M.D Athanassios P Kyritsis

M.D Victor A Levin

An algorithm for chemotherapy treatment of recurrent glioma patients after temozolomide failure in the general oncology setting

Keywords: glioma, glioblastoma, chemotherapy, temozolomide

An algorithm for chemotherapy treatment of recurrent glioma patients after temozolomide failure in the general oncology setting

Introduction

High-grade gliomas (WHO grade 3 and 4 gliomas), and especially glioblastoma, are among the most difficult cancers to treat with short survival and poor response to chemotherapeutic drug intervention [START_REF] Levin | Chemotherapy for brain tumors of astrocytic and oligodendroglial lineage: the past decade and where we are heading[END_REF]. The reasons for the poor efficacy of these agents include the bloodbrain barrier which forms a pharmacological sanctuary, the expression of multidrug resistance proteins in glioma tumor cells and associated capillaries [START_REF] Alexiou | Fotopoulos AD Influence of Glioma's Multidrug Resistance Phenotype on (99m)Tc-Tetrofosmin Uptake[END_REF], the genetic, molecular, and metabolic heterogeneity of glioma tumor cells [START_REF] Kyritsis | Mutations of the p16 gene in gliomas[END_REF], resistance mechanism to commonly used alkylating agents, and, possibly, defects of both humoral and cellular immunity [START_REF] Gousias | Frequent abnormalities of the immune system in gliomas and correlation with the WHO grading system of malignancy[END_REF]. Tumor stem cells that exist within malignant gliomas, may provoke glioma development, perpetuation, and resistance to chemotherapy and radiation, accounting for the failure of conventional therapies and emergence of tumor recurrence [START_REF] Hadjipanayis | Tumor initiating cells in malignant gliomas: biology and implications for therapy[END_REF]. In addition, radiation therapy, the primary treatment modality for gliomas, triggers several signal transduction pathways and toxic events to the tumor, such as direct DNA damage which can lead either to cell death or to mutations in the surviving tumor cells that might result in subsequent emergence of resistant clones and recurrence [START_REF] Kargiotis | Effects of irradiation on tumor cell survival, invasion and angiogenesis[END_REF].

In general, postoperative adjuvant radiation therapy and chemotherapy with the second-generation monofunctional alkylating agent temozolomide (TMZ) became the standard care for newly diagnosed malignant gliomas soon after its introduction [START_REF] Yung | Temozolomide in malignant gliomas[END_REF][START_REF] Stupp | Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma[END_REF][START_REF] Stupp | Effects of radiotherapy with concomitant and adjuvant temozolomide versus radiotherapy alone on survival in glioblastoma in a randomised phase III study: 5-year analysis of the EORTC-NCIC trial[END_REF].

Unfortunately, high-grade gliomas usually recur despite treatment. In most cases, the radiographic determination of tumor growth is obvious, but sometimes radiation effect or outright radiation necrosis mimics tumor growth and requires different treatment. As a result, sometimes reoperation and/or biopsy are needed to confirm the diagnosis or to debulk tumor prior to chemotherapy. If surgery is not an option, specific imaging modalities, prior to initiation of secondary therapies, may help to establish the probability of tumor growth versus radiation effect [START_REF] Alexiou | Glioma recurrence versus radiation necrosis: accuracy of current imaging modalities[END_REF][START_REF] Alexiou | Evaluation of brain tumor recurrence by (99m)Tc-tetrofosmin SPECT: a prospective pilot study[END_REF][START_REF] Kaplan | Thallium-201 brain tumor imaging: a comparative study with pathologic correlation[END_REF][START_REF] Kumar | Malignant gliomas: MR imaging spectrum of radiation therapy-and chemotherapy-induced necrosis of the brain after treatment[END_REF][START_REF] Wong | Correlation between dynamic MRI and outcome in patients with malignant gliomas[END_REF].

For relapsed glioma patients that did not receive adjuvant TMZ chemotherapy, therapy with TMZ still remains a good choice [START_REF] Yung | A phase II study of temozolomide vs. procarbazine in patients with glioblastoma multiforme at first relapse[END_REF][START_REF] Yung | Multicenter phase II trial of temozolomide in patients with anaplastic astrocytoma or anaplastic oligoastrocytoma at first relapse. Temodal Brain Tumor Group[END_REF]. Some authorities recommend rechallenge with alternative dosing TMZ for recurrent malignant gliomas even if patients had received previously standard TMZ chemotherapy [START_REF] Brandes | Temozolomide 3 weeks on and 1 week off as firstline therapy for recurrent glioblastoma: phase II study from gruppo italiano cooperativo di neuro-oncologia (GICNO)[END_REF][START_REF] Franceschi | Salvage temozolomide for prior temozolomide responders[END_REF][START_REF] Wick | New (alternative) temozolomide regimens for the treatment of glioma[END_REF]. Perry et al. using continuous dose-intense TMZ for recurrent or progressive gliomas on standard TMZ therapy reported benefit with 6month progression-free survival of 23.9% in patients with GBM and 35.7% in patients with anaplastic glioma [START_REF] Perry | Phase II trial of continuous dose-intense temozolomide in recurrent malignant glioma: RESCUE study[END_REF]. In this review we evaluated all major phase II trials for recurrent gliomas, published after the emergence of TMZ and all active chemotherapies prior to the TMZ era. Our goal was to construct an algorithm for the general oncologist and not the specialist neuro-oncologist, how to treat an individual patient with recurrent glioma after TMZ failure, depending on the individual patient characteristics.

Evaluation of phase II studies

The criteria used to evaluate the various therapies include response rate which is associated with reduction in progression rate [START_REF] Hess | Response and progression in recurrent malignant glioma[END_REF], histology which is a dominant factor in determining outcome [START_REF] Wong | Outcomes and prognostic factors in recurrent glioma patients enrolled onto phase II clinical trials[END_REF], and progression-free survival (PFS) which is a more reliable end-point than overall survival since it reflects the true antitumor benefit of the chemotherapy [START_REF] Levin | Impact of phase II trials with progression-free survival as end-points on survival-based phase III studies in patients with anaplastic gliomas[END_REF] (Table 1).

In addition, a six-month PFS is strongly associated with survival suggesting that it is a valid end point in recurrent glioma trials [START_REF] Ballman | The relationship between six-month progression-free survival and 12-month overall survival end points for phase II trials in patients with glioblastoma multiforme[END_REF][START_REF] Lamborn | Progression-free survival: an important end point in evaluating therapy for recurrent high-grade gliomas[END_REF]. Objective response as an endpoint in phase II glioma trials may have limitations due to the frequent low response rates observed in recurrent glioma trials rendering extremely difficult to assess its predictive value to PFS and overall survival [START_REF] Van Den Bent | End point assessment in gliomas: novel treatments limit usefulness of classical Macdonald's Criteria[END_REF]. Furthermore, the radiologic assessment of response to treatment in patients with recurrent glioma is vulnerable to substantial inter-observer inconsistency [START_REF] Vos | Interobserver variability in the radiological assessment of response to chemotherapy in glioma[END_REF].

However, if the treatment has efficacy and the objective response substantial, it may be linked to survival and could be a reliable predictor of a treatment benefit in recurrent glioma patients [START_REF] Prados | Response as a predictor of survival in patients with recurrent glioblastoma treated with bevacizumab[END_REF].

Typically, the rate of 6-month progression-free survival (PFS-6) is 21% for TMZ naïve patients with recurrent glioblastoma (GBM) after treatment with TMZ [START_REF] Yung | A phase II study of temozolomide vs. procarbazine in patients with glioblastoma multiforme at first relapse[END_REF]. The PFS-6 in studies of patients with anaplastic glioma (AG) ranged from 37 to 41% [START_REF] Levin | Impact of phase II trials with progression-free survival as end-points on survival-based phase III studies in patients with anaplastic gliomas[END_REF]. After primary therapy with radiotherapy and TMZ followed by adjuvant TMZ, no standard salvage therapy exists for patients with high-grade glioma, although the FDA has approved recently an antiangiogenic therapy with bevacizumab for recurrent GBM because of promising response rates and increased median PFS [START_REF] Kreisl | Phase II trial of single-agent bevacizumab followed by bevacizumab plus irinotecan at tumor progression in recurrent glioblastoma[END_REF].

Molecular targeted therapies

What defines this therapeutic approach varies among physician-scientists. On the one hand alkylating agents that "target" DNA could be called targeted therapy, although most consider targeted therapy to be therapy that is directed toward specific signaling pathways, whether cell surface receptors, signaling proteins in the cytosol, or enzymes important to transcription or DNA effects in the nucleus. Unfortunately, many of the new purported targeted therapies are, in fact, drugs that target anywhere from 3 to more than 5 other proteins and thus are far from being specific and targeted therapies. Common among many of the reversible tyrosine kinase inhibitors is unexpected systemic toxicity based on their promiscuous behaviour to receptor and intracellular kinases.

While many of the new targeted drugs have been studied alone, lack of definite activity in many cases has led to their use in combination with alkylating agents such as temozolomide or with other presumed targeting agents. Combination of molecularly targeted therapies, such as anti-angiogenesis through inhibition of the VEGFR pathway (eg, bevacizumab), EGF receptor tyrosine kinase inhibitors (gefitinib and erlotinib), PDGF receptor inhibitors (imatinib), mTOR inhibitors (temsirolimus and everolimus), and protein kinase C-beta and other angiogenesis pathway inhibitors (vatalanib and enzastaurin) may be in theory more efficient therapies however, the lack of specificity of many of the protein tyrosine kinase inhibitors has led to unexpected dose-limiting toxicities [START_REF] Omuro | Exploring multi-targeting strategies for the treatment of gliomas[END_REF]. Unfortunately, most studies with such compounds, apart from few exceptions, have proven ineffective. For example, a phase II trial of combination of imatinib mesylate, an ATP-mimetic, tyrosine kinase inhibitor, plus hydroxyurea, showed good tolerance but only moderate activity in some patients with recurrent AG [START_REF] Desjardins | Phase II study of imatinib mesylate and hydroxyurea for recurrent grade III malignant gliomas[END_REF]. Similarly, administration of gefitinib (250 mg/d), an EGFR inhibitor, after first-line chemotherapy showed limited activity in patients with recurrent gliomas [START_REF] Franceschi | Gefitinib in patients with progressive high-grade gliomas: a multicentre phase II study by Gruppo Italiano Cooperativo di Neuro-Oncologia (GICNO)[END_REF]. However, the maximum tolerated dose of gefitinib could be 500 to 1000 mg/kg depending on the concomitant use of non-EIAEDs or EIAEDs respectively [START_REF] Reardon | Phase 1 trial of gefitinib plus sirolimus in adults with recurrent malignant glioma[END_REF]. Cetuximab, an EGFR antagonist showed limited toxicity and activity against recurrent GBM independently of the EGFR status of the tumors [START_REF] Neyns | Stratified phase II trial of cetuximab in patients with recurrent high-grade glioma[END_REF]. Overall, studies on EGFR inhibitors found no significant activity of these agents in high-grade gliomas [START_REF] Brandes | Epidermal growth factor receptor inhibitors in neuro-oncology: hopes and disappointments[END_REF]. The phase II clinical trials for recurrent glioblastoma and anaplastic glioma after temozolomide failure are depicted in tables 2 and 3 respectively.

