PharmD Fabienne Venet

PhD Orchidée Filipe-Santos

MD Alain Lepape

PharmD Christophe Malcus

PharmD Françoise Poitevin-Later

MSc Audrey Grives

MSc Nadia Plantier

PhD Nicolas Pasqual

PharmD Guillaume Monneret
email: guillaume.monneret@chu-lyon.fr

S

evere sepsis and septic shock are common indications for admission to ICUs and affect millions of individuals around the world each year [START_REF] Hotchkiss | The pathophysiology and treatment of sepsis[END_REF]. They constitute the leading cause of morbidity and mortality for critically ill patients worldwide, although an enormous effort has been expended during the last 20 yrs to find new therapies that decrease the mortality resulting from this syndrome [START_REF] Hotchkiss | The pathophysiology and treatment of sepsis[END_REF][START_REF] Carlet | Sepsis: Time to reconsider the concept[END_REF].

It is generally accepted that septic patients develop immune alterations affecting both innate and adaptive immunity [START_REF] Monneret | Monitoring immune dysfunctions in the septic patient: A new skin for the old ceremony[END_REF]. So far, most work has been devoted to the study of the innate part of these sepsis-induced immune dysfunctions. Importantly, it has been shown that the intensity and duration of these alterations are associated with increased mortality and susceptibility to secondary/nosocomial infections [START_REF] Monneret | Monitoring immune dysfunctions in the septic patient: A new skin for the old ceremony[END_REF]. In contrast, far less work has been dedicated to the study of lymphocyte alterations, although a link between a reduced delayed-type hypersensitivity response (predominantly mediated by T-cells and a hallmark of immune suppression) in ICU patients and increased risk of nosocomial infections and death has been described >30 yrs ago [START_REF] Meakins | Delayed hypersensitivity: Indicator of acquired failure of host defenses in sepsis and trauma[END_REF].

In this study, we investigated a novel aspect of sepsis-induced lymphocyte alterations. Using a novel molecular biology tech-Objective: Septic syndromes are the leading causes of mortality in intensive care units. In patients, the occurrence of sepsisinduced immune suppression is associated with delayed mortality, although the exact role of lymphocyte dysfunctions is not well established. The objective of this study was to investigate T-cell receptor diversity, an important feature of T-cell response, in patients with septic shock. Design: Preliminary prospective observational study. Setting: Adult intensive care units in a university hospital. Subjects: Patients with septic shock (n = 41) sampled twice after the onset of shock (early after inclusion [day 1] and at the end of the first week [day 7]). Measurements and Main Results: Using a novel molecular biology technique, the combinatorial diversity of human T-cell receptor β-chain (TRB locus) was measured in peripheral blood. Patients with septic shock presented with a marked decreased T-cell receptor diversity after the onset of shock in comparison with nor-mal values. Importantly, in paired samples, a very steep recovery slope of T-cell receptor diversity, never described in other clinical situations, was observed between day 1 and day 7 (p < 0.0001, Wilcoxon's paired test). Decreased T-cell receptor diversity was associated with mortality (log-rank test, p = 0.0058; hazard ratio = 4.48; 95% confidence interval 1.96-53.32), and the development of nosocomial infections (p < 0.05, Mann-Whitney U test). Conclusion: Our results show for the first time that septic patients present with a marked decreased T-cell receptor diversity that returned rapidly toward normal values over time. This opens novel cognitive research perspectives that deserve to be investigated in experimental models of sepsis. After confirmation in larger cohorts of these preliminary results, T-cell receptor diversity measurements may become a crucial tool to monitor immune functions in ICU patients. (Crit Care Med 2013; 41:111-119) Key Words: divpenia; lymphocyte; septic shock; T-cell receptor diversity

Decreased T-Cell Repertoire Diversity in Sepsis:

A Preliminary Study* nique covering the whole T-cell receptor (TCR) repertoire (TRB locus), we studied in patients with septic shock the combinatorial diversity of the TCR β-chain directly at the genomic level [START_REF] Derouazi | Optimal epitope composition after antigen screening using a live bacterial delivery vector: Application to TRP-2[END_REF][START_REF] Banerjee | Adult T-cell leukemia/lymphoma development in HTLV-1-infected humanized SCID mice[END_REF][START_REF] Marodon | High diversity of the immune repertoire in humanized NOD.SCID.gamma c-/-mice[END_REF]. Considering the importance of having a large and diverse TCR repertoire for the quality of the immune response to infection, we reasoned that if TCR diversity was decreased after septic shock, this might participate in increased mortality and susceptibility to secondary nosocomial infections in patients.

PATIENTS AND METHODS

Patients

The study group consisted of 41 consecutive patients with septic shock, according to the diagnostic criteria of the American College of Chest Physician/Society of Critical Care Medicine [START_REF] Levy | SCCM/ESICM/ACCP/ATS/ SIS: 2001 SCCM/ESICM/ACCP/ATS/SIS International Sepsis Definitions Conference[END_REF]. The exclusion criteria were patients <18 yrs of age and subjects with aplasia or immunosuppressive disease (e.g., human immunodeficiency virus). Septic shock was defined by an identifiable site of infection, persisting hypotension despite fluid resuscitation requiring vasopressor therapy, and evidence of a systemic inflammatory response manifested by at least two of the following criteria: 1) temperature >38°C or <36°C; 2) heart rate >90 beats/min; 3) respiratory rate >20 breaths/min; and 4) white blood cell count >12,000/ mm 3 or <4000/mm 3 . Severity was assessed by the Simplified Acute Physiologic Score II calculated at inclusion in the protocol [START_REF] Gall | A new Simplified Acute Physiology Score (SAPS II) based on a European/North American multicenter study[END_REF]. The development of organ dysfunction was assessed by the Sequential Organ Failure Assessment score (SOFA; range, 0-24), measured after 24 hrs of ICU stay [START_REF] Vincent | Use of the SOFA score to assess the incidence of organ dysfunction/failure in intensive care units: Results of a multicenter, prospective study. Working group on 'sepsis-related problems' of the European Society of Intensive Care Medicine[END_REF]. Mortality was defined as death occurring within 28 days after the onset of shock. The onset of septic shock was defined as the beginning of vasopressor therapy. None of these patients was infected by superantigen-producing bacteria. Secondary ICUacquired infections were defined according to European definitions of the European Centre for Disease Prevention and Control [START_REF]Definition of ICU-acquired infections[END_REF].

Biological analyses were performed on residual blood after completing routine follow-up performed in the ICU. EDTAanticoagulated blood was collected from patients at two time points: days 1-3 (D1) and days 5-8 (D7) after diagnosis of septic shock. Because the laboratory did not operate on Saturdays and Sundays, patients were included on weekdays during working hours. In addition to patients who died or left the ICU before the second sampling time, this accounted for the majority of missing samples.

This work belongs to a global study on ICU-induced immune dysfunctions. yrs; 11 females and 16 males). Normal median TCR diversity was 68% (95% confidence interval [CI] 65-69). For the purpose of this preliminary study, we defined 56% (mean -2 SD) as the lower limit of normal TCR diversity (Fig. 1A).

