MD, MSc Pierre-Henri Valentin Moury
email: phmoury@chu-grenoble.fr

MD Flora Pasquier

MD Jean-Lionel Greco

MSc Arvieux

MS Marion Silvia Alves-Macedo

MSc Richard

MD Kristina Myriam Casez-Brasseur

MSc Peggy Skaare

MD Michel Jacon

MD Damien Durand

MD. Samir Bedague

MD Jaber

PhD Jean-Luc Bosson

PhD Pierre Albaladejo

MD, PhD P.-H Moury

MD V Pasquier

MD F Greco

MSc J.-L Arvieux

MSc S Alves-Macedo

MSc M Richard

MD M Casez-Brasseur

MD M Durand

MD D Bedague

MD P Albaladejo

MSc K Skaare

MD J.-L Bosson

MD P Jacon

MD S Jaber

l'utilisation peropératoire de la ventilation non invasive par rapport à la supplémentation en oxygène par masque facial pour la sédation procédurale dans un laboratoire d'électrophysiologie

Keywords: Airway, apnea, cardiology, mechanical ventilation, noninvasive ventilation, sedation

teaching and research institutions in France or abroad, or from public or private research centers.

A randomized controlled trial of the intraoperative use of noninvasive ventilation versus supplemental oxygen by face mask for procedural sedation in an electrophysiology laboratory

Une étude randomisée contrôlée sur

A randomized controlled trial of the intraoperative use of noninvasive ventilation versus supplemental oxygen by face mask for procedural sedation in an electrophysiology laboratory

Pierre-Henri Moury, Valentin Pasquier, Flora Greco, Jean-Lionel Arvieux, Silvia Alves-Macedo, Marion Richard, Myriam Casez-Brasseur, Kristina Skaare, Peggy Jacon, Michel Durand, et al.

To cite this version:

Méthode

Dans cette étude randomisée contrôlée, nous avons inclus 195 patients de statut physique III ou IV selon l'American Society of Anesthesiologists pendant des interventions en laboratoire d'électrophysiologie. Nous avons comparé la VNI à l'oxygénothérapie par masque facial pour les patients sous sédation. Le critère d'évaluation principal était l'incidence des complications respiratoires déterminée par une analyse en aveugle assistée par ordinateur et définie par une hypoxémie (saturation périphérique en oxygène \ 90 %) ou une apnée/hypopnée (absence de respiration pendant 20 secondes a` la capnographie). Les critères d'évaluation secondaires comprenaient les variables hémodynamiques, la sédation, la sécurité des patientes (scores composites des évènements indésirables majeurs ou mineurs) et les issues indésirables au jour 7.

Résultats

Un évènement respiratoire est survenu chez 89/98 (95 %) patients du groupe VNI et chez 69/97 (73%) patients ayant un masque facial (risque relatif [RR], 1,29; intervalle de confiance [IC] à 95 %, 1,13 à 1,47; P\0,001). Une hypoxémie est survenue chez 40 (42 %) patients du groupe VNI et chez 33 (34 %) patients ayant un masque facial (RR, 1,21 ; IC a` 95 %, 0,84 a` 1,74; P = 0,30). Une hypoxémie est survenue chez 40 (42 %) patients du groupe VNI et chez 33 (34 %) patients ayant un masque facial (RR, 1,21; IC 95 %, 0,84 a` 1,74; P = 0,30). Les variables hémodynamiques, la sédation, les évènements de sécurité majeurs ou mineurs et les issues pour les patients n'étaient pas différents entre les groupes.

Conclusion

Les complications respiratoires étaient plus fréquentes chez les patients recevant une VNI sans aucun impact sur la sécurité ou les issues. Ces résultats n'appuient pas l'utilisation systématique de la VNI en peropératoire.

Noninvasive ventilation (NIV) is known to treat acute respiratory failure during postoperative complications and in critically ill patients.1,2 The mechanisms of its benefit include mechanical support, which mitigates the exaggerated workload of acute respiratory failure, and positive end-expiratory pressure, which counteracts alveolar flooding secondary to acute pulmonary edema. This benefit has been shown in patients with cardiothoracic diseases.3-5 It is recommended that patients treated with NIV should be conscious; nevertheless, an increasing number of procedures requiring sedation and NIV support have been reported in the intensive care unit (ICU) setting.6,7 Reports of patients who were sedated for surgery and were treated with NIV exist in the literature.8,9 The pitfalls of such data reside in the weakness of the signal reported, such as transient hypoxemia assessed by an external observer. Challenges reside in direct measurements of the respiratory variables and independent analysis that allow robust reporting.

The number of procedures in electrophysiology laboratories has greatly increased in recent years. In a previous study, we reported a high incidence of respiratory events in this specific population.10 A small number of studies have investigated the use of NIV during cardiothoracic procedures performed under procedural sedation and have reported beneficial results mainly in case reports, but none of them were randomized controlled trials (RCTs).8

We hypothesized that NIV would decrease the number of respiratory events during cardiothoracic procedures performed under procedural sedation compared with standard oxygen face mask use. Our primary goal was to decrease the number of respiratory events, defined by a composite outcome based on an independent direct measurement of physiologic variables from the patient monitor. Hence, we used a computerized blinded analysis that consisted of variable collection via patient monitoring measurements with independent software reporting and a blinded analysis of apnea/hypopnea or hypoxemia. We designed this RCT to continuously evaluate the efficacy of NIV to standard oxygen supplementation during procedural sedation through a computerized blinded analysis.

Methods

Ethics

This single-centre RCT was approved before the start of the study by the Comité de Protection des Personnes Sud-Est V on 4 November 2015 (Chairperson: Dr Charlety; Grenoble, France, RCB: 2015-A01594-45). The ClinicalTrials.gov identifier is NCT02779998 (registered 4 November 2015). This study was designed as a Prospective Randomized Open with Blinded Evaluation (PROBE) study. Written informed consent was obtained from all patients before inclusion in this study.

Study population and setting

The two groups were defined according to the ventilator strategy: the noninvasive ventilation group (NIV group) and the usual clinical care with the face mask (Capnomask , A202MX, Mediplus, High Wycombe, UK) oxygen therapy group (or the standard face mask group).

Patients were eligible for the trial if they were outpatients or hospitalized awaiting a procedure in an electrophysiology laboratory (i.e., cardioverter defibrillator or pacemaker implantation or replacement) and if they had an American Society of Anesthesiologists (ASA) Physical Status of III or IV. This was informed by the higher incidence of respiratory events in ASA III or IV patients compared with low-risk patients observed in a previously published study from our institution.10 The exclusion criteria for enrolment were as follows: age\18 yr; ASA Physical Status of I and II; current treatment for obstructive sleep apnea syndrome with continuous positive airway pressure (CPAP) or NIV devices; and contraindications to the use of NIV. As part of our standard of care, patients who were already being treated with CPAP or NIV devices received specific treatment during the procedure and the immediate postoperative period. Consequently, we elected not include these patients to limit minimize bias as this specific population was already educated regarding the devices.

