Mathilde Hendriks
email: mathilde.hendriks@wanadoo.fr

Stéphanie Bartolo

Benjamin Constans

Max Gonzales

Benoit Tavernier

Charles Garabedian

Damien Subtil

Factors related to severe neonatal acidosis in planned cesarean section. A case-control study

INTRODUCTION

Les césariennes programmées à terme représentent entre 2 et 20 % des accouchements [START_REF] Has | Césarienne programmée à terme. Evolution des taux de césariennes à terme en France entre 2011 et[END_REF][START_REF] Blondel | Enquête nationale périnatale 2010. Avec la participation des services de protection maternelle et infantile des conseils généraux[END_REF]. Elles sont généralement prévues du fait de risques importants ou de contre-indications à l'accouchement par voie basse et réalisées avant tout travail, au-delà de 37 SA [START_REF] Has | Indications de la césarienne programmée à terme[END_REF]. Les indications des césariennes programmées peuvent être multiples [START_REF] Blondel | Enquête nationale périnatale 2010. Avec la participation des services de protection maternelle et infantile des conseils généraux[END_REF]. Dans tous les cas où elle est possible, l'analgésie par rachianesthésie est recommandée. Elle constitue le gold standard depuis 2000 [START_REF] Gouez | Anesthésie pour césarienne[END_REF].

Lors d'une césarienne programmée, le risque d'acidose néonatale est théoriquement très faible [START_REF] Batlle | Analysis of the evitability of intrapartum asphyxia with a peers review[END_REF]. Il arrive cependant que le pH artériel au cordon soit inférieur à 7,00, ce qui définit la survenue d'une acidose néonatale sévère. La survenue d'une acidose s'accompagne d'un risque d'asphyxie néonatale, d'encéphalopathie anoxoischémique et de séquelles à long terme, notamment la paralysie cérébrale [START_REF] Simunek | Definition of intrapartum asphyxia and effects on outcome[END_REF][START_REF] Malin | Strength of association between umbilical cord pH and perinatal and long term outcomes: systematic review and meta-analysis[END_REF][START_REF] Goodwin | Asphyxial complications in the term newborn with severe umbilical acidemia[END_REF]. La survenue de cet événement est d'autant plus regrettable que la césarienne est sensée éviter ces complications en réduisant le risque de survenue d'évennements sentinelles per-partum associés à la tentative de la voie basse [START_REF] Badawi | Intrapartum risk factors for newborn encephalopathy: the Western Australian case-control study[END_REF].

Nous avons mené une étude dont l'objectif était d'identifier les facteurs liés à la survenue d'une acidose néonatale sévère en cas de césarienne programmée dans un contexte de grossesse unique à terme sous rachianesthésie. ____ jouent un rôle partiel dans la survenue d'une acidose néonatale.

Concernant l'éphédrine, notre étude n'a pu mettre en évidence de lien statistiquement significatif avec les pH inférieurs à 7 après régression logistique.

Dans la littérature, la dose d'éphédrine est reconnue comme étant un facteur de risque de pH inférieur à 7 à la naissance [START_REF] Kinsella | International consensus statement on the management of hypotension with vasopressors during caesarean section under spinal anaesthesia[END_REF][START_REF] Kee | Placental Transfer and Fetal Metabolic Effects of Phenylephrine and Ephedrine during Spinal Anesthesia for Cesarean Delivery[END_REF]. L'éphédrine était classiquement le vasopresseur utilisé pour la gestion de l'hypotension en obstétrique jusqu'en 2003 [START_REF] Burns | Prevention and management of hypotension during spinal anaesthesia for elective Caesarean section: a survey of practice[END_REF][START_REF] Mercier | Phenylephrine added to prophylactic ephedrine infusion during spinal anesthesia for elective cesarean section[END_REF][START_REF] Levinson | Vasopressors in obstetrics[END_REF]. Cette drogue augmente le débit cardiaque maternel par son effet bétaadrénergique tachycardisant et son effet veino-constricteur. Le fait d'administrer de l'éphédrine augmente les besoins en oxygène du foetus sans en majorer les apports [START_REF] Lee | A Quantitative, Systematic Review of Randomized Controlled Trials of Ephedrine Versus Phenylephrine for the Management of Hypotension During Spinal Anesthesia for Cesarean Delivery[END_REF]. Elle doit être administrée dès le début de la rachianesthésie, cependant plusieurs études ont montré que cette injection d'éphédrine n'était pas suffisante pour compenser l'hypotension artérielle et qu'elle pouvait provoquer des hypertensions réactionnelles [START_REF] Kee | A dose-response study of prophylactic intravenous ephedrine for the prevention of hypotension during spinal anesthesia for cesarean delivery[END_REF]. Elle peut également entraîner des tachycardies maternelles et foetales [START_REF] Kee | Comparison of metaraminol and ephedrine infusions for maintaining arterial pressure during spinal anesthesia for elective cesarean section[END_REF]. Dans une méta-analyse de Lee et al. de 2004, la dose de 14 mg d'éphédrine a été retenue comme étant la dose-seuil pour prévenir l'hypotension sans diminution significative du pH artériel à la naissance [START_REF] Veeser | Vasopressors for the management of hypotension after spinal anesthesia for elective caesarean section. Systematic review and cumulative meta-analysis[END_REF]. C'est pour cela que les nouvelles recommandations internationales de décembre 2017 ne préconisent l'utilisation d'éphédrine que lorsqu'une bradycardie maternelle survient [START_REF] Kinsella | International consensus statement on the management of hypotension with vasopressors during caesarean section under spinal anaesthesia[END_REF]. Notre étude couvre une longue période qui a été marquée par des changements dans les pratiques d'anesthésie, notamment concernant la dose d'éphédrine. En effet, les doses d'éphédrine avant 2005 étaient beaucoup plus élevées qu'après, ce qui a pu impacter nos résultats.

Dans notre étude, l'obésité apparaît finalement comme principal facteur lié au risque d'acidose néonatale et le mécanisme de cette acidose reste à discuter. Sur le plan anesthésique, les patientes obèses sont exposées à un risque d'extension de niveau métamérique céphalique [START_REF] Hood | Anesthetic and obstetric outcome in morbidly obese parturients[END_REF][START_REF] Hodgkinson | Obesity and the cephalad spread of analgesia following epidural administration of bupivacaine for Cesarean section[END_REF][START_REF] Mcculloch | Influence of obesity on spinal analgesia with isobaric 0.5% bupivacaine[END_REF], entraînant une instabilité hémodynamique. La perfusion utérine peut en être directement diminuée, entraînant une baisse de pH néonatal. Sur le plan métabolique et obstétrical, l'hyperlipidémie peut augmenter la sécrétion de prostaglandine et de thromboxane [START_REF] Eldor | Thrombophilia and its treatment in pregnancy[END_REF], augmentant le risque de thrombose et diminuant la perfusion placentaire [START_REF] Stone | Risk factors for severe preeclampsia[END_REF] avec augmentation du risque de décès périnatal [START_REF]Causes of death among stillbirths[END_REF][START_REF] Nohr | Prepregnancy obesity and fetal death: a study within the Danish National Birth Cohort[END_REF].

L

  Copyright 1985-2013 StataCorp LP, StataCorp, College Station, Texas 77845, USA). la rachianesthésie étaient comparables entre les deux groupes, notamment la dose de bupivacaïne injectée, la quantité de remplissage administrée, ainsi que le délai entre la réalisation de la rachianesthésie et l'incision chirurgicale. En ce qui concerne le déroulement de la césarienne, il n'y avait pas de différence entre les cas et les témoins pour la chute de pression artérielle systolique de plus de 20 %, mais la dose d'éphédrine intraveineuse administrée était significativement plus élevée chez les cas que chez les témoins. Par ailleurs des difficultés d'extraction étaient deux fois plus souvent déplorées dans le groupe des cas que dans le groupe des témoins, et le délai incision-naissance était plus élevé dans le groupe des cas. En effet, bien qu'il s'agisse de facteurs liés entre eux, le délai incision-naissance, l'existence de difficultés d'extraction et la dose totale d'éphédrine administrée étaient également liés au risque d'acidose néonatale sévère en analyse univariée mais cette association n'était plus significative après régression logistique, sans colinéarité ni interaction significative entre les facteurs introduits dans le modèle.

	METHODES position assise à l'aide d'une aiguille pointe-crayon de 25 G. Une injection lente d'un DISCUSSION néonatal < 7,20 et des déficits de base > 8 mmol/L [19].
	mélange de bupivacaïne hyperbare 0,5% : 7,5 à 12,5 mg (selon la taille de la
	A notre connaissance, notre étude est la seule à avoir étudié le risque d'acidose
	Il s'agit d'une étude rétrospective de type cas-témoins menée de janvier 1997 à femme), sufentanil : 2,5 µg (0,5 ml) et morphine : 100 µg (0,1 ml de morphine Dans notre étude rétrospective, l'incidence de l'acidose néonatale sévère en cas de néonatale en cas de césarienne programmée. Le seuil de pH artériel néonatal de 7,0
	décembre 2016 au sein d'une maternité de niveau 3 universitaire réalisant environ 1mg/ml) était réalisée après reflux franc de liquide céphalo rachidien. Immédiatement césarienne programmée était de 0,76% IC95 [0,54-1,04]. L'obésité maternelle a été choisi du fait des risques qu'il fait courir aux nouveau-nés de complications
	5000 accouchements par an durant la période. Les patientes étaient éligibles pour après induction de la rachianesthésie, la patiente était repositionnée en décubitus apparaissait comme le seul facteur lié à la survenue d'une acidose néonatale sévère. liées à l'acidose sévère [5]. L'incidence de 0,8% que nous avons observée apparaît
	être des cas si elles avaient une grossesse unique et avaient bénéficié d'une dorsal. L'extension métamérique de la rachianesthésie était surveillée toutes les 3 conforme à celles publiées par d'autres toutes voies d'abord et contexte confondus,
	césarienne programmée à plus de 37 SA pendant la période d'étude, avec minutes par la sensibilité thermique. L'objectif anesthésique était l'obtention d'un comprises entre 0,5 et 3 % [6,20,21]. Cependant, il peut paraître élevé dans un
	niveau entre la 4 e et la 6 e vertèbre thoracique. groupe de femmes ayant été césarisées avant travail, mais le rôle protecteur des
	version	par	manoeuvre	interne,	appel	d'un	confrère).	Enfin,	les
	Pour chaque femme incluse, nous avons recueilli des données obstétricales, césariennes programmées sur le risque d'asphyxie néonatale reste discuté [5].
	anesthésiques et néonatales. Les nouveau-nés inclus dans le groupe des cas et
	dans celui des témoins ont été décrits selon les critères habituellement retenus:
	sexe, poids < 10e percentile ou > 90e percentile [10,11], score d'Apgar à 5 minutes
	[14], pH artériel néonatal, pCO2, déficit de base > 12 mmol/l [14], transfert en unité
	de soins néonataux ou décès néonatal intra-hospitalier.
	Sur le plan obstétrical, nous avons étudié les critères maternels suivants : l'âge de la
	femme, sa parité, l'antécédent de césarienne antérieure, la survenue d'une
	hypertension artérielle gravidique ou d'une prééclampsie, l'existence d'un diabète
	antérieur à la grossesse, la survenue d'un diabète gestationnel, l'indice de masse Dans notre étude, le délai incision-naissance était significativement lié au risque
	corporelle et l'existence d'une obésité maternelle (IMC≥30 kg/m²) ainsi que la prise risque de première espèce p<0,10. L'absence de colinéarité entre les différentes Pour ce qui concerne les risques néonataux des césariennes réalisées chez les d'acidose néonatale mais ne l'était plus en analyse multivariée. Dans leur analyse
	de poids en cours de grossesse. variables a été vérifiée à l'aide du calcul du Variance Inflation Factor (VIF) qui était femmes obèses, une seule étude a étudié spécifiquement ce lien, mais en mêlant rétrospective citée précédement, Conner et al retrouvaient une élévation significative
	Concernant la césarienne, nous avons classé les indications en fonction de l'enquête inférieur à 2 pour toutes les variables. A l'issue de cette régression, seules les césariennes programmées et césariennes en cours de travail. Dans cette étude des délais incision-naissance au fur et à mesure de l'augmentation de l'indice de
	nationale périnatale française (utérus cicatriciel, présentation du siège ou transverse, variables liées au risque d'acidose néonatale sévère avec p<0.05 ont été retenues. rétrospective de 2266 femmes ayant bénéficié d'une césarienne, Conner et al ont masse corporelle, allant de 9 minutes en moyenne en cas d'indice normal à 16
					____ ____ ____
					2 3 8 9

