S Baffert

S Alran

V Fourchotte
email: virginie.fourchotte@curie.fr

M A Traore

C Simondi
email: cecile.simondi@curie.fr

P Mathevet
email: patrice.mathevet@chuv.ch

C Loustalot

MD P F Dupre

G Ferron

Laparoscopic hysterectomy after concurrent radiochemotherapy in locally advanced cervical cancer compared to laparotomy: a multi institutional prospective pilot study of cost, surgical outcome and quality of life

Keywords: Advanced cervical cancer, cost comparison, laparotomy, laparoscopy, quality of life, laparoscopic hysterectomy

Laparoscopic hysterectomy after concurrent radiochemotherapy in locally advanced cervical cancer

compared to laparotomy: a multi institutional prospective pilot study of cost, surgical outcome and quality of life

Introduction

Cervical cancer is the second most frequent cancer in women, with approximately 500,000 new cases diagnosed, and 270,000 deaths annually worldwide [START_REF] Parkin | Chapter 2: The burden of HPV-related cancers[END_REF][START_REF] Munoz | Chapter 1: HPV in the etiology of human cancer[END_REF] . Although the incidence of cervical cancer has decreased in industrialized countries because of screening programs and progress in management of intraepithelial lesions, 60% of cases are at advanced stages at diagnosis. In France, the estimation of new cases in 2015 is 3,060, with the highest incidence among women in their 40s, leading to nearly 1,070 deaths and a 5-year survival rate of 17.2% in advanced stages [START_REF]Les cancers en France en 2013. Collection état des lieux et des connaissances, ouvrage collectif édité par l'INCa[END_REF][START_REF]Données épidémiologiques sur le cancer du col de l'utérus -Etat des connaissances -Actualisation 2008 -InVS[END_REF] .

The gold standard for treating patients with locally advanced cervical cancer (LACC) is concurrent chemoradiotherapy with or without brachytherapy (CRT) [START_REF] Mcneil | New standard of care for cervical cancer sets stage for next questions[END_REF][START_REF] Whitney | Randomized comparison of fluorouracil plus cisplatin versus hydroxyurea as an adjunct to radiation therapy in stage IIB-IVA carcinoma of the cervix with negative para-aortic lymph nodes: a Gynecologic Oncology Group and Southwest Oncology Group study[END_REF] . Significant survival advantages of chemoradiation in cases of LACC, have been demonstrated in a meta-analysis [START_REF] Green | Concomitant chemotherapy and radiation therapy for cancer of the uterine cervix[END_REF] .

The value of completion surgery after CRT in LACC however, still remains debatable [START_REF] Chereau | The role of completion surgery after concurrent radiochemotherapy in locally advanced stages IB2-IIB cervical cancer[END_REF][START_REF] Ferrandina | Completion surgery after concomitant chemoradiation in locally advanced cervical cancer: a comprehensive analysis of pattern of postoperative complications[END_REF][10][11] , and few studies are available [12][13] . Furthermore, hysterectomy after CRT remains a questionable treatment option, in particular in cases of partial response. Results from multicenter studies have demonstrated that residual disease after concurrent chemoradiation therapy and brachytherapy impact on disease free survival [14][15][16] . Because the accuracy of imaging techniques are not sufficient to measure residual disease [17][18] , surgery remains the current practice in many countries. Furthermore, completion surgery reduces residual pathological disease, which represents an important prognostic factor [START_REF] Fanfani | Scambia Pre-hysterectomy cone biopsy is able to predict response in locally advanced cervical cancer patients submitted to neo-adjuvant chemoradiation[END_REF][START_REF] Ferrandina | Long-term analysis of clinical outcome and complications in locally advanced cervical cancer patients administered concomitant chemoradiation followed by radical surgery[END_REF][START_REF] Jurado | Pilot study of concurrent cisplatin, 5fluorouracil, and external beam radiotherapy prior to radical surgery þ/_ intraoperative electron beam radiotherapy in locally advanced cervical cancer[END_REF][START_REF] Ferrandina | Preoperative concomitant chemoradiotherapy in locally advanced cervical cancer: safety, outcome, and prognostic measures[END_REF][START_REF] Distefano | Preoperative chemoradiotherapy in locally advanced cervical cancer: long term outcome and complications[END_REF] .

We previously assessed the consequences of hysterectomy by laparotomy after CRT and brachytherapy, and showed a high rate of grade 2/3 morbidity (26%), particularly due to urinary complications 14 . The feasibility and consequences of laparoscopic hysterectomy after RCT for LACC have not been sufficiently assessed. In a retrospective series of 102 patients, Colombo et al [START_REF] Colombo | Total laparoscopic radical hysterectomy for locally advanced cervical carcinoma (stages IIB, IIA and bulky stages IB) after concurrent chemoradiation therapy: surgical morbidity and oncological results[END_REF] . studied 56 laparoscopic hysterectomies over a period of 8 years. The

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question arises as to whether this intervention improves the quality of life of these patients, and the efficiency cost and surgical outcome. A comparison of laparoscopy to laparotomy in terms of surgical outcome, cost and quality of life has not been prospectively assessed in the context of surgery after CRT in LACC. The treatment of cervical cancer is expensive and is estimated to total 44 million Euros annually in France, corresponding to a mean patient cost of €22,697 for stage III to €26,886 for stage V disease [START_REF] Arveux | Invasive cervical cancer treatment costs in France[END_REF] .

Our study was aimed at assessing the benefit of laparoscopy, in terms of cost, surgical outcome and quality of life.

Materials and methods

Study design and patient details

Between December 2008 and November 2011, 62 consecutive patients (39 laparoscopy and 23 laparotomy) from 13 French institutions were included in a prospective multicenter comparative observational non-randomized study.

Inclusion criteria were invasive cervical cancer proven by a core biopsy before treatment, stage IB2, and IIA, IIB (proximal), M0, preoperative external platinum based radiochemotherapy, +/-utero-vaginal brachytherapy, and extrafascial hysterectomy (+/lymphadenectomy, pelvic and latero aortic), via laparoscopy or laparotomy, with the feasibility of a one year follow-up. The choice of the surgical approach was at the discretion of the surgeon. Each surgeon used one of the two techniques. Surgeons trained in laparoscopy performed laparoscopic extrafascial hysterectomy, plus lymphadenectomy, whilst surgeons less trained in laparoscopy performed theses procedure by laparotomy. Observational study is more suitable to capture current practice in a real-world situation.

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Hysterectomy was proposed in cases when tumor residual at the end of the treatment was suspected.

Initial staging was defined according to the International Federation of Gynecology and Obstetrics (FIGO) staging system. Staging was performed using a clinical pelvic examination, Magnetic Resonance Imaging (MRI), and Computed Tomography (CT).

