Isabelle Mortier

Julie Blanc

Barthelemy Tosello

Catherine Gire

Florence Bretelle

Xavier Carcopino

Is gestational diabetes an independent risk factor of neonatal severe respiratory distress syndrome after 34 weeks of gestation? A prospective study

Keywords: Gestational diabetes, Neonatal severe respiratory distress syndrome, Late preterm and term pregnancy, Maternal obesity

respectively (p < 0.001). Women whose neonates presented neonatal SRDS were significantly more likely to be obese (p = 0.002), to have undergone a caesarean section (p < 0.001) and to have received corticosteroids therapy before 34 0/7 weeks (p = 0.013). In multivariate analysis, GD was identified as an independent risk factor of neonatal SRDS (aOR 3.6; 95% CI 1.5-8.6; p = 0.005). Other risk factors were maternal obesity (aOR 2.8; 95% CI 1.1-7.1; p = 0.029) and assisted vaginal delivery (aOR 5.5; 95% CI 1.9-15.9; p = 0.002). Conclusions GD is an independent risk factor of neonatal SRDS after 34 0/7 weeks.

Introduction

Since its original description by O'Sullivan in 1961, the incidence of gestational diabetes (GD) has increased dramatically worldwide [START_REF] Jb | Gestational diabetes. Unsuspected, asymptomatic diabetes in pregnancy[END_REF]. Evolution of alimentary habits, increase of obesity prevalence, and recent diminution of diagnostic threshold values contributed to this increase [START_REF] Leary | Gestational diabetes guidelines in a HAPO world[END_REF][START_REF] Panel | International association of diabetes and pregnancy study groups recommendations on the diagnosis and classification of hyperglycemia in pregnancy[END_REF]. The incidence of GD nowadays reaches 17.8% worldwide, becoming a major public health concern [START_REF] Legardeur | Screening of gestational diabetes mellitus: a new consensus?[END_REF]. Some materno-foetal complications of GD are now well-identified [START_REF] Hurwitz | Diabetes and pregnancy[END_REF][START_REF] Robert | Association between maternal diabetes and the respiratory-distress syndrome in the newborn[END_REF][START_REF] Schwartz | Effects of diabetic pregnancy on the fetus and newborn[END_REF][START_REF] Svare | Perinatal complications in women with gestational diabetes mellitus[END_REF][START_REF] Gilmartin | Gestational diabetes mellitus[END_REF][START_REF] Weindling | Offspring of diabetic pregnancy: short-term outcomes[END_REF][START_REF] Mitanchez | Fetal and neonatal complications of gestational diabetes: perinatal mortality, congenital malformations, macrosomia, shoulder dystocia, birth injuries, neonatal outcomes[END_REF]. Thus, GD is known to expose to type 2 diabetes in post-partum [START_REF] Bellamy | Type 2 diabetes mellitus after gestational diabetes: a systematic review and meta-analysis[END_REF] and to foetal consequences such as macrosomia, prematurity, and neonatal metabolic disorders such as severe neonatal hypoglycemia, hyperbilirubinemia, and hypocalcemia. Potential long-term and transgenerational effects have also been described [START_REF] Mitanchez | Infants born to mothers with gestational diabetes mellitus: mild neonatal effects, a long-term threat to global health[END_REF]. Although an optimal balance of GD has been shown to significantly reduce neonatal morbidity Abstract Purpose To evaluate if neonates delivered after 34 0/7 weeks from mothers diagnosed with gestational diabetes (GD) are exposed to an increased risk of neonatal severe respiratory distress syndrome (SRDS). Methods Women with singleton pregnancy in labour after 34 0/7 weeks of gestation or admitted for planned caesarean section and who had been systematically screened for GD were eligible to participate to this prospective cohort study. Diagnosis of SRDS was defined by the association of clinical signs of early neonatal respiratory distress, with consistent radiologic features and requiring mechanical ventilation with a fraction of inspired oxygen (FiO 2 ) >0. [START_REF]Hyperglycemia and adverse pregnancy outcome (HAPO) study: associations with neonatal anthropometrics[END_REF] for a minimum of 24 h and admission to neonatal intensive care unit. Results A total of 444 women were included. GD was diagnosed in 60 patients (13.5%). A neonatal SRDS was diagnosed in 32 cases (7.2%). Compared to others, neonatal SRDS was significantly more often observed in neonates from women diagnosed with GD: 12 (20%) vs. 20 (5.2%), and mortality [START_REF] Crowther | Effect of treatment of gestational diabetes mellitus on pregnancy outcomes[END_REF], perinatal complications remain more frequent than in general population [START_REF] Weindling | Offspring of diabetic pregnancy: short-term outcomes[END_REF].

Respiratory distress syndrome occurs in a significant number of late preterm and term neonates and represents a major cause of admission in neonatal units with associated morbidity and mortality [START_REF] Edwards | Respiratory distress of the term newborn infant[END_REF]. Aetiologies are multiple [START_REF] Kjos | Prevalence and etiology of respiratory distress in infants of diabetic mothers: predictive value of fetal lung maturation tests[END_REF], with the two main causes being transient tachypnoea of the newborn related to abnormal lung fluid absorption and deficiency of surfactant. The risk of respiratory distress syndrome in relation with hyaline membrane disease in neonates from patients with GD has been previously reported [START_REF] Robert | Association between maternal diabetes and the respiratory-distress syndrome in the newborn[END_REF]. However, the definition of respiratory distress syndrome throughout literature is not a homogeneous and the associated risk remains unclear after 34 0/7 weeks of gestation (weeks). Although we previously identified GD as an independent risk factor for severe neonatal respiratory distress syndrome (SRDS) after 34 0/7 weeks in a retrospective study with an aOR 11.55 (95% CI 3.9-33.9, p < 0.001) [START_REF] Vignoles | Gestational diabetes: a strong independent risk factor for severe neonatal respiratory failure after 34 weeks[END_REF] and two others retrospective studies gave contradictory conclusions in late-preterm infants from 34 0/7 to 36 6/7 weeks [START_REF] Fung | Does gestational diabetes mellitus affect respiratory outcome in late-preterm infants?[END_REF][START_REF] Bricelj | Neonatal respiratory morbidity in late-preterm births in pregnancies with and without gestational diabetes mellitus[END_REF], to our knowledge, no prospective study has yet demonstrated the association of SRDS and GD in late preterm and term neonates. The aim of this study is to prospectively assess if neonates delivered after 34 0/7 weeks from mothers diagnosed with GD are exposed to an increased risk of neonatal SDRS.