constatation d'un pH néonatal artériel inférieur à 7,0 à la naissance. Une césarienne programmée était définie par une césarienne réalisée hors contexte d'urgence, en semaine (du lundi au vendredi de 8h à 17h), pour laquelle la patiente avait été prévenue de la date de sa réalisation. En cas de césarienne pendant le travail, d'accouchement par voie basse, de grossesse multiple, de décès in utero ou d'interruption médicale de grossesse, les patientes n'étaient pas éligibles pour l'étude. Pour chaque cas, deux femmes témoins étaient éligibles : celles dont la césarienne programmée avait été réalisée immédiatement avant et celle immédiatement après, avec un pH artériel néonatal ≥ 7,0. Les critères d'exclusion étaient les malformations foetales et les césariennes qui n'étaient pas réalisées sous rachianesthésie. L'analyse des données a été déclarée à la Commission Nationale Informatique et Libertés (DEC16-434).

Concernant la réalisation de la rachianesthésie, la procédure était conforme aux protocoles habituels du service. A l'installation de la patiente sur la table opératoire, étaient mis en place les éléments habituels de surveillance (ECG 3 dérivations, saturation pulsée en oxygène, pression artérielle non invasive par un brassard brachial adapté à la circonférence brachiale de la patiente) et deux voies veineuses périphériques (calibre 18 G). La ponction de la rachianesthésie était effectuée en pathologie maternelle ou placenta praevia, macrosomie foetale ou autre)

[START_REF] Blondel | Enquête nationale périnatale 2010. Avec la participation des services de protection maternelle et infantile des conseils généraux[END_REF]

. Nous avons recueilli l'âge gestationnel à la césarienne, l'existence ou non d'un ____ 4 enregistrement du rythme cardiaque foetal avant la césarienne, ainsi que les conditions de réalisation de la césarienne: doses des différentes drogues injectées dans le liquide céphalo-rachidien maternel avant la naissance (comme la bupivacaïne), l'éphédrine et la phényléphrine injectées par voie systémique, quantité de remplissage vasculaire avant la naissance (mL), existence ou non d'une chute de pression artérielle systolique (PAS) de plus de 20 % par rapport à la PAS avant la naissance [15,16]. La technique chirurgicale et l'existence ou non de difficultés d'extraction a été notée (nécessité d'extraction instrumentale, réalisation d'une délais rachianesthésie-incision et incision-naissance ont été relevés. Les pourcentages figurent entre parenthèses et les médianes sont données avec leurs deux interquartiles. Dans un premier temps, nous avons décrit les caractéristiques des nouveau-nés et comparé de manière univariée les caractéristiques des patientes ainsi que celles des césariennes. Les pourcentages ont été comparés entre eux par des tests de Chi 2 ou de Fisher en fonction des effectifs. Les médianes ont été comparées par des tests non paramétriques de Kruskal et Wallis. Dans un second temps, nous avons réalisé une analyse multivariée par régression logistique en incluant les variables liées à la survenue d'une acidose néonatale en analyse univariée par régression logistique, avec un Les odds ratios ajustés sont donnés avec leur intervalle de confiance à 95%. Toutes les données ont été analysées à l'aide du logiciel STATA (STATA software version 13.0.0 (Finalement, parmi les caractéristiques maternelles et les conditions de réalisation de la césarienne, quatre étaient liées au risque d'acidose néonatale avec un risque de première espèce <0,10 : une obésité maternelle (IMC≥30 kg/m²), une dose totale d'éphédrine intraveineuse plus élevée, l'existence de difficultés d'extraction, et un délai incision-naissance plus élevé. Après régression logistique du risque d'acidose néonatale sur ces quatre facteurs, seule l'obésité maternelle restait significativement associée à la survenue d'une acidose néonatale après césarienne programmée (ORa 3,73 IC95 [1,11 ; 12,56]) (Tableau 4). Il n'existait pas d'interaction entre les quatre facteurs étudiés, notamment entre l'obésité et la durée incision-naissance.

Dans la littérature, l'obésité est reconnue comme étant un facteur de risque de complications de l'accouchement, avec un taux de césariennes qui s'élève selon l'indice de masse corporelle, qu'il s'agisse des césariennes programmées ou en cours de travail

[START_REF] Deruelle | Does a body mass index greater than 25 kg/m2 increase maternal and neonatal morbidity? A French historical cohort study[END_REF][START_REF] Madi | Effect of Obesity on Gestational and Perinatal Outcomes[END_REF]

. Quelle que soit la voie d'accouchement, l'obésité est le plus souvent liée à une augmentation du risque de complications néonatales, qu'il s'agisse du risque des scores d'Apgar < 7 à 5 minutes

[START_REF] Deruelle | Does a body mass index greater than 25 kg/m2 increase maternal and neonatal morbidity? A French historical cohort study[END_REF]

, des pH < 7,10

[START_REF] Madi | Effect of Obesity on Gestational and Perinatal Outcomes[END_REF]

, des décès en salle d'accouchement

[START_REF] Deruelle | Does a body mass index greater than 25 kg/m2 increase maternal and neonatal morbidity? A French historical cohort study[END_REF]

ou des transferts en unité de soins intensifs néonataux

[START_REF] Deruelle | Does a body mass index greater than 25 kg/m2 increase maternal and neonatal morbidity? A French historical cohort study[END_REF]

.

montré

une augmentation de la morbidité néonatale en cas d'indice de masse corporelle supérieur ou égal à 40, avec une augmentation significative du pH artériel En cas d'acidose sévère, les risques néonataux sont d'autant plus grands qu'il existe également une asphyxie, définie par la présence d'un déficit de base supérieur ou égal à 12 mmol/L. Dans notre étude, deux tiers des nouveau-nés en acidose sévère après césarienne programmée présentaient une asphyxie, avec un risque théorique d'encéphalopathie néonatale et de complication neurologique à long terme. Un des nouveau-nés de notre série avait même un pH néonatal à 6,85 avec un déficit de base à 12,7 mmol/L ; un autre a été transféré en soins intensifs néonataux. Heureusement, aucun n'a présenté de signes d'encéphalopathie néonatale. minutes en cas d'indice de masse corporelle ≥ 50 kg/m² [19]. Nous ne pouvons pas écarter la possibilité que ces délais incision-naissance plus élevés en cas d'obésité ____ 10

  'une des forces de notre étude réside dans l'analyse des acidoses néonatales sévères spécifiquement au décours de césariennes programmées, avec un recueil exhaustif des cas sur une période consécutive de 19 ans. Le dénombrement de 36 cas d'acidose sévère et l'étude simultanée de paramètres généraux, anesthésiques et obstétricaux ont permis de réaliser une étude multivariée intégrant un grand nombre de facteurs potentiellement liés au risque d'acidose. Le fait d'avoir exclu les

	FIGURE 1. DIAGRAMME DE FLUX TABLEAU 2. CARACTERISTIQUES GENERALES DES FEMMES TABLEAU 3. CONDITIONS DE REALISATION DES CESARIENNES PROGRAMMEES SELON TABLEAU 4. REGRESSION LOGISTIQUE DU RISQUE
	TABLEAU 1. CARACTERISTIQUES DES NOUVEAU-NES INCLUS SELON LE GROUPE D'ETUDE LE GROUPE D'ETUDE D'ACIDOSE NEONATALE SEVERE
		Cas	Témoins		p
	64 122 naissances ≥ 37 SA entre 1997 et 2016 Age gestationnel (semaines) Sexe masculin Petit poids pour l'âge gestationnel (<10e percentile) ORa* Macrosomie (>90 e percentile) Age (années) ≥ 39SA Délai incision / naissance ** 1,20 Nullipare Dose d'éphédrine *** 1,01 Indication Antécédent de césarienne antérieure Utérus cicatriciel Obésité (IMC ≥ 30 kg/m²) Siège ou transverse Non 1 HTA gravidique ou prééclampsie Pathologie maternelle/placenta praevia Oui 3,73 Score d'Apgar < 7 à 5 min Diabète antérieur à la grossesse Macrosomie foetale, autre Difficultés d'extraction	53 173 par voie basse Cas n=36 Cas n=36 Témoins Témoins n=72 n=72 n=36 n=72 38,8 ± 0,95 38,6 ± 0,83 p IC 95% p 18 (50) 31 (43,1) 1 (2,8) 4 (5,6) 11 (30,6) 16 (22,2) 31,9 ± 5,5 31,7 ± 4,9 22 (61,1) 41 (56,9) 0,80 -1,81 0,38 0,80 5 (13,9) 12 (16,7) 1,00 -1,02 0,09 0,70 27 (75,0) 55 (76,4) 18 (50,0) 49 (68,1) 0,87 9 (25,0) 12 (16,7) 1 (2,8) 1 (1,4) 1,00 4 (11,1) 7 (9,7) 1,11 -12,56 0,03 5638 en cours de travail 291 multiples 1 (2,8) 0 (0,0) 3 (8,3) 3 (4,2) 0,40 5 (13,9) 4 (5,6)	0,22 p 0,49 0,66 0,68 0,23 0,35 0,33
	césariennes réalisées sous anesthésie générale a permis de s'affranchir du rôle Chute de pression systolique > 20% 19 (52,8) 33 (45,8) délétère de certaines drogues administrées par voie générale à la mère et de refléter 4 décès in utero 2 interruptions médicales de grossesse 4976 pH néonatal ≥ 7,0 5014 césariennes avant travail de singletons nés vivant pH artériel néonatal 6,96 [6,94-6,98] 7,25 [7,17-7,29] <0,001 0,50 pCO2 artérielle néonatale 95 [89-104] 62 [55-70] <0,001 Base Excess >12 mmol/l 24 (66,7) 3 (4,2) Transfert en unité de soins intensifs* 1 (2,8) 0 (0,0) Décès néonatal intra hospitalier 0 (0,0) 0 (0,0) NA* Prise de poids pendant la grossesse (kg) 12,0 ± 5,9 11,3 ± 5,2 Délai rachianesthésie-incision (min) 17 [14-21] 16 [11-21] 0,45 0,53 0,16 IMC ≥ 30 kg/m² 12 (34,3) 7 (9,9) Quantité de remplissage (mL) 1000 [1000-1250] 1000 [1000-1500] 0,69 0,002 <0,001 Diabète gestationnel 3 (8,3) 5 (6,9) Enregistrement du RCF* en salle d'opération 29 (80,6) 57 (79,2) 0,87 Non 1 1,00 IMC (kg/m²) 26 [23-31] 24 [21-26] > 10 mg 14 (38,9) 26 (36,6) 0,82 0,01 Dose de bupivacaïne intra-rachidienne injectée(mg) 10 [10-12,5] 10 [10-12,5] 0,54 Oui 2,30 0,80 -6,60 0,12
	la pratique courante des césariennes programmées. D'un autre côté, la longue Dose d'éphédrine (mg) 60 [38-90] 42 [26-70,5]	0,03
	période d'étude -19 ans -nous expose à des variations de pratique au cours du 38 avant le cas / 38 après le cas (éligibles pour les témoins) 38 avec pH néonatal < 7,0 (éligibles pour les cas) *non applicable Dose de phényléphrine (µg)** 210 [175-265] 130 [105-245]	0,10
	temps, mais notre étude était unicentrique et nous pensons avoir relevé l'ensemble Difficulté d'extraction 11 (30,6) 11 (15,3)	0,06
	1 malformation Délai incision-naissance (min)	11 [9-15]	1 malformation 10 [5-12]		0,02
	des facteurs potentiellement liés au risque d'acidose néonatale, même si ceux-ci ont
	75 non malformés pu être modifiés avec le temps (durée incision-naissance, dose d'éphédrine 37 non malformés *Rythme cardiaque foetal **A partir de 2005
	administrée). 1 anesthésie générale		2 anesthésies générales 1 analgésie péridurale
	36 cas inclus	72 témoins inclus