Treatment:

All patients received radiochemotherapy. Concomitant Cisplatin (CDDP) was given on the first day of each week of radiotherapy. Laparoscopic surgical staging of lymph node involvement, pelvic and/or aortic, was initially undertaken to set the field of external radiotherapy in some teams, in others it was a radiological stadification. Radiation therapy was administered at 1.8 Gy in 22 to 25 fractions according to international recommendations 26 . Surgery consisted of an extrafascial hysterectomy. Patients undergoing a laparoscopy were positioned in the Trendelenburg position, and a 0°-laparoscope (10-mm umbilicus trocar) and three 5-mm trocars (left and right iliac fossa and upper pubic region) were inserted. The abdominal pressure was maintained at 12 mm Hg. Laparotomies were performed using a Pfannenstiel transverse incision or a midline incision.

Post-operative follow-up occurred from surgery to one month later, and included the hospital stay, and a first post-operative visit.

Studied parameters

Studied Economic data included the treatment modality, resources consumed, including mean personnel time, conversion to laparotomy and its causes, hospitalization, complications, and annual follow-up. We used the cancer QLQ-C30 version 3.0 from EORTC which is a quality of life instrument for use in international clinical trial in oncology and the EORTC QLQ-CX24 module which is dedicated to patients with cervical cancer and validated by the European Organization for research and Treatment of cancer. Quality of life was evaluated using a patient self-completed survey sent by post at four time periods, before surgery (T0), one week after surgery at the first visit (T1), and one (T2) and six months post-surgery (T3)) [START_REF] Aaronson | The European Organization for Research and Treatment of Cancer QLQ-C30: a quality-of-life instrument for use in international clinical trials in oncology[END_REF][START_REF] Greimel | The European Organization for Research and Treatment of Cancer (EORTC) Quality-of-Life Questionnaire Cervical Cancer Module. EORTC QLQ-CX24[END_REF] . Questionnaires were sent by prepaid envelopes to the Institut Curie (within 8 days with respect to T1, T2, or T3).

This study contained no modifications of standard practice in each institution, and informed consent was not required. It was approved by the regional ethical committee (Authorization n° 908075).

Economic assessment

We conducted our analysis from the hospital provider's perspective. The direct costs associated with surgical strategies (laparoscopy or laparotomy) were taken into account in a prospective manner, from the medical visit prior to surgery up to the first month after surgery using unit costs. Costs of complications during hospitalization, costs of re-intervention for complications, and costs associated with longer hospitalization were also considered. The time period covered, 30 +/-5 days from surgery, allowed all important and relevant consequences and costs between two strategies to be measured and compared. Cost computation focused on inpatient follow-up care and the rehabilitation unit. The hospital provider perspectives included hospital stay in the Medicine, Surgery, Obstetrics (MSO) unit

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and follow-up care in the rehabilitation unit. In France, hospital care settings include conventional hospital in charge of an MSO, post-acute care and convalescence homing.

Hospital provider perspectives include the hospital care setting in a global care pathway.

Cost calculations were made with the micro-costing method obtained from detailed observations, for all patients included in the study and quantities of consumable resources.

The following consumable resources, linked to surgery, are counted and quantified for each

Statistical analysis

Cost comparisons were measured using a student's t-test, a Mann-Whitney test or an analysis of variance (ANOVA), and are reported as the mean -/+ standard deviation. Tests for normality were carried out using a Kolmogorov-Smirnov test. Socio-demographic characteristics, clinical information and all categorical variables were compared using a Chisquare test or a Fisher's exact test. All tests were two-sided with a significant level of 5%.

Data was analyzed using the SAS system software (version 9.2, SAS Institute Inc.).

Results

A flow chart of the study population is shown in Figure 1. Patient characteristics are presented in Table 1. The two groups were not statistically different except for cancer staging (FIGO)

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and previous surgical histories. Patients in the laparotomy group had a more locally advanced disease and more previous abdominal surgery. Pathological results are presented in Table 2.

Preoperative treatment

Twenty-two patients in the laparotomy group (95.6%) and 38 patients in the laparoscopy group (95%) underwent brachytherapy. Completion radiotherapy after surgery was carried out for only one patient in each group and was related to residual lymph node disease.

Complications related to pretreatment were the same in the two groups (7/23 (30.4%) in laparatomy vs 12/39 (30.7%) in laparoscopy, p=1.00).

Surgery

Surgical characteristics are summarized in Table 3. Intraoperative transfusion was significantly lower in the laparoscopy group (p=0.04). Similarly, abdominal drain was significantly lower in the laparoscopy group compared to the laparotomy group (78.3% vs. 7.5%, p<0.001).

Urinary catheterization did not differ between the two groups (23/23 vs 38/39, p=1.00) like duration of urinary catheterization (4.5 ± 1.4 d vs 3.9 ± 1.9 d, p=0.07). The time between the end of prior therapy and the surgery is on average within 6 to 8 weeks.

Although the length of induction of anesthesia as well as the length of incision was significantly higher in the laparoscopy group (respectively, 37.6 min vs 29.4 min, p= 0.03 and 168 min vs 210 min, p=0.04), the overall mean operative time did not differ significantly between the two groups (258 min vs. 294 min, p=0.06) (Table 3). In the laparoscopy group, three patients (3/39, 7.5%) had to be switched to the laparotomy group due to technical requirements, and all three had a previous history of abdominal surgery and brachytherapy before hysterectomy.

Intraoperative and postoperative complications are detailed in

Cost at one-month

Resources used for each phase of the procedures are presented in The extra costs of three conversions reached €18,157 ( =€11,944) per converted patient (min=11,131, max=31,949). The difference in cost between the two procedures was not significant, even if the 3 cases of conversion were not included in the analysis (p=0.96).

Quality of life up to 6 months

Assessment of quality of life using EORTC QLQ-C30 at one week after surgery was better for patients having undergone laparoscopy compared to laparotomy. At one month, global health and quality of life, physical functioning and role functioning were better in the laparoscopy group (respectively p=0.01, p=0.01, p=0.05). At six months, the measurable

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benefits remained for patients who underwent laparoscopy, in particular they experienced less fatigue than patients who underwent laparotomy (p=0.04). Using the specific cervical cancer questionnaire, CX24, only sexual activity was significantly better at six months for the laparoscopy group (p=0.01).

Discussion

Our prospective multicenter study compared the feasibility of laparoscopic versus laparotomic hysterectomy after CRT in LACC. Whilst these procedures had equal hospital costs at one month, the quality of life was considerably better for the laparoscopy patients.