Anti-angiogenesis

Recent research and therapy strategies have focused on understanding the mechanisms leading to the origin of tumor angiogenesis in order to develop new approaches that effectively block angiogenesis and promote tumor regression [START_REF] Kargiotis | Mechanisms of angiogenesis in gliomas[END_REF]. Angiogenesis can be targeted by molecules against the vascular endothelial growth factor (VEGF), or its receptor, VEGFR (small molecule tyrosine kinase inhibitors) [START_REF] Chamberlain | Antiangiogenic therapy for high-grade gliomas[END_REF]. Although antiangiogenic treatment is usually well tolerated, life-threatening complications can occur, such as thromboembolism, hemorrhage and gastrointestinal perforation [START_REF] Norden | Colon perforation during antiangiogenic therapy for malignant glioma[END_REF].

Bevacizumab is a humanized monoclonal antibody against VEGF-A, that has been shown to have significant biologic activity in patients with recurrent GBM [START_REF] Kreisl | Phase II trial of single-agent bevacizumab followed by bevacizumab plus irinotecan at tumor progression in recurrent glioblastoma[END_REF] (Table 2).

Bevacizumab seems to suppress the enhancing tumor recurrence more effectively than the non-enhancing infiltrative tumor [START_REF] Norden | Bevacizumab for recurrent malignant gliomas: efficacy, toxicity, and patterns of recurrence[END_REF]. It has been shown that while antiangiogenic therapy improves PFS, it may not prolong overall survival in patients with recurrent malignant glioma [START_REF] Norden | An exploratory survival analysis of anti-angiogenic therapy for recurrent malignant glioma[END_REF]. Other studies have demonstrated that bevacizumab improved survival but it was unclear if it increased the invasiveness of the tumor [START_REF] Narayana | Antiangiogenic therapy using bevacizumab in recurrent high-grade glioma: impact on local control and patient survival[END_REF]. Furthermore, if patients progressed on a regimen consisted of a combination of bevacizumab with chemotherapy, rarely responded to a second bevacizumab-containing chemotherapeutic regimen [START_REF] Quant | Role of a second chemotherapy in recurrent malignant glioma patients who progress on bevacizumab[END_REF].

Irinotecan (CPT-11)

Irinotecan, is an inhibitor of topoisomerase I, crosses the blood-brain barrier and has antitumor activity against gliomas [START_REF] Vredenburgh | Experience with irinotecan for the treatment of malignant glioma[END_REF]. The activity of irinotecan may be limited though with the concomitant administration of enzyme-inducing antiepileptic drugs (EIAEDs), since the pharmacokinetic studies indicated that the total body clearance of irinotecan is markedly enhanced and the maximum tolerated dose is increased 3.5 times in this study population [START_REF] Gilbert | Phase I clinical and pharmacokinetic study of irinotecan in adults with recurrent malignant glioma[END_REF].

Cloughesy et al. reported in two studies with irinotecan (CPT-11) activity against recurrent glioma especially with escalating doses suggesting that patients on EIAEDs may be underdosed [START_REF] Cloughesy | Two studies evaluating irinotecan treatment for recurrent malignant glioma using an every-3-week regimen[END_REF]. However, Prados et al. failed to show such an association but the CPT-11 dose in patients that received EIAEDs was 750 mg/m2, lower to the maximum of 1700 mg/m2 that was used in the previous study [START_REF] Prados | A phase 2 trial of irinotecan (CPT-11) in patients with recurrent malignant glioma: a North American Brain Tumor Consortium study[END_REF].

Thalidomide

Thalidomide, an antiangiogenic agent, has been considered a therapeutic option for patients with recurrent GBM after failure of other more active chemotherapeutic drugs, since it is generally well tolerated [START_REF] Morabito | Thalidomide prolongs disease stabilization after conventional therapy in patients with recurrent glioblastoma[END_REF]. However, thalidomide as a single agent had very limited antitumor activity in patients with recurrent malignant gliomas [START_REF] Fine | Phase II trial of the antiangiogenic agent thalidomide in patients with recurrent high-grade gliomas[END_REF].

Fotemustine

Fotemustine, a third generation chloroethylnitrosourea given to recurrent glioma patients after TMZ failure showed very promising results and limited toxicity in three recent studies from Italy [START_REF] Fabi | Treatment of recurrent malignant gliomas with fotemustine monotherapy: impact of dose and correlation with MGMT promoter methylation[END_REF][START_REF] Fabrini | A multi-institutional phase II study on second-line Fotemustine chemotherapy in recurrent glioblastoma[END_REF][START_REF] Scoccianti | Second-line chemotherapy with fotemustine in temozolomide-pretreated patients with relapsing glioblastoma: a single institution experience[END_REF]. In addition, like other alkylating agents, fotemustine appears more effective in patients with O6-methylguanine-DNA methyltransferase (MGMT) gene promoter methylation [START_REF] Fabi | Treatment of recurrent malignant gliomas with fotemustine monotherapy: impact of dose and correlation with MGMT promoter methylation[END_REF].

Retinoids

Retinoic acid, a synthetic analog of vitamin A, has multiple biologic effects against malignant gliomas and may be suitable as a maintenance therapy for patients that enter a remission after a primary treatment [START_REF] Wismeth | Maintenance therapy with 13-cis retinoid acid in high-grade glioma at complete response after first-line multimodal therapy--a phase-II study[END_REF]. An earlier study of cis-retinoic acid (cRA) in recurrent glioma had indicated activity of this agent especially in GBMs [START_REF] Yung | Treatment of recurrent malignant gliomas with high-dose 13-cis-retinoic acid[END_REF]. However, a retrospective analysis of 82 patients with recurrent GBM, most of whom had been previously heavily treated with chemotherapy showed modest activity with cRA with response and stable disease of 45%, a median PFS of 19 wks, but in the subset of 11% who responded, the median PFS was 59 weeks [START_REF] See | 13-cis-retinoic acid in the treatment of recurrent glioblastoma multiforme[END_REF]. Another synthetic retinoid derivative, fenretinide, has been inactive in a phase II trial of recurrent malignant glioma [START_REF] Puduvalli | Phase II study of fenretinide (NSC 374551) in adults with recurrent malignant gliomas: A North American Brain Tumor Consortium study[END_REF]. Similar results were obtained with all-trans-retinoic acid (tretinoin) [START_REF] Kaba | The treatment of recurrent cerebral gliomas with all-trans-retinoic acid (tretinoin)[END_REF].

Combination chemotherapy 1. Bevacizumab and irinotecan

A combination of bevacizumab at 10-15mg/kg every 3 weeks with irinotecan at 125-340 mg/m2 depended on the use or not of EIAEDs and administered at various schedules showed significant activity in anaplastic glioma patients with PFS6 of 55% and median OS of 65 weeks [START_REF] Desjardins | Bevacizumab plus irinotecan in recurrent WHO grade 3 malignant gliomas[END_REF]. Activity of this regimen with acceptable toxicity was also documented in multiple phase II trials of only GBM patients [START_REF] Vredenburgh | Bevacizumab plus irinotecan in recurrent glioblastoma multiforme[END_REF][START_REF] Friedman | Bevacizumab alone and in combination with irinotecan in recurrent glioblastoma[END_REF] or studies that combined a mixture of recurrent GBM and AG patients [START_REF] Vredenburgh | Phase II trial of bevacizumab and irinotecan in recurrent malignant glioma[END_REF][START_REF] Kang | Irinotecan and bevacizumab in progressive primary brain tumors, an evaluation of efficacy and safety[END_REF][START_REF] Poulsen | Bevacizumab plus irinotecan in the treatment patients with progressive recurrent malignant brain tumours[END_REF][START_REF] Bokstein | Treatment with bevacizumab and irinotecan for recurrent high-grade glial tumors[END_REF].

Other irinotecan-based combinations

Puduvalli et al. reported that a combination of CPT-11 with the anti-angiogenesis agent thalidomide resulted in potentially promising results in 32 previously treated GBM patients, 83.8% of whom had temozolomide-based chemotherapies. The PFS6 in this study was 14%

and the median overall survival 30 weeks. However, in that study there were 4 deaths, 2 of them possibly due to treatment-related toxicity [START_REF] Puduvalli | Phase II trial of irinotecan and thalidomide in adults with recurrent glioblastoma multiforme[END_REF]. A smaller study in 16 patients with recurrent GBM using the same agents failed to show any additional benefit from the addition of thalidomide to irinotecan [START_REF] Fadul | A phase II study of thalidomide and irinotecan for treatment of glioblastoma multiforme[END_REF]. Combination of CPT-11 with VM-26 a topoisomerase II inhibitor showed no synergy between the 2 drugs and the results were comparable to results when each drug used individually [START_REF] Feun | Clinical trial of CPT-11 and VM-26/VP-16 for patients with recurrent malignant brain tumors[END_REF] In a phase I study the combination of BCNU with escalating dose of weekly administered CPT-11 was evaluated. From this study it was determined that, the recommended CPT-11 dose for patients on EIAEDs was 125 mg/m2 and for patients not on EIAEDs it was 225 mg/m2 [START_REF] Quinn | Phase 1 trial of irinotecan plus BCNU in patients with progressive or recurrent malignant glioma[END_REF]. However, a phase II trial employing such a combination resulted in comparable results to that of CPT-11 alone and increased toxicity [START_REF] Reardon | Phase 2 trial of BCNU plus irinotecan in adults with malignant glioma[END_REF]. Similarly, ACNU, another nitrosourea given alone or in combination with teniposide or cytarabine in patients with GBM after TMZ failure resulted in limited activity and considerable toxicity [START_REF] Happold | ACNU-based chemotherapy for recurrent glioma in the temozolomide era[END_REF]. The combination of CPT-11 and celecoxib, a selective COX-2 inhibitor was well tolerated and had a marginal activity against heavily pretreated recurrent glioma (mostly GBM) patients with PFS6 of 25.1% [START_REF] Reardon | Phase II trial of irinotecan plus celecoxib in adults with recurrent malignant glioma[END_REF].

Other chemotherapeutic combinations

Addition of metronomic etoposide in bevacizumab treated patients with recurrent glioma had no additional benefit compared to bevacizumab alone but resulted in increased toxicity [START_REF] Reardon | Metronomic chemotherapy with daily, oral etoposide plus bevacizumab for recurrent malignant glioma: a phase II study[END_REF].