*ORa Odds Ratio ajusté par régression logistique ** Pour une augmentation de 5 minutes *** Pour une augmentation de 1mg d'éphédrine

Parmi les 64122 naissances réalisées pendant la période d'étude, 5014 ont été réalisées par césarienne programmée chez des singletons nés vivants (7,8 %), et 38 nouveau-nés ont présenté un pH artériel néonatal < 7,0 (0,76 % IC95 [0,54-1,04]).Les témoins éligibles étaient les 38 patientes ayant accouché par césarienne programmée immédiatement avant et après les cas, dont le pH néonatal était ≥ 7,0 (Figure1). Après exclusion de deux nouveau-nés porteurs de malformation avant la naissance (fente labio-palatine, dilatation pyélique) et de quatre césariennes non réalisées sous rachianesthésie, 36 cas et 72 témoins ont été inclus dans l'étude.Dans le groupe des cas, deux tiers de nouveau-nés en acidose présentaient un déficit de base > 12 mmol/l, mais seulement l'un d'entre eux a dû être transféré en unité de soins intensifs néonataux (Tableau 1). Aucun décès n'a été constaté dans les deux groupes.En ce qui concerne les caractéristiques générales des patientes, les cas et les témoins étaient comparables, sauf pour leurs indices de masse corporelle qui étaient plus élevés chez les cas que chez les témoins (Tableau 2). De manière corollaire, le pourcentage de patientes ayant un IMC > 30 kg/m² atteignait un tiers des patientes cas, ce qui était significativement plus élevé que chez les témoins. En ce qui concerne les conditions de réalisation des césariennes programmées, elles étaient majoritairement réalisées après 39 SA et leurs indications étaient pratiquement identiques selon le groupe d'étude (Tableau 3). La plupart des cas et des témoins ont bénéficié d'un enregistrement du rythme cardiaque foetal immédiatement avant la césarienne, en salle d'opération. Les données concernant

CONCLUSION

Les résultats de notre étude indiquent que l'incidence des acidoses néonatales n'est pas rare en cas de césarienne programmée, notamment en cas d'obésité maternelle.


Nicolas Mottet
email: ncmottet@gmail.com

Aude Bourtembourg

Astrid Eckman-Lacroix

Olivia Forner

Clémence Mougey

Jean Patrick Metz

Rajeev Ramanah

Didier Riethmuller

Mottet Nicolas

Mise au point sur l'Odon Device™ : amélioration technique, fonctionnement et avancée du programme de recherche clinique

1

Introduction

En France, 12 % des patientes accouchant par voie vaginale nécessitent une assistance instrumentale à la naissance. La ventouse représentait en 2016, comme en 2010, l'instrument le plus utilisé par les obstétriciens lors des accouchements assistés par voie vaginale (49,8 % des naissances par voie basse instrumentale), devant les forceps et les spatules (respectivement 27,6 % et 22,6 %) [START_REF] Inserm | Enquête nationale périnatale. Rapport de 2016 Les naissances et les établissements Situation et évolution depuis[END_REF]. La flexion de la tête foetale favorisée par la ventouse, sa possibilité d'utilisation quelle que soit la variété de présentation occipitale et sa facilité d'apprentissage sont des propriétés qui favorisent la diffusion de cet instrument. L'utilisation des spatules reste stable et rappelle que les obstétriciens français sont également sensibles au principe de la propulsion de la tête foetale.

En raison de son principe de fonctionnement, l'Odon Device TM est un dispositif qui mérite une attention particulière afin de définir sa place dans la pratique de l'accouchement instrumental. Les premières descriptions dans la littérature sont rapportées en 2013, et l'organisation mondiale de la santé (OMS) soutient son développement à travers son programme de recherche « HRP » (Human reproduction program) [START_REF] Requejo | Odon device: a promising tool to facilitate vaginal delivery and increase access to emergency care[END_REF]. Ce dispositif innovant pourrait être plus simple d'utilisation que les instruments habituels et pourrait être une alternative à la césarienne lors du deuxième stade du travail, en particulier dans les pays où l'accès aux soins obstétricaux est limité [START_REF] Bailey | Assisted vaginal delivery in low and middle income countries: an overview[END_REF][START_REF] Bd Odon | From design to adoption: generating evidence for new technology designed to address leading global health needs: A comprehensive clinical evidence generation plan for the BD Odon Device TM[END_REF]. Trois phases sont nécessaires avant d'envisager sa diffusion : une première phase évaluant la faisabilité d'une assistance instrumentale par Odon Device TM lors d'accouchements spontanés, une deuxième phase confirmant la faisabilité et la sécurité du dispositif lorsqu'il existe une indication d'assistance instrumentale lors du deuxième stade du travail, et enfin une troisième phase comparant l'efficacité de l'Odon Device TM à la ventouse ou au forceps au cours d'un essai randomisé.

Description et amélioration du dispositif

C'est Jorges Odon, mécanicien automobile de formation, qui invente le premier prototype d'Odon Device [START_REF] Jr | Car Mechanic Dreams Up a Tool to Ease Births[END_REF]. En 2008, le concept attire l'attention de l'OMS qui décide de lancer un programme de développement et en 2013, la licence d'exploitation est rachetée par la compagnie Becton Dickinson. Depuis la description du premier prototype, le dispositif Odon Device a fait l'objet de plusieurs améliorations techniques afin de faciliter son utilisation (Figure 1). Comme tout instrument, l'utilisation sécurisée de l'Odon Device TM nécessite de respecter les règles suivantes : patiente à dilatation complète, présentation céphalique engagée et membranes rompues. L'Odon Device TM est un dispositif stérile de classe 1 à usage unique composé de trois éléments principaux : un applicateur, un manchon en plastique et un anneau gonflable. Initialement, l'applicateur était composé de trois spatules en polyéthylène qui permettaient de guider la mise en place de l'anneau gonflable autour de la tête foetale. Une coupelle à l'extrémité de l'applicateur permet de faire glisser l'anneau gonflable autour de la tête foetale tout en la protégeant. Afin optimiser sa mise en place, l'applicateur est désormais équipé de quatre spatules souples légèrement incurvées facilitant le guidage du manchon et de l'anneau dans la filière génitale maternelle (Figure 1). Les spatules (25 cm) permettent d'ajuster la position de l'anneau gonflable (fabriqué à partir d'élastomère thermoplastique) lui permettant de prendre appui sur l'occiput foetal en arrière et l'os malaire en avant.

L'applicateur est équipé d'un indicateur de progression gradué permettant à l'opérateur de s'assurer qu'il a atteint la profondeur souhaitée avant de déployer l'anneau gonflable. L'anneau est ensuite gonflé avec une poire de gonflage à une pression d'environ 70 kPa. Une valve de sécurité empêche un gonflage excessif. La poire de gonflage était initialement indépendante du dispositif et a été secondairement intégrée au niveau de la poignée droite pour favoriser l'ergonomie (Figure 1). Le manchon en plastique est un film transparent de polyéthylène et comporte deux poignées rigides à son extrémité permettant la prise en main du dispositif. On retrouve sur le manchon une ligne bleue de dégonflage signalant à l'opérateur le moment où il doit dégonfler l'anneau et libérer la tête foetale du manchon lors du dégagement.

Principe de fonctionnement et mécanique obstétricale

Au premier coup d'oeil, l'aspect de l'Odon Device TM peut paraître étonnant et son principe de fonctionnement pourrait laisser certains accoucheurs dubitatifs. L'objectif de l'assistance instrumentale par Odon Device TM consiste à placer un anneau gonflable autour de la tête foetale en prenant appui sur l'os malaire en avant et sur l'occiput en arrière. Trois principes mécaniciens favorisent la progression de la tête foetale avec l'Odon Device TM : propulsion partielle, flexion limitée et traction.

Lorsque l'opérateur gonfle l'anneau gonflable, ce dernier prend non seulement un appui distal sur les parties osseuses de la tête foetale (occiput et os malaires) mais également un appui proximal sur les parties molles maternelles dans la filière génitale (Figure 2a). La combinaison de ces deux appuis est à l'origine d'une force de propulsion partielle. Lors de la mise en place de l'anneau, l'applicateur qui appuie sur le sommet du crâne via la cupule génère un léger mouvement de flexion qui est maintenue lorsque l'anneau est gonflé (Figure 2b). Enfin, lorsque l'opérateur saisit les poignées du dispositif, il applique une force de traction (qui est le principe mécanique le plus important) qui favorise la progression de la tête, initialement dans l'axe ombilico-coccygien avant de redresser l'axe lors du dégagement. À la différence du forceps, la conception du dispositif limite les forces d'écrasement appliquées sur le crâne foetal (Figure 2c). Ainsi, un des avantages de l'Odon Device TM est de ne pas laisser d'empreintes ou de marques sur la tête foetale contrairement aux forceps ou à la ventouse.

Evaluation clinique préliminaire

Une des craintes lors de l'utilisation d'un tel dispositif pourrait être le risque d'hypoperfusion cérébrale par compression des axes vasculaires carotidiens en cas de malposition de l'anneau gonflable. De Lange et al. ont étudié la perfusion cérébrale par échographie de contraste chez 12 porcelets dont la région cervicale était comprimée par l'anneau gonflable d'un dispositif Odon Device pendant 2, 5 et 10 min [START_REF] De Lange | Assessment of cerebral perfusion with contrastenhanced ultrasound during constriction of the neck mimicking malposition of the BD Odon Device TM : a study in newborn piglets[END_REF]. En cas de positionnement anormal de l'anneau gonflé à une pression de 300 mmHg, les auteurs retrouvaient une diminution modérée de la perfusion cérébrale avec une restitution immédiate du débit après exsufflation du sac.

