James R Goodhand
email: j.goodhand@qmul.ac.uk

William Alazawi
email: w.alazawi@qmul.ac.uk

Professor David S Rampton
email: d.rampton@qmul.ac.uk

SYSTEMATIC REVIEW: CLOSTRIDIUM DIFFICILE & INFLAMMATORY BOWEL DISEASE

Keywords: Inflammatory bowel disease < Disease-based, Ulcerative colitis < Disease-based, Crohn's disease < Disease-based, Enteric infections

come

The number of studies that report the incidence of CDI in IBD is 35 (33 of which are summarised in tables 1-3 (14 in relapse -table 1; 15 with mixed disease activity in table 2 and the 4 using the US database studies in table 3). We have inserted these references into the introductory section P8 L7-12, in order to make this clearer.

The Clayton et al study is not included in the tables (but is now referenced too), because it is the only study including patients exclusively in remission. Sonnenberg's analysis of US insurance databases is also excluded from table 3 as it is not possible to deduce overall incidence rates from the original paper.

Having read and re-read ours and the reviewer's versions of this sentence we would prefer to retain our own. The other 2 reviewers did not seem to find our version difficult to understand.

3. line 25: no significant differences in the weighted mean [weighted 95% CI] incidence of CDI in adult patients with UC or Crohn?s disease by either culture (UC 6.7% [5.6-11.4], Crohn?s disease 8.8% [2.5-15.1]) or toxin assay (UC 3.1% [0.1-6.2], Crohn?s disease 7.0% (1.8-12.2) please quote the p value. A comment should be made on the fact that these studies has not the adequate power to detect a difference between CDI incidence in UC and CD. We do not think that p-values would be helpful since the overlapping confidence limits in each case indicate that they are not different.

We have not changed our comments about sample size and hence power, as it is already stated on L39-41. We have changed the emphasis from "conceivable" to "possible". Page 9: 1.

It might be better to refer to disease activity instead to relapse or remission Done

2.

The text is confusing. Fifteen studies do not mention about the disease activity; among these studies , 4 are pediatric studies, 11 are studies in adults. One out of 11 studies in adults was excluded therefore 10 studies were analyzed. In this order the text is clearer.

Having read and re-read ours and the reviewer's versions of this sentence we would prefer to retain our own. The other 2 reviewers did not seem to find our version difficult to understand.

The 15 studies included did not specify diarrhoeal relapse or remission as specific inclusion criteria and hence were not stratified for disease activity. Of these 3 were conducted in children, and 11 in adults. 1 of the adult studies reported the denominator using the number of stools tested and not the number of patients included and so was excluded from the quantitative analysis (but was included in the tables for completeness).

3.

?Only 4/ 10 19, 28, 34, 35 of the studies that reported the relationship between CDI and disease activity found a positive association? It is not clear if all the 14 studies reported the relationship between CDI and disease activity. If so, this should be explained in the text at the beginning of the section (b). Having read and re-read ours and the reviewer's versions of this sentence we would prefer to retain our own. The other 2 reviewers did not seem to find our version difficult to 4. line 40: It is not clear why the US nationwide diagnostic coding-based case-controlled studies are reported at the end of section (b) . It is confusing.

We have clarified why we chose to deal with these studies separately in the introductory paragraph to section 3 of the results P8 L7-12 as well as in the methods as already stated. Reviewer 3 appears to endorse the arrangement of the text.

Page 10: 1. line 1: the lack of CD NAP1/027 strain in those identified in this selected population of IBD pts in clinical remission should be discussed We don't feel that this negative point needs discussion. It is probable that patients with IBD will have acquired this strain but because of the limited ribotyping testing in IBD it simply has not yet been reported. We have removed reference to this strain from this sentence.

2. line 10: immunosuppressive therapy should not be reported in the comment of section (c) because the study reported excluded patients on immunosuppressive therapy. We agree with this point and have removed the reference to immunosuppressive therapy.

3. line 27: Among the reasons explaining the differences in reported rates of CDI in IBD patients between the large-scale epidemiological studies and the single centre studies, should be stressed that CDI under reporting could be explained in administrative database studies by reimbursement reasons and that the results form single centre studies are very heterogeneous and often based on very small populations. We don't understand the reviewer's first point here. Reimbursement biases would tend to lead to over-diagnosis (and as in our discussion of 'diagnostic creep' to explain why incidence rates may be increasing) as opposed to under-reporting. We accept the point regarding sample size and heterogeneity of the small studies and have added this to the section dealing with publication bias.

Page 11: 1. ?However, the average age of CDI in IBD cohorts is much lower than in the general population controls, suggesting that patients with IBD have different risk profiles (Table 3)? Please check. In table 3 the age is reported only in 2 studies and within these studies, not for all populations of patients. In one study the average age of the mixed population was 47, in the other study in the CDI population with no IBD tha average age was 73. We have checked this and made no changes.

Page 12: 1. ?Based on those studies adequately powered to detect a difference 33, 39-41, 43 and accepting their limitations as outlined above (Sections 3d and 4), patients with UC seem to be more at risk than patients with small bowel if not colonic Crohn?s disease 41? Administrative Database studies do not report the disease localisation if in the ileum or colon. Only the large cohort study by Issa33 report that CDI was detected in 42 colonic disease out of 46 total CDI. This was a composite summary sentence. We have now moved the references round in the sentence so that the studies tally better with what is stated.

2. line 11: it should be specified that only a single study among those adequately powered reported disease extension. Done. 1. line 50 comments: this is the personal opinion of the authors. I would suggest that antibiotics could be associate to iv corticosteroids, while awaiting the culture results , in patients with high risk of CDI infection. We have added this. Page 14: 1. line 22, colectomy rate and length of stay: the text is not clear and difficult to read and to understand , particularly the last sentence of the section. Missing the reference at line 34.

We chose to incorporate the length of stay and colectomy data in one paragraph to avoid repeating the comments to account for the differences, on the grounds of brevity and ease of reading.

We have added the Jodorkovsky reference.

Page 15: 1. line 6: 4.3% mortality?

We have changed the 5% to about 4% as suggested (the mean of all the mortality rates reported is 4.2%).

2. line 10: ? In contrast, taking all the deaths in each of the included cohort studies, only 0.6% (3/797) patients died 45, 48, a rate similar to the background mortality rate for IBD reported in the database studies? Cannot understand from where these figures are from. Reference 45 and 48 refers to studies that are not included in Table 1-2. These should be clearly explained. We have clarified this as suggested.

3. line 23-24: Why cases with IBD as secondary diagnosis were removed? The first diagnosis was CDI or other? Comorbidities? This should be clearly explained. We have clarified this point as requested.

4. The higher rates of CDI infection in IBD patients than in control population reported in the administrative database studies should be stated at the beginning of the conclusions before the comments on the increasing incidence. We have changed this text as requested.

Authors should revise the alignment of the Tables. We will ensure that all the tables are aligned accorded to the house style during editing and typesetting. In Table 2 are reported all the 15 studies, including the study excluded from analysis , instead in Table 1 are reported only the studies included in the analysis. All studies that reported the incidence (using a population based denominator) are included in each table. We chose to leave all the studies in each table to highlight the differences in ways in which studies were reported. Reviewer 3 appears to condone this approach.

Reviewer: 3 Comments for Transmission to the Authors I commend the authors for this excellent review which is presented in a structured, easy to understand manner. The results have been presented very well in a coherent fashion. In particular, care has been taken to avoid pooling of heterogeneous studies. I have no additional comments or suggestions.

We hope that the changes we have made to the manuscript are acceptable. . In recent years, as well as a rising incidence, there has been a skew in the spectrum of illness towards a more severe disease course with more fatalities, in part explained by the emergence of the hypervirulent NAP1/027 strain exemplified by the outbreak in Quebec in 2002 [START_REF] Pepin | Clostridium difficile-associated diarrhea in a region of Quebec from 1991 to 2003: a changing pattern of disease severity[END_REF] and now reported across North America and Northern Europe .

C. difficile and inflammatory bowel disease

The characteristics of diarrhoeal relapse in IBD are often indistinguishable from those of enteric infections, including C. difficile. The marked differences in therapeutic approach, that is, specific antibiotic therapy versus escalation of immunosuppression to treat CDI and active IBD relapse respectively, emphasise the importance of promptly identifying superimposed infection. Recent epidemiological data report more cases of CDI in IBD than controls and a worse outcome in patients with IBD 8-10

.

We have now sought to review systematically the literature, using PRISMA guidelines [START_REF] Moher | Preferred reporting items for systematic reviews and meta-analyses: the PRISMA statement[END_REF] , in order to evaluate not only temporal trends in incidence and associated mortality rates, but also disease-specific risk factors, diagnosis and treatment of C. difficile infection in patients with established IBD.

METHODS

Search criteria

Structured searches of Pubmed were conducted using the medical subject heading terms (MeSH)

"inflammatory bowel disease" and "clostridium difficile"; the search was repeated using "Crohn's disease" and "ulcerative colitis" in place of "inflammatory bowel disease". This search was last performed on 13 th September 2010. Eligible studies as well as relevant review articles were hand searched to identify any additional studies omitted by the database searches. Non-English language articles were included and translated when required.

Eligibility criteria

Published studies of incidence of CDI, its association with relapse of IBD, temporal trends, disease-specific risk factors, diagnosis and endoscopic features, treatments and outcomes, including recurrence of CDI in IBD were eligible. Titles and abstracts were screened for their relevance, and all eligible articles were assessed in full.

Adult and paediatric cross-sectional, case-control and cohort studies, but not case reports or reviews were included. In vitro and animal studies were excluded. We have not attempted to review studies focussing on the pathogenesis of CDI in IBD. JRG and WA independently reviewed all identified citations to determine eligibility for inclusion. Discrepancies were resolved by consensus of the study team.

Data Abstraction

Using a standardized abstraction tool, the following data points were collected for each included study: location and design, sampling time frame, number of patients included with ulcerative colitis (UC) or Crohn's disease, age, method of diagnosis, incidence of CDI by conventional culture and the presence of its toxins, risk factors for acquisition, treatment and outcome (length of stay, colectomy, recurrence, mortality rates).

Case ascertainment

Case definition: Most authors now agree that laboratory tests should be limited to diarrhoeal stool samples, defining a case of CDI by a positive toxin assay from a stool that takes the form of its container 12 .

Unfortunately, no universal definition of a case of CDI associated with IBD has been adopted by researchers publishing on this topic. Therefore, we have grouped the studies according to whether the populations reported included patients exclusively in diarrhoeal relapse (true CDI) (Table 1), patients with quiescent as well as active IBD (Table 2) or quiescent disease only. In order to avoid bias by including duplicate positive samples from a single individual, wherever possible we have presented the number of patients infected and not the number of positive stool samples tested: studies that reported only proportions of positive stool samples were excluded from the summative analyses.