Human ImmunTraCkeRβ Test/Combinatorial Diversity of TCR β-Chain

Human T-lymphocyte repertoire diversity was measured using Human ImmunTraCkeRβ test (ImmunID Technologies, Grenoble, France), a novel molecular biology technique on genomic DNA. Genomic DNA was extracted using standard techniques from total peripheral mononuclear blood cells of patients and control subjects after informed consent. Multi-N-plex polymerase chain reaction (PCR) was performed using an up stream primer specific of all functional members of a given T-cell receptor β V (TRBV) family and a downstream primer specific of a given T-cell receptor β J (TRBJ) segment. This assay allows for the simultaneous detection of several V-J rearrangements in the same reaction. Using this technique, it is possible to detect 276 different TRBV-TRBJ rearrangements covering 100% of the possible combinatorial rearrangements. All V-J1, J2, J3, J4, and Jn products were separated as a function of their size with a maximum amplicon size of ~5 kb. PCR signals were detected and analyzed using the Constel'ID software developed by ImmunID Technologies. PCR products were generated using iProof GC rich Master Mix (Bio-Rad, Hercules, CA). The cycling conditions were as follows: Touchdown PCR cycle protocol was performed beginning with 98°C for 3 mins, 98°C for 20 secs, 72°C for 20 secs, and 72°C for 3 mins 30 secs; the annealing temperature was reduced by 1° for every cycle until it reached 68°C, which was then repeated 23 times. Finally, one cycle of 10 mins at 72°C was performed. To perform semiquantitative analysis, PCR reactions were stopped at the exponential step of the PCR. To normalize DNA quantity in each reaction, the actin gene was amplified in the same PCR run. DNA isolated from kidney and/or human embryonic kidney cell lines, in which no rearrangement takes place, was used as a negative PCR control. PCR products were separated on a 0.8% agarose gel, directly stained with SYBR Green I, and quantified using a charge-coupled device camera equipped with BIO-1D (Vilbert Lourmart, France). Constel'ID software was used for further analytical studies including the generation of threedimensional repertoire illustration. Results are expressed as percentages of detected rearrangements among the total 276 possible combinatorial rearrangements. A decreased percentage (i.e., a decreased TCR diversity) defines a state of divpenia.

Flow Cytometry

The following lymphocyte subsets were analyzed: total T lymphocytes (CD45 + CD3 + ), CD4 + T lymphocytes (CD45 + CD4 + CD3 + ), CD8 + T lymphocytes (CD45 + CD8 + CD3 + ), total B-cells (CD45 + CD19 + ), natural killer cells (CD45 + CD3 -CD56 + ), and regulatory T-cells (CD4 + CD25 high CD127 low ), as previously described [START_REF] Venet | Early assessment of leukocyte alterations at diagnosis of septic shock[END_REF]. Human leukocyte antigen-DR (HLA-DR) expression was monitored on circulating monocytes (mHLA-DR), as previously described [START_REF] Monneret | Persisting low monocyte human leukocyte antigen-DR expression predicts mortality in septic shock[END_REF]. Results are expressed as number of cells per microliter of blood for lymphocyte subsets, as percentages of regulatory T-cells among total CD4 + lymphocyte population, and as percentages of positive cells among total monocyte subset for mHLA-DR measurements. Age-matched reference values were provided by the routine flow cytometry laboratory of our institution.

Statistics

Results are presented as box plots with individual values. Lower limit for the reference values is shown as a dot line. Comparisons between groups were made using the nonparametric Mann-Whitney U test (survivors vs. nonsurvivors).

The nonparametric Wilcoxon's paired test was used to assess variations between time points in septic patients. Spearman's correlation test was used to assess correlations between TCR diversity values and clinical and biological variables. Fisher's exact test was used to compare proportions. A receiver operat-ing characteristic curve was performed to determine the cutoff values for TCR diversity and number of T lymphocytes with regard to prediction of mortality. The best cutoff value was selected based on likelihood ratio and Youden's index. Using these thresholds, Kaplan-Meier survival curves were obtained, and differences in survival between groups were evaluated using log-rank test. Univariate analysis through the Cox model was used for the estimation of the hazard ratio and 95% CI. Statistical analyses were performed using Prism software (version 4.03; GraphPad Software, La Jolla, CA). A p value < 0.05 was considered statistically significant.

RESULTS

Cohort Description

Forty-one patients with septic shock were included. Patients were sampled twice after the onset of shock: first between day 1 and day 3 (D1, n = 38 samples) and then between day 5 and day 8 (D7, n = 26 samples). Twenty-three patients were sampled at the two time points. Clinical and immunological data are presented in Table 1. Twenty-eight-day mortality was 39% (n = 16 nonsurvivors), and four patients developed a nosocomial infection (10%).

As shown in Table 1, patients presented with usual immune alterations as described in the course of septic shock: a major lymphopenia associated with an increased percentage of circulating regulatory T-cells as well as a decreased percentage of monocytes expressing HLA-DR [START_REF] Venet | Early assessment of leukocyte alterations at diagnosis of septic shock[END_REF][START_REF] Monneret | Persisting low monocyte human leukocyte antigen-DR expression predicts mortality in septic shock[END_REF][START_REF] Monneret | Marked elevation of human circulating CD4+CD25+ regulatory T cells in sepsis-induced immunoparalysis[END_REF].

TCR Diversity Is Decreased in Septic Patients

TCR β-chain combinatorial diversity (TCR diversity) was measured on peripheral blood mononuclear cells in patients using a novel molecular biology technique. According to this indicator, the patients can or cannot be qualified as divpenic (low TCR diversity).

As shown in Figure 1A, TCR diversity was markedly decreased in most septic patients (95% values of individual TCR diversity below control values) at D1 and remained low at D7 in the majority of the cohort (77% below control values). Indeed, in patients, median TCR diversity was 15% at D1 (95% CI, 12-25) and 43% at D7 (95% CI, 36-49). Therefore, most septic patients in this cohort were divpenic. Representative examples of TCR diversity data for one septic patient at D1 and D7 and for one healthy volunteer are given in Figure 1B.

Interestingly, in the subgroup of 23 patients who were sampled twice, there was a significant increase in TCR diversity over time (p < 0.0001, Wilcoxon's paired test) (Fig. 1C). In fact, TCR diversity decreased in only one patient between D1 and D7 and the patient died 9 days after the onset of shock (of note, the number of circulating T-cells increased in this patient between D1 and D7). Meanwhile, in these patients, the number of circulating T-cells increased as well between D1 and D7 (Fig. 1D; p = 0.0026; Wilcoxon's paired test). However, in contrast to TCR diversity, this increase was not systematic in every septic patient.

Correlations Between TCR Diversity and Clinical and Immunological Parameters

No significant correlations were found between TCR diversity and SOFA score or Simplified Acute Physiologic Score II either at D1 or at D7 (Spearman's correlation test). No significant differences were observed between patients with or without comorbidities (the Mann-Whitney U test, data not shown). Interestingly, a significant correlation was observed between TCR diversity and percentage of HLA-DR expressing monocytes when measured at D1 (r = 0.3388, p = 0.0066, Spearman's correlation test). However, no significant correlations were found between TCR diversity and percentage of regulatory T-cells either at D1 or at D7.

A correlation was found between TCR diversity and the number of circulating T-cells at D1 (r = 0.5095, p = 0.0011, Spearman's correlation test), but this correlation was lost at D7 showing that TCR diversity evolves independently of T-cell count after septic shock. No correlation was observed in healthy volunteers.

Low TCR Diversity May Possess a Predictive Value on Mortality

Sixteen patients died within 28 days after the onset of shock in this cohort. Five patients died before D8 and 11 patients died after D10. Mean time of death was 13 days postshock.

No significant differences were observed in TCR diversity between survivors and nonsurvivors either at D1 or at D7. The percentage of monocytes expressing HLA-DR was significantly lower in nonsurvivors than in survivors at D7 but not at D1 (p = 0.0109 and 0.2442, respectively, Mann-Whitney U test).

In both groups and in patients who were sampled twice, TCR diversity increased significantly between the two time points (p = 0.0002 in survivors, p = 0.0195 in nonsurvivors, Wilcoxon's paired test). Importantly, although most patients in the survivors' group (91%) and every patient in the nonsurvivors' group (100%) had a decreased TCR diversity at D1, only 67% of survivors remained with a decreased TCR diversity, whereas most of the nonsurvivors still presented with a decreased value at D7 (91%; Table 2).

Subsequently, the best thresholds for prediction of mortality were calculated for TCR diversity at D1, then at D7 using receiver operating characteristic curve analyses. At D1, the area under the curve was close to 0.5, illustrating the absence of predictive value for this parameter at this time point. However, at D7, the area under the curve was 0.7 (95% CI 0.5-0.9). The best cutoff value based on Youden's index and likelihood ratio was found to be 33% (Youden's index = 0.32; likelihood ratio = 3.4). Using this value, the Kaplan-Meier survival curves were obtained. Importantly, based on this threshold, a significantly different survival was observed between patients with or without low TCR diversity (Fig. 2A; log-rank test, p = 0.0058; hazard ratio = 4.48; 95% CI 1.96-53.32).