Combination of carboplatin and high-dose tamoxifen in a mixture of patients with recurrent malignant gliomas of various histologies showed a median OS of 56 weeks similar to other active regimens and equivalent to those found using tamoxifen as monotherapy [START_REF] Tang | A phase II study of carboplatin and chronic high-dose tamoxifen in patients with recurrent malignant glioma[END_REF]. Use of pegylated liposomal doxorubicin in order to improve better penetration through the bloodbrain barrier, alone or in combination with high dose tamoxifen resulted in PFS6 of 15%, and median TTP of 17 weeks, suggesting moderate effectiveness in patients with recurrent highgrade glioma [START_REF] Hau | Pegylated liposomal doxorubicin-efficacy in patients with recurrent high-grade glioma[END_REF]. Combination of carboplatin at a dose of AUC 6mg x ml/min administered every 28 days, and erlotinib 150 mg/d had modest activity with median PFS of 9 weeks, PFS6 of 14%, and median OS of 30 weeks [START_REF] De Groot | Phase II study of carboplatin and erlotinib (Tarceva, OSI-774) in patients with recurrent glioblastoma[END_REF].

Nitrosourea combinations have long been popular and efficacious, although considerably more toxic than temozolomide-based treatments. Lomustine alone [START_REF] Wick | Phase III study of enzastaurin compared with lomustine in the treatment of recurrent intracranial glioblastoma[END_REF] or in combination with other agents [START_REF] Ahluwalia | Society for Neuro-Oncology Annual Meeting: a report of selected studies[END_REF] has demonstrated activity against recurrent GBM after temozolomide failure. Prior to temozolomide, a chemotherapy combination was developed called TPCH (6-thioguanine, procarbazine, lomustine and hydroxyurea) and was created to increase tumor cell kill and, hopefully, to reduce repair of DNA damage. The therapy was active against previously-treated recurrent anaplastic glioma but less so against glioblastoma [START_REF] Kyritsis | Combination of 6-thioguanine, procarbazine, lomustine, and hydroxyurea for patients with recurrent malignant gliomas[END_REF]. More recently, another combination chemotherapy protocol was developed called TCCC (6-thioguanine, lomustine or temozolomide, capectabine, celecoxib) and was designed to be used for patients failing either temozolomide (lomustine arm) or lomustine (temozolomide arm). The vast majority of patients were treated on the lomustine arm and, like the prior TPCH therapy, glioblastoma patients did not do as well as the anaplastic glioma patients that had combined response and stable disease rate of 68% with a PFS at 12 months of 44% [START_REF] Walbert | Combination of 6-thioguanine, capecitabine, and celecoxib with temozolomide or lomustine for recurrent high-grade glioma[END_REF].

Association of certain genetic abnormalities and response to chemotherapy

O(6)-methylguanine-DNA methyltransferase (MGMT), is a DNA repair enzyme that is frequently up-regulated in gliomas, rendering alkylating treatment insufficient [START_REF] Silber | O6methylguanine-DNA methyltransferase-deficient phenotype in human gliomas: frequency and time to tumor progression after alkylating agent-based chemotherapy[END_REF].

Methylation and silencing of MGMT promoter may be a favorable prognostic factor in patients with glioblastoma treated with alkylating agents [START_REF] Hegi | MGMT gene silencing and benefit from temozolomide in glioblastoma[END_REF]. Thus, in glioma patients if antiepileptic medication is needed, use of levetiracetam, a potent inhibitor of MGMT, may sensitize glioma cells to alkylating agents [START_REF] Bobustuc | Levetiracetam enhances p53-mediated MGMT inhibition and sensitizes glioblastoma cells to temozolomide[END_REF]. Furthermore, clinical trials to deplete MGMT by O-6-benzylguanine, a noncytotoxic substrate of this enzyme, in combination with alkylating agents are in progress [START_REF] Quinn | Phase I trial of temozolomide plus O6-benzylguanine for patients with recurrent or progressive malignant glioma[END_REF][START_REF] Quinn | Phase II trial of temozolomide plus o6-benzylguanine in adults with recurrent, temozolomide-resistant malignant glioma[END_REF].

In tumors of oligodendroglial origin the -1p/-19q genotype may predict response to cytotoxic chemotherapy and favorable outcome [START_REF] Walker | Clinical use of genotype to predict chemosensitivity in oligodendroglial tumors[END_REF]. However, the presence of polysomy for chromosomes 1 and 19 in anaplastic oligodendrogliomas with concurrent deletion of 1p/19q is a marker of early recurrence [START_REF] Snuderl | Polysomy for chromosomes 1 and 19 predicts earlier recurrence in anaplastic oligodendrogliomas with concurrent 1p/19q loss[END_REF]. Irinotecan as single agent in patients with recurrent, TMZ-refractory, 1p19q co-deleted, anaplastic oligodendroglioma showed only modest effect [START_REF] Chamberlain | CPT-11 for recurrent temozolomide-refractory 1p19q co-deleted anaplastic oligodendroglioma[END_REF], but improved activity with a combination of bevacizumab and irinotecan [START_REF] Taillibert | Bevacizumab and irinotecan for recurrent oligodendroglial tumors[END_REF], or bevacizumab alone [START_REF] Chamberlain | Bevacizumab for recurrent alkylator-refractory anaplastic oligodendroglioma[END_REF].

Future directions

Use of various dietary and nutraceutical approaches in patients with gliomas as adjuncts to standard therapies might prove to be beneficial for patients with recurrent gliomas. These agents include certain natural dietary components such as phytoestrogens and flavonoids, methionine restriction which may modulate MGMT activity and polyamine activity, antiinflammatory drugs, and certain polyunsaturated fatty acids such as gamma-linolenic acid (GLA) that may exert antineoplastic activity against gliomas without harming normal cells [START_REF] Kyritsis | Modulation of Glioma Risk and Progression by Dietary Nutrients and Antiinflammatory Agents[END_REF]. For example, intratumoral administration of GLA in 9 patients with recurrent GBM showed some effect and no significant side effects suggesting that high doses of GLA should be investigated in subsequent studies [START_REF] Bakshi | Gamma-linolenic acid therapy of human gliomas[END_REF]. Similarly, eflornithine (DFMO), an irreversible inhibitor of ornithine decarboxylase, the first enzyme in polyamine synthesis, has shown activity against gliomas as a single agent and in combination with cytotoxic drugs [START_REF] Levin | Phase I-II study of eflornithine and mitoguazone combined in the treatment of recurrent primary brain tumors[END_REF][START_REF] Levin | Phase III randomized study of postradiotherapy chemotherapy with combination alphadifluoromethylornithine-PCV versus PCV for anaplastic gliomas[END_REF][START_REF] Levin | Phase III randomized study of postradiotherapy chemotherapy with alpha-difluoromethylornithineprocarbazine, N-(2-chloroethyl)-N'-cyclohexyl-N-nitrosurea, vincristine (DFMO-PCV) versus PCV for glioblastoma multiforme[END_REF]. In patients with recurrent glioma and epilepsy, combination of chemotherapy with antiepileptic drugs that demonstrate antineoplastic activities might offer an additional benefit [START_REF] Kargiotis | AP Epilepsy in the cancer patient[END_REF].

Other potential useful approaches may include use of dendritic cell vaccines [START_REF] Wheeler | Dendritic cell vaccines to combat glioblastoma[END_REF], and gene therapy, virotherapy or stem cells as adjuncts to conventional therapies for gliomas [START_REF] Kyritsis | Viruses, gene therapy and stem cells for the treatment of human glioma[END_REF].

Replication-incompetent or competent viruses can be used either as gene delivery vehicles to gliomas or to induce oncolysis and avoid damage of the adjacent normal cells [START_REF] Kargiotis | Adenovirus-mediated transfer of siRNA against MMP-2 mRNA results in impaired invasion and tumor-induced angiogenesis, induces apoptosis in vitro and inhibits tumor growth in vivo in glioblastoma[END_REF][START_REF] Kyritsis | Viruses, gene therapy and stem cells for the treatment of human glioma[END_REF]. Similar approach consists of the use of stem cells for more efficient delivery of genetic material to glioma cells. Continuing research in improvement of gene transfer efficiency and stem cell technology may overcome various safety issues, and open new avenues in the management of gliomas in the future.

Conclusion

The management of newly diagnosed malignant gliomas include maximal surgery, postoperative radiation therapy and, commonly, adjuvant chemotherapy with TMZ. At recurrence, patients that did not receive adjuvant TMZ chemotherapy should probably receive a TMZ-based chemotherapy. Based on our personal objective evaluation of the published phase II trials, for patients that do not participate in phase II trials, we recommend the following chemotherapies after TMZ failure in order of priority (Figure 1): In patients with adequate bone marrow reserve and methylated MGMT status (likely sensitive to alkylating agents), nitrosourea (fotemustine, lomustine), bevacizumab/irinotecan and cis-retinoic acid. In patients with poor bone marrow reserve and methylated MGMT status, bevacizumab alone should be used followed by cis-retinoic acid. In patients with adequate bone marrow reserve and unmethylated MGMT (likely resistant to alkylating agents), bevacizumab/irinotecan followed by cRA, although TPCH or TCCC-like treatments might still have activity in about 30% of patients [START_REF] Kyritsis | Combination of 6-thioguanine, procarbazine, lomustine, and hydroxyurea for patients with recurrent malignant gliomas[END_REF] could be used. In patients with poor bone marrow reserve and unmethylated MGMT, bevacizumab alone followed by cis-retinoic acid is recommended.

Future studies employing dietary and nutraceutical approaches, tumor vaccines, gene therapy, virotherapy and stem cells may be proven to be useful as adjuncts to standard chemotherapies in order to improve the therapeutic benefit and prolong survival.

Figure 1 .

 1 Figure 1. Algorithm for management of glioma recurrence after initial surgery, radiotherapy

Table 1 .

 1 Endpoints for evaluation of phase II clinical trials in recurrent glioma

	Study	No	Age	Karnofsky	Prior	Prior
		patients	(range)		surgeries	chemotherapy

Table 2 .

 2 Results of phase II clinical trials in recurrent glioblastoma patients after temozolomide failure

	Study	Treatment	No	Toxicity levels 3/4	Response + Stable Disease, %	MPFS, weeks	PFS6 (95% CI)	Median OS, weeks	Comment
	Franceschi et al, 2007 [32]	Gefitinib	16	None	17.9		12.5% 38.4%) (1.6-	NC	Limited activity
	Neyns et al, 2009 [34]	Cetuximab	55	13.6%	35.1	7.6	7.3% (0-14%)	20	Limited activity No differences between EGFR + and EGFR -
	Puduvalli et al, 2008 [64]	CPT-11 and Thalidomide	32	56%	65.6	13	25% (14-46%)	36	CPT-11 with thalidomide is promising
	Fadul et al, 2008 [65]	CPT-11 and Thalidomide	16	37.5%	18.8		19% (4-46%)	41	Comparable to CPT-11 alone
	De Groot et al, 2008 [74]	Carboplatin and Erlotinib	43	100%	49.3	9	14% (4-24%)	30	Carboplatin with erlotinib has modest activity
	Fabrini et a,						48.7%
	2009 [50]	Fotemustine	41	14%	62	23.6	(35.2-	NC	Fotemustine was safe and effective
							62.2)
	Scoccianti et al, 2008 [51] Fotemustine	27	14.8%	48.1	22.8	48.1% (39.1-57.1)	36.4	Fotemustine was safe and effective
	Prados et al, 2006 [46]	CPT-11	38	On EIAED = 0 (Not on EIAED) = 19.6%	NC		15.7% (7-31%).		Ineffective
	Reardon et al, 2005 [70]	CPT-11 and celecoxib	33	21.6% On EIAED = 0 (Not on EIAED) =	55.8	11.1	48.6%) 27.5% (15.5-	31.1	CPT-11 and celecoxib had some activity

time of relapse to death; PFS6: progression-free survival at 6 months; EIAEDs: enzyme-inducing antiepileptic drugs; EGFR: epidermal growth factor receptor; EGFR +: amplified EGFR; EGFR -: Normal EGFR; cRA: 13-cis-Retinoic acid; NC: not clear; *: Objective response.