The intervention was conducted under the supervision of a senior consultant anesthesiologist and intensivist. A trained nurse anesthetist was the principal operator, was present throughout the entire operation, and was able to freely adjust the drugs given and the airway management and initiate emergency procedures. The randomization was computer-generated using centralized web-based blocks of a random size that were produced by the Clinical Investigation Centre of Grenoble without stratification before the first patient was included. Allocation concealment according to the randomization was preserved during the computer acquisition. A research fellow independent from the procedure management was responsible for the computer acquisition of physiologic data from the patient monitor during the procedure and database management. To this point, the software analysis of the respiratory events was registered. Then, the computerized results were kept blinded and analyzed by an independent adjudication committee for a final validation of the respiratory event quotations to minimize measurement bias due to inaccurate wave signals. To separate registration and randomization from the analysis period, processing from the database to the adjudication committee was performed by a senior anesthesiologist independent of the randomization and the surgical procedure. None of the participants in the adjudication committee were members of the cardiovascular department.

Interventions

Noninvasive ventilation was administered with the dedicated medical device, ''Élysée TM 150'' (RESMED Ltd., Bella Vista, NSW, Australia). The NIV protocol was specific and allowed patients to be ventilated using CPAP and pressure support ventilation (PSV) in response to changes in the respiratory pattern needing different mechanical ventilation settings.2 The fraction of inspired oxygen (F I O 2 ) was titrated to target a peripheral oxygen saturation (SpO 2 ) [ 94%. We used a double-limb respiratory circuit (Intersurgical Fontenay sous Bois, France). The patients did not have the opportunity to try NIV preoperatively. Patients allocated to the NIV group had to be ventilated with a titrated positive end-expiratory pressure level from 5 to 10 cm H2O. Pressure support ventilation was not compulsory.

Patients could be ventilated with CPAP if needed. The target tidal volume was B 8 mL</=kg-1 (ideally B 6 mL</=kg-1) of the ideal body weight. The PSV was set to achieve this target and maintained below a 20 cm H2O threshold. The inspiratory trigger was set as sensitive as possible in the auto-triggering threshold limit. A full description of the NIV protocol is given in the Electronic Supplementary Material (ESM) eAppendix and eFig. 1.

In the usual care standard face mask group, a Capnomask (A202 MX, Mediplus, High Wycombe, UK) was used as our standard of care to continuously monitor capnography (see ESM eFig. 2). This device is a face mask with a port to enable attachment of a CO 2 sampling line. At the beginning of sedation, the initial oxygen flow was set at 4 L-min -1 . This flow was titrated to obtain an SpO2 greater than 94%, with a maximum of 15 L -min-1.

In both groups, only if required were maneuvers to maintain the airway applied, such as the use of an oropharyngeal airway (''Guedel cannula''), mandibular subluxation, and the use of face mask ventilation as a rescue intervention. The protocol included the use of intravenous titration of nalbuphine and sedation with a continuous intravenous infusion of propofol administered via a syringe pump. The Bispectral Index TM (BISTM; Medtronic/Covidien, Minneapolis, MN, USA) was continuously recorded, which is not the standard procedure at our institution. Hence, we decided to keep the personnel blinded to the BIS during the procedure to avoid biasing sedation management. All procedures were conducted with patients in the supine position. Ropivacaine was used to provide local anesthesia in the infraclavicular region.

Endpoints

The primary endpoint was the incidence of respiratory events, defined by a composite score of the occurrence of hypoxemia or apnea/hypopnea. Hypoxemia was defined by a decrease in SpO 2 \ 90%, and apnea/hypopnea was defined by an absence of a breathing cycle for 20 sec on the capnogram (CO 2 20s).

Our secondary endpoints were the stability of the intraoperative physiologic variables, sedation, overall patient safety (composite scores of major or minor adverse events), safety of NIV use during sedation in the operating room, and adverse outcomes at day 7. We also compared the number of episodes of hypoxemia and apnea/ hypopnea as continuous variables between groups in patients who had at least one episode.

Regarding safety, we compared the two groups in the incidence of major and minor adverse events and adverse outcomes at day 7. A major adverse event was defined as a composite of the following events: pneumothorax, intubation or laryngeal mask airway insertion, any surgical complication requiring a revision procedure, pericardial effusion or cardiac tamponade requiring a surgical procedure, and cardiopulmonary resuscitation. A minor adverse event was defined as the occurrence of any of the following: oropharyngeal airway insertion, mandibular subluxation, manual face mask ventilation, mask-related skin or nasal injury, and nausea or vomiting. Rescue therapy with crossover to NIV and standard face mask oxygen was considered if needed as well as manual face mask ventilation and intubation. All aspiration events were also recorded.

The adverse outcomes at day 7 were defined as a composite of the occurrence of any of the following: death, pulmonary edema, myocardial infarction, acute respiratory failure, and pneumonia. We also noted hospital and ICU length of stay.

Measurements

The primary endpoint was based on a computer analysis using RECAN software (Alpha-2 , Lyon, France) with physiologic variable extraction from Philips TM monitoring (Amsterdam, Netherlands) (see ESM eFigs 3-5). The software acquisition was performed by a research fellow who was not involved in the clinical management of the patients. Disconnections due to technical issues were noted and not taken into account in the final analysis. The results from the software acquisitions were then analyzed by the independent adjudication committee blinded to the patients' group allocation to validate the calculations based on the SpO 2 and capnogram waveforms on each real-time monitor tracing.

The stability of the intraoperative physiologic variables was assessed through the same computer extraction process. The two components of the primary endpoint were separately analyzed as continuous variables. Low blood pressure or bradycardia was defined by the need for catecholamine drugs or volume resuscitation. The oxygen flow rate and fraction of inspired oxygen were noted. The variables from the ventilator measurements were gathered through continuous software monitoring (ResScan TM 3.16 software, RESMED Ltd., Bella Vista, NSW, Australia). A full description of the NIV measurements is given in the ESM eAppendix.

Statistical analysis

The statistical analyses were independently carried out by the Clinical Investigation Center of Grenoble with Stata version 13 software (Stata Corporation, College Station, TX, USA). The analyses were performed on an intent-totreat basis after the usual procedures of data management and database freezing.

The sample size calculation was based on a previously published clinical observational study.10 We expected a 40% incidence of respiratory events in the control group (standard face mask group). Our hypothesis was that there would be a 50% reduction in the NIV group.

With an alpha of 5% and a statistical power of 80%, 82 patients per group were required, and a total sample size of 200 patients was planned.

The descriptive analyses concerned all the variables that were collected, and the data are described as number and percentage for qualitative variables and mean and standard deviation (SD) or median and interquartile range [IQR] for continuous variables, according to the distribution. A missing data replacement procedure was defined for the primary endpoint.

The statistical tests were performed with alpha = 0.05. The primary endpoint and qualitative secondary endpoints were compared with the Chi square test if the validity conditions were met; otherwise, they were compared with Fisher's exact test. The continuous secondary endpoints were compared with Student's t test or the Mann-Whitney U test according to the data distribution. The additional statistics usual used for therapeutic trials were also presented (the relative risk and 95% confidence interval [CI], the relative risk reduction and 95% CI, the odds ratio and 95% CI, and the number of patients needed).

