David Laharie
email: david.laharie@chu-bordeaux.fr

Armel Reffet

Genevieve Belleannée

Edouard Chabrun

Clement Subtil

Sylvie Razaire

Maylis Capdepont

Geneviève Belleannée

Clément Subtil

Victor De Lédinghen

Mucosal healing with methotrexate in

Keywords: Inflammatory bowel disease < Disease-based, Crohn's disease < Disease-based, Inflammation < Topics, Biologics (IBD) < Crohn's disease, mucosal healing, methotrexate, azathioprine, infliximab

come

The natural history of Crohn's disease is characterized by active and remitting periods [START_REF] Solberg | Clinical course in Crohn's disease: results of a Norwegian population-based ten-year follow-up study[END_REF] . For a long time, medical treatment was solely focused on clinical symptoms, aiming to control disease flare and prevent relapse. Healing the gut mucosa was not considered as a therapeutic goal because of the poor correlation between symptom improvement and mucosal healing and the inefficacy of conventional drugs such as steroids to cure endoscopic lesions [START_REF] Modigliani | Clinical, biological, and endoscopic picture of attacks of Crohn's disease. Evolution on prednisolone. Groupe d'Etude Therapeutique des Affections Inflammatoires Digestives[END_REF] .

During the past decade, new data have highlighted the significance of mucosal healing in Crohn's disease. Firstly, the disappearance of endoscopic inflammatory lesions seems to be related with better long-term disease outcomes -less hospitalization and abdominal surgery 3- 5 . Secondly, the absence of mucosal healing in asymptomatic patients is associated with higher relapse rate [START_REF] Schnitzler | Mucosal healing predicts long-term outcome of maintenance therapy with infliximab in Crohn's disease[END_REF] as positively shown in two recent prospective studies including patients treated with infliximab and immunosuppressants [START_REF] Baert | Mucosal healing predicts sustained clinical remission in patients with early-stage Crohn's disease[END_REF][START_REF] Louis | Infliximab discontinuation in Crohn's disease patients in stable remission on combined therapy with immunosuppressors: interim analysis of a prospective cohort study[END_REF] . Lastly, retrospective data suggested that Crohn's disease patients with severe endoscopic colonic lesions had increased rates of penetrating complications and colectomy [START_REF] Allez | Long term outcome of patients with active Crohn's disease exhibiting extensive and deep ulcerations at colonoscopy[END_REF] .

Biologics, and especially anti-TNFs -infliximab, adalimumab and certolizumab pegol have brought a major advance in the treatment of both luminal and fistulizing Crohn's disease [START_REF] Peyrin-Biroulet | Efficacy and safety of tumor necrosis factor antagonists in Crohn's disease: meta-analysis of placebo-controlled trials[END_REF] .

They have demonstrated not only clinical efficacy, but also that mucosal healing was achievable and could be maintained in patients with active Crohn's disease [START_REF] Rutgeerts | Scheduled maintenance treatment with infliximab is superior to episodic treatment for the healing of mucosal ulceration associated with Crohn's disease[END_REF][START_REF] Hebuterne | Endoscopic improvement in patients with active Crohn's disease treated with certolizumab pegol: first results of the MUSIC clinical trial[END_REF][START_REF] Rutgeerts | Adalimumab induces and maintains mucosal healing in patients with moderate to severe ileocolonic Crohn's disease. First results of the EXTEND trial[END_REF][START_REF] Colombel | Infliximab, azathioprine, or combination therapy for Crohn's disease[END_REF] .

Therefore, mucosal healing has emerged as a new therapeutic goal and is now considered as an endpoint in clinical trials [START_REF] Rutgeerts | Adalimumab induces and maintains mucosal healing in patients with moderate to severe ileocolonic Crohn's disease. First results of the EXTEND trial[END_REF][START_REF] Colombel | Infliximab, azathioprine, or combination therapy for Crohn's disease[END_REF] . It has been less studied with conventional immunosuppressants commonly used in Crohn's disease for many years. Few studies have shown the ability of azathioprine to induce mucosal healing, [START_REF] Sandborn | An intravenous loading dose of azathioprine decreases the time to response in patients with Crohn's disease[END_REF][START_REF] Haens | Healing of severe recurrent ileitis with azathioprine therapy in patients with Crohn's disease[END_REF] but recently, complete mucosal healing at week 26 was achieved in 15.6% of the 109 patients assessable in the azathioprine arm of the SONIC trial [START_REF] Colombel | Infliximab, azathioprine, or combination therapy for Crohn's disease[END_REF] . Data with methotrexate are more scarce. According to two small retrospective studies -14 and 8 recruited patients, respectively -mucosal healing was obtained in 36-37.5% of patients responding to methotrexate [START_REF] Kozarek | Methotrexate induces clinical and histologic remission in patients with refractory inflammatory bowel disease[END_REF][START_REF] Manosa | Does methotrexate induce mucosal healing in Crohn's disease?[END_REF] . Therefore, the aim of the present study was to assess the rate of mucosal healing with methotrexate in Crohn's disease, as compared to azathioprine and infliximab. Samples were fixed in formalin and embedded in paraffin, and 3µm thick sections were stained with HES. All biopsy specimens were analyzed independently by one experienced pathologist (G.B.) blinded to the treatment. The goal of this examination was to assess microscopic activity of Crohn's disease according to the Scoring System for Crohn's disease constructed by Geboes and adapted from D'Haens et al [START_REF] Haens | Endoscopic and histological healing with infliximab anti-tumor necrosis factor antibodies in Crohn's disease: A European multicenter trial[END_REF] . The most inflammatory lesions observed within the most inflamed area at colonoscopy were retained for scoring.

During the follow-up period, from inclusion to December 2009, Crohn's disease clinical relapse, defined as worsening of symptoms coupled with endoscopic, radiographic, and/or biological evidence of inflammation, was recorded.

All patients received treatment according to clinical need. Drugs used were those normally employed in Crohn's disease, according to licensed or published doses and frequency (parenteral methotrexate: 15-25 mg/week; azathioprine: 2-3 mg/kg/d; infliximab: 5 mg/kg/8weeks). All patients received treatment and had colonoscopy only after full and informed consent.

Outcomes and objectives

Mucosal healing was defined as the total disappearance of all mucosal ulcerations in all segments [START_REF] Rutgeerts | Adalimumab induces and maintains mucosal healing in patients with moderate to severe ileocolonic Crohn's disease. First results of the EXTEND trial[END_REF][START_REF] Colombel | Infliximab, azathioprine, or combination therapy for Crohn's disease[END_REF] . Two indices of endoscopic activity, the Crohn's Disease Endoscopic Index of Severity (CDEIS) [START_REF] Mary | Development and validation of an endoscopic index of the severity for Crohn's disease: a prospective multicentre study. Groupe d'Etudes Therapeutiques des Affections Inflammatoires du Tube Digestif (GETAID)[END_REF] and the Simple Endoscopic Score for Crohn's Disease (SES-CD) [START_REF] Daperno | Development and validation of a new, simplified endoscopic activity score for Crohn's disease: the SES-CD[END_REF] , were calculated at inclusion. Considering the CDEIS, two cut-off values were retained: <6 as endoscopic remission and <4 as complete mucosal healing [START_REF] Hebuterne | Endoscopic improvement in patients with active Crohn's disease treated with certolizumab pegol: first results of the MUSIC clinical trial[END_REF] . Microscopic Crohn's disease activity was assessed with the Scoring System for Crohn's disease adapted from D'Haens et al, ranging from 1 (no activity) to 13 (table 1) [START_REF] Haens | Endoscopic and histological healing with infliximab anti-tumor necrosis factor antibodies in Crohn's disease: A European multicenter trial[END_REF] . The primary objective of the study was to establish the proportion of mucosal healing in Crohn's disease patients with sustained remission by monotherapy with methotrexate, as compared to azathioprine or infliximab.

Secondary objectives were: i) to compare CDEIS and SES-CD in Crohn's patients in remission receiving methotrexate to those treated with azathioprine or infliximab; ii) to identify factors associated with mucosal healing; iii) to compare Crohn's disease microscopic activity in patients in remission with methotrexate to those receiving azathioprine or infliximab; iv) to look for an association between mucosal healing at baseline and relapse rate during follow-up.

Statistics

Continuous variables are presented as medians and range; categorical variables are presented as percentages. Continuous data were analysed using Mann-Whitney's test. Categorical data were analysed using the Pearson's chi-squared test, or Fisher's exact test if any cell number was <5, for frequencies. Pearson's coefficient of correlation (κ) and their probabilities (p) evaluated the relationship between CDEIS, SES-CD and microscopic Crohn's disease activity.

Analysis of predictive factors of mucosal healing was performed. Univariate and multivariate analyses were performed to assess impact of clinical, disease, and treatment variables on mucosal healing. Continuous variable were dichotomised according to the median. Variables analysed were age at inclusion, gender, disease duration, active tobacco smoking, Crohn's disease location and behaviour, associated perineal disease, extra-intestinal manifestations, previous intestinal resection, current treatment type and duration, indication for colonoscopy, CRP level (mg/L). Difference between mucosal healing and non-healing groups was assessed by χ² or Fisher's exact tests when appropriate. A logistic regression model was created using significantly associated variables (p<0.10), and the odds ratios (OR) for the variables that

Results

Study population

During the study period, 1 796 consecutive colonoscopies were performed in the Gastroenterology department of the Haut-Leveque Hospital, including 266 in patients with Crohn's disease. Two-hundred-and-fifteen patients were excluded: 161 with active disease, 49 in remission not treated with a single immunomodulator at stable dose within the last three months, and 5 in whom colonoscopy aimed to assess mucosal healing. Therefore, 51 patients met the inclusion criteria (38 women; median age: 42 years): 18 receiving methotrexate, 18 azathioprine, and 15 infliximab. Their main characteristics are presented in Table 2. There was no difference between the methotrexate group and the two others except regarding median treatment duration that was shorter with methotrexate as compared to azathioprine (24 and 57 months, respectively, p=0.014).

Mucosal healing

The primary objective of the study -complete disappearance of mucosal ulcerations -was achieved in 2/18 (11%) patients treated with methotrexate, in 9/18 (50%) with azathioprine (p=0.011 vs. methotrexate) and in 9/15 (60%) with infliximab (p=0.008 vs. methotrexate) (Figure 1). Considering the CDEIS, patients receiving methotrexate had a higher median value than those from the two other groups (5.5, range [0.5-11.2] with methotrexate, 1.5, range [0-9.4] with azathioprine -p=0.002 vs. methotrexate -and 1.2, range [0-23.6] with infliximab -p=0.03 vs. methotrexate). Individual CDEIS values are presented in Figure 2A.

According to the pre-specified CDEIS cut-off for endoscopic remission, rates of patients with CDEIS<6 were 50% with methotrexate, 89% with azathioprine (p=0.011 vs. methotrexate) and 80% with infliximab (p=0.074 vs. methotrexate). With the cut-off for complete mucosal healing (CDEIS<4), rates were 39% with methotrexate, 83% with azathioprine (p=0,006 vs. methotrexate) and 67% with infliximab (p=0,112 vs. methotrexate). In univariate analysis, factors associated with mucosal healing were treatment at baseline and Crohn's disease location. In multivariate analysis, three factors were positively associated with mucosal healing: current treatment with azathioprine (OR=7. 13 ileo-colonic location; Table 3).

Histological response

Biopsy samples were available for 40 patients (12 methotrexate, 14 azathioprine, and 14 infliximab). Considering microscopic activity, no significant difference was observed between patients receiving methotrexate (median 2.5, range [1-12]) and those treated with azathioprine

Follow-up

During the follow-up period (median duration: 13.2 months, range [0-25.2]), 8 patients experienced Crohn's disease relapse with a median time of 4 months, range [0-23] (one patient without mucosal healing was lost to follow-up just after the inclusion and was considered to have relapsed at baseline). All but one had endoscopic ulcerations at entry. The last patient with mucosal healing at baseline experienced Crohn's disease worsening 23 months after inclusion. Relapse rates at 6, 12 and 24 months were 9.8%, 12.1% and 23.9%,

Discussion

Until recently, symptom improvement has been considered as the main therapeutic issue in Crohn's disease. With the advent of biologics, new objectives have been determined: mucosal healing, bowel damage, disability or surgical and hospitalisation rates [START_REF] Rutgeerts | Biological therapies for inflammatory bowel diseases[END_REF] . Mucosal healing has become an endpoint in clinical trials and emerges as a desirable goal in current practice [START_REF] De Chambrun | Clinical implications of mucosal healing for the management of IBD[END_REF] .

However, endoscopic response to conventional Crohn's disease treatments has been less studied than with anti-TNFs. When considering methotrexate, the efficacy of which has been proven in controlled trials [START_REF] Feagan | Methotrexate for the treatment of Crohn's disease. The North American Crohn's Study Group Investigators[END_REF][START_REF] Feagan | A comparison of methotrexate with placebo for the maintenance of remission in Crohn's disease. North American Crohn's Study Group Investigators[END_REF] , little is known about its ability to heal inflammatory lesions [START_REF] Kozarek | Methotrexate induces clinical and histologic remission in patients with refractory inflammatory bowel disease[END_REF][START_REF] Manosa | Does methotrexate induce mucosal healing in Crohn's disease?[END_REF] . In this prospective study conducted in Crohn's disease patients with sustained clinical remission, complete healing of endoscopic ulcerations was observed in 11% of methotrexatetreated patients. This rate was significantly lower compared to patients receiving monotherapy with azathioprine or infliximab: 50% and 60%, respectively.

Although mucosal healing is an important goal of treatment in Crohn's disease, no recognized definition exists. The disappearance of all mucosal ulcerations defined mucosal healing in controlled studies [START_REF] Rutgeerts | Adalimumab induces and maintains mucosal healing in patients with moderate to severe ileocolonic Crohn's disease. First results of the EXTEND trial[END_REF][START_REF] Colombel | Infliximab, azathioprine, or combination therapy for Crohn's disease[END_REF] , and should be very simple in practice. However, this binary outcomehealed-not healed -does not reflect the great variety of endoscopic signs of Crohn's disease activity, and two endoscopic quantitative scoring systems are available at this time. Firstly, the Crohn's disease endoscopic index of severity (CDEIS) has been prospectively established in order to detect changes in disease activity [START_REF] Mary | Development and validation of an endoscopic index of the severity for Crohn's disease: a prospective multicentre study. Groupe d'Etudes Therapeutiques des Affections Inflammatoires du Tube Digestif (GETAID)[END_REF] . It has been validated as a reproducible index.

Despite its complexity and the learning curve required, the CDEIS is considered as the gold standard for classifying severity of disease. Furthermore, two cut-offs defining endoscopic remission and complete mucosal healing have been proposed [START_REF] Hebuterne | Endoscopic improvement in patients with active Crohn's disease treated with certolizumab pegol: first results of the MUSIC clinical trial[END_REF] . Secondly, the simple endoscopic score for Crohn's disease (SES-CD) correlates well with the CDEIS [START_REF] Daperno | Development and validation of a new, simplified endoscopic activity score for Crohn's disease: the SES-CD[END_REF] Although the series reported here involved a modest number of patients, mucosal healing rates observed with azathioprine and infliximab are comparable to that previously published in patients in clinical remission with these drugs [START_REF] Rutgeerts | Scheduled maintenance treatment with infliximab is superior to episodic treatment for the healing of mucosal ulceration associated with Crohn's disease[END_REF][START_REF] Lemann | A randomized, double-blind, controlled withdrawal trial in Crohn's disease patients in long-term remission on azathioprine[END_REF] . Therefore, the small number of patients endoscopically healed with methotrexate in the same setting illustrates probably a lower mucosal efficacy of this drug. However, a more refractory Crohn's disease in patients receiving methotrexate cannot be excluded. Indeed, 78% previously received azathioprine which was discontinued because of failure or intolerance, and 83% infliximab. One can argue that this discrepancy can also be related to the treatment duration, which was shorter with methotrexate as compared to azathioprine. Nevertheless, duration of treatment was not associated with mucosal healing in the multivariate analysis. Furthermore, Crohn's disease inflammatory lesions seem to heal rapidly as suggested in a recent study with adalimumab showing similar endoscopic rates of response after 12 and 52 weeks of treatment. In the same study, shorter Crohn's disease duration was related to better endoscopic response [START_REF] Rutgeerts | Adalimumab induces and maintains mucosal healing in patients with moderate to severe ileocolonic Crohn's disease. First results of the EXTEND trial[END_REF] . This correlation was not confirmed in the current study, even if there was a trend for longer disease duration in patients treated with azathioprine and infliximab as compared to those receiving methotrexate.

Methotrexate is a folate anti-metabolite used at low doses in many inflammatory disorders. In Crohn's disease, it is usually proposed after azathioprine failure or intolerance as a secondline immunosuppressant [START_REF] Patel | Methotrexate for maintenance of remission in Crohn's disease[END_REF][START_REF] Dignass | The second European evidence-based consensus on the diagnosis and management of Crohn's disease: current management[END_REF][START_REF] Hausmann | Methotrexate for maintenance of remission in chronic active Crohn's disease: long-term single-center experience and meta-analysis of observational studies[END_REF][START_REF] Ng | Review article: the role of non-biological drugs in refractory inflammatory bowel disease[END_REF] . Indeed, methotrexate has been shown to have a steroid-sparing effect [START_REF] Feagan | Methotrexate for the treatment of Crohn's disease. The North American Crohn's Study Group Investigators[END_REF] and this compound could provide prolonged remission [START_REF] Hausmann | Methotrexate for maintenance of remission in chronic active Crohn's disease: long-term single-center experience and meta-analysis of observational studies[END_REF][START_REF] Lemann | Methotrexate in Crohn's disease: long-term efficacy and toxicity[END_REF] . Although the duration of treatment is not defined, two main reasons can lead to discontinue the drug. Firstly, the longterm duration of treatment with methotrexate may expose patients to side effects related to high cumulative doses leading then to withdrawal [START_REF] Hausmann | Methotrexate for maintenance of remission in chronic active Crohn's disease: long-term single-center experience and meta-analysis of observational studies[END_REF][START_REF] Lemann | Methotrexate in Crohn's disease: long-term efficacy and toxicity[END_REF] . Secondly, an increasing proportion of patients become methotrexate-refractory with time [START_REF] Hausmann | Methotrexate for maintenance of remission in chronic active Crohn's disease: long-term single-center experience and meta-analysis of observational studies[END_REF][START_REF] Lemann | Methotrexate in Crohn's disease: long-term efficacy and toxicity[END_REF][START_REF] Roumeguere | Combined approach with infliximab, surgery, and methotrexate in severe fistulizing anoperineal Crohn's disease: Results from a prospective study[END_REF] . This secondary loss of response to methotrexate could be related to partial efficacy of the drug, characterized by symptom relief without complete mucosal healing, as suggested here. The route of administration probably cannot explain the low rate of complete endoscopic healing observed as all the included patients were treated parenterally (sub-cutaneous and/or intra-muscular). In the same way, the dosage used in the present study was the regimen proven to be effective as maintenance in Crohn's disease (15-25 mg/week) [START_REF] Patel | Methotrexate for maintenance of remission in Crohn's disease[END_REF] . However, we cannot exclude that higher methotrexate doses may promote better endoscopic response.

Overall, the disappearance of all mucosal ulcerations is probably uncommon with methotrexate. However, it does not imply that methotrexate exerts no effect on gut mucosa.

Considering the CDEIS, which is more quantitative and a less stringent criterion than the complete disappearance of all ulcerations, more than half of patients receiving methotrexate achieved Crohn's disease endoscopic remission. Moreover, whatever endoscopic tool used for mucosal healing assessment, the discrepancy observed between patients receiving methotrexate and those treated with azathioprine or infliximab was mostly due to the persistence of few ulcerations. Prospective studies are now required to determine if methotrexate is able to induce and maintain Crohn's disease endoscopic improvement.

Histology is an established procedure for the diagnosis of Crohn's disease. Nevertheless, the evolution of the microscopic intestinal features has been poorly investigated and there is a lack of data regarding histological response to the main medications used in Crohn's disease. Some changes have been observed, mainly on signs of mucosal injury and the presence of neutrophils [START_REF] Geboes | Influence of treatment on morphological features of mucosal inflammation[END_REF] , and in a small placebo-controlled study, significant histological improvement due to the disappearance of the inflammatory infiltrate was seen after a single infliximab infusion [START_REF] Haens | Endoscopic and histological healing with infliximab anti-tumor necrosis factor antibodies in Crohn's disease: A European multicenter trial[END_REF] . Moreover, the correlation between clinical and endoscopic findings with histology is a matter of debate [START_REF] Geboes | Influence of treatment on morphological features of mucosal inflammation[END_REF] . In the present study which enrolled patients in clinical remission, endoscopic and histologic activities were not correlated. Furthermore, there is no validated and standardized microscopic scoring system for Crohn's disease. Targeting mucosal inflammation to achieve histological healing could be a greater objective than mucosal healing in Crohn's disease [START_REF] De Chambrun | Clinical implications of mucosal healing for the management of IBD[END_REF] and should be explored in future prospective studies.

Mucosal healing is an important issue in Crohn's disease patients and can probably predict the course of disease, as observed in the present study. Using quantitative measures such as CDEIS, methotrexate appears less effective than azathioprine or infliximab to heal the gut mucosa. However, further prospective studies have now to determine if this compound is able to induce a significant improvement of Crohn's disease bowel lesions, not only on the mucosal side but also on the entire gut wall in this transmural disease.

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

SES-CD

Methotrexate

SES-CD

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

  and study schedule This was a prospective single centre study conducted in the Gastroenterology department of the Haut-Leveque Hospital (south-western France) from October 2007 to May 2009. Eligible Crohn's disease patients were consecutively recruited when they met the following criteria: age 18 years or more, previous diagnosis of luminal Crohn's disease established on usual criteria including identification of mucosal ulcerations with ileo-colonoscopy, clinical remission off steroids within the last three months with a single immunomodulator (methotrexate or azathioprine or infliximab) as maintenance treatment at stable doses, requiring a colonoscopy based on usual indication in Crohn's disease (dysplasia/cancer screening according to the recommendations from the French endoscopic society, suspected stenosis) excluding assessment for mucosal healing. Subjects with ulcerative colitis, indeterminate colitis, active luminal Crohn's disease (CDAI>150), isolated ano-perineal location or ileal disease not assessable with colonoscopy were excluded. Inclusion date was defined as that of colonoscopy which was routinely performed by three experienced gastroenterologists (D.L., E.C., C.S.) not blinded to the medications taken by the patients. Data were collected at inclusion and the following characteristics were recorded: date of birth, gender, disease duration, age at diagnosis, disease location and behavior according to the Montreal classification 17 , perianal disease, extraintestinal manifestation, smoking history, previous history of intestinal surgery, previous response to steroids, clinical (CDAI) and biological (CRP level in mg/L) disease activity at baseline, past treatment with immunosuppressants and biologics, type of immunomodulator (methotrexate, azathioprine, infliximab) given as maintenance and duration of this treatment, indication for colonoscopic examination. Endoscopic and clinical data were available in all patients.

  taken in the most inflamed areas and for the screening of dysplasia.

  p<0.05) in the model determined. Considering complete mucosal healing at baseline (CDEIS<4), relapse-free survival rates in relation to observation time were estimated using Kaplan-Meier life-table method.

  -CD value was higher in patients receiving methotrexate as compared to the two other groups (6, range[2-18] with methotrexate, 3, range [0-16] with azathioprine -p=0.008 vs. methotrexate -and 2, range [0-27] with infliximab -p=0.032 vs. methotrexate).Individual values are shown in Figure2B. The two indices of endoscopic activity -CDEIS and SES-CD -were closely correlated (κ=0.927; p<0,001).

( 3 ,

 3 range [0-11]; p=0.622) infliximab (3.5, range [0-12]; p=0.583). No correlation was found between microscopic inflammation and CDEIS (Pearson's coefficient: 0.154; p=0.344) or SES-CD.

  complete mucosal healing at baseline according to CDEIS cut-off, patients with CDEIS<4 experienced fewer relapses during the follow up period than patients with CDEIS≥4 at entry (Figure3).

  and has been validated. No threshold defining remission has been established at this time.

16 Financial

 16 support: None.Potential competing interests:David Laharie has acted as speaker for, and has received research support from Schering-Plough.

Figure 1 :

 1 Figure legends

Figure 2 :

 2 Figure 2: Endoscopic activity in Crohn's disease patients in remission with maintenance monotherapy considering the Crohn's Disease Endoscopic Index of Severity (CDEIS) 19 (A) and the Simple Endoscopic Score for Crohn's Disease (SES-CD) 20 (B). Thick points indicate

Figure 3 : 22 Table 1

 3221 Figure 3: Kaplan-Meier curves of survival without relapse according to complete mucosal healing 10 at baseline. Continuous line: CDEIS<4; discontinuous line: CDEIS≥4

  Figure 2B

  , 95% confidence interval (CI): [1.11-45.68] vs. methotrexate), current treatment with infliximab (OR=19.02, 95%CI [2.49-145.52] vs. methotrexate), and pure ileal disease (OR=5.77, 95%CI [1.27-26.28] vs.

Table 2

 2 Main characteristics of the 51 patients included in the study according to the maintenance therapy (methotrexate, azathioprine, or infliximab).

	Methotrexate	Azathioprine	Infliximab

Table 3 :

 3 Factors associated with mucosal healing in multivariate analysis (logistic regression)

	OR	95% CI	p


Samer Riachy
email: samer.riachy@ensea.fr

Mamadou Mboup
email: mamadou.mboup@univ-reims.fr

Jean-Pierre Richard
email: pierre.richard@ec-lille.fr

Numerical differentiation on irregular grids

Keywords: Numerical differentiation, parameter estimation, irregular sampling

In this article, we propose a multidimensional numerical differentiation method from irregularly sampled data, in a noisy environment. The method extends to irregular grids our previous work and it relies on the same spirit as the algebraic numerical differentiators due to M. Mboup et al. ("Numerical differentiation with annihilators in noisy environment," Numerical Algorithm, 50, pp. 439-457, 2009). We provide a least squares interpretation of the developed estimators as well as a discrete implementation scheme. Several numerical simulations are presented to asses the performance of the proposed differentiators.

INTRODUCTION

Numerical differentiation of noisy data is a recurrent and longstanding problem. It has attracted a lot of attention in the monovariable case [START_REF] Anderssen | Numerical differentiation procedures for non-exact data[END_REF], [START_REF] Cullum | Numerical differenciation and regularization[END_REF], [START_REF] Diop | On numerical differentiation algorithms for nonlinear estimation[END_REF], [START_REF] Ibrir | A numerical procedure for filtering and efficient high-order signal differenciation[END_REF], [START_REF] Fliess | Systematic risk analysis: first steps towards a new definition of beta[END_REF], [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], [START_REF] Rangarajana | Lanczos generalized derivative for higher orders[END_REF]) and multivariable case [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF], [START_REF] Riachy | An algebraic method for multi-dimensional derivative estimation[END_REF][START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF]). This communication is devoted to numerical differentiation from irregularly collected data in the presence of noise.

Regular sampling has long been perceived as the most adequate way of collecting numerical data. Moreover, this perception is further strengthened by the nice sampling theorem of Nyquist-Shannon which has allowed the wide development of information theory. Nonetheless, in many practical situations, it may happen that the data are collected on irregular grids. It is, in particular, the case for seismic measurements [START_REF] Herrmann | Non-parametric seismic data recovery with curvelet frames[END_REF] where the data are often irregular and sparse. Another example arises from the coverage control with distributed mobile sensing robots [START_REF] Schwager | Decentralized, adaptive coverage control for networked robots[END_REF] where the goal is to bring a network of robots to spread out over a region to explore and collect information about a physical quantity while aggregating in areas where the quantity of interest is at a high level.

In addition to these situations, which we may qualify as accidental or stemming from physical constrains, the preeminence itself of the regular sampling is now questioned by the emerging field of compressed sensing or compressed sampling [START_REF] Donoho | Compressed sensing[END_REF]. This new paradigm, which fully This work was supported in part by the PEPS project SURFER of the CNRS and by the SyMeR project of ENSEA.

exploits the unnecessary character of the Nyquist criterion for perfect reconstruction, provides a renewed interest for developing numerical method based on irregularly sampled data.

This paper extends the multivariable differentiation techniques developed in [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF], [START_REF] Riachy | An algebraic method for multi-dimensional derivative estimation[END_REF][START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF] to irregular grids. While there exists few studies on numerical differentiation on irregular grids, interpolation of unorganized data (for 3-D shape reconstruction) has been the subject of a lot of studies (see [START_REF] Hoppe | Surface reconstruction from unorganized points[END_REF] for example). So, numerical differentiation could be approached by first resorting to such an interpolation method and then differentiating the resulting interpolant function. However, we consider a more direct method based on pointwise derivative estimation. As in [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF], [START_REF] Riachy | An algebraic method for multi-dimensional derivative estimation[END_REF][START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF], the method is inspired from the recent algebraic differentiators introduced by [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], within the framework of the algebraic estimation theory initiated in [START_REF] Fliess | An algebraic framework for linear identification[END_REF].

For simplicity's sake, the differentiation method is developed for two dimensional signals. The extension to higher dimensions is straightforward. The departure point is the classical assumption that on a small convex domain D ⊂ R 2 , the signal can be approximated by a convergent Taylor expansion. The desired derivative is expressed in terms of integrals of the measured signal. We point out that the synthesis of the estimators reduces considerably if the domain D is rectangular.

The communication starts with a brief review of the numerical differentiation method on regular grid of [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF], [START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF]. In section 3 we generalize the method to irregular grids and provide point-wise estimators. We show that the developed estimators can be written in a truncated Jacobi polynomial basis with a special weighing function leading to a link with least squares. In section 4 numerical simulations point out the validity of our derivative estimators. In the first (resp. second) simulation, we estimate the first (resp. second) order partial derivative of an academic noisy signal. In the third simulation, we consider the estimation of a physical parameter in a first order partial differential equation.

NUMERICAL DIFFERENTIATION ON A

REGULAR GRID: A BRIEF OVERVIEW Let us consider with [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF], Riachy et al.

[2010] a signal I(x, y) measured through an additive noise (x, y). The variables x and y are independent. The measured signal J(x, y) reads:

J(x, y) = I(x, y) + (x, y).

(1) The structured part I(x, y) is assumed to admit a convergent Taylor series expansion:

I(x, y) = i,j I x i y j (0, 0) x i y j i!j! , (2)
where I x i y j (0, 0) denote ∂ i+j ∂x i ∂y j I(0, 0). Our goal is to estimate I x n y m (0, 0). A truncation of (2) to the N th order (N ≥ n) in x and the M th order (M ≥ m) in y gives:

I N,M (x, y) = N i=0 M j=0 I x i y j (0, 0) x i y j i!j! . (3)
Next, apply (for x ≥ 0 and y ≥ 0) two Laplace transforms to (3):

ÎN,M (s, p) = N i=0 M j=0 I x i y j (0, 0))

s i+1 1 s j+1 2 , (4)
where s 1 (resp. s 2 ) is the operational analogue of x (resp. y). Let us extract the desired (n, m) th order derivative I x n y m (0, 0) from (4) by applying the differential operator1 : Π N,n,M,m κ1,µ1,κ2,µ2 = Ξ N,n κ1,µ1 Λ M,m κ2,µ2 , where

Ξ N,n κ1,µ1 = 1 s N +µ1+1 1 d n+κ1 ds n+κ1 1 1 s 1 d N -n ds N -n 1 s N +1 1 , and
Λ M,m κ2,µ2 = 1 s M +µ2+1 2 d m+κ2 ds m+κ2 2 1 s 2 d M -m ds M -m 2 s M +1 2 .
Finally by applying the inverse Laplace transform we obtain a family of estimators of the derivative I x n y m (0, 0) parameterized by the integers κ 1 , µ 1 , κ 2 and µ 2 .The general form of the estimators is given by:

I x n y m (0, 0) = X 0 Y 0 E(X, Y, x, y)J(x, y)dxdy, ( 5
)
where E is a polynomial in x and y while X and Y are any strictly positive constants and [0, X] × [0, Y ] is a rectangular domain on which the integrals are evaluated.

In [START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF], [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF] a detailed study of the structural properties of the estimators (5) was conducted. A least squares interpretation was provided as well as a discrete implementation scheme. Since regularly spaced data was assumed therein, the implementation scheme reduced to FIR filtering. Unfortunately, this is no longer possible on irregular grids, though, we will see that the numerical complexity is still low with our method.

The work in [START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF], [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF] consisted of a multivariable extension of [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] inspired from algebraic techniques initiated in [START_REF] Fliess | An algebraic framework for linear identification[END_REF].

NUMERICAL DIFFERENTIATION ON IRREGULAR GRIDS

We consider the estimation of the (n, m) th order partial derivative of a smooth signal I(x, y) defined on a convex domain D ⊂ R 2 , from some noisy observation J(x, y), as in (1). Without loss of generality, we will devise the derivative estimators around (0, 0). Derivative estimation at points (x, ȳ) different from (0, 0) is realized by translation. To begin, we replace the rectangular domain of section 2 by any convex domain. Let then D ⊂ D be any such domain around (0, 0). Setting a = inf

(x,y)∈D {x}; b = sup (x,y)∈D {x}, (6)
then there exists two continuous functions f 1 and

f 2 : [a, b] → R such that ∀(x, y) ∈ D, a x b, and f 1 (x) y f 2 (x)
. Of course, one can replace x by y and obtain similar description of D. Let then denote by c, d the infimum and supremum of D relative to the y axis and by g 1 (y) and g 2 (y) the analogue of f 1 and f 2 respectively.

Derivative estimators

In order to derive our estimators we ignore the noise for a moment and rewrite (3) for N = n, M = m with the simplified notation I(x, y) for I N,M (x, y) :

I(x, y) = n i=0 m j=0 I x i y j ( 0) x i y j i!j! , (7)
where 0 stands for (0, 0). Differentiate ( 7) n times with respect to x and m times with respect to y :

∂ n+m ∂x n ∂y m I(x, y) = I x n y m ( 0). ( 8
)
Let α 1 ≥ n, β 1 ≥ n, α 2 ≥ 2m and β 2 ≥ 2m be any four integers. Then, by multiplying equation ( 8) by

G α1,α2 β1,β2 (x, y) = (b -x) α1 (a -x) β1 (f 2 (x) -y) α2 (f 1 (x) -y) β2
and integrating over D, we isolate the desired partial derivative:

I x n y m ( 0) = b a f2(x) f1(x) G α1,α2 β1,β2 (x, y) ∂ n+m I(x, y) ∂x n ∂y m dxdy b a f2(x) f1(x) G α1,α2 β1,β2 (x, y)dxdy . (9)
An estimator I x n y m of I x n y m ( 0) is therefore obtained by replacing the unavailable truncated Taylor series I(x, y) by the actual noisy observation J(x, y):

I x n y m ( 0; •) = b a f2(x) f1(x) G α1,α2 β1,β2 (x, y) ∂ n+m J(x, y) ∂x n ∂y m dxdy b a f2(x) f1(x)
G α1,α2 β1,β2 (x, y)dxdy .

(10) Formula (10) provides a pointwise derivative estimator at (0, 0). Note that this estimator varies with the choice of the parameters α i and β i , i = 1, 2. This dependance is quoted by the notation above where (•) refers to (α 1 , α 2 , β 1 , β 2 ).

As mentioned earlier, a pointwise derivative estimator at (x, ȳ) = (0, 0) is accomplished by translation, that is by replacing the signal J(x, y) in ( 10) by its translated version J(x + x, y + ȳ). When there is no risk of confusion, the superscripts and the subscripts will be omitted from G α1,α2 β1,β2 (x, y) and write G(x, y) for short. Next, we show that (10) corresponds to an orthogonal projection in a properly chosen orthogonal polynomial basis function.

Least squares interpretation

Inspired by [START_REF] Mboup | Parameter estimation for signals described by differential equations[END_REF], let us consider the Jacobi polynomials {P

(α1,α2,β1,β2) q,p (x, y)}, p, q = 0, 1, . . ., orthogonal on the interval [a, b] × [f 1 (x), f 2 (x)
] with respect to the weighing function G(x, y). They can be derived using the following (Rodriguez) differential equation adapted from [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]:

G(x, y)P (α1,α2,β1,β2) q,p = (-1) q+p q!p! ∂ q P (x) ∂x q ∂ p Q(x, y) ∂y p
where P (x) = (b -x) α1 (a -x) α2 and Q(x, y) = (f 1 (x)y) β1 (f 2 (x) -y) β2 . Note that P 0,0 = 1. Let the (q, p) th order least square approximation of a function F(x, y) be defined by:

F LS,(q,p) (ξ, ζ) = q i=0 p j=0 P (•) i,j (x, y), F(x, y) P (•) i,j 2 P (•) i,j (ξ, ζ), (11)
where (•) still stands for (α 1 , α 2 , β 1 , β 2 ). We have: Proposition 3.1. The derivative estimator (10) is given by

I x n y m ( 0; •) = F LS,(1,1) (ξ 0 , ζ 0 ) + ∆ ,n,m (0, 0) (12) for F(x, y) = ∂ n+m I(x,y)
∂x n ∂y m and where ∆ ,n,m (0, 0) represents both the noise and the truncation error contributions and where (ξ 0 , ζ 0 ) is the zero of

P (•) 1,1 .
Proof. The proof follows the same lines as in [START_REF] Bachalany | Diffrentiation numérique multivariable 2 : projection orthogonale et filtrage rif[END_REF], [START_REF] Riachy | Diffrenciation numrique multivariable 1 : estimateurs algébriques et structure[END_REF] 2 This proposition shows that the estimator (10) corresponds to the derivative estimation at a delayed point given by (ξ 0 , ζ 0 ), although it is derived from a Taylor expansion around zero (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for more details).

Remark 1. When α 1 = β 1 and α 2 = β 2 , we have

ξ 0 = a+b 2 and ζ 0 = f1( a+b 2 )+f2( a+b 2 ) 2
. Therefore, it is preferable to have a domain D symmetric with respect to (0, 0) in order to reduce the delay or even cancel it when a = -b and f 1 (x) = -f 2 (x).

Higher order approximations and interpolation

In the previous subsection we showed that the pointwise derivative estimator is equivalent to a delayed first order least squares approximation, with a fixed delay. Here, we show that the delay can be set to any point (ξ, ζ) ∈ D by a judicious linear combination of the previous estimators. We also show that a higher order approximation can be achieved by a proper selection of the delay.

For this, let us consider the following linear combination:

H = λ 1 I( 0; α 1 + 1, β 1 , α 2 , β 2 ) + λ 2 I( 0; α 1 , β 1 + 1, α 2 , β 2 ) + λ 3 I( 0; α 1 , β 1 , α 2 + 1, β 2 ) + λ 4 I( 0; α 1 , β 1 , α 2 , β 2 + 1) + λ 5 I( 0; α 1 , β 1 , α 2 , β 2 ); λ 1 , • • • , λ 5 ∈ R. ( 13
) Let G α1,α2 β1,β2 = b a f2(x) f1(x) G(x, y)dxdy and set λ 1 = λ1 G α 1 +1,α 2 β1,β 2 , λ 2 = λ2 G α 1 ,α 2 β1+1,β 2 , λ 3 = λ3 G α 1 ,α 2 +1 β1,β 2 , λ 4 = λ4 G α 1 ,α 2 β1,β 2 +1 , λ 5 = λ5 G α 1 ,α 2 β1,β 2 .
Then, equation ( 13) reads as:

H = b a f2(x) f1(x)
H(x, y)G(x, y) ∂ n+m J(x, y) ∂x n ∂y m dxdy with H(x, y) given by:

H(x, y) = λ 1 (b -x) + λ 2 (a -x) + λ 3 (f 2 (x) -y) + λ 4 (f 1 (x) -y) + λ 5 . Proposition 3.2. Let (ξ, ζ) be any point in D. Setting λ 1 = (β1+1)P (•) 1,0 (ξ,ζ) P (•) 1,0 2 , λ 2 = (α1+1)P (•) 1,0 (ξ,ζ) P (•) 1,0 2 , λ 3 = (β2+1)P (•) 0,1 (ξ,ζ) P (•) 0,1 2 , λ 4 = (α2+1)P (•) 0,1 (ξ,ζ) P (•) 0,1 2 , λ 5 = 1 P (•) 0,0 2 , we have: H = F LS,(1,1) (ξ, ζ) + ∆ ,n,m (ξ, ζ) (14)
for F(x, y) = ∂ n+m I(x,y) ∂x n ∂y m . The noise and truncation error contributions are represented by ∆ ,n,m (ξ, ζ).

As a direct consequence, if we choose (ξ, ζ) in ( 14) to correspond to a zero of the second order Jacobi polynomial we get a second order least squares approximation. Recall that traditionally, numerical differentiation is approached via a least square interpolation followed by a differentiation of the resulting interpolant function. It is important to point out that our approach directly provides a least square approximation of the derivative. Moreover, the derivatives are approximated locally and pointwise.

Implementable estimators

Examining the estimator (10) we can see that partial derivatives of the noisy signal appear under the integrals. To get rid of the derivatives integrations by parts are performed. Note that G(x, y) was intentionally chosen such that it vanishes on a, b, f 1 and f 2 in order to get rid of the boundary conditions. Next, we put the first and second order derivatives into implementable forms.

Estimator for I x ( 0) and I y ( 0):

Let us consider the estimator for I y ( 0), corresponding to n = 0, m = 1, with the simple choice α 1 = β 1 = 0 and α 2 = β 2 = 1. By integrating by parts the numerator of (10), we obtain:

I y ( 0; •) = b a f2(x) f1(x) (f 1 + f 2 -2y)J(x, y)dxdy b a f2(x) f1(x) (f 1 -y)(f 2 -y)dxdy . (15)
Similarly an estimator for I x ( 0) is obtained by:

I x ( 0; •) = d c g2(y) g1(y) (g 1 + g 2 -2x)J(x, y)dydx d c g2(y) g1(y) (g 1 -x)(g 2 -x)dydx . (16)
Estimator for I x 2 ( 0) and I y 2 ( 0) : Taking 10) and integrating by parts twice we obtain, for n = 0, m = 2:

α 1 = β 1 = 0, α 2 = β 2 = 2, in (
I y 2 ( 0; •) = b a f2(x) f1(x) G 1 (x, y)J(x, y)dxdy b a f2(x) f1(x) (f 1 -y) 2 (f 2 -y) 2 dxdy , ( 17
)
where

G 1 (x, y) = 2(f 1 + f 2 -2y) 2 + 4(f 1 -y)(f 2 -y).
An estimator for I x 2 ( 0) is deduced similarly.

Estimator for I xy ( 0): Let us first introduce the new variables

u = x, v = y -f 1 (u) f 2 (u) -f 1 (u) , (18)
and define J by J(x, y) = J (u(x, y), v(x, y)). The partial derivative ∂ 2 J(x,y) ∂x∂y may be expressed as:

∂ 2 J(x, y) ∂x∂y = ∂ 2 J (u, v) ∂u∂v 1 f 2 (u) -f 1 (u) - ∂ 2 J (u, v) ∂v 2 f 1 (u) + v(f 2 (u) -f 1 (u)) (f 2 (u) -f 1 (u)) 2 - ∂J (u, v) ∂v f 2 (u) -f 1 (u) (f 2 (u) -f 1 (u)) 2 (19)
where f 1 (u) = df1 (u) du and f 2 (u) = df2(u) du . Let us also define

K(u, v) = (a -u)(b -u)v 2 (v -1) 2 (f 2 (u) -f 1 (u))
. With these settings, the numerator of ( 10) with n = m = 1 and using α 1 = β 1 = 1 and α 2 = β 2 = 2 becomes:

N = b a 1 0 K(u, v)(f 2 -f 1 ) ∂ 2 J (u, v) ∂v∂u dudv - b a 1 0 K(u, v)(f 1 + v(f 2 -f 1 )) ∂ 2 J (u, v) ∂v 2 dudv - b a 1 0 K(u, v)(f 2 -f 1 ) ∂J (u, v) ∂v dudv (20)
To the first double integral of (20), we apply the double integration by parts formula from [START_REF] Ghorpade | A course in multivariable calculus and analysis[END_REF] that we recall hereafter for completeness. Proposition 3. • The functions h and k are continuous in the first variable.

• h x , h y , h xy , k x and k y exist and are integrable on R.

• The functions h x and k x are continuous in the second variable. • k xy exists and k xy = g on R.

Then

R hg = ∆ (a,c) (b,d) (hk) - R (h x k y + h y k x + h xy k) (21) where ∆ (b,d) (a,c) (hk) := (hk)(b, d) -(hk)(b, c) -(hk)(a, d) + (hk)(a, c).
Applying (21) to the first integral of equation ( 20) yields:

N = - b a 1 0 K 1 (f 2 -f 1 )(2u -a -b)J (u, v)dudv - b a 1 0 K 2 (12v 2 -12v + 2)[f 1 + v(f 2 -f 1 )]J (u, v)dudv -6 b a 1 0 K 1 K 2 (f 2 -f 1 )J (u, v)dudv with K 1 = 2v(v-1)(2v-1) and K 2 = (a-u)(b-u)(f 2 -f 1 ).
Finally, the estimator reads as:

I xy ( 0; •) = N b a f2 (x) f1(x) G(x, y)dxdy . (22)
Remark that in order to estimate I xy ( 0, •), f 1 (x) and f 2 (x) should be at least differentiable.

Rectangular domain

In many situations, the integration domain D can be rectangular. The derivative estimator formulae become considerably simpler. In fact the pointwise estimator formula (10) evaluated on the rectangular domain D = [a, b]×[c, d] reads as:

I x n y m ( 0, •) = b a d c ∂ n+m G(x, y) ∂x n ∂y m J(x, y)dxdy b a f2 (x) f1(x)
G(x, y)dxdy .

Note, for example, that for α 1 = β 1 = α 2 = β 2 = 1 and n = m = 1, the following estimator:

I x,y ( 0; •) = b a d c (a + b -2x)(c + d -2y)J(x, y)dxdy b a d c (a + b -2x)(c + d -2y)dxdy
can be used in the place of ( 22).

Implementation

The integrals involved in the estimators are discretized and numerically evaluated. For the pointwise derivative estimation at a given point in D, all measurements points within a given distance are considered. The convex hull of these points defines the convex domain D ⊂ D. Then a Delaunay triangulation algorithm is applied on the points of interests. At each triangle the trapezoidal rule is applied to approximate the integral. More precisely the Delaunay triangulation consists of, say, L triangles {T i } L i=1 such that D = ∪ L i=1 T i and T i ∩ T j = ∅ ∀i = j. The estimator in equation ( 15), for example, is computed as:

I y ( 0; •) = L i=1 Ti (f 1 + f 2 -2y)J(x, y)dxdy L i=1 Ti (f 1 -y)(f 2 -y)dxdy . ( 23
)
For each integral in (23) the integrand is evaluated at the vertices of the corresponding triangle T i , then summed up, divided by three and multiplied by the area of T i .

NUMERICAL SIMULATIONS

In this section, we provide numerical simulation of the first and second order derivative estimators as well as a parameter identification example. An extensive numerical study will be published in a an extended future work.

First order derivative estimation

As a first example, we are interested in estimating the first order derivative of: T (x, y) = cos(2(x + y)).

(24) A slice of ( 24) at y = 0 is depicted in Figure 1. The square domain D = [-2.5, 2.5] × [-2.5, 2.5] is considered and 10000 points are generated randomly to simulate an irregular sampling. The derivative estimate is evaluated, however, on the regular grid 23) is evaluated using the convex domain: D = {(x, y); (x-x o ) 2 +(y-y o ) 2 < 0.15}. A white noise corrupts the signal as shown in the Figure 1. The estimation of ∂T ∂x (x, 0), x ∈ [-2, -1.9, • • • , 1.9, 2] is shown in the Figure 2. An example of a grid used to evaluate the estimators is given in the Figure 3.

O = [-2, -1.9, • • • , 1.9, 2] ×[-2, -1.9, • • • , 1.9, 2]. At each point (x o , y o ) of O equa- tion (

Second order derivative estimation

We are interested here in estimating the second order derivative of ( 24) with an additive noise as shown in Figure 1 on D = [-2.5, 2.5] × [-2.5, 2.5]. We randomly generate 100000 points in D. The derivative is calculated on the regular grid given by O above. At each point (x o , y o ) ∈ O a discretization of equation ( 16) is evaluated using D = {(x, y); (x -x o ) 2 + (y -y o ) 2 < 0.2}. The estimation of ∂ 2 T ∂x 2 (x, 0), is shown in the Figure 4.

Parameter estimation example

In this section, we consider an example of a fluid mechanics problem. We are interested in estimating the thermal conductivity κ(x, y) of a fluid, from the measurements of the heat flow rate q(x, y) and the temperature R.

The three quantities are related by the partial differential equation q(x, y) = κ(x, y)( ∂R(x,y) ∂x + ∂R(x,y) ∂y ). We assume that some (inevitably noisy) measurements of q and R are available on an irregular grid. The coefficient κ(x, y) indeed may vary because the material can be heterogeneous. However, we suppose that κ(x, y) is slowly varying such that we can consider it constant on the small domain D. An estimation of κ at a particular point (x, ȳ) is given by : κ(x, ȳ) = q0 (x, ȳ)

R x (x, ȳ) + R y (x, ȳ) , (25)
where q0 (x, ȳ) is some denoised version of q at (x, ȳ), ( R x , R y ) is an estimated temperature gradient. Equation ( 25) can be evaluated provided that R x (x, ȳ) + R y (x, ȳ) = 0. The function R is simulated by R(x, y) = cos(x + y) while q is given by q = -2 sin(x + y) cos(x/10). Thus the exact value of κ is κ(x) = cos(x/10). A white noise is added to the simulation. A plot of the noisy signal q is shown in figure 5. The estimated thermal conductivity is plotted (solid line) in the figure 6, in comparison with the exact solution (dashed line). The peak at 0 visible on figure 6 is due to the fact that both q and R x + R y vanishes at 0. Except for the numerical singularity at the origin, the estimates are in good agreement with the actual values.

CONCLUSION

In this paper we developed a method for derivative estimation on irregular grid. We showed that the estimators can be expressed in a Jacobi polynomials basis yielding a link with least squares local approximation. A discrete, implementable version is proposed. Several numerical simulations attested of the performances of our estimators.

This work laid the foundations of a new numerical differentiation method on irregular grids. Several points will be treated in future works, namely:

• the choice of the domain D (rectangular, triangular...), • robustness to various noises, • application to mobile sensing robotics.

  3. Double integration by parts Ghorpade and Limaye [2010] Let R := [a, b] × [c, d] and let h(x, y), g(x, y), k(x, y) : R → R be integrable functions satisfying the following:

  Fig. 1. A slice of the noisy signal.

Fig. 2 .

 2 Fig. 2. Exact (dashed line) and estimated (solid line) values of ∂T ∂x .

Fig. 3 .Fig. 4 .

 34 Fig.3. A randomly generated grid used for the estimation of T x with (x -2) 2 + (y -2) 2 < 0.15.

Fig. 5 .

 5 Fig. 5. Noisy plot of q(x, 0).

Fig. 6 .

 6 Fig.6. Exact (dashed line) and estimated (solid line) thermal conductivity.

Lack of space prenvents us from reproducing detailed computations.


Mohammad Ali Nazari
email: mohammad.nazari@gipsa-lab.grenoble-inp.fr

Pascal Perrier

Matthieu Chabanas

Yohan Payan

A 3D Finite Element Muscle Model and its Application in Driving Speech Articulators 1,3

A finite element muscle model has been designed with the Usermat functionality of ANSYS ® multiphysics software. It is represented by a transversely isotropic material which its strain energy is a function of I 4 and I 5 , invariants of the Cauchy-Green strain tensor due to muscle fibers. Muscle activation is accounted for by adding an active stress to the Cauchy stress in the direction of fibers. An additional isochoric term is included in the strain energy function, to account for lateral normal stiffness. This element has been tested in an existing finite element model of the human face, which has been designed to study interactions between motor control and physical properties of the face in speech production. Some typical facial gestures of speech are studied.

INTRODUCTION

Developing realistic biomechanical models of the articulators used in speech production (jaw, tongue, lips, velum and vocal folds) is a fruitful methodological approach to contribute to study speech motor control and the link between the units of language and their physical realizations in speech communication. The challenges of this research are as follows. What is the potential influence of the physical properties of the vocal tract and facial articulators in the emergence of the language, its evolution and its variability [START_REF] Ladefoged | Out of the chaos comes order; Physical, biological, and structural patterns in phonetics[END_REF][START_REF] Lindblom | Economy of speech gestures[END_REF][START_REF] Macneilage | The frame/content theory of evolution of speech production[END_REF][START_REF] Stevens | On the quantal nature of speech[END_REF][START_REF] Schwartz | The dispersion-focalization theory of vowel systems[END_REF]? To what extent do these physical properties influence the characteristics of the speech signals that are exchanged between speakers and listeners [START_REF] Löfqvist | Lip and jaw kinematics in bilabial stop consonant production[END_REF][START_REF] Perrier | Influences of tongue biomechanics on speech movements during the production of velar stop consonants: A modeling study[END_REF][START_REF] Fuchs | Is there a biological grounding of phonology? Determining factors, optimization and communicative usage[END_REF]? How are speech motor control strategies elaborated? How do they adapt to variations in speech communication conditions, and to possible pathological perturbations [START_REF] Nelson | Physical principles for economies of skilled movements[END_REF][START_REF] Perkell | A theory of speech motor control and supporting data from speakers with normal hearing and with profound hearing loss[END_REF]? For the biomechanical models to be efficient tools in this research approach, it is a requirement to be as realistic as possible, in order to provide a true and reliable account of the impact of the physics in the speech characteristics. A number of biomechanical models of speech articulators have been designed in the past (e.g. [START_REF] Wilhelms-Tricario | Physiological modeling of speech production: methods for modeling soft-tissue articulators[END_REF][START_REF] Sifakis | Simulating speech with a physics-based facial muscle model[END_REF][START_REF] Buchaillard | A biomechanical model of cardinal vowel production: Muscle activations and the impact of gravity on tongue positioning[END_REF][START_REF] Fang | A model-based investigation of activations of the tongue muscles in vowel production[END_REF][START_REF] Nazari | Simulation of dynamic orofacial movements using a constitutive law varying with muscle activation[END_REF]). In these models, muscles play a major role, since they are the main driving components of movements (that can be very fast for speech production) and they constitute a significant part of the structure that they deform (e.g. the tongue and face tissues). Hence, working at improving mechanical muscle models is a necessity. In this context, using a nearly incompressible hyperelastic material, which includes an account of a transversely isotropic behavior, is well suited to model passive mechanical properties [START_REF] Blemker | A 3D model of muscle reveals the causes of non uniform strains in the biceps brachii[END_REF]. An active component is added to model force generation mechanisms and their consequences on movements. This active component should introduce a force which varies as a function of a time varying activation parameter and depends on length and length variation rate of the muscle [START_REF] Van Leeuwen | Functional design of tentacles in squid: linking sarcomere ultrastructure to gross morphological dynamics[END_REF].

METHOD

The muscle model has been created using the Usermat functionality of ANSYS ® multiphysics software. Since muscles are composed of bundles of fibers embedded in a matrix of tissues, a transversely isotropic material fits well their mechanical behavior. The strain energy in this material is dependent on the following invariants, as shown in [START_REF] Spencer | Continuum theory of the mechanics of fiberreinforced composites[END_REF],

I 4 =a 0 .Ca 0 , I 5 =a 0 .C 2 a 0 C is the right Cauchy-Green strain tensor and a 0 the initial direction of fibers. This dependence is interesting in the design of a finite element muscle model, since it enables the fiber direction to be specified independently of the mesh geometry and topology. The Cauchy stress (σ f ) in fiber direction, as extracted from the strain energy function (W) for an incompressible material, becomes:

a) (a σ f ⊗ ∂ ∂ = 4 4 I I W 2
in which a is the fiber direction in the deformed configuration. Hence, an active stress in fiber direction can be easily added to the Cauchy stress in fiber direction such that:

a) (a σ f ⊗ + ∂ ∂ = ) σ I I W (2 active 4 4
A constitutive law implemented in a nonlinear finite element method is based on the co-rotated stress. In this theoretical context, the objective stress rate is Green-Naghdi stress rate which has been approximated by Jaumann stress rate [START_REF] Miehe | Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity[END_REF]. Jaumann elasticity tensor is derived numerically by a perturbation method [START_REF] Belytschko | Nonlinear Finite Elements for Continua and Structures[END_REF]. The strain energy (W) for the muscle is considered as in [START_REF] Blemker | A 3D model of muscle reveals the causes of non uniform strains in the biceps brachii[END_REF]. Since the proposed energy only takes into account the shear property of the muscle, an additional isochoric term C 1 (I 1 -3) is added to account for muscle lateral normal stiffness. This muscle model is used in a face mesh that has been developed recently [START_REF] Nazari | Simulation of dynamic orofacial movements using a constitutive law varying with muscle activation[END_REF]. In this mesh, elements are associated with muscles on the basis of anatomical fact and of information extracted from original CT scans of the subject's face. Figure 1 shows, as an example, the implementation of the orbicularis oris peripheralis (OOP). For the surrounding passive tissues in the mesh, a five parameter Mooney-Rivlin hyperelastic model is assumed to apply [START_REF] Nazari | Simulation of dynamic orofacial movements using a constitutive law varying with muscle activation[END_REF].

RESULTS AND DISCUSSION

A preliminary qualitative evaluation of the muscle model is provided. First, muscle elements embedded in a hexagonal bar are activated with two assumptions for the fibers directions, a lengthwise one (Fig. 2a) and a diagonal one (Fig. 2b). As expected, the diagonal activation produces a torsional deformation of the bar, while the lengthwise one acts along the main muscle course, with an upward bending due to the top implementation of the muscles fibers. In a second test, the muscle model is used to implement the OOP in the face model [START_REF] Nazari | Simulation of dynamic orofacial movements using a constitutive law varying with muscle activation[END_REF]. Muscle activation produces lip protrusion (Fig. 3).

CONCLUSIONS

This muscle model is a significant improvement in terms of physical realism. Since the active part in this model can be changed easily, it provides a suitable tool for examining different muscle theories. The model can be used easily in each FE model, and it will be used also in our existing tongue model [START_REF] Buchaillard | A biomechanical model of cardinal vowel production: Muscle activations and the impact of gravity on tongue positioning[END_REF] to unify all our models of vocal tract articulators.

Figure 1

 1 Figure 1 Orbicularis oris peripheralis muscle (a) on top of first layer (b) surrounded by elements in middle layer

Figure 2 .

 2 Activation of muscle elements embedded on top of a hexagonal bar for 2 fibers orientations; (a) lengthwise (b) diagonal.

Figure 3

 3 Figure 3 Lip protrusion following OOP activation


Majdi Elhajji

Brahim Attia

Abdelkrim Zitouni

Rached Tourki

Samy Meftali

Jean-Luc Dekeyser

FERONOC:FLEXIBLE AND EXTENSIBLE ROUTER IMPLEMENTATION FOR DIAGONAL MESH TOPOLOGY

Keywords: SoC, NoC, RTL, FeRoNoC

Networks on Chip (NoCs) can improve a set of performances criteria, in complex SoCs, such as scalability, flexibility and adaptability. However, performances of a NoC are closely related to its topology. The diameter and average distance represent an important factor in term of performances and implementation. The proposed diagonal mesh topology is designed to offer a good tradeoff between hardware cost and theoretical quality of service (QoS). It can contain a large number of nodes without changing the maximum diameter which is equal to 2. In this paper, we present a new router architecture called FeRoNoC (Flexible, extensible Router NoC) and its Register Transfer Level (RTL) hardware implementation for the diagonal mesh topology. The architecture of our NoC is based on a flexible and extensible router which consists of a packet switching technique and deterministic routing algorithm. Effectiveness and performances of the proposed topology have been shown using a virtex5 FPGA implementation. A comparative performances study of the proposed NoC architecture with others topology is performed.

INTRODUCTION

Modern applications of specific SoCs in signal audio and video processing require the increase of computation capabilities. Thus, Intellectual properties (IPs) have been more and more integrated which makes communication very complex in these systems. Consequently, according to a set of works in state of the art [START_REF] Shashi | A network on chip architecture and design methodology[END_REF][START_REF] Mikael | The nostrum backbone communication protocol stack for networks on chip[END_REF][START_REF] Luca | Powering networks on chips energy-efficient and reliable interconnect design for socs[END_REF][START_REF] Luca | Networks on chips: A new soc paradigm[END_REF][START_REF] Pierre | Ageneric architecture for onchip packet-switched interconections[END_REF] interconnection architecture based on shared busses shows its limits for the communication requirements of future SoC. In this context NoCs appear as a best solution to provide communication in the Chip. Due to the following characteristics; reliability, scalability of bandwidth and energy efficiency NoCs are emerging to replace busses.

However many applications, especially video encoder like H.264, have some performances requirements. Thus, a major goal in the SoC design is therefore,to ensure performances and QoS required by the application using the minimum available resource.

NoCs seems to be today the most appropriate communication solution for integrating many cores in a system and guaranteed QoS for applications. Indeed, the implementation of high NoC performances has become one of the most important challenges of designers. NoC is generally composed of three basic components: network interfaces (NIs), routers and links. Router which is an element of the NoC topology, implements routing function, switching technique and the flow control algorithms.Topology of NoC defines the connectivity or the routing possibilities between nodes, thus having a fundamental impact on the network performance as well as the switch structure (number of ports and port width). The tradeoff between generality and customization becomes then an important issue when choosing a network topology.

Another major factor that has an important impact on the NoC is the performance of router. The router is characterized by its degree, frequency, power consumption and latency. The degree of a router determines the number of its neighbors. Obviously, designing a router with a higher degree leads to the difficulties of VLSI (Very Large Scale Integration) implementation that can affects the performance of the SoC in term of used resources.

The motivation of this work has been addressing the demand for optimized infrastructure communication by proposing an optimal topology that provides a good performances. In the proposed diagonal mesh topology an even number of routers are connected by a links to the neighboring routers in clock wise and counter clock wise direction plus a central connection. The key characteristics of this topology include good network diameter that equal to 2, vertex symmetry, deterministic routing, generic number of routers and low degree for peripheral routers that equal to 3. High router degree reduces the critical path length but increases complexity.

This paper is organized as follows: Next section presents some related works. Section three presents the architectures and the routing algorithm of FeRoNoC. In section four, simulation and Implementation results for router on FPGA are presented and then some comparison and discussions takes place in section five. Finally we conclude the work.

RELATED WORKS

There are many works which offer new architectures of NoC like STNoC [START_REF] Marcello | Design of Cost-Efficient Interconnect Processing Units: Spidergon STNoC[END_REF] and GeNoC [START_REF] Schmaltz | A generic network on chip model[END_REF]. These NoCs are based on flexible and evolutionary packets. Moreover, they are based on the Spidergon and the Octagon topologies respectively and they have low costs of silicon implementation for the router and network interface. On the other side, most of the NoC use 2DMesh topologies [START_REF] Liang | An architecture and compiler for scalable on-chip communication[END_REF][START_REF] Zeferino | A parametric and scalable network-on-chip[END_REF][START_REF] Goossens | Aethereal network on chip: Concepts, architectures, and implementations[END_REF][START_REF] Uriel | Exact analysis of hot-potato routing[END_REF]. In [START_REF] Liang | An architecture and compiler for scalable on-chip communication[END_REF] SoCIN is presented, which is extensible based on XY routing. The basic router called RASoC(Router architecture for SoC) have a configurable FIFO. A SoC(adaptive system on chip) [START_REF] Zeferino | A parametric and scalable network-on-chip[END_REF] is a scalable architecture, flexible and modular communication between routers. The AEtheral NoC [START_REF] Goossens | Aethereal network on chip: Concepts, architectures, and implementations[END_REF] is based on the ATM network and adopts a fixed size packet technique which is oriented to a real time application. The disadvantage of this NoC is the fact that reception of the packets is not guaranteed when flits take different paths. HERMES [START_REF] Goossens | Hermes: an infrastructure for low area overhead packet-switching networks on chip[END_REF] is a 2D-Mesh NoC topology satisfies the requirement of implementing a low area and low latency communication for system on chip modules. Deterministic routing and Wormhole switching [START_REF] Dally | Deadlock-free message routing in multiprocesor interconnection networks[END_REF] are the dominant approach for NoCs researches.Deterministic routing algorithms are usually used because they require a low cost in term of logic compared to the adaptive algorithms. In Wormhole technique a packet is divided into flits, it implements the functionality with lower buffer requirement [START_REF] Dally | Route packets, not wires: Onchip interconnection networks[END_REF].Thus, it is an interesting solution compared to packet-based circuit switching and virtual cutthrough.Others switch technique like packet switching and virtual cut-through switching require enough buffers for saving a whole packet at each intermediate router. In [START_REF] Partha | Performance evaluation and design trade-offs for network-on-chip interconnect architectures[END_REF] authors have compared different NoC architectures. Based on different point of views including performances such as latency, area and power consumption. In another work, Bononi and Concer [START_REF] Bononi | Simulation and analysis of network on chip architectures: Ring, spidergon and 2d mesh[END_REF] compared ring, mesh and Spidergon topologies. Their paper showed that the spidergon topology outperforms the mesh and the ring topologies.

ROUTER ARCHITECTURE

The proposed Mesh diagonal topology generalizes and improves performance of the well known STOctagon network processor topology by a simple bidirectional ring with a central router. This network is constituted by (N+1) routers including a central element which is connected with all peripheral routers via links. Each peripheral router provides four input/output ports enable to connect left/right neighbors (routers), central router and the local IP. The central router contains N ports for the connection of each peripheral router and additional one permitting the connection with its local IP as is shown in figure 1: The communication packet used in our NoC is composed by three basic messages called flits which consist of:

• Header: composed by one bit (BOP)indicating the beginning of a packet,

• Body: contain the data to be transmitted on 32 bits,

• Tail: one bit indicating the end of a packet (EOP).

Both of the two kinds of routers composing the diagonal mesh topology use packet switching technique and Wormhole flow control mechanism.In packet switching, packets are transmitted without any need for connection establishment procedure.It requires the use of a switching mode, which defines how packets move through the switches.The described router has a controlled logic and bidirectional ports. Each port has an input buffer for temporary storage of information and a local port enabling communication between router and its local IP core. This module contains a set of components that are described below. Figue 2 shows a general block diagram for the router.

In the diagonal mesh, topology is presented in this work, each switch has a different number of ports, depending on its peripheral position or central router.

Peripheral Routers

In our NoC, we used a synchronous router with four input/output ports (local, clockwise, counter clockwise and across). Each port is connected to a bi-directional exchange bus. Each switch has a unique address and the switching technique used is packet switching. The data flowing through the network is a Wormhole routing. We made this choice to reduced number of buffers required per node and the simplicity of the communication mechanism. The diagonal mesh topology uses credit based flow control strategies. This later presents interesting advantages over handshake. In creditbased protocol, when the receiver is free, the transmitter sends a new data at each cycle of the clock and the receiver indicates its availability by a signal named credit. The peripheral router is composed by a bidirectional ports numbered starting from zero. They connect the router with its local IP and these neighboring bidirectional ports. The connection is defined as follows:

1. The first port is connected to neighbor in the clockwise direction.

2. the second port connected to the central router.

3. the third port connected to neighbor in counter clockwise.

The internal architecture of the peripheral router is composed by several components, such as input/output controller ), routing function and switch allocator. We use in each input port of our router an input component. This latter is divided into three components named FIFO, input controller and output controller. The main function of the input controller (IC) is to create an interface between output controller (OC) of the source router and IC of the current router. The IC is vated when the current router receives the beginning of packets (BOP). Thus, it allows the reception of packet flits from the output ports of the adjacent routers, storage in the FIFO buffer and the management of the flow control (credit based) between adjacent routers. The IC can accept a new packet when the previous one is not entirely switched. This part of the router allows the reception of packets sent by the neighboring OC and writes them in the FIFO. The block diagram of this component is shown in figure 3 4) credit out : control signal indicating that FIFO is not full; (5) data: indicating data to be received; [START_REF] Marcello | Design of Cost-Efficient Interconnect Processing Units: Spidergon STNoC[END_REF]data in : data to be written in the FIFO, [START_REF] Schmaltz | A generic network on chip model[END_REF] write: control signal indicating the writing in the FIFO. Thus as explained above, the IC aims to establish a connection between entities of initiator and destination routers. To complete the description of input block, an explanation of the functionality of the output controller (OC) is mandatory in order to finish the construction of the input block of peripheral router.

The output controller (OC) is the last block of the peripheral router. Their main roles consist in communicating with the IC, reading data, storing data in the FIFO and sending them after getting the control signal credit. When the output port, indicated by the routing function, is allocated in the output controller, the OC starts sending packet. This component is composed by control and data signals. The functional description of these signals is similar to that of the IC. Signal Grant indicates that the OC must send the packet to the corresponding output port. The association of an input and output controller constitutes the input module of the peripheral router. The FeRoNoC input module is described in VHDL and validated by a functional simulation. Its behavior can be summarized by the following steps:

1. When BOP=1, sending flits begins.

2. EOP=1, when the last flit is received.

3. Req=1 Sending.

4. Credit=1, the OC reads the data.

5. Full=1 the IC saves data in the FIFO.

Routing function and arbitration

A routing function defines the path followed by each message or packet through the NoC. It describes also how data are forwarded from sender to receiver by interpreting the destination address field of the header flit. The choice of a routing algorithm depends on several metrics like power,logic and routing table,increasing performance and maximizing traffic utilization of the network. In the FeRoNoC router there are two modules implementing the control logic which are routing and arbitration. The routing module is presented in figure 4. In our case, routing is deterministic and the communication between ports is not usually established. Indeed the local port can communicate with all ports. The central router has a direct connection with all the routers of the NoC. The routing function extracts the destination address of the packet, calculates the path to follow and generates a req p signal, where p indicates the number of destination of ports as represented in figure 4.

In our design, each input port of peripheral routers has its specific routing function. The routing algorithm is distributed inside the router as described in the following steps: The routing function of the local port can request three other output ports clock wise, Counter clock wise and across. To provide a correct message transfer, the following algorithm describes how to calculate the direction of packet.

1: numberof jump = (@dest -@curr)mod(N + 1) 2: if number of jump=0 then In the other side the clockwise port can request only two output ports Counter clockwise and local, thus the algorithm can be described as follow:

1: numberof jump = (@dest -@curr)mod(N + 1) 2: if number of jump=0 then 1: numberof jump = (@dest -@curr)mod(N + 1) 2: if number of jump=0 then 1: numberof jump = (@dest -@curr)mod(N + 1) 2: if number of jump=0 then 3:

Local port direction 4: end if

The EOP control signal allows the activation of the arbitration. When it is equal to 1 the arbitration is executed. Indeed the output port is allocated to an input port, thus the correct transfer is provided. The routing function sends control signal to the Switch Allocator of the suitable router. The hardware description of the Switch Allocator is presented in Figure 5. This module contains four arbitration components that decide the connection between input port and the output port. Its behavior is described by the following algorithm:

1: if R 0 allocate the output port then 2:

priority will be R 1 > R 2 > R 0 3: else if R 1 allocate the output port then 4:

priority will be R 2 > R 0 > R 1 5: else if R 2 allocate the output port then 6:

priority will be R 0 > R 1 > R 2 7: end if
As is shown in figure 5 the switch allocator has many control signals such as Req, grant and index. The index signal in-dicates the number of local input port, grant signal: indicates that the request is achieved. Indeed this information can activate the Crossbar element. The crossbar is a physical switch connecting the inputs to the outputs. It can be represented as a module with N inputs and N outputs. The switching in the crossbar can be achieved via multiplexers. The selection bit of each MUX could be generated by the index signal. Generally the transfer of the packet is performed in many steps. Firstly the state of the FIFO must be sent, secondly, the input controller starts data storage in the FIFO. Then, the routing function sends request signal to allocate the output port and finally follows the arbitration technique. Figure 6 presents a mapped architecture of the routing function and the switch allocator. The interaction between modules was described in VHDL and validated by functional simulation as is presented in Figure 7:

The simulation steps are described as:

1. The second port will send to local port.

2. the BOP=1, REQ=1 the IC module starts storage of data.

3. At the same clock the routing function receives the BOP signal and the destination address then specify the suitable output port.

Central router

The main objective of this router is to minimize the traffic and the load imposed on the peripheral router. Moreover, it allows the routing of data in the network to their destination with a number of required jumps equal to two. The importance of this router appears when the SoC contains a set of IP core. Hardware architecture of this router is the same as the peripheral routers. The only difference appears in the number of input/output port. Figure 8 shows the block diagram of this router.

Each input port of the central router has its specific routing function. The routing algorithm is distributed inside the

SYNTHESIS AND SIMULATION RESULTS

This section presents some simulation and synthesis results. It also explains how to find a compromise between latency and diameter. The router in the FeRoNoC was validated by functional simulation and by the synthesis results. We started by a VHDL simulation, in figure 9 a packet transmission in router is illustrated. The local port sends data to the second port after having sent the flits and the request. The simulation shows some of internal signals defining interconnection.

The target address is the peripheral router number 8. This later is located far from the source, thus the packet crosses the central router. The simulation scenario can be explained as follow:

1. Peripheral Router sends the first flit of the packet (address of the target switch) to the data out signal at it's across port and asserts the req and bop signals in this port.

2. Central Router detects the req signal asserted in its port number 0 and gets the flit in the data in signal. It takes 2 clock cycles to route this packet. Next flits are routed with 1-clock cycle latency.

3. Central Router puts the flit in data out signal and asserts the req and Bop signals of its output port number height. It takes 2 clock cycles to route this packet.

4. Router 8 detects asserted req signal of it's across port. The first flit of the packet is routed to the local port of the router 8 and the source to target connection is now established.

5. The remaining flits contain the payload of packet. However, the main objective of the central router is to improve latency in the NoC depending on the distance. In our case study the peripheral router 0 sends data to the peripheral router 8. The simulation shows that minimal latency to switch packet from source at target depends on the diameter of the network. Latency is given by:

Latency = ( n ∑ i=1 (R i )) + P × clockcycles ( 1
)
where n is the number of routers in the communication path, R i is the required time of the routing algorithm at each switch, P is the reference packet size and clock cycle presents a required time to send flits. Based on this equation, for our diagonal Mesh topology, the n=2 target and source routers are involved. In this paragraph some synthesis results are presented and a cost analysis of area and power consumption is realized. The router performance has been evaluated in terms of speed, latency and estimated peak performance. The FeRoNoC router was synthesized on Xlinx virtex 2pr xc2vp device using Xilinx ISE 9.1. The simulation was performed using modelSim 6.5 SE tool. The proposed router has been prototyped on 2 different FPGA technologies: Xilinx Virtex5 xc5vlx50-3ff676 and Xilinx virtex 2 pro. Table I presents our synthesis router results with Xilinx virtex 5 in which: area, operating frequency and power consumption results are shown. Table II presents implementation works targeting the same FPGA. The maximum running frequency is about 264 Mhz and the power consumption is 33mw.

Our approach provides performances in terms of latency, speed and area. The router design is highly modular and adaptable without the use of explicit handshake signal for communication between sub modules of the router. Other metrics evaluating our design is the peak performance which depends on the maximal clock frequency F max , the flit size (f lit size ) and the time (T) in clock cycle for transmitting a flit.

P M perport = (F max /T ) × f lit size (2)
The credit based control flow used by our router requires one clock cycle for transmitting one flit then T = 1 and f lit size = 32bits 6.9 Gbit/s per port 8.44 Gbits/s per port

COMPARAISON AND DISCUSSION

The work on flexible and extensible network on chip is an emerging topic. There are many proposed topologies in the literature, as cited in previous section each topology designs offers a different set of tradeoffs in terms of metrics, such as network degree, network extendibility notice in which a 2D mesh [START_REF] Bartic | Topology adaptive network-on-chip design and implementation[END_REF] topology provides very good theoretical metrics. Nevertheless, due to the increasing complexity of application, this topology cannot provide a good performance. On the other hand simple topology like ring provides a low cost in term of area but poor performance where the number of cores increases. The diameter and average distance represents an important factor in terms of performance and implementation. The proposed NoC is designed to deliver a good tradeoff between hardware cost and theoretical performance. Due to its higher connectivity our topology out performs ring, mesh and STspidergon in terms of diameter and average distance.

In addition, our diagonal Mesh topology can contain a large number of nodes without changing the diameter. Indeed it can deliver a good latency for multimedia application like video encoder (H.264). Thus, it represents a solution for on chip communication in next SoC. The major inconvenience of the proposal appears in the required link. However, based on the evolution of semi conductor technology this problem can be solved.

On the other side, many works are presented in the literature to implement NoC. Authors in [START_REF] Ehliar | An fpga based open source network-on-chip architecture[END_REF] describe a 2D meshes Network on chip implementation based on virtex4 and virtex2 FPGA. They describe an open source FPGA based NoC architecture with a high throughput and low latency. In this work, a generic bridge based on packet switching with Wormhole routing, was proposed. As is shown in table II, the data width of this implementation is about 36 bits and the maximum frequency is less that our frequency. This work provides a low cost area. In [START_REF] Kapre | Packet switched vs. time multiplexed fpga overlay networks[END_REF] authors present a packet switched NoC running on a virtex2. Moreover, they show an implementation of packet switched and time multiplexed FPGA overlay networks running at 166 Mhz. The aim of this work was to support designers in choosing between time multiplexing and packet switching. They use a 32 bits data width, as is shown in table II. Our implementation outperforms this work in term of area, latency and maximum frequency. and high speed (264 Mhz) communication for on chip modules. This paper presents all the details of our NoC such as topology, routing algorithm, dynamic arbiter and router module.This architecture offers a variety of SoC communication services owing to its flexibility and adaptability.The physical parameters of the designed router consist on the width and the depth of the FIFO, the number of the input/output ports, the valence and the maximal diameter of the NoC. Compared with other NoC, the advantage of this architecture, resides in its capacity to handle a suitable cost/performance compromise in the field of NoC. This is due to its wide constant and low diameter, latency of the router, frequency and power consumption. The simulation and implementation of this architecture show its performances and effectiveness. Our next objective is to prove the use of the diagonal mesh topology, mapped H.264 encoder will be investigated in a future work. Also the dynamic reconfiguration (in terms of number of nodes which depends on the number of IP of the application) will be studied. Adopting this technology, it is possible to obtain application specific NoCs. Moreover, we are going to design a low power SoC, based on the diagonal mesh topology and H.264 encoder, which includes dynamic reconfiguration.

Fig. 1 .

 1 Fig. 1. The NoC topology.

Fig. 2 .

 2 Fig. 2. The router architecture.

  . This figure describes signals and recommended hardware implementation for the router. The physical data bus width is 32 bits. The IC in the figure is composed by the following signals: (1) BOP control signal indicating the beginning of received packet, (2) EOP: indicating the end of packet;(3) REQ: control signal indicating data availability, (

Fig. 3 .

 3 Fig. 3. The Input module.

Fig. 4 .

 4 Fig. 4. Routing module.

4 :

 4 else if number of jump =14 v 15 then 5: counter clockwise direction 6: end if The routing function of the counter clockwise port can request only two ports (clock wise and local).

  end if Finally, routing function of across port can request only the local port.

Fig. 5 .

 5 Fig. 5. Switch Allocator.

Fig. 6 .

 6 Fig. 6. Interaction between routing function and Switch allocator.

Fig. 8 .

 8 Fig. 8. Block diagram of the central router.

Fig. 7 .

 7 Fig. 7. Simulation of the router.

Fig. 9 .

 9 Fig. 9. Simulation of the proposed diagonal Mesh topology

Table 1 .

 1 Synthesis

	results

Table 2 .

 2 Comparison with other worksNetwork on chip presents a most adapted technology to perform communication in complex SoCs. In this work, we present a novel topology named diagonal mesh and related router called FeroNoC.It offers a low latency (2 clock cycles)

	Perf./Works	[18]	[19]	Our
	Data width	36bits 32bits	32bits
	Latency	3	6	2
	Slice	431	1464	989
	Frequency(Mhz)	166	166	218
	Topology	Mesh Mesh Diagonal Mesh
	6. CONCLUSION