Medical outcome and quality of life

Few studies have assessed the clinical outcomes and complications of hysterectomy after CRT in LACC. Colombo PE et al. [START_REF] Colombo | Total laparoscopic radical hysterectomy for locally advanced cervical carcinoma (stages IIB, IIA and bulky stages IB) after concurrent chemoradiation therapy: surgical morbidity and oncological results[END_REF] , included 56 laparoscopy patients after RCC over a 8 year period, and Chereau E et al. [START_REF] Chereau | The role of completion surgery after concurrent radiochemotherapy in locally advanced stages IB2-IIB cervical cancer[END_REF] , studied 42 laparoscopy patients over a 10 year period.

Regarding complications, our study revealed no major intraoperative, early postoperative, or late postoperative complications within 30 days in the two groups. In the retrospective cohort, of Colombo PE et al [START_REF] Colombo | Total laparoscopic radical hysterectomy for locally advanced cervical carcinoma (stages IIB, IIA and bulky stages IB) after concurrent chemoradiation therapy: surgical morbidity and oncological results[END_REF] , morbidity rates and urinary complications were significantly reduced in the laparoscopy group compared to the laparotomy group for radical hysterectomy and not simple extrafascial hysterectomy. Other recent published studies showed total laparoscopic radical surgery is feasible in patients with LACC receiving preoperative CT/RT. [START_REF] Gallotta | Laparoscopic Radical Hysterectomy After Concomitant Chemoradiation in Locally Advanced Cervical Cancer: A Prospective Phase II Study[END_REF][START_REF] Gallotta | Minilaparoscopic nerve sparing radical hysterectomy in locally advanced cervical cancer after neoadjuvant radiochemotherapy[END_REF] As expected, from a patient perspective, hysterectomy by laparoscopy after concurrent radiochemotherapy with or without brachytherapy seems to result in better quality of life compared to open surgery. We measured the quality of life for a cohort of 62 patients over a six-month follow-up period. Radical pelvic surgery via laparotomy and chemoradiation are associated with a significant impairment of sexual function in cervical cancer patients [START_REF] Park | Quality of Life and Sexual Problems in Disease-Free Survivors of Cervical Cancer Compared with the General Population[END_REF][START_REF] Mantegna | Long-term prospective longitudinal evaluation of emotional distress and quality of life in cervical cancer patients who remained disease-free 2-years from diagnosis[END_REF] . We

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observed a clinically relevant improvement in the overall quality of life and sexual function scores which was significantly better at six months for the laparoscopy group. This information can be used to guide medical decision making, and highlights that surgical approaches should always be tailored to minimize the negative impact of surgery.

Cost evaluation

Using the microcosting method, our prospective series demonstrated that laparoscopy hospital Existing publications report heterogeneous data for the type of surgery, types of cost components and inclusion criteria. No costs studies using the microcosting method with a one month follow up after surgery have been previously reported. Dennis et al. [START_REF] Dennis | Study of surplus cost of robotic assistance for radical hysterectomy, versus laparotomy and standard laparoscopy[END_REF] found that the cost for radical hysterectomy was highest for robotic, followed by standard laparoscopy, and lowest for laparotomy. In our study, only surgical and anesthetic instrumentations have been included in the cost calculation; the cost calculation did not include staff costs, hospital stay, follow-up, or resources pertaining to medical data aquisition.

Wright et al. [START_REF] Wright | Comparative effectiveness of minimally invasive and abdominal radical hysterectomy for cervical cancer[END_REF] found that both laparoscopic and robotic radical hysterectomies were associated with lower transfusion requirements and shorter hospital stays than abdominal hysterectomy (p<0.05). However, they did not use a prospective and observational design, and did not report direct per-patient costs. In their study, costs were estimated using a national database, which is less powerful for comparing population sub-groups or for matching economic data to clinical outcomes at the individual patient level. Observational studies

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provide accurate and detailed information on health care consumption at an individual patient level. Significantly, such data allow modeling studies of healthcare costs to be refined, and are critical in guiding decision-making with regard to healthcare resource allocation.

The length of stay for patients undergoing laparoscopy in our cohort was longer than the average length reported in the literature for other countries, such as in USA for example. Bell et al. 36 in 2008 reported an average length of stay of 2 +/-1.2 days for laparoscopy in the USA, while Lachance et al. 37 reported a 4 day stay for patients undergoing hysterectomy in the USA. Wright et al. [START_REF] Wright | Comparative effectiveness of minimally invasive and abdominal radical hysterectomy for cervical cancer[END_REF] reported a median length of stay of 3 days for abdominal radical hysterectomy, and 2 days for laparoscopic surgery in the USA. This could be explained by the differences in national health care and reimbursement systems. In the past few years, trends in France are to reduce the length of stay. In France, Colombo PE et al. [START_REF] Colombo | Total laparoscopic radical hysterectomy for locally advanced cervical carcinoma (stages IIB, IIA and bulky stages IB) after concurrent chemoradiation therapy: surgical morbidity and oncological results[END_REF] , reported a 5 day hospital stay after laparoscopy versus 8 days for laparotomy.

Once the relevant range of costs has been identified, the individual items must be measured and valued. The level of accuracy of cost studies is determined by the identification of cost components, (gross costing and/or microcosting) and valuation of cost components (top-down and/or bottom-up costing). In the microcosting approach, all relevant cost components are defined at the most detailed level and in the bottom-up approach. Cost components are valued by identifying resources used directly for a patient, resulting in patient specific unit costs. 36 Our study used the combination of microcosting and the bottom-up costing approach, which is generally believed to be the gold standard methodology for the costing of healthcare services.

This reporting care pathway with a one-month follow-up including transitional care such as rehabilitation makes our study original and relevant. Rehabilitation care represents an important potential benefit for minimally invasive surgery as it represents a large and relevant

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cost component. Our analysis was conducted from the hospital provider perspective, including inpatient hospitalization in MSO and stays in follow-up and rehabilitation care.

One limitation of our study is the small sample size, which is common to many similar studies in this specific field. Ferrandina et al. [START_REF] Ferrandina | Long-term analysis of clinical outcome and complications in locally advanced cervical cancer patients administered concomitant chemoradiation followed by radical surgery[END_REF] included 174 patients over a ten year period, Colombo et al. [START_REF] Colombo | Total laparoscopic radical hysterectomy for locally advanced cervical carcinoma (stages IIB, IIA and bulky stages IB) after concurrent chemoradiation therapy: surgical morbidity and oncological results[END_REF] , 102 patients over 8 years (including 56 laparoscopies), and Chereau et al. [START_REF] Chereau | The role of completion surgery after concurrent radiochemotherapy in locally advanced stages IB2-IIB cervical cancer[END_REF] 80 patients over 10 years (including 42 laparoscopies). Another limitation is that our study was nonrandomized. However, even if randomized studies represent the standard practice in clinical research, our observational study describes current treatment in representative centers. These issues had to be raised in our conclusion. Compared to previously published studies using retrospective data in single institutions or databases, our study is prospective, consecutive and multi institutional. We provide more comprehensive and accurate individual and primary data per patient, in a setting that is relevant to current treatment protocols.