Results

Patients

From 19 January 2017 to December 2018, 1,369 patients underwent a procedure in the electrophysiology laboratory and were screened for inclusion; of these, 200 were included, and 195 underwent randomization. The mean (SD) age of the patients was 72 [START_REF] Auriant | Noninvasive ventilation reduces mortality in acute respiratory failure following lung resection[END_REF] yr and 28% of the patients were women (Figure, flow chart). Five patients were excluded before randomization; three met the exclusion criteria, one withdrew his consent, and one did not meet the inclusion criteria (ASA II). There were 98 patients in the NIV group and 97 patients in the standard face mask group.

Respiratory curves were not interpretable for seven (4%) of the patients removed after randomization (four in the NIV group and three in the standard face mask group). This percentage was below the limit of 5% predefined for imputation of missing data, and statistical analysis of the primary outcome was performed using data from all the remaining patients (n = 188). The secondary outcomes were analyzed using the available data for each endpoint.

The characteristics of the patients at baseline were similar in the two groups (Table 1). The mean left ventricular ejection fraction was 44% ( 14) in the overall studied population, and there was a high incidence of associated comorbidities (diabetes, 24%; hypertension, 41%; chronic heart failure, 26%; coronary artery disease, 36%; and history of cardiac surgery, 12%).

The procedures were mainly represented by dual chamber pacemaker implantation (32%), cardioverterdefibrillator implantation (23%), and implantable device replacement (23%). The median [IQR] duration of the interventions was 66 [50-80] min (Table 2).

The intraoperative sedative, analgesic (Table 2) or hemodynamic support (Table 3) did not vary according to the randomization group.

Trial treatment

Among the 98 patients allocated to the NIV group, four patients had an early failure of NIV, two refused the device, and four had uninterpretable curve signals. In the standard face mask group, two patients had an early failure of the face mask, and three patients had no interpretable signal. Rescue therapy was possible in each group using the alternative treatment. Consequently, seven patients were eventually treated with the standard face mask, and two patients had NIV as a rescue therapy in response to deep and/or repeated hypoxemia despite their original allocation group.

The maximum F I O 2 in the NIV population was 87% [75-97], and the median O 2 maximal flow was 6 L-min -1 [START_REF] Clouzeau | Target-controlled infusion of propofol for sedation in patients with noninvasive ventilation failure due to low tolerance: a preliminary study[END_REF][START_REF] Deletombe | Dexmedetomidine to facilitate noninvasive ventilation after blunt chest trauma: a randomised, double-blind, crossover, placebo-controlled pilot study[END_REF][START_REF] Cabrini | Intraoperative prophylactic and therapeutic noninvasive ventilation: a systematic review[END_REF]. Among the NIV patients, 30 patients received CPAP alone, 29 received PSV alone, and 39 patients had a mix of both settings. The NIV ventilator results are described in the ESM eAppendix.

Primary outcomes

A The hemodynamic variables were similar in both groups (Table 3). The use of sedation and analgesic drugs did not differ between the groups (Table 1). The BIS index was 71 (95% CI, 58 to 76) in the NIV group and 69 (95% CI, 61 to 76) in the standard face mask group (P = 0.82, Mann-Whitney) and was below a threshold of 60 during 16% (95% CI, 3 to 67) of the time in the NIV group and during 17% (95% CI, 1 to 58) of the time in the standard face mask group.

Regarding the safety endpoint assessments (Table 4), the occurrence of a major adverse event was reported in two patients in the NIV group vs one patient in the standard face mask group: one patient had a pneumothorax, and one patient had a procedural complication in the NIV group vs one patient needing cardiopulmonary resuscitation in the standard face mask group. One patient needed medical intervention (for sedation purposes) in the standard face mask group. Aspiration during the procedure was reported in one patient in the NIV group. A minor adverse event was reported in 62/98 patients in the NIV group (63%; 95% CI, 53 to 73) and in 63/97 patients in the standard face mask group (65%; 95% CI, 55 to 74), mainly represented by the need for mandibular subluxation 60% (95% CI, 50 to 70) in the NIV group vs 44% (95% CI, 34 to 55) in the standard face mask group (P = 0.03; Chi square) (Table 4). The median leakage percentage during NIV was 69 [52-85]% (ESM eTable).

A day 7 outcome defined by the composite endpoint of the occurrence of death, pulmonary edema, myocardial infarction, acute respiratory failure and pneumonia occurred in three NIV recipient patients, while none were observed in the standard face mask group. The length of hospital stay, length of ICU stay, and mortality rate were not significantly different between the two groups (Table 5).

Discussion

The main result of our study was that routine use of NIV in the electrophysiology laboratory was associated with an increase in respiratory events. This increase in respiratory events was mainly represented by an excess in the number of absent breathing cycles for 20 sec detected on the capnogram to evaluate apnea/hypopnea. The use of NIV did not prevent the patients' desaturation.

Literature reports have described inconsistent effects of NIV. A benefit of NIV was reported for acute exacerbation of chronic obstructive pulmonary disease, but studies in postextubation respiratory failure showed that NIV did not improve mortality or the need for reintubation in the ICU setting.1,11 Moreover, positive effects on mortality or reintubation rate have been reported in the treatment of acute respiratory failure in the perioperative setting.2,12,13

Nevertheless, NIV trials remained inconsistent and failed in other specific perioperative conditions to significantly prevent postoperative pulmonary complications such as pneumonia, reintubation, and death.14, [START_REF] Hui | Routine postoperative noninvasive respiratory support and pneumonia after elective surgery: a systematic review and meta-analysis of randomised trials[END_REF] We expected a preventive effect on respiratory events, as the number of patients who underwent cardiac surgery was high in our study, and this surgery is known to cause diaphragmatic dysfunction.16 Moreover, the use of NIV with CPAP or pressure support is recommended in postoperative hypoxemic cardiac surgery patients.17 The main difference between such ICU patients and the patients in this study is that very few of them were acutely ill, resulting in the absence of an impaired balance between respiratory workload and demand. Concerning the comparison with surgical patients, most of the postoperative studies reporting positive results were conducted during major surgeries when the thoracic and pulmonary mechanics were affected, notably with atelectasis, whereas patients in the present study were only affected by a wound in the upper chest wall. [START_REF] Jaber | Postoperative noninvasive ventilation[END_REF] Finally, all patients underwent the procedure in the supine position because venous access was needed. This position is known to worsen sleep apnea disorders and might similarly predispose to functional airway obstruction in sedated patients. [START_REF] Ambrogio | Sleep and noninvasive ventilation in patients with chronic respiratory insufficiency[END_REF] Overall, we may conclude that NIV, while potentially beneficial in a selected ICU population,6,20 did not prove to be an appropriate solution for this study population.