Table 3 .

 3 Phase II clinical trials in recurrent anaplastic glioma patients after temozolomide failure -6 previous chemotherapies, including temozolomide; Anaplastic glioma (AG): included patients with anaplastic astrocytoma, anaplastic oligodendroglioma and anaplastic mixed glioma; AO: anaplastic oligodendroglioma; MPFS: median progression-free survival; OS: overall survival, measured from time of relapse to death; PFS6: progression-free survival at 6 months; PFS12: progression-free survival at 12 months; EIAEDs: enzyme-inducing antiepileptic drugs; TTP: thrombotic thrombocytopenic purpura; MGMT: O6-methylguanine-DNA methyltransferase; NC: not clear; ND: Not determined; NR: Not reached.

	Study	Treatment	No	Toxicity levels	Response	MPFS,	PFS6,	PFS12,	Median OS,	Comment
				3/4	+ Stable	weeks	%	%	weeks
					Disease,
					%
	Desjardins et	Imatinib	32	23%	43	11	24	14	33	Anti-tumor activity in
	al, 2007 [31]	mesylate plus								some AG patients
		hydroxyurea
	Chamberlain	CPT-11	22 (AO)	14%	59	18	33	4.5	22	Modest activity in
	and Glantz,									recurrent 1p19q co-
	2008 [86]									deleted AO
	Fabrini et a,	Fotemustine	9	14%		37	66	NR	NC	Fotemustine was safe
	2009 [50]									and effective
	Fabi et al,	Fotemustine	26	35%	54	NC	NC	NC	NC	Fotemustine was
	2009 [49]									effective especially with
										MGMT promoter
										methylation
	Prados et al,	CPT-11	13	Grade 4 (EIAED)	ND	ND	23.1	ND	ND	Ineffective
	2006 [46]			= 0
	Desjardins et	CPT-11 plus	33	1 CNS bleed	94	30	55	39	65	Active regimen
	al, 2008 [57]	bevacizumab		1 TTP
	Taillibert et	CPT-11 plus	25 (AO)	24% intratumoral	88	20	42	ND	NR	Effective therapy for
	al, 2009 [87]	bevacizumab		bleed						recurrent AO
				symptomatic						No relation to 1p19q co-
				bleed 4%						deletion
	Vredenburgh	CPT-11 plus	9	NC	100	30	56	ND	NR	Active regimen with
	et al, 2007	bevacizumab								acceptable toxicity
	[58]
	Chamberlain	Bevacizumab	22 (AO)	41 %	73	27	68	23	34	Efficacy and acceptable
	and Johnston,									toxicity
	2009 [88]
	Walbert et al,	6-thioguanine	23 of 31	18% grade 3 and	68	24	44	44	54	Effective with acceptable
	2010 [78]	followed by	on	6% grade 4						toxicity
		lomustine or	lomustine	myelotoxicity

Algorithm for management of recurrent malignant glioma after temozolomide failure

Response

Survival Comment

Hess et al, 1999 [


B Machet

MIXING AND 1-LOOP FLAVOR STRUCTURE OF FERMIONIC CURRENTS IN THE STANDARD MODEL OF ELECTROWEAK INTERACTIONS

Keywords: 11, 30, Hv 11, 40, -q 12, 15, Ff 12, 15, Hh 12, 15, Mm 14, 60, Pq

Mixing and 1-loop flavor structure of fermionic currents in the Standard Model of electroweak interactions

Bruno Machet

Introduction

We wish to combine two fairly old concepts which have long been proven fruitful: Current Algebra [START_REF] Adler | Current Algebras and Applications to Particle Physics[END_REF] and Gell-Mann's use of "strong SU (3)" symmetry [2]): * much physics is conveyed by currents and is strongly constrained by their algebraic properties; * the laws of transformations of terms in a Lagrangian which break a given symmetry carry important physical information; this is how Gell-Mann deduced his famous mass formulae for hadrons. The currents under concern here are the gauge currents of the Standard Model of electroweak interactions, and the symmetry group flavor SU (2) f and its U (1) subgroup of flavor rotations between pairs of fermions carrying the same electric charge. 2 The Standard Model in flavor space for two generations of quarks

The classical Lagrangian

The basics are more easily explained with two generations of quarks only, (u, d) and (c, s). These are the usual SU (2) L doublets, that we immediately reshuffle into the two flavor doublets (u f , c f ) and (d f , s f ). We use the subscript " f " to denote (bare) flavor eigenstates. No transition exists at this order between c f and u f , nor between s f and d f : these states are (classically) orthogonal. It is convenient to embed the (global part of the) electroweak group into the chiral flavor group U (4) L × U (4) R . The electroweak SU (2) L generators write then at the classical level (1 is the 2 × 2 unit matrix)

T 3 = 1 2   1 -1   , T + =   1   , T -=   1   . (1)
Gauge currents form an SU (2) L triplet: they are obtained by sandwiching the generators in eq. ( 1) between ((u, c), (d, s)) fermions and inserting the appropriate γ matrices. Neutral currents are controlled by T 3 ; they form two singlets of SU (2) f : classically there is no flavor changing neutral current (FCNC) (non-diagonal terms are vanishing) nor any violation of universality (diagonal terms are two by two identical).

The standard Cabibbo phenomenology makes use of Yukawa couplings to induce two mass matrices for the fermions, one for (u f , c f ), one for (d f , s f ). After they are diagonalized, which introduces two mixing angles θ u and θ d the SU (2) L generators write in the space of mass eigenstates (u m , c m ), (d m , s m )

T 3 = 1 2   1 -1   , T + =   C   , T -=   C †   , (2)
in which C is the unitary Cabibbo matrix

C =   cos θ c sin θ c -sin θ c cos θ c   , θ c = θ u -θ d .

Its symmetries

  u f c f   → R(θ)   u f c f   ,   d f s f   → R(φ)   d f s f   .
However, preserving SU (2) L symmetry requires that fermions belonging to the same SU (2) L doublet be transformed with the same phase; thus θ must be equal to φ, which leaves a U (1) invariance.

The Cabibbo angle θ c stays arbitrary, also corresponding to an arbitrary U (1) rotation. This symmetry (arbitrariness) gets broken in nature, but there is no hint at the classical level of how it is broken.

Notations

It is convenient for the following to introduce the set of three flavor SU (2) f generators, which depend on a (mixing) angle α:

T x (α) = 1 2   cos 2α sin 2α sin 2α -cos 2α   , T y = 1 2   -i i   , T z (α) = 1 2   sin 2α -cos 2α -cos 2α -sin 2α   . (3)
A 2-dimensional flavor rotation also writes R(α) = e 2iαTy and T x,z (α) satisfy the relation

R † (ω) T x,z (α) R(ω) = T x,z (α + ω). (4)

At 1-loop

s m → d m , c m → u m transitions spoil the orthogonality of bare flavor states and Cabibbo phenomenology [START_REF] Duret | Quark Lagrangian diagonalization versus non-diagonal kinetic terms[END_REF], unless one introduces a series of counterterms [START_REF] Shabalin | Electric dipole moment of the quark in a gauge theory with left handed-fermions[END_REF]. They are both kinetic and mass-like, with both chiralities. In the (d, s) sector, they write (the expressions are analogous in the (u, c) sector) [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF] -

A d dm p /(1 -γ 5 ) s m -B d dm (1 -γ 5 )s m -E d dm p /(1 + γ 5 ) s m -D d dm (1 + γ 5 )s m , (5)
with

A d = m 2 d f d (p 2 = m 2 d ) -m 2 s f d (p 2 = m 2 s ) m 2 d -m 2 s , E d = m s m d f d (p 2 = m 2 d ) -f d (p 2 = m 2 s ) m 2 d -m 2 s , B d = -m s E d , D d = -m d E d , (6)
In the unitary gauge for the W boson and dimensionally regularized

f d = g 2 sin θ c cos θ c (m 2 c -m 2 u ) 1 0 dx 2x(1 -x) ∆(p 2 ) + p 2 x 3 (1 -x) M 2 W ∆(p 2 ) + x + 3x 2 M 2 W ∆(p 2 ) 2-n/2 Γ(2 -n/2) . (7) n = 4 -ǫ is the dimension of space-time, ∆(p 2 ) = (1 -x)M 2 W + x m 2 u +m 2 c 2 -x(1 -x)p 2
, and Γ is the Euler Gamma function which satisfies in particular Γ(ǫ/2) = 2/ǫ -γ + . . . where γ ≈ 0.5772 . . . is the Euler constant. We shall come back later to more specific properties.

1-loop mixing and currents

Adding counterterms to the bare Lagrangian modifies its diagonalization and thus mixing matrices. We define the mixing matrix C d in the left-handed (d, s) sector by

  d f s f   = C d   d m s m   , in which d m , s m
are the new mass eigenstates (similar expressions hold in the (u, c) sector).

Finding the new expressions of gauge currents is simple [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF] thanks to a straightforward consequence of SU (2) L gauge invariance which dictates that the derivatives in the kinetic-like counterterms be replaced by the SU (2) L covariant derivatives. In a first step one shows that they are controlled by the unit matrix in the new mass basis. Stepping back to the original bare flavor basis, one finds [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF] that they are controlled by the

2 × 2 matrices (C -1 d ) † C -1 d = 1 + 2A d T z (θ d ) in the (d, s) sector, and (C -1 u ) † C -1 u = 1 + 2A u T z (θ u ) in the (u, c)
sector. These expressions exhibit in particular the non-unitarity of the mixing matrices C u,d connecting bare flavor states to 1-loop mass states, which has already been extensively commented upon [START_REF] Duret | Mixing Angles and Non-Degenerate Coupled Systems of Particles[END_REF][5][7] [START_REF] Novikov | Binary systems in QM and in QFT: CPT[END_REF].

Likewise, the charged current writes [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF]

ūf cf [1 + A u T z (θ u ) + A d T z (θ d )] γ µ L   d f s f   ,
which is also controlled, in the bare flavor basis, by a non-unitary matrix 1 .

The back-reaction of introducing 1-loop counterterms is accordingly that the classical mixing angles θ u and θ d now occur inside the gauge currents expressed in the bare flavor basis. Some formal structure starts to appear through the SU (2) f generator T z . It is also important that only the A counterterms occur, because of the renormalization freedom attached to them, that we shall soon focus on.

