MD Souhayl ; Dahmani
email: souhayl.dahmani@rdb.aphp.fr

Daphnee Michelet

MD Paer-Selim Abback

MD Chantal Wood

MD Yves ; Nivoche

MD Jean Mantz

Robert

Debre Robert

Anesthesiology Hospital

Robert Debre

MD Daphnée Michelet

MD Christopher Brasher

MD Daphnée Michelet

A Meta Analysis of Published studies

Keywords: Ketamine, Children, Hyperalgesia, Analgesia Ketamine, Hyperalgesia, Chidren, Analgesia Chirurgie Ketamine Ketamine S+ Local anesthetics used Timing Premedication Peroperative analgesia Postoperative analgesia Preventive antiemetics

come

Material and Methods: A comprehensive literature search was conducted to identify clinical trials that used ketamine as a perioperative analgesic compound in children and infants.

Outcomes measured were: postoperative analgesic consumption, pain intensity and duration of sensory block (when ketamine was used by caudal route) during the PACU (Postoperative Care Unit) stay and the early postoperative period (6 to 24 hours after leaving the operative room). The data from each trial were combined to calculate the pooled odds ratios or standard mean differences and their 95% confidence intervals.

Results: Thirty five randomized, blinded controlled studies were retrieved from the literature. Systemic ketamine was effective in decreasing PACU pain intensity and analgesic requirement but failed to influence early (6 to 24 hours) pain intensity and analgesic requirement. Ketamine administered locally during tonsillectomy, decreased PACU and early (6 to 24 hours) pain intensity and PACU analgesic requirements. Used as an adjuvant for caudal analgesia, ketamine increased the duration of sensory block and PACU analgesic requirement without impacting PACU pain intensity. Ketamine failed to exhibit a postoperative opioid-sparing effect.

Conclusions: This meta analysis found that administration of ketamine was associated with decreased PACU postoperative pain intensity and non-opioid analgesic requirement.

However, ketamine failed to exhibit a postoperative opioid-sparing effect.

Introduction

Postoperative pain treatment has become a major concern in pediatric anesthesia [START_REF] Astuto | Methodologies for the treatment of acute and chronic nononcologic pain in children[END_REF][START_REF] Morton | Management of postoperative pain in children[END_REF] . Although effective analgesics and age specific pain evaluation tools are available [START_REF] Ko | Assessment tools for the evaluation of pain in the oncology patient[END_REF][START_REF] Hain | Pain scales in children: a review[END_REF] , postoperative pain is still undertreated in children [START_REF] Astuto | Methodologies for the treatment of acute and chronic nononcologic pain in children[END_REF][START_REF] Morton | Management of postoperative pain in children[END_REF] .

Many studies have found pain to cause central nervous system sensitization. Memory of pain has been shown to exacerbate both present and future pain experiences. Moreover, some analgesic medications used in the perioperative setting, such as opioids, have also been found to cause postoperative sensitization [START_REF] Lavand | 'homme P: Perioperative pain[END_REF] . Amongst neurobiological events involved in this phenomenon, the long term potentiation, initiated by NMDA receptor activation is one of the most important [START_REF] Clivatti | Review of the use of gabapentin in the control of postoperative pain[END_REF][START_REF] Kock | homme PM: The clinical role of NMDA receptor antagonists for the treatment of postoperative pain[END_REF] . Consequently, the use of NMDA antagonists, such as ketamine, has been hypothesized to counteract sensitization and produce an analgesic-sparing effect. Intravenous ketamine has been found to produce both analgesic and anti-sensitization effects in adult patients [START_REF] Bell | Remifentanil vs fentanyl/morphine for pain and stress control during pediatric cardiac surgery[END_REF][START_REF] Elia | Ketamine and postoperative pain--a quantitative systematic review of randomised trials[END_REF] . However, its efficacy in children is still controversial. The analgesic efficacy of ketamine has also been studied both locally, during tonsillectomy and during epidural analgesia in children. However, no study has been undertaken to review the evidence from these studies. Meta-analysis of published studies is a statistical methodology allowing aggregation and quantification of the therapeutic effects from multiple studies. It is particularly interesting when conflicting results arise from multiple studies. To date no meta analysis has been specifically performed on the efficacy of ketamine in the pediatric population. Thus, the goal of the present work was to perform a meta-analysis of the efficacy of ketamine in the management of postoperative pain in children when used by systemic, local or epidural routes.

Material and Methods

Bibliographic search and analysis

We conducted this meta-analysis according to the guidelines of the Cochrane Handbook for Systematic Reviews a of intervention and the QUORUM statements [START_REF] Moher | Improving the quality of reports of meta-analyses of randomised controlled trials: the QUOROM statement. Quality of Reporting of Meta-analyses[END_REF] .

Literature databases included Pubmed and Embase. The following queries were used: "ketamine or ketalar and children or infant ". In addition, a manual search of the references found in the selected articles (including reviews and meta-analysis) was also performed. The articles obtained from these queries were independently analyzed by four anesthesiologists and those meeting the following criteria were included in the analysis: randomized controlled study, double blinded, absence of neurological and/or psychiatric diseases, standardized anesthesia protocol, standardized analgesic administration (including rescue analgesics), and presence of a control group. In addition, congress abstracts were not included in the analysis in order to assess the quality of studies. These strong criteria allowed selections of articles with lowest methodological bias. The date of the most recent search was February 2010.

Readers assessed articles quality and extracted data. These data consisted of: patient's ages, performed surgery, sedative premedication (dose, timing and route of administration), ketamine administration (route, doses, and timing), hypnotic agents used, intraoperative analgesia, postoperative analgesia. Outcomes analyzed were: postoperative pain (pain scores or percentage of patients experiencing moderate to severe pain), postoperative analgesic requirements (either doses or percentage of patient receiving postoperative treatments), postoperative (24 hours) opioid consumption (morphine or codeine), duration of regional analgesia (defined by the time to first complaint of pain when ketamine was administered by caudal route), time to first analgesic administration (opioids or non opioids analgesics a http://www.cochrane-handbook.org/. Last access December 2010. postoperative nausea and vomiting (percentage of patients), and the occurrence of psychomimetic manifestations such as nightmares or hallucinations (percentage of patients). When conflicting results were found, the article was checked twice by the two anesthesiologists.

Statistical analysis

Statistical analyses were performed using the Review Manager 5 software (RevMan 5, The Cochrane Collaboration, Oxford, United Kingdom). When original data were expressed as continuous variables, meta analysis was performed using the standardized mean difference (SMD). This method allowed aggregation of outcomes measured using different tools. This statistical method is especially pertinent to pediatric pain which is assessed using different pain scales such as the OPS or EDIN scales [START_REF] Hannallah | Comparison of caudal and ilioinguinal/iliohypogastric nerve blocks for control of postorchiopexy pain in pediatric ambulatory surgery[END_REF][START_REF] Debillon | Development and initial validation of the EDIN scale, a new tool for assessing prolonged pain in preterm infants[END_REF] . In all other cases, outcome incidence analysis was performed using the odds ratio (OR) calculated using the Mantel-Haenszel method. In order to include a maximum of appropriate studies, incomplete data were managed by contacting the corresponding author or estimation of the mean and the SD on the basis of the sample size, median, and range according to the method described by Hozo SP and collaborators [START_REF] Hozo | Estimating the mean and variance from the median, range, and the size of a sample[END_REF] . The absence of a validated method to convert median, interquartile ranges to means, sd, lead us to discard all the data expressed in such a way. In articles where outcomes were expressed as continuous values, a partial standardized mean ratio was initially computed for each study, than transformed to partial ORs using Chinn's formulae [START_REF] Chinn | A simple method for converting an odds ratio to effect size for use in metaanalysis[END_REF] : LnOR = 1.814 x SMR (Ln : logarithm). The data were than entered as Ln(OR) and SD(LnOR) in the software.