The experimental approach of this study may provide answers to the issues of intraoperative NIV. Oxygen titration was possible in both groups using two different methods. Nevertheless, oxygenation titration did not reduce the incidence of hypoxemia. Oxygen delivery may have not been the key issue. As a consequence, the exploratory data from the airway management and respiratory pattern with NIV during sedation might be more relevant. The occurrence of apnea/hypopnea events emphasizes the importance of an appropriate interface for airway management. Hence, we cannot exclude the possibility that some patients would have benefited from different interface types, such as nasal masks or helmets. Another important point is the high incidence of interventions necessary to maintain patency of the upper airway tract in patients undergoing this degree of sedation. Mandibular subluxation was needed for most of the patients in the NIV group. Despite this maneuver, the measured expiratory volume and pressure reported were low because of leaks. Therefore, we hypothesize that individualized airway management with an appropriate and wider choice of interfaces might be essential for patients under sedation and NIV. One type of NIV mask with three sizes was insufficient to address each patient's specific need. We also observed that the quantity or depth of sedation or analgesia did not vary between the two groups. Bispectral Index values, total drug use, and achievement of clinical goals were similar in both groups. Nevertheless, as respiratory patterns change with the ventilation modes, the same sedation strategy may not produce the same outcomes. We hypothesize that patients at this level of sedation may benefit from intermittent positive pressure ventilation, while a patient with lighter sedation would benefit more from PSV.

There are several strengths to this study, including the randomized design, the selected population, the blinded analysis of the endpoints through a computer-driven approach, the explicit criteria for respiratory events, the postoperative pulmonary complications, and the patient outcomes with a seven-day follow-up period. Additionally, the safety of NIV, as shown in this randomized controlled trial, is encouraging. Our results showed a low incidence of intraoperative and early postoperative complications and an absence of impaired prognosis at day 7 post intervention. Moreover, our experimental approach provided responses to the challenges of intraoperative NIV.

Our findings are subject to several limitations. First, we designed this study based on the incidence of respiratory events reported in a previous study that only used clinical assessment and not an independent analysis from continuous monitoring.10 Consequently, we may have underestimated the number of respiratory events compared with a computer-based analysis in the sample size calculation. Additionally, we may have overestimated respiratory events with some unrelated clinical meaning for the clinician. Although this fact might be true for apnea/ hypopnea events, the number of hypoxemia events remained comparable in the control group (34%) to the historical cohort.10

Second, our primary endpoint was a composite endpoint of hypoxemia and the absence of breathing cycles for 20 sec detected on the capnogram as surrogate criteria for apnea/hypopnea. Capnography is efficient for evaluating the respiratory rate and is sensitive to upper airway closure.21 Despite its potential usefulness, respiratory rate measurement by the impedance method was inaccurate because of the surgical draping during pacemaker implantation that does not provide an appropriate location. Nevertheless, to limit the measurement bias, the primary endpoint assessment was made by a computer driven analysis that was reviewed by an independent adjudication committee and that was blinded to the randomization groups.

Third, the present study was not designed to show a significant decrease in the number of adverse patient outcomes in the NIV group. The lack of significance is mainly due to the low incidence of adverse events in this stable medico-surgical population who had scheduled procedures.

Fourth, our choice was to include patients in the electrophysiology laboratory for pacemaker implantation. Such interventions are minimally invasive in terms of respiratory mechanics. Our conclusion may not be applicable to other types of procedures.

Fifth, careful registration of monitoring curves and reading from the adjudication committee may not be sufficient to avoid measurement bias due to NIV disconnections or CO2 washout due to high oxygen flow. Although the respiratory rate measured by capnometry with a face mask was efficient at high flows, the absolute values of end-tidal CO2 were lower as the flow increased.21

Sixth, the study was a single-centre study, so the generalizability of the results is limited by a single-centre effect.

Among patients undergoing a procedure in a stimulation laboratory with moderate to deep procedural sedation, NIV was associated with the occurrence of more respiratory events than with the use of standard face mask oxygen supplementation. Although the present study supports the safety of NIV, the patient outcomes were not improved. Overall, these findings do not support the routine use of NIV in this setting.

  Pierre-Henri Moury, Valentin Pasquier, Flora Greco, Jean-Lionel Arvieux, Silvia Alves-Macedo, et al.. A randomized controlled trial of the intraoperative use of noninvasive ventilation versus supplemental oxygen by face mask for procedural sedation in an electrophysiology laboratory. Canadian Journal of Anesthesia / Journal canadien d'anesthésie, 2023, 70, pp.1182-1193. 10.1007/s12630-023-02495-2. hal-04116447 Résumé Objectif L'efficacité de la ventilation non invasive (VNI) pendant les interventions nécessitant une sédation et une analgésie n'a pas été établie. Nous avons évalué si la VNI réduisait l'incidence des complications respiratoires.

  respiratory event (hypoxemia or apnea/hypopnea) was observed in 89 patients in the NIV group (94.7%; 95% CI, 88 to 98) and in 69 patients in the usual care standard face mask Chi squared test). In patients who presented at least one apnea/ hypopnea event, the median [IQR] number of episodes was 10[START_REF] Deletombe | Dexmedetomidine to facilitate noninvasive ventilation after blunt chest trauma: a randomised, double-blind, crossover, placebo-controlled pilot study[END_REF][START_REF] Cabrini | Intraoperative prophylactic and therapeutic noninvasive ventilation: a systematic review[END_REF][START_REF] Pieri | Noninvasive ventilation during endoscopic procedures: rationale, clinical use, and devices[END_REF][START_REF] Trouvé-Buisson | Anesthesiological support in a cardiac electrophysiology laboratory: a single-centre prospective observational study[END_REF][START_REF] Esteban | Noninvasive positive-pressure ventilation for respiratory failure after extubation[END_REF][START_REF] Auriant | Noninvasive ventilation reduces mortality in acute respiratory failure following lung resection[END_REF][START_REF] Squadrone | Continuous positive airway pressure for treatment of postoperative hypoxemia: a randomized controlled trial[END_REF][START_REF] Pearse | Postoperative continuous positive airway pressure to prevent pneumonia, reintubation, and death after major abdominal surgery (PRISM): a multicenter, open-label, randomized, phase 3 trial[END_REF][START_REF] Hui | Routine postoperative noninvasive respiratory support and pneumonia after elective surgery: a systematic review and meta-analysis of randomised trials[END_REF] per procedure in the NIV group (ESM eFigs. 8 and 9) and 3[START_REF] Brochard | Noninvasive ventilation for acute exacerbations of chronic obstructive pulmonary disease[END_REF][START_REF] Jaber | Effect of noninvasive ventilation on tracheal reintubation among patients with hypoxemic respiratory failure following abdominal surgery: a randomized clinical trial[END_REF][START_REF] Zarbock | Prophylactic nasal continuous positive airway pressure following cardiac surgery protects from postoperative pulmonary complications: a prospective, randomized, controlled trial in 500 patients[END_REF][START_REF] Peter | Effect of noninvasive positive pressure ventilation (NIPPV) on mortality in patients with acute cardiogenic pulmonary edema: a meta-analysis[END_REF][START_REF] Liu | Prophylactic noninvasive ventilation versus conventional care in patients after cardiac surgery[END_REF] per procedure in the standard face mask group (P \ 0.001, Mann-Whitney).