Diagnostic testing: Stool culture that allows antibiotic sensitivity testing and strain typing is widely available and, until recent advances in polymerase chain reaction (PCR), has been considered the most sensitive method of C. difficile detection. Importantly in view of the above definition of CDI, it cannot detect toxin production.

Cell cytotoxicity assays are both sensitive and specific but are labour intensive, expensive and require 2 days to complete: despite being able to detect toxin at much lower levels, they have been largely superseded by immunological assays. We have reported separately the proportions of patients with culture and toxin-positive infections, outlining the method of diagnosis used in each instance. For patients with toxin-positive infections, we have reported, where available, the proportions with colonic disease, exposure to preceding antibiotics, and with suspected or proven nosocomial acquisition. We have considered separately more recent studies that have relied on diagnostic coding to identify CDI cases from nationwide insurance databases (Table 3).

Quantitative analysis

The marked heterogeneity in study designs and definitions of CDI used, as well as diagnostic techniques employed, limited the overall quantitative summaries to mean (95% confidence intervals [CI]) proportions weighted according to the sample size [START_REF] Bland | Statistics notes. Weighted comparison of means[END_REF] ; more complex analysis of the data was considered futile.

STUDIES

In the interests of brevity and convenience for the reader, we have followed each of the main results sections (3-8 below) with relevant comments, rather than combining them in the Discussion.

Search results

A total of 407 studies were returned using the Pubmed search criteria described above; an additional 2 studies were identified by searching the other papers' reference lists (Figure 1). 195 studies were excluded on the basis of the title and abstract as shown in Figure 1. 54 studies were assessed in full for eligibility, of which 12 54-57, 59-62, 64-67 were excluded for the reasons shown in Figure 1: overall 42 records were included in the qualitative synthesis.

Overview of included studies

The majority of the studies included were single-centred (81% : not only the site of the underlying disease but also the immunosuppression and antibiotics used to treat it may be a risk factor for acquisition (see section 5 below).

b. Studies of patients with mixed disease activity. Fifteen further studies, three of which were retrospective, reported the incidence of CDI in patients who were not stratified for disease activity ( Comments: As with the studies of patients in relapse there was no significance difference in the incidence of CDI in UC and Crohn's in the studies of patients with mixed disease activity: the reasons for this are likely to be similar to those above (see comments 3a). The larger cohort studies by Issa [START_REF] Issa | Impact of Clostridium difficile on inflammatory bowel disease[END_REF] and Rodemann [START_REF] Rodemann | Incidence of Clostridium difficile infection in inflammatory bowel disease[END_REF] contained enough cases of CDI to allow identification of risk factors including disease type. They reported that the incidence of CDI in UC (6% and 4% respectively) was approximately double that in Crohn's disease (4% and 2% respectively). However, care has to be taken when interpreting these studies: both were retrospective and may have been subject to bias because of under-testing for C. difficile and missing data.

Because patients with both active and inactive disease were included, it is unclear from the studies shown in Table 3 whether or not there is an association between CDI and IBD activity. Indeed, this heterogeneous group of studies may have been under-powered to detect one. Alternatively, in the absence of criteria specifying diarrhoea at inclusion, the lack of association between CDI and IBD activity may be explained by C. difficile being a frequent commensal in IBD, a conclusion supported by a study of C. difficile colonisation in patients with inactive IBD [START_REF] Clayton | The vexed relationship between Clostridium difficile and inflammatory bowel disease: an assessment of carriage in an outpatient setting among patients in remission[END_REF] (see Section c below).

c. Studies of patients in remission

One prospective study has compared the colonization rate of adults with quiescent IBD with that of healthy controls [START_REF] Clayton | The vexed relationship between Clostridium difficile and inflammatory bowel disease: an assessment of carriage in an outpatient setting among patients in remission[END_REF] 3). In these studies, the co-incident hospitalisation rate of CDI and UC is about 3%, and of CDI and Crohn's disease 1%, both higher than in the control population (0.5%), but lower than in most of the single centre studies described above (Tables 1 and2).

Comments: The differences in reported rates of CDI in IBD patients between these large-scale epidemiological studies of hospitalised patients and the single centre studies outlined in Tables 1 & 2 may have several explanations. Firstly, the large database studies because of their retrospective design may have underestimated the incidence of C. difficile as a result of miscoding, missed diagnoses and under-testing for the organism. The accuracy of ICD coding for CDI has been assessed and although it is 99% specific when compared to microbiological data, its sensitivity was only 82%

75

. Secondly, these studies may conceivably have overestimated the denominator (IBD) in these calculations as the cause of the relevant hospital admission: this could be a consequence of the coding phenomenon labelled diagnostic creep

76

. Thirdly, the studies shown in Tables 1 and2 included a mix of in-and out-patients, and it is possible that these two patient groups have different CDI rates. Fourthly, including only patients who are insured will tend to exclude patients of lower socio-economic status, a factor which although unproven might lead to underestimation of CDI rates. Finally, publication bias in the heterogeneous and mostly small cohort studies may account for a relatively higher rate in incidence reported by the single centre studies, since small cohort studies showing low CDI rates may not have been accepted or even submitted for publication; in contrast, studies reporting higher incidence rates, for example following a significant single centre outbreak, are more likely to have been considered favourably by journal editors. Differences in study design may also explain why the database studies, unlike those listed in Tables 1 and2, have shown a higher rate of CDI in UC than in Crohn's disease.

Sonnenberg's analysis of data from the HCUP NIS database [START_REF] Sonnenberg | Similar geographic variations of mortality and hospitalization associated with IBD and Clostridium difficile colitis[END_REF] suggests a north-south divide in the USA, with states in the north-east having a higher prevalence of CDI and associated mortality than those in the south. It is not clear whether there is an environmental risk factor predisposing IBD patients to CDI in the north, or if C.

difficile is shaping the geographic patterns of IBD admission. , we examined the literature for temporal changes of the incidence of CDI complicating IBD.

There was no temporal trend over the years 1980-2010 in the published incidence rates for CDI in the fourteen studies (Table 1) that reported it in patients in relapse of either UC or Crohn's disease. In contrast, the data from the two large studies of Issa et al [START_REF] Issa | Impact of Clostridium difficile on inflammatory bowel disease[END_REF] and Rodemann et al [START_REF] Rodemann | Incidence of Clostridium difficile infection in inflammatory bowel disease[END_REF] that confirmed toxin-positive disease, and from those relying on diagnostic coding (Table 3), all suggest a steady increase in incidence in the last 10 years, particularly in UC and colonic Crohn's disease.

Comments: Why is there an apparent steady increase in the incidence of CDI complicating IBD in the very large studies but not in the smaller ones? It is difficult to determine from the former the extent to which the observed trend is due to increased clinical awareness of and therefore testing for CDI in patients presenting with active IBD, or to diagnostic creep 76 . Conversely, given the different populations studied and diagnostic methods used, it is not possible to amalgamate data from all the studies to evaluate temporal trends in CDI incidence.

Risk factors for acquisition of C. difficile in IBD

One of the major advantages of the database (Table 3) and other large studies 36, 39 is that their large sample sizes allows identification of risk factors associated with, if not causal for, acquisition and associated mortality of CDI in IBD. Against the usefulness of the database studies in this respect is that only a limited number of potential risk factors are entered onto the HCUP database: perhaps most importantly it is not possible to determine IBD activity from these studies.

a. Age & co-morbidity

Nguyen et al [START_REF] Nguyen | A national survey of the prevalence and impact of Clostridium difficile infection among hospitalized inflammatory bowel disease patients[END_REF] reported in adults that, like in the general population, increasing age and co-morbidity increase the risk of acquisition of CDI in IBD. However, the average age of CDI in IBD cohorts is much lower than in the general population controls, suggesting that patients with IBD have different risk profiles (Table 3).

b. Community versus nosocomial infection

C. difficile is the most common nosocomial infection in the UK. However, community-acquisition is recognised in the general population and is estimated at running between 10.7% [START_REF] Riley | Community-acquired Clostridium difficile-associated diarrhea[END_REF] Based on those studies adequately powered to detect a difference and accepting their limitations as outlined above (Sections 3d and 4), patients with UC seem to be more at risk 36, 42-44, 46 than patients with small bowel if not colonic Crohn's disease [START_REF] Ricciardi | Epidemiology of Clostridium difficile colitis in hospitalized patients with inflammatory bowel diseases[END_REF] . Furthermore, in UC, patients with more extensive disease appeared to be at greater risk than those with distal disease alone, in the one adequately-powered study that has reported it 36 .

d. Drug Usage

The main pharmacological risk factor for CDI in the general population is use of broad spectrum antibiotics 80 .

Patients taking immunosuppressants and proton pump inhibitors (PPIs) are also frequently affected 54, 55 .

Antibiotic use in IBD, however, is less of a prerequisite for CDI than in the non-IBD population. Where reported, and taken together (Table 1), only 43% (23/54) toxin-positive relapse samples were associated with antibiotic use (up to 3 months before inclusion). Similarly, for those patients with mixed IBD activity (Table 2), only 52% (51/98) toxin-positive samples were associated with preceding antibiotic use.

Single centre cohort studies have infrequently reported immunosuppression as a risk factor, perhaps because most were conducted before the widespread use of immunosuppressants, and were therefore underpowered to detect small effect sizes. .

Comments: Why pseudomembranes are infrequently seen in IBD is unclear, but flexible endoscopy cannot be recommended for diagnosis of CDI in patients with IBD.

Treatment and outcome of CDI in IBD

There is very little data, and in particular there are no controlled trials of the efficacy of antibiotics or other treatments for CDI in IBD.

Medical treatment:

Where reported in the studies shown in Tables 1 &2 , it suggests that, at least in children with IBD, there is an increased risk of recurrence of CDI.

Colectomy rates and length of stay: Contradictory data are reported concerning length of hospital stay (LOS).

In the studies using NIS HCUP data, the LOS was significantly longer (up to 3 days longer) in the IBD patients than in controls (Table 3). In contrast, reports from most single centre cohorts suggest that the LOS of IBD patients with CDI is similar to or shorter than that of CDI controls 36, 51

.

Variable rates of colectomy have been reported in IBD complicated by CDI (Tables 123). Analysis of the NIS HCUP for colectomy rates is unhelpful, since IBD patients' admissions are frequently for elective surgery, a fact likely to make a substantial contribution to the 6 fold increase reported over the colectomy rate for patients with CDI unrelated to IBD (Table 3). 1.