Concernant les forces appliquées sur le crâne foetal par l'Odon Device TM , une étude sur 84 accouchements sur simulateurs retrouvait une pression sur la face foetale moins importante qu'avec les forceps (83 contre 146 kPa) [START_REF] O'brien | Pressure and traction on a model fetal head and neck associated with the use of forceps, Kiwi TM ventouse and the BD Odon Device TM in operative vaginal birth: a simulation study[END_REF]. De même, en cas de non progression de la tête foetale, l'anneau gonflable glissait le long de la tête et ressortait de la filière génitale avec une force de traction inférieure à celle du forceps (208 contre 270 N). Dans des conditions normales d'utilisation sur simulateurs, les opérateurs étaient capables de mettre en place l'anneau gonflable correctement, soit environ 40 mm au-dessus du menton, et le déplacement de l'anneau gonflable ne dépassait pas 10 mm lors de la traction [START_REF] O'brien | Fetal head position and perineal distension associated with the use of the BD Odon Device TM in operative vaginal birth: a simulation study[END_REF]. Dans cette étude comprenant 400 accouchements sur simulateurs, la distension périnéale générée par l'Odon Device TM était équivalente à celle de la ventouse Kiwi et inférieure à celle d'un forceps sur le périnée postérieur, respectivement 21, 21 et 26 mm [START_REF] O'brien | Fetal head position and perineal distension associated with the use of the BD Odon Device TM in operative vaginal birth: a simulation study[END_REF]. Enfin, une étude de 1 845 accouchements assistés sur simulateurs concluait que dans des conditions normales d'utilisation, l'Odon Device TM se comportait de manière reproductible entre les opérateurs et concluait que son utilisation n'était pas plus à risque que la ventouse ou le forceps [START_REF] O'brien | Design and development of the BD Odon DeviceTM : a human factors evaluation process[END_REF].

Etat d'avancement du programme de recherche clinique

Tableau 1 : Critères d'inclusion et d'exclusion au moment du consentement et lors de l'allocation de l'intervention

Figure 1 : Le dispositif Odon Device TM actuel L'applicateur dispose d'une cupule venant prendre appui sur le sommet de la tête foetale lors de la pose. Il est composé de 4 spatules souples permettant de guider l'anneau gonflable dans la filière génitale puis autour de la présentation. Les poignées du manchon sont rigides et la poire de gonflage est intégrée à la poignée droite. Le liseré bleu sur le manchon indique le moment où l'anneau peut être dégonflé lors du dégagement de la tête.

CNGOF : Collège National des Gynécologues Obstétriciens Français

Critères d'inclusion Critères d'exclusion

Au moment du consentement

  photothérapie pour un ictère secondaire à des traumatismes des tissus mous jusqu'au jour 28 après la naissance, anémie nécessitant une transfusion jusqu'au jour 28 après la naissance, encéphalopathie du nouveau-né nécessitant une hypothermie thérapeutique dans les 28 jours suivant la naissance, défaillance multi viscérale dans les 28 jours suivant la naissance, décès du nouveau-né dans les 28 jours suivant la naissance

Figure 2 :

 2 Figure2: Principes de progression de la tête foetale avec l'Odon Device TM 2a : la force de propulsion résulte de l'association d'un appui distal sur les parties osseuses de la tête foetale (occiput et os malaires) et un appui proximal sur les parties molles maternelles. 2b : La force exercée par l'applicateur sur le sommet du crâne favorise la flexion du crâne maintenue lorsque l'anneau est gonflé. 2c : Lorsque les poignées sont saisies, l'opérateur exerce une force de traction maximale de 208 N.

Figure 3 :Tableau 1 :

 31 Figure 3 : Diagramme consort de l'étude « Besancon Assist Study »


G Michy

Chêne

Erdogan Nohuz 1©
email: erdogan.nohuz@chu-lyon.fr

Géry Lamblin

Thierry Michy

Gautier Chêne

Astuce technique facilitant la suture coelioscopique extracorporelle (avec vidéo)

Keywords: Suture coelioscopique extracorporelle Coelioscopie, suture, technique chirurgicale Laparoscopy, suture, surgical technique

Introduction

La solidité des sutures coeliochirurgicales extracorporelles et l'ergonomie quant à leur exécution demeurent à la fois un préalable et un objectif [START_REF] Pouliquen | Gestes de base en chirurgie laparoscopique de base. EMC Technique chirurgicale[END_REF]. Leur réalisation relève habituellement de la confection de demi-clés qui sont acheminées à quai à l'aide d'un pousse noeud. Ainsi, une demi-clé est effectuée puis immédiatement poussée jusqu'à sa destination finale. Ces gestes sont répétés jusqu'à ce que le nombre souhaité de demi-clés soit effectué. Nous avons recours à une technique très simple qui s'affranchit de la prise en mains et de la dépose répétitives du porte-aiguille. Nous décrivons le procédé tel que réalisé au cours de la fermeture vaginale après hystérectomie totale ainsi que la fixation prothétique lors d'un cerclage cervical de Benson chez une patiente présentant une béance cervico-isthmique.

Procédure préopératoire

Un trocart ombilical de 10 mm pour le coelioscope, deux trocarts opérateurs latéraux de 5 mm en regard des épines iliaques antéro-supérieures droite et gauche et un trocart opérateur sus-pubien de 10 mm sont insérés (Fig. 1). Une fois les tissus chargés, l'aiguille sertie est retirée au travers du trocart sus-pubien. Une série de 4 à 5 demi-clés à la suite, Suture coelioscopique extracorporelle 2 espacées d'environ 1 à 2 centimètres (Fig. 2 et 3A) est alors effectuée, de façon à obtenir un « chapelet ». Cet espacement servira à y immiscer le pousse-noeud qui prend appui sur le brin axial (comportant l'aiguille) afin que chaque demi-clé parvienne à quai, pas à pas (Fig.

3B, 3C et 3D).

Les points essentiels de la technique sont donc les suivants :

-Retrait de l'aiguille sertie au travers du trocart sus-pubien.

-Réalisation d'une série de 4 à 5 demi-clés à la suite, espacées de 1 à 2 centimètres.

-Poussée successive de chaque demi-clé à l'aide du pousse-noeud.

La technique est mise en lumière au travers d'une vidéo jointe à cet article (vidéo)

Conclusion

D

  'exécution simple et facilement reproductible, ce procédé s'affranchit des séquences répétitives de prise en main et dépose du porte-aiguille. L'économie gestuelle octroie gain de temps et ergonomie à la confection des sutures coelioscopiques extracorporelles. Les auteurs déclarent ne pas avoir de liens d'intérêts.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1 : Placement des trocarts de coelioscopie


Elyes Ben Hamida
email: elyes.ben-hamida@insa-lyon.fr

Guillaume Chelius
email: guillaume.chelius@inria.fr

Anthony Busson
email: anthony.busson@u-psud.fr

Eric Fleury
email: eric.fleury@ens-lyon.fr

Neighbor discovery in multi-hop wireless networks: evaluation and dimensioning with interference considerations †

Keywords: sensor networks, radio modeling, neighbor discovery, interference modeling, stochastic geometry

In this paper, we study the impact of collisions and interference on a neighbor discovery process in the context of multi-hop wireless networks. We consider three models in which interference and collisions are handled in very different ways. From an ideal channel where simultaneous transmissions do not interfere, we derive an alternate channel where simultaneous transmissions are considered two-by-two under the form of collisions, to finally reach a more realistic channel where simultaneous transmissions are handled under the form of shot-noise interference. In these models, we analytically compute the link probability success between two neighbors as well as the expected number of nodes that correctly receive a Hello packet. Using this analysis, we show that if the neighbor discovery process is asymptotically equivalent in the three models, it offers very different behaviors locally in time. In particular, the scalability of the process is not the same depending on the way interference is handled. Finally, we apply our results to the dimensioning of a Hello protocol parameters. We propose a method to adapt the protocol parameters to meet application constraints on the neighbor discovery process and to minimize the protocol energy consumption.

Introduction

Context

Neighbor discovery is a key component in the communication protocol stack of multi-hop wireless networks. It is a basic service on which most of the communication protocols rely. In the context of ad hoc networks, proactive and reactive routing protocols generally use the knowledge of the nodes neighborhood to build or manage routes, e.g. Optimized Link State Routing (OLSR [START_REF] Clausen | Optimized link state routing protocol (olsr)[END_REF]) or Ad hoc On-demand Distance Vector (AODV [START_REF] Perkins | Ad hoc on-demand distance vector (aodv) routing[END_REF]). In sensor networks, most of the stateless routing protocols, e.g. Most Forward within Radius (MFR [START_REF] Kleinrock | Optimal transmission ranges for randomly distributed packet radio terminals[END_REF]) or Geographic Face Routing (GFR [START_REF] Bose | Routing with guaranteed delivery in ad hoc wireless networks[END_REF]), require the knowledge of a node neighborhood to select a next hop. Neighbor discovery is also intensively used in the context of Delay Tolerant Networks (DTN [START_REF] Jain | Routing in a delay tolerant network[END_REF]) or Pocket Switched Networks [START_REF] Hui | Pocket switched networks and human mobility in conference environments[END_REF]. Based on the knowledge of contacts between entities/individuals, packet forwarding or gossiping is performed. Finally, neighbor discovery can be very useful for the study of human mobility [START_REF] Hui | Pocket switched networks and human mobility in conference environments[END_REF] or social interactions. The interest in logging individual interactions is to compile traces in order to build models of human mobility [START_REF] Kim | Extracting a mobility model from real user traces[END_REF] or to characterize and study dynamic graphs. Dynamic graph modeling offers a vast field of applications, covering multiple disciplines such as sociology, epidemiology, etc.

The neighbor discovery process is generally performed by a Hello protocol. A Hello protocol builds neighborhood tables through the periodic exchange of Hello messages in which a node advertises its position. The protocol generally involves several parameters such as the Hello packet period, its transmission power, etc. Some complex mechanisms like activity scheduling and the introduction of inactivity periods also interfere with the protocol characteristics and introduce new parameters to consider, like the activity/inactivity period durations. All these parameters have a real impact on the protocol performances and the discovery process: for example, neighbors can be discovered more or less rapidly and the maximum distance at which nodes can be discovered can change drastically. Energy consumption of the neighbor discovery process is also a crucial point to consider, especially in energy constraint devices such as sensor nodes or contact loggers. Finally, the radio environment has a non-negligible impact on the neighbor discovery process, particularly on the probability to discover nodes. Indeed, in real environments, radio links are never reliable and the level of interference as well as other propagation phenomena can largely reduce the protocol performances.

Contributions

In this paper, we take a particular look at the impact of the radio channel on the neighbor discovery process. We consider three models in which interference and collisions are handled in very different ways. From an ideal channel where simultaneous transmissions do not interfere, we derive an alternate channel where simultaneous transmissions are considered two-by-two under the form of collisions, to finally reach a more realistic channel where simultaneous transmissions are handled under the form of shot-noise interference. We prove that if the neighbor discovery process is asymptotically equivalent in these three models, it offers very different behaviors locally in time. For that, we analytically compute the link probability success between two neighbors as well as the expected number of nodes that correctly receive a Hello packet. Results in the three models diverge, offering different asymptotic behaviors regarding scalability for example.

Then we apply these results to the dimensioning of a Hello protocol parameters. The goal of dimensioning is to choose correctly the protocol parameters in order to adapt the link probability success between a node and its neighbors according to the application constraints, while minimizing the protocol energy consumption. Examples of application constraints are: "all nodes at distance at most 10 meters must be discovered as neighbors with high probability" or "node at distance more than 15 meters must not be seen as neighbors. We show how to use the analytical results to derive parameter values that respect these constraints. As a numerical application, a case study is introduced.