	group (73.4%; 95% CI, 63 to 82) (risk ratio [RR], 1.29; 95% CI, 1.13 to 1.47; P\0.0001; Chi
	square test) (Table 2).
	Secondary outcomes
	A hypoxemia event occurred in 40/96 patients in the NIV group (42%; 95% CI, 32 to 52) and
	in 33/96 patients in the standard face mask group (34%; 95% CI, 25 to 45) (RR, 1.21; 95%
	CI, 0.84 to 1.74; Chi square test, P = 0.33). In patients who presented at least one hypoxemia
	event, the median [IQR] number of episodes was 2 [1-2.5] per procedure in the NIV group
	(ESM eFigs. 6 and 7) and 2 [1-4] per procedure in the standard face mask group (P = 0.54,
	Mann-Whitney). Apnea/hypopnea occurred in 83 patients (92%; 95% CI, 85 to 97) in the
	NIV group and in 65 patients (69.9%; 95% CI, 59 to 79) in the standard face mask group (RR,
	1.32; 95% CI, 1.14 to 1.53; P \ 0.001,

Acknowledgements

The authors would like to thank l'école d'infirmier anesthésiste of the CHU Grenoble Alpes, the nurse anesthetists, the anesthetists, and the intensivists of the cardiovascular and thoracic unit. The authors would like to thank the neurosurgery anesthesia and intensive care unit teams for their involvement and independent assessment in the blinded analysis: Drs Perrine Boucheix, Lara Puthon, Marie-Cécile Fèvre, and Clotilde Schilte.

Funding statement

This study was supported by an institutional grant from CHU Grenoble Alpes. The foundation Agiradom provided masks and the respirator. Alpha2 (Lyon, France) provided the CO 2 analysis. Prior conference presentations The abstract of the article was accepted for presentation at the 2020 congress of the Société Française d'Anesthésie-Réanimation (24-26 Spetember 2020, Paris, France).

Editorial responsibility

This submission was handled by Dr. Philip M. Jones, Deputy Editor-in-Chief, Canadian Journal of Anesthesia/Journal Canadien d'Anesthésie.

Author contributions

Pierre-Henri Moury, Flora Greco, and Pierre Albaladejo designed the study; Pierre-Henri Moury, Valentin Pasquier, Flora Greco, Jean-Lionel Arvieux, Silvia Alves-Macedo, Marion Richard, and Myriam Casez-Brasseur enrolled the patients; Pierre-Henri Moury, Jean-Lionel Arvieux, Silvia Alves-Macedo, Damien Bedague, Michel Durand, and Marion Richard conducted the study; Kristina Skaare and Jean-Luc Bosson analyzed the data; Pierre-Henri Moury, Valentin Pasquier, Jean-Luc Bosson, and Pierre Albaladejo analyzed all the data; Pierre-Henri Moury and Valentin Pasquier interpreted the results; Pierre-Henri Moury and Valentin Pasquier drafted the manuscript; all authors had full access to all of the study data, contributed to drafting of the manuscript or critically revised it for important intellectual content, and took responsibility for the integrity of the data and the accuracy of the data analysis.

Disclosures

There are no conflicts of interest to disclose related to the subject of the article.


Othman Cherkaoui Dekkaki
email: othman.dekkaki@um5r.ac.ma

Walid Djema
email: walid.djema@inria.fr

Optimal control of a bioeconomic model applied to the recovery of household waste*

An improved mathematical model describing the process of generating energy from household waste treatment is proposed and analyzed. It is a three-dimensional nonlinear system that illustrates the process of transforming household waste stored in a landfill into energy that flows to a user's network. More precisely, the state of the system describes at a broad scale a process of generating energy E by treating a quota of a waste stock x through K-valorization units that may also consume a part of the produced energy for their operation. Our main objective is to maximize the energy produced and transmitted to the user's network. In particular, we investigate the issue of determining an optimal investing strategy that monitors the deployment of treatment plants. Using Pontryagin's maximum principle (PMP), we characterize, over a fixed time-frame [0, T ], the optimal investment that maximizes the produced energy while limiting the overall production costs. In addition, the efficiency of the suggested strategy is validated and illustrated throughout this work using a direct optimization method.

I. INTRODUCTION

Population growth leads to a significant increase in energy demand and generates a rapidly growing waste stream. Producing energy from the generated waste appears as a reasonable alternative to limit the systematic use of highly polluting fossil fuels to satisfy the worldwide demand [START_REF] Neri | Sustainability of a bio-waste treatment plant: Impact evolution resulting from technological improvements[END_REF]. However, in practice, many initiatives promoting the use of less polluting energies have a rather limited impact, mainly because their business models are not sufficiently competitive with conventional fossil fuels. Thus, knowing how to better evaluate these bioeconomic models and optimize their use for the production of less expensive alternative energy would be useful to improve their profitability (see, e.g., [START_REF] Escamilla-García | Technical and economic analysis of energy generation from waste incineration in Mexico[END_REF], [START_REF] Mohammadi | Performance analysis of waste-to-energy technologies for sustainable energy generation in integrated supply chains[END_REF] and references therein).

Although waste-to-bioenergy conversion has clear environmental advantages, the current production levels are far too low to meet global energy needs. However, waste treatment projects are still gaining more investment due to the rising cost of storing untreated waste caused by increasingly stringent environmental policies (see, e.g., [START_REF] Ahring | Perspectives for anaerobic digestion[END_REF]). Thus, the need to better control, improve, and optimize the waste transformation process -while making it more economically profitable-is growing.

The concept of household waste treatment involves the retrieval and utilization of energy that can be directly or indirectly produced from waste. The energy recovered or extracted is primarily in the form of heat, biofuels, or electricity. In the literature, we can distinguish two primary types of energy recovery1 : 1-by thermal treatment: incineration or co-incineration, gasification, and pyrolysis (i.e., heating of biomass in the absence of oxygen) [START_REF] Lombardi | A review of technologies and performances of thermal treatment systems for energy recovery from waste[END_REF]. 2-Recovery of biogas naturally released from household waste storage facilities and from industrial anaerobic digesters [START_REF] Ahring | Perspectives for anaerobic digestion[END_REF].

Specifically, the incineration process with energy recovery involves converting the heat released by the combustion of stored waste into pressurized steam. The steam is then expanded in a turbo-generator to produce electricity, which is used directly to supply the heating network or at least locally to supply neighboring industries (Fnade reports). On the other hand, biogas from organic waste fermentation processes in landfills or industrial anaerobic digesters can be used either as electricity, heat, or biofuel. This biogas is mainly used to run vehicles that consume natural gas, and can even supply the natural gas network if the produced quantities are sufficiently large. In the literature, there is a broad scientific community focusing on the diverse technical recovery processes (by combustion, anaerobic digestion, etc., see e.g., [START_REF] Ascher | Machine learning methods for modelling the gasification and pyrolysis of biomass and waste[END_REF], [START_REF] Chuka-Ogwude | Developing food waste biorefinery: using optimized inclined thin layer pond to overcome constraints of microalgal biomass production on food waste digestate[END_REF], [START_REF] Ozgun | Anaerobic Digestion Model No. 1 (ADM1) for mathematical modeling of full-scale sludge digester performance in a municipal wastewater treatment plant[END_REF], [START_REF] Sialve | Anaerobic digestion of microalgae as a necessary step to make microalgal biodiesel sustainable[END_REF]). Generally, the developed models are limited to the technical study of the biological and physical phenomena describing waste transformation processes, involving sets of biochemical reactions (for mechanistic models, but see also, e.g., [START_REF] Ascher | Machine learning methods for modelling the gasification and pyrolysis of biomass and waste[END_REF] for alternative machine learning-based approaches), without considering the economic viability of the overall transformation processes.

For deterministic mechanistic biotechnological models, control theory offers strong tools for investigating fundamental system properties, including identification of bioprocess models, sensitivity and stability analysis, stabilization, and control, etc. ( [START_REF] Dochain | Automatic control of bioprocesses[END_REF], [START_REF] Harmand | Microbial ecology and bioprocess control: Opportunities and challenges[END_REF], [START_REF]Control theory and systems biology[END_REF], [START_REF] Murray | Mathematical biology: I. An introduction[END_REF]), with a large diversity of applications, such as bioreactor control [START_REF] Mairet | Hybrid control of a bioreactor with quantized measurements[END_REF], anaerobic digestion [START_REF] Nguyen | Automatic process control in anaerobic digestion technology: A critical review[END_REF], wastewater treatment [START_REF] Iratni | Advances in control technologies for wastewater treatment processes: status, challenges, and perspectives[END_REF], gene expression, and cancer [START_REF] Agrawal | Mathematical modeling of RNA-based architectures for closed loop control of gene expression[END_REF], [START_REF] Djema | Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues[END_REF]. In addition, many studies have highlighted the crucial need for optimization in biological and biotechnological systems ( [START_REF] Alford | Bioprocess control: Advances and challenges[END_REF], [START_REF] Bernard | Hurdles and challenges for modelling and control of microalgae for CO2 mitigation and biofuel production[END_REF], [START_REF] Wang | Industrial bioprocess control and optimization in the context of systems biotechnology[END_REF]). In this paper, we focus on a generic model depicting a recovery of household waste, with an emphasis on its profitability through the transformation chain: from waste storage, to treatment by a specific process, until energy production and utilization (Fig. 1). The generic model (which is an improved version of the one introduced in [START_REF] Cherkaoui Dekkaki | Bioeconomic modeling of household waste recovery[END_REF]; see Remark 1 in Sect. II) is associated with an optimal control problem (OCP) that aims to maximize the produced energy while minimizing the costs of waste treatment/storage and limiting the investment in the mobilized treatment units. The resulting OCP is investigated using the Pontryagin's maximum principle (PMP; [START_REF] Pontryagin | Mathematical Theory of Optimal Processes[END_REF]). A direct optimization method is also performed to derive and illustrate the optimal control strategies. Notice that optimal control theory, and particularly Pontryagin's principle ( [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF], [START_REF] Harmand | Optimal Control in Bioprocesses: Pontryagin's Maximum Principle in Practice[END_REF], [START_REF] Kamien | Dynamic optimization: the calculus of variations and optimal control in economics and management. 2nd Edition[END_REF], [START_REF] Pontryagin | Mathematical Theory of Optimal Processes[END_REF], [START_REF] Vinter | Optimal Control, Systems and Control: Foundations and Applications[END_REF]), have proven their great efficiency to address similar dynamic optimization problems in the biotechnological field: see, e.g., biogas production [START_REF] Haddon | Optimal and Sub-optimal Feedback Controls for Biogas Production[END_REF], microbial strains selection [START_REF] Djema | Optimal Darwinian Selection of Microorganisms with Internal Storage[END_REF], [START_REF] Djema | Turnpike features in optimal selection of species represented by quota models[END_REF], microbial metabolite production [START_REF] Caillau | Turnpike property in optimal microbial metabolite production[END_REF]; and it is also widely used in the context of economic and bioeconomic optimization problems: [START_REF] Aseev | The Pontryagin maximum principle and optimal economic growth problems[END_REF], [START_REF] Clark | Mathematical Bioeconomics: The Mathematics of Conservation, 3rd Edition[END_REF], [START_REF] Kamien | Dynamic optimization: the calculus of variations and optimal control in economics and management. 2nd Edition[END_REF], [START_REF] Reed | Optimal harvesting of a fishery subject to random catastrophic collapse[END_REF], [START_REF] Shell | Applications of Pontryagin's maximum principle to economics[END_REF].

This paper is organized as follows. In Sect. II, the bioeconomic model of interest is introduced. In Sect. III, the OCP maximizing the energy produced from waste is formulated. Then, in Sect. IV, the PMP is applied to the studied OCP in order to determine necessary optimality conditions for the optimal controls and trajectories. A complementary direct optimization approach is also performed in Sect. V. Next, in Sect. VI, a specific case in which a part of the produced energy is used to run the treatment units is introduced and analyzed. The obtained results are also validated and illustrated through a direct optimization method. Finally, a concluding discussion is outlined in Sect. VII.

II. THE MATHEMATICAL MODEL

In the process representation provided in Fig. 1, we can distinguish three main compartments:

i) The (constant) waste stream w ≥ 0 flows directly into the landfill. A stock of waste x(t) is thus constituted by accumulation. It may undergo partial loss due to natural bio-degradation with time at a rate β ∈]0, 1]. A quota q ∈ [0, q max ] of the available waste stock x is treated in K-processing units, i.e., a given amount q(t)x(t) undergoes at time t ≥ 0 a waste treatment process (e.g., incineration, see Sect. I), to recover energy, inside K-processing units, thus resulting in K(t)q(t)x(t) consumption of x(t) at time t ≥ 0.

ii) The treatment units K can be equivalently interpreted either as some physical units (incineration furnace, digesters, etc.), or as a capital invested in physical units. In both cases, K depreciates with respect to time at a constant rate γ ∈ ]0, 1], while it grows with the amount of investment I(t) ≥ 0 supporting the valorization activity at time t ≥ 0.

iii) Energy is generated from the waste poured in the Kprocessing units at a constant conversion rate waste-to-energy µ, and it depreciates over time at a constant rate α ∈]0, 1].

Therefore, based on the previous description, we end up with the following dynamics,

       ẋ(t) = ω -(β + K(t)q(t))x(t), K(t) = I(t) -γK(t), Ė(t) = µK(t)x(t)q(t) -αE(t).
(1)

Remark 1: The system ( 1) is an upgraded version of the model studied in [START_REF] Cherkaoui Dekkaki | Bioeconomic modeling of household waste recovery[END_REF]. Firstly, we note that in [START_REF] Cherkaoui Dekkaki | Bioeconomic modeling of household waste recovery[END_REF] the x-dynamics does not depend on the variable K (the coupling between states in [START_REF] Cherkaoui Dekkaki | Bioeconomic modeling of household waste recovery[END_REF] occurs only in the dynamics of E). Moreover, the model ( 1) has an extra-parameter µ quantifying the rate of conversion waste-to-energy, and we also adjust the free waste uptake-rate (the term Kqx) that depends now on K. Furthermore, in Sect. VI, we extend the model ( 1) to a typical case where a part of the energy is redeployed to run the K-units, by introducing an extra term α K in the E-dynamics, leading to the system [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF]. Thus, the OCP in Sect. III will also be adapted accordingly.