Methods

Patients

We conducted a prospective cohort study among women who delivered in our tertiary care-level maternity unit in Marseille between January 1st 2011 and July 31st 2012. Women were recruited at the time of their admission in labour ward. All women with a monofoetal pregnancy in beginning of labour after 34 0/7 weeks of gestation or planned for a caesarean section and who had been systematically screened for GD, according to our institution guidelines, were eligible to participate to the current study. Pregnant women with multiple pregnancy, existence of a pre-gestational diabetes, absence or poor pregnancy follow-up, uncertain dating of pregnancy, absence of systematic GD screening, and in utero foetal death were not included. The study protocol received approval from the Ethics Committee on Research in Obstetrics and Gynecology (CEROG no. 2010-OBS11-02). Written informed consent was obtained from all participants.

Collected data

Age, previous history, weight gain during pregnancy (kg), body mass index (BMI) at the moment of delivery (kg/m 2 ), and complications of pregnancy and time of occurrence were recorded for all participants. Maternal obesity was defined as BMI ≥ 30 kg/m 2 . We also recorded the exact term and mode of delivery (spontaneous vaginal delivery, assisted vaginal delivery, and caesarean section), the foetal gender, the neonatal weight, the Apgar score at 5 min, and any diagnosed foetal malformation. Gestational age calculation was based on foetal biometry of cranio-caudal length at first-trimester systematic ultrasound scan performed between 11 0/7 and 13 6/7 weeks. Foetal growth restriction was defined as a neonatal weight lower than the 10th percentile and macrosomia as a neonatal weight over the 90th percentile according to the reference curves from the AUDIPOG study [START_REF] Mamelle | Fetal growth from the AUDIPOG study. I. Establishment of reference curves[END_REF][START_REF] Mamelle | Fetal growth from the AUDIPOG study. II. Application for the diagnosis of intrauterine growth retardation[END_REF]. The prenatal administration of corticosteroid therapy before 34 0/7 weeks consisting in two intramuscular injections of 12 mg of Betamethasone repeated over 24 h was also systematically recorded.

Systematic screening of gestational diabetes

According to our institution guidelines, screening of diabetes was systematically performed in all patients. Screening was performed using glucose tolerance test between 24 0/7 and 28 0/7 weeks independently of the existence of any identified maternal risk factors for GD. Overnight fasting plasma glucose was determined just before the administration of 75 g of glucose. Blood for glucose values was drawn at 1 and 2 h. Diagnose of GD was made if at least one abnormal glucose value(s) was identified: glucose value after an overnight fast ≥0.92 g/L; 1-h post-prandial level ≥1.80 g/L; and 2-h post-prandial level ≥1.53 g/L. In case of GD, the type of therapy (dietetic alone or associated with insulin therapy) and glucose level (6 capillary glycaemia per day) were monitored and documented. After 10 days of monitoring, controlled GD was defined by fast glucose values <0.95 g/L and post-prandial glucose values <1.20 g/L.

Neonatal outcomes and diagnosis of SRDS

The primary outcome was the diagnostic of neonatal SRDS. As the definition of respiratory distress syndrome is quite heterogenic throughout literature, we chose to precisely define the SRDS using both clinical and radiological criteria. Thus, SRDS was defined by the association of clinical signs of early neonatal respiratory distress occurring within the first 2 h following birth, with consistent radiologic features and oxygen dependence requiring invasive and/or noninvasive mechanical ventilation with a fraction of inspired oxygen (FiO2) >0.25 for a minimum of 24 h and admission 1 3 to neonatal intensive care unit (NICU). Duration of admission in NICU and of ventilation was systematically recorded.

Number of subjects required and power of the study

On a preliminary review of 300 deliveries from our institution, GD was diagnosed in 10.9% of women and was associated with 34.4% of SDRS in the GD group compared with 5.4% in control group (p < 0.001) (unpublished data). Based on these results, the number of subjects required was calculated to detect a four-fold increased risk of SRDS in women with GD to achieve a 95% powered study with a 10% incidence of GD. We thus calculated the sample size to be at least of 429 women.

Statistical analysis

Statistical analysis was performed using the SPSS 17.0 software (SPSS Inc. Chicago, IL). Data were expressed as mean ± SE or median (min-max). Groups were compared using the χ 2 or Fisher exact test for categorical characteristics and using the Student t test for continuous ones, as appropriate. A two-sided p value <0.05 was considered as statistically significant.

Results

During the study period, a total of 444 pregnant women were included in the current study. GD was diagnosed in 60 patients (13.5%).