Flavor rotations

Like in the classical case, we perform flavor rotations, respectively with angles θ on (u f , c f ) and φ on (d f , s f ). Since the classical Cabibbo angle θ c is unchanged, one easily shows that the counterterms also stay identical. The neat effect of these arbitrary transformations is finally the one obtained naively by transforming the neutral and charged currents given above and using the relations (4). The neutral currents get now controlled in bare flavor space by 1 + 2A d T z (θ d + φ) in the (d, s) sector, and 1 + 2A u T z (θ u + θ) in the (u, c) sector. The charged current becomes (omitting the fermions and the γ matrix)

1 + A u T z (θ u + (1/2)(θ + φ)) + A d T z (θ d + (1/2)(θ + φ)).
Like before, going through all the steps, one can show that the Cabibbo angle stays unchanged, whatever θ and φ, and so do the mass eigenvalues. So, "physics is unchanged". Arguing with SU (2) L symmetry to impose φ = θ, the arguments of all T z generators become θ u,d + φ. The arbitrariness of φ reflects an up-to-now unbroken U (1) invariance. At 1-loop, neutral currents are no longer flavor singlets of SU (2) f : there exist, in bare flavor space, both FCNC's and violations of universality (differences among diagonal terms); flavor rotations continuously transform FCNC's into violation of universality and vice-versa. 2 . This shows in particular that at least one non unitary mixing matrices occur in one of the two channels (u, c) and (d, s). After these manipulations, the neutral currents are respectively controlled by 1 in the (u, c) channel, (d,s) channel, and the charged currents by 1 + A d T z (θ d -θ u ), such that they all only depend on the arbitrary Cabibbo angle through the T z (θ c ) generator.

Left-over freedom

(m 2 c -m 2 u )A u -(m 2 s -m 2 d )A d is finite
1 + 2A d T z (θ d -θ u ) in the

How flavor rotations get broken

Once all freedom has been used, the arbitrariness of φ has become that of the Cabibbo angle.

We noticed that classical neutral gauge currents only project on SU (2) flavor singlets. The simplest extension is that quantum corrections add to it a part proportional to a triplet of SU (2) f , that is Comparing with the expressions obtained in subsection 2.5 and matching the term proportional to T z 3 with the formulae above, one gets the relation tan 2θ c = ±1/2, which corresponds with very good accuracy to the measured value of the Cabibbo angle. The choices that we made in order to align at 1-loop (u, c) flavor and mass states ensure that the same angle that occurs in the (d, s) neutral current also occurs inside charged currents, which get similar structures 4 .

d s     0 ±1 0 0   ,   0 0 ±1 0   , 1 2   1 0 0 -1     γ µ L   d s   ,

The last generator: T x and the mass matrix

Of the three SU (2) f generators, we have up to now only used T y , which, by exponentiation, triggers flavor rotations, and T z which controls gauge currents. It is natural to look for T x in the mass terms since it is the third element of the process: diagonalization → mixing matrix → gauge currents. And, indeed,

2 It is g 2 sin 2 θc (m 2 c -m 2 u )(m 2 s -m 2 d ) m 4 W 27 12 (m 2 c + m 2 u ) + 23 24 (m 2 s + m 2 d ) 1 + O(m 2 /m 2 W
) . 3 The general expression of Tz is given in eq. ( 3). 4 The U (1) breaking occurs in such a way that the violation of universality 1/2 -(-1/2) = 1 is identical to the "amount" of FCNC which is also 1. So, flavor rotations, which, as we stressed, continuously transform the former into the latter, are broken in such a way that the two violations occur with the same strength.

supposing, to simplify, that the mass matrix is symmetric, it writes

M =   a c c b   = m 2 +m 1 2 + (m 2 -m 1 )T x (θ)
, where m 1 and m 2 are the eigenvalues of M and tan 2θ = 2c a-b . This corresponds to the decomposition of the symmetric non-degenerate matrix M : M = m 1 P 1 + m 2 P 2 , where P 1 = 1 2 (1 -2T x (θ)) and P 2 = 1 2 (1 + 2T x (θ)) are the projectors on the sub-spaces of eigenvectors of M . So, the whole set {masses, currents} of the Standard Model is controlled by the SU (2) f flavor group with generators given by (3).

Most efforts have been devoted to establishing connections between the mass spectrum (m 1 , m 2 ) and the Cabibbo angle (see for example [START_REF] Giunti | Fundamentals of Neutrino Physics and Astrophysics[END_REF]). This is probably not the appropriate way to proceed for at least two reasons: * any homographic transformation on a mass matrix M : M → αM +β δM +γ leaves unchanged the eigenvectors of M and thus the mixing angles: an infinite number of different mass matrices, with different eigenvalues, therefore correspond to the same mixing angle; * the structure that we found within gauge currents stays hidden at the level of the mass matrix. Indeed, the (d, s) mass matrix corresponding to

tan 2θ d = 1 2 is M = ms+m d 2 + ms-m d √ 5   1 1 2 1 2 -1   =   m s ( 1 2 + 1 √ 5 ) + m d ( 1 2 -1 √ 5 ) ms-m d 2 √ 5 ms-m d 2 √ 5 m s ( 1 2 -1 √ 5 ) + m d ( 1 2 + 1 √ 5 )   . Though its part proportional to (m s - m d
) is obtained from the triplet part of neutral currents by a rotation θ → θ + π 4 , it has as a whole no conspicuous property.

Three generations of quarks

The simplest generalization of what has been done for two generations is to consider, inside each flavor triplet of quarks, (u, c, t) and (d, s, b), the three 2 × 2 flavor rotations associated with the three quark pairs, and the corresponding three sets of four currents, for example [

dγ µ L s, sγ µ L d, sγ µ L s, dγ µ L d], [ dγ µ L b, bγ µ L d, bγ µ L b, dγ µ L d] and [ bγ µ L s, sγ µ L b, sγ µ L s, bγ µ L b].
One then requests that each set decomposes into a trivial singlet part + a triplet of the corresponding SU (2) f . This is akin to requesting that the matrix (C -1 d ) † C -1 d which controls neutral currents, be of the form 5 [5]

(C -1 d ) † C -1 d = 1 +      α ±(α -β) ±(α -γ) ±(α -β) β ±(β -γ) ±(α -γ) ±(β -γ) γ      . ( 8
)
Decomposing the 3 × 3 mixing matrix C d as a product of three "quasi-rotations" acting in each of the three 2-dimensional flavor sub-spaces, and introducing, like in the 2 × 2 case, two mixing angles for each 5 Eq. ( 8) also writes

(C -1 d ) † C -1 d = 1 + (α -β)      1/2 ±1 0 ±1 -1/2 0 0 0 0      + (β -γ)      0 0 0 0 1/2 ±1 0 ±1 -1/2      + (α -γ)      1/2 0 ±1 0 0 0 ±1 0 -1/2      + α 2      0 1 1      + β 2      1 0 1      + γ 2      1 1 0     
, making explicit its decomposition on SU (2) f singlets and triplets.

of them, this yields a set of trigonometric equations which have been investigated and solved in [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF]. Even if the whole set of solutions was not produced, it was shown that it includes values of the mixing angles in remarkable agreement with observation.

This accordingly supports the conjecture that 2 × 2 flavor rotations get spontaneously broken such that the corresponding neutral currents decompose onto a singlet + a (small) triplet component of SU (2) f . At this point it is useful to note that the decomposition ( 8) is more restrictive than a singlet + octet of SU (3) f , because the Gell-Mann matrices λ 2 , λ 5 and λ 7 do not appear.

It is still more hopeless than for two generations to look for any particular structure in the mass matrix.

Neutrinos

4.1 Where do they take their maximal mixing from?

• The perturbative vacuum of the Standard Model is unstable while the true vacuum, away from which one can safely quantize small variations, cannot be reached perturbatively from the former. Let us transpose this type of consideration to mixing angles. The usual "classical solution" corresponds here to a unitary C d = R(θ d ) mixing matrix with an arbitrary θ d . We call it "Cabibbo-like". Can there be other classical solutions? If we parametrize the mixing matrix by a rotation R(θ d ), we have no chance whatsoever to eventually find them. We should instead parametrize it in a general non-unitary form, since we know that this is how it is ultimately realized, then find all possible solutions at the "classical limit" C † d C d → 1. This has been done in [START_REF] Duret | Mixing Angles and Non-Degenerate Coupled Systems of Particles[END_REF] where we parametrized (for two generations)

C d =   e iα c 1 e iδ s 1 -e iβ s 2 e iγ c 2   , c 1 = cos θ 1 , s 2 = sin θ 2 etc.
The first type of solutions are "Cabibbolike" and correspond to * θ 2 = θ 1 + kπ associated with e i(α-δ) = e i(β-γ) , or to * θ 2 = -θ 1 + kπ associated with e i(α-δ) = -e i(β-γ) ; the second type of solutions are sets of discrete values

6 * [θ 1 = (k -n) π 2 , θ 2 = (k + n) π 2 ] or [θ 1 = π 4 + (n -k) π 2 , θ 2 = π 4 + (n + k) π 2 ], both associated with e i(α-δ) = e i(β-γ) , * [θ 1 = (n -k) π 2 , θ 2 = (n + k) π 2 ] or [θ 1 = -π 4 + (k -n) π 2 , θ 2 = π 4 + (k + n) π 2 ]
, both associated with e i(α-δ) = -e i(β-γ) , which include in particular the maximal mixing ± π 4 . • In the case of three generations, the measured values of the neutrino mixing angles point at two "Cabibbo-like" classical mixing angles θ 12 , θ 13 , combined with a maximal θ 23 . It is again simple to see how maximal mixing springs out there. Let us do this in the case where the third mixing angle θ 13 is, for the sake of simplicity, taken to vanish. One can even take a parametrization of the mixing matrix without the phases that were introduced in the case of two generations [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF]:

C -1 =      1 0 0 0 c 23 s 23 0 -s 23 c23           c 12 s 12 0 -s 12 c12 0 0 0 1     

Mixing angles

Solving the corresponding system of trigonometric equations in full generality is quite an involved task but it is not our goal. We can instead keep the value of θ 23 maximal as it has been obtained in the "classical" (unitary) limit, and solve in this limit the equations for θ 12 and θ 13 given by eq. ( 8). It is also convenient to start with θ 13 = 0. With these two constraints, one obtains [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF][5] for θ 12 the "golden ratio value" [START_REF] Duret | The emergence of the Cabibbo angle in non-degenerate coupled systems of fermions[END_REF][11] tan 2θ 12 = 2, which, associated with tan 2θ c = 1/2 ensures the so-called "quarksleptons complementarity" [12]. Imposing then this value for θ 12 and still keeping θ 23 maximal, one solves the equations for θ 13 coming from eq. ( 8) in the approximation that it is small (sin 2 θ 13 < .1). This gives [START_REF] Duret | Mixing angles of quarks and leptons in Quantum Field Theory[END_REF] two possible values sin 2 2θ 13 = 1.3 10 -4 and sin 2 2θ 13 = .267. They lie within the recent 90% c.l. interval of the T 2K experiment sin 2 2θ 13 < .5 [START_REF] Hartz | for the T2[END_REF]. In this framework accordingly perfectly accommodate a value of θ 13 not so small as it had long been expected.

