Charlotte Andrieux

S Biasutti

Josette Barrieu

Philippe Morganx

Mireille Morisson

Vincent Coustham

Stéphane Panserat

Marianne Houssier
email: marianne.houssier@univ-pau.fr

Identification of different critical embryonic periods to modify egg incubation temperature in mule ducks

Keywords: Poultry, embryogenesis, incubating, mortality, hatchability Implications

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

Introduction

The technology of egg incubation is already well known in poultry field and is a key to obtain good performances at hatch and post-hatch [START_REF] Wilson | Physiological requirements of developing embryo: temperature and turning[END_REF][START_REF] Mueller | The Physiology of the Avian Embryo[END_REF]. Conditions of incubation depend not only on the temperature and the relative humidity but also on gas exchanges and several others parameters to mimic the natural phenomena, such as rotation of eggs or light exposure. These conditions are the center of many studies to optimize zootechnical performances at birth [START_REF] Landauer | The Hatchability of Chicken Eggs as Influenced by Environment and Heredity[END_REF][START_REF] Yahav | Perinatal thermal manipulations in poultry , does it cause longlasting thermoregulatory memory[END_REF][START_REF] El-Hanoun | Effect of incubation humidity and flock age on hatchability traits and posthatch growth in Pekin ducks[END_REF][START_REF] Boleli | Poultry egg incubation: Integrating and optimizing production efficiency[END_REF][START_REF] El-Hack | Impact of environmental and incubation factors on hatchability of duck eggs[END_REF][START_REF] El-Shater | Embryonic thermal manipulation of Japanese quail : effects on embryonic development, hatchability, and post-hatch performance[END_REF]. The most published measures are weight, internal temperature, sex ratio and also hatchability, defined as the percentage of fertile eggs that produce a live animal [START_REF] King'ori | Review of the factors that influence egg fertility and hatchability in poultry[END_REF]. The best conditions of incubation are based on the optimum of hatchability [START_REF] Barott | Effect of temperature, humidity and other factors on the hatch of hens' eggs and on energy metabolism of chick embryos[END_REF][START_REF] Bruzual | Effects of relative humidity during incubation on hatchability and body weight of broiler chicks from young breeder flocks[END_REF][START_REF] Harun | Artificial incubation of Muscovy duck eggs: Why some eggs hatch and others do not[END_REF].

In recent decades, increasing incubation temperature has emerged as a promising way to program the physiology of avian species, resulting in improved thermal tolerance of adults [START_REF] Piestun | Thermal manipulations during broiler embryogenesis: Effect on the acquisition of thermotolerance[END_REF], and increased meat production in chickens [START_REF] Piestun | Thermal treatments prior to and during the beginning of incubation affect phenotypic characteristics of broiler chickens posthatching[END_REF] or increased fatty liver production after a period of overfeeding in ducks [START_REF] Massimino | Impact of thermal manipulation during embryogenesis on hepatic metabolism in mule ducks[END_REF]. However, since the temperature of incubation is the factor with the strongest impact on growth and development of embryos (Al [START_REF] Sardary | Effect of Thermal Manipulation during Embryogenesis on Hatching Traits[END_REF][START_REF] Maatjens | Temperature during the last week of incubation. I. Effects on hatching pattern and broiler chicken embryonic organ development[END_REF], its modulation can easily impair hatchability. The relative humidity (RH) is a parameter that can be used to avoid egg dehydration during a rise in temperature and to improve hatching performances [START_REF] Barott | Effect of temperature, humidity and other factors on the hatch of hens' eggs and on energy metabolism of chick embryos[END_REF] but it is also one of the most difficult to control. Nevertheless, depending on the duration, period, level and direction of temperature change applied, the consequences of survival on the hatching rate are highly variable and difficult to predict [START_REF] Halle | Influence of temperature stimulation during the last 6 days of incubation on hatching results and later performance in Pekin ducks[END_REF][START_REF] Loyau | Thermal manipulation of the embryo modifies the physiology and body compositiob of broiler chickens reared in floor pens without affecting breast meat processing quality[END_REF][START_REF] Piestun | Thermal treatments prior to and during the beginning of incubation affect phenotypic characteristics of broiler chickens posthatching[END_REF][START_REF] Al-Rukibat | Thermal manipulation during late embryogenesis: Effect on body weight and temperature, thyroid hormones, and differential white blood cell counts in broiler chickens[END_REF][START_REF] Narinç | Effects of thermal manipulations during embryogenesis of broiler chickens on developmental stability , hatchability and chick quality[END_REF][START_REF] Massimino | Impact of thermal manipulation during embryogenesis on hepatic metabolism in mule ducks[END_REF]. In addition, the species studied may also greatly influence performance after a change in temperature during a certain incubation period, since opposite results were measured after a relatively similar short-term heat treatment during the last days of incubation between chickens (Galliform) [START_REF] Halle | Influence of temperature manipulation during the last 4 days of incubation on hatching results, post-hatching performance and adaptability to warm growing conditions in broiler chickens[END_REF] and Pekin ducks (Anseriform) [START_REF] Halle | Influence of temperature stimulation during the last 6 days of incubation on hatching results and later performance in Pekin ducks[END_REF]. These differences may be due to a different timing of the ontogeny of thermoregulatory mechanisms between these two species of different orders [START_REF] Tzschentke | Development of hypothalamic neuronal thermosensitivity in birds during the perinatal period[END_REF][START_REF] Halle | Influence of temperature stimulation during the last 6 days of incubation on hatching results and later performance in Pekin ducks[END_REF], and could therefore also be seen within the same subspecies (animals of the same order and family) with different incubation times.

In fatty liver production, two species of male's duck are used: Muscovy and mule.

Mule duck is a sterile hybrid from the crossbred between a female Pekin (Anas platyrhynchos) and a male Muscovy (Cairina moschata) and is used for more than 99 % of the production with around 16 000 tons of fresh fatty liver per year [START_REF] Litt | Conditions d'élevage des palmipèdes à foie gras : des demandes sociétales à une démarche de progrès[END_REF].

The hatchability of duck is lower than hatchability of other poultry species, ranging from 65 to 82 % against 79 to 85 % for the other birds [START_REF] Hodgetts | Current hatchabilities in species of domestic importance and the scope for improvement[END_REF], but the literature is not as broad as for other species such as broilers. The period of incubation of crossbred eggs is around 32 days for the mule duck, compared to 28 days for the Pekin and 35 days for the Muscovy [START_REF] Brun | Le canrd mulard: déterminisme génétique d'un hybride intergénérique[END_REF]. Conditions of incubation of the three species are relatively close, with an optimal ambient temperature comprised between 37 to 38°C and a relative humidity between 60 to 80 %, according to the embryogenesis period [START_REF] El-Hack | Impact of environmental and incubation factors on hatchability of duck eggs[END_REF].

The purpose of this study is therefore to measure, for the first time, a set of zootechnical parameters around the birth of mule ducks after a discontinuous (16h/24h) increase of 1.5°C of the incubation temperature with an adjusted targeted RH of 65 %.

Six 10-day developmental windows were tested, from embryonic day 9 (E9) to embryonic day 14 (E14), to assess the impact of temperature on these parameters by measuring zootechnical performance before, during and after hatching.

Material and Methods

Experimental design

Three experimental trials were done to test seven conditions of egg incubation (Figure 1), including the control condition. Each condition was performed using 299, 300 and 289 eggs per incubator for trials 1, 2 and 3 respectively. In each trial, a control group was added, with an average temperature of 37.6°C and an average relative humidity (RH) of 61.5 %, 62.8 % and 63.6 % for the trial 1, 2 and 3 respectively. The six others conditions differed from control groups by the temperature (+1.5°C, 16h/24) and RH (+5 % targeted) during 10 days, but at different developmental stages during the incubation. In the trial 1, we applied the changes in conditions from embryonic day 11 (E11) until E21, first window chosen according to the high expression level of genes involved in lipid and carbohydrate metabolism measured during this period [START_REF] Massimino | Ontogeny of hepatic metabolism in mule ducks highlights different gene expression profiles between carbohydrate and lipid metabolic pathways[END_REF]. In the trial 2, windows E10 to E20 and E12 to E22 were tested. In the trial 3, windows E9 to E19, E13 to E23 and E14 to E24 were tested in order to cover a wide incubation period and to see if the periods are more or less traumatic for the embryos.

The target of temperature has been reached for all the conditions but the targeted RH was more difficult to attain, and most of the time remained close to the control group (E9-E19: 62.5 %, E10-E20: 64.3 %, E11-E21: 59.8 %, E12-E22: 61.9 %, E13-E23: 63.0 % and E14-24: 62.3 %).

For the experimental trials 1, 2 and 3, eggs obtained from the same females, aged of 36, 42 and 49 weeks respectively. The genotype H85 was provided by Grimaud Frères Selection company (Roussay, France). Eggs were collected on a day and stored at ambient temperature three days before incubation. Eggs were dispatched randomly between treatments to obtain an equal number of eggs in each group with 299, 300 and 289 eggs in trial 1, 2 and 3 respectively. In each trial, eggs were incubated in identical automated commercial incubators with regulation of temperature, humidity and ventilation (SOLOGNE model of LA NATIONALE, reconditioned model). Incubation parameters were recorded during the whole incubation period and temperature and RH were confirmed by two independent sensors (LoRa® SPY U). Each incubator had its own pair of sensors fixed on incubator inner wall closed to the doors. All eggs were subjected to an automatic rotation of 90° every three hours and were sprayed manually the afternoon before the changing of conditions of incubation.

Two light controls, called candling, were done during the incubation period in order to estimate the viability of the egg. The first was done at the 7 th embryonic day (E7) and the second, after conditions of incubation change, at the 28 th embryonic day (E28). Unfertile eggs were removed from incubators and the remaining eggs were pooled to avoid local temperature disturbance.

At the 28 th day of incubation, all eggs were placed in the same hatchery (BRETAGNE model of LA NATIONALE, reconditioned model) at 37.3°C and 80 % of RH. During four days, the number of newly hatched ducks was recorded once a day.

Egg incubation measurements

The temperature and the relative humidity, in incubators, were recorded every 15 minutes by sensors (LoRa® SPY U), and results were available on a software. Targeted at 60 % for control conditions and 65 % for treated conditions, means of RH throughout the 10-days incubation period were 61.5 % for control and 59.8 % for changes conditions in trial 1; 62.8 % for control, 64.3 % for E10-E20 and 61.9 % for E12-E22 conditions in trial 2; 63.6 % for control, 65.5 % for E9-E19, 63.0 % for E13-E23 and 62.3 % for E14-E24 in trial 3.

At the first candling (E7), the number of non-fertile eggs and dead embryos were recorded. At the second candling (E28), only the number of dead embryos was recorded. Fertility, early (up to E7), medium (between E7 and E28) and late (in hatchery) mortality rates were therefore measured. Fertility was the ratio of the number of fertilized eggs, observed at E7, to the number of eggs in the incubator. Mortalities were the ratio of the number of died eggs, recorded at E7, E28 and at hatch, to the number of eggs fertilized in the incubator.

Hatch and post-hatch measurements

Once a day, the number and gender of hatched ducklings were recorded. The identification of the sex was based on two criteria: black spots on the duckling head and black eyes for males and no black spot and red eyes for females, completed by vent sexing when it is not clear. The weight of male and female ducklings was recorded using a scale (SARTORIUS Signum 1, model SIWRDCP.1.3.1) during the peak hatching day corresponding to the day with the highest proportion of hatched ducklings (n = 129 to 214 excepted trial 1 n = 10). For each group, hatchability, sex ratio, mean weight and internal temperature, by using digital express thermometers (Digital Express Thermometer from Gilbert), were measured.

Around 50 male ducklings and 50 female ducklings, born at the hatching peak, were arbitrary kept to do measurements of weight (Automatic weighing system) and internal temperature after six days of life. Internal temperatures were measured using digital express thermometers (Digital Express Thermometer from Gilbert), inserted into the cloaca. During the rearing period, water and food were supplied ad libitum.

Because of the health crisis related to COVID19, the trial 1 (March 2020 to April 2020) had to be stopped at D1. Only the weight and internal temperature of 10 animals at hatching were recorded due to staff limitations.

Statistical analysis

Within a single trial, statistical tests were done to compare the control group and groups with a change in incubation conditions. Hatchability, mortalities and sex ratio were compared by using a Chi-squared test. Weights and internal temperatures were compared by using one-way Anova or student and Wilcoxon tests depending on normality and homoscedasticity. For these tests, the individual was used as the experimental unit, and the variable response was studied according to the treatment. All statistical analyses were performed with R software version 3.6.2. Statistical testing was performed on datasets at a statistical significance of 5%.

Results

In each trial, an incubation control was done with constant conditions of incubation. Even if conditions were similar, we observed a variation between controls of the different trials, named the inter-trial dispersion (Figure 2). Inter-trial dispersions were estimated by the difference between the highest value and each value of the two other trials. Regarding mid-embryonic mortalities, the maximum inter-trial dispersions were 2.0 % between trial 1 and 2 and 3.8 % between trial 3 and 2, although not significant (Figure 2a). In contrast, for the late embryonic mortalities, differences were significant (Figure 2b) and these maximal inter-trial dispersions were 3.5 % between trial 1 and 2 and 5.7 % between trial 3 and 2. Interestingly, the most variable factor also proved to be the most critical: hatchability. The maximum inter-trial dispersions were as high as 8.0 % between trial 2 and 3, thus becoming significant with a p-value lower than 0.01 (Figure 2c). Between trials 1 and 3, the dispersion was 3.7 % and was not significantly different.

Similarly, we did not measure a significant difference between the three trials for the sex ratio (Figure 2d), despite a maximum inter-trial dispersion of 3.8 % between trials 2 and 3.

For all the conditions, fertility of eggs was around 93 % and the embryonic mortality at the first candling varied around 1 % (data not shown). The second candling was done after the change of incubation conditions and reflected the impact of this change on the embryo mortality. The results of three trials conducted at different time with their own variability can only be compared statistically with their internal control (within the same row, Table 1). However, even considering the maximal inter-trial dispersion (ITD), it is possible to distinguish two groups among the experimental the early change groups (with E9-E19 and E10-E20) and the late change groups (with E11-E21, E12-E22, E123-E23, and E14-E24) that appeared to respond differently to incubation changes. The average mid-embryonic mortality of the control groups was 8.3 % while the increase of 1.5°C (16h/24, with the adjusted RH) in the early stages of development (from the 9th or 10th embryonic day), resulted in a strong increase of this rate in the second candling, reaching 24.5 % for trial 2. This observation was then confirmed by the statistical tests performed within each trial (table 1, first panel). In trial 3, the mid-embryonic mortality was significantly higher for the condition E9-E19 (reaching 20.5 %) compared to the control of the trial 3, and conditions E13-E23 and E14-E24 (6.4 %, 13.0 % and 13.1 % respectively; P < 0.05). In trial 2, the mid-embryonic mortality was significantly higher for the condition E10-E20 (24.5 % mortality) compared to the control of the trial 2 (10.2%), and the condition E12-E22 (15.9 %) (P < 0.05).

Changes in condition from E11 to E14 led to a significantly lower increase in mortality than earlier stages and one condition (E12-E22) did not show a significant difference compared to its control from trial 2.

The measurement of late mortality was done inside the hatchery after the incubation period and all results were grouped in the second panel of the Table 1. However, no statistical difference has been detected at this stage.

For all the conditions, the hatching peak occurred on embryonic day 30 (E30), with an average hatchability of 84.2% for control groups (Table 1, third panel). Again, the two earliest groups (trial 2 and 3) presented the strongest impact on hatchability, with a significant decrease compared to their own control (P < 0.001), dropping below 70 %. On the other hand, among the four late conditions changes, only the E14-E24 differed significantly from its control.

In the last panel of the table 1, the proportion of total males born appeared to be higher with earlier temperature change. If we looked at groups with early condition changes, we observed a significantly higher sex ratio in favor of males, compared to their own control. Groups E9-E19 and E10-E20 raised indeed a rate of 67.6 % and 65.5 % of males compared to controls with an average around 52 % (P < 0.05). Three out of the four other experimental groups have the same sex ratio as the controls.

Post-hatch body weight and internal temperature measurements are listed in Table 2 all groups with condition of incubation changes presented lower body weights than their control group on the day of hatching (Table 2, first panel). The condition E11-E21 appeared to have a lower average birth weight than the other groups but this result was not statistically different, due to the fact that only 10 animals were sampled.

However, six days later (D6), most of the experimental groups had caught up to the weight of their control group (Table 2, second panel). At this stage, only the condition E12-E22 remained significantly below its own control (P < 0.001). At D6, the internal temperature was also lower for all groups with changes in incubation conditions, compared to their own controls (Table 2, third panel) and these differences were significant (P < 0.01).

Discussion

In ducks, the incubation is one of the most delicate periods in terms of survival [START_REF] Wilson | Physiological requirements of developing embryo: temperature and turning[END_REF][START_REF] Mueller | The Physiology of the Avian Embryo[END_REF]. In recent years, the concept of embryonic thermal programming has implied that changes in egg incubation conditions could optimize animal resistance and even production [START_REF] Yahav | Thermal manipulations during broiler chick embryogenesis: Effects of timing and temperature[END_REF][START_REF] Piestun | Thermal manipulations during broiler embryogenesis improves post-hatch performance under hot conditions[END_REF][START_REF] Massimino | Impact of thermal manipulation during embryogenesis on hepatic metabolism in mule ducks[END_REF][START_REF] Carvalho | Embryonic thermal manipulation has short and long-term effects on the development and the physiology of the Japanese quail[END_REF]. In this study, we tested six embryonic periods during which we increased temperature and relative humidity (RH), in a discontinuous way (16h/24h) to measure the consequences of such variations on early performances.

Evidence of variability between trials

Many external parameters may explain differences in hatchability within the same species, such as age of females, quality of transport, storage of eggs before incubation, incubation conditions (El-Hanoun et al., 2012)… However, the contribution of these parameters should be minimal in your trials since we were careful to reproduce the same control conditions between trials. Besides, the same incubator was used for all the control conditions in the three trials to avoid an incubator effect, but an inter-trial dispersion was still observed. The age of the females providing the eggs changed between trials, but the results do not show a correlation between this factor and hatching, since the oldest females were not the ones that gave the worst yields. Only the relative humidity was difficult to control and varied between 61.5 % to 63.6 % despite a targeted of 60 %. This difference could be at least partially responsible for the variability in hatching measured at the end of the trial, since RH was a very important parameter for the proper conduct of embryogenesis [START_REF] Barott | Effect of temperature, humidity and other factors on the hatch of hens' eggs and on energy metabolism of chick embryos[END_REF][START_REF] Bruzual | Effects of relative humidity during incubation on hatchability and body weight of broiler chicks from young breeder flocks[END_REF][START_REF] Peebles | Effects of incubator humidity and hen age on yolk composition in broiler hatching eggs from young breeders[END_REF]. The RH variation could be related to the manual part of its control by spraying, and therefore to the spraying duration, the amount of water sprayed or the duration of door opening… Finally, the observed inter-trial dispersion could also be due to random but significantly different inter-individual variability between groups in different trials.

Consequently, the actual impact of the treatment was tempered by this inter-trial dispersion which was allowed by the realization of a control in each of the trials.

Early embryonic stages are more sensitive to temperature changes

Three critical periods have been described in poultry to explain embryonic mortality [START_REF] Romanoff | Critical Periods and Causes of Death in Avian Embryonic Development[END_REF][START_REF] Sauveur | Développement embryonnaire et incubation[END_REF]. The early mortality was linked to difficulties of setting up the first embryonic structures. The mid-mortality was mainly due to the start of essential functions as the respiration or the kidney function. The mortality in the hatchery, also called late mortality, involved the difficulty of getting out of the eggshell. In consequence, a change of environmental factors during these critical periods could be lethal for embryos [START_REF] Romanoff | Critical Periods and Causes of Death in Avian Embryonic Development[END_REF][START_REF] Ono | Responses of developing chicken embryos to acute changes in ambient temperature: Noninvasive study of heart rate[END_REF][START_REF] Loyau | Cyclic variations in incubation conditions induce adaptive responses to later heat exposure in chickens: A review[END_REF].

Here, the changes in incubation conditions were applied for 10 days, over six different periods from E9 to E14, thus over a fairly wide range of time corresponding to the middle of the incubation and known to have high plasticity at a molecular level in duck [START_REF] Massimino | Ontogeny of hepatic metabolism in mule ducks highlights different gene expression profiles between carbohydrate and lipid metabolic pathways[END_REF]. We measured early embryonic mortality (E7), midmortality (E28) and late mortality in the hatchery. The early embryonic mortalities were identical between all groups, due to the lack of change in incubation conditions at this time. In contrast, we showed an increase in mid-mortality in almost all treated groups compared to their own control. These results confirmed that incubation must respect very strict parameters to obtain good hatch performance [START_REF] Wilson | Physiological requirements of developing embryo: temperature and turning[END_REF][START_REF] Mueller | The Physiology of the Avian Embryo[END_REF] and that the application of new incubation conditions for the purpose of embryonic programming requires careful analysis of its impact on survival. However, we also found an interesting result regarding the period of change, with two groups standing out. Indeed, trials 2 and 3 allowed us to statistically compare early change conditions (starting at E9 and E10) with later changes (starting at E12, E13 and E14), and these comparisons showed us that the earlier the changes, the greater the impact on embryonic mortality.

These results suggest that our incubation parameter changes were more likely to disrupt the early settlement of embryonic structures, than the later functionality of essential organs. Because several systems are organized during the first two weeks of embryogenesis, (the digestive system, the body temperature regulation system, the organization of embryonic membranes, the immune system, the nervous system...), this period appeared to be more critical and the embryo appeared to be more vulnerable to external factors [START_REF] Dibner | Early feeding and development of the immune system in neonatal poultry[END_REF][START_REF] King | Ontogenetic development of intestinal digestive functions in white pekin ducks[END_REF][START_REF] Jenkins | Ontogeny of the hypothalamo-pituitary-adrenocortical axis in the chicken embryo: A review[END_REF][START_REF] Baarendse | Ontogeny of avian thermoregulation from a neural point of view[END_REF][START_REF] De Oliveira | Important metabolic pathways in poultry embryos prior to hatch[END_REF][START_REF] Baggott | Development of extra-embryonic membranes and fluid compartments[END_REF]. Consistent with previous studies in poultry, these results in ducks confirm that early embryonic stages are more sensitive to changes in environmental factors, and that a modification in condition can therefore become lethal [START_REF] Yan | Evidence in duck for supporting alteration of incubation temperature may have influence on methylation of genomic DNA[END_REF][START_REF] Sardary | Effect of Thermal Manipulation during Embryogenesis on Hatching Traits[END_REF][START_REF] Carvalho | Embryonic thermal manipulation has short and long-term effects on the development and the physiology of the Japanese quail[END_REF].

Changes in incubation conditions provide a first challenge before hatching

A few days later, almost all late change conditions (after E11) showed no significant decrease in hatchability compared to their own control, unlike the two earliest conditions. These data demonstrated that although mid-term mortality was increased by the later changes in incubation condition, hatching performance was the same as the control groups. Finally, this assessment indicated that total mortality during embryogenesis was less impacted by a discontinuous temperature increase of 1.5°C and 5 % RH, when applied from the 11 th embryonic day, confirming previous study in chicken [START_REF] Collin | The effect of duration of thermal manipulation during broiler chick embryogenesis on body weight and body temperature of post-hatched chicks[END_REF][START_REF] Piestun | Thermal manipulations during broiler embryogenesis: Effect on the acquisition of thermotolerance[END_REF]. On the contrary, when these increases in temperature and RH were applied from E9 or E10, hatchability was definitely lower compared to the control condition. This decrease, visible even considering the theoretical maximum dispersion, was probably directly related to the high mortality measured at the second candling. This information could be a valuable aid for the design of new embryonic thermal programming conditions in mule ducks.

These results led us to hypothesize a pre-elimination of weak embryos during the late changes of incubation conditions, and a proportional reduction in hatchery mortality.

Unfortunately, we did not measure a significant decrease in late mortality, certainly because of the low number of deaths at this period for all groups. However, it can be assumed that in the control groups, mortality spread over time because the weakest embryos were not exposed to environmental changes until hatching, but ultimately the final survival rate was the same as for the late treated groups. This observation also suggests that there was no selection of animals despite the high mid-mortality observed with these new incubation conditions, but potentially a time lag, which could therefore be worth further investigating in the context of embryonic programming. On the contrary, the earlier changes (starting from E9 or E10) actually induced a higher overall mortality, and not a time lag.

An unbalanced sex ratio linked to the crossbred but also influenced by environmental factors

The unbalance in the sex ratio of mule duck is well known with a preponderance of males at hatching (Batellier et al., 2004 a), explained by an alteration of female embryo development [START_REF] Byerly | Sex Ratio and Embryonic Mortality in the Domestic Fowl[END_REF]. This unbalanced sex ratio follows the rule of Haldane [START_REF] Haldane | Sex ratio and unisexual sterility in hybrid animals[END_REF], explaining the greater fragility of the heterogamous sex (the female in ducks) during cellular division.

In contrast to a recent study that showed no effect of the incubation temperature on sex ratio in Mallard ducks [START_REF] Koláčková | Effect of incubation temperature on sex-dependent embryo mortality and morphological traits in Mallard[END_REF], our results suggest that increasing temperature and RH at the earliest stages of embryogenesis may further increase female mortality in mule ducks. In chicken, last days of incubation seem to be also critical and impact the sex ratio in favor of males [START_REF] Tzschentke | Influence of temperature stimulation during the last 4 days of incubation on secondary sex ratio and later performance in male and female broiler chicks[END_REF][START_REF] Elmehdawi | Low-intensity, short-duration thermal stimulation during the late phase of incubation alters secondary sex ratio in favour of males[END_REF]. These data tend to confirm the greater fragility of heterogamous sex in hybrid species, and could be investigated as a precocious method of selection of males for the fatty liver production (Batellier et al., 2004 b).

The catch-up growth, a mechanism that could influence long-term physiology

At hatch, all groups with incubation conditions changes had lower weights than control groups. Numerous studies on chickens [START_REF] Piestun | Thermal manipulations during broiler embryogenesis: Effect on the acquisition of thermotolerance[END_REF][START_REF] Al-Rukibat | Thermal manipulation during late embryogenesis: Effect on body weight and temperature, thyroid hormones, and differential white blood cell counts in broiler chickens[END_REF][START_REF] Zaboli | Thermal manipulation during pre and post-hatch on thermotolerance of male broiler chickens exposed to chronic heat stress[END_REF] and mule ducks [START_REF] Massimino | Impact of thermal manipulation during embryogenesis on hepatic metabolism in mule ducks[END_REF] have already shown that an increase in incubation temperature leads to a decrease in hatching body weight.

These observations can be explained in part by an acceleration of embryonic development [START_REF] Sardary | Effect of Thermal Manipulation during Embryogenesis on Hatching Traits[END_REF][START_REF] Boleli | Poultry egg incubation: Integrating and optimizing production efficiency[END_REF][START_REF] Maatjens | Temperature during the last week of incubation. I. Effects on hatching pattern and broiler chicken embryonic organ development[END_REF] resulting in increased yolk sac consumption prior to the first meal [START_REF] Lamot | Effects of moment of hatch and feed access on chicken development[END_REF]. Although here all groups had their peak hatch on the same day (E30), hatcher surveys were only done once a day at 8:00 am, leaving a 24-hour window for the animals to emerge from their shells. It was therefore possible that animals treated with the new incubation conditions were born on average a few hours earlier than the control group resulting in a decreased hatch weight. To support this hypothesis, it has been shown in quail that when hatchings are synchronized taking into account the acceleration of development induced by a rise in temperature, the birth weight of thermally treated individuals is not different from controls [START_REF] Carvalho | Embryonic thermal manipulation has short and long-term effects on the development and the physiology of the Japanese quail[END_REF].

However, this growth delay was totally recovered in only a few days by most of the treated groups, since at D6, only the E12-E22 condition still presented a significant difference with its control. This catch-up growth has already been observed in poultry exposed to increased temperature during embryogenesis [START_REF] Piestun | Thermal treatments prior to and during the beginning of incubation affect phenotypic characteristics of broiler chickens posthatching[END_REF][START_REF] Zaboli | Thermal manipulation during pre and post-hatch on thermotolerance of male broiler chickens exposed to chronic heat stress[END_REF][START_REF] Massimino | Impact of thermal manipulation during embryogenesis on hepatic metabolism in mule ducks[END_REF], but also in mammals with delay of growth at birth for different reasons (McLaughlin et al., 2020). This phenomenon is not yet fully understood, but two main theories seem to emerge [START_REF] Wit | Catch-up growth : Definition , mechanism , and models[END_REF]: the neuroendocrine (involving Growth Hormone, thyroid, Insulin-like Growth Factor-I axis) and the growth plate hypothesis. Whatever the mechanism, the long-term consequences on metabolism are no longer to be demonstrated [START_REF] Bieswal | The importance of catch-up growth after early malnutrition for the programming of obesity in male rat[END_REF][START_REF] Singhal | Long-Term Adverse Effects of Early Growth Acceleration or Catch-Up Growth[END_REF][START_REF] Berends | Programming of central and peripheral insulin resistance by low birthweight and postnatal catch-up growth in male mice[END_REF] and may be of interest to study in the context of embryonic programming.

Finally, as previously shown in chicken [START_REF] Loyau | Cyclic variations in incubation conditions induce adaptive responses to later heat exposure in chickens: A review[END_REF] and duck [START_REF] Massimino | Impact of thermal manipulation during embryogenesis on hepatic metabolism in mule ducks[END_REF] all groups treated with increased incubation temperature had lower body temperatures compared to their own control after birth. In chickens, this decrease in body temperature is often associated with a better thermotolerance of the animals [START_REF] Al-Zghoul | Hsp90, Hsp60 and HSF-1 genes expression in muscle, heart and brain of thermally manipulated broiler chicken[END_REF][START_REF] Al-Rukibat | Thermal manipulation during late embryogenesis: Effect on body weight and temperature, thyroid hormones, and differential white blood cell counts in broiler chickens[END_REF], and can be explained by the application of a temperature increase during the embryonic development of the axes responsible for thermal regulation [START_REF] Jenkins | Ontogeny of the hypothalamo-pituitary-adrenocortical axis in the chicken embryo: A review[END_REF][START_REF] Piestun | Thermal manipulations during broiler embryogenesis: Effect on the acquisition of thermotolerance[END_REF]. Our results in mule ducks suggest that incubation changes from E9 to E14 during 10 days may alter the same axes of thermal regulation, but no differences were measured between the earliest and latest treated groups.