Laparoscopic radical hysterectomy after radiochemotherapy for the treatment of LACC is feasible, results in lower intraoperative transfusion and abdominal drain interventions, a shorter hospital stay, less convalescent time, and results in a better quality of life with similar costs at one-month compared to laparotomy. For these reasons, when hysterectomy is indicated for the treatment of LACC after chemoradiation and brachytherapy, the laparoscopic approach must be the first choice.

  , tumor size) and complications according to the Chassagne glossary[START_REF] Chassagne | A glossary for reporting complications of treatment in gynecological cancers[END_REF] .

  patient, from economic items integrated into the case report form, and per center through the center survey as specific devices and annual activities. Direct costs include different surgery techniques, pathology requirements, supplies, depreciation for equipment, hospital stay, medical visits and surgical costs. Depreciation of the laparoscopic video System is included in the direct costs, and is proportional to the time of use in surgery. Depreciation costs were calculated based on a five-year straight-line depreciation. Costs are presented in Euros in 2010. A sensitivity analysis was performed to assess the potential effects of uncertainty inherent in the study.

  re-operation for post-operative intraperitoneal abscesses and post-operative bowel obstruction secondary to adhesion. There were no post-operative deaths within 30 days after surgery in either group.

  costs were not significantly different at one month compared to the open procedure. The longer length of operating time, the costly single-use consumables, equipment and materials used for laparoscopy and the extra cost of conversion (failure of laparoscopy) are offset by a shorter length of stay and a reduction in the use of a convalescence home following hospitalization.

Table 4

 4

. Complications leading to re-hospitalization and/or re-intervention are presented in Table

5

. Two patients in the

Table 5

 5 , and detailed costs in Table6. The direct cost of each procedure was not statistically different at one month:

	€10,991 ( =3616) for laparotomy versus €11,267 ( =4237) for laparoscopy, p=0.76.

Hospital stays for patients were significantly lower in the laparoscopy group (8.3 d vs 6.7 d, p<0.001). Post-hospital stay in a convalescence home was required for 17.4% (4/23) of the patients after laparotomy (21, 23, 24 and 30 days respectively), but was not required for any patients in the laparoscopy group who return home after surgery. The one-month follow-up step includes the additional cost convalescence for the laparotomy group, while all patients in laparoscopy group have hospital discharge at home. The one-month follow-up step results in significantly lower costs for the laparoscopic group compared to the laparotomy group (€929 vs. €1,739, p=0.05).

Table 1 :

 1 Characteristics of patientsCategorical data N(%) statistical test is only on documented data. Non parametric tests: Wilcoxon or exact test

	10. Sun L, Sheng X, Jiang J, et al. Surgical morbidity and oncologic results after concurrent
	chemoradiation therapy for advanced cervical cancer. Int J Gynecologic Obstet. 2014
	May;125(2):111-5.
	11. Favero G, Pierobon J, Genta ML, et al. Laparoscopic extrafascial hysterectomy (completion
	surgery) after primary chemoradiation in patients with locally advanced cervical cancer: technical
	aspects and operative outcomes. Int J Gynecol Cancer. 2014 Mar;24(3):608-
	12. Houvenaeghel G, Lelievre L, Gonzague-Casabianca L, et al. Long-term survival after concomitant
	chemoradiotherapy prior to surgery in advanced cervical carcinoma.Gynecol Oncol. 2006
	Feb;100(2):338-43.
	13. Delpech Y, Haie-Meder C, Rey A, et al. Para-aortic involvement and interest of para-aortic
	lymphadenectomy after chemoradiation therapy in patients with stage IB2 and II cervical
	carcinoma radiologically confined to the pelvic cavity. Ann Surg Oncol. 2007 Nov;14(11):3223-
	31
	14. Classe JM, Rauch P, Rodier JF et al. Surgery after concurrent chemoradiotherapy and
	brachytherapy for the treatment of advanced cervical cancer: morbidity and outcome: results of a
	multicenter study of the GCCLCC (Groupe des Chirurgiens de Centre de Lutte Contre le
	Cancer).Gynecol Oncol. 2006;102(3):523-9.
	15. Hequet D, Marchand E, Place V, Fourchotte V, De La Rochefordière A, Dridi S, Coutant C,
	Lecuru F, Bats AS, Koskas M, Bretel JJ, Bricou A, Delpech Y, Barranger E. Evaluation and
	impact of residual disease in locally advanced cervical cancer after concurrent chemoradiation
	therapy: results of a multicenter study. Eur J Surg Oncol. 2013 Dec;39(12):1428-34.
	16. Poujade O, Morice P, Rouzier R, et al. Pathologic response rate after concomitant neo-adjuvant
	radiotherapy and chemotherapy for adenocarcinoma of the uterine cervix: a retrospective
	multicentric study. Int J Gynecol Cancer. 2010 Jul;20(5):815-20
	17. Ferrandina G, Petrillo M, Restaino G, et al. Can radicality of surgery be safely modulated on the
	basis of MRI and PET/CT imaging in locally advanced cervical cancer patients administered
	preoperative treatment? Cancer. 2012;118:392-403.

Table 2 :

 2 Pathology resultsCategorical data N(%) statistical test is only on documented data. Non parametric tests: Wilcoxon or exact test

		Laparotomy group	Laparoscopic group	P value
		(n=23)	(n=39)
	Lymphovascular invasion	2 (8.7%)	5 (12.8%)	1.0000
	Free margins	21 (91.3%)	38 (97.4%)	1.0000
	Missing	1	0
	No. of patients > 1 Positive pelvic	17	25	0.3160
	lymph node involvement
	No. of patients > 1 Positive aortic	3	3	1.0000
	lymph node involvement
	Preoperative complications N (%)	7 (30)	12 (30)	1,0000
	Type of complications N (%)			0.84
	chemotherapy	3 (43)	3 (25)
	radiotherapy	1 (14)	2 (17)
	surgery		3 (25)
	others	2 (29)	1 (8)
	chemotherapy and radiotherapy	1 (14)	2 (17)
	chemotherapy and surgery		1 (8)

Table 3 :

 3 Surgical characteristics

	Laparotomy group Laparoscopic group P value

Categorical data n(%) statistical test is only on documented data. Non parametric tests: Wilcoxon or exact test *No echec pneumoperitoneum

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O Bernard

B Chachuat
email: benoit.chachuat@epfl.ch

J.-Ph Steyer
email: steyer@ensam.inra.fr

STATE ESTIMATION FOR WASTEWATER TREATMENT PROCESSES

In this chapter, we provide the key ideas on how to build software sensors (also called observers) for wastewater treatment plants (WWTPs). We give an overview of the existing linear and nonlinear observers and discuss criteria that help to identify which observer is best suited with respect to the amount of information being available for the WWTP. Depending on the model reliability, the available measurements and the level of uncertainties associated to the influent concentrations, different class of observers can be considered. We distinguish between those that rely on a full model description (e.g., the 1 2 STATE ESTIMATION FOR WASTEWATER TREATMENT PROCESSES extended Kalman filter), and those based on a mass-balance model wherein the biological kinetics are assimilated to unknown inputs (e.g., the asymptotic observer). Moreover, if bounds are known for the uncertainties, then interval observers can be designed. We discuss the principles of each class of observers and illustrate them through a number of examples.