Of note, similar analyses were performed to determine the prognostic value for the number of circulating T-cells. The best cutoff value at D7 was found to be 730 cells/µL. However, based on this threshold, no significantly different survival was observed between the groups (data not shown). This illustrates that divpenia by itself possesses predictive value on mortality independently of the number of circulating lymphocytes.

TCR Diversity May Present With Predictive Value on Nosocomial Infections

Only four patients developed a nosocomial episode in our cohort. Every nosocomial infection was diagnosed later than 14 days after the onset of shock. Average diagnosis time was 16 days after shock. Interestingly, and despite the less number of patients in our cohort, patients who will develop a nosocomial episode present with significantly reduced TCR diversity at D1 in comparison with the patients who remain free of secondary infection (Fig. 2B). Furthermore, at D7, every patient who developed a nosocomial episode remained with a reduced diversity, whereas only 73% of the patients free of any nosocomial infection present with a low TCR diversity (Table 2).

Of note, neither at D1 nor at D7 did we observe any difference regarding the number of circulating T-cells between patients who will or will not develop a nosocomial episode (data not shown).

Predictive Value of Numeration Diversity Lymphocyte Scoring

Finally, to test whether the combination of TCR diversity and number of circulating T-cells was more informative than each parameter considered alone, patients were stratified based on these parameters at D1 and at D7 using thresholds previously calculated through receiver operating characteristic curve analyses (TCR diversity < 33%, number of circulating T-cells < 730 cells/µL). Based on these two parameters, patients sampled at D1 were classified into four groups (Fig. 3A; n = 38). The numeration diversity lymphocyte (NDL)1 group (presenting with low TCR diversity and low number of T-cells, n = 26) was compared with NDL2 (higher TCR diversity and low number of T-cells, n = 5), NDL3 (low TCR diversity and higher number of T-cells, n = 4), and NDL4 (higher TCR diversity and higher number of T-cells, n = 3). At this time point, the majority of patients belonged to group NDL1 (68% patients), whereas group NDL2 represented 13%, group NDL3 11%, and group NDL4 8% of the total population. The NDL1 group showed a significantly lower percentage of mHLA-DR expressing monocytes than the other three groups (median, 26%; 95% CI 21-39 for group NDL1; median, 55%; 95% CI 35-67 for the other groups; p < 0.05, Mann-Whitney U test). However, no differences between NDL1 and the other groups were observed for Simplified Acute Physiologic Score II and SOFA scores. Interestingly, every nosocomial infection occurred in the NDL1 group, and mortality was 38% vs. 25% in the other pooled groups.

At D7 (n = 26; Fig. 3B), patients' distribution according to these NDL groups was NDL1 (n = 6), NDL2 (n = 11), NDL3 (n = 1), and NDL4 (n = 8). Therefore, the majority of patients now belonged to group NDL2 (42% patients), whereas group NDL1 represented 23%, group NDL3--4%, and group NDL4--31% of the total population. Interestingly, at this time point, initial SOFA score was significantly higher in group NDL1 as compared with the other three groups (median, 13; 95% CI 11-15 in NDL1; median, 9; 95% CI 9-11 in the other groups; p < 0.01, Mann-Whitney U test), whereas the percentage of mHLA-DR expressing monocyte was lower (median, 25%; 95% CI 12-41 in NDL1; median, 51; 95% CI 39-58 in the other groups; p < 0.05, Mann-Whitney U test). Furthermore, mortality in the NDL1 group was 67% (four nonsurvivors among six patients), whereas only seven patients among 20 died in the other groups (mortality = 35%).

DISCUSSION

As a result of their ability to interact with the cells of the innate immune system and with the other cells of the adaptive immune system, lymphocytes play a central role in the anti-infectious immune response not only as effectors, but also as regulators of this response. This has been illustrated by the observation of a dysregulated, proinflammatory immune response after polymicrobial septic challenge in mice lacking T-cells and B-cells [START_REF] Hotchkiss | Caspase inhibitors improve survival in sepsis: A critical role of the lymphocyte[END_REF][START_REF] Shelley | Interaction between the innate and adaptive immune systems is required to survive sepsis and control inflammation after injury[END_REF]. Without showing any clear association with the increased risk of death or nosocomial infections in patients, some alterations of the lymphocytic response have been previously reported after septic shock. They comprise the following: 1) phenotypic alterations including an increase in inhibitory coreceptor expression associated with a decreased expression of activating coreceptors and CD3; 2) functional alterations mainly characterized by a decreased ex vivo lymphocyte proliferation associated with decreased Th1 and increased Th2 immune responses; 3) induction of regulatory T-cell populations; and 4) increased apoptosis participating in the initial lymphopenia [START_REF] Hotchkiss | The pathophysiology and treatment of sepsis[END_REF][START_REF] Monneret | Monitoring immune dysfunctions in the septic patient: A new skin for the old ceremony[END_REF]. Such lymphocyte dysfunctions were recently observed in patients' lymphoid organs in a postmortem study [START_REF] Boomer | Immunosuppression in patients who die of sepsis and multiple organ failure[END_REF]. The authors observed that patients who died of sepsis present with features consistent with the development of immunosuppression after sepsis (reduced cytokine production after TCR stimulation of splenocytes, increased expression of inhibitory receptors and ligands as well as expansion of suppressor cell populations in both lung and spleen cells). These results illustrate that sepsis-induced immune suppression is a profound mechanism that can be observed not only in circulating blood cells, but also in lymphoid organs.

In the current study, we reinforce and expand these results by investigating at the molecular level a novel aspect of sepsisinduced lymphocyte alterations: the diversity of the TCR β-chain repertoire. To the best of our knowledge, we observe for the first time that septic patients present with a dramatic decrease of peripheral T-cell repertoire. This was associated with other usual immune dysfunctions described in such cohorts of patients (lymphopenia, decreased percentage of HLA-DR expressing monocytes) and most importantly with increased risk of death and nosocomial infections. The diversity of T-cells plays a central role in the host's ability to mount protective anti-infectious immune responses [START_REF] Naylor | The influence of age on T cell generation and TCR diversity[END_REF][START_REF] Nikolich-Zugich | The many important facets of T-cell repertoire diversity[END_REF] and an optimal level of TCR diversity allows for efficient protection against pathogens [START_REF] Nikolich-Zugich | The many important facets of T-cell repertoire diversity[END_REF]. Indeed, because the organism cannot predict the precise pathogen-derived antigens that will be encountered, the immune system relies on the generation and maintenance of a diverse TCR repertoire [START_REF] Nikolich-Zugich | The many important facets of T-cell repertoire diversity[END_REF]. It follows that the size and diversity of the available T-cell repertoire are crucial in shaping the immune response to a given antigen.

One physiological (aging) and several pathological/therapeutic (HIV infection/highly active antiretroviral therapy, T-cell leukemia, and bone marrow transplantation) conditions have yielded reduced TCR diversity in humans [START_REF] Nikolich-Zugich | The many important facets of T-cell repertoire diversity[END_REF][START_REF] Gorochov | Perturbation of CD4+ and CD8+ T-cell repertoires during progression to AIDS and regulation of the CD4+ repertoire during antiviral therapy[END_REF]. Importantly, decreased TCR repertoire diversity has been associated with viral escape in chronic viral infections such as HCV [START_REF] Meyer-Olson | Limited T cell receptor diversity of HCV-specific T cell responses is associated with CTL escape[END_REF], and changes in the TCR repertoire have been correlated with outcomes of renal transplantation [START_REF] Naesens | Harnessing the diversity of the human T-cell repertoire: A monitoring tool for transplantation tolerance?[END_REF][START_REF] Miqueu | Analysis of the peripheral T-cell repertoire in kidney transplant patients[END_REF]. In ICU patients, such reduced TCR repertoire, as observed in our study, might participate in increased susceptibility to nosocomial infections in the context of use of invasive devices (intubation, urinary catheter, central venous catheter, etc.). Furthermore, this might play a role in the reactivation of viral infections (cytomegalovirus, Epstein-Barr virus), normally solely occurring in immunocompromised hosts and also observed in ICU patients [START_REF] Heininger | Cytomegalovirus reactivation and associated outcome of critically ill patients with severe sepsis[END_REF][START_REF] Kalil | Prevalence and mortality associated with cytomegalovirus infection in nonimmunosuppressed patients in the intensive care unit[END_REF].