Overall OR (and its 95% confidence interval) was then calculated using the inverse variance method b . Heterogeneity was assessed using I² statistics. This describes the percentage of the variability b http://www.cochrane-handbook.org/ (Section 9.4.6). Last access December 2010. in effect estimates (OR or SMD) that is due to heterogeneity rather than sampling error.

According to the Cochrane review guidelines c , an I² > 40 % and a p < 0.1 were considered as the threshold for heterogeneity and indicated the use of a random effect in OR and SMD computation rather then a fixed effect model. Random effect model assumes that the observed effects are estimating different intervention effects while a fixed effect model estimates the same "true" intervention effect. This difference makes the weight given for each study different. In the random effect model all studies are equally weighted while in the fixed effect model each study is weighted according to the number of included patients. In addition, subgroup analyses for ketamine efficacy were performed (when at least two studies included the considered outcome) according to the administration regime (bolus or continuous) and the use of perioperative opioid agents. The effects of the racaemic or the S(+) forms of ketamine were also investigated.

In studies with more than one intervention arm, each one was considered as a study and compared to the control group. Finally, to avoid calculation problems related to zero values, a 1 was added to all groups d . Statistical methodologies are available to assess the effects of unpublished studies on the results of meta-analysis (published bias). This type of bias is assessed by plotting the odds ratio (OR), or the logarithm of the OR, against a measurement of the precision of the OR such as the standard error of the OR. This plot is named the Funnel plot. An asymmetry of the Funnel plot could indicate that some studies might have been unpublished because their negative results c, [START_REF] Sterne | Systematic reviews in health care: Investigating and dealing with publication and other biases in meta-analysis[END_REF] . This asymmetry can also indicate the heterogeneity of results or the poor methodology of included studies [START_REF] Sterne | Systematic reviews in health care: Investigating and dealing with publication and other biases in meta-analysis[END_REF][START_REF] Sutton | Recent developments in meta-analysis[END_REF] . Some published studies, due to specific design, can produce large positive results that can lead to asymmetry in the Funnel plots interpreted as "publication bias". Methodological bias of studies can also exhibit large positive results c http://www.cochrane-handbook.org/ (Section 9.5.2). Last access December 2010. d http://www.cochrane-handbook.org/ (Section 16.9.2). Last access December 2010. [START_REF] Sterne | Systematic reviews in health care: Investigating and dealing with publication and other biases in meta-analysis[END_REF][START_REF] Egger | Bias in meta-analysis detected by a simple, graphical test[END_REF] . According to the Cochrane collaborative guideline e publication bias can be assessed when analysis aggregating at least 10 studies. Two tests were used in the current analysis the Begg-Mazumdar [START_REF] Begg | Operating characteristics of a rank correlation test for publication bias[END_REF] and Egger tests [START_REF] Egger | Bias in meta-analysis detected by a simple, graphical test[END_REF] . When these tests are significant, asymmetry is assumed to be present and publication bias is highly suspected.

Results were expressed as OR, MD or SMD [95 % confidence interval], I², p value for I² statistic. Funnel plots were not displayed in this article. Publications with no control groups, cohort observational studies, studies comparing ketamine to other therapeutics and finally studies with non exploitable results (data expressed as median and interquartiles ranges) were excluded from this meta analysis (figure 1). This meta analysis included 35 articles. The administration route for ketamine was systemic in 18 articles (567 patients received ketamine versus 418 did not received it) [START_REF] Aydin | Pain prevention with intraoperative ketamine in outpatient children undergoing tonsillectomy or tonsillectomy and adenotomy[END_REF][START_REF] Dal | The efficacy of intravenous or peritonsillar infiltration of ketamine for postoperative pain relief in children following adenotonsillectomy[END_REF][START_REF] Da Conceicao | Effect of an intravenous single dose of ketamine on postoperative pain in tonsillectomy patients[END_REF][START_REF] O'flaherty | Does ketamine or magnesium affect posttonsillectomy pain in children?[END_REF][START_REF] Elhakim | Ketamine reduces swallowing-evoked pain after paediatric tonsillectomy[END_REF][START_REF] Murray | Prevention of posttonsillectomy pain with analgesic doses of ketamine[END_REF][START_REF] Kararmaz | Oral ketamine premedication can prevent emergence agitation in children after desflurane anaesthesia[END_REF][START_REF] Umuroglu | Analgesia for adenotonsillectomy in children: A comparison of morphine, ketamine and tramadol[END_REF][START_REF] Abu-Shahwan | Ketamine does not reduce postoperative morphine consumption after tonsillectomy in children[END_REF][START_REF] Batra | Intraoperative small-dose ketamine does not reduce pain or analgesic consumption during perioperative opioid analgesia in children after tonsillectomy[END_REF][START_REF] Tarkkila | Comparison of remifentanil versus ketamine for paediatric day case adenoidectomy[END_REF][START_REF] Zahra | Intramuscular ketamine for prevention of postanesthesia shivering in children[END_REF][START_REF] Dix | Double-blind randomized placebo-controlled trial of the effect of ketamine on postoperative morphine consumption in children following appendicectomy[END_REF][START_REF] Engelhardt | Intraoperative low-dose ketamine does not prevent a remifentanil-induced increase in morphine requirement after pediatric scoliosis surgery[END_REF][START_REF] Butkovic | Comparison of a preincisional and postincisional small dose of ketamine for postoperative analgesia in children[END_REF][START_REF] Darabi | Ketamine has no pre-emptive analgesic effect in children undergoing inguinal hernia repair[END_REF][START_REF] Inanoglu | Intravenous ketamine and local bupivacaine infiltration are effective as part of a multimodal regime for reducing post-tonsillectomy pain[END_REF][START_REF] Bazin | Effects of perioperative intravenous low dose of ketamine on postoperative analgesia in children[END_REF] (table 1) ; local or topical during tonsillectomy in 4 studies (125 patients received ketamine versus 100 did not received it) [START_REF] Dal | The efficacy of intravenous or peritonsillar infiltration of ketamine for postoperative pain relief in children following adenotonsillectomy[END_REF][START_REF] Honarmand | The preventative analgesic effect of preincisional peritonsillar infiltration of two low doses of ketamine for postoperative pain relief in children following adenotonsillectomy. A randomized, double-blind, placebocontrolled study[END_REF][START_REF] Canbay | Topical ketamine and morphine for post-tonsillectomy pain[END_REF][START_REF] Erhan | Ketamine in post-tonsillectomy pain[END_REF] (table 2) and as an adjuvant to local anesthetics or opioid during caudal analgesia in 13 articles (392 patients received ketamine versus 323 did not received it) [START_REF] Akbas | Comparison of the effect of ketamine added to bupivacaine and ropivacaine, on stress hormone levels and the duration of caudal analgesia[END_REF][START_REF] De Negri | How to prolong postoperative analgesia after caudal anaesthesia with ropivacaine in children: S-ketamine versus clonidine[END_REF][START_REF] Gunes | Comparison of caudal ropivacaine, ropivacaine plus ketamine and ropivacaine plus tramadol administration for postoperative analgesia in children[END_REF][START_REF] Kumar | Caudal additives in pediatrics: A comparison among midazolam, ketamine, and neostigmine coadministered with bupivacaine[END_REF][START_REF] Lee | Caudal ropivacaine and ketamine for postoperative analgesia in children[END_REF][START_REF] Locatelli | Analgesic effectiveness of caudal levobupivacaine and ketamine[END_REF][START_REF] Ozbek | The comparison of caudal ketamine, alfentanil and ketamine plus alfentanil administration for postoperative analgesia in children[END_REF][START_REF] Weber | Caudal bupivacaine and s(+)-ketamine for postoperative analgesia in children[END_REF][START_REF] Pathania | Comparison of efficacy and duration of caudal analgesia produced by bupivacaine alone and in combination with adrenaline and ketamine in children[END_REF][START_REF] Choudhuri | Comparison of caudal epidural bupivacaine with bupivacaine plus tramadol and bupivacaine plus ketamine for postoperative analgesia in children[END_REF][START_REF] Naguib | Ketamine for caudal analgesia in children: comparison with caudal bupivacaine[END_REF][START_REF] Pan | Caudal analgesia in paediatrics: Comparison between bupivacaine alone and in combination with ketamine, midazolam and ketamine-midazolam[END_REF][START_REF] Nafiu | Comparison of caudal ketamine with or without bupivacaine in pediatric subumbilical surgery[END_REF] (table 3).