INTRODUCTION

A major bottleneck in the application of advanced monitoring and optimization strategies for WWTPs lies in the difficulty of measuring chemical and biological variables. Even though considerable progress in on-line sensors technology has been made over the recent years, it is still often difficult to measure many of the key state variables such as biomass concentration, specific bacterial activity or intermediate products concentration. Hence, the idea of using observers, also called software sensors, that combine a number of readily available on-line measurements (gaseous flow rates, pH, dissolved gases, etc.) with a process model for estimating the values of (unmeasured) state variables.

Quite of few methods have been proposed to design such observers. In this chapter, we shall only focus on those approaches that are relevant to the field of WWTPs. It is worthwhile noting that the principles underlying the design of observers can be quite different. Therefore, the choice of an observer inherently depends on the specificities of the problem at hand. In practice, this choice is strongly guided by the reliability of the process model as well as the amount and accuracy of the data. If a reliable process model is available and if this model has been thoroughly identified and validated, either an (extended) Kalman filter or a high gain observer can be developed. When the process model is not accurate enough, an asymptotic observer relying on mass-balance principles, but not on the uncertain kinetics, shall be used instead. Finally, If bounds are known for the uncertain inputs and/or parameters, an interval observer can be used for predicting intervals in which the unmeasured variable are guaranteed to belong (instead of pointwise estimates).

The type of observer to be constructed should not only be based on the model quality, but it must also account for the objectives to be achieved. Indeed, an observer can have other purposes than simply monitoring a WWTP. It can be developed with the objective of applying a control action that needs an estimate of some internal state; it can also be used for diagnosing whether a failure occurred during process operation or not.

The remainder of this chapter is organized as follows. A number of useful definitions and results are given in section 1.2. Observers that require a full-model description of the process are presented in section 1.3, with emphasis placed on linear systems. The design of observers relying on the mass-balance principles is discussed in section 1.4. Interval observers that exploit knowledge of bounds on the model uncertainty are presented in section 1.5. Finally, section 1.6 concludes the chapter.

PRELIMINARIES

This section gives an overview of the main theoretical concepts in system observability.

These concepts are useful in the analysis conducted later on. The interested reader is referred to [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF][START_REF] Luenberger | Introduction to Dynamic Systems: Theory, Models and Applications[END_REF] for additional information.

It is first fundamental to study the observability property of a system prior to designing an observer. Intuitively, observability consists of determining whether the measured signals contain sufficiently rich information to estimate the unmeasured state variables; a system is then said to be observable if it satisfies this property from a theoretical point of view. The problem to address next is to derive an observer for the problem at hand, i.e. an auxiliary dynamic system that provides the state estimates. At this point, it should be noted that the problems of observability and observer design are very different in nature. In particular, the observability property does not give any clue on how to build an observer.

The theory of observation has been extensively developed in the linear case. Several methods also exist in the nonlinear case, but are tailored to specific classes of models.

Notion of Observability

We consider the following general model driving the process dynamics:

     dx(t) dt = f x(t), u(t) ; x(0) = x 0 y(t) = h x(t) (S)
where u ∈ IR m is the input vector; y ∈ IR p , the output vector; x ∈ IR n , the state vector made-up of the concentrations of the various species inside the liquid phase; and x 0 , the vector of initial conditions. The applications f and h provide the dynamics of the state variables and the links between the state variables and the measurements, respectively.

The objective is to estimate x(t) from the measurements y(t). Observability is a structural property of a system that states whether this is possible or not. The reader is referred to [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF][START_REF] Luenberger | Introduction to Dynamic Systems: Theory, Models and Applications[END_REF] for more details.

General Definition of an Observer

Once a system has been shown to be observable, the next step is to design an observer that estimates the state variable x based on a model and a set of input/output measurements.

The principle of an observer is presented in Fig. 1.1. Roughly speaking, an observer is an auxiliary dynamic system coupled to the original system via the measured inputs and outputs. This is formalized in the following definition. Definition 1 An observer is an auxiliary system (O) coupled to the original system (S) as:

     dz(t) dt = f z(t), u(t), y(t) ; z(0) = z 0 x(t) = ĥ z(t), u(t), y(t) (O)
where z ∈ IR q denotes the state of the observer; f is the observer dynamics; and ĥ relates z to the estimate x of the real system. An observer has the property that the observation error converges to zero asymptotically:

lim t→∞ x(t) -x(t) = 0.
A desirable property for an observer is the ability to tune the convergence rate in order for the estimates to converge more rapidly than the original dynamics of the system. Another desirable property is that the estimate x(t) should remain equal to x(t) under proper initialization, i.e. when it is initialized with the true value x(0). This easily justifies that following structure is often used to design observers in practice:

     dx(t) dt = f (x(t), u(t)) + k[z(t), h(x(t)) -y(t)] dz(t) dt = f (z(t), u(t), y(t)) with k(z(t), 0) = 0.
This observer consists of a replica of the original dynamics corrected by a term that depends on the discrepancy between both the measured and predicted outputs. Note also that the correction amplitude is tuned via the function k that is often referred to as the observer gain (internal tuning of the observer).

OBSERVERS FOR LINEAR SYSTEMS

For time-invariant, linear systems, the general system (S) simplifies to

     dx(t) dt = Ax(t) + Bu(t) y(t) = Cx(t) (S )
with A ∈ IR n×n (n ≥ 2) and C ∈ IR p×n . A well known observability criterion for (S ) is

given by the rank condition:

rank          C CA . . . CA n-1          = n.

Luenberger Observer

Theorem 1 If the pair (A, C) is observable, a Luenberger observer for (S ) is obtained as

[16] dx(t) dt = Ax(t) + Bu(t) + K(C x(t) -y(t))
where K is a n × n gain matrix that can be used for tuning the convergence rate of the observer, and can be chosen in order for the observation error to converge to zero arbitrarily fast.

