 5.

  The McGill Pain Questionnaire (McGPQ) The McGill Pain Questionnaire was created in 1975. Classically described as a 78-item tool , this multidimensional pain scale is separated into four parts: Pain Rating Index (PRI) containing 20 subclasses, each containing 2-6 words finally grouped in 4 pain dimensions (sensory, affective, evaluative, and a miscellaneous category); a 5-point pain intensity scale (present pain intensity [PPI]); a pictoral body to locate the pain and a 3-point verbal rating scale where patients choose the word corresponding to the pattern of their pain. See Table 16 on APPENDIX 5.

  The Short-Form McGill Pain Questionnaire (SF-McGPQ) is the main variant of the McGPQ. It was created in 1987 as a rapid evaluation tool and includes 15 pain descriptors (11 sensory and 4 affective), a VAS and a present pale intensity scale. See Table 17 on APPENDIX 5.

  ABBREVIATIONS LIST

A tous les copains du rugby,

  Toto, Nico (enfin Rouennais), Tang, Xa, Migou, Seser, Scho, Po, Cha et Cha, Nico et Marie, Victor et Mathilde, Mam's, Elise et Marie, Milou, Audrey, Bapt, Ju, Ju Rondy ; merci pour toutes ces années passées en votre compagnie. #SMC#malins#dubliners#GDB Vince, Vinzi, Nico, Mickick, Mémé, Momo, Malo, Augustin va marquer, Antoine et tous les autres, j'aime à penser à cette finesse qui vous sied bien, à cette délicatesse que vous respirez. Vous êtes tous extraordinnaires. Je vous aime. Allez Plab A mes colocataires rennais, Lena, Marina, Antonio et Stephane, merci de m'avoir intégré si facilement. J'ai hate de voir ce que la suite nous réserve !

TABLE OF CONTENT

 OF

20 In Table 1 ,

 In1 are listed the 88 included studies. APPENDIX 2 exhibits different off topics.

	Title	Author	Year	Revue
	Comparative Responsiveness of the PROMIS	Kean J, Monahan PO,		Medical care
	Pain Interference Short Forms, Brief Pain	Kroenke K, Wu J, Yu Z,
	Inventory, PEG, and SF-36 Bodily Pain	Stump TE, Krebs EE
	Subscale(14)
	Establishing a common
	Figure 1: Flow Diagram
		21

metric for self-reported pain: linking BPI Pain Interference and SF-36 Bodily Pain Subscale scores to the PROMIS Pain Interference metric(15)

		Cook KF, Schalet BD,	Qual Life
		Kallen MA, Rutsohn JP,	Res
		Cella D.
	Associations between pain drawing and	Hayashi K, Arai YC,	Pain Pract
	psychological characteristics of different body	Morimoto A, Aono S,
	region pains(16)	Yoshimoto T, Nishihara
		M, Osuga T, Inoue S,
		Ushida T.
	Validity, reliability,

and assessment sensitivity of the Japanese version of the short-form McGill pain questionnaire 2 in Japanese patients with neuropathic and non-neuropathic pain(17)

		Maruo T, Nakae A,	Pain Med
		Maeda L, Shi K,
		Takahashi K, Morris S,
		Hosomi K, Kanatani H,
		Matsuzaki T, Saitoh Y.
	Using multiple daily pain ratings to improve	Heapy A, Dziura J, Buta	The Journal
	reliability and assay sensitivity: how many is	E, Goulet J, Kulas JF,	Of Pain
	enough?(18)	Kerns RD.

verbal numeric rating scale with the visual analogue scale for the measurement of acute pain(63)

		Spadoni GF, Stratford	J Orthop
		PW, Solomon PE,	Sports Phys
		Wishart LR.	Ther
	Development and validation of a brief,	Perkins FM1, Werner	Acta
	descriptive Danish pain questionnaire	MU, Persson F, Holte K,	Anaesthesiol
	(BDDPQ)(59)	Jensen TS, Kehlet H.	Scand
	Validation of the Brief Pain Inventory for	Tan G, Jensen MP,	J Pain
	chronic nonmalignant pain(60)	Thornby JI, Shanti BF.
	Minimum clinically significant VAS differences	Yamamoto LG, Nomura	American
	for simultaneous (paired) interval serial pain	JT, Sato RL, Ahern RM,	journal of
	assessments(61)	Snow JL, Kuwaye TT	emergency
			medicine
	Signaling does not adequately improve diary	Broderick JE, Schwartz	Ann Behav
	compliance.(62)	JE, Shiffman S, Hufford	Med
		MR, Stone AA.
	Comparison of a Holdgate A1, Asha S,	Emerg Med
		Craig J, Thompson J.	(Fremantle)

Cross-cultural adaptation of the Nordic musculoskeletal questionnaire(64)