Maternal, obstetrical, and neonatal characteristics according to the diagnosis of gestational diabetes

Patients and neonatal characteristics are summarized in Table 1. The groups were statistically different in maternal age and maternal BMI at the time of delivery (p < 0.001). Compared to others, women diagnosed with GD were statistically older (mean age 32.5 years vs. 28.7 years, respectively, p < 0.001), and significantly more likely to be obese at the moment of delivery (66% vs. 35%, respectively, p < 0.001). Although mean gestational age at delivery was no statistically different between the two groups (39.23 ± 1.61 in women with GD vs. 39.45 ± 1.44 in control group, p = 0.31), women with GD were significantly more likely to have undergone induced labour (p = 0.03). When considering the different modes of delivery (spontaneous vaginal delivery, vaginal-assisted delivery, or caesarean section), no statistical difference was observed between the two groups.

Neonates delivered from women diagnosed with GD presented higher neonatal weight (p = 0.001) and higher rate of macrosomia (p = 0.02). A neonatal SRDS was diagnosed in 32 cases (7.2%). No statistical difference was found in the median length of admission in NICU of neonates diagnosed with SRDS and in the median duration of mechanical ventilation. No case of neonatal death was recorded. Compared to others, neonatal SRDS was significantly more often observed in neonates from women diagnosed with GD 12 (20%) vs. 20 (5.2%), respectively (p < 0.001). A higher proportion of neonates with an Apgar score <7 at 5 min was observed in neonates from women diagnosed with GD (p = 0.001), but no statistical difference was found in the mean arterial umbilical pH (p = 0.06).

Maternal, obstetrical, and neonatal characteristics according to the occurrence of SRDS

Maternal, obstetrical, and neonatal characteristics according to the occurrence of SRDS are summarized in Table 2. We found no statistically difference between the two groups regarding the following maternal characteristics: age, smoking status, nulliparity, and gestational age at delivery. GD was more frequently observed in women, whose neonates were diagnosed with SRDS 12 (38%) vs. 48 (12%), respectively (p < 0.001). Our study failed to identify any statistical difference in case of controlled GD 7 (58.3%) vs. 35 (72.9%), respectively (p = 0.481). Women whose neonates presented SRDS were more often obese: 18 (56%) vs. 132 (32%), respectively (p = 0.002). Furthermore, women whose neonates were diagnosed with SRDS were significantly more

Mode of delivery and SRDS

Compared to others, neonates diagnosed with SRDS were significantly more often born by caesarean section: 92 (22%) vs. 13 (40%), respectively (p < 0.001). Caesarean sections were performed before 39 0/7 weeks in 6 cases (46%) in SRDS group versus in 19 (21%) (p = 0.07). Furthermore, neonates diagnosed with SRDS were significantly more often born after assisted vaginal delivery: 37 (9%) vs. 7 (22%), respectively (p < 0.001).

Identification of independent risk factors for SRDS

In multivariate analysis (Table 3), GD was identified as an independent risk factor of neonatal SRDS (aOR 3.6; 95% CI 1.5-8.6; p = 0.005). Other identified risk factors were obesity at time of delivery (aOR 2.8; 95% CI 1.1-7.1; p = 0.029) and assisted vaginal delivery (aOR 5.5; CI 95% 1.9-15.9; p = 0.002). Foetal macrosomia was not identified as an independent risk factor of neonatal SRDS.

Discussion

To our knowledge, this study is the first prospective study to report GD as an independent risk factor of neonatal SRDS after 34 0/7 weeks of gestation. We believe these results to be of genuine importance in daily obstetrical practice as it might help for optimal neonatal care in anticipating the need for potential urgent neonatal management.

With a global rate up to 7% of SRDS after 34 0/7 weeks of gestation, our results are consistent with those of the previous studies [START_REF] Dani | Risk factors for the development of respiratory distress syndrome and transient tachypnoea in newborn infants. Italian Group of Neonatal Pneumology[END_REF][START_REF] Kumar | Epidemiology of respiratory distress of newborns[END_REF]. Although using a rigorous definition of SRDS combining clinical and radiological criteria, the prevalence of SRDS reached up to 20% of neonates from patients diagnosed with GD. It is noticeable that, differently from the previous studies, we did not observe any statistical association between the diagnosis of foetal macrosomia and the occurrence of SRDS [START_REF] Esakoff | The association between birthweight 4000 g or greater and perinatal outcomes in patients with and without gestational diabetes mellitus[END_REF][START_REF]Hyperglycemia and adverse pregnancy outcome (HAPO) study: associations with neonatal anthropometrics[END_REF][START_REF] Das | Neonatal outcomes of macrosomic births in diabetic and non-diabetic women[END_REF]. Because macrosomia is known as a major complications of GD and controlled GD was demonstrated to be associated with a diminution of the incidence of macrosomia [START_REF] Langer | Gestational diabetes: the consequences of not treating[END_REF][START_REF] Horvath | Effects of treatment in women with gestational diabetes mellitus: systematic review and meta-analysis[END_REF], we can hypothesize our results to be explained by the small size of our study. Furthermore, we identified operative vaginal delivery as an independent risk factor of SRDS. We believe this result to be easily explained by the fact that more than the half of operative vaginal deliveries was indicated by non-reassuring foetal heart rate, being the genuine cause of poor neonatal outcome. Caesarean delivery even planned or during labour was not associated with SDRS in multivariate logistic regression contrary to results reported in a recent retrospective study of late-preterm births [START_REF] Bricelj | Neonatal respiratory morbidity in late-preterm births in pregnancies with and without gestational diabetes mellitus[END_REF] and lack of power or different period of inclusion (late preterm and term) can explain this difference.