Only two studies previously investigated the TCR repertoire in sepsis by using flow cytometry: one in patients and one in the murine model of cecal ligation and puncture. The first study included nine patients with Staphylococcus aureus septic shock [START_REF] Ferry | Analysis of superantigenic toxin Vbeta T-cell signatures produced during cases of staphylococcal toxic shock syndrome and septic shock[END_REF], and no specific T-cell repertoire modifications were observed. In the second study, Unsinger et al [START_REF] Unsinger | Differential lymphopenia-induced homeostatic proliferation for CD4+ and CD8+ T cells following septic injury[END_REF] showed that after cecal ligation and puncture, mice presented with lymphopenia but that the TCR repertoire measured 21 days after cecal ligation and puncture was not skewed toward any Vβ type but resembled the repertoire found in normal mice. The authors concluded that, after sepsis, T-cells were not primed to any antigen resulting from the infection. Importantly, in both studies, TCR repertoire was measured by flow cytometry. Using this technique, a maximum of 25 different Vβ clones could be detected [START_REF] Van Den Beemd | Flow cytometric analysis of the Vbeta repertoire in healthy controls[END_REF]. In contrast, 276 TCR Vβ-Jβ rearrangements could be identified by the present molecular biology technique. Furthermore, in both previously published studies, TCR repertoire was evaluated at late time points after the onset of shock (after 21 days in mice and within 10 days in patients). Considering the fast recovery slope observed in our cohort, this could explain the discrepancy with our results. However, a study comparing these two techniques of TCR repertoire measurement should be conducted. To our knowledge, such study has never been done yet.

This fast increase in TCR repertoire observed in patients between D1 and D7 is of the utmost importance in this study. Indeed, to our knowledge, such a rapid recovery of TCR diversity has not been described previously in any other clinical context. This argues against a novel thymus-based lymphocyte generation but rather suggests a systemic liberation of lymphocytes trapped in peripheral compartments. This assumption is in agreement with the reduced percentage of CD45RA + (naive) but not of CD45RO + (memory) T-cells previously described in septic patients [START_REF] Monneret | Marked elevation of human circulating CD4+CD25+ regulatory T cells in sepsis-induced immunoparalysis[END_REF][START_REF] Monserrat | Clinical relevance of the severe abnormalities of the T cell compartment in septic shock patients[END_REF][START_REF] Roth | Susceptibility to programmed cell death in T-lymphocytes from septic patients: A mechanism for lymphopenia and Th2 predominance[END_REF]. This is further confirmed by our observation that lymphocyte reconstitution after septic shock occurs mainly through a switch from NDL1 to NDL2 group (i.e., through increased diversity) and not through a switch from NDL1 to NDL3 group (i.e., through an increase in cell number). This suggests a uniform reconstitution of TCR diversity through rapid mobilization from lymphocyte peripheral stores rather than immune reaction-based active oligoclonal expansion. These results open novel cognitive research perspectives that deserve to be further investigated in experimental models of sepsis. Similarly, the evaluation of TCR diversity of memory vs. naive T-cells and its overtime evolution in patients with septic shock should be evaluated in a specifically designed study. Indeed, the dynamic measurement of TRB diversity and the composition of T-cell subpopulations (i.e., naive/memory/regulatory) over time in peripheral tissues as well as in spleen and thymus would help in understanding the mechanisms of the rapid recovery of TCR diversity in patients with septic shock and of sepsis-induced immune suppression. More generally, this fast increase in TCR diversity after the onset of shock suggests that this temporal immune suppression belongs to the homeostatic response that takes place after septic shock. It is the lack of (or slow) recovery of immune functions that is associated with unfavorable outcome (as observed with decreased mHLA-DR). This is in favor of the initiation of immunotherapy in sepsis to facilitate this restoration in selected patients with persistent immune alterations.

Our study has some limitations. First, this preliminary study included a small cohort of patients. Therefore, no multivariate analysis could be performed to test the independent predictive value of TCR diversity on mortality vs. clinical variables (Simplified Acute Physiologic Score II and SOFA scores) or other immune parameters (mHLA-DR or number of circulating T-cells). Similarly, the low number of nosocomial infections clearly weakens the statistical analysis of our data because it is quite difficult to obtain significant results with such unbalanced groups of patients. This may be explained by the rather high mortality rate (i.e., 39%) in our cohort knowing that, by definition, patients who died cannot develop nosocomial infections. Second, the impact of therapeutic interventions on TCR diversity was not evaluated. Finally, the absence of an exact agematched control group prevents us to definitively conclude on the sepsis-induced decrease in TCR diversity. With that said, such a low percentage of TCR variability observed at D1 in septic patients could not solely be explained by their increasing age, considering results for TCR repertoire diversity in elderly published in the literature [START_REF] Naylor | The influence of age on T cell generation and TCR diversity[END_REF] as well as the fact that no correlation with age was measured at D7 in patients or within the cohort of healthy volunteers (maximum age = 70 yrs, data not shown).

In total, in a cohort of patients with well-documented immune dysfunctions (lymphopenia, decreased HLA-DR expression on monocytes), we describe for the first time an additional aspect of sepsis-induced immune suppression. Our results add to the current knowledge of lymphocyte dysfunctions in sepsis and of their association with increased mortality and risk of nosocomial infections [START_REF] Monneret | Monitoring immune dysfunctions in the septic patient: A new skin for the old ceremony[END_REF][START_REF] Skrupky | Advances in the management of sepsis and the understanding of key immunologic defects[END_REF][START_REF] Hotchkiss | Immunotherapy for sepsis-A new approach against an ancient foe[END_REF]. Furthermore, our results show that the measurement of TCR repertoire diversity, as a novel immune response indicator, may provide predictive information regarding risk of death and nosocomial infection independently of the number of circulating lymphocytes. As a consequence, immunostimulating therapies are now proposed in the treatment of sepsis. One potential candidate could be recombinant human IL-7 because, in addition to its capacity to increase circulating lymphocyte number and function [START_REF] Mackall | Harnessing the biology of IL-7 for therapeutic application[END_REF], its in vivo injection in healthy volunteers was able to increase TCR diversity without inducing any major side effect [START_REF] Sportès | Administration of rhIL-7 in humans increases in vivo TCR repertoire diversity by preferential expansion of naive T cell subsets[END_REF]. Furthermore, in mice, initial studies showed that recombinant human IL-7 was able to reduce mortality after cecal litigation and puncture [START_REF] Unsinger | IL-7 promotes T cell viability, trafficking, and functionality and improves survival in sepsis[END_REF][START_REF] Kasten | Interleukin-7 (IL-7) treatment accelerates neutrophil recruitment through gamma delta T-cell IL-17 production in a murine model of sepsis[END_REF].

CONCLUSION

In this study, we describe for the first time at the molecular level a novel aspect of sepsis-induced lymphocyte dysfunctions. We observed that patients with septic shock present with a severely decreased TCR β-chain diversity after the onset of shock. Importantly, independent of the number of circulating T-cells, patients who did not restore rapidly their diversity presented with both increased risk of death and of nosocomial infections. On validation in a large multicentric clinical study, divpenia assessment may become an interesting tool for immunomonitoring. Furthermore, on a more general basis, the observation of a rapid recovery of TCR diversity in peripheral lymphocytes, never described in other clinical situations, opens a novel and interesting cognitive research avenue to understand the underlying mechanisms of immune homeostasis after septic shock.

  It has been approved by our Institutional Review Board for ethics (Comité de Protection des Personnes) and registered at French Ministry of Research and Enseignement (#DC-2008-509). It is also recorded at the Commission Nationale de l'Informatique et des Libertés. Normal values for TCR β-chain combinatorial diversity were defined based on a cohort of 27 healthy volunteers extracted from a larger study (SuSa Study, AFSSAPS clinical trial number 2010-A00428-31, ImmunID; age range, 39-70

Figure 1 .