Conclusion

We have used 1-loop counterterms to motivate explicit calculations of mixing angles at this order. However the results that we have obtained are general and only rely on two properties: * Quantum corrections always induce non-unitarity for the mixing matrices connecting classical flavor eigenstates to 1-loop mass eigenstates of non-degenerate coupled fermions [START_REF] Duret | Mixing Angles and Non-Degenerate Coupled Systems of Particles[END_REF] [5] [START_REF] Novikov | Binary systems in QM and in QFT: CPT[END_REF]; * Flavor rotations get broken in such a way that quantum corrections induce SU (2) f triplets in 1-loop neutral currents (expressed in terms of classical flavor eigenstates) while they only involve singlets at the classical level.

Among open issues one can mention: * Why do, at the classical (unitary) limit, quarks exhibit three continuous "Cabibbo-like" mixing angles while neutrinos get a discrete value for one of them? We have seen that, if one pays enough attention, all these choices are perfectly legitimate (and also ± π 2 values), but we have no dynamical explanation for them. In particular, why does, seemingly, only θ 23 get a finite "classical value" ? * The system of trigonometric equations resulting from the symmetry constraints that we impose has many solutions. That this set includes the subset of observed values gives no hint concerning why this subset is chosen by nature and why other allowed solutions are not. * Why does SU (2) seems to play again an important role? Is it more than an "opportunistic" (low-energy, at the best) symmetry only introduced because it is the simplest that comes to mind? Should it be given a more fundamental status like that of a (spontaneously broken or not) gauge symmetry?

Both the non-unitarity of mixing matrices in bare flavor space and the generation of SU (2) f triplets in neutral currents, which seemingly control mixing angles, originate from the existence of mass splittings. However, there is no direct connection between masses and mixing angles: the gap stays wide open and the mass spectrum remains apart. Gell-Mann's achievement to find relations among hadron masses from a specific breaking pattern of SU (3) (u, d, s) flavor symmetry still waits for its equivalent for fundamental fermions. There lies most probably the realm of physics beyond the standard model.

  Let R(ω) =   cos ω sin ω -sin ω cos ω   . The Cabibbo matrix C and the fermion masses stay unchanged if one makes arbitrary and independent flavor rotations

  Since an arbitrary rotation by φ does not change "physics", we can choose φ = -θ u , which aligns classical flavor and mass (u, c) fermions. The counterterms A d , B d , D d , E d satisfy the conditions B d = cst (thus E d = cst and E d = cst and there is no freedom on B, E, D). A u and A d have both a pole in n -4 = ǫ but the combination

4 ≈ 1 .

 41 One has the freedom to renormalize to 0 one among the two counterterms A u or A d . Let us choose A u = 0; the alignment of flavor and mass states that we imposed classically by choosing φ = -θ u is maintained at 1-loop. Then, A d can only be non-vanishing A d ≈ -g 2 sin 2 θ c 24 10 -7 .

  three components of the triplet are coupled to the same gauge boson, unlike what happens for the SU (2) L triplet of electroweak currents (1)).

LPTHE tour 13-14, 4 ème étage, UPMC Univ Paris 06, BP 126, 4 place Jussieu, F-75252 Paris Cedex 05 (France), Unité Mixte de Recherche UMR 7589 (CNRS / UPMC Univ Paris 06)

machet@lpthe.jussieu.fr

It is important to recall that the mixing matrices C d , C d and C which connect 1-loop mass eigenstates to 1-loop flavor states (instead of bare flavor states) are unitary and satisfy the relation C = C † u C d [5]. The standard Cabibbo (CKM) phenomenology is indeed restored at 1-loop by the counterterms.

which still satisfy θ2 = ±θ1 + kπ and are thus superposed to the continuous sets of Cabibbo-like solutions.

Acknowledgments: these notes outline the main ideas contained in the works [3], [5], [6] and [10] done in collaboration with Q. Duret and M.I. Vysotsky, to which I address my warmest thanks.


Vincent Savaux

Yves Louët

Moïse Djoko-Kouam

Alexandre Skrzypczak

An Iterative and Joint Estimation of SNR and Frequency Selective Channel for OFDM Systems

This paper proposes an iterative method for a joint estimation of the signal-to-noise ratio (SNR) and the frequency selective transmission channel in OFDM systems. The noise variance is estimated using a pilot preamble scheme, with the minimum mean square error (MMSE) criterion which requires the channel estimation. The channel estimation uses the Linear MMSE (LMMSE) method which requires the noise variance estimation. As each estimatior feeds the other one, an iterative algorithm is proposed. We prove that this algorithm converges for any non-null initialization value. Simulations show the validity of the method with a very low number of iteration for both SNR and channel estimation. We show that the number of pilot symbols in the preamble with respect to the performance of SNR estimation is improved compared to existing SNR estimation methods. Our approach also requires only one pilot symbol. Furthermore, for a given BER value, the SNR gap between perfect estimation and our proposed method is less than 0.5 dB.

I. INTRODUCTION

The performance quality of a communication system is strongly dependent on the design of the transmitter. The knowledge of the signal-to-noise ratio (SNR) is then very useful in order to adjust transmitter parameters as mapping constellation size or robustness of encoding. Moreover, in many algorithms such as linear minimum mean square error (LMMSE) channel estimation [START_REF] Van De Beek | On Channel Estimation In OFDM Systems[END_REF], or adaptive modulations [START_REF] Keller | Adaptive Multicarrier Modulation: A Convenient Framework for Time-Frequency Processing in Wireless Communications[END_REF], the SNR or the noise power is required, but most of the time, the SNR is only supposed to be known at receiver side. The context of this paper is orthogonal frequency division multiplexing (OFDM) broadcast systems (as digital television [START_REF] Etsi | Digital video broadcasting (DVB) framing structure, channel coding and modulation for digital terrestrial television[END_REF] or digital radio [START_REF]Digital Radio Mondiale (DRM);System Specification[END_REF]) in frequency selective channels.

Reference [START_REF] Xu | A Novel SNR Estimation Algorithm for OFDM[END_REF] covers the usual maximum likelihood (ML), minimum mean square error (MMSE) and second-and forth-Order moment (M 2 M 4 ) algorithms performance for SNR estimation in OFDM systems and makes a comparison with its proposed method, based on the use of the autocorrelation function given by the model of the channel (Jakes or Ricean models). ML, MMSE and M 2 M 4 algorithms are depicted and their efficiencies are compared in the case of AWGN channel for single carrier systems in [START_REF] Pauluzzi | A Comparison of SNR Estimation Techniques for the AWGN Channel[END_REF], [START_REF] Ren | A New SNR's Estimator for QPSK Modulations in an AWGN Channel[END_REF] and [START_REF] Li | A Low Bias Algorithm to Estimate Negative SNRs in an AWGN Channel[END_REF]. Some iterative algorithms are presented in [START_REF] Li | A Low Bias Algorithm to Estimate Negative SNRs in an AWGN Channel[END_REF] and [START_REF] Wiesel | Non-Data-Aided Signal-to-Noise-Ratio Estimation[END_REF], but they are not suitable for frequency selective channels. ML SNR estimator, whose developments are given in [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF], presents a prohibitive calculation complexity in the case of frequency selective channels. MMSE estimator, from which we derive our proposed method, requires the estimation of the transmission channel.

Mainly for the same requirement, only theoretical expressions of the MMSE SNR estimator are proposed in the literature [START_REF] Xu | A Novel SNR Estimation Algorithm for OFDM[END_REF], [START_REF] Pauluzzi | A Comparison of SNR Estimation Techniques for the AWGN Channel[END_REF], [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF], and no practical algorithm. M 2 M 4 , firstly mentioned in [START_REF] Benedict | The Joint Estimation of Signal and Noise from the Sum Envelope[END_REF], presents a low computational complexity and does not require any channel estimation. However, its efficiency is degraded in frequency selective channels. Simplifications of ML estimator have been brought by the recursive expectation maximisation EM algorithm [13] [14], wich also allows to estimate both noise variance and channel perturbations [START_REF] Krisjansson | Joint estimation of noise and channel distortion in a generalized EM framework[END_REF], [START_REF] Baranski | EM-based Iterative Receiver for Space-Time Coded Modulation with Noise Variance Estimation[END_REF]. However, each step of the EM algorithm necessitates a maximisation wich requires the estimation of all the parameters from the previous step. Furthermore, EM is used when the observed datas are not complete.

In order to avoid the need of the channel estimation, [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF] develops a method for a 2×2 multi input multi output (MIMO) OFDM system assuming that the channel is invariant during a two-symbols preamble and for two consecutive carriers. Reference [START_REF] Ren | SNR Estimation Algorithm Based on the Preamble for OFDM Systems in Frequency Selective Channels[END_REF] presents a method using the estimation of the variance of the noise thanks to a two-pilots preamble and combines it with the estimation of the second moment ( 2 ) of the received signal. These two methods require a twopilot preamble, what reduces the data rate of the transmission, especially if the preamble must be frequently repeated. In [START_REF] Xu | Subspace-Based Noise Variance and SNR Estimation for OFDM Systems[END_REF], an estimation of the SNR is presented using the properties of the channel correlation matrix, estimated thanks to a one-pilot preamble. This method is limited by an insufficient statistic on the channel, what degrades the performance, in particular for low SNR.

In this paper, we propose a method to estimate both SNR and frequency selective channel with MMSE criterion. The SNR is estimated using the noise variance estimation combined with the second moment of the received signal. The noise variance estimation method, based on the MMSE criterion, requires the channel estimate. In its turn, the channel estimation, performed with LMMSE method, needs the noise variance in addition to the channel correlation matrix. As a consequence, we propose an iterative approach in which both noise variance and channel estimations are computed for each iteration, using the results of the previous one. Here is the similarity with the EM, although in our algorithm, the estimation of each parameter only requires the estimation of the other one and not both. We show in the cases of perfect and approximated covariance matrix, that our method quickly converges for both noise variance and channel estimation. Furthermore, as our method only requires a one-pilot preamble by frame, the trade-off between the number of pilot and the performance is improved compared to existing methods.

The continuation of this paper is organized as follows : Section II presents the OFDM system model and the noise variance and channel estimation methods. In Section III, we present our iterative estimation method, and we prove in Section IV the convergence of the proposed algorithm. With the simulations in Section V, one can firstly verify the convergence of the method, secondly compare it with two existing methods of the literature [START_REF] Ren | SNR Estimation Algorithm Based on the Preamble for OFDM Systems in Frequency Selective Channels[END_REF], [START_REF] Xu | Subspace-Based Noise Variance and SNR Estimation for OFDM Systems[END_REF]. We draw the conclusions in Section VI.

II. BACKGROUND

This section aims at describing the transmission system model and the techniques of estimation (noise variance, SNR, channel) required for the proposed method presented in Section III.