III. STATEMENT OF THE OPTIMAL CONTROL PROBLEM

Let T > 0 be a fixed finite-time. In practice, T is rather large since it is more interesting to study long-term behavior of investment2 . Moreover, the instantaneous yield energy E(t) is supposed to be sold at a given constant unit price denoted p. Taking into account a given actualization rate δ, a constant unit cost of production c, and a quadratic cost of investment (as in [START_REF] Cherkaoui Dekkaki | Bioeconomic modeling of household waste recovery[END_REF], [START_REF] Moser | A non-autonomous optimal control model of renewable energy production under the aspect of fluctuating supply and learning by doing[END_REF]), the optimal control problem (OCP) of interest reads as follows. OCP: find admissible controls (I, q) maximizing,

J := T 0 e -δt J (t)dt, (2)
where,

J (t) = pE(t) -cK(t)x(t)q(t) -I(t)(c 1 + c 2 I(t)), (3)
over a finite fixed-time horizon [0, T ]. The set of admissible controls is defined by,

I = {I : [0, T ] → [0, I max ] meas., I(•) ∈ [0, I max ] a.e.} Q = {q : [0, T ] → [0, q max ] meas., q(•) ∈ [0, q max ] a.e.}
(meas. stands for measurable) where I max > 0 is the maximum possible amount of instantaneous investment, and q max > 0 is the maximum waste recovery amount. For later use, we put U = (I, Q). In fact, OCP can be interpreted as follows. The investor's issue consists of choosing the best control strategy (I, q) in order to maximize over [0, T ] the net income generated by the sale of the energy produced by the activity (pE), while limiting the costs cKxq+I(c 1 +c 2 I).

Note that similar criteria can be found in [START_REF] Moser | A non-autonomous optimal control model of renewable energy production under the aspect of fluctuating supply and learning by doing[END_REF], [START_REF] Reed | Optimal harvesting of a fishery subject to random catastrophic collapse[END_REF].

IV. PMP APPLICATION Let H be the current-value Hamiltonian (as formulated in [START_REF] Kamien | Dynamic optimization: the calculus of variations and optimal control in economics and management. 2nd Edition[END_REF], Chap. 8) corresponding to the studied OCP, using the system's dynamics and the criterion (2), i.e.,

H = pE -cKxq -I(c 1 + c 2 I) +λ 1 (ω -(β + qK)x) +λ 2 (I -γK) + λ 3 (µxKq -αE) , (4)
or, equivalently,

H = h(X, λ) + hKxq + h † (I), (5)
where,

• h(X, λ) = pE + λ 1 (w -βx) -λ 2 γK -λ 3 αE, • h = -c -λ 1 + µλ 3 , • h † (I) = -I(c 1 + c 2 I) + λ 2 I
, and λ = (λ 1 , λ 2 , λ 3 ) is the so-called pseudo-covector [START_REF] Kamien | Dynamic optimization: the calculus of variations and optimal control in economics and management. 2nd Edition[END_REF].

A. The pseudo-costates and transversality conditions

According to [START_REF] Kamien | Dynamic optimization: the calculus of variations and optimal control in economics and management. 2nd Edition[END_REF], the pseudo-costates λ 1 , λ 2 , and λ 3 are slightly different from the classical formulation of the costates λ x , λ K and λ E , since they satisfy,

     λ1 = δλ 1 -∂H ∂x = (δ + β)λ 1 -hKq, λ2 = δλ 2 -∂H ∂K = (δ + γ)λ 2 -hxq, λ3 = δλ 3 -∂H ∂E = (δ + α)λ 3 -p. (6)
In addition, since in the studied OCP the final-state is free, the transversality conditions in this case are given by,

λ i (T ) = 0, ∀i = 1, 2, 3. (7)

B. PMP maximization condition

The PMP aims to determine admissible controls satisfying, (I(t), q(t)) ∈ argmax

I∈I, I(t)∈[0,Imax]; q∈Q, q(t)∈[0,qmax] H, (8)
for almost all t ∈ [0, T ]. Using (5), one can readily deduce that the maximization condition (8) leads to the following result.

Proposition 1: For almost all t ∈ [0, T ], where T is the fixed final-time, we get, i) The optimal control I * (t) satisfies,

I * (t) =    0, if λ 2 (t) ≤ c 1 , min λ2(t)-c1 2c2 , I max , if λ 2 (t) > c 1 . (9)
ii) The optimal control q * (t) satisfies,

q * (t) =        0, if h < 0, q max , if h > 0, q s (t), if h ≡ 0, over [t 1 , t 2 ], t 1 < t 2 , (10)
where we recall that, h = -c -λ 1 + µλ 3 .

1) About the single arcs in the optimal quota of recovered waste: From ( 10), it appears that a singular arc q s may occur if the associated switching functions h vanishes over a time interval. However, this situation is excluded under certain conditions as discussed in the sequel. So, let us assume that over a time interval I = [t 1 , t 2 ], s.t. t 1 < t 2 , the singular arc q s (t) occurs, i.e.,

h(t) = 0 ⇐⇒ -c-λ 1 (t)+µλ 3 (t) = 0, ∀t ∈ [t 1 , t 2 ]. (11)
Thus, if h ≡ 0 over I, then its first derivative also satisfies, ḣ(t) = 0, i.e., -λ1 (t) + µ λ3 (t) = 0. Through a process of successive derivation, we get for all n ≥ 1

h(n) = -(δ + β) n λ 1 + µ(δ + α) n λ 3 -µ(δ + α) n-1 p. ( 12
)
One notices that h(n) does not involve the control q. In the general case, we can prove that the optimal control q * cannot have a singular phase if c ̸ = µp/(δ + α) or α ̸ = β. In the following sections, using direct optimization methods, we show that singular arcs does not appear in the optimal control structure of q.

2) Characterization of the final-time: By combining the transversality conditions [START_REF] Betts | Practical Methods for Optimal Control and Estimation Using Nonlinear Programming[END_REF] with the PMP-maximization conditions expressed in ( 9)-( 10), we can prove that the optimal controls q * and I * will vanish when the time t approaches the final time T . In other words, both controls end with a bang-0 arcs over [T -ε, T ], ε > 0. This statement is also illustrated in the next section (see Figs. 5-6 and 8-9).

V. DIRECT OPTIMIZATION

A direct-optimization approach ( [START_REF] Betts | Practical Methods for Optimal Control and Estimation Using Nonlinear Programming[END_REF]) is performed in order to solve the studied OCP. The numerical direct methods that we use are implemented in Julia (using the JuMP interface and Ipopt software library, as performed in [START_REF] Caillau | An algorithmic guide for finite-dimensional optimal control problems[END_REF]). Direct methods transform the OCP into a nonlinear programming problem (NLP) in finite-dimension, through a discretization process of the controls and the state variables. Thus, in this section, the states and the controls of the studied model [START_REF] Agrawal | Mathematical modeling of RNA-based architectures for closed loop control of gene expression[END_REF] are discretized in the JuMP framework following a Crank-Nicolson scheme. The main settings used in JuMP in all the performed examples are given in Table I. Example A. We consider the model parameters and the weighting constants of the criterion (2)-(3): p, c, c 1 and c 2 , listed in Table II. In this example, the final-time is fixed to T = 30. The initial condition is given by, (x 0 , K 0 , E 0 ) = (1, 3, 0). The optimal controls are given in Figs. 56, and their associated optimal trajectories are illustrated in Fig. 2. The costate trajectories, and then the resulting pseudocovector are given respectively in Figs. 3 and4. We note that the pseudo costates λ i , i = 1, 2, 3 satisfy the transversality

w 12 β 0.2 α 0.1 γ 0.2 δ 0.2 µ 0.6 p 1 c 0.4 c 1 0.1 c 2 0.1 T (final time) 30
α K 0 (Ex. A) 0.5 (Ex. B)
conditions stated in [START_REF] Betts | Practical Methods for Optimal Control and Estimation Using Nonlinear Programming[END_REF]. We also highlight that the behaviors derived from the PMP maximization conditions, summarized in Proposition 1 (Sect. IV), are verified in this example, as illustrated in Figs. 56.