The direct effect of maternal GD on the risk of neonatal SRDS may have physiopathological and clinical explanations. First, GD has been experimentally shown to induce both insufficient and an abnormal maturation of surfactant due to immaturity of type II alveolar cells from rats, with particularly important effects at the end of pregnancy [START_REF] Gewolb | Surfactant secretion by type II pneumocytes is inhibited by high glucose concentrations[END_REF]. Furthermore, phosphatidyl glycerol, an essential component of surfactant, may be diminished, due to a delayed secretion in case of unbalanced GD [START_REF] Piper | Lung maturation in diabetes in pregnancy: if and when to test[END_REF]. Second, induced labour or planned caesarean section is more likely to be performed in cases of GD, independently of suspected macrosomia, resulting in an increased risk of neonatal respiratory distress syndrome due to a delayed resorption of pulmonary liquid or "wet lung" [START_REF] Ehrenberg | The influence of obesity and diabetes on the risk of cesarean delivery[END_REF][START_REF] Al-Agha | Caesarean section and macrosomia increase transient tachypnoea of the newborn in type 1 diabetes pregnancies[END_REF][START_REF] Al-Qahtani | Diabetes is associated with impairment of uterine contractility and high Caesarean section rate[END_REF]. Our study reported that women diagnosed with GD were more likely to have undergone induced labour and neonates diagnosed with SRDS were more often born after the induction of labour. Finally, GD is known to be associated with an increased risk of prematurity. A prospective cohort study on 46,230 pregnancies reported this risk to depend on the levels of maternal glycemia, independently of perinatal complications that could have triggered early delivery such as gestational hypertension, preeclampsia, hydramnios, or macrosomia [START_REF] Hedderson | Gestational diabetes mellitus and lesser degrees of pregnancy hyperglycemia: association with increased risk of spontaneous preterm birth[END_REF]. Another study comforted these results in women with impaired glucose tolerance [START_REF] Ostlund | Maternal and fetal outcomes if gestational impaired glucose tolerance is not treated[END_REF]. We found in our study a trend of increasing risk of preterm birth in women with GD but our study failed to demonstrate a significant difference, possibly because of a lack of power, with small effectives in each group.

We found obesity to be an independent risk factor of SRDS. This finding is consistent with the previous literature as maternal obesity has already been shown to be associated with a significantly higher risk of perinatal death [START_REF] Chu | Maternal obesity and risk of stillbirth: a metaanalysis[END_REF]. Obesity in women with GD has also been documented to be associated with a higher proportion of adverse neonatal outcomes, including the need respiratory support [START_REF] Langer | Overweight and obese in gestational diabetes: the impact on pregnancy outcome[END_REF].

Our study failed to identify a benefit in the achievement of optimal glycaemic level during pregnancy. Such optimal balance of glucose level by insulin therapy has, however, been previously shown to be the sole factor associated with a reduction of neonatal complications in women with GD [START_REF] Crowther | Effect of treatment of gestational diabetes mellitus on pregnancy outcomes[END_REF][START_REF] Landon | A multicenter, randomized trial of treatment for mild gestational diabetes[END_REF]. Although early optimal balance of GD seems to be crucial to reduce neonatal complications, the association of controlled GD and of decreasing risk of SRDS remains undemonstrated. However, the small effectives in each group and a non-dedicated design of our study do not allow for any genuine conclusion and further large prospective studies remain needed to properly answer to that question.

Our results raise concerns regarding the potential benefit of corticosteroids therapy performed between 34 0/7 and 36 6/7 weeks in case of GD with the risk of imminent delivery to reduce the risk of SRDS. This question was asked in a retrospective study with a higher proportion of infants from the GD group receiving corticosteroids but without impact on logistic regression analysis [START_REF] Fung | Does gestational diabetes mellitus affect respiratory outcome in late-preterm infants?[END_REF]. A randomized trial recently suggested that the administration of betamethasone to women at risk for late-preterm delivery (from 34 0/7 to 36 5/7 weeks) could significantly reduce the rate of neonatal respiratory related morbidity [START_REF] Gyamfi-Bannerman | Antenatal betamethasone for women at risk for late preterm delivery[END_REF]. In this trial, the proportion of women with GD was the same in both betamethasone and placebo groups (around 10%). It is, however, noticeable that the primary outcome was a neonatal composite of treatment in the first 72 h and did not specifically describe the severity of SRDS. Although that outcome decreased from 14.4% (placebo) to 11.6% (betamethasone), the incidence of neonatal hypoglycaemia was greatly increased in the steroid group. At this stage, the benefit of corticosteroids therapy in patients with GD and threatened late-preterm delivery remains, however, questionable; more studies are needed to investigate this point.

Finally, in showing GD to be an independent risk factor of neonatal SRDS after 34 0/7 weeks of gestation, our results suggest the need of reorganizing the management of pregnant women diagnosed with GD and to decide, where these patients should deliver, especially since the incidence of GD reaches 13.5% in our study. This incidence is lower than reported in the French national perinatal surveys in 2010 (7.2%) in general population but concordant with incidence in high risk population in France (10-20%) [START_REF] Galtier | Definitions, epidemiology, risk factors[END_REF]. However, this incidence is lower than the worldwide incidence probably because of difference in incidence of obesity, maternal age, ethnicity, and other factors influencing the occurrence of GD. As recently highlighted by the European Consensus Guidelines on the Management of Respiratory Distress Syndrome [START_REF] Sweet | European consensus guidelines on the management of respiratory distress syndrome-2016 update[END_REF], "optimizing the outcome for babies with respiratory distress syndrome includes good obstetric practice and consideration of whether transfer to a perinatal centre is necessary and safe". Nowadays, GD still does not justify management in a specific level of care maternity unit. Most will deliver in low level of care maternity units, with paediatrician not being systematically present at the time of delivery. Our results bring additional arguments for a genuine reflection about these recommendations.

Conclusion

GD is an independent risk factor of neonatal SRDS after 34 0/7 weeks. This risk is also increased in obese women. These results confirm the need for careful GD screening during pregnancy, with antenatal prevention of maternal obesity. They also suggest the need for anticipation of the management of pregnant women diagnosed with GD and specially to transfer them in perinatal centre with the possibility of systematic immediate neonatal paediatric management if required.

Table 1

 1 Maternal, obstetrical, and neonatal characteristics according to the diagnosis of maternal gestational diabetes

	All values are expressed as n (%) otherwise specified
	SD standard deviation, SRDS severe respiratory distress syndrome * Body Mass index >30 (kg/m 2 )
	** <37 weeks
	*** Macrosomia defined as a neonatal weight ≥4000 g

Table 2

 2

	Maternal, obstetrical, and neonatal characteristics according to the occurrence of severe respiratory distress syndrome

Table 3

 3 Identification of factor having a significant and independent impact on the risk of severe neonatal respiratory distress syndrome in multivariate logistic regression

		AOR	95% CI	p
	Obesity*	2.8	1.1-7.1	0.029
	Gestational diabetes	3.6	1.5-8.6	0.005
	Planned c-section	1.9	0.6-6.0	0.284
	Emergency c-section during labour	2.9	0.9-9.2	0.071
	Operative vaginal delivery	5.5	1.9-15.9	0.002
	Macrosomia**	0.9	0.3-2.9	0.848
	CI confidence interval, AOR adjusted odds ratio * BMI > 30 (kg/m 2 )
	** Defined as a neonatal weight ≥4000 g

Author contributions MI: data collection and manuscript writing, BJ: manuscript writing, TB: reviewing, GC: supervision and methodology, BF: supervision and methodology, CX: Project development, reviewing, and supervision.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.

Ethical approval

All procedures performed in this study were in accordance with the ethical standards of the national research committee and the study protocol received approval from the Ethics Committee on Research in Obstetrics and Gynecology (CEROG no. 2010-OBS11-02).

Informed consent Written informed consent was obtained from all participants included in the study.


Dr T Loulou
email: loulou@enstimac.fr

S Salvador

J L Dirion

DETERMINATION OF REACTION PARAMETERS FOR CARDBOARD THERMAL DEGRADATION USING EXPERIMENTAL DATA

Keywords: parameter estimation, inverse problem, sensitivity coefécient, kinetic model

T his work is part of an ongoing research effort in which parameter estimation techniques have been utilized in recovering the kinetic parameters of a given reacting model using measurements collected from TGA devices. The goal of this research work is to develop a useful and universal estimation procedure to determine simultaneously the kinetic parameters involved in the chemical modelling under study such as combustion, pyrolysis, waste stabilization, etc. The present parameter estimation problem is solved with the Levenberg-Marquardt method of minimization of the least square norm representing the square difference between the measured mass variations during cardboard pyrolysis and the mass responses obtained with numerical solution of the model. An analysis of the linear dependency of the pyrolysis parameters needed to design a robust estimation tool is presented. In order to perform this analysis, the sensitivity coefécients and the sensitivity matrix determinant were examined. Experimental data obtained with TGA during the pyrolysis process of cardboard are analysed in this paper and the unknown parameters involved in the kinetic modelling are estimated.

INTRODUCTION

In building the reaction schemes for solids decomposition, the estimation of kinetic parameters represents a crucial step in chemical modelling (Font et al., 2001). Indeed, once a kinetic scheme has been established, two main methods can be used to estimate the corresponding Arrhenius parameters: a graphical method and numerical étting. The érst method is usually limited to a global (single) reaction. When the reacting scheme involves more than one reaction, the graphic method is not adequate and the numerical étting represents the only alternative. However, the second method needs to be implemented with precautions and eféciency to avoid the determination of physically unacceptable reaction schemes.

In a recent paper (David et al., 2003), the authors presented the determination of a reacting scheme of thermal degradation of cardboard. As mentioned in many references, they have underlined some diféculties in estimating the kinetic parameters involved in the reaction schemes. In the present study the authors present a numerical tool developed to alleviate the estimation procedure in terms of the computing time and simplicity of use. Physical and chemical considerations, such as positiveness of kinetic parameter, continuity in reaction rate values, are also taken into consideration.

The literature is rich in this éeld and one can énd several commercial and educational softwares such as Themokinetics [START_REF] Opfermann | Kinetic analysis using multivariate non-linear regression. (i) Basic concepts[END_REF], Imsl (IMSL, 1987), AKTS-TA [START_REF] Roduit | Inèuence of experimental conditions on the kinetic parameters of gas-solid reactionsparametric sensitivity of thermal analysis[END_REF] which help in the solution of such difécult parameter estimation problems. In this sense, the presented algorithm does not present any originality with respect to the existing software, except for the detailed sensitivity analysis shown in the next section and the incorporation of the constraints on the parameters to be estimated. In fact the sensitivity analysis plays an important part in understanding the effect of each parameter present in the kinetic schemes developed here.

This work is part of an ongoing research effort to develop a practical, universal and minimally time-consuming computing tool that will be helpful in the determination of kinetic parameters occurring in the modelling of any chemical and=or physical problem such as combustion, pyrolysis, stabilization and other phenomena investigated in our laboratory. The estimation of unknowns is formulated as an inverse problem of simultaneously estimating the kinetic parameters involved in the modelling of the physical problem under investigation.

This paper presents an application of this estimation tool with data obtained, during pyrolysis of cardboard, by a thermal gravimetric analysis device (TGA). The minimization of the least square norm is achieved using the Levenberg -Marquardt method (Levenberg, 1944;[START_REF] Marquardt | An algorithm for least square estimation of nonlinear parameters[END_REF][START_REF] More | Numerical Analysis[END_REF][START_REF] Bard | Nonlinear Parameter Estimation[END_REF]. In order to investigate the estimation feasibility, the sensitivity coefécients and the sensitivity matrix determinant are examined. This analysis helps in quantifying the correlation degree among the parameters of interest.

DIRECT PROBLEM

In a previous paper (David et al., 2003), the authors demonstrated that the pyrolysis of cardboard can be described with a two steps reaction scheme, i.e.

Cardboard ¡!

F1

Intermediate ¡!

F1

Char

In a érst reaction, the initial cardboard mass m 1 (t), is converted into an intermediate pseudo-species denoted m 2 (t) and gas. Then, the pseudo-species is converted into char, denoted m 3 (t), and gas through a second reaction. This scheme was chosen from eight proposed kinetic schemes (David et al., 2003). The best étting of experimental data recorded with three different heating rates was obtained with this scheme. At a given time t, the total mass of the sample is the sum of the masses of cardboard, intermediate pseudospecies and char, i.e.

M (t) ˆm1 (t) ‡ m 2 (t) ‡ m 3 (t) (1)
The mathematical model, describing the time evolution of the different mass fractions during the pyrolysis process, is based on set of érst-order differential equations. The weight loss phenomenon of the cardboard pyrolysis is described by:

dm 1 dt ˆ¡k 1 m 1 (2) dm 2 dt ˆ ‡a[k 1 m 1 ¡ k 2 m 2 ] (3) dm 3 dt ˆ ‡bk 2 m 2 (4)
with the following initial boundary conditions

m 1 (0) ˆ1: m 2 (0) ˆ0: m 3 (0) ˆ0 ( 5 )
The decomposition rule gives:

dm i dt ˆdm i dT £ dT dt ˆdm i dT b i ˆ1, 2, 3 (6)
where b represents the heating rate, which is considered constant in this study. The kinetic reaction rates depend on the absolute temperature and are given by the two Arrhenius functions:

k i ˆAi exp ¡ E i RT µ ¶ i ˆ1, 2 (7)
where A i is the Arrhenius factor, E i is the activation energy for the pyrolysis, R is the ideal gas constant, and T is the absolute temperature of the reaction. a and b represent the mass stoichiometric coefécients.

In the direct problem associated with the kinetic reaction described above, the kinetic parameters A 1 , E 1 , A 2 , E 2 , as well as the coefécients a, b, initial boundary conditions, are known. The objective of the direct problem is then to determine the time variation of mass fractions, m i (t), i ˆ1, 2, 3, and the total mass M(t) during the pyrolysis process.

There exists no analytical solution for the direct problem given in equations ( 2)-( 5). As we are dealing with more than one érst-order differential equation, the possibility of stiffness of the set of equations can arise. For these two reasons the numerical solution of the direct problem is obtained with the Kaps-Rentrop algorithm in terms of the subroutine stiff developed in [START_REF] Press | Numerical Recipes in Fortran[END_REF].

INVERSE PROBLEM

Once the kinetic scheme has been established, two principal methods are usually used to estimate the corresponding Arrhenius parameters: the graphical method or numerical étting. The érst method is limited only to single reaction (global). The case of simultaneous reactions is treated through a numerical procedure formulated as an inverse problem.

For the inverse problem considered here, the parameters

A 1 , E 1 , A 2 , E 2 , a
, and b are regarded as six unknowns, while the other quantities appearing in the formulation of the direct problem described above are assumed to be known precisely. Thus, the vector of the unknown parameters is

U T ˆ[A 1 , E 1 , A 2 , E 2 , a, b] ˆ[u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ] (8)
The additional information needed in the simultaneously estimation of the kinetic parameters is available from the experimental data obtained with the TGA apparatus.

Generally, inverse problems are solved by minimizing a residual functional J based on the ordinary least square norm. The sum of the squared residuals between the measured data and the responses of a model simulating the physical problem under investigation deénes the least square norm. For discrete measured data, the residual functional is written as follows:

J (U) ˆX N iˆ1 [Y (t i ) ¡ M (t i )] 2 (9)
where M(t i ) and Y (t i ) are, respectively, the computed and measured total mass of chart at time t i . N is the total number of measurements. In vectorial form, the above expression can be written as

J (U) ˆ[Y ¡ M(U)] T [Y ¡ M(U)] (10)
Here,

Y T ˆ[Y 1 , Y 2 , . . . , Y N ] is the vector of measured mass, M T (U) ˆ[M 1 (U), M 2 (U), . . . , M N (U)
] is the vector of estimated total mass at time t i , (i ˆ1, 2, . . . , N ) obtained from the solution of the direct problem with an estimate of vector U, U T ˆ[u 1 , u 2 , . . . , u P ] is the vector of unknowns parameters, N is the total number of measurements, and P is the number of unknown parameters, which is equal to 6 in this case.

A version of Levenberg-Marquardt method was applied for the solution of the presented parameter estimation problem. This method is quite stable, powerful and straightforward and has been applied to a variety of inverse problems. This method belongs to a general class of damped least square methods. The solution for vector U is achieved using the following iterative procedure:

U (k ‡1) ˆU(k) ‡ [(X (k) ) T X (k) ‡ m (k) O (k) ] ¡1 £ (X (k) ) T [Y ¡ M(U (k) )] ( 11
)
where the superscript (k) deénes the iteration number and X represents the sensitivity matrix evaluated at the iteration (k).

The sensitivity matrix is given by:

X ˆ@M T (U) @U µ ¶ T ˆ@M 1 @u 1 . . . @M 1 @u m . . . . . . . . . @M N @u 1 . . . @M N @u m 2 6 6 6 6 6 4 3 7 7 7 7 7 5 (12)
The elements of the sensitivity matrix X, denoted X ij , are known as the sensitivity coefécients. They can provide considerable insight to the estimation problem and in the design of the experiment for optimum accuracy in the estimates.