Comments: The conflicting rates in LOS and colectomy in the above studies could conceivably be accounted for by differences in the interests and experience of the primary caring physician as well as in the threshold for surgery. The single centre cohort experiences were reported by gastroenterologists with interests not only in IBD but also C. difficile. It is likely that the index of suspicion for diagnosis of, rates of testing for, and use of empiric treatment of CDI would be higher in these specialist centres than those associated with the practice of less specialist physicians sampled in the NIS HCUP databases. Furthermore, although specific ribotypes were not reported by Jodorkovsky or Issa et al, the hypervirulent strain NAP1/027 was endemic in the USA during their sampling time frames and might have contributed to the higher colectomy rates occurring in the USA than Europe Mortality: Similar discrepancies exist for data regarding mortality. The NIS HCUP database studies suggest that about 4% patients with CDI complicating IBD will die on that admission, the risk factors for mortality being increasing age, co-morbidity, treatment in a teaching hospital, intestinal surgery and lack of an insurance policy. In contrast, taking all the deaths in each of the included cohort studies (Tables 1&2 andreferences 48,

51

), only 0.6% (3/797) patients died, a rate similar to the background mortality rate for IBD reported in the database studies.

Comments: Possible explanations for these contrasting mortality rates include the facts that in the database studies all of the patients were in hospital and therefore presumably more unwell than many of those in the cohort studies reporting on both out-and in-patients, and that their average age was higher. The database studies are restricted to all-cause mortality on that admission whereas the deaths reported in the single centre cohorts were more likely to be C. difficile-related. Furthermore, at least in the Anathankrishnan dataset 42 , when the mortality rates were re-analysed by primary (IBD) versus secondary diagnosis (other co-morbidity), after removal of the cases where IBD was listed as a secondary diagnosis, the association between mortality and IBD was no longer statistically significant 42 .

CONCLUSIONS

C. difficile infection has become a major public health problem. Although the US administrative database studies suggest that the incidence of CDI in IBD, as in the general population, is increasing, their design makes it hard to distinguish the extent to which this reflects growing awareness of CDI in IBD, with consequent detection bias. Infection without prior healthcare contact or antibiotic exposure is common in IBD patients.

IBD appears to be an independent risk factor for commensal carriage compared with the general population, and the difference in risk profile between IBD patients and the general population warrants further investigation. Endoscopy is rarely helpful in diagnosing CDI in patients with IBD. Unfortunately, there have been no prospective studies of treatment of CDI specifically in patients with IBD but these are urgently needed given the apparently adverse outcome associated with the use of combined antibiotic and immunosuppressant therapy compared with antibiotics alone. Therapeutic decision-making would be easier with the development of more rapid, for example PCR-based, methods of detection of C difficile. For the moment, however, a high index of suspicion for CDI in all patients presenting with active colonic IBD is essential.

F

F o r P e e r R e v i e w

Section/topic # Checklist item Reported on page # Risk of bias across studies 15 Specify any assessment of risk of bias that may affect the cumulative evidence (e.g., publication bias, selective reporting within studies).

-15

Additional analyses 16 Describe methods of additional analyses (e.g., sensitivity or subgroup analyses, meta-regression), if done, indicating which were pre-specified.

N/A

RESULTS

Study selection 17 Give numbers of studies screened, assessed for eligibility, and included in the review, with reasons for exclusions at each stage, ideally with a flow diagram.

& 7

Study characteristics 18 For each study, present characteristics for which data were extracted (e.g., study size, PICOS, follow-up period) and provide the citations.

-28

Risk of bias within studies 19 Present data on risk of bias of each study and, if available, any outcome level assessment (see item 12).

-15

Results of individual studies 20 For all outcomes considered (benefits or harms), present, for each study: (a) simple summary data for each intervention group (b) effect estimates and confidence intervals, ideally with a forest plot.

-28

Synthesis of results 21 Present results of each meta-analysis done, including confidence intervals and measures of consistency.

-15

Risk of bias across studies 22 Present results of any assessment of risk of bias across studies (see Item 15).

-15

Additional analysis 23 Give results of additional analyses, if done (e.g., sensitivity or subgroup analyses, meta-regression [see Item 16]).

N/A DISCUSSION Summary of evidence 24 Summarize the main findings including the strength of evidence for each main outcome; consider their relevance to key groups (e.g., healthcare providers, users, and policy makers).

-15

Limitations 25 Discuss limitations at study and outcome level (e.g., risk of bias), and at review-level (e.g., incomplete retrieval of identified research, reporting bias).

-15

Conclusions 26 Provide a general interpretation of the results in the context of other evidence, and implications for future research.

  18-19 check the quote. Reference 52 seems to be about antibiotics. We have re-checked ref 52 (now reference 55, Dial et al 2008) and it discusses not only antibiotics but also PPIs. 4. line 34: cannot find reference 60 in the text e within the studies in the tables. Page 13 The Sonnenberg analysis of the US database [Ref 60, now Ref 63] is dealt with on page 10 L54-59; it is not in the tables for reasons given above. Page 13

3 BACKGROUND 2 . 3 . 4 .

 3234 Clostridium difficile is a gram positive spore-forming bacterium whose effects range from asymptomatic carriage through mild diarrhoeal disease to fulminant colitis. Although pseudomembranous colitis was first described in 1893, the microbiological characterisation of C. difficile did not occur until 1935 1, Despite this, the causal association between the two was established only in the 1970s, when Tedesco et al reported C.difficile-induced diarrhoea in 21% patients given clindamycin, half of whom had pseudomembranes at endoscopy Since then nearly all broad spectrum antibiotics have been implicated in the acquisition of C. difficile infection (CDI), which is now the most common nosocomial infection in the UK The changing face of C. difficile infection Historically, C. difficile was considered an infrequent nuisance complicating necessary antibiotic usage in mostly elderly hospitalised patients. Even allowing for the change to mandatory case reporting in 2004, it seems remarkable now that in the UK in 1990 there were apparently only 1172 cases of CDI nationwide: in 2008 there were over 50,000 5

7

 7

  Systematic Review: Clostridium Difficile & IBD

4

 4

  Systematic Review: Clostridium Difficile & IBD 5

  Systematic Review: Clostridium Difficile & IBD 9

50 .

 50 Systematic Review: Clostridium Difficile & IBD 10 criteria of this study, indicates acquisition in the community rather than from a limited number of healthcare sources Comments: The reason why patients with IBD are more frequently colonised than controls has not yet been elucidated. d. Association of C. difficile with IBD in studies using diagnostic coding Utilising data from the US Healthcare Cost and Utilisation Project (HCUP) Nationwide Inpatient Sample (NIS), 4 studies using International Classification of Disease (ICD) diagnostic coding to identify cases have employed a retrospective nested case-controlled design to compare CDI rates and outcomes in hospitalised IBD patients and controls (Table

4 .

 4 Temporal changes in the incidence of CDI in IBDIn view of recent data from the general population showing that the incidence of C. difficile is increasing over time5, 77

  Systematic Review: Clostridium Difficile & IBD 15

15 FUNDINGFunding 27

 1527 Describe sources of funding for the systematic review and other support (e.g., supply of data); role of funders for the systematic review.N/AFrom: Moher D, Liberati A, Tetzlaff J, Altman DG, The PRISMA Group (2009). Preferred Reporting Items for Systematic Reviews and Meta-Analyses: The PRISMA Statement. PLoS Med 6(6): e1000097. doi:10.1371/journal.pmed1000097

  The studies analysed in sections a-c below reported the incidence of CDI, measured by culture and/or toxin assay, in cohorts of patients with IBD which was active14-16, 19-21, 23-26, 28-30, 32 (Table1), of mixed activity

			Systematic Review: Clostridium Difficile & IBD
	3. Incidence of CDI in IBD
	407 records identified through database search 27, 31, 33-41, 45, 68 (Table 2) or quiescent Identification 50 . In section d the results of United States (US) nationwide diagnostic 2 records identified through other sources 17, 18, 22,
	coding-based case-controlled studies (Table 3), that did not require laboratory based confirmation of CDI are
	described	42, 43, 46, 63, 69 .
	a. Studies of patients in diarrhoeal relapse
	249 records screened after duplicates removed Of the 35 studies reporting the incidence of CDI in IBD (Tables 1-3), only 40% [14/35] included patients Records excluded (195) Screening exclusively in relapse and therefore satisfied the criteria for CDI. They included 640 patients with UC, 480 with
		F o r F o r
	Eligibility		P e 54 full text articles assessed for eligibility P e
			e r e r
	Inclusion		R e v i e 42 records included in qualitative synthesis R e v i e
			w w
	The significantly higher rates of CDI complicating IBD in paediatric studies compared with those in adults could
	be due to several factors. Firstly, faeco-oral ingestion of spores may be more common in children	45, 58 than in
	adults. Secondly, and in relation to IBD, children are more likely than adults to have colonic Crohn's disease
	and to have extensive UC	72-74
			8

[34/42]

), and conducted in North America (48%

[20/42]

) or Europe (45%

[19/42]

). Patients were recruited to these studies between 1979 and 2008. Overall, 55%

[23/42]

were prospective cohort studies. The remainder were retrospective observational studies. The primary aim in 86%

[36/42]

studies was to report the incidence of CDI in IBD. The remainder were designed to define risk factors for acquisition and/or to report outcomes following infection. Systematic Review: Clostridium Difficile & IBD 7 Figure 1: PRISMA 11 flow diagram describing the identification, screening, eligibility and inclusion of studies. • IBD & CDI Case reports (39) • Non IBD &/or CDI Case reports (19) • IBD & CDI Reviews (20) • Non IBD &/or CDI reviews (85) • Non IBD &/or CDI original papers (18) • In vitro & animal studies (14)

Records excluded

(12)

• IBD reported as a risk factor in population-based cohort studies (4)

• Case report/series (3) • Non IBD or mixed denominator (2) • Letters to Journals (2) • Study of cultured mucosal biopsies (1)

Crohn's disease, and 40 with colitis of uncertain type or aetiology (CUTE) (Table

1

). The median [range] number of IBD patients in each study was 60

[18-284]

. One study was excluded from the following analysis because the authors reported the incidence of CDI using the number of positive stool samples tested 28 . There were no significant differences in the weighted mean [weighted 95% CI] incidence of CDI in adult patients with UC or Crohn's disease by either culture (UC 6.7%

[5.6-11.4

], Crohn's disease 8.8% [2.5-15.1]) or toxin assay (UC 3.1% [0.1-6.2], Crohn's disease 7.0% [1.8-12.2]). Two studies 19, 25 reported incidence rates of CDI in IBD in children. The weighted mean incidence of CDI in IBD by toxin was higher than in adult patients 26.1% [16.1-36.1]. Only one of these studies reported the incidence in UC and Crohn's disease separately, so no further comparative analysis was made. Comments: The observation that CDI seems to be as common in active UC as in active Crohn's is surprising given C. difficile's predilection for the colon 70 , and the fact that about half of adults with Crohn's have terminal ileal disease with little or no colonic involvement 71

. It is possible that because of the relatively small sample sizes of each study, differences in the incidence rates between UC and Crohn's may have been missed. It is also possible that the apparent incidence of CDI in Crohn's disease was boosted by the preferential testing for CDI of patients with Crohn's colitis by clinicians believing the infection to be unlikely in patients with small bowel disease. Further elucidation of this explanation is impossible because too few studies reported Crohn's disease phenotypes.

Table 2 )

 2 . They included 1810 patients with UC and 3311 with Crohn's disease as well as a group of 28 patients with undefined IBD. The median [range] number of IBD patients in each study was 54[10-3397]. Based on sample size, two outlying studies were identified, Issa et al

	36 (n=999) and

19, 28, 34, 35

studies that reported the relationship between cdi and disease activity found a positive association.

  using polymerase chain reaction (PCR) and pulsed field gel electrophoresis to detect and type C. difficile. To eliminate confounding influences and study the effect of IBD alone, patients hospitalised and/or

treated with antibiotics or immunosuppressants within 6 months of recruitment were excluded. Adults with IBD were more frequently colonised with toxigenic strains of C. difficile than healthy controls (UC 9% [6/64], Crohn's 7%

[4/58]

, controls 1% [1/99]). The marked heterogeneity of strains identified and the strict inclusion

  While there have been no studies comparing different microbiological methods of diagnosis CDI in patients with IBD, flexible endoscopy has been assessed as a diagnostic method in this setting. Pseudomembranes,

					Systematic Review: Clostridium Difficile & IBD
	7.	Endoscopic features of CDI in IBD
	comprised of neutrophil degradation products, fibrin, and bacterial remnants filling the intestinal crypt, are
	characteristic of C. difficile infection and at endoscopy are seen in approximately 50% of patients without
	IBD 83 . However, in 7 studies in patients with IBD	31, 32, 35-37, 47, 49 only 9% (13/150) patients were found to have
	pseudomembranes at sigmoidoscopy or colonoscopy. In one study, fever at onset predicted
	pseudomembranes at subsequent endoscopy	47
				F o r
				P
				e
				e r
	1.53 -4.57)	52 .		R e	3 95%CI
	PPI use has been reported in only 2 studies in IBD to detect significant effects.	38, 48 and the numbers of patients taking them were too small v i e
	6.	CDI in specific IBD-related circumstances	w
	Pouchitis and enteritis: In a study omitted from the above analyses because it restricted its inclusion to 115
	patients with an ileo-anal pouch for UC, C. difficile was detected by ELISA in the stools in 18% (21/115)
	patients. There was a positive association with pouch disease activity scores and endoscopic inflammation,
	which improved with treatment with rifamixin or tinidazole	53 . On multivariate analysis, risk factors associated
	with CDI included male gender (OR 5.1 [95% CI, 1.4-20.5) and distal disease preoperatively (OR 8.4 [95%CI1.3-
	56.4]. C. difficile infection has been reported to cause an acute enteritis in IBD patients with ileostomies in
	several case studies (for review 81 ) and colitis in the rectal stump of a patient with ileostomy and sub-total
	colectomy for UC	82 .
					13

Issa et al reported that immunosuppression, defined as any of thiopurines, methotrexate and/or steroids, doubles the risk of CDI (OR 2.58 95%CI 1.28 -5.12) in patients with IBD 36 . The database studies 42-44, 46, 63 do not include pharmacological data and so were unable to report on drug risk factors. Schneeweiss et al, by combining insurance healthcare records and a pharmacological database from admissions in British Columbia, found no association with infliximab or immunomodulator use and admission of IBD patients with C. difficile infection: corticosteroids, however, increased the risk three fold (OR 3.

  In the absence of prospective randomised controlled data, antibiotic therapy alone for CDI occurring in patients with acute severe IBD cannot yet be recommended. In sick inpatients, it remains entirely reasonable to start patients on intravenous corticosteroids and metronidazole or vancomycin while awaiting the results of C. difficile assay. Conversely in less unwell outpatients, decision making can often await the result of stool testing.Systematic Review: Clostridium Difficile & IBD 14 experienced a recurrence, significantly more than in the non-IBD controls (7.5%). No difference in recurrence rates were seen according to the use of acid suppression or immunosuppressants, or to whether metronidazole or vancomycin was used to treat CDI 58 . Wultanska et al in their prospective cohort of Polish children reported that 17% of their 58 patients experienced a recurrence, but did not insist on microbiological evidence of CDI clearance between the first and subsequent infection. Indeed, when strains were genotyped

	using PCR ribotyping, similar proportions of patients had acquired a new strain (re-infection) as had persistent
	carriage of the original one 45 .
	Comments: Although this data is limited, the former by being retrospective and subject to bias in relation to
	how controls were identified	58 , and the latter by the imprecise definition of recurrence	45
	F o r
		P
		e
		e r
		R
		e
		v i e
		w

, 88% (300/340) patients were treated with metronidazole, vancomycin or both. In most cases (88% [258/292 patients]), this treatment was apparently successful. Since the symptoms of active IBD are usually indistinguishable from those of CDI, CDI in patients with IBD is frequently treated with both antibiotics and immunosuppression, particularly corticosteroids. In a retrospective cohort study from centres across Europe 48 , 12% [11/104] patients given antibiotics together with immunomodulators reached a composite endpoint of death or colectomy within 3 months of admission or inhospital megacolon, bowel perforation, haemodynamic shock or respiratory failure, compared with none of 51

given antibiotics alone. This difference did not appear to be explained by the patients given combined therapy being sicker at the outset than those given antibiotics alone. Use of more than one immunomodulator increased the risk of having an adverse outcome independently of disease severity at presentation (OR 17 95%CI 3.2-91).

Comments:

Recurrence of CDI in IBD: We could find no relevant studies in adults. One retrospective nested case control study in children defined recurrence as a separate episode of CDI within 60 days of the index infection, with microbiological evidence of CDI clearance between the two infections. 34%

[38/111]

children with IBD

  Jodorkovsky et al

	51 , describing the long term outcome of 46 UC patients
	with superimposed CDI treated with appropriate antibiotics, reported that over the following year 44% of the
	UC -CDI patients had come to colectomy compared with only 25% in the non-infected UC control group; these
	figures are similar to the colectomy rates reported by Issa et al in their IBD cohort in 2004 (45%) and 2005
	(25%)

36

. These US colectomy rates are much higher than the immediate rates reported from Europe by Bossuyt et al

[START_REF] Bossuyt | Increasing incidence of clostridium difficile-associated diarrhea in inflammatory bowel disease[END_REF]

[5% (1/15)], and Ben-Horin et al

[START_REF] Ben-Horin | Combination immunomodulator and antibiotic treatment in patients with inflammatory bowel disease and clostridium difficile infection[END_REF]

[6% (9/152)]. Indeed colectomy was reported in only 1 of 49 patients in whom outcome was detailed in the single centre studies shown in Table

Table 1 :

 1 Studies reporting the incidence of C. difficile infection in association with relapse of IBD; n/r, not reported; UC, ulcerative colitis; CD, Crohn's disease; CUTE,

	o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 2 :

 2 Studies reporting the incidence of C. difficile infection in populations with mixed disease activity. n/r, not reported; UC, ulcerative colitis; CD, Crohn's disease;

	criteria

CUTE, colitis of uncertain aetiology; CCA, cell cytotoxicity assay; n/a, not applicable; C , colectomy; EIA, Enzyme immunoassay; a , toxin A; b , toxin B; OIA, optical immunoassay; ELISA, enzyme-linked immunosorbant assay; LA, latex agglutination; ≈ incidence reported per stool sample tested. Systematic Review: Clostridium Difficile & IBD 30

Table 3 :

 3 Studies that used diagnostic coding to report the association of C. difficile infection in hospitalised patients with IBD. All studies involved retrospective nested case control analysis of nationwide hospital discharge databases. UC, ulcerative colitis; CD, Crohn's disease; MIP, mixed inpatient population; n/r, not reported; n/a, not applicable.

F o r P e e r R e v i e w

Section/topic # Checklist item Reported on page # TITLE Title 1 Identify the report as a systematic review, meta-analysis, or both.

N/A

Eligibility criteria 6 Specify study characteristics (e.g., PICOS, length of follow-up) and report characteristics (e.g., years considered, language, publication status) used as criteria for eligibility, giving rationale.

4

Information sources 7 Describe all information sources (e.g., databases with dates of coverage, contact with study authors to identify additional studies) in the search and date last searched.

3/4

Search 8 Present full electronic search strategy for at least one database, including any limits used, such that it could be repeated.

3/4

Study selection 9 State the process for selecting studies (i.e., screening, eligibility, included in systematic review, and, if applicable, included in the meta-analysis).

& 8

Data collection process 10 Describe method of data extraction from reports (e.g., piloted forms, independently, in duplicate) and any processes for obtaining and confirming data from investigators.

4

Data items 11 List and define all variables for which data were sought (e.g., PICOS, funding sources) and any assumptions and simplifications made.

Risk of bias in individual studies

12 Describe methods used for assessing risk of bias of individual studies (including specification of whether this was done at the study or outcome level), and how this information is to be used in any data synthesis.

4,

Summary measures 13 State the principal summary measures (e.g., risk ratio, difference in means).

5

Synthesis of results 14 Describe the methods of handling data and combining results of studies, if done, including measures of consistency (e.g., I

2 ) for each meta-analysis.

5

Page 1 of 2


Teissandier Denis
email: d.teissandier@lmp.u-bordeaux.fr

Operations on polytopes: application to tolerance analysis

Keywords:

This article presents numerical methods in order to solve problems of tolerance analysis. A geometric specification, a contact specification and a functional requirement can be respectively characterized by a finite set of geometric constraints, a finite set of contact constraints and a finite set of functional constraints. Mathematically each constraint formalises a n-face (hyperplan of dimension n) of a n-polytope (1 ≤ n ≤ 6). Thus the relative position between two any surfaces of a mechanism can be calculated with two operations on polytopes : the Minkowski sum and the Intersection. The result is a new polytope: the calculated polytope. The inclusion of the calculated polytope inside the functional polytope indicates if the functional requirement is satisfied or not satisfied. Examples illustrate these numerical methods.