Proof. The dynamics of the observation error e(t) = x(t) -x(t) is given by

de dt = (A + KC)e,
and is independent of the input u(t). The result follows from the pole placement theorem which guarantees that the error dynamics can be chosen arbitrarily.

In principle, the gain matrix K can be chosen in such a way that the observation error converges to zero as quickly as desired. However, the larger the gain of the observer, the more sensitive it becomes to external perturbations (measurement noise for example). A good compromise must thus be sought that ensures both stability and accuracy at the same time. The Kalman filter, discussed in subsection 1.3.3 below, proposes a way of achieving such a compromise.

The Linear Case up to an Output Injection

A particular situation wherein a linear observer can be designed for a nonlinear system arises in the simple case where the nonlinearities depends on the output y only:

     dx(t) dt = Ax(t) + φ(t, y(t)) + Bu(t) y(t) = Cx(t)
with φ being a (known) nonlinear function in IR n . The following "Luenberger-like" observer has linear dynamics with respect to the observation error:

dx(t) dt = Ax(t) + φ(t, y(t)) + Bu(t) + K(C x(t) -y(t)).
In particular, the error dynamics can be chosen arbitrarily, provided that the pair (A, C) is observable. Here again, however, an adequate choice for the gain vector K is one that guarantees a fast enough convergence of the observer, while keeping it stable.

Kalman Filter

The Kalman filter is notorious in the field of linear systems (see e.g. [START_REF] Lewis | Optimal Estimation (with an Introduction to Stochastic Control Theory[END_REF]). Loosely speaking, a Kalman filter can be seen as a Luenberger observer with a time varying gain.

More specifically, the gain is chosen in such a way that the variance of the observation error is minimized (or, equivalently, the integral between t 0 and t of the squared errors is minimized); for this reason the Kalman filter is often referred to as the optimal estimator.

Consider an observable continuous-time system in the following stochastic representation:

dx(t) dt = Ax(t) + Bu(t) + Gw(t) ; x(t 0 ) = x 0 (1.1)
where w(t) ∼ (0, Q(t)) is a white noise process with zero mean and covariance Q(t).

Suppose that initial state x 0 is unknown, but there is available a priori knowledge that x 0 ∼ (x 0 , P 0 ). Suppose also that measurements are given at discrete times t k according to

y k = Cx(t k ) + v k (1.2)
where v k ∼ (0, R k ) is uncorrelated with w(t) and x 0 .

Besides initialization, a continuous/discrete Kalman filter for system (1.1,1.2) consists of two steps: a propagation step (between two successive measurements), followed by a correction step (at measurement times):

Initialization (t = t + 0 ): P (t 0 ) = P 0 , x(t 0 ) =¯x 0 Propagation (t + k-1 ≤ t ≤ t - k , k ≥ 1): dP (t) dt = AP (t) + P (t)A T + GQ(t)G d x (t) dt = Ax(t) + Bu(t) Correction (t = t + k , k ≥ 1): K k = P (t - k )C T CP (t - k )C T + R k -1 P (t + k ) = (I -K k (t)) P (t - k ) x(t + k ) = x(t - k ) + K k z k -Cx(t - k )
At this point, we shall emphasize on several points. Note first that the foregoing Kalman filter can be applied to time-varying linear system, i.e. with matrices A, B, C and G depending on time. One should however keep in mind that observability must be proven for such systems prior to constructing the observer. Note also that Kalman filters can be extended by adding a term -θP (t), θ > 0, in the propagation equation of P . This exponential forgetting factor allows us to consider the case where Q = 0. Finally, estimating the positive definite matrices R, Q and P 0 often proves to be tricky in practice, especially when the noise properties are not known precisely.

The Extended Kalman Filter

Consider a continuous-time nonlinear system of the form

dx(t) dt = f (x(t)) + G(t)w(t) ; x(t 0 ) = x 0 , (1.3)
with measurements at discrete time t k given by

y k = h(x(t k )) + v k . (1.4)
The idea behind Extended Kalman Filter (EKF) is to linearize the nonlinear system (1.3,1.4) around its current state estimate x(t) [START_REF] Lewis | Optimal Estimation (with an Introduction to Stochastic Control Theory[END_REF]. By doing so, the problem becomes equivalent to building a Kalman filter for a non-stationary linear system (1.1,1.2) with A and C taken as

A(t) = ∂f ∂x x(t) , C(t k ) = ∂h ∂x x(t k ) .
The EKF is used routinely and successfully in many practical applications, including WWTPs, even though few theoretical guarantees can be given as regards its convergence [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF][START_REF] Lewis | Optimal Estimation (with an Introduction to Stochastic Control Theory[END_REF]. Note also that multirate versions of the EKF have been developed to handle those (rather frequent) situations where measurements are available at different samplings rates [START_REF] Gudi | Adaptive multirate state and parameter estimation strategies with application to a bioreactor[END_REF]. An application of EKF in alternating activated sludge WWTPs is detailed hereafter.

Other applications to the activated sludge process can be found, e.g. in [START_REF] Lukasse | A recursively identified model for short-term predictions of NH4/NO3 concentrations in alternating activated sludge processes[END_REF][START_REF] Zhao | State and parameter estimation for phosphorus removal in an alternating activated sludge process[END_REF].

Application to an Alternating-Activated-Sludge Plant

We consider an alternating-activated-sludge (AAS) WWTP similar to the one shown in A multirate EKF is developed based on the reduced nonlinear model given in [START_REF] Chachuat | Reduction of the ASM1 model for optimal control of small-size activated sludge treatment plants[END_REF]. This 5-state model describes the dynamics of COD, NO, NH, organic nitrogen (ND) and DO, and was shown to be observable under both aerobic and anoxic conditions (with the aforementioned measurements).

Numerical simulations have been performed by using a set of synthetic data produced from the full ASM1 model [START_REF] Henze | Activated Sludge Model No. 1[END_REF] corrupted with white noise. The DO, NO and NH measurements are assumed to be available every 10 sec., 10 min. These results show satisfactory performance of the EKF for COD estimation. However, it should be noted that the COD estimates are very sensitive to model-parameter mismatch, which is hardly compatible with the fact that some parameters are time-varying and/or badly known in real applications. This motivates the development of mass-balance-based observer that are independent of the uncertain kinetic terms.