 1 Figure 1. Reduced T-cell receptor (TCR) diversity in patients with septic shock. A, T-cell receptor β-chain diversity (TCR diversity) was measured by molecular biology in 41 patients with septic shock between day 1 and day 3 (D1, n = 38) and day 5 and day 8 (D7, n = 26) after the onset of shock (gray boxes) and in 27 healthy volunteers (HV, open box). Results are presented as box plots and individual values for the percentage of TCR diversity. Dotted line represents inferior limit of normal TCR diversity value (56%) defined based on values measured in HV. B, Representative examples of TCR diversity measurements are shown for one septic patient at D1 and D7 and one HV. x-axis represents 23 T-cell receptor β V (TRBV) genes (Vβ); y-axis represents 12 T-cell receptor β J (TRBJ) genes (Jβ); and z-axis represents the relative intensity of each Vβ-Jβ combinatorial TCR chain amplified. C, TCR diversity was monitored in 23 septic patients with paired samples at D1 and D7. Individual values and medians are shown. D, The number of circulating T-cells (10 3 cells/µL) in 23 septic patients with paired samples at D1 and D7 are shown. Individual values and medians are presented. Nonparametric Wilcoxon's paired test was used to compare overtime evolution of TCR diversity and number of T-cells in patients.

a

  Lower limit of normal T-cell receptor diversity based on the evaluation of 27 healthy volunteers. T-cell receptor diversity (%) was evaluated on peripheral blood mononuclear cells extracted from 41 patients with septic shock at day (D) 1 (between D1 and D3 after the onset of shock, n = 38 samples) and D7 (between D5 and D8 after the onset of shock, n = 26 samples).

Figure 2 .

 2 Figure 2. Reduced T-cell receptor (TCR) diversity in patients with septic shock is associated with increased risk of death and nosocomial infection. A, T-cell receptor β-chain diversity (TCR diversity) was measured by molecular biology in 41 patients with septic shock between day 1 and day 3 (D1, n = 38) and day 5 and day 8 (D7, n = 26) after the onset of shock. The Kaplan-Meier survival curves were obtained after patients with septic shock were stratified based on TCR diversity value at 33% at D7 determined by receiver operating characteristic curve analysis. Log-rank test was used to analyze the difference in 28-day survival between patient groups. B, TCR diversity was monitored in 41 septic patients sampled at D1 and/or D7. Results are presented as box plots and individual values for patients who remained free of any nosocomial episode (open boxes, n = 34 at D1, n = 22 at D7) and patients who developed a nosocomial infection (gray boxes, n = 4 at D1 and at D7). The nonparametric Mann-Whitney U test was used to compare TCR diversity between groups of patients.

Figure 3 .

 3 Figure 3. Numeration diversity scoring at day (D1) and day 7 (D7). T-cell receptor (TCR) diversity and number of circulating T-cells (10 3 /µL) were monitored in 38 septic patients at D1 (A) and 26 septic patients at D7 (B). Based on thresholds previously determined (33% TCR diversity, 730 T-cells/µL), four groups were created: numeration diversity lymphocyte (NDL)1 score corresponding to lymphodivpenic patients (low TCR diversity, low number of T-cells, n = 26), NDL2 score corresponding to lymphopenic patients (higher TCR diversity, low T-cells, n = 5), NDL3 score corresponding to divpenic patients (low TCR diversity, higher T-cells, n = 4), and NDL4 score corresponding to normality (higher TCR diversity, higher T-cells, n = 3). Results are presented as individual values.

TABLE 1 . Demographic, Clinical, and Immunological Data for Patients With Septic Shock Clinical Data

 1

	Age at admission (yrs) a	71 [57-77]
	Male gender	29 (71)
	Simplified Acute Physiologic Score II	57 [44-67]
	at diagnosis of shock a
	Sequential Organ Failure Assessment	9.5 [8-12]
	score a
	Main diagnosis category
	Medical	17 (41)
	Surgery	24 (58)
	Comorbidities
	0	21 (51)
	≥1	20 (49)
	Infection
	Bacilli Gram-negative	14 (34)
	Cocci Gram-positive	10 (24)
	Fungi	3 (7)
	Others	10 (24)
	Type of infection
	Community-acquired	22 (54)
	Hospital-acquired	15 (37)
	ICU-acquired	4 (10)
	Site of infection
	Pulmonary	12 (29)
	Abdominal	13 (32)
	Other	16 (39)
	Mortality
	Nonsurvivors	16 (39)
	Nosocomial infections
	Occurrence	4 (10)
		(Continued)

TABLE 1 (Continued). Immunological Data

 1 Continuous data as well as biological parameters are presented as medians and interquartile ranges [Q1-Q3]. Simplified Acute Physiologic Score II calculated at inclusion in the protocol; Sequential Organ Failure Assessment score measured after 24 hrs of intensive care unit stay. Blood samples were obtained from 41 patients with septic shock at two time points after the onset of shock. Patients were sampled within the first 3 days (D1) and between day 5 and day 8 (D7) after the onset of shock. For clinical parameters, categorical data are presented as number of cases and percentages respective to the total patient's population, within brackets. The following leukocyte subsets were analyzed: total lymphocyte number, CD3 + T lymphocytes, CD3 + CD4 + T lymphocytes, CD3 + CD8 + T lymphocytes, B lymphocytes, and natural killer cells, CD4 + CD25 high CD127 low regulatory T-cells (Treg). Results are expressed as number of cells/μL and as percentages of Treg among total CD4 + lymphocyte population.

	Total lymphocytes a
	D1	0.6 × 10 3 [0.2-0.9]
	D7	0.8 × 10 3 [0.6-1.3]
	B lymphocytes a
	D1	80 [40-200]
	D7	110 [70-260]
	Natural killer cells a
	D1	60 [30-100]
	D7	80 [50-130]
	CD3 + T lymphocytes a
	D1	400 [100-500]
	D7	500 [300-900]
	CD3 + CD4 + T lymphocytes a
	D1	200 [100-400]
	D7	400 [200-600]
	CD3 + CD8 + T lymphocytes a
	D1	70 [40-200]
	D7	160 [70-260]
	Regulatory T-cells (%) a
	D1	8.5 [6-12.5]
	D7	11 [8-13]
	mHLA-DR (%) a
	D1	31 [18-53]
	D7	42 [26-60]
		Expression of human
	leukocyte antigen-DR (HLA-DR) was monitored on circulating monocytes
	(mHLA-DR). Results are presented as percentages of monocytes express-
	ing HLA-DR. Normal values from our laboratory for age-matched individuals
	are 1.3 × 10 3 -3.2 × 10 3 total lymphocytes/µL, 150-400 B-cells/µL, 50-600
	natural killer cells/µL, 900-1900 T-cells/µL, 500-1250 CD4+ T-cells/µL,
	350-900 CD8+ T-cells/µL, 5%-7% Treg, and >90% mHLA-DR.

D = day.

a

TABLE 2 . T-Cell Receptor Diversity in Patients With Septic Shock

 2

		Survivors	Nonsurvivors	Noninfected	Infected
		D1	D7	D1	D7	D1	D7	D1	D7
	Number of samples	23	15	15	11	34	22	4	4
	Median	13	49	18	40	18	41	1	49
	Quartile 1	2	37	2	20	2	29	1	31
	Quartile 3	29	64	34	52	32	58	12	51
	Divpenia (proportion <56%) a	91	67	100	91	94	73	100	100

D = day.

www.ccmjournal.org January 2013 • Volume 41 • Number 1

www.ccmjournal.org

ACKNOWLEDGMENTS

We thank Anne Portier and Caroline Guignant, PharmD, PhD, from the Immunology Laboratory of Hopital E. Herriot, Lyon, France, for their help in performing preanalytical handling of samples; Hélène Thizy and Marion Provent (Clinical Research Center, Lyon-sud) for their work on patients' inclusion and clinical data acquisition; and Gilles Parmentier, PhD, Tyoka Rabéony et Anais Couturier, for assistance in statistical analyses.