INTRODUCTION

The variational classes by Requicha were introduced at the beginning of the 80's and propose a model of tolerances of form, orientation, position and dimension [Requicha, 1983]. A generalization of specifications by volume envelopes is based on the works of Requicha [START_REF] Srinivasan | Conditional tolerances[END_REF]. Fleming presents a model for geometric tolerances and constraints from contacts [Fleming, 1988]. Among the dimension-chains models based on Small Displacements Torsor concept [START_REF] Clément | A Study of Optimal-Criteria Identification Based on the Small-Displacement Screw Model[END_REF], we note the Clearance Deviation Space by [START_REF] Giordano | Clearance Space and Deviation Space, Application to three-dimensional chains of dimensions and positions[END_REF]. The Clearance Deviation Space purposes an assembly method according to maximum material condition limit. This article presents numerical methods in order to solve problems of tolerance analysis. These methods deal with Tolerance Zones constructed by offsetting as in [Requicha, 1983]. We use the Small Displacements Torsor [START_REF] Bourdet | The concept of the small displacement torsor in metrology[END_REF]. Geometric specification and contact specification are characterised by the same model as in [START_REF] Giordano | Clearance Space and Deviation Space, Application to three-dimensional chains of dimensions and positions[END_REF].

EXPRESSION OF CONSTRAINTS

A Tolerancing Tool manipulates three sources of essential information: a. the geometric specifications (between associated surfaces of the same part). b. the contact specifications (between associated surfaces of two distinct parts), c. the functional requirements of an assembly (between any associated surfaces).

Geometric constraints

An associated surface is a surface of perfect form (i.e. an ideal surface: surface described with a finite number of geometric features). A nominal surface is an ideal surface by definition.

A geometric specification is formalised by geometric constraints of position between a nominal surface S0 and an associated surface S1 : see figure 1.

The tolerance zone (ZT) limit an area of space around S0 within which S1 must be situated: they are constructed by two (positive and negative) offsettings on S0 . We define geometric constraints of position between S1 and S0 as follows [START_REF] Teissandier | Threedimensional Functional Tolerancing with Proportioned Assemblies Clearance Volume: application to setup planning[END_REF]:

( ) t . k z . t . 1 k 1 k 0 with S N ZT S N 0 / 1 , N 0 1 ≤ ε ≤ - ≤ ≤ ∈ ∀ ⇔ ⊂ (1)
A unit vector zN is constructed such as zN is parallel to the local normal at any point N of S0 (see figure 1). Vector zN is oriented in such a way that the positive direction corresponds to the side exterior of the material.

3 With:

( )

0 / 1 0 / 1 , M 0 / 1 , N 3 3 0 1 0 / 1 0 1 0 / 1 , N ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ × ρ + ε = ε ∈ ∀ ρ ε
We use the property of linearization of displacements into small displacements [START_REF] Bourdet | The concept of the small displacement torsor in metrology[END_REF], [Clément et all., 1988]. We get according to (1):

( ) ( )t k 1 z . MN kt 1 k 0 with E M , S N ZT S N 0 / 1 0 / 1 , M 3 0 1 - ≤ × ρ + ε ≤ - ≤ ≤ ∈ ∀ ∈ ∀ ⇔ ⊂ (2)
In a base ( )

z , y , x
, the normal vector N z of S0 at point N can be written as follows:

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ c b a z N N N N and ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ d d d MN MNz MNy MNx . ( ) ( ) ( ) ( ) k.t c . d . d . b . d . d . a . d . d . t 1 k : 1 k 0 with E M , S N ZT S ) 2 ( N MNx 1/0y MNy 1/0x M 1/0z, N MNx 1/0z MNz 1/0x M 1/0y, N MNy 1/0z MNz 1/0y M 1/0x, 3 0 1 ≤ ρ - ρ + ε + ρ + ρ - ε + ρ - ρ + ε ≤ - ≤ ≤ ∈ ∀ ∈ ∀ ⇔ ⊂ ⇒ (3)
We obtain an infinity of equations ( 3). The unknowns are the six components written at point M:

ε ε ε ρ ρ ρ z 0 / 1 , M z 0 / 1 , M y 0 / 1 , M z 0 / 1 y 0 / 1 x 0 / 1 , , , , , .
Any surface can be discretised into n points Ni . So, it is possible to express a set of n equations (3). The n equations (3) characterize the n geometric constraints induced by the tolerance zone associated with S0 .

The vertices of this polytope correspond to the maximum and the minimum values of

ε ε ε ρ ρ ρ z 0 / 1 , M z 0 / 1 , M y 0 / 1 , M z 0 / 1 y 0 / 1 x 0 / 1 , , , , , .
This method can be applied on any ideal surface. We consider five types of surfaces: plane, cylindrical, conic and toric surfaces.

Contact constraints

Amongst the five main types of surfaces considered in the previous paragraph : plane, cylindrical, conic and toric surfaces, figure 2 summarizes the possible cases of joint. Since complex surfaces are not used in a joint between two parts (with the exception of a few particular cases i.e. gearing ) they will not be considered. Each case in the above table can be sub-classified into several other cases according to the relative position of nominal surfaces. Any joint is characterized by the types of two considered surfaces with a set of mating conditions. A mating condition is a set of constraints between geometric features of two ideal surfaces [Teissandier, 1995]. An exhaustive list of the cases has been compiled by [START_REF] Clément | The TTRSs : 13 Constraints for Dimensioning and Tolerancing[END_REF]: in order to define the relative position between two any surfaces 13 constraints has been identified. Following the same method as in the previous paragraph, a contact specification can be formalised by a set of n contact constraints of position between two associated surfaces of two distinct parts. Let us consider a joint made up of two planes S1 and S2 . S1 and S2 are nominally parallel and separated by a distance « d » (see figure 3). Vectors z1 and z2 are respectively constructed such as z1 and z2 are normal vectors of S1 and S2 (see figure 3) oriented in such a way that the positive direction corresponds to the side exterior to the material. Let us define surface S such as :

0 d with S S S 2 1 = ∩ =
. The set of mating conditions is:

{ } surface plane a is S , 0 z . z , 0 z z 2 1 2 1 < = ×
(4) If the set of mating conditions is satisfied, we can express a constraint of positioning :

D d 0 ≤ ≤
(5) A permanent contact between S1 and S2 is such as:

0 d 0 d 0 0 D = ⇒ ≤ ≤ ⇒ =
. If S is not a plane (line, point or empty hole), the previous constraint can not be defined: the set of mating condition is not verified. Let us consider the boundary (C) of S. At any point M of the Euclidean space E 3 , it is therefore:

( ) ( ) ( ) D z . MN 0 d z . MN 5 , 4 E M , S N N 2 / 1 2 / 1 , M N 2 / 1 2 / 1 , M 3 ≤ × ρ + ε ≤ ⇒ ≤ × ρ + ε ⇒ ∈ ∀ ∈ ∀ (6) (S 1 ) (S 2 ) (S) M d (C) N z N 1 z 2 z

Figure 3: Contact specification between two nominal parallel planes.

Following the same method as geometric constraints, we can write:

( ) ( ) ( ) D c . d . d . b . d . d . a . d . d . 0 ) 6 , 5 , 4 ( : E M , S N N MNx 1/0y MNy 1/0x M 1/0z, N MNx 1/0z MNz 1/0x M 1/0y, N MNy 1/0z MNz 1/0y M 1/0x, 3 ≤ ρ - ρ + ε + ρ + ρ - ε + ρ - ρ + ε ≤ ⇒ ∈ ∀ ∈ ∀ (7)
If z z N = , this corresponds to the particular case where: 1 c and 0 b a

N N N = = =
. Thus (7) can be written as follows:

D d . d . 0 ) 6 , 5 , 4 ( : E M , S N MNx 1/0y MNy 1/0x M 1/0z, 3 ≤ ρ - ρ + ε ≤ ⇒ ∈ ∀ ∈ ∀ (8)
Contact constraints (8) traduce the three degrees of freedom of the studied joint: 1 displacement in rotation and 2 displacements in translation at point M. As in the previous paragraph, we can obtain a finite set of n equations (7). The figure 4 illustrates an example of two surfaces S2,1 and S1,1 of any mechanism. S2,1 is an associated surface of nominal surface S2,0 and S1,1 is an associated surface of nominal surface S1,0 : see figure 4. The relative position between S2,1 and S1,1 is the result of a combination of sets of contact specifications and geometric specifications on different parts: see figures 4 and 5. In the figure 4 S2,0 and S1,0 are two nominal parallel planes. A functional requirement between S2,1 with regards to S1,1 is defined by a tolerance zone ZT which limits an area of space around 0 ZT within which S2,1 must be situated: see figure 4. ZT is composed of two parallel planes separated by the dimension t of ZT. These two surfaces are parallel to ZT 0 , a parallel plane to S1,1 . The relative position between S1,1 and ZT 0 is specified by the dimension d. A functional requirement is formalised by a finite set of n functional constraints between any associated surface with regards to any surface of the same mechanism. The method is the same as geometric specification.

Functional constraints

( )

( ) ( ) ( ) ( ) k.t c . d . d . b . d . d . a . d . d . t 1 k k.t z . t 1 k : 1 k 0 vec a E M , S N ZT S N MNx 2,1/1,1y MNy 2,1/1,1x M 2,1/1,1z, N MNx 2,1/1,1z MNz 2,1/1,1x M 2,1/1,1y, N MNy 2,1/1,1z MNz 2,1/1,1y M 2,1/1,1x, N M 2,1/1,1x, 3 2,0 1 , 2 ≤ ρ - ρ + ε + ρ + ρ - ε + ρ - ρ + ε ≤ - ⇒ ≤ ε ≤ - ≤ ≤ ∈ ∀ ∈ ∀ ⇔ ⊂ (9)
We obtain a finite set of n equations ( 8). The surfaces S1,1 and S2,1 can be (or can not be) specified from the same part. That means that a functional requirement can be reduced to a combination of two geometric specifications.

OPERATIONS ON POLYTOPES

Definition of a polytope

Geometric specifications, contact specifications and functional requirements can be respectively characterized by a finite set of geometric constraints (3), a finite set of contact constraints (8) and a finite set of functional constraints (9). Each constraint of (3), ( 8) and ( 9) corresponds to a n-face (hyperplan of dimension n: 0≤ n ≤6) in the real affine space R d . Mathematically, (3) defines a geometric npolytope (polytope of dimension n). By analogy, (8) defines a contact n-polytope and (9) defines a functional n-polytope.

Convex hull of a finite set points in R 2

Intersection of finitely many closed halspaces in R 2 which can be presented either as [Ziegler, 1995]: -a bounded intersection of finitely many closed halfspaces in some R d , -a convex hull of a finite set of points in some R d . The figure 6 illustrates the two definitions of a polytope. The dimension of a polytope is the dimension of its affine hull. A n-polytope is a polytope of dimension n in some R d . For example, we have defined a finite set of contact constraints (7) specified on two nominal parallel planes. ( 7) is a finite set of 3-faces in R 6 and ( 8) is a finite set of 3-faces in R 3 . It is possible to give a graphic representation of a n-polytope if 1 ≤ n ≤ 3. A 0-polytope is a point, a 1-polytope is a segment line and a 2-polytope is a polygon. For example, the finite set of contact constraints (7) (i.e. finite set of 3-faces) between two nominal parallel planes such as S is rectangular plane can be illustrated in R 3 by figure 7.

3.2.

Minkowski sums of polytopes Let us consider the mechanism of figure 8. The contact specification between S2,1 and S1,1 is a permanent contact. The corresponding contact polytope is a 0-polytope (i.e. a point). We can express the relative position of S2,2 with regards to S1,2 . We have according to the properties of small displacements:

2,2 2,0 2,1 1,1 1,0 1,2 Part 2 Part 1 Geometric specification Contact specification Functional requirement d kt (k-1)t S 1,2 S 1,1 S 2,1 2,2 z y x S 2,2
( ) t . k z . t . 1 k ZT S S N N 2 , 1 S / 2 , 2 S , N 2 , 2 2 , 2 ≤ ε ≤ - ⇔ ⊂ ∈ ∀ (10) ( ) ( ) ( ) ( ) ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎫ ≤ ε ≤ - ≤ ≤ ∈ ∀ ≤ - ε ≤ - ≤ ≤ ∈ ∀ = ε ∈ ∀ ≤ - ε ≤ - ≤ ≤ ∈ ∀ ≤ ε ≤ - ≤ ≤ ∈ ∀ ε + ε + ε + ε + ε = ε

Figure 1 :

 1 Figure 1: Geometric specification -geometric constraints.We define geometric constraints of position between S1 and S0 as follows[START_REF] Teissandier | Threedimensional Functional Tolerancing with Proportioned Assemblies Clearance Volume: application to setup planning[END_REF]]:

Figure 2 :

 2 Main cases of joints.

Figure 5 :

 5 Figure 4: Functional requirement. geometric specification functional requirement

Figure 6 :Figure 7 :

 67 Figure 6: Definition of a 2-polytope in R 2 .

Figure 8 :

 8 Figure 8: Association of specifications in series.

Figure 10 :∩Figure 11 :

 1011 Figure 10: Association of specifications in parallel.

  Figure 11: Intersection of 3-polytopes.Each relation of (12) can be formalised by a Minkowski sum of three 3-polytopes. The result of (12) is an Intersection of two Minkowski sums of 3-polytopes: see figure11.

  and Intersections of geometric d-polytopes and contact d-polytopes, the relative position between any surfaces of a mechanism can be calculated. The result is a new polytope: the calculated polytope. If the calculated polytope is included inside the functional polytope, the functional requirement is satisfied: see figure12.

th CIRP Seminar on Computer Aided Tolerancing -Enschede, The Netherlands,March 22-24, 1999

, 1

Figure 9: Minkowski sum of 3-polytopes in R 3 . (11) characterizes the Minkowski sum of five polytopes. These five polytopes are the geometric specifications of S2,2 , S2,1 , S1,1 , S1,2 and the contact polytope between S1,2 and S2,2 (in this case this polytope is a 0-polytope). The Minkowski sum of a two polytopes P1 and P2 is a polytope P P 2 1 + [START_REF] Gritzmann | Minkowski addition of polytopes: computational complexity and applications to Gröner bases[END_REF]:

(12) The association of geometric specifications and contact specifications in series are mathematically formalised by Minkowski sums of d-polytopes [Srinivasan, 1993]. In the example of figure 8, we can illustrate the Minkowski sum of five 3-polytopes in R 3 : see figure 9. The Minkowski sum is an commutative and associative operation.

Intersection of polytopes

In the example presented in figures 10 and 11, the relative position between S1,0 and S2,0 must satisfy the two following relations:


Fabien Leurent

Ektoras Chandakas

Alexis Poulhès

User and service equilibrium in a structural model of traffic assignment to a transit network

Keywords: Transit capacity, Line model, Platform model, Dwell time, Frequency reduction, Zip algorithm

A structural model is provided for traffic assignment to a transit model. It deals with a hierarchy of layers from passenger to network passing by vehicle, service route and line, while taking advantage of the spatial structure of service routes and lines. A range of capacity effects are addressed: in-vehicle passenger capacity, access-egress capacity in relation to dwelling time, platform occupancy by vehicles which may reduce service frequency. The model treatment involves a line sub-model, which amounts to a sophisticated cost-flow relationship, and platform submodels first for passenger storage and flowing, second for vehicle operations. Traffic equilibrium is addressed in a hyperpath framework on the basis of leg links.

Introduction

In the transit network of a large urban area, it frequently occurs that a high duty line is submitted to heavy congestion at the peak hours on working days, especially so at the morning peak in the central part of the urban area. Under that circumstance, not only may the passengers experience the discomfort of crowding, delay and unreliability, but also the operation of services may be disrupted by increased dwelling times, vehicle bunching and delays, leading to reductions in service frequency. Thus transit traffic is inconvenienced and disrupted at both levels of mobile units, passenger and vehicle.

Although many transit engineers are well aware of these issues, which are discussed at length in the Transit Quality of Service and Capacity Manual (Trb, 2003), little attention has been given by the scientific community of transportation modellers to the interplay of passenger and vehicle traffic. On one hand, passenger traffic is addressed by models of traffic assignment to a transit network, typically in planning studies where the operating conditions are described through the service routes and operation frequencies, section run time, station dwelling time, maybe vehicle capacity [START_REF] Thomas | Traffic Assignment Techniques[END_REF][START_REF] Ortuzar | Modelling Transport[END_REF]; in some advanced models, one or several of these features are taken as flow-responsive. However little consideration has been given to the dynamic conditions of realtime operations: the flow-responsive features and policies are described mostly by heuristic constructs rather than physical and technical models. Route frequency has been related by [START_REF] Lam | A stochastic user equilibrium assignment model for congested transit networks[END_REF] to fleet size and cycle time by vehicle, taking both station dwelling time and section running time as flow dependent. In fact however, the scarce resource pertains to platform availability rather than fleet size. [START_REF] Cepeda | A frequency-based assignment model for congested transit networks with strict capacity constraints: characterization and computation of equilibria[END_REF] associated to each station along a service route a fictive frequency adapted from nominal to account for the effects of the incoming flow and the residual capacity: but the relationship is an artefact, with neither account of the egress flow on the dwelling time nor forward propagation of the reduction in service frequency. The macroscopic dynamic assignment models proposed so far (eg. [START_REF] Sumalee | Dynamic stochastic transit assignment with explicit seat allocation model[END_REF]) take the service timetable as exogenous, in contradiction with the unreliability that appears under severe congestion. Micro-simulation is still restricted to local issues of station design (eg. Hoogendorn and Daamen, 2005).

On the other hand, service operations are planned at the line level on the basis of a model of vehicle running along the service trajectory, taking into account the local conditions such as line geometry and station dwelling time. The passenger traffic is considered only through its influence on the dwelling times and the vehicle load (hence mass) (e.g. [START_REF] Vuchic | Urban Transit : Operations, Planning and Economics[END_REF][START_REF] Lai | Development of Analytical Capacity Models for Commuter Rail Operations with Advanced Signaling Systems[END_REF].

This paper is purported to bridge the gap between these two families of models. It brings out a model of traffic assignment to a transit network that is sensitive to the interplay between passenger and vehicle traffic at the line level. Platform occupancy by successive vehicles influences service frequency. Dwelling time depends on passenger access and egress flows. A local decrease in frequency implies a decrease in service residual capacity that is delivered downstream, all the more so as it comes along with a decrease in the residual capacity in each vehicle. On the passenger side, local queuing may occur on station platform and contribute to leg cost, the leg being the trip part along the transit line from access to egress station. Route choice at the network level depends on leg costs.

Our approach to the physical and economic features is static, in order to deal with service frequency conveniently. Transit modes such as bus or train are distinguished in their travel conditions, including platform layout and the potential location of route choice within a line station. Route services are grouped into a line of operation inasmuch as they share resources of section track and station platform, which makes their operation interdependent. The tree-like pattern of a line provides our main line of attack, as in Leurent's model of seat capacity (2006,2010): from upstream to downstream the passenger flow constraints are progressively tied up along the successive stations, with influence on service frequency and residual capacity. Conversely to this 'zip' treatment an 'unzip' treatment from downstream to upstream yields the travel conditions by trip leg, thus providing the basis for passenger route choice by origin-destination pair at the network level. The rest of the paper is organized into five sections. First, the model assumptions are set out in Section 2. Then, Section 3 provides the line sub-model, including the 'zip' and 'unzip' algorithms. The model composition and overall solution scheme is sketched out in Section 4. Section 5 deals with a classroom case adapted from the busiest railway line in the Paris metropolitan area (line A of the RER, the so-called Regional Express Network). Lastly, Section 6 concludes about the model contents and points to potential developments.

Model assumptions

As stated by Leurent (2011a) a transit system involves five main subsystems: Line, Vehicle, Station, Passenger and Management. Each one deserves special consideration in the modelling.

Network topology, from infrastructure to services

The infrastructure network is made up of a set N of nodes n , together with a set A of arcs a (unidirectional links) with endpoints in N . Basically, an arc represents a state transition for a trip-maker: set A is split between subsets B A of boarding links, A A of alighting links, I A of interstation links and S A of sojourn links within vehicle and station and P A of other private links for pedestrian access or transfer between two services routes. Such a route is modelled as an oriented sequence of sojourn and interstation links, alternately.

In the basic hyperpath model of transit assignment [START_REF] Spiess | Optimal strategies: a new assignment model for transit networks[END_REF], a passenger routing structure is in fact a bundle of paths from origin to destination: the structure may combine service routes at given choice nodes, i.e. platform nodes (or information nodes) at which point several routes are available and attractive to get the trip-maker closer to his destination. This is a weak interaction between service routes, as compared with strong interaction on the operator side where given physical nodes and arcs -notably sojourn and interstation -are shared by several services. Here it is assumed that such related services belong to the same "service line", i.e. a sub-network that connects platform user nodes from entry to exit.

Thus the trip-maker is faced to a service network with one leg arc per couple of entry-exit stations along any service route. Such a leg arc represents one or several interstation and sojourn links along the service route in the infrastructure network that is still considered on the supply side. On the demand side, it is assumed that the passengers consider only leg arcs and private arcs for route boarding or route alighting, station access and transfer from route to route at a given interchange station (e.g. De Cea and Fernandez, 1993); [START_REF] Leurent | On capacity and congestion in passenger transit systems: Systems analysis and implications for network assignment models[END_REF].

Overall, the service network is made up of the set

L A B P A A A A U U U ≡ S A
, L A being the set of service legs, with the associated node sets.

Vehicle features

By service route z , the average vehicle has an on-board capacity of z k passengers. At station n , the time for a passenger to pass a vehicle door is ω be the minimum inter-vehicular gap prior to a vehicle servicing z (on average). The resource occupancy must satisfy the following constraint:

H T f n z zn zn zn ≤ ω + ∑ ∩ ∈l ) ( .
The minimum inter-vehicular gap is set up by the operator on the basis of the station and its upstream interstation, their geometry, track features, signaling system and so on.

Line traffic

A tree-like structure is assumed for each line l , with branches connecting on a main trunk providing service between pairs ) , ( j i of stations that are oriented along the line: this is denoted as j i l > ( i is upstream of j ). The set of stations serviced by route z between i and j is denoted as

[ , [ j i z I : l S j i ∈ , .
At the vehicle level, let za t denote the running time along section a for route z . Then the vehicle time by route z from i to j is

∑ ∑ ∩ ∈ ∩ ∈ + = [ , [ [ , [ j i z a za j i z n zn z ij t T V . Notation [ , [ j i z a I ∈
is an unambiguous shorthand to encompass the sections travelled along z from i to j . The traffic of passengers along the line is analyzed at two levels external vs. internal. The internal perspective involves the line sub-model that is addressed in the next Section. On the external perspective, during a given time period the line carries a vector ] :

[ j i q ij l l l > = q
of passenger flows by leg and yields leg costs ] :

[ j i c ij l l l > = c
. The gist of the line sub-model is to relate l c to l q , thus making up a cost-flow relationship at the line level.

Platform issues: the passenger side

At a platform a passenger has to wait for a route service to become available and eventually to choose a service. Let us here restrict the concept of a line to a subset of services that cannot overtake one another, especially so because they share station platforms and section tracks. Then if a passenger uses a line he will take the first service with available vehicle capacity. The line frequency on a given leg is the sum of those of the service routes that connect the leg endpoints:

∑ ∩ ∈ = ) , ( j i z zi ij f f l l .
Thus service routes available along a given line are bundled together. Further combination may occur between alternative lines: assuming that if neither is saturated, then the standard condition for line attractiveness with respect to the trip destination applies. If the alternative option is a bundle b with frequency b s i f ) ( and minimum cost b s i c ) ( up to destination s , then line l is attractive if, denoting by js u the minimum cost from j to s , and by α the discomfort coefficient of waiting time compared to in-vehicle time:

b s i b s i js ij f c u c ) ( ) ( / α + ≤ + l .
Conversely the alternative bundle is attractive if

l l ij js ij b s i f u c c / ) ( α + + ≤ . If both options are attractive, they are combined in an extended bundle l U b b ≡ ′ with joint frequency l ij b s i b s i f f f + ≡ ′ ) ( ) ( and joint minimum cost b s i js ij ij b s i b s i b s i f u c f c f c ′ ′ + + = ) ( ) ( ) ( ) ( / )] ( [ l l .
In the unsaturated case the average waiting time is

l l ij ij f w / α = by using l only, or b s i b s i f w ′ ′ α = ) ( ) ( / if the bundle is attractive.
The saturated case occurs when there is a waiting queue that does not dissipate during the period so that a passenger is unable to access the first vehicle that arrives after his instant of joining the queue. Then the attractiveness condition for line bundling should be modified: an ad-hoc treatment is proposed in Section 4. Overall, the extended cost-flow relationship at the line level relates ) , ( l l w c to l q .

Passenger traffic

Along a network path every passenger spends time and money; he experiences state transitions e.g. between invehicle and pedestrian states and he is submitted to discomfort by transition as well as by state. Let us assume that all kinds of costs can be aggregated into a generalized cost which is added up along the path, including the costs of pedestrian arcs, of waiting at platform nodes and the leg costs.

Route choice along the network yields a user-optimized path of minimum generalized cost -or an optimal hyperpath if bundling occurs.

The passenger flows along the network result from the assignment to the optimal paths of the OD matrix of trip flows by origin-destination pair, ] , :

[ D d O o q od ∈ ∈ = q
with O (resp. D ) the set of origin nodes (resp. destination).

Line problems and algorithms

The line sub-model takes a set of exogenous flows Before stating the flow loading 'zip' algorithm ( §3.3) and the leg costing 'unzip' algorithm ( §3.4), let us introduce first a sub-model of transit bottleneck ( §3.1) and, then, a sub-problem of platform flowing ( §3.2).

Transit bottleneck model

At a given station i along l , consider a group of routes i Z ′ that serve a set of destinations i J ′ downstream of i , such that these routes and stations make up a connected component in the bipartite graph in

l l i i J Z ×
that links the routes to the stations that they service.

Given exogenous flows + j q by egress station, route frequencies z f ′ with available capacity z k′ by vehicle, the objective is to yield vehicle loads z ij y by vehicle servicing route z and waiting times l ij w by passenger boarding at i to alight at j . Define j v the stock of passengers destined to j that wait on the station platform, and

∑ ∈ = z j j z v n
the number of passengers that are candidates to board on a vehicle of route z when it arrives. Then the probability to board is

} , 1 { min z z z n k′ ≡ π . (1)
Assuming queuing of mingled passengers in a bottleneck, the stock variables ] : [

j j Z j v ′ ∈ = v satisfy a Fixed
Point Problem (FPP) as follows (Leurent, 2011b):

j j j j j f v q H q v ) ( 2 2 π - = + + , in which ∑ ∈ π ′ ≡ π j z z z j f f ) ( ) ( v .
(2) Some destinations may not be queued, if the exit flow

j j j v f q ) ( π =
matches the entry flow + j q : in such a case the FPP is only an approximation and yields

+ << j j Hq v .
The FPP has a solution which is unique. It can be solved using a Newton algorithm.

From the stock variables j v , the z π and - j q follow, as well as the vehicle inflows

j z z ij v y π ≡
. Furthermore, for destination j the queue duration is

- + = j j j q H q H
/ , yielding waiting cost as follows:

j j j j j j ij f q v H q H v w ) ( 1 π = = = - + l .
(3)

Lastly, H H j > induces l ij j f f < π) (
, which may hold also when H H j = but some routes are saturated.

Platform flowing

The flowing of both service vehicles and passengers at station i along line l involves, first, the vehicle arrivals at upstream frequency

+ i z f
and the discharge from each vehicle of a number z si y of passengers from upstream station s to current station; second, the revision of route frequency; third, the waiting and eventual boarding of those passengers coming in at i ; fourth, the propagation to downstream of the refilled vehicles and the revised route frequency.

The second step deals with the constraint of platform occupancy by the dwelling and gap times of the service vehicles. The assignment period, H , should suffice to accommodate the boarding and alighting times of passengers plus the inter-vehicle gaps:

H g y g y T T f f i z zi zi zi zi zi zi i z i z zi i z ≤ + + + ω ∑ ∑ ∈ + + - - + + } . . , { max 1 0 via (4)
Enforcing the constraint may require to reduce frequency, which is accomplished by multiplying the left hand side by a reduction factor 1 ≤ η i . To simplify the evaluation of (4), let us substitute the residual capacity by vehicle after passenger discharge, o zi k , to the still unknown incoming flow

+ zi y . If constraint (4) is satisfied for 1 = η i then frequency + -≡ i z i z f f is unchanged from upstream to downstream, whereas if 1 < η i then + -η ≡ i z i i z f f , in which (5) ] } . . , { max /[ 1 0 via ∑ ∑ ∈ + - - + + + + + = η i z zi o zi zi zi zi zi i z i z zi i z i g k g y T T f h f H . (6)
Remark that vehicle gaps must be summed over all services that pass through that platform at station i , either stopping or not. All the "through" routes are submitted to frequency revision. The third step requires solving as many sub-problems of transit bottlenecks as there are "clusters" of egress stations and routes.

The fourth and last step simply consists in propagating, for each service route z passing through i , the revised variables

- i z f
to the next station along z ahead of i , together with the completed vehicle loads ) ( z sj y if i is a dwelling station for z .

The line flow loading problem and zip algorithm

To load both flows of passengers and vehicles onto the line is the problem of setting out the local route frequencies and the local vehicle loads of passengers by route and access-egress pair in a coherent way, by meeting the local capacity constraints at platforms and in-vehicle while giving priority to conditions upstream onto those downstream.

This problem of flow loading can be solved by a zip algorithm that deals with every station in turn, from upstream to downstream in the topological order of the line. At station i , solve the platform flowing problem on the basis of the local conditions

+ zi f , ] , : [ i j i s y z sj ≤ >
inherited from all upstream stations s , and propagate the adapted conditions

- zi f , ] , : [ i j i s y z sj < ≥
to the downstream stations.

The complexity in computation time of the zip algorithm is a product of the number of stations by the number of routes, times the number of operations required to solve any transit bottleneck model -the latter depends on the number of egress stations and of routes dwelling at i . The complexity in memory space may be limited to the number of routes times the number of egress stations, since only the passenger load by egress station and service route (not by entry station) need to be kept in memory.

The leg costing problem and unzip algorithm

To cost the line legs is the problem of evaluating the travel conditions to a passenger, by pair of access-egress stations. The travel conditions include the eventual money expenses, the time spent either in-vehicle or on platform, perhaps evaluated as generalized rather than physical time to account for local discomfort.

Concerning physical time, the vehicle times by service route and the passenger waiting times at stations stem from the line loading problem. The in-vehicle section time may be taken as exogenous, i.e. za t , or state dependent, for instance by including a time penalty at any station at which the service frequency is reduced.

Generalized time involves physical time multiplied by discomfort coefficients -say 1 for sitting in-vehicle and about 2 for waiting on an uncongested platform or for standing in a crowded vehicle. Discomfort functions with respect to passenger density on platform or in-vehicle may be considered, eventually distinguishing between sitting and standing riders as in [START_REF] Leurent | On capacity and congestion in passenger transit systems: Systems analysis and implications for network assignment models[END_REF].

Leg costing can be addressed by an unzip algorithm that deals with every egress station in reverse order from downstream to upstream, after evaluation by network element of the local generalized time on the basis of the physical time and the flow density. By egress station s :

• Initialize variables of physical time zs T and generalized time zs G to zero for all routes z that pass through s or to infinity otherwise. • Enumerate the line stations i upstream of s in reverse topological order from s , (i) to cumulate zs T and zs G from i , (ii) to evaluate the share of passenger flow incoming at i and destined to s between the available routes, and the leg cost, on the basis of the following formulas:

l l l i zi zi s i z f f p ) ( , / π π = + in which ∑ ∈ + π ≡ π ) , ( ) ( s i z zi zi i f f l l . (7) ∑ ∈ ≤ + π π = ) , ( , ) ( 1 s i z i zs zi zi i s i G f f c l l l .
(8)

The computational complexity both in time and space amounts to the squared number of stations times the number of routes.

On the static assumption and the continuity of flow

The static framework is convenient to deal with the constraint of platform occupancy by the passage and dwelling of vehicles. However the line model is likely to disrupt the conventional assumptions in static assignment in two ways: first, the transit bottleneck model may turn H into l ij H for access-egress pair ) , ( j i , yielding period flow of

- ) (l ij Hq equal to ) ( 2 ) ( / l l ij ij H H q + instead of + ) (l ij

Hq

; second, the local frequency revision may decrease a frequency of zi f at leg entry down to

+ zj f at leg exit, which provides some ground for an additional delay of

) / 1 ( 2 1 - + - zi zj f f H
by vehicle that is operated during the reference period but also disrupts the conservation of the service flow during the reference period H . As the main purpose of the line model is to establish the leg costs, these disruptions are considered by the authors to be significant mainly as results internal to the line model -which may be of separate interest to a study analystwhereas in the context of network assignment the travel conditions should be applied to leg flows ) (l ij q taken only at their line entry value,

+ ) (l ij q .

Network model

Let us turn to the general problem of network assignment, with passenger route choice from origin to destination pair of nodes.

Traffic state and hyperpath representation

Hereafter the service network with private and leg arcs is considered: the arc set is S A . A Traffic state is a vector of arc flows, ] :

[ S A a x a A ∈ = x
. The arc travel cost function:

) ( A A A A x C c x = a
yields, on pedestrian arcs, the local cost with respect to local passenger flow a x and, on leg arcs, the leg costs that stem from the line assignment of the leg flows in A x (by transit line). This function can be made continuous by enforcing on each route and at each station a strictly positive though arbitrarily small minimal residual capacity. This is both innocuous and realistic enough since any passenger would be ready to board even in a very crowded vehicle rather than to wait indefinitely long.

] / ) 1 [( ) 1 ( a a a a a b b b f w u w u β + β - α + ≤ β - α + . (9)
Line bundling would occur between attractive lines similarly to the uncongested model but for the revised frequency,

a a a f f β ≡ / ˆ, yielding:
• Combined frequency of attractive bundle B :

∑ ∈ ≡ B a a B f f ˆ.
• Passenger flow share by line of

B a f f / ˆ
if attractive or zero otherwise.

• Bundle average cost of

B B a a a a a B f w u f u / ] ) ( [ ∑ ∈ αβ + + α ≡ .
Let us define a hyperpath A h ⊂ as an oriented, connected, acyclic sub-graph that directs any of its nodes to a given destination node say s . Based on arc costs and leg waits and nominal frequencies, by applying the bundling treatment in recursive order from s , routing proportions

∈ ∈ = q is a pair ) , ( ζ A x
such that (i) ζ is admissible with respect to q , (ii) ζ is user-optimized with respect to A

x and (iii) A x results from the network assignment of ζ based on A x , i.e.

A a ∈ ∀ : ∑ ∑ ∑ ∑ ∈ ∈ η ∈ ∩ ∈ γ ζ = D s N n s n h h a r A h r h ns A x ) , ( ) (x , (11) Wherein ∏ ∈ γ ≡ γ r a h a h r
is the routing proportion of path r along hyperpath h . This definition amounts to a fixed point problem. It is equivalent to a quasi variational inequality problem in the space of hyperpath flow vectors. As the cost function is continuous and the space has finite dimension, there must exist an equilibrium state.

MSA Algorithm and convergence criterion

A Method of Successive Averages (MSA) can be used to solve the traffic equilibrium problem of network assignment in the following way, using a decreasing sequence of positive numbers x . This involves notably to deal with each line by applying the zip algorithm to load the leg flows and then the unzip algorithm to cost the legs.

• Search for shortest hyperpaths for all origin-destination pairs and load the OD flows along them (and with respect to A x ) onto the network elements, yielding auxiliary traffic state A y . By destination, a shortest hyperpath is built on the service network from each node recursively, in the classical way save for the adaptation of bundle cost as in Section 4.1.

Application instance

A classroom instance was built to demonstrate the application of the model. The case mimics the busiest railway line in the Paris metropolitan area, named RER A -the RER being the Regional Express Network. At the morning peak the directional passenger flow through the central trunk amounts to about 50,000 persons per hour; the nominal frequency of 30 trains per hour is frequently decreased to 25 on average, due to congestion which is particularly acute at central stations where many transfers take place.

In the East to West direction, there are two main service routes which link the two Northern (resp. Southern) branches to the central trunk (Fig. 1). At the morning peak hour the operation frequency is of 18 /h and 12/h respectively, yielding a nominal frequency of 30 trains per hour on the central trunk. Between North-East and Centre another railway line, RER E, competes with RER A. The passenger capacity by train is about 2,000 depending on the route. The minimum dwelling time is planned as 40 s on RER A and 50 s on RER E which is less congested.

Six stations were selected as origin or destination zones. The model of infrastructure network includes 28 nodes and 35 arcs (Fig. 2). On this small network convergence was achieved in fifty iterations (Fig. 3).

Here are some selected assignment results: • At the Nation station, vehicle dwell time amounts to 54s (resp. 58s) on the North route (resp. South). These are reduced to 43s (resp. 42s) at Auber station where there is less residual capacity. • From Nation station westwards, the operation frequency is reduced to 17.1/h (resp. 11.4/h) on the North (resp.

South) route: this is a 5% reduction on the nominal capacity of the line. • Individual wait time is increased from 3.3' (resp. 5.0') on the North (resp. South) route in the absence of congestion, to 5.6' (resp. 6.7') at Nation station and 11.6' (resp. 17.0') at Auber station, owing to the interplay of incoming flow and residual capacity. • On the OD pair from Nation to Saint Germain (South West), at equilibrium only the South route is used, not the Northern one because the wait time at Auber is dissuasive to transfer there. This induces a significant change in the share of flow incoming at Nation between the North and South routes westwards, from 2 versus one thirds without congestion to 57%-43% under congestion. • The travel cost from Nation to Auber is increased from 12.0' without congestion to 14.0' with congestion. From Auber to Cergy (resp. Saint Germain) it is increased from 43.5' (resp. 30') to 50.8' (resp. 42'). The variation in generalized time would be even larger by penalizing the time spent in-vehicle in order to capture the discomfort of vehicle crowding.

North-East

North-West South-West South-East

Conclusion

A transit assignment model has been provided, which is structural because it deals with a hierarchy of layers or scales -from passenger at the bottom to network at the top passing by vehicle, service route and line -in an explicit way, along with the spatial structure of routes and lines. Spatial features are considered at each scale, with emphasis on capacitated resources of vehicle platform as well as access-egress of passengers to vehicles. In spite of the static assumption, time features are involved in resource occupancy, frequency reduction and the associated delay, physical and generalized passenger travel times. There are model variables standing for passenger stocks on platform and in-vehicle.

The model treatment involves a series of innovations: first, the transit bottleneck model for passenger flowing at platform to vehicles; second, the influence of passenger flows on dwelling times; third, the local revision of service frequency based on platform occupancy and its downstream propagation; fourth, a revised condition for attractiveness for line combination at choice nodes; fifth, a hyperpath framework with state-dependent bundling.

The model addresses a wide range of traffic phenomena: in-vehicle passenger capacity, vehicle access-egress capacity, platform capacity on the vehicle side. It would be straightforward to include seat capacity by vehicle, corridor pedestrian capacity, track capacity in vehicles. These features are expected to be useful in transit planning applications. On-going work is focused on congestion assessment on the Paris network.

Research topics include: the estimation of passenger behaviour under congestion; the refinement of operating policy -beyond proportionate frequency reduction; the stochasticity in passenger stocks and flows, dwelling times and section run times.

  + l ij q from access to egress stations as inputs to yield cost outputs l ij c and l ij w . Internal variables include, by route service l ∈ z , local frequencies + i z f before station i and i z f after it, as well as vehicle loads z sj y by access-egress pair ) , ( j s . It operates on the basis of a topological ordering of the stations, either from upstream to downstream or reverse from downstream to upstream.

.

  If several lines are available at a given node n , each leg arc a with post-boarding cost a u and platform waiting time a w and ex-ante frequency a f (Cf. frequency l ij f in Section 2), then line combination may occur in a revised way of the uncongested model: argument ε . Let us assume that line a delivers a minimum time of Then the condition for line b to be attractive relative to line a would be

  Cost evaluation: by network element based on A

Fig. 1 .

 1 Fig. 1. Abstraction of selected network, emphasizing line legs.

Fig. 2 .

 2 Fig. 2. Underlying infrastructure network.

  The station platform n of a given line, l , is a resource to be shared between the vehicles of the service routes l ∈ z. During a period of given time length, say H , let zn f denote the frequency of operation of route z at node

	-zn t on egress and + zn t on access. Denoting by za y the number of passengers on-y (resp. + board on link a and by -zn zn y ) the number of passengers alighting (resp. boarding) at station n , then the
	time for passenger discharge and refill cannot be less than
														T	zn + -	≡	(	y	zn -	t .	zn -	+	y	zn +	t .	zn +	)	/	X	z	,
	X being the number of passengers that can simultaneously traverse the vehicle door(s). Let us define z z zn zn X t g / --≡ and z zn zn X t g / + + ≡ . The z k and z X parameters depend on the vehicle design, whereas -zn t and + zn t
	also involve the station design.
	2.3. Platform issues: the vehicle side
	n ,	T	zn	≡	{ max T	1 zn	+	T	zn + -	,	T	0 zn	}	be the dwell time of a vehicle (denoting by 1

zi

T a dead time for door operation etc) and zn

•

  Next state. Let

	x	A ′	: =	λ	k	y	A	+	1 (	-	λ	k	)	x	A	.
	• Evaluate a convergence criterion between A x′ and A x . If it is sufficiently small then Stop with solution A x′ ,
	else increment k , replace A x by A x′ and go to Cost Evaluation.
	A convenient yet crude convergence criterion may be	A x x -′	A	. A Matlab implementation is available on
	request from the authors.

Acknowledgement. This work is supported by a Research and Education Chair operated by Ecole des Ponts

ParisTech in partnership with the Transport Organizing Authority in the Paris region (STIF). The Chair is targeted to the socio-economic analysis and modeling of passenger transport in the urban setting.