A. Transmitted Signal Model

We consider the transmission of OFDM symbols over a multipath channel. After the removal of the cyclic prefix (CP) and the Discrete Fourier Transform (DFT), [START_REF] Van De Beek | On Channel Estimation In OFDM Systems[END_REF] gives the formulation of the ℎ received OFDM symbol in the frequency domain:

U = C H + W . (1)
U = [ 0, , ..., -1, ] , H = [ 0, , ..., -1, ] and W = [ 0, , ..., -1, ] are the × 1 complex vectors of the received signal, the multipath channel and the gaussian white noise on the ℎ time-slot respectively.

is the size of the DFT, which also tallies with the number of carriers per symbol in our model. C is the × diagonal matrix of the transmitted signal containing the vector [ 0, , ..., -1, ]. , are either data elements or pilots, whose gains, phases and positions are perfectly known at emission and reception sides. In this article, pilots are dedicated to channel estimation and noise variance estimation. We consider a pilot-preamble scheme with only one OFDM pilot symbol by frame and assume a constant channel between two consecutive preambles. In the rest of the paper, the pilot symbols are noted with the subscript .

, are the components of the vector H given by

, = -1 ∑ =0 ℎ , exp ( -2 ) , (2)
where denotes the sub-carrier subscript, the length of the impulse response and ℎ , the zero-mean complex process of the ℎ path of the channel. All paths are considered independent.

B. Noise Variance Estimation

We note 2 the noise variance (or noise power), equal to

2 = { | , | 2 }
, or in the same way in its vector form

2 = 1 { ||W || 2 }
. We here consider the matrix Frobenius norm, defined for a matrix A as ||A|| 2 = (AA ), where is the trace and (.) is the Hermitian transpose. We use in this article the Minimum Mean Square Error (MMSE) criterion to estimate the noise variance (see [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF]), noted ˆ 2 and given by ( 3)

ˆ 2 = 1 { ||U -C Ĥ || 2 } , (3)
where Ĥ is the channel estimation performed on the pilot symbol. In practice, the expectation can only be approximated by the mean on a sufficiently large number of sub-carriers, leading to

ˆ 2 = 1 -1 ∑ =0 | , - , ˆ , | 2 , ( 4
)
where ˆ , is the estimation of the ℎ subcarrier. Equation ( 4) then shows that the efficiency of the MMSE noise variance estimation depends on the quality of the channel estimation.

C. SNR Estimation

The SNR noted is basically obtained from the second moment 2 of the received signal and the noise variance:

2 = 1 { ||U || 2 } = + 2 , (5)
where is the power of the useful transmitted signal. We then get the SNR:

= 2 2 -1. (6)
In practice, we estimate the SNR ˆ in the same way:

ˆ = ˆ 2 ˆ 2 -1, (7)
where ˆ 2 is defined in (4) and ˆ 2 by

ˆ 2 = 1 -1 ∑ =0 | , | 2 . ( 8
)

D. Channel Estimation

The two basic estimation methods are the Least Square (LS) and Linear Minimum Mean Square (LMMSE) presented in [START_REF] Edfords | OFDM Channel Estimation by Singular Value Decomposition[END_REF] and [START_REF] Biguesh | Training-Based MIMO Channel Estimation: A Study of Estimator Tradeoffs and Optimal Training Signals[END_REF]. [START_REF] Wiesel | Non-Data-Aided Signal-to-Noise-Ratio Estimation[END_REF] gives the LS channel estimate:

Ĥ = C -1 U = H + C -1 W . ( 9
)
LS estimation is very simple but sensitive to the noise. Furthermore, this estimation cannot be used for the noise variance estimation in (4) as we obtain ˆ 2 = 0 for Ĥ = Ĥ . [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF] gives the efficient LMMSE channel estimation:

Ĥ = R (R + 2 (C C ) -1 ) -1 Ĥ , ( 10
)
where R is the channel covariance matrix. LMMSE channel estimation is more efficient than LS, but requires a matrix inversion. We assume in the rest of the paper that: ∀ = 0, ..., -1, , = 1 on a given preamble position . Consequently, the pilot matrix C is equal to the identity matrix noted I. Thus, we get from [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF]: Ĥ = R (R + 2 I) -1 Ĥ . R is usually unknown at receiver, we then proposed in [START_REF] Savaux | Near LMMSE Channel Estimation Performance with Artificial Channel at Receiver for OFDM Systems[END_REF] a LMMSE-based estimation method which can replace [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF]. Furthermore, this estimation method requires the noise variance knowledge generally unknown and estimated by ( 4). This problem leads to our proposed algorithm for both noise variance and channel iterative estimation.

III. PROPOSED ALGORITHM FOR NOISE VARIANCE AND CHANNEL ESTIMATION

From (4), we notice that the accuracy of the noise variance estimation is directly linked to the channel estimation technique that is used. As, by combining (3) and ( 9), a LS channel estimation gives a noise variance estimation equal to zero, we then consider a solution that is based on the efficient LMMSE channel estimation. However, from its expression given in [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF], the knowledge of the noise variance is necessary and also determines the quality of the channel estimation. Consequently, as the variance estimation feeds the channel estimation and vice versa, we here propose an iterative technique allowing a joint estimation of the noise variance and the channel coefficients, whose principle is described in Fig. 1. In this figure, we then observe that the iterative improvement of the noise variance estimation (respectively the channel coefficients estimation) allows a better estimation of the channel coefficients (respectively the noise variance). Let us then consider the th iteration of our algorithm, with ≥ 1. At this step, due to the iterative nature of our algorithm, the noise variance ˆ 2 ( -1) , estimated at the iteration ( -1), can be used for the LMMSE estimation of the channel by using [START_REF] Kay | Fundamentals of statistical signal processing : estimation theory[END_REF], giving then:

Ĥ ( ) = R (R + ˆ 2 ( -1) I) -1 Ĥ . (11)
Consequently, by using (3) with Ĥ = Ĥ ( ) , we can propose a new estimation of the noise variance. Recalling that C = I, this new estimation is expressed by

ˆ 2 ( ) = 1 { ||U -C Ĥ ( ) || 2 } = 1 { ||H + W -Ĥ ( ) || 2 } ˆ 2 ( ) = 1 { || Ĥ -Ĥ ( ) || 2 } . (12)
This algorithm finally necessitates a non-null value of ˆ 2 (0)

as initialization step in order to be completely described. Indeed, if we choose ˆ 2 (0) = 0, the channel estimation in ( 11) is equivalent to the LS one. Consequently, applying this result in [START_REF] Benedict | The Joint Estimation of Signal and Noise from the Sum Envelope[END_REF], we then get a noise variance estimation equal to zero and the algorithm enters an endless loop. These both expressions [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF] and ( 12) are obtained for pilots in the preamble equal to 1. In a different case, the pilot matrix C (respectively the pilot total energy) has to be taken into account in [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF] (respectively in [START_REF] Benedict | The Joint Estimation of Signal and Noise from the Sum Envelope[END_REF]). Note also that for practical issues, ( 12) is replaced by

ˆ 2 ( ) = 1 -1 ∑ =0 | ˆ , -ˆ , ( ) | 2 . ( 13
)
Then if 0 is the final algorithm iteration, from ( 7), the SNR is estimated from the noise variance ˆ 2 ( 0) by the expression:

ˆ = ˆ 2 ˆ 2 ( 0 ) -1. (14)
In practice, the covariance matrix R in ( 11) is unknown. However, if ( ) is the power intensity profile of the channel and if we use the notations of (2), the covariance matrix may be approximated by the matrix Ȓ , whose elements ( Ȓ ) , are derived by the expression given in [START_REF] Edfords | OFDM Channel Estimation by Singular Value Decomposition[END_REF]:

( Ȓ ) , = -1 ∑ =0 ∫ 0 ( ) -2 ( -) . ( 15
)
Consequently, our proposed algorithm can be described as follows:

1) Calculate the matrix Ȓ from [START_REF] Krisjansson | Joint estimation of noise and channel distortion in a generalized EM framework[END_REF] or consider R if the channel is perfectly known. 2) Initialize the noise variance so that ˆ 2 (0) > 0. 3) For ≥ 1, performe a LMMSE estimation of the channel by using [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF]. 4) For ≥ 1, performe the noise variance estimation ˆ 2 ( ) from ( 12). 5) Back to step 3 with ← + 1 or go to step 5. 6) Estimate the SNR ˆ by ( 14) from the final noise variance estimation ˆ 2 ( 0 ) . 7) end of the algorithm. In the case of an unknown channel, where neither Ȓ nor R can be computed, the covariance matrix is estimated by R = Ĥ ( Ĥ ) . This matrix must be then regularly updated according to the channel fluctuations. An adaptation of our algorithm in this context is currently under development.

The similarity of the proposed method with EM algorithm comes from its iterative character. However, our method is MMSE-based, and not ML-based. Furthermore, the iterations of our technic are only made up of two steps of estimation [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF] and ( 13), without step of maximisation. Lastly, EM is used when the observed data vector dimension is lower than the wanted estimated vector dimension [START_REF] Dempster | Maximum Likelihood from Incomplete Data via the EM Algorithm[END_REF], [START_REF] Wautelet | Comparison of EM-Based Algorithms for MIMO Channel Estimation[END_REF]. This is not the case here, the estimation being performed on a complete one-symbol pilot preamble.

IV. CONVERGENCE OF THE ALGORITHM

In this section, it is shown that our proposed algorithm converges for both noise variance ˆ 2 ( ) and channel Ĥ ( ) estimations. From [START_REF] Boumard | Novel noise variance and SNR estimation algorithm for wireless MIMO OFDM systems[END_REF], it is easy to prove that Ĥ ( ) converges to a given channel estimation if the sequence (ˆ 2 ( ) ) admits a limit for an infinite number of iterations. Then, after the derivation of a recursive expression of ˆ 2 ( ) , the proof of the convergence of this sequence is given.

A. Scalar Expression of the Iterative Noise Variance Estimation

In the following, the different mathematical formulations are based on the covariance matrix R . However, as Ȓ and R have the same properties than R , these formulations remain valid if one of these matrices is considered. Consequently, from [START_REF] Benedict | The Joint Estimation of Signal and Noise from the Sum Envelope[END_REF], we get As a consequence, (26) ensures the existence of an ∈ ] 0, -2 ] verifying ( ) ≥ . In addition, as

∀ > 0, ′ ( ) = 2 -1 ∑ =0 ( + 2 ) ( + ) 3 > 0, (27)
we prove that is a strictly growing function.

From ( 25) and ( 26), we easily obtain that ([ , +∞ [ ) ⊂ [ , 2 ]. In addition, as is a strictly growing function and upper-bounder by 2 , we get the following inclusion:

([ , 2 ]) ⊂ [ , 2 ], (28)
proving then that has at least one fixed point on [ , 2 ].