Conclusion

Although changes in temperature and RH during incubation might improve certain animal performances. This study also confirms that incubation parameters are critical to the zootechnical performance of male ducks, a sterile hybrid from the crossbred between a female Pekin and a male Muscovy, at hatch and that the timing of changes is an important factor in maintaining optimal hatchability.

Ethic approval

Tables Table 1. Impact of changing incubation conditions on embryonic mortality and birth parameters in mule ducks. The results of mid mortality (first panel), late mortality (second panel), hatchability (third panel) and proportion of males born (fourth panel) are listed according to the embryonic window (E). The maximum of inter-trial dispersion (ITD) and the average +/-SD of controls complete the table. Average ± SD 51.8 ± 1.9

a-c Within a row, means values without a common superscript letter differ significantly (P < 0.05).

Table 2. Impact of changing incubation conditions on rearing parameters in mule ducks.

The results of weight at hatch (first panel), weight at 6 days (D6) (second panel) and internal temperature at 6 days (third panel) are represented according to the embryonic window (E). The condition E11-E21 has no data at D6 for BW and internal temperature. Changes in conditions (+1.5°C 16h/24 and target +5 % RH) applied for 10 days are illustrated by the horizontal lines representing 10-day intervals. The period from E11 to E21 was tested on trial 1 (light gray line), periods E10-E20 and E12-E22 on trial 2 (dark gray lines) and periods E9-E19, E13-E23 and E14-E24 on trial 3 (black line). A control group was present for each trial with a temperature of 37.6°C and a mean relative humidity (RH) of 62.6 %.

  Within a row, means values without a common superscript letter differ significantly (P < 0.05).

  Figure captions

Figure 2 .

 2 Figure 2. The inter-trial dispersion between controls for 4 parameters: mid embryonic mortality, late embryonic mortality, hatchability and sex ratio in mule ducks Measurement of mid-embryonic mortality (n = 265 to 279) (a), late embryonic mortality (n = 247 to 256) (b), hatching rate (c) (n = 265 to 279), and sex ratio (d) (n = 221 to 238) for the control conditions of trials 1, 2, and 3. Chi² statistical tests were used. Bars with different superscript letters are significantly different (P < 0.05).

Acknowledgements

We thank all the technicians of the experimental station of Artigueres for the daily care given to the animals and the monitoring of the good conditions for the experiments.

Financial support statement

This work was supported by grants from the "Comité Interprofessionnel des Palmipèdes à Foie Gras (CIFOG agreement 2020/05) and from "Comité Départemental des Landes" (CD40-CA salary for three years).

Data and model availability statement

None of the data were deposited in an official repository. The data that support the study findings are available upon request.

Declaration of interest

The authors state that this work was supported by a grant from the Comité Interprofessionnel des Palmipèdes à Foie Gras (CIFOG) but that these funders played no role in the design of the study and in the collection, analysis and interpretation of the results.


Mathieu Lacroix

Michele Barbato

Roland Grappe

Emiliano Lancini

Roberto Wolfler Calvo

Roberto Woler Calvo

The Schrijver system of the flow cone in series-parallel graphs

Keywords: polyhedra and equimodular matrices, arXiv Total dual integrality, Box-total dual integrality, Schrijver system, Hilbert basis, Flow cone, Multicuts, Series-parallel graphs

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

Introduction

Total dual integral systems were introduced in the late 70's and are strongly connected to min-max relations in combinatorial optimization [START_REF] Schrijver | Theory of linear and integer programming[END_REF]. A rational system of linear inequalities Ax ≤ b is total dual integral (TDI) if the minimization problem in the linear programming duality: max{cx : Ax ≤ b} = min{yb : y ≥ 0, yA = c} admits an integer optimal solution for each integer vector c such that the maximum is nite. Such systems describe integer polyhedra when b is integer [START_REF] Edmonds | A min-max relation for submodular functions on graphs[END_REF]. Schrijver [START_REF] Schrijver | On total dual integrality[END_REF] proved that every full-dimensional polyhedron is described by a unique minimal TDI system Ax ≤ b with A integerits Schrijver system [START_REF] Cook | Linear systems for constrained matching problems[END_REF].

A stronger property is the box-total dual integrality, where a system Ax ≤ b is box-total dual integral (box-TDI) if

Ax ≤ b, ≤ x ≤ u
is TDI for all rational vectors and u (with possible innite components). General properties of such systems can be found in Cook [START_REF] Cook | On box totally dual integral polyhedra[END_REF] and Chapter 22.4 of Schrijver [START_REF] Schrijver | Theory of linear and integer programming[END_REF]. Note that, although every rational polyhedron {x : Ax ≤ b} is described by a TDI system 1 k Ax ≤ 1 k b, for some integer k, not every polyhedron is described by a box-TDI system. A polyhedron described by a box-TDI system is called a box-TDI polyhedron. As proved by Cook [START_REF] Cook | On box totally dual integral polyhedra[END_REF], every TDI system describing such a polyhedron is actually box-TDI.

In the last decade, several new box-TDI systems were exhibited. Chen, Ding, and Zang [START_REF] Chen | A characterization of box-mengerian matroid ports[END_REF] characterized box-Mengerian matroid ports. In [START_REF] Chen | The box-TDI system associated with 2-edge connected spanning subgraphs[END_REF], they provided a box-TDI system describing the 2-edge-connected spanning subgraph polyhedron for series-parallel graphs. Ding, Tan, and Zang [START_REF] Ding | When is the matching polytope box-totally dual integral[END_REF] characterized the graphs for which the TDI system of Cunningam and Marsh [START_REF] Cunningham | A primal algorithm for optimum matching[END_REF] describing the matching polytope is actually box-TDI. Ding, Zang, and Zhao [START_REF] Ding | On box-perfect graphs[END_REF] introduced new subclasses of box-perfect graphs. Cornaz, Grappe, and Lacroix [START_REF] Cornaz | Trader multiow and box-TDI systems in seriesparallel graphs[END_REF] provided several box-TDI systems in series-parallel graphs. Recently, Chervet, Grappe, and Robert [START_REF] Chervet | Principally box-integer polyhedra and equimodular matrices[END_REF] gave new geometric characterizations of box-TDI polyhedra.

As mentioned by Pulleyblank [START_REF] Pulleyblank | Total dual integrality and b-matchings[END_REF], it is not uncommon that the minimal integer system and the Schrijver system of a polyhedron coincide. This is the case of the matching polytope and matroid polyhedra. However, this does not hold in general, as shown by Cook [4] and Pulleyblank [14] for the b-matching polyhedron, and by Seb® [START_REF] Seb® | The schrijver system of odd join polyhedra[END_REF] for the T -join polyhedron.

In this paper, we are interested in TDI, box-TDI, and Schrijver systems for the ow cone of series-parallel graphs. Given a graph G = (V, E), a ow of G is a couple (C, e) with C a circuit of G and e an edge of C. In a ow (C, e), the edge e represents a demand and C \ e represents the path satisfying this demand. The incidence vector of a ow (C, e) is the 0/ ± 1 vector χ C\e -χ e . The ow cone of G is the cone generated by the ows of G and the unit vectors χ e of R E .

A cut δ(W ) is the set of edges having exactly one endpoint in a subset W of V . A bond is an inclusionwise minimal nonempty cut. Note that a nonempty cut is the disjoint union of bonds. The cut cone of G is the cone generated by the incidence vectors of the cuts of G, or, equivalently, by the incidence vectors of bonds of G.

When G has no K 5 -minor, the ow cone of G is the polar of the cut cone and is described by x(C) ≥ 0, for all cuts C of G [START_REF] Seymour | Sums of circuits[END_REF]. Chervet, Grappe, and Robert [START_REF] Chervet | Principally box-integer polyhedra and equimodular matrices[END_REF] proved that the ow cone is a box-TDI polyhedron if and only if the graph is series-parallel. Moreover they provided the following box-TDI system:

1 2
x(B) ≥ 0 for all bonds B of G.

(

) 1
Quoting them, they leave open the question of nding a box-TDI system with integer coefcients, which exists by [START_REF] Schrijver | Theory of linear and integer programming[END_REF]Theorem 22.6(i)] and [5, Corollary 2.5].

Contribution. The goal of this paper is to answer the question of [START_REF] Chervet | Principally box-integer polyhedra and equimodular matrices[END_REF] mentioned above.

Throughout, the main concept that we use is that of Hilbert basis, whose denition and connection with TDIness are given at the end of the introduction.

We rst prove that

x(M ) ≥ 0 for all multicuts M of G,

January 14, 2020 is a TDI system describing the ow cone if and only if the graph is series-parallel. As the ow cone is a box-TDI polyhedron for such graphs, this implies that System (2) is a box-TDI system if and only if the graph is series-parallel. We then rene this result by providing the corresponding Schrijver system, which is composed of the so-called chordal multicutssee Corollary 3.4.

This completely answers the question of [START_REF] Chervet | Principally box-integer polyhedra and equimodular matrices[END_REF].

Outline. In the next paragraph, we provide denitions and notation. In Section 2, we rst characterize the graphs for which multicuts form a Hilbert basis. It follows that System [START_REF] Chen | The box-TDI system associated with 2-edge connected spanning subgraphs[END_REF] is box-TDI precisely for series-parallel graphs. In Section 3, we provide a minimal integer Hilbert basis for multicuts in series-parallel graphs. This gives the Schrijver system for the ow cone in series-parallel graphs.

Denitions. Given a nite set S and a subset T of S, we denote by χ T ∈ {0, 1} S the incidence vector of T , that is χ T s equals 1 if s belongs to T and 0 otherwise, for all s ∈ S. Since there is a bijection between sets and their incidence vectors, we will often use the same terminology for both.

Let G = (V, E) be a loopless undirected graph. Given a subset U of V , the cut δ(U ) is the set of edges having exactly one endpoint in U . A bond is a cut containing no other nonempty cut. Given a partition {V 1 , . . . , V k } of V , the set of edges having endpoints in two distinct V i 's is called multicut and is denoted by δ(V 1 , . . . , V k ). A set of edges M is a multicut if and only if |M ∩ C| = 1 for all circuits C of Gsee e.g. [START_REF] Cornaz | Max-multiow/min-multicut for G+H series-parallel[END_REF]. The reduced graph of a multicut M is the graph G M obtained by contracting all the edges of E \ M . Note that a multicut of G M is also a multicut of G. We denote respectively by M G and B G the set of multicuts and the set of bonds of G. A subset of edges of G is called a circuit if it induces a connected graph in which every vertex has degree 2. Given a circuit C, an edge of G is a chord of C if its endpoints are two nonadjacent vertices of C. A graph is 2-connected if it remains connected whenever a vertex is removed.

A graph is series-parallel if its 2-connected components can be constructed from the circuit of length two C 2 by repeatedly adding edges parallel to an existing one, and subdividing edges, that is, replacing an edge by a path of length two. Series-parallel graphs are those having no K 4 -minor [START_REF] Duffin | Topology of series-parallel networks[END_REF]. A graph is chordal if every circuit of length 4 or more has a chord.

The cone C generated by a set of vectors {v 1 , . . . ,

v k } of R n is the set of nonnegative combinations of v 1 , . . . , v k , that is, C = k j=1 λ j v j : λ 1 , . . . , λ k ≥ 0 . A set of vectors {v 1 , . . . , v k } is a
Hilbert basis if each integer vector in their cone can be expressed as a nonnegative integer combination of v 1 , . . . , v k . A Hilbert basis is integer if it is composed of integer vectors, and it is a minimal integer Hilbert basis if it has the smallest number of vectors among all integer Hilbert basis generating the same cone. Each pointed rational cone has a unique minimal integer Hilbert basis [START_REF] Schrijver | On total dual integrality[END_REF]Theorems 16.4]. The link between Hilbert basis and TDIness is in the following result. Theorem 1.1 (Corollary 22.5a of [START_REF] Schrijver | Theory of linear and integer programming[END_REF]). A system Ax ≥ 0 is TDI if and only if the rows of A form a Hilbert basis. [START_REF] Seymour | Sums of circuits[END_REF] 2. When Do Multicuts Form a Hilbert Basis?

A Characterization

The following result characterizes the graphs for which the multicuts form a Hilbert basis.

Theorem 2.1. The multicuts of a graph form a Hilbert basis if and only if the graph is series-parallel.

Proof. First, let us show that the incidence vectors of the multicuts of a non series-parallel graph do not form a Hilbert basis. Suppose that G = (V, E) has K 4 as a minor. Then, V can be partitioned into four sets {V 1 , . . . , V 4 } such that V i induces a connected subgraph and at least one edge connects each pair V i , V j for i, j = 1, . . . , 4. We subdivide δ(V 1 , V 2 , V 3 , V 4 ) into E 1 , . . . , E 6 as in Figure 1.

E 1 E 2 E 3 E 4 E 5 E 6 V 3 V 1 V 2 V 4
Figure 1: Edges in the gure represent sets of edges of G having endpoints in distinct V i 's.

Let Ê = {e 1 , . . . , e 6 } where e i ∈ E i for all i = 1, . . . , 6, and let w ∈ Z E be as follows:

w e =    2 if e ∈ E 1 , 1 if e ∈ E 2 , . . . , E 6 , 0 otherwise. Since w = 1 2 δ(V 1 ) + 1 2 δ(V 2 ) + 1 2 δ(V 1 ∪ V 3 ) + 1 2 δ(V 1 ∪ V 4
), it belongs to the cut cone of G. Moreover, w χ Ê = 7. Any conic combination of multicuts yielding w involves only multicuts contained in δ(V 1 , . . . , V 4 ). Each of these multicuts contains between 3 and 6 edges of Ê. Hence, if w is an integer combination of such multicuts, it is the sum of two multicuts containing 3 and 4 edges of Ê, respectively. This means that w is the sum of δ(V i ) and δ(V i , V j ) for some i = j. Since w e 1 = 2, we have i ∈ {1, 2} and j ∈ {3, 4}. But then δ(V i ) ∩ δ(V i , V j ) contains an edge among e 2 , . . . , e 5 , a contradiction with w e 2 = w e 3 = w e 4 = w e 5 = 1.

Therefore, w is not an integer combination of multicuts, implying that the set of multicuts of G is not a Hilbert basis. [START_REF] Seymour | Sums of circuits[END_REF]

Acknowledgments

We are indebted to the anonymous referees for their valuable comments which greatly helped to improve the presentation of the paper.

 [START_REF] Chen | A characterization of box-mengerian matroid ports[END_REF]Michele Barbato participated in this work as a member of FCiências.ID (University of Lisbon) and was nancially supported by Portuguese National Funding under Project PTDC/MAT-NAN/2196/2014. 2 Supported by ANR DISTANCIA (ANR-17-CE40-0015).

For the other direction, remark that each multicut of a series-parallel graph is the disjoint union of multicuts of its 2-connected components. Since they belong to disjoint spaces, if the set of multicuts of each 2-connected component forms a Hilbert basis, then so does their union. Hence, it is enough to prove that the multicuts of a 2-connected series-parallel graph form a Hilbert basis. From now on, assume the graph to be 2-connected.

We prove the result by induction on the number of edges of G. When G = ({u, v}, {e, f }) is the circuit of length two, the only nonempty multicut is {e, f }, and its incidence vector forms a Hilbert basis. Now, let G = ( Ṽ , Ẽ) be obtained from a 2-connected series-parallel graph G = (V, E) by either adding a parallel edge or subdividing an edge. By the induction hypothesis, M G is a Hilbert basis.

Suppose rst that G is obtained from G by adding an edge f parallel to an edge e of E. A subset of edges M of G containing (respectively not containing) e is a multicut if and only if M ∪ f (respectively M ) is a multicut of G. Thus, the incidence vector of each multicut of G is obtained by copying the component associated with e in the component of f . Since the incidence vectors of the multicuts of G are a Hilbert basis, so are the incidence vectors of the multicuts of G.

Suppose now that G is obtained from G by subdividing an edge ē ∈ E. We denote by u the new vertex and by f and g the edges adjacent to it. A multicut M of G can be expressed as the half-sum of the bonds of G. Moreover, as each bond is a multicut, bonds and multicuts of G generate the same cone: the cut cone. Since System (1) is TDI in seriesparallel graphs [3, end of Section 6.4], the set of vectors { 1 2 χ B : B ∈ B G} forms a Hilbert basis.

Let v be an integer vector in the cut cone. There exist λ B ∈ 1 2 Z + for all B ∈ B G such that v = B∈B G λ B χ B . The vector v is an integer combination of multicuts of G if and only if v -λ δ(u) χ δ(u) is, thus we may assume that λ δ(u) ∈ {0, 1 2 }. Dene w ∈ Z E by:

Remark that (B \ē)∪f and (B \ē)∪g are bonds of G whenever B is a bond of G containing ē. Moreover, a bond B of G which does not contain ē is a bond of G. Since δ(u) is the unique bond of G containing both f and g, we have:

Thus, w belongs to the cut cone of G. Moreover, as λ δ(u) is half-integer, w is integer. By the induction hypothesis, M G is a Hilbert basis, hence there exist

Consider the family N of multicuts of G where each multicut M of G appears µ M times. Suppose rst that λ δ(u) = 0. Then, v f + v g multicuts of N contain ē. Let P be a family January 14, 2020

Hence v is a nonnegative integer combination of multicuts of G. (ii.⇔iii.) If G is series-parallel, then System (1) is box-TDI [3, end of Section 6.4]. Hence, the ow cone of G is box-TDI. Since a TDI system describing a box-TDI polyhedron is a box-TDI system [START_REF] Cook | On box totally dual integral polyhedra[END_REF], point ii. implies point iii.. A box-TDI system being TDI by denition, point iii. implies point ii.. A multicut is chordal when its reduced graph is 2-connected and chordal. Note that bonds are chordal multicuts. [START_REF] Seymour | Sums of circuits[END_REF] Theorem 3.1. The chordal multicuts of a series parallel graph form a minimal integer Hilbert basis.

Proof. Let G = (V, E) be a series-parallel graph. By Theorem 2.1, the multicuts of G form an integer Hilbert basis. Hence, the minimal integer Hilbert basis is composed of the multicuts which are not disjoint union of other multicuts. These multicuts are characterized in the following lemma, from which stems the desired theorem. Lemma 3.2. A multicut of a series-parallel graph G is chordal if and only if it can not be expressed as the disjoint union of other nonempty multicuts.

Proof. Let M be a multicut of G. Recall that every multicut of G M is a multicut of G.

Beside, since the disjoint union of multicuts is a multicut, a disjoint union of nonempty multicuts is actually the disjoint union of two nonempty multicuts.

We rst prove that, if G M is 2-connected and chordal, then M is not the disjoint union of two nonempty multicuts. By contradiction, suppose that G M is 2-connected and chordal, and M = M 1 ∪ M 2 where M 1 , M 2 are disjoint multicuts of G M . If C is a circuit of length at most three in G M , then C ⊆ M i for some i = 1, 2. Indeed, the edges of C are partitioned by M 1 and M 2 , and a multicut and a circuit intersect in either none or at least two edges.

Since G M is 2-connected and M i is nonempty for i = 1, 2, there exists at least a circuit containing edges of both M 1 and M 2 . Let C be such a circuit, of smallest length. Then, C has length at least 4, as otherwise it would be contained in one of M 1 and M 2 . Since G M is chordal, there exists a chord c of C. Denote by P 1 and P 2 the two paths of C between the endpoints of c. For i = 1, 2, the circuit P i ∪ {c} is strictly shorter than C. Since C is the shortest circuit intersecting both M 1 and M 2 , we get that P i ∪ {c} ⊆ M i for i = 1, 2. But then c ∈ M 1 ∩ M 2 , a contradiction.

To prove the other direction, rst suppose that G M is not 2-connected. Then, the set of edges of each 2-connected component of G M is a multicut of G, and M is the disjoint union of these multicuts. Now, suppose that G M is not chordal, that is, G M contains a chordless circuit C of length at least 4. We will apply the following. Claim 3.3. Let C be a circuit of length at least 4 in a series-parallel graph G. Then, there exists a pair of vertices nonadjacent in G[V (C)] whose removal disconnects G.

Proof. We can assume that there are two nonadjacent vertices u and v of G[V (C)] such that there exists a path P between u and v that has no internal vertex in C. Indeed, otherwise, removing any two nonadjacent vertices of G[V (C)] would disconnect G.

Let us show that removing u and v disconnects G. Denote by Q and R the two paths of C between u and v. By contradiction, suppose that G \ {u, v} is connected. Then, there exists a path containing neither u nor v between an internal vertex of R and an internal vertex of either P or Q. Let S be a minimal path of this kind. Then, no internal vertex of S belongs to P , Q, or R, and the subgraph composed of P , Q, R and S is a subdivision of K 4 . This contradicts the hypothesis that G is series-parallel. [START_REF] Seymour | Sums of circuits[END_REF] By Claim 3.3 there exist two vertices u and v of C, nonadjacent in G[V (C)], whose removal disconnects G. Denote by V 1 , . . . , V k the sets of vertices of the connected components of G \ {u, v}, and let G i = G[V i ∪ {u, v}], for i = 1, . . . , k. Note that, since C is chordless, E(G i )∩E(G j ) = ∅ for all distinct i and j. Thus, M is the disjoint union of E(G 1 ), . . . , E(G k )

Let us prove that E(G i ) is a multicut of G M , for i = 1, . . . , k. Consider a circuit D of G M . If D is contained in one of the G i 's, then |D ∩ G j | = 1 for j = 1, . . . , k. Otherwise, D is the union of two paths from u to v, these paths being contained in two dierent G i 's. Without loss of generality, let these paths be P 1 ∈ G 1 and P 2 ∈ G 2 . Then, we have D ∩ G i = P i if i = 1, 2, and ∅ otherwise. Since C has no chord, the shortest path from u to v in each G i is of length at least two, hence

Therefore, E(G i ) is a multicut of G M , and hence of G, for i = 1, . . . , k. Hence, M is the disjoint union of multicuts of G.

The Schrijver System of the Flow Cone in Series-Parallel Graphs

Theorem 2.2 provides an integer box-TDI description of the ow cone in series-parallel graphs. However, this box-TDI description is not minimal: there are redundant inequalities whose removal preserves box-TDIness. Here, we provide the minimal integer box-TDI system for this cone. This completely answers the question of [3, end of Section 6.4].

Corollary 3.4. The Schrijver system for the ow cone of a series-parallel graph G is the following:

x(M ) ≥ 0 for all chordal multicuts M of G.

(3) Moreover, this system is box-TDI.

Proof. By Theorems 1.1 and 3.1, System (3) is TDI. Since every bond is a chordal multicut, this system describes the ow cone for series parallel graphs. Therefore, by [START_REF] Cook | On box totally dual integral polyhedra[END_REF]Corollary 2.5] and by the ow cone being box-TDI for series-parallel graphs, System (3) is box-TDI.

We mention that, by planar duality, Corollary 3.4 provides the Schrijver system for the cone of conservative functions [START_REF] Schrijver | Combinatorial optimization: polyhedra and eciency[END_REF]Corollary 29.2h] in series-parallel graphs.


Adèle Brotons
email: adelebrotons5@gmail.com

Floriane Remy
email: floriane.remy.ymd@gmail.com

Bruno Foti
email: bruno.foti@univ-amu.fr

Camille Philip-Alliez
email: camille.philip@ap-hm.fr

Concordances and correlations between chronological, dental and bone ages: A retrospective study in French individuals

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Introduction

Estimating the age of a child is crucial to understanding his or her overall development. Chronological age is the most used age because the subjects' dates of birth are known in most cases. However, it may not be a reliable factor to reflect the maturation of an individual [1]. Moreover, this information is not present in the case of children found without valid identity papers or found deceased without identity [2]. Therefore, the growth of a child can be estimated by evaluating the biological age. Estimation of this age is based on several parameters as height, weight or sexual, skeletal and dental maturations [START_REF] Demirjian | Interrelationships among measures of somatic, skeletal, dental, and sexual maturity[END_REF]. Thus, evaluating skeletal maturation is useful in several fields treating the living subject, as in orthodontics or in (dentofacial) orthopedics: to take advantage of the patient's growth changes for extra oral tractions, orthodontic retentions and orthognathic surgeries or to treat different types of malocclusions related to maxillofacial growth [START_REF] Satoh | Bone age: assessment methods and clinical applications[END_REF][START_REF] Flores-Mir | Correlation of skeletal maturation stages determined by cervical vertebrae and hand-wrist evaluations[END_REF][START_REF] Koshy | Dental age assessment: The applicability of Demirjian's method in South Indian children[END_REF]. This assessment is also of interest in endocrinology and pediatric medicine [START_REF] Martin | The Use of Bone Age in Clinical Practice -Part 1[END_REF][START_REF] Chaudhry | Interrelationship among Dental, Skeletal and Chronological Ages in Urban and Rural Female Children[END_REF]: it allows health professionals to anticipate treatment plans and types that are more appropriate to the child's overall development [START_REF] Durka-Zając | Radiological Indicators of Bone Age Assessment in Cephalometric Images[END_REF].

These assessments are also necessary in the forensic field, where victims are in certain situations found without known identity: [START_REF] Kieser | Lessons Learned from Large-scale Comparative Dental Analysis Following the South Asian Tsunami of 2004[END_REF][START_REF] Kieser | Automated dental aging for child victims of disasters[END_REF][12][START_REF] Schuller-Götzburg | Forensic odontologists successfully identify tsunami victims in Phuket, Thailand[END_REF][START_REF] Tsokos | Experiences in tsunami victim identification[END_REF][START_REF] Valenzuela | The application of dental methods of identification to human burn victims in a mass disaster[END_REF]. Estimations of bone and dental ages are also made in context of criminal proceedings, leading to reductions in sentence and criminal responsibility when the offender is a minor. These biological ages are also operated in unaccompanied minor migrant's problematic [START_REF] Mishori | The Use of Age Assessment in the Context of Child Migration: Imprecise, Inaccurate, Inconclusive and Endangers Children's Rights[END_REF]. In addition, new methods of estimation are emerging each year because of the rising level of infanticide crimes [START_REF] Cericato | Correlating skeletal and dental developmental stages using radiographic parameters[END_REF][START_REF] Moirangthem | Child sexual abuse: Issues & concerns[END_REF].

In 2012 and 2018, the SCEP and EASO reports recommended a multidisciplinary holistic approach for age estimations with a global harmonization through a gradual implementation of non-medical and medical methods [START_REF]SCEP Position Paper on Age Assessment in Context of Separated Child in Europe[END_REF][START_REF][END_REF]. In a context of separated child, non-medical methods (albeit subjective) therefore became a priority in age estimation because they should provide the same results as those obtained with bone and dental maturation assessments by medical procedures.

However, while there is some controversy about the accuracy of dental x-ray examinations [START_REF] Aynsley-Green | Medical, statistical, ethical and human rights considerations in the assessment of age in children and young people subject to immigration control[END_REF], it is reported that medical methods, although intrusive, result in more reliable and objective findings [START_REF] De Micco | Ethical issues in age assessment by the third molar development[END_REF][START_REF] Roberts | Dental age assessment (DAA): a simple method for children and emerging adults[END_REF]. Psychological tests undertaken as a priority in child assessment have margins of error that are not validated and are subject to environmental or social pressures that influence them. As Cummando et al. said, "medical radiological methods […] have fairly large error ranges but these are better known and quantifiable than those of psychosocial assessments" [START_REF] Cummaudo | The "forensic paradox" of aging unaccompanied minors in the migration crisis: Why medicine and forensics are a must[END_REF].

Morphological methods based on radiological examinations of skeletal and dental development are then recommended for biological age estimations in immatures [START_REF] Ritz-Timme | Age estimation: The state of the art in relation to the specific demands of forensic practise[END_REF]. The scientific literature shows anyway that analysis of dental calcification allows to estimate an age more reliably than with dental eruption or with skeleton, because it is less influenced by intrinsic and extrinsic variations (endocrinological disorders or environmental alterations) [START_REF] Mollabashi | The relation between dental age and cervical vertebral maturation in orthodontic patients aged 8 to 16 years: A cross-sectional study[END_REF].

Demirjian's is the one most widely used method for dental age estimation [START_REF] Demirjian | A new system of dental age assessment[END_REF]. He proposed this method in 1973 based on dental calcification of the seven permanent left mandibular teeth excluding third molars. Each tooth is assigned a stage of bone development, ranging from A (early calcification) to H (complete tooth) with an additional stage of 0 indicating no evidence of mineralization. A score is then assigned to each stage from numerical tables distinguishing girls and boys. The sum of all the scores obtained results in an estimation of dental age.

During growth, bones undergo changes that can be detected by X-rays. Bone age estimation is then used to assess the growth and maturation of young individuals, and to diagnose and treat pediatric disorders [START_REF] Satoh | Bone age: assessment methods and clinical applications[END_REF]. Previously, there was a preponderance of estimations of skeletal maturation by observation of wrist and hand radiographs, particularly based on the Greulich and Pyle atlas [START_REF] Greulich | Radiographic atlas of skeletal development of the hand and wrist[END_REF]. Only very recently, the assessment of maturation stages of the cervical vertebrae on orthodontic cephalometries has been integrated to estimate bone age without the need for additional radiation of the hand and wrist [START_REF] Flores-Mir | Correlation of skeletal maturation stages determined by cervical vertebrae and hand-wrist evaluations[END_REF][START_REF] Mollabashi | The relation between dental age and cervical vertebral maturation in orthodontic patients aged 8 to 16 years: A cross-sectional study[END_REF][START_REF] Hassel | Skeletal maturation evaluation using cervical vertebrae[END_REF][START_REF] Heil | Lateral cephalometric analysis for treatment planning in orthodontics based on MRI compared with radiographs: A feasibility study in children and adolescents[END_REF][START_REF] Gandini | A comparison of hand-wrist bone and cervical vertebral analyses in measuring skeletal maturation[END_REF]. Since the publication of the Lamparski's atlas, orthodontists have been using changes in vertebral morphology to estimate a child's skeletal age and to predict the most optimal treatment plan [START_REF] Lamparski | Skeletal age assessment ustilizing cervical vertebrae[END_REF]. Baccetti et al. proposed an improved version of the Cervical Vertebral Maturation (CVM) method to assess skeletal development from the morphology of the C2, C3 and C4 vertebrae. [START_REF] Baccetti | An improved version of the cervical vertebral maturation (CVM) method for the assessment of mandibular growth[END_REF]. This method associates a stage of maturation based on the observation of the morphological variations established on these vertebrae over time on lateral cephalometric radiographs [START_REF] Franchi | Mandibular growth as related to cervical vertebral maturation and body height[END_REF]. Recently, Decocq et al. published a computerized version of this method to overcome the subjective interpretation of vertebral morphology [START_REF] Decocq | Méthode informatisée pour le calcul de l'âge osseux vertébral[END_REF].

The aim of the present study was to explore the relationship between cervical maturation stages and those of dental calcification to determine if biological age can be estimated from panoramic and cephalometric radiographs on a French population.

Materials and methods

Materials

This retrospective study was based on a sample of 192 randomly selected individuals (101 females and 91 males) aged between 9 to 19 years and treated for dentofacial anomalies (Table 1). For each child, panoramic and lateral cephalometric radiographs taken the same day were digitally retrieved from the Dentofacial Orthodontic Unit of the Timone Hospital (Marseille). Both the radiographs were taken in the Radiology Department between 2014 and 2021 for treatment forecasting and diagnosis. They were selected according to the following exclusion criteria: patients with dental agenesis, pathologies (genetic or systemic) altering the maturation of the cervical vertebrae or the eruption and calcification of teeth, with previous surgeries and orthodontic treatments, patients undergoing treatment and wearing orthodontic appliances and poor-quality radiographs that do not show the teeth and C2, C3 and C4 vertebrae in their entirety. Patients were anonymized: only the sex and the age of each patient at the time of radiographic examinations were recorded (computed from the date of birth and the exam date). The entire sample was analyzed by a forensic science student majoring in odontology and the ethical framework was thus respected and followed the PADS20-342 GDPR's protocol (General Data Protection Regulation).

Dental age estimation

Dental calcification was evaluated using the Demirjian's method [START_REF] Demirjian | A new system of dental age assessment[END_REF] where a score ranging from A to H is assigned to the first seven left mandibular teeth. All panoramics were evaluated and scored in the same order: central and lateral incisors, canine, first and second premolars, first and second molars.

Skeletal age estimation

Bone development was evaluated on three cervical vertebrae following the Cervical Vertebral Maturation (CVM) method described by Baccetti et al. [START_REF] Baccetti | An improved version of the cervical vertebral maturation (CVM) method for the assessment of mandibular growth[END_REF] and computerized by Decocq et al. [START_REF] Decocq | Méthode informatisée pour le calcul de l'âge osseux vertébral[END_REF]. The inferior borders of the C2, C3, and C4 vertebrae were analyzed on lateral cephalometric radiographs in addition to their general shape. The ODRADE® cephalometric analysis extension of the ORTHOKIS® software was used to place the eight points mentioned by Bacetti et al. to obtain precise measurements of the three vertebrae.

Statistical analysis

Statistical analyses were performed with R Studio® software (1.4.1106 version). P-values less than 0.05 were considered as statistically significant. Quantitative variables were tested for a normal distribution using the Shapiro-Wilk test and were expressed as means associated with standard deviations. Qualitative variables were expressed as percentages.

Spearman's correlation coefficient was calculated to statistically assess the relation between chronological age, dental calcification stages and cervical vertebrae's maturation stages. Lin's concordance correlation coefficient and the Bland & Altman' methods were applied to investigate the concordance between estimated and chronological ages. The accuracy of the methods used was assessed by calculating the residuals between dental and bone age estimations and chronological ages.

Reproducibility and repeatability were assessed by forensic sciences students trained by radiologists and orthodontists who analyzed a second time 20% of the total sample within 30 days of the initial analysis. Lin's concordance correlation coefficient (CCC) method was performed to assess the reliability of the methods used and the results were interpreted using the classifications proposed by Partik et al. [START_REF] Partik | 3D versus 2D ultrasound: accuracy of volume measurement in human cadaver kidneys[END_REF].

Results

Table 2 shows the repartition of chronological, dental and bone ages in the CVM stages for females and males. It confirms that cervical vertebral maturation stages (CS) and thus bone age were increasing with chronological and dental ages. A higher proportion of females was found in late maturation stages (21.8% for girls in CS5 versus 12.1% for males); girls reached maturation stages earlier than boys (13.18 ± 2.24 years in CS4 for girls versus 14.37 ± 1.50 for boys, 14.85 ± 1.70 in CS5 for girls versus 16.58 ± 2.05 for boys). Overestimations of dental age can be seen in all CS in both girls and boys (mean of differences between CA and DA in girls: 0,72 ± 0.41 years, mean of differences between CA and DA in boys: 0.78 ± 0.84 in boys). On the contrary, bone age was underestimated for both girls and boys for all CS (mean of differences between CA and BA in girls: 1.69 ± 1.2 years, mean of differences between CA and BA in boys: 2.45 ± 1.15 years). The relative distributions of each dental development stages in each CS for females and males are presented in Table 3 andTable 4. It indicates that the relative frequencies of teeth belonging to high dental maturation stages increased when the maturation stages of the cervical vertebrae increased.

The correlation coefficients between dental development and cervical vertebral maturation stages are also presented in the tables 3 and 4. Coefficients vary from 0.383 (first premolar in females) to 0.618 (second molars in males and females). Unless for the canine and first premolar in boys, all correlations were statistically significant (p<0.001). Tables 5,6 and 7 present the concordance coefficients between bone, dental and chronological ages for both sexes. Differences between estimated and chronological ages are presented in Fig. 1 Fig. 2. represents the reliability of the methods used to estimate dental and bone ages. Lin's concordance correlation coefficients varied from fairly good to excellent according to Partik's classification [START_REF] Partik | 3D versus 2D ultrasound: accuracy of volume measurement in human cadaver kidneys[END_REF]. For the Demirjian's method, CCCs were 0.92 for reproducibility and 0.97 for repeatability. For the Decocq's method, CCC's were 0.89 for reproducibility and 0.88 for repeatability. Cb coefficients were also calculated. They quantify the difference between the regression line estimated from the two series of measurements and the line at 45°. For both methods, the values varied between 0.96 and 0.90. It indicates excellent alignments of the two lines and thus excellent concordances.

Discussion

The precision and reliability of age estimations are determinant in clinical and forensic sciences. An erroneous assessment of biological age can lead to serious consequences when it is used to plan treatments or to find an identity. Moreover, contemporary migratory movements linked to globalization and ongoing belligerent conflicts have led to new issues where authorities are forced to make decisions about the fate of thousands of people based on morphological criteria to arrive at civil ages and partially reconstructed identities [START_REF] Moirangthem | Child sexual abuse: Issues & concerns[END_REF][START_REF] Franco | Applicability of Willems model for dental age estimations in Brazilian children[END_REF]. To respond to these specific new demands, more and more estimation methods are published every day. Most of these methods are based on the morphologies and anatomical structures that can be observed in radiography, such as carpals [START_REF] Greulich | Radiographic atlas of skeletal development of the hand and wrist[END_REF], cervical vertebrae [START_REF] Lamparski | Skeletal age assessment ustilizing cervical vertebrae[END_REF][START_REF] Baccetti | An improved version of the cervical vertebral maturation (CVM) method for the assessment of mandibular growth[END_REF] or teeth [START_REF] Demirjian | A new system of dental age assessment[END_REF][START_REF] Nolla | The development of permanent teeth[END_REF]. Since more available anatomical structures provide more reliable results [START_REF] Cericato | Correlating skeletal and dental developmental stages using radiographic parameters[END_REF], the combination of different methods leads to better age estimations.

The aim of the present study was to observe the relationships between dental and bone maturation stages in order to evaluate the possibility of combining the Demirjian' and Decocq' methods to estimate more reliable and accurate biological ages. Previous studies have already investigated the relationships between these two maturity stages [2, [START_REF] Demirjian | Interrelationships among measures of somatic, skeletal, dental, and sexual maturity[END_REF][START_REF] Chaudhry | Interrelationship among Dental, Skeletal and Chronological Ages in Urban and Rural Female Children[END_REF][START_REF] Cericato | Correlating skeletal and dental developmental stages using radiographic parameters[END_REF][START_REF] Mollabashi | The relation between dental age and cervical vertebral maturation in orthodontic patients aged 8 to 16 years: A cross-sectional study[END_REF][START_REF] Rozylo-Kalinowska | Relationship between dental age according to Demirjian and cervical vertebrae maturity in Polish children[END_REF][START_REF] Vaida | Correlations between morphology of cervical vertebrae and dental eruption[END_REF][START_REF] Perinetti | Diagnostic performance of dental maturity for identification of skeletal maturation phase[END_REF] and prove that the combination of dental and skeletal maturity stages represents a useful tool for age estimation of children.

According to our study results, the correlations between dental calcification (Demirjian's method variables) and cervical vertebral maturation stages (Decocq's method variables) were significant in both males and females, except for the canine and first premolar in boys. The mean correlation coefficients for all teeth are 0.523 for females and 0.333 in males but higher correlation was found in the second molar for both sexes (R=0.618). Similar to our study, Chen et al. [START_REF] Chen | Correlation between dental maturity and cervical vertebral maturity, Oral Surg Oral Med Oral[END_REF], Başaran et al. [START_REF] Başaran | Cervical vertebral and dental maturity in Turkish subjects[END_REF], Cericato et al. [START_REF] Cericato | Correlating skeletal and dental developmental stages using radiographic parameters[END_REF], Chaudhry et al. [START_REF] Chaudhry | Interrelationship among Dental, Skeletal and Chronological Ages in Urban and Rural Female Children[END_REF], Gulati et al. [START_REF] Gulati | Inter-relationship between dental, skeletal and chronological ages in well-nourished and malnourished children[END_REF], Mollabashi et al. [START_REF] Mollabashi | The relation between dental age and cervical vertebral maturation in orthodontic patients aged 8 to 16 years: A cross-sectional study[END_REF], Rai et al. [START_REF] Rai | Dental and skeletal maturity-a biological indicator of chronologic age[END_REF] and Rozylo-Kalinowska et al. [START_REF] Rozylo-Kalinowska | Relationship between dental age according to Demirjian and cervical vertebrae maturity in Polish children[END_REF] obtained significantly positive correlations between dental and bone ages. Mollabashi et al. (0.724 and 0.641 in boys and girls respectively) and Chen et al. (0.496 and 0.582 in boys and girls respectively) also reported stronger correlations for the mandibular second molar between dental and bone ages [START_REF] Mollabashi | The relation between dental age and cervical vertebral maturation in orthodontic patients aged 8 to 16 years: A cross-sectional study[END_REF][START_REF] Chen | Correlation between dental maturity and cervical vertebral maturity, Oral Surg Oral Med Oral[END_REF]. In our case, the weakest correlations between dental and cervical stages were observed in the first premolars for both sexes (0.383 for girls, 0.090 and non-significant for boys). It can be explained because the later the eruption and calcification of the tooth occur, the better predictor of maturation it will be [START_REF] Mollabashi | The relation between dental age and cervical vertebral maturation in orthodontic patients aged 8 to 16 years: A cross-sectional study[END_REF]. In the studies of Rozylo-Kalinowska et al. [START_REF] Rozylo-Kalinowska | Relationship between dental age according to Demirjian and cervical vertebrae maturity in Polish children[END_REF], Başaran et al. [START_REF] Başaran | Cervical vertebral and dental maturity in Turkish subjects[END_REF], and Cericato et al. [START_REF] Cericato | Correlating skeletal and dental developmental stages using radiographic parameters[END_REF], stronger correlations were present for the second premolars while the strongest correlation was noted between skeletal age and the calcification stage of first premolars in the study of Valizadeh et al. [START_REF] Valizadeh | Correlation Between Dental and Cervical Vertebral Maturation in Iranian Females[END_REF].

The present study applied a previously validated method for the dental ages' estimations [START_REF] Gungor | Validity of the Demirjian method for dental age estimation for Southern Turkish children[END_REF][START_REF] Nykänen | Validity of the Demirjian method for dental age estimation when applied to Norwegian children[END_REF]. Reproducibility and repeatability studies of the measurements made to estimate bone ages from Decocq's computerized method led to the hypothesis that this method is also reliable and precise. However, some limitations can be noted.

Evaluation of the methods concordance by Lin's and Bland and Altman's methods indicate overestimations of dental ages by Demirjian's method and underestimations of bone ages by Decocq's method (Fig. 1). The overestimations obtained with Demirjian were also seen in other studies [START_REF] Da Silva | Estimating age of Brazilians using the methods of Demirjian and Nicodemo, Moraes and Médici: A comparative analysis[END_REF][START_REF] Staaf | Age determination with the aid of tooth development: a test of the reliability and validity[END_REF]. Indeed, da Silva et al. [START_REF] Da Silva | Estimating age of Brazilians using the methods of Demirjian and Nicodemo, Moraes and Médici: A comparative analysis[END_REF] highlighted an average overestimate of 7 months. Moreover, Decocq's method is based on that of Baccetti et al. which was itself established from that proposed by Hassel and Farman. These are not commonly validated throughout the literature [START_REF] Cericato | Correlating skeletal and dental developmental stages using radiographic parameters[END_REF]. Therefore, even though the correlations were positive and statistically significant, the results should be interpreted with caution as they were moderate and over/under-estimations are present. It is also important to note that the method for estimating dental age was based on a Canadian population and that the method for estimating bone age was based on an American population. As Cunha et al. said, "although Demirjan's method is one of the most suitable for forensic purposes, the high ethnic variability requires particular caution in discussion of the final result." [START_REF] Cunha | The problem of aging human remains and living individuals: A review[END_REF] Regarding concordance coefficients, they ranged from 0.30 to 0.80. According to the Partik et al. classification [START_REF] Partik | 3D versus 2D ultrasound: accuracy of volume measurement in human cadaver kidneys[END_REF], they vary from fairly good to excellent. The strongest concordances were observed between chronological and dental age in both sexes (C=0.79-0.80). Regarding the relationships between dental and bone age estimations, they were fairly good (C=0.40 in females, C=0.30 in males).

Despite these moderate values, the study has the advantage of making its observations on a larger population compared to most of the previous studies. These results also validated Decocq's computerized CVM method, which was based on a population of 30 subjects.

Although ADFAD (International Study Group on Forensic Age Diagnostics) recommends estimating bone age from hand and wrist radiographs [START_REF] Schmeling | Age estimation[END_REF], the simultaneous use of Demirjian' and Decocq' methods from panoramic and cephalometric radiographs would reduce ionizing radiation exposure for children. Indeed, aging the living requires the use of non-invasive methods [START_REF] Cunha | The problem of aging human remains and living individuals: A review[END_REF]. However, the exposure level related to panoramic or cephalometric X-ray can be classified as negligible risk to a person's health. As DeMicco et al. said, "dental imaging can bring significant benefits with a risk of limited damage" [START_REF] De Micco | Ethical issues in age assessment by the third molar development[END_REF].

Conclusion

The correlation between dental calcification and cervical vertebral maturation stages (especially in the calcification stages of the second molar) and the concordance between chronological, dental and skeletal ages indicated moderate and statistically significant results. Girls reached late stages of maturation earlier than boys.

From panoramic and lateral cephalometric radiographs, it is possible to combine the methods of Demirjian and Decocq to obtain precise biological ages for orthodontic treatments or forensic identifications.

Further investigation of this study could be done by involving more operators to obtain more reliable results.

Fig. 1 .

 1 Fig. 1. Bland and Altman plots showing differences between a) CA and DA in females b) CA and DA in males c) CA and BA in females d) CA and BA in males e) DA and BA in females f) DA and BA in males. The middle bold dashed line represents the mean difference, and the two extreme bold dashed lines define the limits of agreement.

Fig. 2 .

 2 Fig. 2. Reliability of Demirjian's and Decocq's methods. a) c) e) and g) show reproducibilities for the methods of Demirjian and Decocq respectively. b) d) f) and h) show repeatabilities for the methods of Demirjian and Decocq respectively. a) b) e) and f) plots the correlations between variables. c) d) g) and h) are the Bland and Altman plots and show the differences between variables. Blue lines represent the regression lines and red lines represent the reference lines where x=y.

Table 1

 1 Number of subjects, mean and median for each sex.

	Sex	Number	Mean	Median	p*
		of	age	age
		subjects	±SD	(years)
			(years)
	Females	101	12.57	12.16	0.471
	(p=0.002)**		± 2.44
	Males	91	12.80	12.40
	(p=0.002)**		± 2.57

SD, standard deviation; IQR, interquartile range; *χ2 test; **Shapiro-Wilk test.

Table 2

 2 Mean chronological, dental and bone ages in each sex and Cervical Vertebral Maturation (CVM) stages.

	CVM		Females		Males
			Number	Mean	Number	Mean
	CS0	Chronological age		10.94 ± 1.35		10.95 ± 1.23
		Dental age	6 (5.9%)	11.57 ± 1.13	12 (13.2%)	12.15 ± 2.22
		Bone age		7.71 ± 0.22		7.58 ± 0.77
	CS1	Chronological age		10.44 ± 1.47		11.07 ± 1.28
		Dental age	9 (8.9%)	11.22 ± 1.81	11 (12.1%)	11.97 ± 1.96
		Bone age		8.94 ± 0.18		8.90 ± 0.23
	CS2	Chronological age		11 ± 1.58		11.07 ±1.35
		Dental age	15 (14.9%)	11.93 ± 1.82	19 (20.9%)	12.24 ± 2.28
		Bone age		9.89 ± 0.30		9.60 ± 0.25
	CS3	Chronological age		11.21 ± 1.32		12.20 ± 1.70
		Dental age	23 (22.8%)	12.33 ± 1.64	14 (15.4%)	11.88 ± 1.74
		Bone age		10.84 ± 0.33		10.80 ±0.36
	CS4	Chronological age		13.18 ± 2.24		14.37 ± 1.50
		Dental age	20 (19.8%)	13.47 ± 1.51	20 (22%)	15.09 ± 1.19
		Bone age		11.81 ± 0.31		11.76 ± 0.33
	CS5	Chronological age		14.85 ± 1.70		16.58 ± 2.05
		Dental age	22 (21.8%)	14.50 ± 0.88	11 (12.1%)	15.95 ± 0.18
		Bone age		12.79 ± 0.31		12.83 ± 0.30
	CS6	Chronological age		16.20 ± 0.66		16.25 ± 1.70
		Dental age	6 (5.9%)	15.22 ± 0.88	4 (4.4%)	15.68 ± 0.65
		Bone age		13.98 ± 0.27		13.85 ± 0.46

CVM: Cervical Vertebral Maturation; CS: Cervical Stage.

Table 5 .

 5 Concordances between chronological (CA), dental (DA) and bone (BA) ages in girls and boys.

		CA	DA	BA
	CA	1	0.79	0.47
	DA	0.79	1	0.32
	BA	0.47	0.32	1

Table 6 .

 6 Concordances between chronological (CA), dental (DA) and bone (BA) ages in girls.

		CA	DA	BA
	CA	1	0.78	0.50
	DA	0.78	1	0.40
	BA	0.50	0.40	1

Table 7 .

 7 Concordances

			between chronological (CA),
	dental (DA) and bone (BA) ages in boys.
		CA	DA	BA
	CA	1	0.80	0.44
	DA	0.80	1	0.30
	BA	0.44	0.30	1

Table 3 .

 3 Relative distribution (%) of Cervical Vertebral Maturation stages (CS) in face of dental development stages (DS) and Spearman's correlation coefficient (r) for females.

	Tooth	DS	CS

Table 4 .

 4 Relative distribution (%) of Cervical Vertebral Maturation stages (CS) in face of dental development stages (DS) and Spearman's correlation coefficient (r) for males.

	Tooth	DS	CS

Acknowledgments