Supported, in part, by funds from the Hospices Civils de Lyon, by DHOS-Inserm "Recherche Clinique Translationnelle 2009" (to Dr. Monneret) and by the French Ministry of Health (PHRC 2008, to Dr. Monneret and Dr. Lepape). These funding sources had no role in the study design; in the collection, analysis, and interpretation of data; in the writing of the report; and in the decision to submit the paper for publication.


Damien Kuss

Vincent Laurain

Hugues Garnier

Mathieu Zug

José Vasquez

J Vazquez

Data-based mechanistic rainfall-runoff continuous-time modelling in urban context

Keywords: data-based mechanistic, contiunous-time models, rainfall-runoff modelling, urban hydrology

This paper presents a data-based mechanistic modelling (DBM) approach to rainfallrunoff modelling based on the direct identification and estimation of continuous-time models from discrete-time series. It is argued that many mechanistic model parameters are more naturally defined in the context of continuous-time, differential equation models. As a results, there are advantages if such models are identified directly in this continuous-time form rather than being formulated and identified as discrete-time models. An illustrative example based on the analysis of rainfall-flow data from the Kerinou catchment in Brest demonstrates the relevance of the proposed modelling approach.

INTRODUCTION

The integrated urban wastewater system has a major impact on the quality of the receiving waters, especially during wet weather. During this period, in the case of combined systems, part of wastewaters collected are discharged at the level of combined sewer overflows, causing a degradation of the quality of receiving waters (rivers, sea,...).

The Water Framework Directive (WFD) adopted on October 23, 2000 by the European Union defines quality objectives for all water masses. A good ecological and chemical status is required before 2015 for all surface waters, including rivers and coastal waters. For good chemical status, this means that the concentrations of pollutants in the water body must not exceed the limit values specified in the directive.

Moreover the new directive on bathing waters (2006/7/EC) is more restrictive in terms of water quality than the previous one (76/160/EEC). Bathing water quality is assessed on the basis of two bacteriological criteria. As raw wastewaters are the major source of fecal contamination of coastal waters, the discharge of wastewaters at the level of combined sewer overflows (CSOs) is very harmful for the quality of the receiving waters.

Many recent studies have shown the necessity to implement an integrated management of urban wastewater systems in order to minimize the load of discharged pollutants, (see e.g. [START_REF] Vanrolleghem | Modelling and real-time control of the integrated urban wastewater system[END_REF], [START_REF] Butler | Integrating simulation models with a view to optimal control of urban wastewater systems[END_REF]). The Integrated Management of Coastal Wastewater Discharges project was born in 2006 in this context. This project is managed by Veolia water c and involves some French environmental companies and research institutes. Two sites have been selected to carry out the experimental studies; both are located in France: the first is in Brittany (Saint-Malo and Brest) while the second is in the Mediterranean sea. The main objective is to minimize the impacts of the urban wastewaters discharged in coastal wasters thanks to an integrated real-time control of the urban wastewater system.

Before minimizing the load of discharged pollutants, a first study consists in assessing the volume, frequency, and flow rate of the wastewaters discharged in the receiving waters and their corresponding impacts. In order to reach this goal, measurements of flow rate and pollution are crucial all along the urban wastewater system. Those measurements permit to design urban drainage network models that will be used to test new control and management strategies. The paper presents a data-based approach to rainfall-runoff modelling in urban context.

In most urban hydrology modelling softwares, the rainfallrunoff relationship is approximated with lumped conceptual continuous-time models (Ministère de l'écologie et du développement durable (MEDD) and CERTU [2003]).

Here, the a priori conceptual model structure is effectively a (normally simple) theory of hydrological behaviour based on the perception of the hydrologist/modeller. The models are formulated on the basis of natural laws and are often expressed in terms of continuous-time linear or nonlinear differential equations. However, the resulting model is strongly conditioned by assumptions that derive from current hydrological paragidms.

An alternative modelling approach known as data-based mechanistic modelling (DBM) is used here (see e.g. [START_REF] Young | Data-based mechanistic modelling of environmental, ecological, economic and engineering systems[END_REF]). In DBM modelling, the most parametrically efficient (parsimonious) model structure is first inferred statistically from the available time series data in an inductive manner. After this initial back-box modelling stage is complete, the model is interpreted in a physically meaningful, mechanistic manner based on the nature of the system under study. Paradoxically, even if most mathematical models of hydrological systems are formulated in continuous-time, the model to be identified are almost always presented in the alternative discrete-time form. One reason for this paradox is that hydrological data are normally sampled at regular intervals over time, forming discrete time series that are in a more appropriate form for DT modelling. Another ground is that most of the technical literature on the statistical identification and estimation (calibration) of transfer function models deals with these discrete-time models. Closer review of this literature, however, reveals apparently less well-known publications (see e.g. [START_REF] Young | Identification and estimation of continuous-time, data-based mechanistic (dbm) models for environmental systems[END_REF], [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF], [START_REF] Sinha | Identification of Continuous-Time Systems: Methodology and Computer Implementation[END_REF]) dealing with estimation methods that allow the direct identification of continuous-time models from discrete-time data.

The present paper outlines an approach to DBM of urban rainfall-runoff data based on the recent advanced direct method of continuous-time model identification and estimation [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF]). Previous DBM modelling of rainfall-flow data has confirmed many aspects of earlier hydrological research and identified that the rainfall-flow model can be stated as an 'Hammerstein' nonlinear form, with an input nonlinearity that converts the measured rainfall into effective rainfall (i.e. the rainfall that is effective in causing variations in flow). This effective rainfall then passes through the linear transfer function to yield the river flow.

A DBM-based approach is developed to jointly identify both linear and nonlinear parts of the continuous-time model. This method is then used to investigate the nonlinear relationship within the context of rainfall and flow time series from the Kerinou catchment in Brest, France.

DATA-BASED MECHANISTIC CONTINUOUS-TIME MODELLING

Every rainfall-runoff model requires two essential components [START_REF] Desbordes | Modélisation en hydrologie urbaine[END_REF], [START_REF] Beven | Rainfall-Runoff Modelling: The Primer[END_REF]):

• the runoff production component which determines how much of the rainfall becomes part of the hydrograph; • the runoff routing component which determines the distribution and the transfer time of the surface runoff up to the outlet of the catchment.

A schematic representation of the components of an urban rainfall-runoff model is depicted in Figure 1. If we assume that continuous losses are proportional to the intensity of the rainfall, the rainfall-runoff relationship can be broken down in a non-linear part including only the initial losses and a linear part including the continuous losses and the runoff routing component.

Thus, the objectives of the proposed approach are:

• to use a simple model for the estimation of the initial losses compatible with real-time modelling; • to identify and estimate the continuous losses and the runoff routing model component by using recent advanced direct continuous-time modelling approaches.

The measured data used to identify a rainfall-runoff model consists of time-series data of total rainfall intensities (mmh -1 ) for the input(s) and time-series data of flow rates (m 3 s -1 ) at the outlet of the catchment.

According to the size of the catchment, it can be useful to identify multiple input single output (MISO) rainfallrunoff models including more than one input (i.e. one rain gauge) in order to take into account the spatial distribution of the rainfall. Indeed, modelling a rainfallrunoff relationship on the basis of only one input is in some cases dangerous because it is thus assumed that the rainfall fallen in the whole catchment is the same than the one measured in one discrete point. For this reason, referring to the literature, the distance between two rain gauges that have to be installed in urban areas should be comprised between 1 to 4 km [START_REF] Einfalt | On the spatial validity of heavy point rainfall measurements[END_REF], [START_REF] Berne | Temporal and spatial resolution of rainfall measurements required for urban hydrology[END_REF]). In urban hydrological models, a possible method to take into account the nearest rain gauges is the method of the Thiessen polygons which delimit the area of influence of each rain gauge. This method consists in drawing on a base map polygons around each rain gauge such that any location within a particular polygon is nearer to the rain gauge within the polygon than to any other rain gauge.