Fig. 2. The optimal trajectories x(t), K(t) and E(t), associated with the optimal controls given in Figs. 56.

Furthermore, let us highlight that the optimal controls in Figs. 56verify the feature discussed in Sect. IV, since we observe that the last phases in the structure of both controls are bang-0 arcs. Next, we can notice that the control I * in Fig. 6 exhibits a turnpike-like behavior (see, e.g., [START_REF] Caillau | Turnpike property in optimal microbial metabolite production[END_REF], [START_REF] Djema | Turnpike features in optimal selection of species represented by quota models[END_REF], [START_REF] Trélat | The turnpike property in finitedimensional nonlinear optimal control[END_REF]), since I * (t) stays most of the time (approximately between t = 5 and t = 25) close to a static value I * (t) = I ∈ (2; 3). In addition, we note that the optimal control q * given in Fig. 5 is most of the time at its maximum value, i.e., an arc bang-q max for all t ∈ [0, T -ε], which represents the most profitable phase for generating energy. In this case, the K-treatment units process the maximum possible amount of waste at time t. Then, when time t approaches the final-time T , a bang-0 phase occurs. A direct consequence is that the Fig. 3. The costate trajectories derived from JuMP. Using these costates, we can readily deduce the pseudo-costates defined in [START_REF] Bernard | Hurdles and challenges for modelling and control of microalgae for CO2 mitigation and biofuel production[END_REF]. Fig. 4. Optimal pseudo costate trajectories (λ 1 , λ 2 , λ 3 ) derived from the current-value Hamiltonian as defined in [START_REF] Bernard | Hurdles and challenges for modelling and control of microalgae for CO2 mitigation and biofuel production[END_REF]. These trajectories are reconstituted using the optimal costates (λx, λ K , λ E ) in Fig. 3.

produced energy E(t) in Fig. 2 shows a stable growth during [0, T -ε]. It is also worth noting that the optimal trajectories of x and K given in Fig. 2 have a clear turnpike behavior.

VI. STRATEGIC REDEPLOYMENT PLAN FOR THE PRODUCED ENERGY

Now, we highlight the case where the processing units K consume a part of the produced energy E. A simple way to include this feature in the model ( 1) can be achieved by adding an extra-parameter α K > 0 in the E-dynamics, Fig. 5. The obtained optimal control q * (t) satisfies the necessary optimality conditions derived from the PMP in Sect. IV. In particular, q * (t) is bangqmax when h(t) is positive, while q * (t) is bang-0 when h(t) is negative. We also notice that when t approaches T , the optimal control is a bang-0 as stated in Sect. IV-B.2. Fig. 6. The optimal control I * (t) (in blue) satisfies the necessary optimality conditions derived from the PMP, formulated in Proposition 1 in Sect. IV. As for q * , the last phase of the optimal control I * is a bang-0.

leading to the following dynamic system,

     ẋ(t) = ω -(β + K(t)q(t))x(t), K(t) = I(t) -γK(t), Ė(t) = µK(t)q(t)x(t) -αE(t) -α K K(t). ( 13
)
The objective is the same as in the previous OCP, i.e., maximizing the criterion (2)-(3) under similar considerations.

Example B. Let us consider the model parameters of System (13) with the weighting constants of the criterion p, c, c 1 and c 2 , given in Tab. II, where the final-time is fixed to T = 30. In this example, the initial conditions of the states is given by, (x 0 , K 0 , E 0 ) = (1, 3, 0). We consider similar JuMP settings as in Example A. (Tab. I). Note that Example A. can be recovered in the case α K = 0 in [START_REF] Clarke | Functional Analysis, Calculus of Variations and Optimal Control[END_REF].

1) Numerical results in Example B. (α K > 0): The obtained optimal q * in Fig. ( 9) has a bang-bang structure, similar to the one in Example A. in Fig. 5. However, the optimal control I * in Example B. (Fig. 8) is different from the previous case (Fig. 6). Indeed, in Example B., the control I * has three distinct phases over [0, T ]: a bang-0 (which does not exist in Example A.), followed by a turnpike phase where I * (t) is close to a steady-state I ∈]0, I max [ (note Fig. 7. The optimal trajectories x(t), K(t) and E(t) in Example B., associated with the optimal controls in Fig. 89. however that the value of I has changed). Finally, a bang-0 occurs over [T -ε, T ]. One notices that for the same T the produced energy achieved in Example B. is lower than the one in Example A. This follows naturally from the fact that a part of the produced energy is consumed by the K units. It is also worth mentioning that in Example B, even if it is not clearly displayed in Fig. 7, the trajectory of E is slightly negative at the beginning of the process. A negative E reflects the fact that the K-units are consuming energy (taken from the external network). The investment I * is also lower at steady state, which reduces the processing capacity (K has a turnpike around 10 in Example A., while it is around 2 in Example B.).

VII. CONCLUSION

In this work, we introduced and studied a generic OCP that investigates the issue of cost-effectiveness of waste-to-energy processes, with the objective of maximizing the energy, while minimizing overall costs (including waste storage fees and investment charges). The issue of energy redeployment to run the treatment units has also been considered. A PMPbased study, as well as a direct optimization, were performed to fully-characterize the optimal investment-control strategy over a fixed time horizon. Future work will focus on the case of perturbed waste flow-input (ω in System (13)), in order better represent a more realistic situation where a waste supply is not perfectly constant but rather time-varying.

Fig. 1 .

 1 Fig. 1. Illustration of a generic household waste treatment process (see Sect. II for the definition of the model variables).

Fig. 8 .

 8 Fig. 8. The optimal control I * (t) obtained in Example B.

Fig. 9 .

 9 Fig. 9. The optimal control q * (t) obtained in Example B.

TABLE I DISCRETIZATION

 I SCHEME AND JuMP SETTINGS.

	Discretization method Crank-Nicolson
	Time steps	4000 (Ex. A, B)
	NLP tolerance	10 -14 (Ex. A, B)

Adapted from Fnade reports: French National Federation of Depollution and Environmental Activities.

T specifically designates the end of the limited-term agreement between the investor and the legal authority managing the landfill.

ACKNOWLEDGMENT

The authors are grateful to Nadia Raïssi and Noha El Khattabi (Mohammed V University in Rabat), as well as Jean-Luc Gouzé (Inria, University Côte d'Azur).

*This work was partially supported by BIOMSA project, funded by ADEME (France). It has been initiated during the visit of Othman Cherkaoui Dekkaki to the BIOCORE project-team, at