OBSERVERS FOR MASS-BALANCE-BASED SYSTEMS

The underlying structure of many WWTP models consists of two parts [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF]: (i) a linear part based on mass-balance considerations; and (ii) a number of nonlinear term that describes the biological reaction rates (kinetics). These latter kinetic terms are often poorly known in practice, and there is little hope to construct a reliable observer by accounting for such uncertain terms. In contrast to the previous section wherein a full-model structure was used in the observer design, we shall show, in this section, how to take advantage of the foregoing twofold structure to design observers that are independent of the uncertain reaction rates, via a linear change of variables. As we shall see, the main condition to design a so-called asymptotic observer is that enough variables are measured. Note also asymptotic observers fall into the broad class of observers with unknown inputs [START_REF] Darouach | Full-order observers for linear systems with unknown inputs[END_REF][START_REF] Hou | Design of observers for linear systems with unknown inputs[END_REF][START_REF] Kudva | Observers for linear systems with unknown inputs[END_REF], whose principle relies on cancellation of the unknown part via a change of variables.

Preliminaries

Throughout this section, we consider mass-balance models for WWTPs of the following form [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF]:

dx dt = Kr(x) -D(t)x + D(t)x in (t) -Q(x) (1.5)
where x, x in ∈ IR n represent the concentrations in the reactor and the influent, respectively; D ∈ IR n×n , the dilution rate matrix; Q(x) ∈ IR n , the gaseous exchange between the reaction medium and the environment; r(x) ∈ IR p , the reaction rates; and K ∈ IR n×p , a constant pseudo-stoichiometric coefficient matrix. In this representation, K r(x) stands for the biological and biochemical conversions in the reactor (per unit of time) according to the underlying macroscopic reaction network.

Suppose that the set of available measurements y corresponds to is partitionned into y 1

and y 2 such that:

• y 1 is a set of q ≤ n measured state variables; without loss of generality, we assume that y 1 corresponds to the first q components of x, y 1 = [x 1 , . . . , x q ] T .

• y 2 consists of the measured gaseous flow rates, y 2 = [q 1 (x), . . . , q n (x)] T .

dx 1 dt = K 1 r(x) -Dx 1 + Dx in 1 -Q 1 (x) dx 2 dt = K 2 r(x) -Dx 2 + Dx in 2 -Q 2 (x) (1.6)
where matrices K 1 and K 2 , vectors x in 1 , x in 2 , q 1 and q 2 are such that

K =    K 1 K 2    , x in =    x in 1 x in 2    , q =    q 1 q 2    .

Asymptotic Observers

Constructing an asymptotic observers for system (1.5) requires the following two technical assumptions to hold.

Assumption 1 (i)

There are more measured quantities than reactions, i.e. q ≥ p; (ii)

Matrix K 1 has full rank.

These assumptions guarantee that a non-zero r cannot cancel the term K 1 r and the q × p matrix K 1 has a left inverse. Accordingly, there exists a p × q matrix G such that:

GK 1 = I p .
Let us denote A = -K 2 G and M = (A I n-p ). An observer for subsystem (1.6) can be obtained as indicated in the following lemma.

Lemma 1 (see [START_REF] Bastin | On-line Estimation and Adaptive Control of Bioreactors[END_REF]) If D is positive definite, the solution x2 of the auxiliary system

d ζ2 dt = -D( ζ2 -M x in ) -M y 2 x2 = ζ2 -Ay 1
converges to the solution x 2 of subsystem (1.6), asymptotically.

Application to an Anaerobic Digester

We consider a very simple model of the anaerobic digestion process that accounts for a single "global" degradation step of the soluble COD (S t ) by the biomass X t [START_REF] Andrews | A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrate[END_REF]:

k t S t µt(.)Xt -→ X t + k m CH 4
where k t and k m are the yield coefficients associated to COD degradation and methane production, respectively; µ t (•) stands for bacterial growth rate. The corresponding massbalance model reads:

dS t dt = -k t µ t (•)X t -D(S t -S in ) (1.7) dX t dt = µ t (•)X t -αDX t q CH 4 = k m µ t (•)X t (1.8)
where q CH4 stands for the methane outflow rate; D, the dilution rate; and α, the fraction of bacteria not attached onto a support (i.e., being affected by the dilution rate D in the reactor).

The objective is to design an asymptotic observer for the biomass X t , based on COD measurements, without knowing the reaction rates µ t (•). In this case, we have

x 1 = y 1 = S t , x 2 = X t and Q(x) = 0. Moreover, K 1 = -k t and K 2 = 1, hence a possible choice for matrices A and G is: G = -1/k t , A = 1/k t .
Finally, lemma 1 provides the following auxiliary differential system:

d ζ2 dt = -αD( ζ2 - S in αk t ) -(1 -α)D y 1 k t (1.9) Xt = ζ2 - y 1 k t . (1.10)
Eqs. (1.9)-(1.10) is an observer for the biomass concentration in the digester. However, this observer may provide poor estimates in practice, because of the large errors made on S in and k t . A more appropriate solution is then to use an interval observer for coping with uncertainty, as discussed in the following section.

INTERVAL OBSERVERS

Usual observers rely on the implicit assumption that the process model is a good approximation of the real plant. Nevertheless, we have seen that WWTP models are often corrupted.

In such situations where large modeling and measurement errors prevail, one can no longer construct an exact observer (i.e., with the guarantee that the observation error converges to zero asymptotically and that the convergence rate can be tuned). Instead, the observation principle must be weakened. In this section, we explain how to derive rigorous bounds enclosing the estimated states by accounting for the uncertainty in the process model [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF][START_REF] Rapaport | Practical observers for uncertain affine outputs injection systems[END_REF].

Principle

Interval observers require knowledge of time-varying bounds enclosing the uncertainty,

v -(t) ≤ v(t) ≤ v + (t)
. These bounds are used to calculate time-varying bounds enclosing the state variable to be estimated.

Consider the following general system:

     dx(t) dt = f (x(t), u(t), w(t)) ; x(t 0 ) = x 0 y(t) = h(x(t), v(t)) (S I )
where known lower and upper bounds are available for the uncertain quantities w ∈ IR r and v ∈ IR s ,

w -(t) ≤ w(t) ≤ w + (t) ∀ t ≥ t 0 v -(t) ≤ v(t) ≤ v + (t) ∀ t ≥ t 0 .
Based on the fixed model structure (S I ) and on the set of measured data, an auxiliary dynamic system (O I ) can be designed such that it provides a lower bound and an upper bound for the state variables:

                   dz - dt = f -(z -, z + , u, y, w -, w + , v -, v + ) ; z -(t 0 ) = g -(x - 0 , x + 0 ) dz + dt = f + (z -, z + , u, y, w -, w + , v -, v + ) ; z + (t 0 ) = g + (x - 0 , x + 0 ) x -= h -(z -, z + , u, y, w -, w + , v -, v + ) x + = h + (z -, z + , u, y, w -, w + , v -, v + ). (O I )
Definition 2 (interval estimator) System (O I ) is said to be an interval estimator of system (S I ) if for any pair of initial conditions x -(t 0 ) ≤ x(t 0 ) ≤ x + (t 0 ), there exists bounds z -(t 0 ), z + (t 0 ) such that the solutions of the coupled system (S I ,O I ) verifies:

x -(t) ≤ x(t) ≤ x + (t) ∀ t ≥ t 0 .
Interval estimators result from the coupling of two estimators which provide both an underestimate x -(t) and an overestimate x + (t) of x(t). Of course, such bounds can be very large, thus making interval observers practically useless in some cases. However, for particular classes of systems (e.g., linear systems up to an output injection), theoretical guarantees can be given that the time-varying intervals [x -(t), x + (t)] converge to a "limit" interval of finite magnitude [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF]. Moreover, the convergence rate towards this limit interval can be tuned if certain properties hold [START_REF] Rapaport | Parallelotopic and practical observers for non-linear uncertain systems[END_REF]. Note that these ideas find their origin in the theory of positive systems [START_REF] Smith | Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems[END_REF]. More recently, probabilistic observers have been formulated for a class of uncertain biological processes [START_REF] Chachuat | Probabilistic observers for a class of uncertain biological processes[END_REF]; these observers take advantage of the knowledge of probability density functions (PDFs) for the uncertain parameters to calculate the PDFs of the unmeasured sate variables.

An application of interval observers to an anaerobic WWTP is presented next; another application to an activated sludge process can be found in [START_REF] Hadj-Sadok | Estimation of uncertain models of activated sludge process with interval observers[END_REF].

Application to an Anaerobic Digester

We consider the same reduced model of an anaerobic digestion plant as in subsection 1.4.3.

The objective is to design an interval observer that estimates the COD concentration S t from on-line methane measurements, with µ t (•) and α being unknown.

Using Eq. (1.8) of the methane flow rate, Eq. (1.7) can be rephrased as:

dS t dt = -γq CH 4 -D(S t -S in t )
where the uncertain parameter γ = k t /k m is such that γ -≤ γ ≤ γ + ; the inlet COD concen- Finally, an interval observer is derived from Eq. (1.11) as

     dS - t dt = φ (S - t ) -γq CH 4 -D(S - t -S in ) + λ(S in -S - t ) q CH 4 -qCH 4 + (S - t ) dS + t dt = φ (S + t ) -γq CH4 -D(S + t -S in ) + λ(S in -S + t ) q CH4 -qCH4 -(S + t )
where the function φ (S - t ) = S - t /(S - t + ) (with 0 < inf S t (t)) is used to enforce S t ≥ 0; and the bounds q-CH4 and q+ CH4 on qCH 4 are calculated by considering the bounds µ - t and µ + t on µ t . This observer is applied to a real process in the next paragraph.

Application to Real Measurements

The interval observer given in Eq. (1.11) was implemented on a pilot-scale fixed bed up-flow anaerobic digester used for wine wastewater processing. Both the dilution rate D and methane outflow rate q CH 4 were measured on-line at a high frequency. Besides on-line measurements, a COD sensor was also used to validate the observer predictions. Details on the plant configuration and the experiments can be found in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF][START_REF] Steyer | Evaluation of a four year experience with a fully instrumented anaerobic digestion process[END_REF]. The predictions of the interval observer are presented in Fig. 1.4., together with the off-line COD measurements.

CONCLUSIONS

In this chapter, we have presented a number of methods to design observers for WWTPs.

Of course, other techniques exist and we did not pretend to be exhaustive. The observers presented herein assumed constant parameter values in the models. In some cases however, the parameters can evolve during process operation. Algorithms to estimate the parameters must then be used, hence leading to adaptive observers.

For bioprocesses and WWTPs in particular, the type of observer to be considered must be chosen by considering the reliability of the process model at hand, as well as the available measurements. Moreover, designing an observer always results from a trade-off between its robustness towards modeling uncertainties and disturbances, and its convergence rate.

Finally, the implementation of an observer requires a discretization to be performed as regards the continuous-time equations, e.g. an Euler type algorithm. Although not difficult, discretization must be performed carefully. In the case of low measurement frequency, for example, the use of a continuous/discrete observer shall be preferred to full discretization of a continuous-time observer.

To conclude, it is worth insisting on the fact that an observer should always be validated prior to using it in a real treatment plant. In particular, systematic and extensive comparisons should be made between the observer predictions and direct on-site measurements (other than those used to calibrate the observer).

Figure 1 . 1 .

 11 Figure 1.1. Observer principle

Fig. 1 . 2 .Figure 1 . 2 .

 1212 Fig. 1.2. AAS plants degrade both organic and nitrogenous compounds by alternating aerobic and anoxic phases in the bioreactor. Besides dissolved oxygen (DO) concentration that is routinely measured in activated sludge WWTPs, both nitrate (NO) and ammonia (NH)concentrations can also be measured on-line (at a lower frequency than DO, though). The objective here is to estimate the concentration of COD in the bioreactor based on these measurements.

Figure 1 . 3 .

 13 Figure 1.3. Estimated COD concentration (left plot) and observation error (right plot).

Figure 1 . 4 .

 14 Figure 1.4. Methane flow rate (left plot) and interval observer (--) for soluble COD (right plot).

  In practice, it was found that the upper bound S + t (t) is weak when large uncertainties are considered for γ and S in t (t). These considerations motivate the following improvements.

	where Xt =	S in t -St αkt . Then, these estimates are used in the following robust observer [3],
		dS t dt	= -γq CH 4 -D(S t -S in t ) + λ(q CH 4 -qCH 4 (S t )).	(1.11)
	Property 2 The auxiliary dynamic system
	     = -γ Improvements The idea for reducing prediction intervals consists of using structured dS -t = -γ + q CH 4 -D(S -t -S in t -) dt dS + t dt
	kinetic models for µ(•), despite uncertainty. We suppose here that the process does not
	operate in a region where inhibition phenomena occur, and use a Monod kinetic model for
	obtaining an estimate of q CH4 as
			qCH 4 (S t ) = k m µ t	S t S t + K St	Xt

tration S in t (t) fluctuates between known bounds as S in t -(t) ≤ S in t (t) ≤ S in t + (t), ∀t ≥ t 0 ; and the initial COD concentration is bounded as S - t (t 0 ) ≤ S t (t 0 ) ≤ S + t (t 0 ). -q CH 4 -D(S + t -S in t + ) is an interval observer for S t , i.e. guarantees that S - t (t) ≤ S t (t) ≤ S + t (t) at each t ≥ t 0 . Proof. It is easily verified that both lower bound e -= S t -S - t and upper bound e + = S + t -S t on the observation error remain positive. (See [3] for details.)