Initial loss model

Initial losses refer to hydrological processes occurring at the beginning of a rainfall event, such as interception by the canopy or retention in surface depressions. Models of initial losses used in urban areas are generally function [START_REF] Chocat | Un système d'aide à la gestion, la prévision et la conception des équipements d'assainissement[END_REF], [START_REF] Falk | Characteristics of the above-ground runoff in sewered catchments[END_REF]):

• of a parameter independent of the rainfall and representative of the surface of the catchment, for example the slope, • or of a parameter representing the initial wetness of the catchment at the beginning of the rainfall event.

Nevertheless, the use of such models requires to preliminarily divide the rainfall chronicle in independent rainfall events in order to identify the beginning of each rainfall event. This step carried out manually is time-consuming and is not suited to real-time modelling.

In urban areas, values of initial losses may not exceed 2 to 3 mm [START_REF] Brulé | Study of runoff losses on impervious surfaces in an urban environment[END_REF]) and can sometimes be neglected because interception losses due to canopy and retention in surface depressions are less important than in natural areas.

Initial losses introduce non-linearities in the rainfall-runoff relationship. If their effects are important, for example in the case of mixed or rural catchments, the proposed approach considered here might not be the most appro- priate and two of the authors of this paper are currently investigating some new approaches which aim to identify directly a non-linear Hammerstein continuous-time Box-Jenkins model [START_REF] Laurain | Refined instrumental variable methods for identification of Hammerstein continuous-time Box-Jenkins models[END_REF]).

For urban areas we suggest a simple algorithm that transforms the total rainfall intensity i T (t) in an "effective" rainfall intensity i E (t) where the initial losses are deducted from the total rainfall.

As rainfall events are not manually detected, we have to find an automatic procedure that realizes this task.

The automatic detection of the beginning of a rainfall event is difficult, even impossible to realize. To get around this issue, for a given time-step j we defined a sliding window of duration d, starting at time-step j -d and ending at time step j. In this sliding window, if the total rainfall intensity of the current time-step i T (j) (mmh -1 ) is positive, reference initial losses L in (mm) are compared with the sum of the rainfall height S(j) (mm) fallen between time-step j -d and time-step j :

               if i T (j) > 0 and S(j) < L in ⇒ i E (j) = 0 if i T (j) > 0, S(j) < L in and S(j + 1) ≥ L in ⇒ i E (j + 1) = (S(j + 1) -L in ) ∆t else i E (j) = i T (j) (1)
with:

S(j) = j j-d i T (j)∆t , where ∆t is the sampling time (2)
This approach has a physical meaning but requires the specification of two user-parameters: L in which represents the reference initial losses and d which denotes the length of the sliding window necessary for the evaporation of reference initial losses stored in soil depressions or in the canopy. An approach to automatically estimate these two user parameters will be presented in Section 2.4.

Direct continuous-time linear model estimation

Dynamic systems in the physical world are naturally described in continuous-time (CT), differential equation terms because the physical laws have been evolved mainly in this form. Paradoxically, however, the best known system identification schemes have been based on discretetime (DT) models, without much concern for the merits of natural continuous-time model descriptions and their associated identification methods. In fact, the development of CT system identification techniques occurred in the last century, before the development of the DT techniques, but was overshadowed by the more extensive DT developments [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF]).

Moreover, identification of continuous-time models can be done either with a direct or indirect approach. In the direct approach, a continuous-time model is directly identified from the sampled data. In the indirect approach a discretetime model is first identified and estimated. This estimated DT model is then converted to a CT model, using some assumption about the nature of the input signal over the sampling interval.

Direct identification of CT models proved to have many advantages [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF], [START_REF] Rao | Numerical illustrations of the relevance of direct continuous-time model identification[END_REF]):

• The model parameters are unique: unlike discretetime models, they are not a function of the data sampling interval. • The model is in an ordinary differential equation form that can be related directly to the formulation of physically meaningful models, such as those derived from mass, energy and momentum conservation. • Model parameter estimation is superior over a wider range of sampling intervals, particular at fast sampling intervals (e.g. 5 min. for rainfall-flow measurements).

These advantages, together with the parsimony that is a natural consequence of DBM transfer function modelling, should mean that any relationships between the CT model parameters and physical measures of the catchment characteristics should be clearer and better defined statistically, as required for hydrological system applications.

The theoretical basis for the statistical identification and estimation of linear, continuous-time models from discretetime data can be outlined by considering the case of a linear, single input, single output system:

q(t) = B(s) A(s) i E (t -τ ) (3)
i E (t) denotes the input signal, i.e. the intensity of the total rainfall minus the initial losses (mmh -1 ). q(t) is the output signal, i.e. the unmeasured noise free runoff flow rate (m 3 s -1 ). τ is a time delay in time units representing the delay between the input and the output signal. Note consequently, that the continuous-time model that has to be identified also includes the continuous losses. A(s) and B(s) are polynomials in the derivative operators (s n x(t) = d n x(t) dt n ):

A(s) = s n + a 1 s n-1 + • • • + a n-1 s + a n (4) B(s) = b 0 s m + b 1 s m-1 + • • • + b m-1 s + b m (5)
n and m can take on any positive integer values with m ≤ n. Of course, the model can also be written in the usual following differential equation form:

d n q(t) dt n + a 1 d n-1 q(t) dt n-1 + • • • + a n q(t) = b 0 d m i E (t -τ ) dt m + • • • + b m i E (t -τ ) (6)
In the case of uniform sampling, at a constant sampling interval T s , the sampled signals will be denoted by i E (t k ) and Q(t k ) and the output observation equation can be written in the following form:

Q(t k ) = q(t k ) + ξ(t k ) k = 1, . . . , N (7)
where ξ(t) is a discrete-time colored noise associated with a sample output measurement Q(t k ). It is assumed to be independent of the input signal i E (t) and represents the aggregate effect of all the stochastic inputs to the system, including distributed unmeasured inputs, measurement errors and modelling error. The structure of the CT model is denoted by the triad [m n τ ]. The objective is then to identify a suitable model structure ([m n τ ]) for ( 6) and to estimate the parameters that characterize this structure, based on the observed sampled measurement of the input (rainfall intensity) and output signal (runoff flow rate)

i E (t k ) and Q(t k ) (k = 1, . . . , N ).
Various statistical methods of identification and estimation have been proposed to implement both direct and indirect approaches to continuous-time model identification outlined above [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF]). We have chosen to use the Simplified Refined Instrumental Variable algorithm (SRIVC) [START_REF] Young | Refined instrumental variable methods of recursive time-series analysis -part III. extensions[END_REF]). This algorithm is available in both CAPTAIN (see www.es.lancs.ac.uk/ cres/captain/) and CONTSID (see www.cran.uhp-nancy.fr/contsid/) Toolboxes for Matlab.

The SRIVC method can be interpreted in optimal statistical terms, so providing an estimate of the parametric error covariance matrix and, therefore, estimates of the confidence bounds on the parameter estimates when the measurement noise ξ(t k ) is assumed to be white.

Model structure identification is based on two statistical criteria. First, the coefficient of determination, R 2 T , based on the error between the sampled output and the simulated CT model output se:

R 2 T = 1 - σ 2 ξ σ 2 Q (8)
where σ 2 ξ is the variance of the estimated noise ξ(t k ) and σ 2 Q is the variance of the measured output Q(t k ). Second, the YIC statistic, which is a measure of model identifiability and is based on how well the parameter estimates are defined statistically, (see e.g. [START_REF] Young | Identification and estimation of continuous-time, data-based mechanistic (dbm) models for environmental systems[END_REF], [START_REF] Garnier | Identification of continuous-time models from sampled data[END_REF]).

YIC = log e σ2 σ 2 y + log e {NEVN}; NEVN = 1 n θ n θ i=1 pii θ2 i (9)
Here, n θ is the number of estimated parameters; pii is the ith diagonal element of the block-diagonal covariance matrix P θ , and so is an estimate of the variance of the estimated uncertainty on the ith parameter estimate. θ2

i is the square of the ith parameter estimate in the θ vector, so that ratio pii / θ2

i is a normalized measure of the uncertainty on the ith parameter estimate.

Note that the closer R 2

T is to 1, the better the accuracy of fit between the model and the measured output is. In the same manner, the more negative Y IC is, the more parsimonious the model is. The best structure is thus the result of a compromise between accuracy and parsimony.

Equivalence between a first order model and a linear tank model

It can be seen that a first order model denoted by the triad [1 1 0] (10) is close to the linear tank model widely used for the modelling of the runoff routing component in urban rainfall-runoff models (11) (see [START_REF] Beven | Rainfall-Runoff Modelling: The Primer[END_REF] or Ministère de l'écologie et du développement durable (MEDD) and CERTU [2003]) :

dQ(t) dt + a 1 Q(t) = b 0 i E (t) (10) dQ(t) dt + 1 K Q(t) = 1 K Q E (t) (11)
In the linear tank model, K denotes the lag-time, i.e. the lag-time between the center of gravity of the effective hyetograph and the center of gravity of the hydrograph. Note also that the term i E (t) in ( 10) is replaced by:

Q E (t) = C R • A • i E (t) = A E • i E (t), (12)
where Q E (t) is the flow rate generated by the effective rainfall on the whole catchment, C R is the runoff coefficient, and A E is the effective area of the catchment, i.e. the area that contributes effectively to runoff production. Thus, the link between the physical parameters A E and K and the estimated parameters of (10) can be stated in the form:

     a 1 = 1 K • 24 • 60, b 0 = A E 360 • K . ( 13
)
Equation ( 13) shows that the physical parameters of the linear tank model (K, C R ) can be automatically and optimally determined from a 1 and b 0 for a first order model structure.

Proposed iterative methodology

As we saw earlier, in some cases, the initial losses can be neglected. Here, the problem simply consists in identifying a DBM model from the total rainfall.

In other cases, an initial losses model is required to improve the results. Thus, an iterative algorithm was developed in order to estimate at the same time the user-parameters of the proposed initial losses model and those of the linear part of the continuous-time transfer function model:

(1) The algorithm begins with the identification of the mathematical structure of the DBM model from the total rainfall. The best mathematical structure, e.g. the best triad [m n τ ], is chosen on the basis of both R 2 T and Y IC criteria, (2) In the second step, we search for the best couple (L in , d) that maximizes the coefficient of determination between the simulated and the measured flow rate.

(3) Finally, we check that the best structure [m n τ ] chosen from total rainfall in step 1 is still the same from the total rainfall calculated in step 2. If not, we start again the algorithm from step 2.

Note that this iterative algorithm is simple but not optimal since the best couple (L in ,d) is not estimated for all linear model structures. Nevertheless, due to the low influence of initial losses in urban areas, this algorithm gives good results and prevents us to test all the possible combinations.

AN ILLUSTRATIVE EXAMPLE

The developed methodology was tested on a part of the Kerinou catchment located in Brest, in France. An overview of the catchment is given in Figure 2. The urban drainage network system mainly consists of combined sewers which collect both dry and wet weather wastewaters. The total area of the catchment Kerinou is about 500 ha but the area drained by the subcatchment where the measurement of flow rate was carried out is about 107 ha. A pumping station also discharges the wastewaters collected from another catchment in the main sewer. The estimated effective area of the other catchment was estimated to at least ten or so ha. The runoff coefficient of the Kerinou catchment is about 40% and was calculated on the basis of measurements of rainfall and runoff volumes. The catchment is mainly residential and characterized by steep slopes from 3 up to 10%. This indicates that lag-times may be low.

The measurements of rainfall and flow rate were carried out during 49 days with a sampling interval of respectively 1 and 5 minutes. The flow rate was assessed in the main sewer with an ultrasonic probe for the measurement of the water level and a Doppler velocimeter for the velocity. For the measurement of rainfall intensity, no rain gauge was available within the catchment. The Thiessen polygons are represented in Figure 2. We used the first half of the data for the calibration and the other part for the validation of the identified model.

Referring to the criteria R 2 T and Y IC and according to the proposed methodology, we identified some SISO models and MISO (with two rain gauges as inputs) models. Nevertheless, only first order SISO and one MISO models have real poles that can be interpreted in a physically meaningful manner. The other ones have complex or negative values that do not have any sense for an hydrological system. The estimation results obtained for the two models are summarized in Table 1.

The second column of the table indicates the rain gauge taken into account for the identification of the model. No model including 3 rain gauges could have been interpreted physically because of negative flow rate generated by one of the three processes. This shows that the influence of one of the three rain gauges (rain gauge N1 "Bellevue") is negligible. Thus, in this case, the method of the Thiessen polygons is not relevant. Moreover, it can be observed that for both SISO and MISO models R 2

T and Y IC are very close. This suggests that taking into account 2 rain gauges only improves slightly the results. It was previously checked that the 2 signals are not linearly correlated (r 2 = 0.23 for the calibration dataset).

Nevertheless, a physical interpretation can be given for both models. According to (13), the effective area of the subcatchment is 67 ha. As the measured output also includes a flow rate collected from another subcatchment, the value of 67 ha is totally realistic. The values of lag-time found are also relevant.

The MISO model provides another information concerning the area of influence of each rain gauge. Referring to (13), the effective areas influenced by the rain gauges N2 and N3 are respectively 36.2 and 36.1 ha and the corresponding lag-times have values of 12 and 39 min. Those values seem also relevant. It can be observed in Figure 2 that the rainfall-runoff process occurring in the area influenced by the rain gauge N3 which is further from the outlet of the subcatchment has a larger lag-time. The measured and the simulated flow rate obtained from the MISO model are plotted in Figure 3.

Finally, the proposed methodology makes it possible to identify the two user-parameters of the model of initial losses in an automatic way.

CONCLUSION

This paper has outlined an approach to data-based mechanistic (DBM) modelling for urban hydrological sys- A practical example based on the analysis of rainfall-flow data from the Kerinou catchment in Brest has been used to illustrate the relevance of the proposed modelling approach. The differences between this approach and the one carried out in most urban drainage network modelling softwares can be summarized as follows:

• No a priori assumption is made on the mathematical structure of the model. This is a major advantage even if the example studied showed that the best mathematical structure identified correspond to a linear tank. • There is no need to fully describe the catchment, and break it down into sub-units because the model is directly identified at the catchment scale. Thus, the approach makes it possible to simplifiy the model. • The model is automatically calibrated. This is also a great advantage of the developed method, because in urban drainage network modelling softwares, a lot of time is spent to choose the appropriate models and to estimate the corresponding parameters. • The mechanistic validation of the model allows us to evaluate if the estimated parameters are relevant. • The method directly identifies a continuous-time model (i.e. a linear differential equation). Thus, the estimated parameters do not depend on the sampling interval.

Fig. 1 .

 1 Fig. 1. Schematic representation of the components involved in urban rainfall-runoff

#Fig. 2 .

 2 Fig. 2. Schematic representation of the urban drainage network of the Kerinou catchment. The red circles represent the location of the rain gauges and include their labels. The subcatchment is delimited by the dotted line. The bold lines delimits the Thiessen polygons.

Fig. 3 .

 3 Fig. 3. Comparison of the measured and MISO model flow rate tems based on the direct identification and estimation of continuous-time (transfer function or differential equation) models.A practical example based on the analysis of rainfall-flow data from the Kerinou catchment in Brest has been used to illustrate the relevance of the proposed modelling approach. The differences between this approach and the one carried out in most urban drainage network modelling softwares can be summarized as follows:

Table 1 .

 1 Characteristics of the two best data-based identified continuous-time models

		Model type	Rain gauge	[m n τ ]	R 2 T	YIC	L in	d	A E	K
			SISO	N2		[1 1 0]	0.85	-11.87	0.4 mm	3 h	67.4 ha	18 min
			MISO	N2 N3		[1 1 0] [1 1 0]	0.87	-10.32	0.4 mm 0.2 mm	3 h 3 h	36.2 ha 31,1 ha	12 min 39 min
		0.8
			measured flow rate
		0.7	MISO model flow rate
		0.6
	flow rate (m 3 /s)	0.3 0.4 0.5
		0.2
		0.1
	09/01 08:00 0	10/01 05:20	11/01 02:40	12/01 00:00
				Time