Systemic route ketamine

All included studies used racemic ketamine as either a single bolus or as a bolus plus an intraoperative continuous administration, except in two studies [START_REF] Engelhardt | Intraoperative low-dose ketamine does not prevent a remifentanil-induced increase in morphine requirement after pediatric scoliosis surgery[END_REF][START_REF] Bazin | Effects of perioperative intravenous low dose of ketamine on postoperative analgesia in children[END_REF] in which ketamine was administered continuously both intra and postoperatively (table 1). Efficacy outcomes included pain intensity scores and analgesic requirements, both during the PACU stay (the first two hours) and the early postoperative period (between 6 and 24 hours). These two ranges were defined based on individual study results to maximize the data included in the meta analysis. Ketamine was associated with a decrease in PACU postoperative pain scores and analgesic requirements (figures 2A and 2B, respectively). However, it failed to improve early postoperative pain intensity and analgesic requirements (figures 3A and 3B, respectively). Analyzing studies that reported cumulative opioid administration during the F o r P e e r R e v i e w 10 first 24 hours postoperatively found that ketamine was not associated with an opioid-sparing effect during this period (figure 4).

The Results of the overall analysis did not take in account interventions performed, single bolus versus continuous administration of ketamine or preoperative or postoperative analgesics used (opioids versus non opioids). This is reflected by the high heterogeneity of the results obtained by pooling all studies. Consequently, subgroup analyses were indicated.

Analysis of ketamine efficacy according to the intraoperative, postoperative or perioperative (intra and postoperative periods) opioid use, showed this factor to influence analyzed outcomes (table 4). When opioid agents were used, results were similar to the overall analyses (decrease in PACU pain intensity and analgesic requirement, without impacting upon early outcomes). Conversely, when anesthesia and/or postoperative analgesia did not involve opioid agents, ketamine decreased early postoperative pain scores and early analgesic requirements without impacting nor PACU pain scores neither PACU analgesic requirements (table 4). All the included studies used intraoperative Nitrous Oxide (N 2 O). Consequently, the effect of this agent on ketamine efficacy could not be analyzed.

Analyzing studies involving tonsillectomy, found an efficacy of ketamine on pain scores and analgesic requirements during the PACU and the early postoperative stages (table 4).

Ketamine was only effective in decreasing the PACU postoperative pain when used during abdominal, orthopedic or urologic surgeries (table 4).

Local or topical ketamine during adenotonsillectomy

All the included studies used the racemic form of ketamine either by local infiltration in the surgical field or by a topical application during adenotonsillectomy (table 2). Efficacy outcomes analyzed were: PACU (the two first hours) pain scores and analgesic requirements and early pains scores and analgesic requirements (between 6 and 24 hours). Ketamine decreased PACU pain scores and analgesic requirements (figure 5A and 5B, respectively).

Concerning early outcomes, ketamine decreased pain scores without impacting early analgesic requirements (figure 6A and 6B, respectively).

Given the number and characteristics of the included studies, subgroup analysis could not be performed. All the included studies used intraoperative N 2 O. Consequently, the effect of this agent on ketamine efficacy could not be analyzed.

Ketamine as an adjuvant during caudal analgesia

When given as an adjuvant during caudal analgesia, racemic and S(+) ketamine increased the block duration (figure 7A) and decreased analgesics requirement in the PACU (figure 7B). However, it failed to decrease pain intensity at admission to the PACU and at 60, 180 and 360 minutes postoperatively (figures 8A, 8B, 8C and 8D, respectively).

Subgroup analysis according to the racemic forms of ketamine, found no influence of the racemic or S+ form on the results (table 5). However, when analysis included studies using intraoperative, postoperative or perioperative opioids, postoperative analgesic requirements were equivalent in the ketamine and the control arms (table 5). Conversely, when studies involving only non-opioid analgesics were analysed, ketamine exhibited superiority over controls (table 5). All the included studies used intraoperative N 2 O. Consequently, the effect of this agent on ketamine efficacy could not be analyzed.

Neither the occurrence of PONV nor that of psycho-mimetic manifestations were influenced by ketamine administration (OR = 1.17 The main finding of this meta analysis can be summarized as follows: used systemically, ketamine was effective in decreasing pain intensity and analgesic requirements during the PACU stay but not during the early postoperative period (6 to 24 hours). However, used locally during tonsillectomy, ketamine decreased both PACU and early (6 to 24 hours) postoperative analgesic requirement. Finally, used as caudal analgesia adjuvant, it increased the duration of sensory block and improved pain relief at emergence without influencing pain scores in PACU.

The subgroup analysis of studies using opioid analgesics either intra and/or postoperatively, found systemic ketamine to have no impact upon early postoperative pain intensity or analgesic requirement. In contrast, studies without intra and/or postoperative opioid administration showed systemic ketamine to reduce early pain intensity and analgesic requirement (table 4).

