De Rosso

Camille Riera

Camille Schwartz

Counseling Parents About Child Feeding: a Qualitative Evaluation of French Healthcare and Childcare Professionals' Experiences and Their Perception of a Brochure Containing New Recommendations

Keywords: feeding recommendations, guidelines, health communication, health promotion, complementary feeding, qualitative research, parental feeding practices, breastfeeding, public health, pediatricians

Background: Parents are crucial in establishing their children's eating habits, and healthcare (HCPs) and childcare professionals (CCPs) can provide meaningful and trusted guidance on feeding, especially in the 0-3-year-old period. With the upcoming release of the o cial brochure containing the new child feeding recommendations in France, this study aims to: 1) assess professionals' practices and perceptions regarding their communication with parents on child feeding and 2) evaluate their impressions of the new brochure.

Methods: A 15-page brochure containing updated child feeding recommendations for children 0-3 years old was developed by the French public health agency. Online semi-structured interviews were conducted with professionals (n=21), including 13 pediatricians and general practitioners (HCPs), nine childcare assistants or professionals (CCPs) two weeks after they were provided with this brochure to read. The interview guide was developed and piloted with other professionals (n=3) prior to these interviews. Interview data were transcribed verbatim and analyzed thematically using an inductive approach.

Results: While HCPs and CCPs mostly communicate orally with parents, both acknowledged that the brochure might be a helpful supplement, especially for CCPs to legitimize their advice to parents. For HCPs, giving the brochure to parents may help provide systematic advice and save time during consultations. Professionals serving parents of lower socioeconomic status would prefer a supplement with less text and more illustrations. In general, the messages were perceived to be easily understandable but providing detachable cards to distribute according to the child's age would facilitate information dissemination and might be more useful to parents. Professionals reported that lack of training, the circulation of contradictory information, and language barriers were common challenges. Professionals suggested that short texts with more references and visual aids could improve the brochure.

Conclusions: French professionals welcome the new o cial brochure as a means to spread updated child feeding recommendations, but it could be modi ed to be better adapted to their needs of communication with parents and to facilitate the relay of information. Providing updated and consistent information to parents should be considered a priority for public health stakeholders toward increased adherence to new recommendations.

Background

The rst thousand days of life from conception constitute a crucial period in child development; in fact, this stage is characterized by rapid growth in terms of physical evolution and behavior establishment (1).

Interventions targeting early childhood provide a unique window of opportunity to establish optimal infant nutrition and healthy eating behaviors (2,3). Moreover, promoting healthy lifestyles from infancy is paramount for the prevention of chronic conditions such as childhood obesity, of which the increasing prevalence in developed and developing countries is growing into a signi cant global health concern (4).

Childhood obesity, in addition to being a strong risk factor for adult overweight or obesity (5), is associated with an increased risk of comorbidities throughout life, including the development of diabetes, certain forms of cancer, premature all-cause and cause-speci c mortality, and cardiovascular diseases (6-8).

Diet and physical activity are factors directly associated with the development of overweight and obesity [START_REF] Lioret | Dietary and Physical Activity Patterns in French Children Are Related to Overweight and Socioeconomic Status[END_REF][START_REF] Lobstein | Obesity in children and young people: a crisis in public health. Obesity reviews: an o cial journal of the International Association for the Study of Obesity[END_REF]. These factors are linked to behaviors that are sensitive to the effects of the different environmental stimuli that in uence child development during early life, as shown by Bronfenbrenner's ecological systems theory [START_REF] Bronfenbrenner | The ecology of human development: Experiments in nature and design[END_REF]. Among the external components that could affect children, parents play a decisive role in shaping the eating behaviors of their offspring, among other behaviors [START_REF] Wood | Caregiver In uences on Eating Behaviors in Young Children: A Scienti c Statement From the American Heart Association[END_REF]. For example, parents could modify children's food regulation capacities [START_REF] Hetherington | Understanding infant eating behaviour -Lessons learned from observation[END_REF][START_REF] Monnery-Patris | Development of a new questionnaire to assess the links between children's self-regulation of eating and related parental feeding practices[END_REF][START_REF] Brugaillères | Caloric compensation ability around the age of 1 year: Interplay with the caregiver-infant mealtime interaction and infant appetitive traits[END_REF]. Lack of responsiveness in parentchild interaction during meal times could impair the ability of the child to adjust his or her energy intake, often resulting in accelerated weight gain (16-18). Parental feeding practices related to introducing vegetable are also likely to in uence how much children like these foods [START_REF] Bell | Supporting strategies for enhancing vegetable liking in the early years of life: an umbrella review of systematic reviews[END_REF]. Addressing lifestyle behaviors and feeding practices with parents early in children's lives could help in the prevention of chronic diseases such as obesity and its health-related consequences [START_REF] Haire-Joshu | Preventing Obesity Across Generations: Evidence for Early Life Intervention[END_REF][START_REF] Smith | Prevention and Management of Childhood Obesity and Its Psychological and Health Comorbidities[END_REF]. Public policies, for child feeding in this case, should account for the latest scienti c contributions when addressing public health initiatives aimed at shifting the management of health education of the population toward evidencebased best practices [START_REF] Schwartz | Development of healthy eating habits early in life. Review of recent evidence and selected guidelines[END_REF][START_REF] Fewtrell | Complementary Feeding: A Position Paper by the European Society for Paediatric Gastroenterology, Hepatology, and Nutrition (ESPGHAN) Committee on Nutrition[END_REF].

Parents might worry about not knowing how to behave to establish healthy diet habits for their children, which can drive them to search for any kind of advice about feeding. The need for guidance might alter parents' ability to look for factual information [START_REF] Norton | Co-creating healthful eating behaviors with very young children: The impact of information overload on primary caregivers[END_REF]. Parents seek information about child feeding from many different sources, including the internet, media, books and magazines, family, friends, and health and childcare professionals [START_REF] Chouraqui | The French national survey on food consumption of children under 3 years of age -Nutri-Bébé 2013: design, methodology, population sampling and feeding practices[END_REF]26). The nature of the information conveyed by these different sources can vary widely, and the content of information can even be contradictory [START_REF] De Rosso | The Perceptions and Needs of French Parents and Pediatricians Concerning Information on Complementary Feeding[END_REF], resulting in the spread of disinformation and misinformation. Considering that early eating habits generally have a lasting impact throughout childhood and continue into adulthood (28), it is essential to support parents with the appropriate tools and guide them to promote the healthy development of children. In France, o cial recommendations for feeding children aged zero to three were updated and published in 2019 [START_REF] Anses | AVIS de l'Anses relatif à l'actualisation des repères alimentaires du PNNS -Jeunes enfants (0-3 ans)[END_REF][START_REF] Hcsp | AVIS relatif à la révision des repères alimentaires pour les enfants âgés de 0-36 mois et de 3-17 ans[END_REF]. A paper brochure intended for parents and professionals was produced by the French public health agency to make the recommendations relating to infant and young child feeding as accessible as possible to parents. From a public health perspective, a precise and systematic evaluation of communication tools produced by policy-makers and public health stakeholders at the design stage is useful to account for the needs of public health intervention recipients. In the present case, these recipients are parents and professionals, who are judged by parents as a natural intermediary for the circulation of such information [START_REF] De Rosso | Information seeking of French parents regarding infant and young child feeding: practices, needs and determinants[END_REF].

In France, when the child is between zero and sixteen years old, parents are encouraged to take her or him to 20 free and compulsory consultations with health care professionals (HCPs), and the majority of visits (13 visits) take place within the rst three years of life [START_REF]Suivi médical de l'enfant: examens médicaux obligatoires 2020[END_REF]. Therefore, health and childcare professionals (CCPs) play an important role in France for the support of harmonious child development: they interact with parents during the different phases of child feeding, and they are well placed to positively in uence parenting decisions. The majority of parents in France trust pediatricians; as the 2013 Nutri-Bébé study showed, 58% of mothers of children between 15 days and 35 months seek advice to feed their child [START_REF] Bocquet | Nutri-Bébé 2013 Study Part 2. How do French mothers feed their young children? Archives de pediatrie: organe o ciel de la Societe francaise de pediatrie[END_REF].

Moreover, a recent study explored French parents' information-seeking behaviors on infant and young child feeding. In terms of in uence on parental feeding practices, HCPs are the most used and trusted source of advice (primary source of information for 81% of parents with a level of in uence of 7.7 ± 1.7/10), followed by CCPs (in uence 7.3 ± 1.8, but less used, 30%) [START_REF] De Rosso | Information seeking of French parents regarding infant and young child feeding: practices, needs and determinants[END_REF]. For these reasons, it is relevant to address HCPs' and CCPs' perceptions of public health material to support parental feeding decisions.

Interventions given via HCPs have an impact on obesity prevention, as shown by quantitative and qualitative studies [START_REF] Toomey | A collaborative approach to developing sustainable behaviour change interventions for childhood obesity prevention: Development of the Choosing Healthy Eating for Infant Health (CHErIsH) intervention and implementation strategy[END_REF][START_REF] Stender | The Role of Health Care Providers in the Prevention of Overweight and Type 2 Diabetes in Children and Adolescents[END_REF](36). However, little research has been conducted on understanding professionals' practices and needs and integrating those needs to design effective prevention material for feeding children aged zero to three years. This material could be used to e ciently disseminate relevant messages and up-to-date information about child feeding to parents. Thus, the main objectives of this study were: (1) to investigate, via semi-structured interviews, the practices and needs of HCPs and CCPs when giving guidance to parents on child feeding (zero to three years old) and ( 2) to explore the rst impressions of professionals on the brochure produced by the French public health agency, together with their level of understanding and acceptance of the new recommendations.

Methods

Spreading child feeding recommendations via the mean of a brochure targeting parents

In France, there was a recent update of o cial guidelines for feeding children aged zero to three years, which were published by the French national agency for Food, Environmental and Occupational Health Safety (ANSES) in 2019 [START_REF] Anses | AVIS de l'Anses relatif à l'actualisation des repères alimentaires du PNNS -Jeunes enfants (0-3 ans)[END_REF]. The process was supervised by the Ministry of Health, and in October 2020, the High Council of Public Health released a report re ecting these benchmarks [START_REF] Hcsp | AVIS relatif à la révision des repères alimentaires pour les enfants âgés de 0-36 mois et de 3-17 ans[END_REF]. The main changes in the new recommendations (tested with this study) compared to the previous recommendations [START_REF] Ineps | La santé vient en mangeant et en bougeant[END_REF] are summarized in Table 1. Subsequently, in 2019-2020, the content for a paper brochure intended for parents was developed by the French public health agency (Santé publique France). Santé publique France organized a group of experts with the intent of designing material to make recommendations relating to feeding children aged zero to three years as accessible as possible to parents, who are the ultimate users of this communication strategy. The process of reformulating the recommendations and designing the brochure spanned November 2019 to July 2021, with the document becoming available to the general public in September 2021 (the version used in this study was one of the latest drafts).

The draft version of the brochure, comprising the core text without pictures or illustrations, was provided to the professionals interviewed in this study. The document contained 15 pages plus a table summarizing the recommendations based on the age of the child and for the quantities of each food group. The brochure consisted of eight chapters. The rst three chapters addressed generic topics pertaining to the development of eating behaviors of children and good practices for parents (including responsive feeding, discovering textures, screen exposure); the remaining ve chapters covered the development of feeding practices based on the age of the child.

Study design

Healthcare and childcare professionals' experiences with communication with parents about child feeding were undertaken using a qualitative inductive approach (38). Face-to-face, semi-structured interviews were considered the best methodology to address the aim of the study, as they could allow us to address speci c questions in line with the research aims, but they could still give the freedom to the participant to freely talk about his or her experiences [START_REF] Draper | Qualitative research in nutrition and dietetics: data collection issues[END_REF]. Given the context of the COVID-19 pandemic, interviews were scheduled online and conducted via videoconference when possible. In this way, it was easier to access participants in distant and different geographical locations, which allowed us to interview professionals from all over France [START_REF] Whitehead | Methodological and ethical issues in Internet-mediated research in the eld of health: An integrated review of the literature[END_REF]. Moreover, the advantage of seeing the person and capturing the body language of participants was maintained and could allow researchers to make considerations on that, if needed. Written notes were taken after each interview to determine whether the interviewer deemed it necessary to highlight the attitudes or reactions of participants. The consequences of the restrictions imposed by COVID-19 health management on professionals' practices and their interactions with parents (doing medical appointments via video or phone calls) were also assessed in this study but will not be discussed in the present article. The consolidated criteria for reporting qualitative research (COREQ) checklist was applied to the description of methods and results [START_REF] Tong | Consolidated criteria for reporting qualitative research (COREQ): a 32item checklist for interviews and focus groups[END_REF].

Participant recruitment

In a previous quantitative study exploring the sources of parental information on child feeding in France, it was highlighted that HCPs and CCPs are the most in uential sources of advice on child feeding for parents [START_REF] De Rosso | Information seeking of French parents regarding infant and young child feeding: practices, needs and determinants[END_REF]. Following what emerged in that study, we aimed to recruit a sample composed of two-thirds of HCPs, including pediatricians, general practitioners (GPs), and physicians working in maternal and child protection (PMI centers: protection maternelle infantile), and one-third of CCPs, including professionals working in childcare centers, childcare assistants, and nurses working in PMI centers. Professionals were eligible for the study if they were older than 18 years of age, practicing in France, and regularly interacting with young children (and consequentially their parents) as part of their professional practice. Retired professionals were excluded. To recruit participants, a purposive sampling method was initially used to ensure sample variety diversi cation: gender and age, profession and duration of professional experience, whether the participant has children, working in rural, urban, or semi-urban areas, and poverty rate of the working area. The objective was not to reach any quotas but rather to obtain a sample that was as balanced and diverse as possible. This initial method of recruitment was based on random calls or emails to HCPs and CCPs in departments that were pre-selected according to poverty and urbanization rates, [START_REF]Toujours plus d'habitants dans les unités urbaines[END_REF][START_REF]Taux de pauvreté selon l'âge du référent scal en 2018[END_REF] to include participants who dealt with parents of different socioeconomic status. As per the protocol, if enough participants could not be recruited in due time, then we planned to use the snowball sampling method. Participants were also recruited by means of Facebook groups.

Considering other qualitative studies interviewing healthcare professionals (44-46), 21 participants was considered su cient to ensure a diverse sample to explore the objectives of this study. However, a possible data saturation effect was evaluated, i.e., the recruitment of participants would stop if new interviews did not provide any additional information. Recruitment and data collection took place simultaneously. The recruitment outcome was determined if further recruitment was needed to ensure representatives from different professional categories.

Data collection

Data collection took place in April and May 2021. An online demographic questionnaire was developed, and an informed consent form was also included. The link to complete the questionnaire and the informed consent were sent to those professionals whom, after rst being contacted by the researchers (CRN or SDR) with a brief explanation of the study, volunteered to participate. No other links were established between researchers and participants before the study commenced. The eligibility questionnaire was sent to 118 professionals, of which 40 completed the questionnaire (25 HCPs, 15 CCPs), although one HCP was unreachable after having completed the eligibility questionnaire, and thus was not included in the study. After an eligibility check, the participants who corresponded to the required pro les were selected and included in the study (n=21), and none of the included participants refused to participate in the study. Once the inclusion phase was completed, individual, semi-structured interviews were then scheduled for a date and time that suited the participants. The brochure was sent to participating professionals by email at least two weeks before the interview to allow adequate time to read it.

Interview guide and interview procedure

The interview guide was rst developed based on data found in the literature [START_REF] Van Der Maas | Discussing overweight in children during a regular consultation in general practice: a qualitative study[END_REF] and then was tailored to the objectives of the study. This guide was checked by a reference group of public health stakeholders: three members of Santé publique France with backgrounds in nutrition and policy communication convened to advise the research team for the purposes of this study. This group worked on revising the brochure before national dissemination in September 2021. They reviewed and re ned the study protocol and the interview guide. The interview guide was pilot tested with three preliminary interviews involving one HCP and two CCPs. A preliminary analysis of these interviews allowed modi cations and re nements of the script and provided an estimate of the duration of interviews. This nal version of the script ensured that the ow of the interviews could develop according to the research objectives. However, there was a margin of exibility to enable participants to express their ideas and deepen their responses. The interview guide comprised three main topic sections: (1) perceptions and practices related to the communication needs of professionals about child feeding; (2) changes in communication with parents due to the COVID-19 health crisis; and (3) comprehension of a brochure intended for parents and acceptance of the new recommendations. For each section, there were several core questions and probes. The interview guide used for this study is presented in original language (French) as Additional le 1.

Semi-structured interviews took place in April and May 2021 in the professionals' native language (French). Interviews lasted an average of 54 ± 9 minutes. They were performed by the second author (CRN), a 25-year-old female, for the purpose of her thesis for her Master in Physiological and Psychological Food Choice Determinants. The interviewer had experience in health services research and engaging with HCPs as a result of her prior studies in medicine. The interviews were conducted either using a videoconference system (Skype for Business) or via telephone (two interviews). During the interviews, the participants were at their home (n=18) or workplace (n=3), and only the participant and the interviewer were present during the recording phase. At the beginning of each interview, CRN introduced herself (name and workplace) and gave a short introduction to the study and the data protection policy.

The participants did not know anything else about the interviewer. Professionals had to verbally con rm that they agreed to be audio recorded. After being asked all the questions regarding the interview topics, professionals were invited to share any additional information that they considered important. When the participants had not read the brochure prior to the interview (n=2), the interviewer gave them time to read it, either by postponing the interview or by calling for a break during the session, according to participant's preference. When there was nothing to add, participants were thanked for their time, and the following steps of the study were elucidated. The interviewer took some notes after each interview and had regular discussions with SDR, the PhD student (31-year-old female) with a background in public health and nutrition who oversaw this study and who contributed to the conception of the brochure and could participate in the previous phases of the brochure evaluation (e.g., focus groups with parents, evaluation of effect of the brochure on parental knowledge). All interviews were audio recorded and transcribed verbatim. After all the interviews concluded, the participants were sent a brief summary of the whole analysis (not a transcript) and were asked to refer to the researchers if they had anything to add or disprove. Participants did not provide additional feedback on the report to the researchers.

Ethical consideration

All the participants received a written description of the project and were informed of their right to withdraw at any time without prejudice. Professionals agreed to participate in the study by giving their informed consent and selecting the appropriate case in the online form. To ensure con dentiality of the information collected and anonymity, the participants were assigned a code that only the interviewer could match to their names. These codes were erased once the interviews were completed, and only sociodemographic characteristics were retained. Each participant received a voucher worth 40 Euros as a token of appreciation. Ethical approval (n°21-788) for this study was obtained from the Institutional Review Board (IRB00003888, IORG0003254, FWA00005831) of the French Institute of Medical Research and Health. This study was registered by the data protection service involved.

Data analysis

The responses of the professionals were audio-recorded and transcribed verbatim by a professional transcription service (11/21 interviews) or by the interviewer (CRN) with the help of a research technician with experience in transcribing interviews. All verbatim data were double checked for accuracy by the interviewer and the rst author (CRN and SDR). The software NVivo was used to analyze the interviews in their original language to avoid loss of meaning. A thematic analysis was conducted in accordance with the six steps proposed by Braun and Clarke (48): (1) data familiarization, (2) initial coding generation using a data-led approach, (3) searching for themes based on initial coding, (4) review of the themes, (5) themes nal de nition and labeling, and (6) report writing. The familiarization step was performed throughout the data collection phase by the rst two authors (SDR, CRN), who both listened to recorded interviews and read the transcripts. The rst author discussed with the interviewer on a regular basis during the interview phase to brainstorm about the themes based on emerging patterns (main ideas) in the data. The rst three steps were performed manually and independently by SDR and CRN. Initially, eight interviews were selected for independent initial coding. Coding was performed with a data-led approach, with particular attention to include the topics addressed in the interview guide. At the completion of this coding phase, the emerging themes and subthemes were discussed with the other authors until a consensus was reached and a nal coding scheme was created. The coding scheme was used to complete the coding of all the other interviews, which were coded independently by SDR and CRN.

The nal coding scheme was also used to readjust the initial coding of the rst eight interviews. SDR and CRN subsequently compared the coding of all the interviews and evaluated whether there were doubts or di culties regarding the placement of quotes. Then, a discussion took place until an agreement was reached, and all the codes were homogenized. At the end of this process, SDR and CRN once again reviewed the themes and subthemes and de ned those that were relevant for answering the research questions of the current study. A nal decision was made after discussion with SN. Data were analyzed in two parts: 1) discourse dealing with HCP and CCP practices and needs in terms of communication with parents about child feeding and 2) discourse dealing with their perception and impressions of the brochure and its evaluation.

Results

Table 2 summarizes the characteristics of the 21 participants. Interviews were undertaken with a total of 13 HCPs (nine women) and eight CCPs (seven women) from a range of disciplines: ve pediatricians, ve general practitioners, and three PMI physicians (working in maternal and child protection centers) for a sub-total of 13 HCPs, and two childcare assistants, three professionals working in childcare centers, one childminder, and two PMI nurses, for a sub-title of eight CCPs. Eleven of the 21 professionals were aged 25-34 years (8/13 HCPs, 3/8 CCPs), eight were aged 35-49 years and two were aged 50-64 years. Fourteen professionals out of 21 (including 11/13 HCPs and 3/8 CCPs) had from zero to ten years of professional experience. Four professionals had 11-20 years of experience (1/13 HCP, 3/8 CCPs), two professionals had between 21 and 30 years of experience (one HCP, one CCP), and one CCP had over 30 years of experience. Fifteen professionals had children (8/13 HCPs and 7/8 CCP). Nine out of the 21 professionals were working in areas with low poverty rates (< 13.3%), four in areas with high poverty rates (>17.2%) and eight in areas with middle poverty rates (between 13.3% and 17.2%). Fourteen out of the 21 professionals were working in rural areas (urbanization rate over 80%) and seven in semi urban areas (urbanization rate between 40% and 80%). 1 HCPs = Healthcare professionals.

2 CCPs = Childcare professionals.

*

Poverty rate: the monetary poverty rate corresponds to the proportion of individuals (or households) in a given department in a situation of monetary poverty. An individual (or a household) is considered poor when he lives in a household with a standard of living below the poverty line. In France and in Europe, the threshold is most often set at 60% of the median standard of living.

Two main themes and nine subthemes were developed through the thematic analysis. The main themes were: 1) practices and needs in terms of communication with parents on child feeding, and 2) perceptions of the brochure. Regarding practices and needs, four subthemes were identi ed: communication practices, communication barriers between professionals and parents, needs of professionals and needs of parents to improve communication. Regarding perceptions of the brochure, subthemes included new recommendations, general impressions, suggestions to improve the brochure, perceptions of the utility of the brochure in HCP and CCP practices, and of the use that parents can make of it. Detected differences between HCPs' and CCPs' discourse were reported in the results. The themes and subthemes are illustrated in Table 3 and described in detail below.

Table 3 Overview of themes and subthemes. Professionals mentioned that they mostly provided information on child feeding orally to parents. HCPs also suggest websites to parents where they could search for information and the list of the most cited websites is reported in Table 4. HCPs explained that they systematically spoke with parents about child feeding at regular consultations the rst year of life, especially on complementary feeding (CoF) when the child is from four or ve months of age. They mentioned that they do not have a rigid approach on how and when parents should start CoF. However, almost all HCPs reported that they liked to provide parents with a framework (starting with fruits and vegetables and introducing proteins after six months of age).

"So, I recommend starting at ve months but only with vegetables, and then introducing fruits. After six months, introduce animal proteins such as meat, sh, and eggs." (P2, HCP, General practitioner)

HCPs mentioned that exchanges about child feeding become less frequent and less systematic after one year. The media used by HCPs to support oral communication are the child health record booklet, paper documents adapted to the child's age range, and websites that parents can visit (see Table 4). Professionals working with the most disadvantaged families (especially in PMIs) explained using Google during consultations to show pictures on the computer screen and to translate their speech.

"We adapt to families, for example when we recommend whole milk, we put pictures on the computer, we say "the red cap", or "the pink cap" for follow-on milk; nally, we manage to nd solutions." (P20, CCP, PMI nurse)

Regarding CCPs, very practical information on child feeding is exchanged with parents (what the child eats at home and at the childcare center), especially orally during the morning (when parents drop off their child) and evening "transmission" times (when parents pick up their child) but also with the support of paper documents. It was also reported that parents' meetings are occasionally organized in some childcare centers.

Communication barriers between professionals and parents

Language was mentioned by professionals as a barrier to communication with parents, especially in PMIs, where professionals meet the most disadvantaged families (often with a foreign origin).

"One of the di culties we have at the PMI is that we have a lot of very precarious families [...] therefore they face many di culties other than the problems linked to their daily diet." (P12, HCP, PMI physician).

The lack of time and training (inconsistent training done on a voluntary basis) were also mentioned as barriers. It was also stated that the perpetual change in the child feeding recommendations confuses parents and impairs communication, since it induces the circulation of con icting information, even among professionals. Socioeconomic insecurity was reported to impact communication on child feeding.

"It's supposed to be our role, but we don't necessarily have the training, support or the time." (P7, HCP, General practitioner)

Needs of professionals to improve communication

Professionals mentioned that they needed "o cial documents" and nutrition training to improve their communication with parents to standardize the knowledge related to infant and young child feeding.

"There are a lot of different speeches that are not necessarily very serious and not very embarrassing, but I think that by standardizing the speeches, we would have more credit and it would be easier for parents." (P8, HCP, General practitioner) Professionals working in PMIs emphasized their need for more adapted media (in particular, a short document with many illustrations and few texts and/or in various languages).

"In fact, we would need either adapted documents in several languages, or much simpler things. We really need things that are quick and simple to show to parents. And that's kind of what we're missing." (P20, CCP, PMI nurse)

Needs of parents

Professionals reported that parents are asking for information on feeding, especially CoF. They noted that some parents need very speci c guidance from HCPs, while other parents prefer in the information they receive. Parents ask for details on the recommended quantities of food and milk as well as practical advice (e.g., how to introduce an allergen).

2) Perceptions of the brochure

New recommendations

Professionals were aware that it is now recommended to start CoF between four to six months of age to prevent allergies. However, when a mother was breastfeeding, professionals were more likely to recommend starting CoF at six months of age in accordance with the World Health Organization (WHO) recommendations (encouraging exclusive breastfeeding until six months of age).

"Sometimes breastfeeding mothers want to continue exclusive breastfeeding up to six months. In general, I am not really against it because the WHO recommends that. So, I don't have too many reasons to go against it especially if it's desired on their part." (P2, HCP, Pediatrician)

Recommending introducing all types of food between four to six months of age was still not grounded in practice among the interviewed professionals. Professionals were used to advising introducing proteins from six months of age, as suggested in the previous recommendations. Some doubts about the early introduction of allergens and proteins were expressed.

"Being able to introduce all the foods is the biggest novelty. We can introduce everything, taste more or less anything, in particular and at the same time, without respecting... waiting the six months." (P8, HCP, General practitioner) "What's the point of wanting to absolutely start protein right away between four and six months? […]. The info on the introduction of peanuts, which I used to do previously, but is it really scienti cally validated?

Because they don't all have the same indications about it between allergology organizations." (P5, HCP, Pediatrician)

According to the new brochure, pulses and whole grain starches can be introduced to children from four months of age. This recommendation sounded relatively new to professionals, but it appeared to be well accepted despite some doubts about their digestibility were raised. Some CCPs noticed that feeding practices in childcare centers do not meet this recommendation.

"I discovered that at four months you could give that: lentils and mashed beans. I did not know. It was complicated for me because the ber and the lentil, bean, or chickpea skins have to be very well mixed.

But I nd it good, it's interesting." (P21, CCP, PMI nurse)

Most of the professionals knew that it was possible to alternate follow-on milk and full-fat cow milk after the child turned one years old. They found that this was a good option to propose to disadvantaged families. However, there was a tendency to emphasize follow-on formula to avoid iron de ciency. The purpose of this recommendation seemed to be less understood by CCPs.

"And I had to do it before because I know that you can give whole cow's milk instead of follow-on milk. [...]

On the other hand, it is true that where I insisted, it was that it should not be exclusively cow's milk. They need a lot of iron and that is not covered by cow's milk." (P9, HCP, General practitioner) "I think there is ambiguity, and we are not sure if we are completely marketing or if this follow-on milk really brings more to the children. And again, from one pediatrician to another, we can see that there are two schools, there are really those who switch to UHT milk without worry and others who are reluctant."

(P14, CCP, professional working in a childcare center).

Professionals were familiar with non-recommended foods (e.g., plant-based alternatives to formula, sugar sweetened beverages and foods). Some of them asked for more explanation, especially regarding the new recommendation about avoiding chocolate until three years old.

Professionals mentioned that they particularly enjoyed one section of the brochure on baby-led weaning as they face an increasing number of questions about this topic and need more evidence-based information.

"As I read, I also thought about the baby-led weaning, I wondered if it would be covered when I saw that there was a little passage on it. It's true that sometimes it's not very common, but it so happens that there are parents who say they want to do the baby-led weaning when in fact, frankly, I have no knowledge of it. I do not know what to advise them, as it is really outside the framework of the recommendations that I've read." (P2, HCP, Pediatrician)

"Then, we have more and more questions about baby-led weaning. This is something I had never heard of... I have the feeling that a year ago we didn't talk about it, and in the last year we have talked about it a lot. There is a little insert on it, that's good." (P17, CCP, Childcare assistant) General impressions of the brochure "These are somewhat formal documents; we still say to ourselves that it was thoughtfully studied by professionals. It gives us legitimacy and technical consistency." (P14, CCP, Professional working in a childcare center)

Suggestions to improve the brochure Most of the participants recommended making the brochure more entertaining (more pictures and less text) as they were provided with a preliminary version of the brochure with no illustrations. Participants suggested translating the brochure in different languages or to make one document with visual explanations and many pictures adapted to PMI populations. Participants also proposed making different detachable sheets adapted for each age range of the child to easily distill the information to parents. Including the brochure in the child health record booklet and adding more explanations and/or scienti c references (e.g., why starting CoF between four to six months of age help prevent the development of allergies; why is it recommended to avoid salt until three years of age; scienti c references to justify the introduction of gluten and nuts, etc.) were also popular suggestions.

"In fact, we should be able to cut down the number of items to give just the one that corresponds to the needs of the families in the moment. After that there is a need to put images so that we can adapt it to our families." (P20, CCP, PMI nurse)

Perceptions of the use that parents can have of the brochure Professionals believe that the brochure is a comprehensive and useful written tool for parents and can serve as a reference for discussing child feeding with doctors. They believe that the brochure will be more useful for parents who are already invested in the topic of child feeding and that its usefulness will be more limited for parents of the most disadvantaged backgrounds. The brochure could encourage parents to change certain habits (e.g., to cook more homemade dishes).

Discussion

In this qualitative study, we interviewed French healthcare and childcare professionals who serve young children and their parents with the aim to explore their practices and needs when communicating with parents about child feeding from zero to three years of age. We also aimed to explore professionals' impressions regarding a brochure containing the newly updated child feeding recommendations and focused on understanding acceptance of the new nutritional recommendations. We decided to focus speci cally on HCPs and CCPs in recognition of their central role in guiding and in uencing parents in the early life of children [START_REF] De Rosso | Information seeking of French parents regarding infant and young child feeding: practices, needs and determinants[END_REF].

The results of this study suggest that overall, the brochure meets professionals' needs. The paper format was well appreciated by professionals. This is in line with the results of a recent double survey regarding the perceived needs of French parents and pediatricians concerning information on CoF, in which 59% of pediatricians and 42% of parents considered paper brochures to be an appropriate tool to spread information regarding CoF [START_REF] De Rosso | The Perceptions and Needs of French Parents and Pediatricians Concerning Information on Complementary Feeding[END_REF]. Professionals indeed reported they would use the brochure as a medium to communicate with parents, except in serving PMI populations where a simpler and more colorful document was requested (it worth noting that the draft version of the brochure that was being tested at the time did not present any graphical adaptation yet). There are some differences in terms of communication practices between CCPs and HCPs, especially concerning the content of exchanges with parents. While CCPs tend to discuss more practical topics such as daily issues related to the diet of the child, HCPs tended to discuss more feeding-related topics. On average, pediatricians and GPs perceived the brochure as more useful in their practice than CCPs. Those dissimilarities in practices between CCPs and HCPs and their perceptions of the utility of the brochure could re ect the different roles of employment between these two professional categories. Speci cally, while HCPs reported they could use the brochure in a systematic way with parents during consultations, CCPs would rather use it as an instrument to inform themselves, legitimize their knowledge on the subject, or to assuage the doubts of curious parents. Nevertheless, both types of professionals agreed that this brochure would enable them to update their knowledge about nutrition for children from zero to three years of age.

In PMI centers, the story is completely different from that in pediatricians' o ces. In fact, the professionals dealing with more disadvantaged families expressed strain in having to use the brochure with their patients. Parents visiting PMI professionals are often immigrants or those of foreign origin who do not uently speak or understand French; in those cases, an adapted tool with more pictures and images is required to facilitate communication. The lack of tools that account for the needs of those parents was previously noted in another study in which 35% of pediatricians thought that communication materials available to parents do not su ciently consider the different economic and cultural situations of families [START_REF] De Rosso | The Perceptions and Needs of French Parents and Pediatricians Concerning Information on Complementary Feeding[END_REF]. It was already demonstrated that public health campaigns focused on diet and nutrition struggle to reach those of modest means [START_REF] Régnier | Une affaire de goût ? Réception et mise en pratique des recommandations nutritionnelles[END_REF][START_REF] Dhuot | La genèse précoce des différences sociales dans les habitudes alimentaires[END_REF]. This aligns with data showing that chronic conditions such as obesity and diabetes are more prevalent within the lowest socioeconomic classes of society [START_REF] Tanaka | Income, wealth and risk of diabetes among older adults: cohort study using the English longitudinal study of ageing[END_REF][START_REF] Bonaccio | Low income is associated with poor adherence to a Mediterranean diet and a higher prevalence of obesity: crosssectional results from the Moli-sani study[END_REF]. The challenges of involving a disadvantaged population in public health campaigns could exacerbate social health inequalities [START_REF] Barker | Developmental biology: Support mothers to secure future public health[END_REF]. To tackle this issue, the campaign regarding the new recommendations, which launched in September 2021, also included other mediums of communication, such as videos. These tools can overcome language barriers and may increase parents' capabilities to turn recommendations into practice by providing concrete advice and examples for feeding children.

Disseminating health messages through HCPs and CCPs was also a means to e ciently reach all parents (including those who are disadvantaged), since a previous study has shown that HCPs and CCPs are the most in uential sources of information to parents regardless of their nancial status [START_REF] De Rosso | Information seeking of French parents regarding infant and young child feeding: practices, needs and determinants[END_REF]. In addition, in the near future, other tools are planned to be designed to speci cally address the needs of disadvantaged populations as identi ed by professionals. Moreover, in France, the program MALIN (www.programme-malin.com) is now disseminated through the family allowance with the objective of promoting healthier food practices for children under the age of three who come from families in precarious situations [START_REF] Cavalli | Challenges encountered for implementing a public health intervention: The ECAIL study Di cultés rencontrées pour la réalisation d'une recherche interventionnelle en santé publique: l'étude ECAIL[END_REF]. Disadvantaged families who meet the eligibility criteria (e.g., income) could be provided vouchers to buy baby food or groceries and given advice, to empower them to make better nutrition-related choices for themselves and for their children.

This study did not reveal important differences between the new recommendations and the advising practices of professionals (especially HCPs) but still pointed out particular advice given by professionals that could be updated and improved in light of the new recommendations, in particular with regard to the introduction of allergens and of all types of food between four to six months of age. Our results are in line with those of a study conducted in France in 2016, in which the interviewed general practitioners and pediatricians reported giving advice on child feeding in line with recommendations [START_REF] Chouraqui | Physicians advice, parental practice and adherence to doctor's advice: an original survey on infant feeding[END_REF]. For instance, in this study, professionals acknowledged that the right age for introducing complementary food was between four and six months of age and gave the appropriate advice to parents in 85.2% of cases. However, in the same study, the given advice was not in line with recommendations as was highlighted regarding the introduction of allergens and some foods. For example, 44.5% of professionals were advising parents to introduce gluten to their child after six months of age [START_REF] Chouraqui | Physicians advice, parental practice and adherence to doctor's advice: an original survey on infant feeding[END_REF]. Additionally, in the case of parental allergy history, there was a tendency to tell parents to postpone the introduction of some speci c foods. Some of the professionals interviewed in our study also experienced doubts regarding the introduction of allergens. An uncertain opinion from a professional could create confusion regarding the time of introduction of complementary foods and could introduce non-compliance in parents due to reasons such as lack of con dence in the doctor and preference to follow advice from family or friends (56, 57). The uncertainty experienced by some professionals facing the newest recommendations underscores the time required for scienti c, evidenced-based, knowledge to reach all the intended recipients. A consensus among professionals on child feeding recommendations could improve parental con dence to follow the given advice, but this remains a di cult objective to achieve. The French Public Health Agency has been moving in this direction with a communication campaign targeted toward professionals with the goal of sensitizing them to the new recommendations prior to the launch of the campaign targeted toward parents (August 2021).

The brochure produced by Public Health France is an o cial document that formalizes and standardizes the latest recommendations for feeding children from zero to three years of age. It is interesting to note that the brochure itself did not ful ll the HCPs' need for explaining some recommendations, since they expressed the need to verify some of the information themselves. As was the case in this study, GPs and pediatricians surveyed by Chouraqui et al. reported their need to obtain more information about certain infant feeding practices, more generally on speci c pathology-related recommendations (such as the prevention of allergies, the age of gluten introduction, and the treatment of food allergies) as well as weaning and the introduction of foods according to the child's age [START_REF] Chouraqui | Physicians advice, parental practice and adherence to doctor's advice: an original survey on infant feeding[END_REF]. Interestingly, in our study, GPs and pediatricians also reported their need to obtain more information regarding the quantities of milk according to the child's age to answer parents' questions, which suggested that this recommendation about milk quantities still creates confusion. Professionals also needed more explanations regarding why some foods are not recommended to young children (e.g., certain types of sh, chocolate). French pediatricians surveyed by Banti et al. in 2015 to understand whether their practices followed o cial recommendations stated that frequent changes could be perceived as errors in the original or latest recommendations (58). This might reduce the level of trust in updated o cial documents, and consequently, the given advice may no longer align with the latest evidence. Clear explanations are requested to detail why recommendations had changed in comparison to the previous ones. In this case, scienti c references are requested, especially by HCPs.

Our results showed that professionals believed that they have a role to play in the dissemination of the brochure. Indeed, the discourse of HCPs suggested that the brochure could help overcome some of their communication barriers. In fact, providing the brochure to parents would allow HCPs to save time during consultation (parents would have all the necessary information in the document so HCPs focus on talking about the most important topics and save time to answer questions) and to overcome the lack of practical advice (since the brochure contained practical advice such as recipes, which were appreciated by HCPs). In addition, using the brochure as a support to communicate with parents would help HCPs and CCPs provide more complete and systematic information about child feeding and bring more legitimacy to their communication with parents. In serving PMI populations, the main barrier to communication with parents was language. However, a written support with images and few words could facilitate communication with foreign or immigrant families. The development of other means of dissemination of information might be needed to foresee an inclusive strategy at every population level. The perspective of creating digital solutions has been discussed previously in the literature, but doubts still exist on how to deal with social disparities in terms of digital health [START_REF] Régnier | Social Appropriation of "Diet and Health" Information: From Public Health Campaigns to Digital Tools[END_REF].

Strengths and limitations must be analyzed when considering the results of our study. For the design of the study, we aimed to include unequal numbers in different professional categories. This decision was based on another recent study, as explained above, but this consideration should be taken when interpreting and generalizing the ndings of our study. A potential bias could emerge when interpreting ndings by giving undue prominence to views of certain professions (mostly GPs and pediatricians).

Notwithstanding, during the analysis and presentation of the results, efforts were made to ensure that points at which views and perceptions differed across professional groups was explicitly stated. Additionally, as often happens with qualitative studies, social desirability bias could be a limitation; in fact, professionals might have had the instinct to say something just because it was perceived to be the "correct" answer according to their perception of their role. In addition, an effort was made to include a high number of doctors (GPs, pediatricians, PMI physicians) across a spread of different departments in France. In fact, this population is often di cult to involve in research studies due to work schedules that often cannot accommodate activities other than outpatient practice. Moreover, while we included a wide variety of professionals, other categories of HCPs and CCPs, such as midwives, could have been involved. Our inclusion method occurred mostly via social media, which yielded heterogeneity in terms of the age of the participants, which was not representative of the national population.

Conclusions

Early life is an important period for the development of healthy behaviors in children, especially regarding eating. Involving health professionals in a structured strategy of disseminating child feeding

  utility of the brochure in HCPs and CCPs practice Perceptions of the use that parents can have of the brochure 1) Practices and needs in terms of communication with parents on child feeding Communication practices

Table 1

 1 Previous (2004) vs. new (2021) recommendations on child feeding contained in the o cial public health material (brochure) released in France.

	Topic	Previous recommendation	New recommendation
	Start of CoF	After 6 months	Between 4 and 6 months
	Introduction of all food	After 6 months	Between 4 and 6 months
	types
	Introduction of different	After 6 months	Between 6 and 8 months
	textures
	Introduction of pulses	Between 15 and 18 months	Between 4 and 6 months
	and whole grain
	starches
	Introduction of	After 12 months	Between 4 and 6 months
	allergens (including
	gluten)
	Alternance of follow-on	Between 1 and 3 years, but	After the child turns 1 year
	milk and full fat cow	to prefer follow-on milk
	milk
	Baby-led weaning	/	No recommendation but this topic is
			addressed
	Non-recommended	To limit, and not before 7	To limit, to be introduced as later as
	foods and beverages *	months	possible
	Fat	Never before 6 months	From the start of CoF
	Responsive feeding	Not really addressed, from	Trust the baby's appetite at all the ages;
		6/8 months trust baby's	examples of baby's hunger and satiety
		appetite	cues
	CoF: complementary feeding
	* E.g.: plant-based alternatives to formula, sugar sweetened beverages and foods, coffee, tea.

Table 2

 2 Characteristics of the sample of French professionals (n=21).

	Characteristics

Table 4

 4 Websites cited by HCPs (from top to bottom, the most to the least cited).

	Websites
	Mpedia.fr
	Pediadoc.fr
	MangerBouger.fr
	Pédiatre-online.fr
	Ameli.fr
	Pap-pédiatrie.fr
	Alimentationdutoutpetit.fr

Acknowledgements

The authors thank the professionals for their time and willingness to participate in the study. We also acknowledge Corinne Delamaire, Laurence Noirot and Anne-Juliette Serry (Santé publique France) for their advice when designing the interview guide and Courtney Dow (INSERM) for the translation of the verbatim.

Funding

The study was conducted as part of the project "Edulia-Bringing down barriers to children's healthy eating", which has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 764985. This work also supported by grants from the Conseil Régional Bourgogne, Franche-Comte (PARI grant) and the FEDER (European Funding for Regional Economic Development).

Availability of data and materials

The datasets used and/or during the current study are available from the corresponding author on reasonable request.

Positive impressions of the brochure outweighed negative impressions (see Table 5). In fact, all 21 professionals expressed at least one positive impression, and overall, they seemed enthusiastic about the brochure (some of them wanted to use it forthwith). Perceptions of the utility the brochure in HCP and CCP HCPs considered the brochure as a supplement that could facilitate communication with parents. Using the brochure could help save time during consultations (parents could read the brochure at home and return to the next consultation with additional and more precise questions). Moreover, the brochure could enable HCPs to have a more systematic approach in their information delivery to parents about child feeding, and help HCPs update their knowledge of the new recommendations.

"I nd that supports our speech by saying: "It's not just my personal opinion and what I personally believe.

This is really what is recommended"." (P8, General practitioner)

"That [reading the brochure] clari es the recommendations, so that's great. I am very happy with it. [...] I learned some things." (P9, HCP, General practitioner)

In PMIs, the brochure could be used only with a few parents because it is too long and complex for serving the most disadvantaged populations. CCPs generally stated that they would give the brochure to parents who have speci c questions about infant and young child feeding. The brochure would help CCPs feel more legitimate and precise in giving advice to parents.

recommendations can be important given the high frequency of routine contact that they have with parents in their child's early years. This qualitative analysis conducted with health and childcare professionals allowed us to explore their practices and needs in terms of communication with parents on child feeding. Professionals reported that while communication with parents primarily occurs orally, they also often refer to other supports (e.g., pamphlets, online sources). Lack of time and language were major barriers, especially in PMI centers that are serving disadvantaged populations. Moreover, this study investigated whether professionals found the brochure developed by Santé publique France suitable to meet their needs, whether they could easily integrate the new recommendations when advising parents, and whether they considered that additional adaptations are needed. Our results showed that professionals generally welcomed the brochure, but this medium was still considered as a tool that was not well-suited for disadvantaged populations, for whom a different support that includes more visual representations should be developed. A strategy that entails the use of the same document by both parents and professionals has the advantage of harmonizing the information circulating between them.

Professionals would feel more legitimate in advising parents with the help of a document enclosing recommendations supported by all the newest scienti c evidence. Moreover, providing relevant and consistent information to parents could help reduce the ow of con icting information and misinformation and ultimately increase parental support for adopting new recommendations in their child feeding practices. To reduce health inequalities, further actions are needed from the public health sector to more speci cally address the disadvantaged population. In the long run, this may contribute to reducing social disparities in nutrition and health.

Declarations

Ethics approval and consent to participate

An informed consent obtained from all the participants. Ethical approval (n°21-788) for this study was obtained from the Institutional Review Board (IRB00003888, IORG0003254, FWA00005831) of the French Institute of Medical Research and Health. See the paragraph regarding "Ethical considerations" for more details. All methods were carried out in accordance with relevant guidelines and regulations.

Consent for publication

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

SDR was a major contributor in writing the manuscript, the study design and conception of the interview guide. SN, CRN, PD and CS contributed and validated the methodology and critically edited the interview guide. SDR and CRN nalized the interview guide. SDR and CRN analyzed the data, with constant feedback from the SN. All authors critically reviewed and commented on subsequent drafts of the manuscript and approved the nal version.

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Keywords: Groupage de trafic, algorithme distribué, trafic en ligne

Groupage de traffic en ligne et en distribué sur le chemin Résumé : Le facteur de groupage C d'un réseau optique WDM est le nombre de connexions pouvant partager la bande passante d'une longueur d'onde. Le fait de pouvoir grouper les requêtes qui vont partager une longueur d'onde est appelé groupage de trafic. L'objectif du groupage de trafic est d'une part de reduire le coût de transmission en réduisant le nombre de longueurs d'ondes utilisées et d'autre part le coût des équipements en réduisant le nombre de multiplexeurs à insertions/extractions (ADM). Dans ce rapport, nous étudions le groupage de trafic sur le chemin orienté avec du trafic en ligne et un algorithme de routage distribué. Lorsque les connexions sont en ligne, la topologie virtuelle construite en affectant les ADMs aux longueurs d'ondes n'est pas modifiée. La conception d'une topologie virtuelle performante qui minimize le nombre d'ADMs nécessaire pour soit satisfaire toute instance de trafic ou pour réduire la probabilité de blocage dépend fortement de l'algorithme de routage. Nous montrons comment concevoir la meilleure topologie virtuelle possible, indépendemment de l'algorithme de routage, lorsque chaque noeud est équippé avec le même nombre d'ADMs. Puis, nous analysons les performances d'un algorithme de routage distribué.

Introduction

In connection-oriented networks, such as wavelength division multiplexing (WDM) optical networks, traffic grooming is used to improve the usage of the bandwidth and components by combining connections (low speed traffic streams) onto high speed wavelengths. The goal is to minimize the network-wide electronic switching cost. Typically, nodes of the network are equipped with add drop multiplexers (ADMs) to insert and extract the traffic streams on a wavelength. One ADM is needed in a node for each wavelength to which traffic is added or dropped. The grooming factor C of a WDM network is the number of connections that can share the bandwidth of each wavelength, and an ADM is able to add up to C and drop up to C unitary traffic streams on a given wavelength. Thus, the traffic grooming problem is to minimize the total number of ADMs to be installed in a network to accommodate all of the traffic streams.

For a given fixed set of connection requests and single-hop routing (a connection uses a single wavelength from source to destination), the general traffic grooming problem of minimizing the total number of ADMs is NP-complete [CM00, APS07, SUZ07] and hard to approximate. The corresponding network design and routing problems have been considered by many authors. See the surveys [ML01, DR02, ZM03, BC06], the books [ZZM05, Som06, DKR08], and many articles [GRS00, Hu02, WCLF00, ZQ00].

The traffic grooming problem with multi-hop routing is also NP-complete [START_REF] Huang | Traffic grooming in path, star, and tree networks: Complexity, bounds, and algorithms[END_REF]. With such routing algorithm, a traffic stream can be switched from one wavelength to another at an intermediate node by using an optical-electronic-optical conversion. This requires one ADM for each wavelength at each intermediate node where switching can occur. Multi-hop routing allows further improvement of the usage of the bandwidth.

When the connection requests arrive online, the network design problem is different [K 06, CRSL08] because it is impossible to modify the assignment of ADMs to wavelengths for each new connection request. Therefore, the problem is to design an efficient virtual topology that minimizes the blocking rate by permanently assigning ADMs to wavelengths at nodes. There is a link in a virtual topology between two nodes for a particular wavelength if each node has an ADM associated with that wavelength.

In this paper, we investigate traffic grooming for directed path networks with online connection requests and distributed routing algorithms. Our objective is to design the best possible virtual topologies with the constraints that each node (i) is equipped with the same number of ADMs, (ii) is connected to wavelengths in a uniform way, (iii) is the source or destination of at most k connection requests and any traffic pattern that satisfies this constraint can be routed without blocking. We first give upper bounds on the size of the network, independently of the routing algorithm. Then, using the distributed greedy routing algorithm proposed in [START_REF] Crouser | On-line distributed traffic grooming[END_REF], we give optimal constructions that improve upon the constructions in [START_REF] Crouser | On-line distributed traffic grooming[END_REF].

Notation

P N : unidirectional path with N nodes C: grooming factor k: maximum number of connection requests leaving (origianting) at a node and the maximum number of connectio requests arriving (terminating) at a node. A family of requests that satisfies the constraint that each node originates or terminates at most k requests is said to be k-allowable.

r: maximum in-degree and out-degree of the directed virtual topology V : set of nodes of the physical and logical topologies. For convenience, we associate an integer in the range [-p..q] to each node of the physical topology. We have

|V | = N = p + q + 1.
l 1 , l 2 , . . . , l r : lengths of the links of the virtual topology, such that l j < l j+1 . (Note that

l j = j in [CRSL08].)
S 1 , S 2 , . . . , S r : sum S i = l 1 + l 2 + . . . + l i of the i smallest lengths of the links of the virtual topology A: set of arcs of the virtual topology. We have

A = {(i, i + l j ) | i ∈ [-p..q], j ∈ [1..r], i + l j ≤ q}, and |A| = r j=1 j = r(r + 1)/2.
We say that an arc (u, v) of the virtual topology crosses 0 if u ≤ 0 and v > 0. In other words, arc (u, v) of the virtual topology requires capacity C on arc (0, 1) of the physical topology. We say that a virtual arc is saturated when it is involved in C connections and available otherwise.

Problems and examples

Problem 1 (General Problem) Given k ≥ 1, find the admissible values of N , C, l 1 , l 2 , . . . , l r such that any k-allowable set of requests can be routed online in the virtual topology in a way that respects the traffic grooming constraint (at most C connections use the same arc of the virtual topology) and is independent of the routing algorithm.

Example 2 Let k = 1, C = 2, r = 2, l 1 = 1, and l 2 = 4. If N ≥ 6 we cannot route the three requests (0, 3), (-1, 2), and (-2, 1) as the only possible routes are, respectively, (0, 1); (1, 2); (2, 3), (-1, 0); (0, 1); (1, 2) and (-2, -1); (-1, 0); (0, 1). This routing uses arc (0, 1) three times, contradicting C = 2.

INRIA

Distributed Greedy Algorithm, DGA [START_REF] Crouser | On-line distributed traffic grooming[END_REF] . A new request (s, t) is routed online as follows. The current vertex v is initially the start vertex. At the current vertex, v = t, the available arc of maximum possible length is chosen. More precisely, if the distance from the current vertex v to t is d(v, t) ≥ l r and the arc (v, v + l r ) is available, then the request is routed from v to v + l r and the current vertex becomes v + l r . Otherwise, the arcs (v, v + l r-1 ), (v, v + l r-2 ), . . . are tried until an available arc is found. If an available arc is not found, then the connection is blocked and the request cannot be routed.

Example 3 Let k = 1, C = 2, r = 3, and l 1 = 1, l 2 = 3, l 3 = 6.
The following table shows how four requests are routed using DGA:

Request Route Request Route (-12, 3) → (-12, -6); (-6, 0); (0, 3) (-6, 4) → (-6, 0); (0, 3); (3, 4) (-9, 1) → (-9, -3); (-3, 0); (0, 1) (-3, 2) → (-3, 0); (0, 1); (1, 2)
If a fifth request (0, 5) arrives, it cannot be routed because the arcs (0, 1) and (0, 3) have already been used twice and are not available, and d(0, 5) < l 3 = 6. Note that the network does have the capacity to accommodate all five requests and a different algorithm could find a successful routing.

Problem 4 (General Greedy Problem) Given k ≥ 1, find the admissible values of N , C, l 1 , l 2 , . . . , l r such that any k-allowable set of requests can be routed by DGA in the virtual topology.

To address Problems 1 and 4, we concentrate on the following problems:

Problem 5 Given k = 1 and an admissible set of values of the parameters C, and l 1 < l 2 < . . . < l r , find the maximum possible number of vertices N max (C; l 1 , l 2 , . . . , l r ) that can route any k-allowable set of requests.

Problem 6 Given k = 1, C and r, find the maximum possible number of vertices N max (C, r) that can route any k-allowable set of requests.

Problem 7 Given k = 1, C and r, find the maximum possible number of vertices N greedy max (C, r) that can route any k-allowable set of requests using DGA.

Universal counter-examples

We now provide examples that are independent of the routing algorithm and that will serve as counter-examples.

Lemma 8 If l 1 > 1, then N < 2 for any choice of the l p , 2 ≤ p ≤ r.
If l 1 > 1, we cannot route the request (0, 1). In the remainder of this paper we assume that l 1 = 1.

Lemma 9 If l 2 > C + 1, then N ≤ 2C + 1 for any choice of the l p , 3 ≤ p ≤ r.

Proof. Suppose that N ≥ 2C + 2 and consider the set of C + 1 requests (-i, -i + C + 1) for 0 ≤ i ≤ C. The length of each request is C + 1 and l p > C + 1 for 2 ≤ p ≤ r, so only arcs of length l 1 = 1 can be used. So, each request requires the arc (0, 1) and only C of the C + 1 requests can be routed. Example 2 illustrates the case C = 2, l 2 = 4 and N = 2(C + 1) = 6.

Lemma 10 If l r > CS r-1 + 1, then N ≤ 2CS r-1 + 1.
Proof. The proof is similar to the proof of Lemma 9. Suppose that N ≥ 2CS r-1 + 2 and consider the set of CS r-1 + 1 requests (-i, -i + CS r-1 + 1) for 0 ≤ i ≤ CS r-1 . The length of the request is CS r-1 + 1 and l r > CS r-1 + 1, so only arcs of length l 1 = 1, l 2 , . . . , l r-1 can be used. There are only l p arcs of length l p for each 1 ≤ p ≤ r -1, l p , so at most CS r-1 of the CS r-1 + 1 requests can be routed.

Lemma 11 N max (C; 1, l 2 ) ≤ 2C(l 2 + 1) + 1.
Proof. The proof is similar to the proof of Lemma 9. Suppose that N ≥ 2C(l 2 + 1) + 2 and consider the set of C(l 2 + 1) + 1 requests (-i, -i + C(l 2 + 1) + 1) for 0 ≤ i ≤ C(l 2 + 1). Each of these requests uses an arc crossing 0, but there are at most C(l 2 + 1) available arcs crossing 0, namely C times (0, 1) and C times the l 2 arcs (-i, i + l 2 ) for 0 ≤ i < l 2 .

Lemma 12 N max (C; 1, l 2 , . . . , l r ) ≤ 2CS r + 1.

Proof. The proof is similar to the proof of Lemma 11. Suppose that N ≥ 2CS r + 2 and consider the set of CS r + 1 requests (-i, -i + CS r + 1) for 0 ≤ i ≤ CS r . Each of these requests uses an arc crossing 0, but there are at most CS r available arcs crossing 0, namely C times the l p arcs (-i, i + l p ) for 0 ≤ i < l p for each l p , p = 1, 2, . . . , r, .

3 Case r = 2 Theorem 13 For l 2 > C + 1, N max (C; 1, l 2 ) = 2C + 1.
Proof. By Lemma 9 we have N max (C; 1, l 2 ) ≤ 2C + 1. Let us now show that any counterexample has at least 2C + 2 nodes. Apply DGA to the set of requests of a counter-example. As it does not find a solution, that means that at some point of the algorithm we have a request R routed until a vertex we denote 0 and that we cannot furthermore route it (0 can be the source of R) and in particular the arc (0, 1) is no more available ; therefore the arc (0, 1) has been already used for routing C requests. But these C requests and the request INRIA R have all a starting node i ≤ 0 and a destination node j ≥ 1. As all the sources and all the destinations are different, we have at least 2C + 2 nodes.

Theorem 14 For l 2 ≤ C + 1, N max (C; 1, l 2 ) = 2C(l 2 + 1) + 1.

Proof. Similar as for theorem 13.

Theorem 15 N max (C, r = 2) = 2C 2 + 4C + 1 and this value is attained for l 2 = C + 1

Proof. From Lemma 9 we know that if l 2 > C + 1, then N max (C; 1, l 2 ) ≤ 2C + 1 < 2C 2 + 4C + 1 and from Lemma 11 we know that if l 2 ≤ C + 1, then N max (C; 1, l 2 ) ≤ 2C(l 2 + 1) + 1 ≤ 2C 2 + 4C + 1. The maximum is attained for l 2 = C + 1.

Note that for C ≥ 2 this value is strictly better than the value of [START_REF] Crouser | On-line distributed traffic grooming[END_REF] where they had only 6C + 1.

4 Case r = 3 Theorem 16 When l 2 ≤ C + 1 and l 3 ≤ 2C + 1, then N max (C; 1, l 2 , l 3 ) = 2C(1 + l 2 + l 3 ) + 1
Theorem 17 N greedy max (C, r = 3) = 6C 2 + 6C + 1 and this value is attained for l 2 = C + 1 and l 3 = 2C + 1.

Proof.

• If l 2 > C + 1, then by Lemma 9 we have N ≤ 2C + 1.

• If l 2 ≤ C+1 and l 3 ≤ 2C+1, then by Theorem 16, N ≤ 2C(1+l 2 +l 3 )+1 ≤ 6C 2 +6C+1.

The value is attained for l 2 = C + 1 and l 3 = 2C + 1.

• If l 2 ≤ C + 1 and l 3 > C(1 + l 2 ) + 1, then by Lemma 10, N ≤ 2C(1 + l 2 ) + 1 ≤ 2C 2 + 4C + 1 < 6C 2 + 6C + 1.
• If l 2 ≤ C + 1 and 2C + 1 < l 3 ≤ C(1 + l 2 ) + 1, then the result follows from the counter-examples involved in Proposition 18.

Proposition 18 Let l 2 ≤ C + 1 and 2C + 1 < l 3 ≤ C(1 + l 2 ) + 1, then there exist an N < 6C 2 + 6C + 1 and a set of requests R that cannot be routed by DGA.

s a,b,d is routed till v a,b,0 = -(bl 2 + a), if no arc of length l 3 is used more than C times. The arc the more loaded is the arc (v a,b,1 , v a,b,0 ) loaded δ-1 times except for the arc (v 0,0,1 , v 0,0,0 ) loaded δ times if the sender s 0,0,δ is used. So, no arc of length l 3 is used more than C times if :

δ ≤ C (2)
Now we route the requests arrived in the nodes v a,b,0 = -(bl 2 + a) with a ≥ 0. We want to force them to arrive in the node v a,0,0 = -a by using uniquely arcs of length l 2 . For that we need first that the distance from v a,b,0 to the receiver is strictly less than l 3 .That is the following inequalities (3) should be satisfied for every triple a, b, d :

bl 2 + a + r a,b,d < l 3 (3)
We should also insure that the arcs of length l 2 are available. To realize that (and also for other reasons), we will do in sort that exactly C requests arrive in v a,1,0 = -(l 2 + a). Therefore, C requests use the arc (-l 2 -a, -a) and less than C requests use the arcs (v a,b,0 , v a,b-1,0 ) for b > 1. In the node v a,1,0 = -(l 2 + a), arrive all the requests originating at the s a,b,d with 1

≤ b ≤ β -1. For 1 ≤ b ≤ β -2 we have δ such requests as 0 ≤ d ≤ δ -1.
For b = β -1 we will have δ -g such requests as 0 ≤ d ≤ δ -1 -g. So we choose β and g such that the equation ( 4) is satisfied.

(β -1)δ -g = C ; 0 ≤ g < δ (4)
At that point we have exactly C requests arrived in v a,1,0 = -(l 2 + a). which are routed to node -a. Furthermore,if a > 0, in node -a arrive directly δ -1 -h a requests from the s a,0,d with 0 ≤ d ≤ δ -1 -h a . In each node -a, with a > 0, C requests will be routed via the arc of length l 2 to -a + l 2 . The remaining requests in number δ -1 -h a are routed to -a + 1 via the arc of length l 1 = 1 and so on till node 0.

In summary we have on each arc (-a, -a + l 2 ), with 1 ≤ a ≤ α -1, C requests and furthermore, in node 0 are arriving • C requests originating at the s 0,b,d , with b > 0, and transiting via the vertex v 0,1,0 = -l 2

• plus Σ a (δ -h a ) = αδ -h requests from the other s a,b,d (this includes the requests with a = b = 0).

• plus the l 2 -2 requests (-i, -i + l 2 -1) of the first type.

To insure there are exactly 2C requests arriving in node 0 we need to have the equation (5) which defines h:

l 2 -2 + αδ -h = C (5)
The idea for that case is that the maximum will be M 0 and so in order that all the inequalities (3) are satisfied we should have l 3 > M 0 = (α + 1)C + 1. More precisely, let us choose α as follows:

(α + 1)C + 1 < l 3 ≤ (α + 2)C + 1 (9)
As

l 3 > 2C + 1, then α ≥ 1. As l 3 ≤ C(1 + l 2 ) + 1, then α ≤ l 2
Now we choose the smallest δ such that αδ ≥ C + 2 -l 2 ; that is

α(δ -1) < C + 2 -l 2 ≤ αδ (10)
So equation ( 5) is satisfied and as

l 2 ≤ c+2 2 , αδ ≥ C + 2 -l 2 ≥ c + 2 2 ≥ l 2 . ( 11
)
Therefore by equation (7

), for b > 1, M b-1 -M b = αδ -l 2 ≥ 0 and so M 1 ≥ M b , for b > 1. Also by equation (8) M 0 -M 1 = αδ -h -l 2 = C + 2 -2l 2 ≥ 0.
So the maximum is M 0 , and as l 3 > M 0 = (α + 1)C + 1 by ( 9) all the inequalities (3) are satisfied As l 2 ≥ 2 equation (10) implies δ -1 < C and so δ ≤ C (inequality (2)). Now we choose β in order to satisfy equation (4) that is

δ(β -2) < C ≤ δ(β -1) (12)
Using inequality (11, 12, 9) we get (β -2)l 2 ≤ (β -2)αδ < αC ≤ l 3 -(C + 2) So βl 2 = (β -2)l 2 + 2l 2 ≤ l 3 -(C + 3) + 2l 2 . But we are in the case l 2 ≤ c+2 2 and so βl 2 ≤ l 3 implying the last inequality of (1). So, with these choices, all the conditions are satisfied. By inequality (6), N ≤ δl 3 + (α + 1)C + 2.

By inequality (9) N ≤ δ(α+2)C +δ +(α+1)C +2 = α(δ -1)C +αC +2δC +(α+1)C +2. Then by inequality (10) α(δ -1) ≤ C + 1 -l 2 and so N ≤ C(C + 1 -l 2 + 2α + 2δ + 1) + δ + 2 Finally α ≤ l 2 ≤ C+1 and δ ≤ C imply N ≤ C(C+1+C+1+2C+1)+C+2 ≤ 4C 2 +4C+2 proving the theorem in that case.

Case 2: l 2 > C+2 2 . We first choose β such that:

(β + 3)l 2 -3 < l 3 ≤ (β + 4)l 2 -3 (13)
So the second inequality of (1) is satisfied. Furthermore, as l 3 > 6C + 3 (we deal with the other case in the special case), then β ≥ 3. Now we choose δ such that equation (4) is satisfied

(β -1)(δ -1) < C ≤ (β -1)δ (14)
The first part of (14) and β ≥ 3 imply C > (β -1)(δ -1) ≥ 2δ -2 implying a stronger inequality than (2):

δ < C + 2 2 (15)
Finally, we choose α such that equation ( 5) is satisfied.

(α -1)δ < C + 2 -l 2 ≤ αδ (16)
As l 2 > C+2 2 , then α ≤ l 2 and the first part of (1) is satisfied. Now it remains to show that all the inequalities (3) are satisfied. Equation (8) and l 2 > C+2 2 imply that M 0 < M 1 . Equations (7) imply that the maximum of the M b is either M 1 or M β-1 . So it suffices to verify that we have both l 3 > M β-1 and l 3 > M 1 .

We have

M β-1 = (β + 1)l 2 -1 + α(δ -g) ≥ (β + 1)l 2 -1 + αδ.
Using the first part of (16), we get M β-1 ≤ (β + 1)l 2 -1 + C + 1 -l 2 + δ. As C + 3 ≤ 2l 2 and by (15 we obtain M β-1 ≤ (β + 3)l 2 -4. Therefore, by (13) M β-1 < l 3 .

We have M 1 = (α + 1)C + 1 + 2l 2 -(C + 2). Using the first part of (16), we get:

M 1 δ < (C + 2 -l 2 )C + 2δC + δ(1 + 2l 2 -(C + 2)) ≤ (C + 2 -l 2 )C + 2δl 2 + δ(C -1).
On the other side, by the first part of (13) l 3 δ > (β + 3)δl 2 -3δ. Then, by the second part of ( 14) l 3 δ > Cl 2 + 4δL 2 -3δ. Finally, l 2 > C+2 2 implies Cl 2 > (C + 2 -l 2 )C and 2δl 2 -3δ > δ(C -1) and so l 3 δ > M 1 δ that is l 3 > M 1 .

So, all the conditions are satisfied for this choice of values. Let us now compute N . By ( 6) N ≤ δl 3 + (α + 1)C + 2. As l 3 > M 1 > M 0 = (α + 1)C + 1, we get N ≤ (δ + 1)l 3 .

By the second part of ( 13), we get N ≤ (δ + 1)((β + 4)l 2 -3) ≤ (δ + 1)(β -1)l 2 + 2(β + 4)l 2 + 5(δ -1)l 2 -3(δ + 1).

The inequality l 3 ≤ C(1 + l 2 ) + 1 of the hypothesis of the theorem and first part of ( 13) give (β + 3)l 2 -3 < l 3 ≤ C(1 + l 2 ) + 1 that is as l 2 ≤ C + 1 : (β + 3) ≤ C + 1. Using that, δ < C+2 2 ( 15) and the first inequality of ( 14) we obtain N ≤ Cl 2 + 2(C + 2)l 2 + 5 C-3 2 l 2 -3 C+3 2 = 11C 2 2 + C 2 -10 implying the theorem.

Discussion and future work

In [START_REF] Crouser | On-line distributed traffic grooming[END_REF], we have N ≤ C k r(r + 1) using r(r + 1)/2 wavelengths. So in some sense, adding one wavelength allows to add 2C k nodes. In this work, we have N ≤ 2CS r + 1 using S r = 1 + l 2 + • • • + l r wavelengths. So, each extra wavelength contribute to 2C nodes. So the usage of wavelength is more efficient in our work than in [START_REF] Crouser | On-line distributed traffic grooming[END_REF].

Our results are independent of the routing algorithm. It would be interesting to refine them for specific distributed routing algorithms.

INRIA

RR n°6833

Proof.

Idea of the proof: We will construct a set of requests containing 2C requests, each with a negative sender and a positive receiver in the set {1, 2, . . . , l 3 -1} -{r 0 }, such that when we apply DGA, these 2C requests are forced to transit in node 0. As the distance from 0 to the receiver is strictly less than l 3 , these 2C requests use to reach their destination the 2C arcs (0, 1) and (0, l 2 ) and so it will be impossible to route the request (0, r 0 ) if r 0 < l 3 . We will choose r 0 = l 2 -1, except for the case l 3 ≤ 6C + 3 where will choose r 0 = 2C + 1. We will first put small requests forced to use arcs of length 1 and define the other requests, dealing first with the special case l 3 ≤ 6C + 3. I. Requests of first type: First we have the l 2 -2 requests (-i, -i + l 2 -1) for 1 ≤ i ≤ l 2 -1. All these requests are of length l 2 -1 and so use only arcs of length 1 and therefore transit in node 0.

II. Requests of second type: II.a. Special case l 3 ≤ 6C + 3: We add the requests

If we apply DGA, the requests R d and R d use first arcs of length l 3 and each arc carries at mots C requests; so they arrive respectively in vertex -l 2 and 0. For the requests arriving in -l 2 , the distance to their receiver is at most l 2 + C as the largest receiver is C; therefore they cannot use an arc of length l 3 as l 3 > 2C + 1 ≥ l 2 + C. So we have exactly 2C requests transiting in 0, whose receiver is at most 2C < l 3 and which use all the C arcs of length 1 and all the C arcs of length l 2 . Then it is impossible to route the request (0, 2C + 1).

II.b. Special case l 3 > 6C + 3:

The requests of the second type have a sender s a,b,d < 0 and a receiver r a,b,d > 0. Senders are of the following form :

In order all these nodes are different, we suppose

We do not have the triple a = b = d = 0 (the sender 0 being reserved to the last request (0, l 2 -1)), and some triples with b = β -1 (see bellow). In fact, we will miss for b = β -1 and any a exactly g triples. More precisely we will have only the s a,β-1,d , with 0 ≤ d ≤ δ -1 -g.

We will also not use h triples with b = 0. More precisely we will have only the s a,0,d with 0

In contrary, when h 0 = 0, we might add the sender s 0,0,δ = -δl 3 .

All the parameters α, β, δ, g, h will be defined after in order to force the requests to transit in 0 when using the greedy algorithm and to bound the number of vertices N .

Following the greedy algorithm, if a request is arrived in v a,b,d with d > 0, we route this request to v a,b,d-1 via the arc of length l 3 if that is possible. Therefore a request starting in

INRIA

Note that altogether we have (α -1)C + 2C + 1 = (α + 1)C + 1 requests (the +1 comes from the request (0, l 2 -1). Therefore we will take the values of the r a,b,d in the interval [l 2 , (α + 1)C]. Indeed recall that among the positive values we already used the values from 1 to l 2 -2 for the requests of first type and we reserved l 2 -1 for the request (0, l 2 -1). In particular the maximum of the r a,b,d is (α + 1)C + 1. The smallest value of the s a,b,d is obtained for s 0,0,δ = -δl 3 if it exists. So altogether we obtain a bound for the number of vertices

Now we will fix the values of the r a,b,d and see what are the conditions in order inequalities (3) are satisfied. The idea is give to the r a,b,d small values when bl 2 + a is large (in view of having the smallest sum bl 2 + a + r a,b,d in inequality 3). The values are given in increasing order from l 2 to (α + 1)C + 1 in the opposite order of variation of the bl 2 + a. More precisely, for the δ -g requests originating in s α-1,β-1,d the values of r α-1,β-1,d are in the interval [l 2 , l 2 -1+δ -g]; then for the δ -g requests originating in s α-2,β-1,d the values of their destinations are in the interval [l 2 +δ-g, l 2 -1+2(δ-g)] and so on for the δ-g requests originating in s a,β-1,d the values of r a,β-1,d are in the interval [l 2 +(α-1-a)(δ-g), l 2 -1+(α-a)(δ-g)].

In particular the maximum value of s 0,β-1,d is l 2 -1 + α(δ -g). For the values 0 < b < β -1, there are δ requests for given a and b and the intervals are translated by δ. Therefore, for the δ requests originating in s a,b,d , with 0 < b < β -1, the values of r a,b,d are in the interval of length δ ending in l 2 -1 + α(δ -g) + α(β -b -2)δ + (α -a)δ. Note that for a given b when a decreases by 1 , then bl 2 + a decreases by 1 and the maximum value of r a,b,d increases by δ

the maximum of the right part of inequality (3) bl 2 + a + r a,b,d is attained for a = 0 and its value is

For b = 0, the maximum is also attained for a = 0, as all the existing requests s a,0,d go to 0 and as h α-1 ≥ h α-2 ≥ . . . ≥ h 0 . In that case the maximum is M 0 = (α + 1)C + 1 (the number of requests).

Note that there are αδ -h requests with b = 0 and so the maximum of r 0,1,d is (α + 1)C + 1 -(αδ -h) M 1 = l 2 + (α + 1)C + 1 -(αδ -h) and using equation (5) we get:

Now to finish we deal with two cases according the values of l 2 .

Case 1: l 2 ≤ C+2 2 .

INRIA


Christelle Planche

Frédéric Mercier

Johanna Rivas

Magaly Angénieux

Hanli Wang

Benjamin Calmont

Sébastien Crépieux

Jérémy Ratel

Erwan Engel

Fate of polychlorobiphenyls in the insect Tenebrio molitor: consequences for further use as food and feed

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Fate of polychlorobiphenyls in the insect Tenebrio molitor: consequences for further use as food and feed

Christelle Planche 1 , Frédéric Mercier 1 , Johanna Rivas 1 , Magaly Angénieux 1 , Hanli Wang 1 , Benjamin Calmont 2 , Sébastien Crépieux 3 , Jérémy Ratel 1 and Erwan Engel Background/Objectives: Given the world population increase, there is a growing need to find alternative and sustainable protein sources for food and feed. Edible insects are an interesting solution with low greenhouse gas emissions, high feed conversion efficiency, low land use, and their ability to transform low value organic co-products into high value protein products (1) . In counterpart, one of the potential risks associated with their consumption is the chemical contamination. In urban areas, this chemical risk may be linked to emission sources of persistent organic pollutants (POPs) (2) such as polychlorobiphenyls (PCBs), dioxins (PCDD/Fs), polycyclic aromatic hydrocarbons (PAHs). . .inducing their ubiquitous presence at trace levels in food supply (3) . In this context, this study explored the ability of yellow mealworm (Tenebrio molitor), the most widely bred and traded insect species in Europe (4) , to bioaccumulate PCBs from their feeding substrate.

Methods: Tenebrio molitor larvae were reared during 20 days in a temperature and humidity-controlled incubator and fed with wheat bran artificially contaminated with PCBs at a concentration of 0.67 ppb, 4 ppb or 24.4 ppb (n = 3 for each concentration). The larvae PCB content was then measured based on an extraction method inspired by Planche et al. (5) and an analysis by gaz chromatography coupled with an micro-Electron Capture Detector (μECD).

Results: Whatever the PCB contamination level, no significant impact was observed on the larval growth. The bioaccumulation factors (BAF (%) = 100 × concentration of PCBs in larvae / concentration of PCBs in wheat bran) obtained with fresh larvae are between 77% and 82%. With dried larvae, the most common commercially available form, the bioaccumulation factors ranged between 216% and 233%.

Discussion / Conclusion: This study highlights the ability of Tenebrio molitor to accumulate PCBs from their diet during rearing and the significant impact of the drying process on the bioaccumulation factors. This demonstrates the importance of considering the quality of the substrates used for farming insects as food and feed in terms of content in chemical contaminants including POPs like PCBs.

Proceedings of the Nutrition Societyhttps://doi.org/10.1017/S0029665122000945 Published online by Cambridge University Press