An iterative procedure is required due to the non-linear nature of the estimation problem because the coefécients of the sensitivity matrix depend on the unknown thermophysical properties to be recovered. Iteration continues until convergence of the estimated parameter is reached, i.e. when there is negligible change in any component of U. One criterion to indicate this is deéned as:

jU (k ‡1) ¡ U (k) j jU (k) j 4 e ( 13
)
where e is a small number to quantify convergence, such as 10 ¡5 . Different versions of the Levenberg-Marquardt method can be found in the literature, depending on the choice of the diagonal of the damping matrix O (k) and of the form chosen for the variation of the damping parameter m (k) .

The most used forms of matrix O (k) are:

O (k) ˆI and O (k) ˆdiag[(X (k) ) T X (k) ] ( 14
)
where I is the identity matrix.

RESULTS

In this section, we present the results obtained with the developed algorithm as applied to the solution of our inverse problem. In the érst part of this section, we present a detailed sensitivity analysis to show the feasibility of the estimation. The second part will be dedicated to a numerical test case with its statistical analysis. In the énal part we present the kinetic parameters issued from the use experimental data obtained with the pyrolysis of char.

Sensitivity Analysis

There are several different approaches for the computation of the sensitivity coefécients [START_REF] Beck | A method for the solution of certain non-linear problems in least square[END_REF]. In the present inverse problem the central énite-difference approximation is used to calculate the sensitivity coefécients in the form:

X ij ˆM(t i , u 1 , . . . , u j ‡ eu j , . . . , u 6 ) ¡M (t i , u 1 , . . . , u j ¡ eu j , . . . , u 6 ) 2eu j (15)
where e ˆ10 ¡5 . The sensitivity coefécient X ij , as deéned in the previous equation, and in the sensitivity matrix deénition given in equation ( 12), is the measure of the sensitivity of the estimated M i with respect to changes in the parameter u j . A small value of the magnitude of X ij indicates that large changes in u j yield small changes in M i . The estimation of the parameter u j is extremely difécult in such a case, because basically the same value for total mass would be obtained for a wide range of values of u j . In fact when the sensitivity coefécients are small, we have jX T Xj º 0 and the inverse problem is ill-conditioned. It can also be shown that jX T Xj is null if any column of X can expressed as a linear combination of the other columns [START_REF] Beck | A method for the solution of certain non-linear problems in least square[END_REF]. Therefore, it is desirable to have linearly independent sensitivity coefécients X ij with large magnitudes, so that the inverse problem is not very sensitive to measurement errors and accurate estimates of the parameters can be obtained.

Since the unknown parameters can assume different values, the analysis of the sensitivity coefécients is much simpliéed by using their relative versions deéned as:

X ‡ ij ˆuj @M i @u j j ˆ1, . . . , P (16)
Figure 1 presents the transient behavior of the relative sensitivity coefécients for the six components of U. The exact values of the parameters, used in the sensitivity study, are reported in Table 1. Except for the parameter E 1 , the relative sensitivity coefécients of the remainder parameters are approximately of the same order of magnitude. The smallest relative sensitivity coefécient is observed with the coefécient A 2 , which can have major diféculties in its estimation and a relatively high estimation error. As displayed in Figure 1, the sensitivity coefécients looks slightly linearly dependent, but a careful examination of the different ratios between all the relative coefécients shows that they are not linearly dependent and therefore their simultaneously estimation is feasible. Finally, the analysis of the temperature variation of the determinant D of the matrix X T X reveals that the simultaneous estimation is feasible when the énal temperature T max is equal or greater than 850 K. Indeed, the maximum value of D is reached at T ˆ850 K.

Numerical Simulation

The simulated measurements of total mass M(t) are obtained from the solution of the direct problem, by using a priori prescribed values for the unknown parameters to be recovered simultaneously. The exact values of the parameters are reported in Table 1. The other quantities are taken as b ˆ10 K min ¡1 , T min ˆ450 K, T max ˆ850 K.

The solution of the direct problem, obtained using known parameters, provides the exact total mass measurements

M

x (t i ), i ˆ1, . . . , N (errorless). Measurements containing random errors are simulated by adding a white noise (error term) to M

x (t i ) in the form:

M (t i ) ˆMx (t i ) ‡ os (17)
where M(t i ) is the simulated measurements of total mass, M

x (t i ) is the exact total mass (errorless), s is the standard deviation of the measurement errors, and o is a random variable with normal distribution, zero mean and unitary standard deviation. For the 99% conédence level we have ¡2.576 < o < ‡2.576. This variable is generated with the subroutine DRRNOR of the IMSL library (IMSL, 1987). Figure 2 shows the solution of the direct problem using the exact parameters given in Table 1.

We now consider the inverse problem of estimating simultaneously the components of the vector U by the Levenberg-Marquardt method. The initial guesses for the unknown parameters are displayed in Table 1. The computations were performed using a Pentium computer, under the Fortran PowerStation platform. The relative error of the estimated parameters is deéned as

e i ˆui ¡ • u u i • u u i ≠ ≠ ≠ ≠ ≠ ≠ ≠ ≠ £ 100% i ˆ1, . . . , P ( 18
)
where the overbar designates the exact parameter under hand.

Table 1 summarizes the results obtained for the simultaneously estimation of six parameters. It shows the initial guess of the unknowns, their recovered and exact values, and their estimation error, respectively. Two levels of measurement errors are considered for numerical analysis including s ˆ0 (errorless) and s ˆ0.001, which corresponds approximately to 0.1% of the maximum of total mass M(t) during the pyrolysis process.

By using the errorless measurements (s ˆ0), the six parameters are recovered exactly in 47 s of CPU time and the estimated loss weight matches precisely the measured one. We should mention here that, beyond the displayed initial guess, we cannot get convergence of the estimation algorithm. We observe that, except for the parameter E 2 , the initial guess of the remain parameters is wide enough with respect to their énal values (exact values).

Also, Table 1 shows the estimated parameters when they are obtained with measurements containing random errors according to equation ( 17). Generally, the estimated parameters are in good agreement with their exact values. The highest error is observed with parameter A 2 and this results from its low sensitivity as underlined in the sensitivity analysis section. The estimation error of parameter A 1 is relatively high with respect to the noise level added to the exact data in generating the simulated measurements. Also, this can be explained by its low sensitivity, as shown in Figure 1.

Real Test Case

Pyrolysis of cardboard

Experiments were performed using a SETERAM ATG-DSC-111 apparatus. To generate dynamic data, different cycles were realized for which the temperature increased linearly with time and at different rates, b ˆdT=dt (K min ¡1 ). The upper temperature value was chosen to ensure that pyrolysis process was complete, as mentioned in the sensitivity analysis.

Figure 3 shows the experimental and model predicted mass evolutions versus temperature obtained with the developed algorithm. Small differences can be observed around the curvature of the measured total loss weight and the étting can be considered as acceptable and efécient. Also, Figure 3 displays the residual of the weight loss, i.e. M (t) ¡ Y (t). The biggest difference is observed over the temperature domain [600, 700] and the loss mass residual is still less than 2%. The residual is scattered around zero line during the whole experimental temperature change. If the étting was perfect, i.e. the étting matches precisely the mean measured proéle, we would get the measurement errors (noise) instead of the model èuctuations. Figure 4 displays the computed total mass vs the total measured mass loss. As mentioned in the residual analysis, the estimation is acceptable since this égure shows approximately a straight line representing the bisection of the principal axis of Figure 4. By using the experimental data of cardboard pyrolysis, the values of estimated parameters are: A 1 ˆ1.07 £ 10 8 min ¡1 , E 1 ˆ101,107 J mol ¡1 , A 2 ˆ19,824 min ¡1 , E 2 ˆ67,648 J mol ¡1 , a ˆ0.41 and b ˆ0.32.

These parameters are obtained in less than 50 s CPU time and with the same initial guesses as displayed in Table 1. In comparison with the estimation procedure developed in our previous paper (David et al., 2003) around the function fmins of Matlab software [START_REF] Matlab | Library Edition 5.3, User's Manual[END_REF], where the estimation time takes over 2 h of computing time, the presented method is more than 100 times faster. This tool will be used in the future to facilitate model building. Finally, the results shown were obtained with a personal computer powered by an Intel 1 Pentium 4 processor of 2 GHz, and using 256 MB of RAM, under the Fortran Powerstation platform.

CONCLUDING REMARKS

A numerical procedure is presented for the simultaneous estimation of kinetic parameters characterizing the pyrolysis process of cardboard. The minimization procedure is conducted by minimizing the square difference between experimental measured total mass and the corresponding calculated values from a mathematical model.

A comparison between recovered and exact data showed good agreement. The obtained results underline the robustness of the algorithm and its capabilities to recover simultaneously the kinetic parameters using less a priori information and wide deviation in the initial guess of parameters. This tool is developed by taking into consideration some physical constraints such as positiveness of kinetic parameters, and continuity of reaction schemes.

Despite a relatively wide deviation in the initial guess of parameters to be recovered, the presented tool is still sensitive to the initial values. This deéciency is due to the strong non-linearity of the kinetic reacting schemes. Efforts are currently underway to address this problem by considering the genetic algorithms in obtaining the best initial guess. Also, the direct estimation of the Arrhenius functions instead of Arrhenius parameters will be analysed by means of function estimation tools.

Figure 1 .

 1 Figure 1. Evolution of relative sensitivity coefécients as temperature function (T bt).

  Interscience, New York, USA). David, C., Salvador, S., Dirion, J. and Quintard, M., 2003, Determination of a reaction scheme for cardboard thermal degradation using thermal gravimetric analysis, J Anal Appl Pyrol, 67(2): 307-323. Font, R., Martin-Gullon, I., Esperanza, M. and Fullan, A., 2001, Kinetic law for solids decomposition. Application to thermal degradation of heterogeneous materials, J Anal Appl Pyrol, 58-59: 703-731. IMSL, 1987, Library Edition 10.0, User's Manual, Math=Library (IMSL Houston, TX, USA).

Figure 3 .

 3 Figure 3. Comparison between the experimental and the computed loss mass evolution. Weight loss residual evolution as temperature function. The heating rate was taken as b 10 K min 1 .

Figure 4 .

 4 Figure 4. Computed total mass loss versus measured total mass loss.

Table 1 .

 1 Exact, and estimated parameters and their respective estimation error for two simulated test cases.

					Initial
	s	Parameter	Unit	Exact	guess	Estimated e i (%)
	0	A 1 E 1 A 2 E 2	min ¡1 1.07 £ 10 8 1.7 £ 10 6 1.07 £ 10 8 J mol ¡1 101,000 70,000 101,000 min ¡1 20,000 1 20,000 J mol ¡1 67,500 35,000 675,000	0.0 0.0 0.0 0.0
		a		0.40	0.10	0.40	0.0
	0.001	b A 1 E 1 A 2 E 2	0.30 min ¡1 1.07 £ 10 8 1.7 £ 10 6 1.09 £ 10 8 0.10 0.30 J mol ¡1 101,000 70,000 101,092 min ¡1 20,000 1 22,121 J mol ¡1 67,500 35,000 68,101	0.0 1.90 0.09 10.6 0.89
		a		0.40	0.10	0.399	0.01
		b		0.30	0.10	0.299	0.02

Figure

2

. Solution of the direct problem with exact data (s 0) shown in Table

1

.

ACKNOWLEDGMENT

The authors would like to express their deep acknowledgements to the reviewers for their constructive critic and remarks in improving the quality of this paper.