Insufficient ketamine dosage and/or administration regimen might explain the lack of efficacy of ketamine when co administered with opioid agents or when administered continuously (table 4). Most studies performed in adults have used ketamine as a continuous administration during the perioperative period [START_REF] Elia | Ketamine and postoperative pain--a quantitative systematic review of randomised trials[END_REF][START_REF] Subramaniam | Ketamine as adjuvant analgesic to opioids: a quantitative and qualitative systematic review[END_REF] . Perioperative continuous administration was found in two studies included in our meta analysis [START_REF] Engelhardt | Intraoperative low-dose ketamine does not prevent a remifentanil-induced increase in morphine requirement after pediatric scoliosis surgery[END_REF][START_REF] Bazin | Effects of perioperative intravenous low dose of ketamine on postoperative analgesia in children[END_REF] . Additionally, ketamine doses were similar to, and in some cases lower than, those used in adults [START_REF] Subramaniam | Ketamine as adjuvant analgesic to opioids: a quantitative and qualitative systematic review[END_REF] , whilst pharmacokinetic and pharmacodynamic studies in children and adults indicate a faster pharmacokinetics (The context-sensitive half-time in children (10 kg) after 1.5 h, rising from 30 min at 1 h to 55 min at 5 h at 3 mgxk -1 gxh -1 , the context-sensitive half-time in adults was 23 min at 1 h and 83 min at 5 h at the same infusion rate [START_REF] Dallimore | Ketamine anesthesia in children-exploring infusion regimens[END_REF] ) and a lower sensitivity in pediatric populations when compared to adults (concentrations of ketamine at awakening in children and adults were 0.9 to 3.8 mgxL -1 and 0.5 mgxL -1 , respectively [START_REF] Idvall | Ketamine infusions: pharmacokinetics and clinical effects[END_REF][START_REF] Grant | Ketamine disposition in children and adults[END_REF] 14 doses of ketamine adapted to the pharmacology of this agent in children. These pharmacologic considerations might also explain the absence of an opioid-sparing effect of ketamine. Alternatively, this result can also be explained by studies reviewed that included few cases of major intracavitary or major orthopedic surgeries (table 1). Another concern about the absence of an opioid-sparing effect observed in our study was the possibility of underestimation of pain and systematic pre emptive analgesia that might misestimates the true opioid requests. However, this is very unlikely while postoperative opioids were given by care givers according to pain evaluation scales (table 1). Pain scales used in the included studies (OPS 11 , EDIN [START_REF] Debillon | Development and initial validation of the EDIN scale, a new tool for assessing prolonged pain in preterm infants[END_REF] , CHEOPS [START_REF] Suraseranivongse | Cross-validation of a composite pain scale for preschool children within 24 hours of surgery[END_REF] scores, analog visual and verbal numerical scales [START_REF] Bailey | Validation and properties of the verbal numeric scale in children with acute pain[END_REF] ) were another concern in the interpretation of our results. These scales were not validated during ketamine administration either alone or in association with other analgesics and they can not discriminate the analgesia from the sedation observed after ketamine administration [START_REF] Hannallah | Comparison of caudal and ilioinguinal/iliohypogastric nerve blocks for control of postorchiopexy pain in pediatric ambulatory surgery[END_REF][START_REF] Debillon | Development and initial validation of the EDIN scale, a new tool for assessing prolonged pain in preterm infants[END_REF] .

Consequently, the effects of ketamine observed in the absence of opioid administration might be due to the sedative effect of this agent rather than to its analgesic effects.

Ketamine has been shown experimentally to antagonize NMDA and opioid receptors [START_REF] Sarton | The involvement of the mu-opioid receptor in ketamine-induced respiratory depression and antinociception[END_REF][START_REF] Hirota | Stereoselective interaction of ketamine with recombinant mu, kappa, and delta opioid receptors expressed in Chinese hamster ovary cells[END_REF][START_REF] Minami | Analysis of the effects of anesthetics and ethanol on mu-opioid receptor[END_REF] . The absence of ketamine effects on early (6 to 24 hours) pain intensity and analgesic requirement when opioid were used might be explained by the absence of ketamine opioid receptors modulation in the presence of opioids. This hypothesis is supported by the higher affinity of opioids agents to the opioids receptors (for exemple, K i values of ketamine and morphine to inhibit the binding of [ 3 H]diprenorphine to µ-opioid receptors were 2.6x10 6 and 24.80 nM respectively [START_REF] Hirota | Stereoselective interaction of ketamine with recombinant mu, kappa, and delta opioid receptors expressed in Chinese hamster ovary cells[END_REF][START_REF] Amiche | Characterisation and visualisation of [3H]dermorphin binding to mu opioid receptors in the rat brain. Combined high selectivity and affinity in a natural peptide agonist for the morphine (mu) receptor[END_REF][START_REF] Yu | Thienorphine is a potent long-acting partial opioid agonist: a comparative study with buprenorphine[END_REF] ). However, this hypothesis seems very unlikely in this context while it is inconsistent with previous works that found NMDA receptors antagonism as the major mechanism of ketamine antihyperlgesia [START_REF] Lutfy | Antinociceptive effects of NMDA and non-NMDA receptor antagonists in the tail flick test in mice[END_REF][START_REF] Hingne | Blockade of NMDA receptors prevents analgesic tolerance to repeated transcutaneous electrical nerve stimulation (TENS) in rats[END_REF] . In addition, this hypothesis cannot give a satisfactory explanation about the difference in analgesic properties of ketamine in children and adults [START_REF] Bell | Remifentanil vs fentanyl/morphine for pain and stress control during pediatric cardiac surgery[END_REF][START_REF] Elia | Ketamine and postoperative pain--a quantitative systematic review of randomised trials[END_REF] . Our results showed that caudal ketamine was effective in reducing the emergence pain and in increasing the duration of sensory block without influencing pain intensity (until the 360 postoperative minutes). This fact suggests that in this type of surgery, the effect of caudal ketamine was relatively weak on epidural analgesia. Interestingly, the administration of S(+) ketamine, a more potent form than the racemic ketamine [START_REF] Pees | Comparison of analgesic/sedative effect of racemic ketamine and S(+)-ketamine during cardiac catheterization in newborns and children[END_REF][START_REF] Joo | The effects of ketamine and its enantiomers on the morphine-or dexmedetomidine-induced antinociception after intrathecal administration in rats[END_REF] , failed to decrease analgesic requirement in the PACU. Its rapid pharmacokinetics, in comparison to the racemic product (clearances of S(+) ketamine and R(-) ketamine are 19.1 ± 7.2 and 16.5 ± 4.8 mlxmin -1 xKg -1 , respectively [START_REF] Geisslinger | Pharmacokinetics and pharmacodynamics of ketamine enantiomers in surgical patients using a stereoselective analytical method[END_REF] ), could explain this result. Finally, postoperative systemic opioid administration totally blunts the analgesic sparing effect of caudal ketamine (table 5). This result can be explained by the weak effect of caudal ketamine. Otherwise, the competition of opioid agents and ketamine on opioid receptors, described experimentally, might also account in this result.

However, this hypothesis is very unlikely in this context while it implies a spinal effect of systemically injected opioids. In addition, experimental studies have shown ketamine to potentiate the analgesia produced by opiates when both drugs were administered intrathecally [START_REF] Boettger | Spinally applied ketamine or morphine attenuate peripheral inflammation and hyperalgesia in acute and chronic phases of experimental arthritis[END_REF][START_REF] Kosson | Intrathecal antinociceptive interaction between the NMDA antagonist ketamine and the opioids, morphine and biphalin[END_REF] .

Meta-analysis quality relies on three major factors. Firstly, the quality of selected studies; secondly the heterogeneity of results; and lastly the detection of publication bias.

Studies included in this analysis met very strict criteria as detailed in Material and Methods. These strict selection criteria were very likely to decrease the methodological bias of each study included in the analysis [START_REF] Sterne | Systematic reviews in health care: Investigating and dealing with publication and other biases in meta-analysis[END_REF] .

The methodological plan used in this review was applied to take into account the heterogeneity of studies. Subgroup analyses were performed to decrease the heterogeneity and to identify factors influencing the results. This goal was partially achieved with the analysis of the systemic ketamine administration. Conversely, despite subgroup analyses, this was not possible for the caudal administration of ketamine. One factor that might have caused the high heterogeneity was the variability of the sensory block.

Using two statistical tools, we found no significant publication bias. In addition, publication bias was unlikely to have been a problem in the present analysis because it is usually the case that unpublished studies are those with negative outcomes. The conclusions of this review were in large part negative, so the absence of more negative data might have little effect on the outcome of this analysis.

In conclusion, this meta analysis demonstrated the safety of ketamine. During systemic, local or caudal administration of ketamine, the analgesic effect of this compound was observed in the PACU period and was not associated with an opioid-sparing effect. Ketamine decreased early (6 to 24 hours) pain intensity and non-opioid analgesic requirements only when no opiates were given.

F o r P e e r R e v i e w

  Ketamine for Perioperative Pain Management in Children. A Meta-Analysis of Published Studies.Keywords: Ketamine, Hyperalgesia, Chidren, Analgesia analgesia, using both opioid and non-opioids agents, has become the standard care for postoperative pain management. Ketamine, a compound with analgesic and antihyperalgesic properties, has been shown to decrease postoperative pain and opioid requirements in adults. The goal of the present meta analysis was to investigate postoperative analgesic properties of ketamine in pediatric patients.

  defined pain intensity target during caudal ketamine administration),

  e http://www.cochrane-handbook.org/ (Section 10.4.3.1). Last access December 2010.

  identified using the selected criteria. Analysis allowed the selection of 53 relevant articles. The details of the selection process are summarized in figure1.

  Bolus mode was effective in decreasing PACU postoperative pain intensities and analgesia requirements and early postoperative pain intensities, while bolus plus continuous infusion failed to improve the postoperative pain or decrease analgesic requirement during neither the PACU nor the early postoperative stages (table 4). Concerning the adverse effects, ketamine given intravenously was not associated with postoperative nausea and vomiting (during the first 24 postoperative hours) or psycho-OR = 1.35 [0.99, 2.09], I² = 0 %, p = 0.52; OR = 1.52 [0.72, 3.24], I² = 0 %, p = 0.96; respectively). Analysis found no evidence of publication bias for all the following outcomes: PACU pain intensities (Begg-Mazumbar test : Kendall's Tau = -0.09 p = 0.65 or Egger's test : bias = -2.3, p = 0.58), early pain intensities (Begg-Mazumbar test : Kendall's Tau = -0.09 p = 0.71 or Egger's test : bias = -1.12, p = 0.), early analgesic consumption (Begg-Mazumbar test : Kendall's Tau = -0.2 p = 0.32 or Egger's test : bias = -1.7, p = 0.39), postoperative nausea and vomiting (Begg-Mazumbar test : Kendall's Tau = -0.09 p = 0.70 or Egger's test : bias = -0.2, p = 0.87) or psycho-mimetic manifestations (Begg-Mazumbar test : Kendall's Tau = -0.27, p = 0.29 or Egger's test : bias = -1.17, p = 0.18).

  Locally administered ketamine was not associated with an increased incidence of PONV or psycho-mimetic manifestations (OR = 0.60 [0.28, 1.29], I² = 0 %, p = 0.49; 1.00 [0.28, 3.55], I² = 0 %, p = 1; respectively).

  [0.69, 1.98], I² = 0 %, p = 0.83 and 1.7 2 [0.69, 4.26], I² = 0 %, p = 0.97, respectively). Analysis found no evidence of publication bias for all the following outcomes: block duration (Begg-Mazumbar test : Kendall's Tau = 0.3, p = 0.19 or Egger's test : bias = 11.5, p = 0.06), analgesic consumption (Begg-Mazumbar test : Kendall's Tau = -0.3, p = 0.17 or Egger's test : bias = -3.99, p = 0.07), postoperative nausea and vomiting (Begg-Mazumbar test : Kendall's Tau = -0.06, p = 0.73 or Egger's test : bias = -1.15, p = 0.41), motor block (Begg-Mazumbar test : Kendall's Tau = 0.08, p = 0.75 or Egger's test : bias = -1.16, p = 0.4) or urinary retention (Begg-Mazumbar test : Kendall's Tau = 0, p = 0.9 or Egger's test : bias = 0.02, p = 0.9).

  analyze separately the local ketamine effects, although, its analgesic efficacy could result from a systemic action. There has been a recent criticism of the local use of ketamine because of the possibility of an increased risk of postoperative hematoma and bleeding after peritonsillar ketamine infiltration[START_REF] Shah | Re: the efficacy of intravenous or peritonsillar infiltration of ketamine for postoperative pain relief in children following adenotonsillectomy[END_REF] . Local administration of ketamine during tonsillectomy decreased both the PACU and the early postoperative pain and produced an analgesic-sparing effect during the PACU period. No other benefice of local ketamine during this surgery could be analyzed.

1 :

 1 Characteristics of included studies using systemic ketamine. (IH : Inguinal Hernia, Circ : Circumcision; PCA : Patient Controlled Analgesia).Table 2: Characteristics of included studies using local or topic ketamine. Table 3: Characteristics of included studies using caudal ketamine. (IH : Inguinal Hernia, Circ : Circumcision; Orch : orchidopexia).Table 4: Subgroup analysis of the effects of surgeries, opioid analgesics and the administration regime on the analgesic efficacy of intravenous ketamine. Data are expressed as odds ratio (OR) or standard mean difference (SMD) with the 95 % confidence interval (CI), I², p for I². NP: not performed. Results in bold indicate significant effect of ketamine on the studied outcome. * indicate the use of a random effect model in OR or SMD computation. Table 5: Subgroup analysis of the effects of the racemic form of ketamine and the opioïds analgesia on the analgesic efficacy of caudal ketamine. Data are expressed as odds ratio (OR) or standard mean difference (SMD) with the 95 % confidence interval (CI), I², p for I². Results in bold indicate significant effect of ketamine on the studied outcome. * indicate the use of a random effect model in OR or SMD computation.

Figure 1 :

 1 Figure 1: Meta analysis flowchart. RTC: randomized controlled trials.

Figure 2 :

 2 Figure 2: Forest plot of meta-analysis of the effects of intravenous ketamine on postoperative

Figure 3 :

 3 Figure 3: Forest plot of meta-analysis of the effects of intravenous ketamine on early (6 to 24

Figure 4 :

 4 Figure 4: Forest plot of meta-analysis of the effects of intravenous ketamine on postoperative

Figure 5 :

 5 Figure 5: Forest plot of meta-analysis of the effects of local or topic ketamine on

  of each study (first author and year of publication) is the odds ratio (OR) or the standard mean difference (SMD) for individual trials, and the corresponding horizontal line is the 95% confidence interval (CI). The lozenge back in the figure is the pooled OR or SMD with the 95 % confidence interval (CI). Studies with more than one intervention group were starred (Author, year of publication* and Author, year of publication **).

Figure 6 :

 6 Figure 6: Forest plot of meta-analysis of the effects of local or topic ketamine on early (6 to

Figure 7 :

 7 Figure 7: Forest plot of meta-analysis of the effects of caudal ketamine on sensory block

Figure 8 : 1 :

 81 Figure 8: Forest plot of meta-analysis of the effects of caudal ketamine on pain intensities at

Table 4 :Table 5 :

 45 Subgroup analysis of the effects of surgeries, opioid analgesics and the administration regime on the analgesic efficacy of intravenous ketamine. Data are expressed as odds ratio (OR) or standard mean difference (SMD) with the 95 % confidence interval (CI), I², p for I². NP: not performed. Results in bold indicate significant effect of ketamine on the studied outcome.* indicate the use of a random effect model. F o r P e e r R e v i e w 0.26 [0.10, 0.66], I² = 85 %, p < 0.00001* 2.26 [1.53, 2.98] , I² = 93 %, p < 0.00001* Overall 0.37 [0.10, 1.40], I² = 84 %, p = 0.003* 0.21 [0.05, 0.82], I² = 86 %, p < 0.00001 * 2.35 [1.02, 3.67], I² = 95 %, p < 0.00001* 2.25 [1.34, 3.15], I² = 91 %, p < 0.00001* S+ Ketamine Racemic 0.80 [0.27, 2.32], I² = 68 %, p = 0.04* 0.18 [0.05, 0.60], I² = 85 %, p < 0.00001* 3.50 [1.63, 5.37], I² = 91 %, p < 0.00001* 2.03 [1.17, 2.88], I² = 92 %, p < 0.00001* Postoperative Opioids No postoperative Opioids 0.56 [0.19, 1.69], I² = 83 %, p < 0.00001* 0.09 [0.04, 0.23], I² = 53 %, p = 0.08* 3.07 [2.02, 4.12], I² = 94 %, p < 0.00001* 1.05 [0.16, 1.93], I² = 89 %, p < 0.00001* Intra and/or Postoperative Opioids No intra and/or postoperative Opioids 0.36 [0.10, 1.29], I² = 89 %, p < 0.00001* 0.15 [0.08, 0.31], I² = 0 %, p = 0.44* 2.35 [1.52, 3.19], I² = 91 %, p < 0.00001* 2.33 [0.92, 3.75], I² = 95 %, p < 0.00001* Intraoperative Opioids No intraoperative Opioids Analgesic consumptions at 24h (OR) Duration of analgesia (SMD) Subgroup analysis of the effects of the racemic form of ketamine and the opioïds analgesia on the analgesic efficacy of caudal ketamine. Data are expressed as odds ratio (OR) or standard mean difference (SMD) with the 95 % confidence interval (CI), I², p for I². Results in bold indicate significant effect of ketamine on the studied outcome. * : indicate results computed using a random effect model.

  ). Consequently, future studies have to focus on

	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 3 :

 3 Characteristics of included studies using caudal ketamine. (IH : Inguinal Hernia, Circ : Circumcision; Orch : orchidopexia).

Pediatric Anesthesia

Acknowledgement

The authors thank Mrs. The Professor Isabelle Murat for her help in improving the manuscript and Amal Sahli (PhD in statistics) in her help in statistics.

F o r P e e r R e v i e w


D T Nguyen

C Voisin

Ph Roussignol

C Roquelet

J S Lauret

G Cassabois

Phonon-induced dephasing in single-wall carbon nanotubes

Keywords: numbers: 78.67.Ch, 71.35.-y, 78.47.nd

We report original nonlinear spectral hole-burning experiments in ensembles of single-wall carbon nanotubes. We have measured the temperature dependence of the excitonic homogeneous linewidth from 5K up to room temperature. We show that the phonon-induced broadening in the weak coupling regime predominantly consists in pure dephasing. We present theoretical calculations of the exciton scattering rate due to phonon absorption within a one-dimensional framework for both excitons and phonons. We obtain a fair agreement with our experimental data and we discuss the specific properties of the exciton-phonon coupling in single-wall carbon nanotubes.

Single wall carbon nanotubes (SWNTs) are pure carbon-based nanostructures characterized by a nanometric radial quantum confinement and a translation invariance along the tube axis [START_REF] Saito | Physical properties of carbon nanotubes[END_REF]. The corresponding onedimensionality of carriers results in strong Coulomb correlations and in particular to a non-perturbative binding of an electron-hole pair into an exciton [START_REF] Rohlfing | [END_REF][3][4][5]. Compared to epitaxially grown semiconductor quantum wires with much lower excitonic binding energies [6], SWNTs appear as a model system with quasi-1D excitonic properties. The originality of SWNTs is even more pronounced when considering the exciton-phonon coupling. In SWNTs the vibrational excitations are also quasi-1D whereas quantum wires, and more generally epitaxially grown nanostructures, are embedded in a semiconductor crystalline matrix and thus interact with 3D phonons [7][8][9][10]. The exciton-phonon coupling is at the heart of the thermallyassisted carrier relaxation, and its understanding is of great interest both from a fundamental point of view and for the development of carbon-based optoelectronic devices operating at room-temperature.

The existence of phonon sidebands was identified as a specific feature of the strong exciton-phonon coupling in semiconducting SWNTs [11,12]. This interaction reflects in a dynamical structural distortion which contributes in binding energies of the order of hundreds of meV [11]. The variety of the exciton-phonon interaction in SWNTs is further illustrated by the evidence of a broad zerophonon line at low temperature, which strongly contrasts with the phenomenology observed in zero-dimensional nanostructures [13]. In fact, the study of the temperature dependence of the excitonic homogeneous linewidth provides a direct approach for accessing the intrinsic characteristics of the exciton-phonon coupling in SWNTs. However, optical measurements in ensembles of SWNTs always suffer from inhomogeneous broadening due to chiral variety or environment fluctuations. Such an issue can be experimentally addressed by performing measure-ments either at the scale of a single SWNT [13][14][15][16][17][18][19], or in SWNTs ensembles by means of nonlinear techniques such as four-wave mixing [20]. There is nevertheless a large dispersion in the quantitative analysis of the different temperature-dependent experiments [17,18,20]. Moreover, no theoretical calculation was reported so that there is still a lack for a quantitative interpretation of the phonon-induced broadening and its diameter dependence [18].

In this paper, we report original nonlinear spectral hole-burning experiments in SWNTs ensembles that allow us to study the temperature dependence of the excitonic homogeneous linewidth from 5K up to room temperature. We compare our measurements to the specific contribution due to phonon-assisted population relaxation and we find that pure dephasing dominates the phonon-induced broadening in SWNTs. In order to reach a quantitative interpretation of our data, we have calculated the exciton scattering rate by phonon absorption within a 1D framework for both excitons and phonons. From the fair agreement with our experimental values, we discuss the peculiar 1D properties of the exciton-phonon interaction in SWNTs.

Our sample consists of SWNTs synthetized by laser ablation of a doped graphite target and purified by ultrasonic and chemical treatment [21,22]. The linear absorption spectrum is composed of inhomogeneously broadened lines centered at 0.75 and 1.3 eV corresponding to the S 11 and S 22 excitonic transitions in semiconducting SWNTs, respectively [23]. Our work is focused on the fundamental S 11 transition (Fig. 1, inset). In order to suppress the influence of external non-radiative relaxation processes as much as possible, the SWNTs are first isolated by the micelle-wrapping technique which allows the observation of a bright photoluminescence signal [24,25]. The individual SWNTs are further embedded in a gelatin matrix following the procedure described in Ref. [26], in order to perform temperature-dependent measurements.

Homogeneous linewidth measurements are perfomed by means of nonlinear spectral hole-burning experiments. Our experimental scheme is based on differential transmission experiments in a spectrally resolved cw pumpprobe setup [27]. The pump laser is a semiconductor distributed feed-back laser seeding an erbium-doped fiber amplifier. Its output is coupled to a monomode fiber providing up to 1W at 0.8 eV (1.55 µm) within a spectral width δγ <100 µeV, and it is focused on a spot of 6 µm-diameter. The probe laser is an external-cavity semiconductor laser tunable from 1.52 µm to 1.6 µm. The pump-induced variation ∆T of the probe transmission T is detected by a lock-in amplifier at the frequency f 1 + f 2 , where f 1 and f 2 are the modulation frequencies of the pump and probe laser intensity, respectively. The orthogonally-propagating laser beams are both incident at 45 • on the sample mounted on the cold finger of a continuous-flow helium cryostat for measurements down to 5K. The narrow pump laser centered at 0.8 eV partially saturates the absorption of the SWNTs that are in resonance (Fig. 1). The corresponding sub-ensemble of SWNTs reduces the overall absorption of the sample in a spectral window determined by the homogeneous linewidth Γ. The photo-induced bleaching of the SWNTs sample is detected by recording the change of transmission of the tunable probe laser. A typical spectrum of the pumpinduced differential transmission ∆T /T of the probe laser is displayed in Fig. 1, for a pump power of 4 kW.cm -2 .

At low temperature, the nonlinear signal exhibits a pronounced bleaching with an amplitude of 2.10 -4 and a full width at half maximum (FWHM) of 7 meV, which is much smaller than the inhomogeneous broadening of the linear absorption spectrum (dashed line, Fig. 1), and much larger than the system response function width (solid line, Fig. 1).

In the case of excitonic transitions, the saturation of absorption occurs for bosonic excitations so that it can not be due to a redistribution of an electronic population among the ground and excited states as in atomic systems, but to a change of the oscillator strength f and homogeneous linewidth Γ of the excitonic transition, as described in Ref. [28]. The analysis of power-dependent measurements performed on the same sample [29] indicated that the major nonlinearity in SWNTs arises from the pump-induced change ∆Γ of homogeneous linewidth and that the oscillator strength renormalization ∆f is marginal. This implies that the width of the spectral hole is given by Γ, instead of the standard 2Γ value as in the case of absorption saturation in atomic systems [START_REF] Demtröder | Laser spectroscopy[END_REF].

The temperature dependence of the excitonic homogeneous linewidth is obtained by measuring the FWHM of the spectral hole-burning signal as a function of temperature. When raising the temperature, the differential transmission ∆T /T spectra become broader (Fig. 1), as expected from the phonon-induced broadening of the excitonic transition. Above 150K, the limited spectral range of our probe laser does not allow us to accurately determine the intensity of the background signal at large pump-probe detuning. This explains the reduction of the relative precision of our measurements of Γ close to room temperature (Fig. 2, inset).

The thermal broadening is fairly reproduced by the expression [17,18,20]:

Γ(T ) = Γ 0 + aT + b exp ( - E 0 k B T ) (1)
where Γ 0 is the residual homogeneous linewidth in the limit of zero temperature (in the range of several meVs [13,[16][17][18]), aT the broadening due to low-energy acoustic phonons, and b exp

( -E0 kB T
) the contribution of highenergy phonons, such as radial breathing modes, that dominates above 75K. While Ma et al. fit their data with a=0, b=254 meV and E 0 =105 meV [20], Yoshikawa et al. take on the contrary b=0 with a ranging from 20 to 35 µeV.K -1 [18]. In our case, the fit displayed as solid red line in Fig. 2 is obtained by taking Γ 0 =6.9±0.2 meV, and a=32±3 µeV.K -1 . Given the reduced precision of our measurements above 150K (Fig. 2, inset), the estimation of E 0 and b is more problematic. Even if the coupling to high-energy phonons is not at the heart of this paper, we have tried to obtain some estimates. Our data can be fitted by varying E 0 from 20 and 75 meV, while b changes from 10 to 250 meV, respectively. By fixing E 0 to the energy E RBM of the radial breathing mode at the zone center, we get b=11±4 meV with E RBM ∼25 meV in our SWNTs for which the diameter is about 1.2 nm [START_REF] Meyer | [END_REF]. In the following, we will specifically discuss the linear temperature dependence below 75K corresponding to low-energy acoustic phonons.

An interesting question is the contribution of puredephasing in addition to population relaxation in the phonon-assisted broadening in SWNTs. The pure dephasing is usually characterized by the T 2 time given by 1/T 2 = 1/2T 1 + 1/T 2 where T 1 and T 2 are the population and coherence relaxation times, respectively. Timeresolved photoluminescence showed that the intrinsic recombination rate increases linearly with temperature up to 300K [26]. Such a process provides a linear contribution ãT to Γ(T ) shown by the red dashed line in Fig. 2, with ã ∼0.04 µeV.K -1 [26]. Since ã a, we conclude that phonon-assisted processes predominantly lead to pure dephasing in SWNTs. Although phonon-induced broadening was experimentally investigated by means of various techniques for SWNTs of different diameters [13,[16][17][18]20], there is so far no theoretical calculation of the exciton-phonon coupling efficiency in these 1D systems. In order to provide a quantitative interpretation of our measurements, we have calculated the temperature-dependence of the excitonic homogeneous linewidth due to the interaction with acoustic phonons. For that purpose we have transposed in 1D the theoretical treatment developped for 3D excitons coupled to 3D phonons [9]. More specifically, we have considered the deformation potential interaction of 1D excitons with 1D phonons that consist of low-energy acoustic phonon modes such as twisting (T) and stretching (S) modes [33]. The assumption of 1D phonon is supported by the acoustical impedance mismatch of one order of magnitude between the carbon nanotubes and the gelatin. In the long-wavelength limit, the coupling constants u e,h q of an electron (e) or a hole (h) to a phonon of wave vector q [9] read in 1D for T and S modes:

u e,h q = √ q 2ρvL D e,h √ F φc (2)
where v is the longitudinal sound velocity [33], ρ the linear mass density, L the nanotube length, D e,h the deformation potential constant, and F φc the chiralitydependent coupling factor of the T and S modes defined in Ref. [33].

The homogeneous broadening due to the excitonphonon coupling is then given by the scattering rate of excitons at k = 0 due to phonon absorption, calculated by the Fermi golden rule [9]. This scattering rate is proportional to the Bose-Einstein occupation factor n(T ) of the involved phonons. In the case of low-energy phonons, n(T ) becomes linear for temperatures larger than T a where k B T a is the characteristic energy of the acoustic phonons involved in the scattering processes. In the present case, the linear term aT in Eq.1 thus accounts for the coupling to the T and S phonon modes. On the contrary, for high-energy phonons, n(T ) is approximated by a classical Boltzmann factor as long as the temperature is much smaller than E 0 /k B where E 0 is the characteristic phonon energy, given in the present case by E RBM equal to 25 meV in our 1.2 nm diameter SWNTs [START_REF] Meyer | [END_REF].

We focus on the linear term aT corresponding to the pure-dephasing due to S and T phonons and we provide below a quantitative interpretation of our estimation of a=32±3 µeV.K -1 . Pure dephasing processes only consist in intraband scattering within the fundamental excitonic band. Energy and momentum conservation laws impose that the mean acoustic phonon energy is given by 2M v 2 where M =m e + m h is the sum of the electron and hole effective masses. With v=20 km.s -1 [32,33] and M =0.2m 0 where m 0 is the free electron mass [START_REF] Saito | Physical properties of carbon nanotubes[END_REF], we obtain a value of 0.9 meV. This justifies the linear approximation of the occupation factor for T 10K. As far as the chirality-dependent factor F φc is concerned, it is difficult to get an estimate since we perform ensemble measurements where the pump laser saturates SWNTs of different chiralities within the excitonic homogeneous linewidth Γ. Since F φc varies from 1.45 to 1.95 as a function of the chiral angle φ c [33], we approximate it by its mean value F ∼1.7 in our calculations. Finally, we obtain for the average broadening efficiency a the following expression:

a = k B F ρ v 3 D e 1 1 + ( 2rB mev ) 2 -D h 1 1 + ( 2rB m h v ) 2 2 (3)
where r B is the exciton Bohr radius. The ratios 2rB m e,h v express cutoffs in reciprocal space for phonons of wavevectors smaller than the Bohr radius. With r B in the range of few nms [3], they are of the order of 0.1 and are neglected in Eq.3. As a matter of fact, the average broadening efficiency takes the simple expression:

a k B F ρ 0 d v 3 |D e -D h | 2 (4)
where we have rewritten the linear mass density as ρ 0 d (with ρ 0 the value for 1 nm SWNTs, and d the SWNT diameter in nm [33]). We would like to highlight that the exciton mass M does not appear in Eq.4 whereas a scales like M 2 in 3D [9] and M in 2D [10]. This effect arises from the modification of the density of states with the exciton dimension p which results in a M p-1 dependence. The striking absence of the exciton mass M in Eq.4 is thus specific to 1D systems and it leads to a situation where the acoustic phonon dephasing efficiency only depends on characteristic constants of the phonon modes.

The quantitative interpretation of our experimental data thus brings a direct estimation of the deformation potential constant D e -D h for low-energy acoustic phonon modes, which is poorly documented in the literature contrarily to the corresponding ones for radial breathing modes [34][35][36]. Note that due to the mirror symmetry of the conduction and valence bands in SWNTs, m e m h , and one expects D h -D e so that D e -D h does not vanish, as expected for the deformation potential interaction where the electron and hole couplings are additive instead of substractive as for the Fröhlich interaction with optical phonons [9]. With ρ 0 =1.9×10 -15 kg.m -1 [33], v=20 km.s -1 [32,33], the fitted value of a=32±3 µeV.K -1 leads to D e -D h of 4 eV, which is smaller than the theoretical value of 8 eV in Ref. [33] (labelled D A in this reference), whereas Galland et al. deduce a larger one of 14 eV from their experiments [13].

Eventually, we discuss the 1/d dependence of the a coefficient which was experimentally reported by Yoshikawa et al. [18]. Their interpretation is based on the analogy with epitaxial semiconductor quantum wells or wires where the a coefficient scales like the inverse of the nanostructure width W [7,8]. We show here that the underlying physics is however radically different in the two cases. In quantum wells or wires, the 1/W dependence arises from the coupling of 2D and 1D excitons to phonons of higher dimensionality (3D) since these epitaxial nanostructures are embedded in a crystalline matrix. Consequently, the characteristic component of the phonon wavevector along the confinement directions is determined by the inverse of the nanostructure width W because of the wavefunction matching condition that ensures a maximum matrix element for the exciton-phonon interaction [10]. In our theoretical model where 1D excitons interact with 1D phonons, the 1/d dependence can not have the same origin. As a matter of fact, the observed 1/d dependence of the a coefficient [18] comes from the linear mass density variation as a function of the nanotube diameter (Eq.4), which is is radically different from 1/W cut-off of the phonon wavevector in epitaxial quantum wells and wires.

In summary, we have studied the temperature dependence of the excitonic homogeneous linewidth by nonlinear spectral hole-burning experiments in ensembles of SWNTs. We find that the phonon-induced broadening predominantly consists in pure dephasing. We have developped theoretical calculations of the exciton scattering rate by phonon absorption within a one-dimensional framework for both excitons and phonons. We provide a quantitative interpretation of our experiments. Moreover, we show that the 1/d dependence of the acousticphonon dephasing efficiency is specific to the excitonphonon interaction in the 1D systems formed by SWNTs and that it originates from the linear variation of the nanotube linear mass density with diameter.

FIG. 1 .

 1 FIG. 1. (Left axis) Differential transmission ∆T /T (squares) of the probe laser as a function of energy with a pump laser of 4 kW.cm -2 at 0.8 eV, for three different temperatures: 5, 75 and 250K (the two upper curves are shifted for a better visibility). System response function (solid line) of the nonlinear spectral hole-burning setup. (Right axis) Optical density spectrum (dotted line) obtained by linear absorption spectroscopy of the carbon nanotubes gelatine sample. Inset: optical density spectrum on a larger scale.

FIG. 2 .

 2 FIG.2. Temperature dependence of the excitonic homogeneous linewidth: data (squares), theoretical fits (red solid line). The dashed red line indicates Γ0 + ãT where ãT is the temperature dependence of the population relaxation rate, from Ref.[26]. Inset: same display on a larger temperature range.

We acknowledge fruitful discussions with T. Michel, M. Paillet, and J. L. Sauvajol. This work was financially supported by the CNano IdF contract TAIGA and the ANR agency-contract BONAFO.

* e-mail: Guillaume.Cassabois@univ-montp2.fr