The remainder of this paper is organized as follows. We first reference related works in Section 2. Then, we present the considered hello protocols in Section 3 and three radio channel models are described in Section 4. The network is modeled in Section 5 and a qualitative study of the impact of interference on the neighbor discovery process is done in Section 6. The radio models are then analytically analyzed in Section 7 and compared in Section 8 . In Section 9, we discuss on the dimensioning of Hello protocols before concluding in Section 10.

Related works

Neighbor discovery

Several works have studied the design of Hello protocols in the context of wireless multi-hop networks.

McGlynn et al [START_REF] Mcglynn | Birthday protocols for low energy deployment and flexible neighbor discovery in ad hoc wireless networks[END_REF] have proposed a family of birthday protocols which use random transmissions and node synchronization to discover the nodes adjacencies in static ad hoc networks. The proposed mathematical models as well as the simulations show the energy efficiency and the robustness of the random protocols in comparison to deterministic or scheduling algorithms.

Alonso et al [START_REF] Alonso | Probabilistic protocols for node discovery in ad-hoc, single broadcast channel networks[END_REF] have provided a general model to study and analyze Hello protocols in ad hoc single broadcast channel networks. The time is slotted, the nodes are synchronized, in either of the two following states: listening or talking. Using this model, the authors have described and compared various Hello protocols. However, the model as well as the studied protocols do not consider the energy consumption. In (1), Alonso et al have extended their model to the case of ad hoc multi-channel broadcast networks.

Jakllari et al [START_REF] Jakllari | An integrated neighbor discovery and mac protocol for ad hoc networks using directional antennas[END_REF] have proposed a polling-based MAC protocol that addresses the problem of neighbor discovery with directional antennas. This type of antenna increases the capacity of the network thanks to the spatial reuse. Their protocol uses a polling strategy wherein a node polls its discovered neighbors periodically. This provides each node with the opportunity to adjust its antenna weighting coefficients in order to track its neighbors. The analytical study as well as the simulations show the efficiency of the protocol in term of capacity enhancement in the context of mobile ad hoc networks.

Vasudevan et al [START_REF] Vasudevan | On neighbor discovery in wireless networks with directional antennas[END_REF] have studied the problem of neighbor discovery in static wireless ad hoc networks with directional antennas. The proposed protocols are classified in two categories: the Direct-Discovery algorithms in which nodes discover their neighborhood only upon reception of a Hello message, and the Gossip-Based algorithms in which nodes also broadcast their neighbor location information. The analysis and the simulations show the efficiency of the second type of algorithms which surpasses the first one in term of adjacency discovery delay.

Most of the studies made on Hello protocols use rather simplistic models which do not take into account the specificities of radio communications. In our knowledge, few studies analyze the impact of interference or collisions, and more generally the impact of the radio channel modeling, on the neighbor discovery process. Few works also regard the dimensioning of the protocol with respect to interference, application constraints and energy efficiency.

Stochastic modeling of sensor networks

The stochastic modeling of wireless networks has already been used in various works. In particular, a general model has been introduced in (3). This model was first used to study the coverage of cellular networks (5) but was then modified and extended (4) to study properties of multi-hop wireless networks such as their capacity [START_REF] Franceschetti | Closing the gap in the capacity of wireless networks via percolation theory[END_REF] or their asymptotic connectivity (11; 12; 10). In these works, stochastic geometry was often mixed with elements of the continuum percolation theory.

In this paper, we consider, apply and extend the model introduced in (4) to the context of neighbor discovery and to the performance study and dimensioning of Hello protocols.

The Hello protocols

As mentioned in Section 1, neighbor discovery is a key component of the communication protocol stack in multi-hop wireless networks such as sensor networks, delay-tolerant networks or pocket-switched networks. In most systems, this task is comitted to a Hello protocol which broadcasts periodically and locally a Hello packet to advertise the node presence.

In a Hello protocol, several parameters are generally involved: for example, the Hello packet period, its transmission power, etc. Some protocols can also include more complex mechanisms such as activity scheduling, thus introducing new parameters like the activity/inactivity period durations. All these parameters have an impact on the protocol performances such as the delay to discovery a neighbor, the communication range of a Hello packet, the energy consumption of the protocol, etc. The radio channel also impacts on the protocol efficiency.

To study the dimensioning of these protocols and the impact of the radio channel on the neighbor discovery process, we consider in this paper two Hello protocol variants. They are based on the Aloha [START_REF] Kuo | The aloha system[END_REF] Medium Access Control (MAC) protocol in the sense that no carrier sensing nor clear channel assessment is performed before the transmission of a Hello packet. In the first Hello protocol, each node successively listens to the medium and transmits a Hello packet whereas in the second protocol, nodes can also be sleeping. This inactivity period is introduced to reduce the protocol energy consumption as the radio interface is turned off. We now present these two protocols in more details.

A basic Hello protocol

In the first variant, a node has two states : listening or talking. These two states occur inside a time frame, f , of duration w, as depicted on Fig. 1. In each occurrence of w (a.k.a, in each round or time frame of the Hello protocol), a node picks randomly an offset t i , such that t i ∈ [0,wτ ]. The Hello message is then transmitted at t i during τ . The Hello packet transmission is performed without any carrier sensing nor clear channel assessment. Each node transmits only one Hello message per time frame and keeps listening to the medium during the rest of the frame, for a cumulative listening duration wτ .

To briefly illustrate the impact of the protocol parameters on the neighbor discovery process, we can make a trivial reasoning. Given τ and w, the medium access probability is τ w and the probability for a transmission not to collide with neighboring transmissions is equal to :

1 - 2τ w -τ n-1
Where n represents the number of nodes within the collision range. A fine tuning of w is thus important to cope with the total number of interfering nodes n.

An energy aware Hello protocol

We now present a modified version of this basic Hello protocol including some activity scheduling. This second variant uses sleep periods to reduce the energy consumption in each node. During this inactivity period of duration s, the node radio interface is turned off. As shown on Fig. 2, a node has three states : listening, talking or sleeping. These states occur inside a periodic time frame f of duration w + s. In each round of the Hello protocol (i.e., in each occurrence of f ), a node picks randomly an instant t w , such that t w ∈ [0,s]. Then, the node picks randomly a second instant t i , such that t i ∈ [t w ,t w +w-τ ]. The Hello message is transmitted at t i with a duration of τ .

Hereby, a node is in the talking state during [t i ,t i +τ ], in the listening state during [t w ,t w +w]\[t i ,t i +τ ] and in the sleeping state during the rest of the frame f . In the sleeping state, the radio interface is off and the node can not receive any message. To sum up, for each occurrence of f , a node transmits only one message with a duration of τ , listens to the medium during wτ and sleeps during s.

To continue with the trivial reasoning started in the previous section, the medium access probability is now τ w+s and the probability to discover a given node is equal to:

w -τ w + s 1 - 2τ w + s n-2
Where n is the total number of nodes within the collision range. In this variant, two parameters, w and s, have to be well tuned in order to increase the probability for a node to discover its neighbors.

The radio channel

From a network point of view, a radio channel model aims at predicting the transmission error probability associated with each radio link. A radio link relies on the transmitter and receiver positions but depends also on many parameters and phenomena such as path-loss, shadowing, fading, modulation or interference, etc. At a given location x (x ∈ IR 2 ), the signal strength for an emission from node X i is S i l( X ix ) where . is the Euclidean norm in IR 2 , S i is the emission power of X i and l(.) is the path-loss function. l(.) is a decreasing function from IR + in IR + which models the attenuation of the signal strength with regard to the distance. Many different functions have been proposed in the literature for l(.). As an example, the path-loss is classically modeled using a power-law function as in the Friis formula [START_REF] Friis | A note on a simple transmission formula[END_REF]:

l(u)=A 0 .u -β
Where A 0 is a constant depending on the frequency and the antenna gain, and β is a parametric path-loss exponent introduced to fit various environments and typically ranging from 2.0 to 6.0. In the original Friis formula, we have β =2.

However, mathematical problems may arise due to the divergence of this function near 0. Basically, this formula is valid in far field only. To avoid any problem occurring near the transmitter when the node intensity increases, several bounded functions have been introduced. As examples, (10; 6) propose the following path-loss functions:

l(u)=A 0 .min(1,u -β ) l(u)=A 0 .(1 + u) -β l(u)=A 0 . 1 1+u β
When it will not be specified, we will consider l(.) to be a generic decreasing function. However, we will also consider l(u)= 1 C+u β , with C ∈ IR + and β ≥ 2.

The ideal model

In a first time, we consider a very simple radio channel model where interference is not considered. We assume that a transmission is received if its signal to noise ratio (snr) is above a given constant θ function of the transmission data-rate. This view is inherited from the field of information theory and the Shannon's law. Given this model, an emission from node X i is correctly received at location x if:

snr(X i ,x)= S i l( X i -x ) W x >θ (1)
Where W x in IR + models the ground noise level at the location x. As l(.) is decreasing and for a constant noise W x , this model results into a perfect circular coverage area around each node with a maximal communication range R(S i ) defined as follows:

R(S i ) = sup{r ∈ IR + | S i l(r) W x >θ} (2)
Thus, a node X i can send frames only to nodes within its communication area B(X i ,R(S i )), i.e., the open ball of radius R(S i ) centered on X i . A node x is said to be covered by X i if it lies within X i communication area.

The collision model

We define a radio channel model where interference is partially considered under the form of collisions.

We still assume that a transmission can be received if its snr is above θ but we also suppose that a simultaneous communication initiated by node X j collides with the transmission of X i if:

snr(X j ,x) > δsnr(X i ,x) (3)
With δ ∈ IR + . When suffering a collision, we assume that a transmission is lost.

We can note that this model is a generalization of the Protocol model described in [START_REF] Gupta | Capacity of wireless networks[END_REF]. Indeed, formula 3 can be rewritten as follows:

snr(X j ,x) > δsnr(X i ,x) ⇔ S j l( X j -x ) W x >δ S i l( X i -x ) W x ⇔ S j l( X j -x ) > δS i l( X i -x )
As l(.) is decreasing and under the assumption of a constant emission power for all nodes, we obtain the condition given by the Protocol model in ( 16).

The SINR model

We finally present a more realistic radio model where perturbations steaming from all concurrent transmissions are considered simultaneously when evaluating the correct reception of a frame. A frame from X i can be correctly received at a location x if and only if the frame Signal to Interference and Noise Ratio, the so-called SINR, at location x is above a given threshold θ.

sinr(X i ,x)= S i l( X i -x ) W x + I(x) >θ (4)
Where W x in IR + still models the ground noise level at the location x and I(x) is the shot-noise interference, i.e., the sum of all simultaneous transmission signal strengths:

I(x)= Xj ∈Γ,Xj =Xi S j l( X j -x )
Where Γ is the set of simultaneous emitters. This model is similar to the CDMA model introduced in (4) with a correlation factor γ =1 . It supposes that the interference is Gaussian. This assumption holds in the case of a high number of interfering nodes and in the absence of a preponderant interferer. In practice, with most systems, 3 to 4 interferers are enough for the interference to be Gaussian. In the other cases, it is not valid ( 28) and a precise study of the interference impact would require to enter into details and precisely specify the radio interface characteristics, e.g. its modulation. In order to keep a high level of abstraction, we rather consider a Gaussian interference. This assumption is also acceptable regarding the two following remarks: first, we consider a large network where the number of simultaneous emissions is important. Second, in the case of a preponderant interferer, the communication would no succeed, whatever the interference model. Indeed, we consider single resource radio systems with no multi-user mechanisms such as CDMA.

Fading & shadowing

The path-loss function l(.) predicts the mean received power at a given distance and leads to disk coverage areas. It is obvious that strong variations can be experienced by real nodes owing to the real environment.

It is usual to classify these effects in two main categories: shadowing and fading. Shadowing relies on large scale signal variations due to obstacles in the environment. These large scale variations are introduced thanks to a multiplicative random noise. This noise is generally modeled using a log-normal law. Note that this function stands for static variations around the isotropic mean function.

In severe environments, with multiple obstacles and walls, wave propagation generates several paths between the transmitter and the receiver. The received signal thus issued from an incoherent summation of several components becomes a random variable and is subject to fading. In this case, the received signal is still corrupted by a multiplicative noise. In the case of the most severe conditions, referred to as a Rayleigh channel, the statistics of the signal's amplitude becomes a random exponential variable of parameter µ.

From a network point of view, the leading difference between shadowing and fading is that shadowing introduces a static alteration of the isotropic determinist power prediction while fading introduces received power variations from frame to frame.

To handle the shadowing and fading phenomena, we do not consider the sequence (S i ) i=1,2,.. as a sequence of deterministic values but rather as a sequence of random variables in IR + independently and identically distributed. If not explicitly specified, we will suppose that the S i follow a generic distribution. However, we will also consider the special case of a Rayleigh channel where the S i follow an exponential distribution.

In reality fading and shadowing are emitters and receivers dependent. It means that for a given emitter, the fading are different with regard to the locations of the receivers. But, for the quantities we are studying, we consider only the signal power received by a typical node (at a given location). Therefore, without loss of generality, fading and shadowing can be modeled by two random variables associated to each node.

Stochastic modeling of the network

In order to study and dimension the neighbor discovery process, we use a stochastic modeling of the network. This approach is interesting as it permits to reduce the complexity of the study and in particular the number of variables, such as the location of each node, to a single parameter, the intensity of the point process used to model the nodes location. Stochastic modeling provides useful hints on the macroscopic and average behavior of the network.

Location model

We consider a network made of nodes dispatched in a two-dimensional geographical region. We assume that location of nodes is modelled by a stationary Poisson point process Φ 0 of constant spatial intensity λ 0 (the mean number of nodes per unit area). With such a process, the number of points lying in a region B of the plane follows a discrete Poisson law, and given the number of points in B, nodes are independently and uniformly distributed in B.

If Φ 0 (B) is the random variable which counts the number of points lying in B, we have: Where k is a positive integer and |.| is the Lebesgue measure in IR 2 . As an illustration, a realization of a Poisson point process of intensity λ =0.0035 is given in Figure 3.

P (Φ 0 (B)=k)= (λ 0 |B|) k k! e -λ0|B|

Application model

Given the description of the Hello protocols presented in Section 3, the nodes of Φ 0 are split in two subsets, the active and the sleeping nodes. Each node is active with probability q independently of the other nodes. The set of active nodes is a thinning of the point process Φ 0 which leads to a stationary Poisson point process Φ 1 with intensity λ 1 = qλ 0 . An active node is emitting with probability p independently of the other nodes. It leads to a new thinning of the point process Φ 1 . Let denote Φ 2 , the point process modeling the emitters. Φ 2 is a stationary Poisson point process with intensity λ 2 = pλ 1 = qpλ 0 . If we inject the parameters associated to the Hello protocols of Section 3, we have p = τ w , q =1for the protocol without sleep periods and p = τ w , q = w w+s for the one with sleep periods. The values of p and q correspond to a simplification with regard to the Hello protocol. First, we have neglected the border effect. Indeed, if a node picks its emitting time at the beginning or at the end of the slot, its collision probability gets smaller. Second, Φ 2 represents the nodes which emission overlap with the considered one. In practical, the nodes being non-synchronized, this overlapping is mainly partially and does not last for the whole transmission. However, in order to simplify our model, we do not take into account the variation of the interference stemming from the other nodes. In other words, we assume that each overlap affect the whole transmission of the Hello frame. In a way, it leads to a worst case analysis. These simplifications can be easily changed. For instance, the parameters p and q can also be chosen in such a way they represent the mean number of interfering nodes during a transmission. In fact, there is no particular assumptions on the values of p and q (except that 0 < p, q ≤ 1 and that the process Φ 2 must be an independent thinning of Φ 0 ).

Impact of interference on neighbor discovery

Now that we have modeled both the network and the radio channel, we can start looking at the impact of interference on the neighbor discovery process. To begin with, we make a short qualitative study by giving an illustration of the logical network topology built by the Hello protocol and as observed by the nodes. We introduce the notion of logical connectivity graph in opposition to the physical connectivity graph. Then, we illustrate the impact of the different radio channel models presented in Section 4 on this logical connectivity graph.

Graph of connectivity

During the process of neighbor discovery, i.e., the execution of the Hello protocol, when a Hello packet is received, the identity of the emitter is added to the neighborhood table of the receiving node. From the tables of all nodes, we can extract a graph of logical connectivity as it is observed by the nodes. The graph is composed of the adjacencies that the nodes have discovered. This notion of connectivity is logical as it is built upon the reception of packets as opposed to a physical connectivity which would only rely on the physical medium characteristics. In other terms, interference and collisions affect the logical connectivity as they affect the correct reception of Hello packets.

Contrarily to a physical connectivity graph, the logical connectivity graph is dynamic. Indeed, due to the non-deterministic nature of the Hello protocol, new adjacencies may be observed as the neighbor discovery process keeps on being executed. The logical connectivity graph is thus the sum of all adjacencies observed up to a given instant. In the next section, to simplify the discussion but without loss of generality, we will decompose the time in rounds of the Hello protocol.

Impact of the radio channel model

We have simulated the basic Hello protocol described in Section 3.1 according to the three radio channel models presented in Section 4 (i.e., the ideal model, the collision model and the sinr model). The objective is to qualitatively study the number of nodes discovered during the different rounds of the Hello protocol. As for each node and each round, the instant t i of the Hello packet transmission changes, one can expect to discover new neighbors at each round of the protocol.

The graphs of logical connectivity obtained for the ideal radio model is plotted on Figures 4(a), while in Figures 4(b) and (c), we show the collision radio model. In Figures 4(b) we consider a collision range equal to the transmission range and in Figures 4(c) this range is twice the coverage radius. Finally, results obtained for the sinr radio model are depicted on Figures 4(d).

First, we notice on Figure 4(a) that in the ideal radio model, all nodes within transmission range are discovered during the first round. This resulting graph of logical connectivity is not realistic as no collision nor interference occur. If high level protocols are built assuming an ideal underlaying radio model, we can have doubts about their proper functioning in real environments.

Second, from Figures 4(b), we can observe that collisions affect the logical graph of connectivity as numerous adjacencies are not discovered. If in each round, new neighbors are discovered, their total number remains lower than the number of nodes present within the transmission range of each node, even after 3 rounds of the Hello protocol. In Figures 4(c), the collision range is twice the transmission range. The impact of collisions increases and, as a consequence, the number of discovered adjacencies is largely reduced. However, in both cases, we notice that long links are sometimes established. This suggests that in the collision model, the distance between two nodes does not strongly restrict the neighbor discovery. The graphs of connectivity resulting from the sinr model are shown on Figure 4(d). As for the collision model, the number of discovered neighbors is lower than the number of nodes present within transmission range of each node, even after 3 rounds. Furthermore we observe that in this model, only short links are established. This suggests clearly that in the sinr model, the distance has a stronger impact on the discovery process than in the collision one.

This qualitative study shows the importance of considering interference and collisions while designing and evaluating communication protocols. Thereby, in the rest of this paper, we will study the radio link success probability depending on the considered radio channel model. We will use later this analysis for the dimensioning of the neighbor discover process.

Analysis of the radio link

Now that we have highlighted the impact of the radio channel model on the neighbor discovery process, we perform an analysis of a given radio link depending on the way collisions and interference are handled.

In this section as well as in the following ones, in order to compute the probability that a transmission succeeds as well as the average number of successful receptions for a given packet, we consider a particular emitter and receiver. The emitter is located at the origin and is named O. The emission power for this node is S and follows the same distribution as the random variables (S i ) i=1,2,.. . The receiver is located at y (y ∈ IR 2 ) and W is the ground noise level at location y. The distance between the two nodes is denoted r ( y = r). These two points are added to the points of Φ 0 .

Analysis of the ideal channel

We first study the ideal channel where no collision nor interference occur. Let p(r) be the probability that a frame from O is correctly received by y. In this particular channel, we have:

p(r)= 1 if r < R(S) 0 otherwise
Let us call N the random variable denoting the number of nodes that correctly receive the frame from O. As transmitting and sleeping nodes have their radio interface busy or off, the point process of potential receivers is a Poisson point process of intensity λ 1λ 2 = (1p)λ 1 . Trivially, we have

E [N ] = (1 -p)qλ 0 πR(S) 2
If we consider the particular path-loss function l(u)= 1 C+u β introduced in section 4, we obtain:

p(r)= 1 if r< β S Wθ -C 0 otherwise (5)
The average number of successful receptions can be written as follows:

E [N ] = (1 -p)qλ 0 π S Wθ -C 2 β (6)

Analysis of the collision channel

We now leave the ideal case and lightly refine the radio channel to handle collisions using the collision model given in section 4.2. We also assume that all nodes transmit at a constant power and we do not consider shadowing nor fading. We still consider a transmission between node O and node y.

Link success probability

Given the inequality 3, a collision with another transmission initiated by X j occurs if:

snr(X j ,y) > δsnr(O, y) ⇔ l( X j -y ) > δl(r)
Still considering the particular path-loss function l(u)= 1 C+u β introduced in section 4, the condition becomes:

snr(X j ,y) > δsnr(O, y) ⇔ l( X j -y ) > δl(r) ⇔ X j -y ) <
β (1δ)C + r β δ Thus, p(r) can be expressed as:

p(r)=P Φ 2 B y, β (1 -δ)C + r β δ =0
Where B(y, r) is the ball of radius r centered around y. Finally, p(r) can be written as:

p(r)=    e -pqλ0π( (1-δ)C+r β δ ) 2 β if r< β S Wθ -C 0 otherwise (7)

Mean number of successful receptions

According to the Campbell theorem, the mean number of nodes N which correctly receive the frame is:

E [N ]=E   Xi∈IR 2 1l ∀Xj ∈Φ2, Xj -x ≥ β q (1-δ)C+ x β δ dx   = (1 -p)qλ 0 IR 2 E 0 Φ1 1l ∀Xj ∈Φ2, Xj -x ≥ β q (1-δ)C+ x β δ dx = (1 -p)qλ 0 IR 2 P ∀X j ∈ Φ 2 , X j -x ≥ β (1 -δ)C + x β δ dx = (1 -p)qλ 0 2π +∞ 0 p(r)rdr
Where E 0 Φ1 [.] is the expectation under the Palm measure with regard to the process Φ 1 (see [START_REF] Stoyan | Stochastic Geometry and Its Applications[END_REF] page 119 for more details). Finally E [N ] is written as :

E [N ] = (1 -p)qλ 0 2π β √ S Wθ -C 0 e -pqλ0π( (1-δ)C+r β δ ) 2 β r dr
If we assume that the collision area is equal to the transmission range (i.e., δ =1 ), then E [N ] can be expressed as :

E [N ] = (1 -p)qλ 0 2π β √ S Wθ -C 0 e -pqλ0πr 2 r dr = (1 -p)λ 1 - e -pqλ0πr 2 pqλ 0 β √ S Wθ -C 0 = (1 -p)qλ 0 pqλ 0 1 -e -pqλ0π( S Wθ -C) 2 β = 1 -p p 1 -e -pqλ0π( S Wθ -C) 2 β (8)

Analysis of the SINR channel

We finally analyse the most realistic model where a frame from O is correctly received by a node y if and only if the Signal to Interference and Noise Ratio at y is above a given threshold θ:

p(r)=P Sl(r) W + I Φ2 (y) >θ (9)
The random variable I Φ2 (y) represents the interference generated by the other emitters. I Φ2 (y) is the sum of the signal strengths at y from all simultaneous transmissions:

I Φ2 (y)= Xi∈Φ2 S i l( X i -y )
Note that as the marks S i are identically distributed, they do not depend on location of nodes. Consequently, the distribution of I Φ2 (y) does not depend on y.

Link success probability

In order to increase the realism of the model and to study its impact on the radio link, we consider here a Rayleigh channel. In consequence, the sequence of random variables (S i ) i=1,2,.. follow exponential laws with parameter µ. The mean emission power of a node is then 1 µ . Equation ( 9) can be rewritten as:

p(r)=P S> θ l(r) (W + I Φ2 (y))
As S is exponentially distributed, P (S > x)=e -µx for x>0 and conditioning by the values of W and I Φ2 (y), we get

p(r)=E e -µ θ l(r) (W +IΦ 2 (y)) = E e -µ θ l(r) W E e -µ θ l(r) IΦ 2 (y) (10)
The last expression depends on the Laplace transform of I Φ2 (y). Fortunately, a close form expression for it is known. Without loss of generality we compute the Laplace transform at the origin rather than at y. Remember that the distribution of I Φ2 (y) does not depend on y. The Laplace transform L IΦ 2 (0) of I Φ2 (0) is defined as

L IΦ 2 (O) = E e -sIΦ(O)
When the emission power of a node follows an exponential law and for l(r)= 1 C+r β , we get:

L IΦ 2 (O) = E e -s P X i ∈Φ 2 Sil( Xi ) = E Xi∈Φ2 e -sSil( Xi )
Classical arguments of stochastic geometry (see for instance the generating functional of a Poisson point process in [START_REF] Stoyan | Stochastic Geometry and Its Applications[END_REF] pages 115-116) lead to:

L IΦ 2 (O) = e -λ2 R IR 2 (1-E[e -sS 1 l( x ) ])dx = e -λ2 R +∞ 0 (1-E[e -sS 1 l(u) ])udu = e -λ22π R +∞ 0 (1-µ µ+sl(u) )udu = e -λ22π " R +∞ 0 s µ(C+u β )+s « udu = e -λ22π 2 s(µC+s) 2 β -1 βsin ( 2π β ) µ 2 β (11)
If we suppose that the noise W is constant, and combining equations ( 10) and ( 11), the probability p(r) becomes:

p(r)=e -µ θ l(r) W L IΦ(O) µ θ l(r) = e -µ θ l(r) W e -λ22π 2 µ θ l(r) ( µC+µ θ l(r) ) 2 β -1 βsin ( 2π β ) µ 2 β = e -µθ(C+r β )W e -λ22π 2 θ(C+r β ) ( C+θ(C+r β ) ) 2 β -1 βsin ( 2π β ) (12)
We note that if the noise is nil (W =0 ), the probability p(r) does not depend on the emitting power. Increasing or decreasing the emission power increases or decreases the interference proportionally in such a way that the SINR remains constant.

Mean number of successful receptions

According to the Campbell theorem, the mean number of nodes N which correctly receive the frame is:

E [N ]=E   Xi∈IR 2 1l Sl( x ) W +I Φ 2 (x) >θ dx   = (1 -p)λ 1 IR 2 E 0 Φ1 1l Sl( x ) W +I Φ 2 (x) >θ dx = (1 -p)λ 1 IR 2 P Sl( x ) W + I Φ2 (x) >θ dx = (1 -p)λ 1 2π +∞ 0 p(r)rdr (13)
In the general case, the integration of p(r) is difficult but can be resolved numerically using a Riemann sum.

Discussion on the radio link model

After the qualitative discussion of Section 6 and the analysis done in Section 7, we can now precisely study the impact of the radio model on the neighbor discover process. A first result is that after an infinite number of rounds, the logical connectivity graphs build in the three models are equal. However, if they are asymptotically equivalent, the models offer very different behavior over a short period of time. This will be shown through the comparison of p(r) and E [N ].

Convergence of the logical connectivity graphs

We suppose here that Φ 0 is a homogeneous Poisson point process and (S i ) i=1,2,.. are constant and equal to S. Let G(Φ 0 ,R(S)) be the graph made of the elements of Φ 0 with an edge between x, y ∈ Φ 0 if and only if xy <R (S). Let G i t (Φ 0 ), G c t (Φ 0 ) and G s t (Φ 0 ) be the logical connectivity graphs built by the neighbor discovery process after t rounds in the ideal, collision and sinr models respectively. More generally, for a generic radio model m, G m t (Φ 0 ) contains all the adjacencies that have been discovered by the execution of the Hello protocol during t rounds in the radio model m. We define:

lim t→+∞ G m t (Φ 0 )=G m ∞ (Φ 0 )
theorem 1 If 0 <p<1 and 0 <q≤ 1, we have:

G(Φ 0 ,R(S)) = G i ∞ (Φ 0 )=G c ∞ (Φ 0 )=G s ∞ (Φ 0 )
Proof: We consider a particular sample of the homogeneous Poisson point process Φ 0 . Let x, y two points of Φ 0 . Suppose that the edge (x, y) is not in G(Φ 0 ,R(S)). By definition, we have xy ≥R(S) and thus p( xy )=0in all three radio models. Trivially, we can say that whichever t ∈ IN , the edge does not belong to G i t (Φ 0 ), G c t (Φ 0 ) or G s t (Φ 0 ). Suppose now that (x, y) is in G(Φ 0 ,R(S)), i.e. xy <R (S). We want to prove that (x, y) is in G m ∞ (Φ 0 ) with m being one the three considered models. Let A(n) be the event that the adjacency (x, y) is observed during the round n. Clearly the (A(n)) n=1,2... are independent (given Φ 0 and for S i = S ∀i). Moreover, ∀(i, n), P(A n )=P(A i ). Thanks to the Borel-Cantelli lemma, in order to show that

(x, y) is in G m ∞ (Φ 0 ), it is sufficient to show that ∞ n=1 P(A n )=∞.
Whichever model is considered, we have P(A n )=p( xy |Φ 0 )q(1p) where p(r|Φ 0 ) is the probability of success when the distance between the emitter and the receiver is r and given Φ 0 . It is thus sufficient to prove that p( xy |Φ 0 ) > 0. It is clearly the case in the ideal and collision models as xy <R(S). For the sinr model, it is a bit more complicated.

Let I d (Φ 0 (y)) be the interference stemming from all the nodes of Φ 0 located outside the ball of radius d centered on y. For the path-loss presented in Section 4 and since Φ 0 is a Poisson point process, the interference I(Φ 0 (y)) is almost surely finite. Therefore, for all ǫ> 0 it exists almost surely a finite distance d such that the probability that {I d (Φ 0 (y)) <ǫ} is strictly positive.

So we have a strictly positive probability to have an interference level coming from outside of B(y, d) inferior to ǫ. Moreover, we have a strictly positive probability to have no interference coming from inside

B(y, d) as Φ 0 (B(y, d)) is finite (since d is almost surely finite). Since I d (Φ 2 (y)) ≤ I d (Φ 0 (y)), this result also holds for I d (Φ 2 (y)) even if Φ 2 is not an independent thinning of Φ 0 .
As the product of these two probabilities is an inferior bound to p( xy |Φ 0 ), the probability for the adjacency to be observed is thus strictly positive.

This result can be extended to the case where the random variables of the sequence (S i ) i are randomly distributed (but in such a way that I Φ is almost surely finite). We have to prove that, for the sinr model, a link observed in the G i t (Φ 0 ) or G c t (Φ 0 ) graphs has a positive probability to be observed during a slot. If we suppose that S i varies in time but that they are independent in each slot, the same proof holds. If we suppose that S i does not vary in time, the same line as the proof applied to P(A n )=p( xy |Φ 0 , (S i ) i ) leads to the result. This result is interesting as it says that whichever radio model is used, at the end, the neighbor discovery process leads to the same connectivity graph. However, this asymptotic notion is problematic as the number of rounds required to reach this convergence can be infinite. During the real execution of the discover process, i.e. after a finite number of rounds, the obtained logical connectivity graphs can largely vary as observed in Section 6. We will show in the next Section that the radio link properties also largely vary.

Radio link probability

If the different models asymptotically lead to the same logical connectivity graph, they have really different behaviors. As an example, the distributions of p(r) are different depending on the radio model. Figure 5 compares the link success probability for the three models as given by equations 5, 7 and 12. As we can see, in the ideal channel, a link is either up or down, with no failure, whereas in the two other channels, links are unreliable and transmissions are always prone to failure. For the two models where interference or collisions are considered, the success probability gets more and more distant from the ideal step function as the number of interfering nodes increase. This can be seen on Figures 6(a) and 6(b) where p(r) is depicted for various intensities of interfering nodes.

On Figures 6, we can observe the impact of the Rayleigh channel on the link success probability. In the collision channel, a constant transmission power S has been considered and thus, we have p(r)= 0, ∀r ≥ R(S). In the sinr model study of Section 7.3, we have considered an exponential distribution for S to introduce the fading phenomena. The mean of the exponential distribution has been chosen equal to the constant power S considered in the other models. Randomization of S has two impacts. First, it reduces p(r) for values of r inferior to R(S) as the received signal strength may be reduced by the fading. Second, it may create some links of length r ≥ R(S) as the fading phenomena modeled by the exponential distribution may also increase the transmitted power S compared to its mean value. The result is a curve p(r) which does not show the discontinuity property at r = R(S).

Average number of neighbors

As for the link success probability, the expected number of successful receptions for a given frame varies largely depending on the considered radio model. Figure 7 illustrates this value for the collision and sinr models as a function of the transmission power. As we can see, the expected number of correct receptions is larger in the collision model, around twice with the chosen numerical values, than in the sinr one. However, increasing the transmission power has the same effect in both models: the expected number of correct receptions increases and converges towards a constant. This differs from the ideal model where E [N ] increases linearly with S and diverges. From equation 8, we can state that the convergence value is 1-p p for the collision model and thus independent from the node density. It is not trivial to derive this limit from equation 13 in the sinr model except for the particular case where C =0:

E [N ] = lim S->+∞ (1 -p)qλ 0 2π +∞ 0 p(r)rdr = (1 -p)qλ 0 2π +∞ 0 lim S->+∞ p(r)rdr = (1 -p)qλ 0 2π +∞ 0 e -λ22π 2 (θ) 2 β r 2 βsin ( 2π β ) rdr = (1 -p)qλ 0 λ 2 βsin 2π β 2π(θ) 2 β = (1 -p) p βsin 2π β 2π(θ) 2 β
If we now consider the scalability of the neighbor discovery process, i.e., the evolution of the expected number of correct receptions as a function of the node intensity λ 0 , we observe radically different behaviors, as shown on Figure 8(a) and 8(b). If E [N ] first increases with λ 0 in both cases, it then converges towards 1-p p in the collision model whereas it starts decreasing towards 0 in the sinr one. Contrarily to the study of E [N ](S), the asymptotic behavior of E [N ](λ 0 ) is not similar in the two models.

To conclude with this comparison, and as a snapshot of the impact of collisions and interference on the behavior of a radio link, and by extension any communication protocol, Figure 9 illustrates the scalability of E [N ] and its asymptotic behavior in the three studied radio channel models. The observed differences clearly highlight the importance of a correct radio modeling during the design and study of a communication protocol.

Dimensioning of hello protocols

As an example of application, we propose to study the dimensioning of the energy aware Hello protocol described in Section 3. The goal of dimensioning is to choose correctly the protocol parameters in order to adapt the link probability success between a node and its neighbors according to the application constraints, while minimizing the protocol energy consumption. A real scenario stemming from the CAPNET project is proposed as a case study.

Probability of node discovery

In real world deployments, we can expect a sensor network or delay-tolerant network application to have some constraints on the neighbor discovery process. Not necessarily hard constraints, but probabilistic or statistical constraints such as: "all nodes at distance at most 10 meters must be discovered as neighbors with high probability" or "node at distance more than 15 meters must not be seen as neighbors. Mobility may also add some constraints on the neighbor discovery process: "all nodes remaining more than 30s at less than 10 meters should be discovered as neighbors with high probability". The study done in Section 7 offers the opportunity to dimension the protocol to match this type of constraints. Given the use of sleep periods, the link success probability is not enough to determine the correct reception of a Hello packet by a node. Indeed, if the considered receiver is not active, the packet is lost. Thus the probability of link success, p(r), computed in Section 7 has to be refined in order to cope with inactivity periods. Given a transmitter x and a receiver y such that xy = r, we define the probability of node discovery p d (r) as the probability that y correctly receives the Hello packet from x while in listen mode. This probability depends on the probability of link success p(r) and the probability that y is listening to the medium:

p d (r)= w -τ w + s p(r) (14)
Given this definition, if the application requires all nodes within distance at most L to be discovered with probability a least p min , we must select w, s and S in order to satisfy the condition: p d (L) ≥ p min .

In order to cope with constraints bound to mobility, we can also be interested in the number of times a Hello packet must be sent by x before its correct reception by y. We denote by ̺ the random variable representing the first time y correctly receives the Hello packet. If the successive transmissions were independent, ̺ would follow a discrete geometric law defined as follows:

P(̺ = k)=p d (r) (1 -p d (r)) k-1 and E[̺]= 1 p d (r) .
Unfortunately, the successive transmissions may not be independent. Indeed, if a transmission fails, it indicates the presence of some emitting nodes in the neighborhood which have interfered. And if the emitters are the same between the successive transmissions, it induces a correlation between the link success probability of each attempt. This correlation is such that the distribution of ̺ is not a geometric law. However, if the intensity of interfering nodes is not excessively high compare to the intensity of nodes (all the nodes), the geometric law can be used as a good approximation.

Case study: the CAPNET project

As a case study, we consider the example of the CAPNET project. In this project, it is planned to equip all students of the French engineer school INSA de Lyon with contact loggers. The task of these sensor is to periodically discover all the students present in their owner vicinity, to log this information and periodically upload it in a database through the relay of base stations. The collected data should be useful to study graphs of human interaction as well as students mobility. In the big lines, this project is similar to the pocket-switched network experience that took place at the INFOCOM 2005 conference in Miami, USA.

In the CAPNET project, one objective is to have the experiment last for several weeks. It induces a very strong constraint on the energy consumption of the neighborhood discovery process. Moreover, the technical requirements of the project are such that a contact must be logged whenever two sensors are closer than 10 meters for a period superior or equal to 30 seconds.

Description of the sensors and assumptions

The sensors which are used in the CAPNET experiment are built with the CC1100 communication chipset (8). The CC1100 is a RF transceiver characterized by a low power consumption and effective radio performances. The main characteristics of this transceiver are presented in Table 1. We make the following assumptions given by the experiment constraints. We consider the different available transmission powers (from -30 to 10dBm) with a sensitivity threshold equal to -88dBm and a radio frequency of 900M hz. We suppose an average presence of 5 nodes per 20m 2 which yields to a density λ 0 =0 .0125. We assume that the size of the Hello packets is 18 bytes and that the data rate of the sensors is 2.4kbps. The transmission of one hello packet has a duration of τ = 60ms.

Parameters Values

In Table 2, we present the maximal communication range, R(S), associated to the various available transmission powers. The average number of nodes present in the disc of radius R(S) and the energy consumption for the transmit mode are also presented. We assume that the average energy consumption in receive mode is 14, 8mA.

Finally, given that nodes (i.e., the students) are mobile, we wish to dimension the Hello protocol to guarantee the discovery of a node located at a distance below L = 10m during at least 30s with a probability of 90%. According to the results of Table 2, we can already restrict ourselves to a transmission power S ≥-20dBm. The probability of node discovery (see Eq. 14) is depicted on Figure 10 for different sizes of w and s. On Figure 10(a), the probability of node discovery is plotted according to the collision model, while on Figure 10(b), we consider the sinr model. We notice on both figures that the probability of node discovery is raised for a high w duration and a short s period. Indeed, increasing w reduces the number of interfering nodes and thus increases the link success probability. In addition, decreasing s reduces the probability for the receiving node to be in the inactive state and thus increases the probability to discover the emitter.

Hello protocol's dimensioning

On Figure 11(a) the probability of node discovery is depicted for the collision model and the sinr one according to different transmission powers ranging from -20dBm to 0dBm. The collision model is depicted for only one transmission power as its link success probability is not function of this parameter. We can observe from the obtained results that an increase of the transmission power raises the probability of node discovery in the sinr model. For example with a transmission power of -20dBm, this probability remains lower than 0.8. However if we increase the transmission power to -10dBm, this probability is raised above 0.9. Thus, when dimensioning the Hello protocol, both the transmission power and the periods w and s have to be well defined. Finally, the normalized energy consumption is plotted on Figure 11(b) for varying sizes of w and s. This normalized energy is defined as : e(w, s)= w w+s . As expected, we notice that the energy consumption increases with w and decreases as the inactivity period s increases. It is then important to choose the adequate size for w and s to ensure a low energy consumption. In order to dimension the parameters, we resolve numerically the equation 14. We make the size of w vary and for each considered value, we seek the optimal size of s such that the constraint p d (10) ≥ 0.9 is respected. Many configurations are then obtained and the one which minimizes the normalized energy consumption is selected. This process is repeated for the collision model as well as for the sinr model with a transmission power ranging from -20dBm to 20dBm. The obtained results are summarized on Table 3. As already seen in Figure 11(a), the use of a transmission power S = -20dBm does not allow to meet the dimensioning requirements in the sinr model. We also observe that the use of a transmission power higher or equal to -10dBm guarantees a probability of node discovery higher than 0.9. Thus, we must consider S ≥-10dBm.

To choose between the acceptable values of S, we look more closely to the protocol energy consumption. We refine the normalized energy consumption e S (w, s) definition to consider the transmission power: e S (w, s)= P rx w + P tx (S)τ w + s

where P rx = 14, 8mA and P tx (S) values are taken from Table 2. Given he numerical values of Table 3, the set of parameters that finally minimizes the energy consumption of the Hello protocol is S =0dBm , w = 9840ms and s = 1000ms.

Conclusion

In this paper, we have analytically analyzed the impact of collisions and interference on two Hello protocols in the context of multi-hop wireless networks. We have computed the link success probability as well as the expected number of nodes that correctly receive a Hello packet in three radio channel models that handle interference in very different ways. Using this analysis, we have shown that if the neighbor discovery process is asymptotically equivalent in the three models, it offers very different behaviors locally in time. In particular, the scalability of the process varies depending on the way interference is handled. Finally, we have proposed a methodology to dimension Hello protocol parameters regarding both application constraints and energy consumption. Several parts of this study are open to extensions. First, the radio channel model could still be refined. For example, shadowing has not been considered. Fading could also be better modeled using generic laws as the Nakagami functions family. More important, the condition imposed by θ on the sinr for a safe reception corresponds to an ideal model and does not correspond to realistic conditions. In a more realistic perspective, transmissions suffer errors as a function of their sinr and their modulation. This is modeled by a bit error rate (ber) which introduces more uncertainty in the link success probability study. This extensions is very promising as up to now, very few works [START_REF] Gorce | Impact of radio link unreliability on the connectivity of wireless sensor networks[END_REF] have considered the modulation and the ber in the study of network and communication protocol properties.

Another interesting extension corresponds to the consideration of a carrier sense mechanism in the modeling of the Hello protocol execution. This is not trivial as the point process associated to the simultaneous emitters can no more be modeled by a Poisson point process. It can however be modeled by a Matern Hard-core process. This modeling has recently been proposed in [START_REF] Nguyen | A stochastic geometry analysis of dense ieee 802.11 networks[END_REF] in the context of dense 802.11 networks and should be extended to sensor networks.

Fig. 1 :

 1 Fig. 1: A basic Hello protocol.

Fig. 2 :

 2 Fig. 2: An energy aware Hello protocol.

Fig. 3 :

 3 Fig. 3: Realization of a Poisson point process of intensity λ =0.0035 in a 500 * 500 square region.

(Fig. 4 :

 4 Fig. 4: Logical connectivity graphs for various radio channel models (w = 200, τ = 10, s =0 , λ =0 .0035, S = 50000, θ =1, β =3, W =1).

Fig. 5 :

 5 Fig. 5: Link success probability p(r) (S = 50000, β =3, θ =1, C =1, W =1, λ0 =0.0035, τ = 10).

  collision model (b) sinr model

Fig. 6 :

 6 Fig. 6: Link success probability p(r) (S = 50000, β =3, θ =1, C =1, W =1, λ0 =0.0035, τ = 10).

  collision model (b) sinr model

Fig. 7 :

 7 Fig. 7: Average nbr. of discovered neighbors (λ0 =0.0035, β =3, θ =1, C =1, τ = 10).

Fig. 8 :

 8 Fig. 8: Impact of λ0 on the average nbr. of successful receptions (S = 50000, β =3 , θ =1 , C =1 , W =1 , τ = 10, µ = 1 S ).

Fig. 9 :

 9 Fig. 9: Average nbr. of successful receptions (S = 50000, β =3, θ =1, C =1, W =1, λ0 =0.0035, τ = 10).

Fig. 10 :

 10 Fig. 10: Probability of node discovery (S = -20dBm, r = 10m, λ0 =0.0125, θ = -88dBm, τ = 60ms, δ =1, β =3, C =1)

  Probability of node discovery (s =0). (b) Energy consumption.

Fig. 11 :

 11 Fig. 11: Probability of node discovery according to S / SINR versus Collision model. (r = 10m, λ0 =0 .0125, θ = -88dBm, τ = 60ms, δ =1, β =3, C =1)

Tab. 3 :

 3 Hello protocol parameters dimensioning. (r = 10m, λ0 =0 .0125, θ = -88dBm, τ = 60ms, δ =1 , β =3, C =1)

† This work is partly supported by the European Commission, project AEOLUS IST-15964, and by the CAPNET project 1365-8050 c 2008