			Pain Res
			Manag
	Changes in chronic pain severity over time: the	Elliott AM1, Smith BH,	Pain
	Chronic Pain Grade as a valid measure (73)	Smith WC, Chambers
		WA.
	Concordance between ratings using different	Svensson E1.	Stat Med
	scales for the same variable(74)
	Internal	De Barros EN,	Int Nurs Rev
		Alexandre NM.
	Validation of a verbally administered numerical	Bijur PE, Latimer CT,	Acad Emerg
	rating scale of acute pain for use in the	Gallagher EJ.	Med
	emergency department(65)
	Clinical importance of changes in chronic pain	Farrar JT, Young JP Jr,	Pain
	intensity measured on an 11-point numerical	LaMoreaux L, Werth JL,
	pain rating scale(66)	Poole RM.
	Reliability of the visual analog scale for	Bijur PE, Silver W,	Acad Emerg
	measurement of acute pain.(67)	Gallagher EJ.	Med

structure and validity of the multidimensional pain inventory, Italian language version(75)

	Comparative study of methods of measuring	Berthier F, Potel G,	Am J Emerg
	acute pain intensity in an ED.(80)	Leconte P, Touze MD,	Med
		Baron D.
	Does the clinically significant difference in visual	Kelly AM.	Nov Acad
	analog scale pain scores vary with gender, age,		Emerg Med
	or cause of pain? (81)
	Reliability and factor structure of the	Bergström G1, Jensen	Pain
	Multidimensional Pain Inventory--Swedish	IB, Bodin L, Linton SJ,
	Language Version (MPI-S)(82)	Nygren AL, Carlsson
		SG.
	The Chronic Pain Grade questionnaire:	Smith BH, Penny KI,	Pain
	validation and reliability in postal research(83)	Purves AM, Munro C,
		Wilson B, Grimshaw J,
		Chambers WA, Smith
		WC.
	The minimum clinically important difference in	Todd KH, Funk JP.	Acad Emerg
	physician-assigned visual analog pain scores.(84)		Med
	Confirmatory factor analysis of a 4-factor model	De Gagné TA, Mikail	Pain
	of chronic pain evaluation.(85)	SF, D'Eon JL.
	Increasing the reliability and validity of pain	Jensen MP, McFarland	Pain
	intensity measurement in chronic pain	CA.
	patients(86)
	Grading the severity of chronic pain(87)	Von Korff M, Ormel J,	Pain
		Keefe FJ, Dworkin SF.
		Ferrari R1, Novara C,	Pain Med
		Sanavio E, Zerbini F.
	Simple pain rating scales hide complex	de C Williams AC1,	Pain
	idiosyncratic meanings(76)	Davies HT, Chadury Y.
	Some empirical evidence regarding the validity	Masedo AI, Esteve R.	Apr Pain
	of the Spanish version of the McGill Pain
	Questionnaire (MPQ-SV)(77)
	Comparative reliability and validity of chronic	Jensen MP, Turner JA,	Pain
	pain intensity measures.(78)	Romano JM, Fisher LD.
	Psychometric properties of the Multidimensional	Lousberg R1, Van	Behav Res
	Pain Inventory, Dutch language version (MPI-	Breukelen GJ,	Ther
	DLV)(79)	Groenman NH, Schmidt
		AJ, Arntz A, Winter FA.

Questionnaire development: an examination of the Nordic Musculoskeletal questionnaire(88)

	The McGill Pain Questionnaire constructed for	Vanderiet K, Adriaensen	Pain
	the Dutch language (MPQ-DV). Preliminary	H, Carton H,
	data concerning reliability and validity(93)	Vertommen H.
	The measurement of clinical pain intensity: a	Jensen MP, Karoly P,	Pain
	comparison of six methods (94)	Braver S.
	The West Haven-Yale Multidimensional Pain	Kerns RD, Turk DC,	Pain
	Inventory (WHYMPI)(95)	Rudy TE.
	The McGill Pain Questionnaire: a replication of	Reading AE, Everitt BS,	Br J Clin
	its construction.(96)	Sledmere CM.	Psychol
	A pain vocabulary in Finnish-The Finnish pain	Ketovuori H, Pöntinen	Pain
	questionnaire(97)	PJ.
	Reproducibility along a 10 cm vertical visual	Dixon JS, Bird HA.	Ann Rheum
	analogue scale(98)		Dis
	Graphic representation of pain(99)	Scott J, Huskisson EC.	Pain
	The McGill Pain Questionnaire: major	Melzack R.	Pain
	properties and scoring methods(100)
		Dickinson CE, Campion	Appl Ergon
		K, Foster AF, Newman
		SJ, O'Rourke AM,
		Thomas PG.
	The development of a Norwegian pain	Strand LI, Wisnes AR.	Pain
	questionnaire(89)
	Self-monitored pain intensity: psychometric	Kerns RD, Finn P,	Behav Med
	properties and clinical utility(90)	Haythornthwaite J.
	Test-retest reliability of the pain drawing	Margolis RB, Chibnall	Pain
	instrument(91)	JT, Tait RC.
	The German counterpart to McGill Pain	Stein C, Mendl G.	Pain
	Questionnaire(92)

Table 1 : Included articles Type of the 88 studies

 1

	Cross	Randomized	Cohort	Translation Systematic	Construction Meta-analysis
	Sectional	placebo controlled			Review
		paired trial
	55	1		6	15	3		7	1
				Aim of the 88 studies
	Minimum	Clinical	Reliability	Comparison	Daily	Translation	Translation	Construction Construction
	clinically	importance			Assessment			+	+ Reliability
	significant	of changes						Reliability
	differences
	1	5	30	25	1	2		16	5	3

Table 2 :

 2 Type and aim of studies 29

Table 3 :

 3 Algoplus[START_REF] Belhadj | What pain scales assessment are usable in general population ? A systematic literature review of Medline according to the PRISMA statement[END_REF]

Table 4 :

 4 Summary of one dimensional scales parameters c. Multidimensional scalesMultidimensional scales are useful to evaluate chronic pain severity. Composed by several items, they evaluate pain intensity but also pain interferences, disabilities etc.… Short Form-36 Bodily Pain Subscale (SF-36 BPS)

	type of
	evaluation used

Table 4 :

 4 Summary of Multidimensional scales parameters (BPI:Brief Pain Inventory, BSy MPI: Brief screening version of the Multidimensional Pain Inventory, CMDQ fsw: Cornell Musculoskeletal Discomfort Questionnaire for standing worker, CPG: Chronic Pain Grade, Ex NMQ: Extended version of the Nordic Musculoskeletal Questionnaire, GPS: Global Pain Scale, McGPQ: McGill Pain Questionnaire, MPSS: The Mainz pain scales. It didn't allow us to conclude on the best pain scale, for sake of completeness. The second part of the search allowed to include additional original articles. Therefore, many pain scales are missing in this current review. If the 2017 study interrogating the Medline database(11) is excluded, only two other literature reviews were found(106,107), making a difficult comparison. Moreover, there is selection bias. Only one investigator did inclusions of studies despite discussions about eligibility criteria in a research group leading by training professor. The quality assessment was also done by one investigator, with the "Exercer" grid which is not used in the Prisma Statement.

Pain Staging System, NMQ: Nordic Musculoskeletal Questionnaire, NRS: Numerical Rating Scale, PCP: S: Profile of Chronic Pain: Screen, SF-36 BPS: Short-Form-36 Bodily Pain Subscale, SF-McGPQ: Short-Form McGill Pain Questionnaire, SF-McGPQ 2: Short-Forme McGill Pain Questionnaire 2, STIQ on CP: Structured Telephone Interviews Questionnaire on Chronic Pain, UB-3-PEG: Ultra-Brief 3-item PEG, VAS: Visual Analog Scale, VRS: Verbal Rating Scale, WHYMPI: West Haven Yale Multidimensional Pain Inventory.

8

Table 13: West Have Yale Multidimensional Pain Inventory[START_REF] Laliberté | French translation of the Multidimensional Pain Inventory: L'inventaire multidimensionnel de la douleur[END_REF]

Table 14: Mainz Pain Staging System[START_REF] Belhadj | What pain scales assessment are usable in general population ? A systematic literature review of Medline according to the PRISMA statement[END_REF]

Table 15: McGill Pain Questionnaire[START_REF] Belhadj | What pain scales assessment are usable in general population ? A systematic literature review of Medline according to the PRISMA statement[END_REF]

Table 16: Short-Form McGill Pain Questionnaire[START_REF] Belhadj | What pain scales assessment are usable in general population ? A systematic literature review of Medline according to the PRISMA statement[END_REF]

SERMENT D'HIPPOCRATE

« Au moment d'être admis à exercer la médecine, je promets et je jure d'être fidèle aux lois de l'honneur et de la probité.

Mon premier souci sera de rétablir, de préserver ou de promouvoir la santé dans tous ses éléments, physiques et mentaux, individuels et sociaux.

Je respecterai toutes les personnes, leur autonomie et leur volonté, sans aucune discrimination selon leur état ou leurs convictions. J'interviendrai pour les protéger si elles sont affaiblies, vulnérables ou menacées dans leur intégrité ou leur dignité. Même sous la contrainte, je ne ferai pas usage de mes connaissances contre les lois de l'humanité. J'informerai les patients des décisions envisagées, de leurs raisons et de leurs conséquences.

Je ne tromperai jamais leur confiance et n'exploiterai pas le pouvoir hérité des circonstances pour forcer les consciences.

Je donnerai mes soins à l'indigent et à quiconque me le demandera. Je ne me laisserai pas influencer par la soif du gain ou la recherche de la gloire.

Admis dans l'intimité des personnes, je tairai les secrets qui me seront confiés. Reçu à l'intérieur des maisons, je respecterai les secrets des foyers et ma conduite ne servira pas à corrompre les moeurs.

Je ferai tout pour soulager les souffrances. Je ne prolongerai pas abusivement les agonies. Je ne provoquerai jamais la mort délibérément.

Je préserverai l'indépendance nécessaire à l'accomplissement de ma mission. Je n'entreprendrai rien qui dépasse mes compétences. Je les entretiendrai et les perfectionnerai pour assurer au mieux les services qui me seront demandés. J'apporterai mon aide à mes consoeurs et confrères ainsi qu'à leurs familles dans l'adversité.

Que les hommes et mes confrères m'accordent leur estime si je suis fidèle à mes promesses ; que je sois déshonorée et méprisée si j'y manque. »

This scale was studied in 5 articles (34,37,73,83,87) in this current review. It was used once as a reference (42). See Table 8 on APPENDIX 5.

Profile of Chronic Pain: Screen (PCP: S) This 15-items multidimensional pain scale, created in 2005, comprised a 4-items severity scale, a 6-items interference scale and a 5-items emotional burden scale. This scale was studied in 3 articles (26,48,56). It was not used as a reference. See Table 9 on APPENDIX 5.

Structured Telephone Interviews Questionnaire on Chronic Pain (STIQ on CP)

This multidimensional scale, created in England in 2006(105), is a telephone questionnaire, composed of two parts: a screening interview consisting of 12 questions to gather data on age and gender, duration of pain, frequency of pain during the previous week, intensity of pain during last pain episode, location of pain and cause of pain; and 44 questions to gather information about demographics, frequency, duration, intensity and impact of pain on work and quality of life. In Fact, it gathers 90 items divided into 5 groups: assessment of inclusion criteria, evaluation of the global pain context, pain evaluation, assessment about pain interference and, finally, demographic aspect. Verbal scales, numerical scales, yes/no answer, free answers and lists with multiple choices are used in this pain scale to assess many pain dimensions. This scale was studied in only one article in this current review (27). It was not used as a reference. See Table 10 on APPENDIX 5.

Global Pain Scale (GPS)

This 33-items pain scale, created in 2005, assesses 4 pain dimensions with NRS: physical pain (pain intensity), affective effects of pain, specific clinical outcomes (effect of pain on

In the one-dimensional group, scales were classified by type and subscales. VAS and NRPS were the most studied pain scales. Whether they need or not a graphic support, their principal advantage is to be easy to use. They are also quick to realize. These 2 scales are among the most used scales and often employed to compare other pain scales, as a reference. They mainly analysed one pain dimension: pain intensity. Three pain scales used a graphic support: Pain Drawings, the Wong and Baker Faces Pain Scale and the VAS. This dimension is crucial to locate patients' pain but give few information about pain intensity. In this systematic review, a lot of scales evaluating specific populations (children, pregnant women, patients with cancer…) were excluded respecting eligibility criteria to be as general as possible.

There are limitations that should be considered in interpreting these results even if the choice to use the PRISMA method allowed reducing the bias.

First, studies non-published have not been searched. The search was restricted to titles and abstracts of articles because of too many off-topics.

In the first part of the search only Cochrane database was studied. Considered as one of the main medical database with Medline, it's not, however, an exhaustive database. In this current review, despite 1500 screened articles, only 4 could be included with Cochrane database, giving [START_REF] Erdinc | Turkish version of the Cornell Musculoskeletal Discomfort Questionnaire: cross-cultural adaptation and validation[END_REF] COCHET (Laurent) -What pain scales assessment are usable in general population? A systematic literature review of Cochrane library according to the PRISMA statement Th. : Méd. : Brest 2018

APPENDIX 5: MULTIDIMENSIONAL SCALES TABLES

ABSTRACT:

Background: Pain is one of the most common symptom reported in general practice. For this purpose, one-dimensional and multidimensional pain scales have been developed to assess acute or chronic pain. There is no gold standard in pain assessment in general practice. This study aimed to list all pain scales studied for the general population.

Method: A systematic review questioning Cochrane database, according to the PRISMA protocol, was conducted until February 2018. Only English articles assessing pain scales in general population were included. The bibliography of selected studies and recent systematics reviews had been examined to include additional articles in a second search.

Results: On 1431 results found on Cochrane, 4 relevant articles were included. With the second search, 203 additional articles were detected and 84 were added. Finally, 88 articles were selected and 35 pain scales were found: 6 unidimensional-scales with 13 variants and 12 multidimensional scales with 4 variants.

Discussion: Pain scales were different by explored dimensions and the number of item. Some of them are not studied in this current review by a lack of articles available on the database responding to eligibility criteria. Reliability data of available pain assessment scales were recently collected in a systematic review. It will be necessary to consider external validity data in further studies and to choose which pain scale is the most appropriate for a daily practice.

KEYWORDS:

PAIN MEASUREMENT ;


Bérangère Delourme
email: delourme@math.univ-paris13.fr

David P Hewett
email: d.hewett@ucl.ac.uk

Electromagnetic shielding by thin periodic structures and the Faraday cage effect

In this note we consider the scattering of electromagnetic waves (governed by the time-harmonic Maxwell equations) by a thin periodic layer of perfectly conducting obstacles. The size of the obstacles and the distance between neighbouring obstacles are of the same small order of magnitude δ. By deriving homogenized interface conditions for three model configurations, namely (i) discrete obstacles, (ii) parallel wires, (iii) a wire mesh, we show that the limiting behaviour as δ → 0 depends strongly on the topology of the periodic layer, with full shielding (the so-called "Faraday cage effect") occurring only in the case of a wire mesh.

Introduction

The ability of wire meshes to block electromagnetic waves (the celebrated "Faraday cage" effect) is well known to physicists and engineers. Experimental investigations into the phenomenon date back over 180 years to the pioneering work of Faraday [START_REF] Faraday | Experimental researches in electricity[END_REF], and the effect is routinely used to block or contain electromagnetic fields in countless practical applications. (An everyday example is the wire mesh in the door of a domestic microwave oven, which stops microwaves escaping, while letting shorter wavelength visible light pass through it.) But, somewhat remarkably, a rigorous mathematical analysis of the effect does not appear to be available in the literature.

The mathematical richness of the Faraday cage effect was highlighted in an recent article by one of the authors [START_REF] Chapman | Mathematics of the Faraday cage[END_REF], where a number of different mathematical approaches were applied to the 2D electrostatic version of the problem. In particular it was shown in [START_REF] Chapman | Mathematics of the Faraday cage[END_REF] how modern techniques of homogenization and matched asymptotic expansions could be used to derive effective interface conditions that accurately capture the shielding effect. These results were generalised to the 2D electromagnetic case (TE-and TM polarizations) in [START_REF] Hewett | Homogenized boundary conditions and resonance effects in Faraday cages[END_REF], and related approximations for similar problems have also been studied recently by other authors, e.g. [START_REF] Holloway | A homogenization technique for obtaining generalized sheet transition conditions for an arbitrarily shaped coated wire grating[END_REF][START_REF] Marigo | Two-scale homogenization to determine effective parameters of thin metallic-structured films[END_REF]. However, as far as we are aware, an analysis of the full 3D electromagnetic version of the problem with perfectly conducting scatterers (the focus of the current note) has not previously been performed. (We note that related approximations have been presented for thin layers of dielectric obstacles in [START_REF] Delourme | On the well-posedness, stability and accuracy of an asymptotic model for thin periodic interfaces in electromagnetic scattering problems[END_REF][START_REF] Delourme | High-order asymptotics for the electromagnetic scattering by thin periodic layers[END_REF].)

In this note we consider full 3D electromagnetic scattering by a thin periodic layer of small, perfectly conducting obstacles. We derive leading-order homogenized interface conditions for three model configurations, namely where the periodic layer comprises (i) discrete obstacles, (ii) parallel wires, and (iii) a wire mesh. Our results verify that the effective behaviour depends strongly on the topology of the periodic layer, with shielding of arbitrarily polarized waves occurring only in the case of a wire mesh. We note that analogous observations have been made in the related setting of volume homogenization in [START_REF] Schweizer | Effective Maxwell's equations in general periodic microstructures[END_REF].

Our analysis assumes that the obstacles/wires making up the thin periodic layer are of approximately the same size/thickness as the separation between them. The case of very small obstacles/thin wires is expected to produce different interface conditions, analogous to those derived in [START_REF] Chapman | Mathematics of the Faraday cage[END_REF][START_REF] Hewett | Homogenized boundary conditions and resonance effects in Faraday cages[END_REF] in the 2D case. But we leave this case for future work.

Statement of the problem

Our objective is to derive effective interface conditions for electromagnetic scattering by a thin periodic layer of equispaced perfectly-conducting obstacles on the interface Γ = {x = (x 1 , x 2 , x 3 ) ∈ R 3 : x 3 = 0}. Let Ω ∈ R 3 be the canonical obstacle described by one of the following three cases (see Fig. 1):

1. Ω is a simply connected Lipschitz domain whose closure is contained in (0, 1) 2 × (-1 2 , 1 2 ). 2. Ω = [0, 1] × ( 3 8 , 5 8 ) × (-1 8 , 1 8
), i.e. a wire (of square section) parallel to the direction e 1 .

1 (a) Case (i) 1 (b) Case (ii) 1 (c) Case (iii)
Figure 1: The canonical obstacle Ω in the three cases under consideration.

3. Ω = {[0, 1] × ( 3 8 , 5 8 ) × (-1 8 , 1 8 )} ∪ {( 3 8 , 5 8 ) × [0, 1] × (-1 8 , 1 8
)}, i.e. a cross-shape domain made of the union of two perpendicular wires (one parallel to e 1 and the other parallel to e 2 ). We construct the thin layer as a union of scaled and shifted versions of the canonical obstacle Ω. For δ > 0 we define On the domain Ω δ we consider the solution u δ of the Maxwell equations

L δ ⊂ R 2 × [-δ/2, δ/2] by L δ = int   (i,j)∈Z 2 δ Ω + ie 1 + je 2   . Our domain of interest is then Ω δ = R 3 \ L δ (cf.
curl curl u δ -ω 2 εu δ = f in Ω δ , (1)
where ω > 0 and ε ∈ C, subject to the perfectly conducting boundary condition

u δ × n = 0 on Γ δ . (2)
For analytical convenience we avoid any complications arising from far-field behaviour by assuming that Re[ε] > 0 and Im[ε] > 0. The assumption that Im[ε] > 0 could be relaxed to Im[ε] ≥ 0 at the expense of some technical modifications, including the imposition of an appropriate radiation condition. We also assume that the support of f does not intersect the interface Γ. Then, given

f ∈ L 2 (Ω δ ) 3 , the Lax-Milgram Lemma ensures that Problem (1)-(2)
has a unique solution u δ in the standard function space

H(curl; Ω δ ) = v ∈ (L 2 (Ω δ )) 3 : curl v ∈ (L 2 (Ω δ )) 3 , (3)
equipped with the usual graph norm v H(curl

;Ω δ ) = v 2 (L 2 (Ω δ )) 3 + curl v 2 (L 2 (Ω δ )) 3
1/2 . Moreover, one can prove that there exists C > 0, independent of δ, such that

u δ H(curl;Ω δ ) ≤ C f (L 2 (Ω δ )) 3 , for all 0 < δ < 1. ( 4
)
The objective of this work is to identify formally the limit u 0 of u δ as δ tends to 0. This limit solution is defined in the union of two distinct domains Ω ± = {x ∈ R 3 : ±x 3 > 0}, whose common interface is Γ. Our main result is the following: Theorem 2.1 The limit solution u 0 satisfies the Maxwell equations

curl curl u 0 -ω 2 εu 0 = f in Ω + ∪ Ω -, (5)
together with the following interface conditions on Γ:

Case (i): [u 0 × e 3 ] Γ = 0 and [curl u 0 × e 3 ] Γ = 0. Case (ii): u 0 • e 1 = 0 on Γ, [u 0 • e 2 ] Γ = 0, and [(curl u 0 × e 3 ) • e 2 ] Γ = 0.
Case (iii): u 0 × e 3 = 0 on Γ.

Let us make a few comments on this result. First, we emphasize that the nature of the limit problem depends strongly on the topology of the thin layer of obstacles L δ . In case (iii), where L δ comprises a wire mesh, we observe the "Faraday cage effect", where the effective interface Γ is a solid perfectly conducting sheet. Hence if the support of f lies in Ω + (above the layer L δ ), then u 0 = 0 in Ω -. In other words, despite the holes in its structure, the layer L δ shields the domain Ω -from electromagnetic waves of all polarizations. At the opposite extreme, in case (i), where L δ comprises discrete obstacles, the interface is transparent and there is no shielding effect. In the intermediate case (ii), where L δ comprises an array of parallel wires, one observes polarization-dependent shielding.

Fields polarized parallel to the wire axis are shielded, whereas those polarized perpendicular to the wire axis are not. Note that this case (ii) includes as a subcase the simpler two-dimensional situation studied in [START_REF] Hewett | Homogenized boundary conditions and resonance effects in Faraday cages[END_REF][START_REF] Holloway | A homogenization technique for obtaining generalized sheet transition conditions for an arbitrarily shaped coated wire grating[END_REF][START_REF] Marigo | Two-scale homogenization to determine effective parameters of thin metallic-structured films[END_REF] where the fields are invariant in the direction of the wire axis. The remainder of this note is dedicated to the formal proof of Theorem 2.1. The proof is based on the construction of an asymptotic expansion of u δ using the method of matched asymptotic expansions (cf. [START_REF] Maz | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]). To simplify the computation, we work with the first order formulation of (1), introducing the magnetic field h δ = 1 iω curl u δ (see e.g. [START_REF] Monk | Finite element methods for Maxwell's equations[END_REF]) and obtaining

-iωh δ + curl u δ = 0 in Ω δ , -iωu δ -curl h δ = -1 iω f in Ω δ , u δ × n = 0 and h δ • n = 0 on Γ δ . (6)
Far from the periodic layer L δ , we construct an expansion of h δ and u δ of the form

h δ = h 0 (x) + δh 1 (x) + • • • , u δ = u 0 (x) + δu 1 (x) + • • • , x = (x 1 , x 2 , x 3 ), (7)
and, in the vicinity of L δ ,

h δ = H 0 (x 1 , x 2 , x δ ) + δH 1 (x 1 , x 2 , x δ ) + • • • , u δ = U 0 (x 1 , x 2 , x δ ) + δU 1 (x 1 , x 2 , x δ ) + • • • , (8)
where, for i ∈ {0, 1}, H i (x 1 , x 2 , y 1 , y 2 , y 3 ) and U i (x 1 , x 2 , y 1 , y 2 , y 3 ) are assumed to be 1-periodic in both y 1 and y 2 . Near and far field expansions communicate through so-called matching conditions, which ensure that the far and near field expansions coincide in some intermediate areas. Since we are only interested in the leading order terms, it is sufficient to consider only the O(1) matching conditions, namely lim

x3→0 ± h 0 = lim y3→±∞ H 0 and lim x3→0 ± u 0 = lim y3→±∞ U 0 . (9)
Inserting ( 7) into (1) and separating the different powers of δ directly gives [START_REF] Marigo | Two-scale homogenization to determine effective parameters of thin metallic-structured films[END_REF]. To obtain the interface conditions, we have to study the problems satisfied by U 0 and H 0 :

     curl y U 0 = 0 in B ∞ , div y U 0 = 0 in B ∞ , U 0 × n = 0 on ∂B ∞ ,      curl y H 0 = 0 in B ∞ , div y H 0 = 0 in B ∞ , H 0 • n = 0 on ∂B ∞ , B ∞ = Ω 1 = R 3 \ L 1 . ( 10
)
3 The spaces K N (B ∞ ) and K T (B ∞ )

Denoting by B the restriction of B ∞ to the strip (0, 1) 2 × (-∞, ∞), we introduce the spaces

H N (B ∞ ) = {u ∈ H loc (curl; B ∞ ) ∩ H loc (div; B ∞ ) : u is 1-periodic in y 1 and y 2 , u |B 1 + (y 3 ) 2 ∈ (L 2 (B)) 3 , curl u |B ∈ (L 2 (B)) 3 , div u |B ∈ L 2 (B), u × n = 0 on ∂B ∞ , (11)
H T (B ∞ ) = {h ∈ H loc (curl; B ∞ ) ∩ H loc (div; B ∞ ) : h is 1-periodic in y 1 and y 2 , h |B 1 + (y 3 ) 2 ∈ (L 2 (B)) 3 , curl h |B ∈ (L 2 (B)) 3 , div h |B ∈ L 2 (B), h • n = 0 on ∂B ∞ , (12)
both of which include periodic vector fields in H loc (curl; B ∞ ) ∩ H loc (div; B ∞ ) that tend to a constant vector as |y 3 | → ∞. Investigation of [START_REF] Monk | Finite element methods for Maxwell's equations[END_REF] requires the characterization of the so-called normal and tangential cohomology spaces K N (B ∞ ) and K T (B ∞ ) defined by (see [START_REF] Amrouche | Vector potentials in three-dimensional non-smooth domains[END_REF])

K N (B ∞ ) = {u ∈ H N (B ∞ ), curl u = 0, div u = 0} , K T (B ∞ ) = {h ∈ H T (B ∞ ), curl h = 0, div h = 0} . ( 13
)
This characterization involves the representation of elements of K N (B ∞ ) and K T (B ∞ ) as gradients of harmonic scalar potentials, constructed by solving certain variational problems in the space

W 1 (B ∞ ) = p ∈ H 1 loc (B ∞ ) : p is 1-periodic in y 1 and y 2 , p |B 1 + (y 3 ) 2 ∈ L 2 (B), ∇p |B ∈ L 2 (B) , (14)
and variants of it. In each case the existence and uniqueness of the potential follows from the Lax-Milgram Lemma. While we do not reproduce the proofs here, we remark that the unbounded nature of the domain B requires us, when verifying coercivity of the requisite bilinear forms, to appeal to the inequality

p 1 + (y 3 ) 2 L 2 (B+) ≤ 2 ∇p L 2 (B+) , B + = B ∩ {y 3 > 0}, ( 15
) valid if p ∈ C ∞ (B + ), p/ 1 + (y 3 ) 2 ∈ L 2 (B + ), ∇p ∈ L 2 (B +
) and p = 0 in a neighbourhood of {y 3 = 0}, which is an elementary consequence of the Hardy inequality [12, Lemma 2.5.7]

∞ 0 t -2 |ϕ(t)| 2 dt ≤ 4 ∞ 0 |ϕ (t)| 2 dt, ϕ ∈ C ∞ 0 ((0, ∞)). (16)

Characterization of K N (B ∞ )

To characterize K N (B ∞ ) we first define two functions p ± 3 ∈ H 1 loc (B ∞ ), 1-periodic in y 1 and y 2 , such that

-∆p ± 3 = 0 in B ∞ , p ± 3 = 0 on ∂B ∞ , lim y3→±∞ ∇p ± 3 = e 3 , lim y3→∓∞ ∇p ± 3 = 0.
Then, in case (i) we introduce the functions p1 ∈ W 1 (B ∞ ) and p 1 ∈ H 1 loc (B ∞ ), such that -∆p 1 = 0 in B ∞ , p1 = -PRy 1 on ∂B ∞ , and p 1 = p1 + y 1 .

Here, for any function u ∈ L 2 loc (B ∞ ), Ru denotes its restriction to B, while for any function u ∈ L 2 loc (B), Pu denotes its periodic extension to B ∞ . Similarly, in cases (i) and (ii) we introduce the functions p2 ∈ W 1 (B ∞ ) and p 2 ∈ H 1 loc (B ∞ ), such that -∆p 2 = 0 in B ∞ , p2 = -PRy 2 on ∂B ∞ , and p 2 = p2 + y 2 .

We emphasize that it is not possible to construct p1 in cases (ii) and (iii), and it is not possible to construct p2 in case (iii). An adaptation of the proof of [START_REF] Amrouche | Vector potentials in three-dimensional non-smooth domains[END_REF]Proposition 3.18] leads to the following result:

Proposition 3.1 Case (i): K N is the space of dimension 4 given by K N (B ∞ ) = span ∇p 1 , ∇p 2 , ∇p - 3 , ∇p + 3 . Case (ii): K N is the space of dimension 3 given by K N (B ∞ ) = span ∇p 2 , ∇p - 3 , ∇p + 3 . Case (iii): K N is the space of dimension 2 given by K N (B ∞ ) = span ∇p - 3 , ∇p + 3 .

  Fig 2), and we define Γ δ = ∂Ω δ . 1 (a) Case (i) -discrete obstacles 1 (b) Case (ii) -parallel wires 1 (c) Case (iii) -wire mesh

Figure 2 :

 2 Figure 2: The domain Ω δ in the three cases under consideration.

Sketch of the proof in case (ii). First, one can verify directly that the family ∇p 2 , ∇p - 3 , ∇p + 3 is linearly independent (using the limit of ∇p 2 and ∇p ± 3 as y 3 tends to ±∞). Moreover, it is clear that ∇p 2 and ∇p ± 3 belong to K N (B ∞ ). Now, let u ∈ K N (B ∞ ). Since B ∞ is connected, there exists p ∈ H 1 loc (B ∞ ), unique up to the addition of a constant, such that u = ∇p. (This follows e.g. from applying [START_REF] Monk | Finite element methods for Maxwell's equations[END_REF]Theorem 3.37] on an increasing sequence of nested subsets of B ∞ after extension of u by zero inside R 3 \ B ∞ .) Moreover, ∇p is periodic and there exists a real sequence (c j ) j∈Z such that

Because ∇p is periodic and

Since p = c j -α 1 y 1 -α 2 y 2 on ∂B ∞,j , the periodicity of p in y 1 implies that α 1 = 0, while its periodicity in y 2 leads to c j = c 0 + α 2 j. As a result, p = c 0 -α 2 (y 2 -j) on ∂B ∞,j .

Since p is harmonic, we deduce that p = c 0 + α 2 p2 , and hence that p = c 0 + α 2 p 2 + ± α ± 3 p ± 3 , which completes the proof. Cases (i) and (iii) follow similarly.

Characterization of K T (B ∞ )

First, let us define q 3 ∈ H 1 loc (B ∞ ) as the unique function such that

Then for i ∈ {1, 2} we introduce the functions qi ∈ W 1 (B ∞ ) and

lim y3→+∞ qi = 0, and q i = qi + y i .

In case (ii) we introduce a set of 'cuts' Σ defined by

Similarly, in case (iii) we introduce the cuts

In both cases, B ∞ \ Σ is then the union of the two simply connected domains

and we define q

In case (iii) the functions q ± 2 are defined similarly, except that we replace Σ j by Σ ij in the jump conditions. In case (iii) we additionally introduce the functions q±

and we define q ± 1 = q±

Then, adapting the proof of [11, Proposition 3.14] one obtains the following result:

Sketch of the proof in case (ii). As in the proof of Proposition 3.1, it is not difficult to prove that the family {∇q 1 , ∇q + 2 , ∇q - 2 , ∇q 3 } is linearly independent and that its elements belong to

∞ are simply connected, there exists q = (q

) and a real sequence (c j ) j∈Z such that

Since ∇q is periodic and

]y 2 and q is periodic, we find that [α 1 ] = 0, and

for each j ∈ Z. To conclude, it suffices to note that q = q -α 1 q 1 -± α ± 2 q 2,± -α 3 q 3 is periodic and satisfies

which proves that q is constant in each of B ± ∞ .

4 Formal proof of Theorem 2.1

We treat the three cases separately. In case (i), using Propositions 3.1-3.2, we have

The behaviour at infinity of the functions p i and q i and the matching conditions (9) then imply

and, consequently (by ( 6)), that [u 0 × e 3 ] Γ = 0 and [curl u 0 × e 3 ] Γ = 0. In case (ii) we have

which, together with the matching conditions (9), leads to (u 0 )

Finally, in case (iii) we have U 0 = ± a ± 3 ∇p ± 3 , which implies that (u 0 ) ± i (x 1 , x 2 ) = 0 for i = 1 or 2.

Remark 1 We point out that our formal proof can be made rigorous by justifying the asymptotic expansions (7)-( 8). This can be done a posteriori by constructing an approximation of u δ on Ω δ (based on the truncated series (7)-( 8)) and using the stability estimate (4) (see [START_REF] Maz | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF]). However, this would require us to identify the terms of order 1 in the expansions, which is beyond the scope of this note.