As it has been previously shown that is stricty growing on the interval [ , +∞ [ , equivalently on [ , 2 ], the sequence (ˆ 2 ( ) ) is consequently monotonous. From ( 25) and ( 26), the sequence (ˆ 2 ( ) ) is also lower bounded by and upper bounded by 2 . Finally, from the fixed-point theorem, (ˆ 2 ( ) ) converges to one of the fixed point of . These mathematical formulations theoretically prove that our proposed algorithm converges, what is confirmed by simulations in the following section. Fig. 2 depicts the way our algorithm works. The noise variance and channel estimations alternatively feed each other until the algorithm reaches its limit, characterized by the couple (ˆ 2 , Ĥ ). This couple is normally different from the couple ( 2 , H ) leading to the perfect estimation. Section V-A then depicts the performance of our algorithm and finally shows that our estimated couple (ˆ 2 , Ĥ ) is close to the perfect estimation couple ( 2 , H ).

V. SIMULATION RESULTS

Simulations are based on the Digital Radio Mondiale (DRM) standard [START_REF]Digital Radio Mondiale (DRM);System Specification[END_REF] designed as the digital audio broadcasting over the currently AM frequency bands. The OFDM modulation considered uses 201 sub-carriers for a sampling frequency equal to 12 kHz. The added cyclic prefix (CP) of time duration = 2.66 ms is supposed to be longer than the maximum channel delay. Although DRM standard recommends a scattered pilot repartition, we consider for the need of our method a pilot-preamble scheme. Each preamble is then composed of only one pilot symbol. The channel model considered is the , also taken from the DRM standard. This four-paths channel has a maximum delay = 2.2 ms and a maximum Doppler frequency = 2 Hz. All paths are mutually independent and follow a Gaussian power density spectrum.

A. Convergence of noise variance estimation

Fig. 3 and4 show the evolutions of two estimated noise variances compared to the real noise variance as a function of the number of iterations. Case 1 tallies with the channel estimation made with the real covariance matrix R and Case 2 tallies with the channel estimation made with the approximated covariance matrix Ȓ . Fig. 3 shows the evolution of the algorithm for the initialization ˆ 2 (0) = 0.1 and Fig. 4 for ˆ 2 (0) = 2. Furthermore, for both Fig. 3 and4, two fixed values of the SNR are considered : = 0 dB (for high values of

2 ), and = 10 dB (for the lower ones). All the curves are obtained with 4000 simulation runs. Figs. 3 and4 display the convergence of the algorithm, for low and high values of the initializations ˆ 2 (0) and for two values of SNR (0 dB and 10 dB). Although the development of Section IV is not done with the matrix Ȓ , we also observe the convergence of the algorithm in Case 2. Whatever the value of ˆ 2 (0) , we verify that the sequence (ˆ 2 ( ) ) is monotonous and converges to a given limit, as shown in Section IV. Furthermore, for both cases 1 and 2, the convergence is fast : the noise variance estimation ˆ 2 ( ) is constant for ≥ 1 for Case 1 and for ≥ 3 for Case 2. Fig. 3 and 4 also allow to characterize the bias of the noise variance estimation method. For both cases 1 and 2, after a few iterations, we observe a very low bias. Indeed, if we note = |(ˆ 2 ( ) ) -2 |/ 2 the normalized error, given in percentage, we have, in Case 1 (real covariance matrix) = 1.67 % for = 0 dB and = 4.57 % for = 10 dB. In Case 2 (approximated covariance matrix), we get = 1.96 % for = 0 dB and = 5.43 % for = 10 dB.

B. Comparison of SNR estimation with other methods

Fig. 5 shows the curves of the Normalized Mean Square Error (NMSE) of SNR estimations in the case of the frequency selective channel . We remind that estimated SNR is ˆ =

ˆ 2 ˆ 2 -1 (see section II-C). The NMSE of the SNR estimation ˆ is defined by = [ |ˆ -| 2 / 2 ]
. In our simulations, the expectation is performed on 200000 samples. The initialization is ˆ 2 (0) = 0.1 and the number of iterations is = 3. We compare the proposed algorithm to two existing methods using a preamble-based SNR estimation in a frequency selective channel supposed invariant between two consecutive preambles. Ren's method, depicted in [START_REF] Ren | SNR Estimation Algorithm Based on the Preamble for OFDM Systems in Frequency Selective Channels[END_REF], uses a 2 pilot-symbols preamble which allows to remove the effect of the channel in order to estimate the noise variance only. The SNR is then estimated thanks to the second moment 2 , as done in [START_REF] Ren | A New SNR's Estimator for QPSK Modulations in an AWGN Channel[END_REF]. Xu's method, depicted in [START_REF] Xu | Subspace-Based Noise Variance and SNR Estimation for OFDM Systems[END_REF], uses a single pilot-symbol preamble to estimate the covariance matrix. A diagonalization of this matrix allows to estimate the noise variance in one hand and the second moment in the other hand. Unlike our technique, none of these two methods performs a joint channel and SNR estimation. It makes these techniques of SNR estimation less complex to apply than our algorithm, but our method makes the joint estimation of SNR and channel directly usable for the equalization. To perform equalization, Ren and Xu's methods require then a separated channel estimation. Fig. 5 compares the efficiency of our algorithm in terms of NMSE to Ren and Xu's ones [START_REF] Ren | SNR Estimation Algorithm Based on the Preamble for OFDM Systems in Frequency Selective Channels[END_REF], [START_REF] Xu | Subspace-Based Noise Variance and SNR Estimation for OFDM Systems[END_REF]. The shape of the curves of Ren's and Xu's estimation methods matches with those of the simulations in [START_REF] Ren | SNR Estimation Algorithm Based on the Preamble for OFDM Systems in Frequency Selective Channels[END_REF]. Our theoretical case (Case 1) has a lower NMSE than Ren's method (which is better than Xu's method), while Ren's technique requires two pilot symbols (our technique requires only one pilot symbol). In Case 2, Ren's method has a lower NMSE than ours, due to the approximation of the covariance matrix. However, our algorithm has a globally lower NMSE than Xu's method, except for the SNR values between 3 and 7 dB. Xu's method and ours require only 1 pilot symbol, so the proposed algorithm is globally more efficient for the same useful bit rate. Furthermore, our method performs a channel estimation, whose efficiency is studied in next sub-section V-C.

C. Channel estimation

Fig. 6 shows the Bit Error Rate (BER) as a function of SNR of the proposed algorithm in both cases 1 and 2. The convergence of the channel estimation is proved by simulation. We use the frequency selective channel . The initialization is ˆ 2 (0) = 0.1. The BER is computed on 2.5×10 6 bits. We simulate frames of 22 OFDM symbols, including 1 pilot symbol. The channel is then considered as invariant over 22 consecutives symbols. The useful-data symbols are composed of samples mapped with a 16-QAM. For both cases 1 and 2, we study the efficiency of the channel estimation as a function of the number of iteration. The method is compared with the perfect estimation and the regular LS estimation.

The channel estimation made in Case 1 reaches its limit from the first iteration, what tallies with the convergence speed shown in Fig. 3 and4. The error compared to the perfect estimation is less than 0.1 dB for = 25 dB. This estimation matched with the theoretical LMMSE estimation using the real covariance matrix R and the real noise variance 2 . This theoretical LMMSE estimation is not displayed for a better visibility. In the same way for Case 2, the channel estimation reaches its limit from = 3, what tallies with the number of iterations necessary for (ˆ 2 ( ) ) reaches its limit (see Fig. 3 and4). The error compared to the perfect estimation is less than 0.5 dB for = 25 dB. Furthermore, the proposed iterative method is more efficient than the regular LS estimation. Indeed, for = 25 dB, the error of the LS estimation compared to the proposed method in Case 2 is equal to 2.5 dB.

VI. CONCLUSION

This article presents an iterative and joint method for SNR and channel estimation. The SNR is estimated combining the second moment of the received signal and the noise variance estimation made thanks to a pilot-preamble scheme. We use the MSE criterion to estimate the noise variance. To do so, we require an efficient channel estimation, thus we use an LMMSE channel estimation, which requires itself the noise variance estimation. We then propose an iterative estimation of both noise variance and transmission channel. We prove that the algorithm converges whatever the non-null initialization, and the simulations show the low bias of the noise variance estimation. In comparison to two existing SNR estimation methods, our method improves the rate between the number of pilots in the preamble and the efficiency of the SNR estimation, as far as our method requires only one pilot. Furthermore, this pilot is used to perform in the same time the channel estimation. The simulations show that the channel estimation is close to the perfect estimation by less than 0.5 dB. The subjects of our future work is the theoretical proof of the unicity of the convergence of the algorithm and the extension of the algorithm for the case of the estimated channel covariance matrix R .

Fig. 1 .

 1 Fig. 1. Block diagram of the proposed iterative algorithm.

Fig. 2 .

 2 Fig. 2. Illustration of the convergence of the algorithm.

Fig. 3 . 2 Fig. 4 .

 324 Fig. 3. Convergence of the algorithm for ˆ 2 (0) = 0.1.

Fig. 5 .

 5 Fig. 5. NMSE of the SNR estimation of 3 methods.

2 Fig. 6 .

 26 Fig. 6. BER of the iterative channel estimation.

To extend that (R + ˆ 2 ( -1) ) is an Hermitian invertible matrix, (R + ˆ 2 ( -1) ) -1 is also an Hermitian matrix, we obtain

We note D ( ) (respectively, D ) the diagonal matrix obtained by the diagonalization of (R + ˆ 2 ( ) ) (respectively, (R + 2 )). The diagonal elements of D ( ) (respectively, D ) are equal to + ˆ 2 ( ) (respectively, + 2 ), where , = 0, 1, ..., -1 are the eigenvalues of R . We also note Q as the orthonormal transformation matrix of R . Consequently, we can rewrite [START_REF] Ren | SNR Estimation Algorithm Based on the Preamble for OFDM Systems in Frequency Selective Channels[END_REF] as follows:

) .

(18) From ( 18), we obtain a recursive formulation of ˆ 2 ( +1) :

B. Proof of Convergence

As the length of the channel impulse response is equal to , the eigenvalues for = , + 1, ..., -1 are equal to zero, and the eigenvalues for = 0, 1, ..., -1 are positive. If we note the largest eigenvalue and = 0 the lowest one, we express the following upper and lower bounds for ˆ 2 ( +1) :

As

), we have :

We can write ˆ 2 ( +1) = (ˆ 2 ( ) ), where the function is defined by :

As ˆ 2 ( ) > 0, the variable is necessarily positive. From (22), we obtain:

proof of the convergence of the sequence (ˆ 2 ( ) ) is based on the fixed-point theorem. Indeed, if possesses at least one fixed point on an closed interval [ , ] and if the sequence (ˆ 2 ( ) ) is bounded and monotonous, (ˆ 2 ( ) ) necessarily converges to one of the fixed points of . In order to prove that has at least one fixed point on a closed interval [ , ] (i.e. that the equation ( ) = has at least one solution in [ , ]), we first prove that ([ , ]) ⊂ [ , ], with and correctly chosen. Indeed, from (24), we show that ( ) -----→ →+∞ 2 .

(25)

Due to the fact that = 0, is not defined for = 0. However: