Jan Christian Goebel

Martin Roesel

Carsten Heinz

Hartmut Michels

Gerd Ganser

Prof. Dr Arnd Heiligenhaus
email: arnd.heiligenhaus@uveitis-zentrum.de

Azathioprine as a treatment option for uveitis in patients with juvenile idiopathic arthritis

Keywords: immunosuppression, juvenile idiopathic arthritis, azathioprine, uveitis 2497

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Introduction

Juvenile idiopathic arthritis (JIA) is the most common chronic inflammatory rheumatic disease in childhood and begins before 16 years of age. JIA is characterized by a chronic inflammation of the joints, which may lead to joint destruction. [START_REF] Gare | Juvenile arthritis--who gets it, where and when? A review of current data on incidence and prevalence[END_REF][START_REF] Manners | Worldwide prevalence of juvenile arthritis why does it vary so much?[END_REF][START_REF] Petty | International League of Associations for Rheumatology classification of juvenile idiopathic arthritis: second revision, Edmonton 2001[END_REF] The most frequent form of eye involvement in JIA is uveitis, with an estimated incidence between 10 and 18%. [START_REF] Carvounis | Incidence and outcomes of uveitis in juvenile rheumatoid arthritis, a synthesis of the literature[END_REF][START_REF] Heiligenhaus | Prevalence and complications of uveitis in juvenile idiopahic arthritis in a population-based nation-wide study in Germany: suggested modification of the current screening guidelines[END_REF] Uveitis is characterized by chronic inflammation of the iris and ciliary body. The course of the disease varies highly. A severe and vision-threatening course develops in approximately one third of patients, and the complication rate rises during the chronic course of disease. [START_REF] Heiligenhaus | Prevalence and complications of uveitis in juvenile idiopahic arthritis in a population-based nation-wide study in Germany: suggested modification of the current screening guidelines[END_REF][START_REF] Edelsten | An evaluation of baseline risk factors predicting severity in juvenile idiopathic arthritis associated uveitis and other chronic anterior uveitis in early childhood[END_REF][START_REF] Paroli | Prognosis of juvenile rheumatoid arthritisassociated uveitis[END_REF] The aim of treatment is to completely abolish active inflammation in the eye.

Therapy is usually started with topical corticosteroids, occasionally combined with systemic corticosteroids. However, in approximately 30% of patients the uveitis remains active, so that immunosuppressive drugs are required.

Methotrexate is the first-choice immunosuppressive, cyclosporine A, azathioprine or the anti-tumor necrosis factor (TNF) alpha inhibitors represent other options.

Azathioprine is a well-known purine synthesis inhibitor that is widely used in organ transplantation and several autoimmune diseases, e.g., rheumatoid arthritis. A favorable response has also previously been noted in the treatment of arthritis in JIA patients. [START_REF] Kvien | Azathioprine versus placebo in patients with juvenile rheumatoid arthritis: a single center double blind comparative study[END_REF][START_REF] Savolainen | Azathioprine in patients with juvenile chronic arthritis: a longterm followup study[END_REF] Azathioprine has only occasionally been utilized to treat uveitis in JIA patients. Indeed, only small case series have been reported in the literature, describing experience with azathioprine in only few children. [START_REF] Savolainen | Azathioprine in patients with juvenile chronic arthritis: a longterm followup study[END_REF][START_REF] Hemady | Immunosuppressive drugs in the management of progressive, corticosteroid-resistant uveitis associated with juvenile rheumatoid arthritis[END_REF] In this paper, we summarize our observations using azathioprine in treating 41 patients with JIA-associated uveitis.

Patients and Methods

All patients in this study fulfilled the classification of juvenile idiopathic arthritis (JIA) in agreement with the International League of Associations for Rheumatology (ILAR) criteria. [START_REF] Petty | International League of Associations for Rheumatology classification of juvenile idiopathic arthritis: second revision, Edmonton 2001[END_REF] The 41 children were treated with azathioprine for severe chronic anterior uveitis. The cohort included consecutive patients who were started on azathioprine treatment for uveitis between 1995 and 2007. Charts were reviewed retrospectively. The design of the work conforms to the standards currently applied in Germany. No institutional review board approval is required for chart review studies.

Rheumatologic examinations and diagnostic procedures included a review of systems, laboratory tests (e.g., antinuclear antibodies, rheumatoid factor, and HLA B 27), urine analysis, and chest x-ray, if necessary. An infectious etiology was excluded if serological tests for EBV, CMV, HSV, HZV, treponema, toxoplasma, and toxocara and the tuberculosis skin test were negative.

Patients with other uveitis entities, such as intermediate, Fuchs heterochromic cyclitis, and infectious uveitis syndromes, were excluded from the study. A standardized ophthalmic database was applied for the analysis. The ophthalmic assessment included best-corrected visual acuities (BCVA), slitlamp examination, tonometry (Goldmann), and funduscopy.

Uveitis was defined and anatomically classified according to the Standardization of Uveitis Nomenclature (SUN) Working Group recommendations. [START_REF] Jabs | Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] The presence of ≥ 1+ cells in the anterior chamber was defined as active uveitis. We documented the duration of uveitis and whether disease was uni-or bilateral. Dates of first and final azathioprine treatments were noted, as were azathioprine dosage, adverse effects, any topical and antiinflammatory medication before and during azathioprine treatment, and initial manifestation of uveitis in the fellow eye after beginning azathioprine treatment. Additionally, the uveitis-related complications in the eye were analyzed, as central band-keratopathy, new synechiae formation, cataract, vitreous opacity, phthisis and glaucoma. Data were collected from all patients at 6-to 12-month intervals.

Azathioprine monotherapy was instituted in patients with active uveitis, if they were being treated with topical (≥3 drops/day) and possibly also systemic corticosteroids (≥ 10mg, or ≥ 0.15 mg/kg body weight), or in order to spare systemic corticosteroids. Moreover, azathioprine was considered as an additional immunosuppressive if uveitis remained active under treatment with any other immunosuppressive drug.

The dosages of systemic corticosteroids and immunosuppressive drugs were altered or medication discontinued according to the clinical assessment of arthritis and uveitis activity and to the medication-related side effects.

After 1 year of azathioprine treatment and at all visits during the entire treatment period, the response to treatment, as defined by inactivity of uveitis, and nonresponse, as defined by activity of uveitis, were analyzed. Inactivity of anterior uveitis was defined as < 1+ cells in the anterior chamber. [START_REF] Jabs | Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] Response to treatment with azathioprine was additionally judged according to a modified grading system described by Saurenmann et al.. [START_REF] Saurenmann | Tumour necrosis factor alpha inhibitors in the treatment of childhood uveitis[END_REF] Briefly, good response was defined as a decrease ≥ 50% in both corticosteroid use and immunosuppressives (if not on combined immunosuppressive drug and systemic corticosteroid at institution of azathiopine, then only reduction of corticosteroids ≥ 50%); moderate response, reduction ≥50% in either corticosteroid or immunosuppressive drug; and poor response, reduction of <50% in both corticosteroid and immunosuppressive drug. In this study, dose sparing was assessed independently for systemic and topical steroids.

All analyses were performed using MedCalc software (MedCalc Version 9.6.4.0, MedCalc Software, Mariakerke, Belgium). The t test, chi-square test and Fisher´s exact test for categorical data were applied for statistical analysis when appropriate. A p-value < 0.05 was considered as a significant difference.

Results

A total of 41 children were included in this study. All children had JIA and associated active chronic anterior uveitis. Demographic characteristics including JIA subgroups and uveitis are listed in Table 1. Notably, the majority of patients suffered from early-onset arthritis and showed a severe course of uveitis, with complications already present at the initial visit in nearly half of the patients, e.g. central band-keratopathy (n=5), cataract (n=20), vitreous opacity (n=5), and glaucoma (n=12). All of the patients had previously received topical corticosteroids, and 29.3% had immunosuppressive drugs. The follow-up after instituting azathioprine was 26 months (mean, range 3-107). Azathioprine was started at a mean dosage of 2.4 mg/kg body weight (range 1.4 to 3.2). Then, azathioprine was continued, and the mean dosage at the end of the study period was 2.1 mg/kg body weight daily (range 1.0 to 2.8) (p=0.047).

Effect of azathioprine monotherapy on uveitis

Of the 26 patients who initially received azathioprine monotherapy (fig 1) for uveitis, inactivity of uveitis was achieved in 16 (61.5%), while inflammation recurred or persisted in the other 10 (38.5%). Mean follow-up was 26 months (Table 2). Response to treatment did not correlate well with JIA subgroup (p=1.0), age at onset of arthritis (p=0.76), ANA positivity (p=1.0), gender (p=0.12), age at onset of uveitis (p=0.41), bilateral uveitis (p=1.0), and presence of complications at initial presentation (p=1.0).

A significant interpatient variation was observed in the dose-sparing effect of immunosuppressive drugs and topical steroids. New uveitis complications were noted in 3 of 26 patients in at least one eye during the treatment period (Table 2).

Effect of azathioprine in combination with other disease-modifying antirheumatic drugs on uveitis

Of the 41 patients included, azathioprine was given in combination with other immunosuppressive drugs in 15 patients ( The outcome of azathioprine in combination with other immunosuppressive drugs with regard to sparing the dose of the other systemic immunosuppressive agents or corticosteroids was good in 6, moderate in 4, and poor in 5 patients (Table 3). In this respect, the difference between the azathioprine mono-and combination therapy failed to reach the level of significance (p=0.23). In this group, topical steroids could be tapered off in 6 patients, while the dosage was unchanged in another 9. The sparing effect of topical steroids was not better than in the monotherapy group (p=0.42).

The observations with azathioprine given in combination with methotrexate or cyclosporine A, which are commonly used to treat uveitis associated with JIA, are summarized in Table 4 and5. Although the numbers of patients in both of the groups are fairly small, the observations suggest that uveitis inactivity may be achieved with the combined use of methotrexate or cyclosporine A with azathioprine. However, the sparing effect of topical and systemic corticosteroids and immunosuppressive drugs was moderate to poor in the majority of cases.

New complications and side effects under azathioprine

Twenty patients (49%) had already developed uveitis complications before azathioprine therapy was instituted. However, the first or additional uveitis complications were noted in another 7 patients while under azathioprine treatment.

During the treatment period with azathioprine monotherapy, 3 of the 26 patients (11.5%) developed new secondary complications. Four of the 15 patients under systemic combination therapy (26.7%) were reported to have developed new uveitis complications, and these numbers did not differ from the azathioprine monotherapy group (p=0.41).

Azathioprine was withdrawn in 3 patients because of nausea. Apart from this, no other serious side effects were documented during the study period.

Discussion

Only very few small case series have assessed the value of azathioprine to treat uveitis in patients with associated JIA. [START_REF] Kvien | Azathioprine versus placebo in patients with juvenile rheumatoid arthritis: a single center double blind comparative study[END_REF][START_REF] Savolainen | Azathioprine in patients with juvenile chronic arthritis: a longterm followup study[END_REF] The observations from our multi-center study show that this agent may be helpful as systemic monotherapy or in combination with other immunosuppressive agents.

Azathioprine may also be useful for long-term therapy and shows a dosesparing effect for corticosteroids or immunosuppression.

The pathogenesis of uveitis in JIA patients is not well defined. The few published histopathological papers demonstrated that the disease is marked by nongranulomatous inflammation in the iris and ciliary body. [START_REF] Hinzpeter | Ocular histopathology in Still´s disease[END_REF][START_REF] Sabates | Ocular histopathology in juvenile rheumatoid arthritis[END_REF][START_REF] Merriam | Early onset pauciarticular juvenile rheumatoid arthritis: a histopathologic study[END_REF] A recently published paper showed that the cellular infiltration mostly consisted of plasma cells, plasmocytic lymphocytes and monocytes, and few T cells. In light of these findings, the poor response to cyclosporine A that was described most recently is not surprising, suggesting that JIA uveitis may not be primarily a T cell-mediated disease. [START_REF] Tappeiner | Limited value of cyclosporine A for the treatment of patients with uveitis associated with juvenile idiopathic arthritis[END_REF] The focal aggregates mainly consisted of CD20 positive B cells: primarily IgG B cells and only few IgM B cells. [START_REF] Parikh | Immunhistochemical study of chronic nongranulomatous anterior uveitis in juvenile idiopathic arthritis[END_REF] However, as the infiltrate at nucleation may not reflect that during early active disease, the therapeutic implications of end-stage histology are limited.

Azathioprine is a purine nucleoside analog. It interferes with adenine and guanine ribonucleotides by suppressing inosinic acid synthesis, which in turn interferes with DNA replication and RNA transcription. [START_REF] Elion | Azathioprine. In: Handbook of experimental pharmacology[END_REF] Immunologically, azathioprine decreases the numbers of peripheral T and B lymphocytes and reduces mixed lymphocyte reactivity, interleukin-2 synthesis, and IgM production. [START_REF] Mckendry | Purine antagonists[END_REF][START_REF] Bacon | Modes of action of second line agents[END_REF] In addition, the production of cytokines such as IFN-y and TNF in T lymphocytes is inhibited and apoptosis of these cells is initiated. [START_REF] Hildner | Azathioprine, mycophenolate mofetil and methotrexate specifically modulate cytokine production by T cells[END_REF] Azathioprine is well absorbed orally and cleaves to 6-mercaptopurine, which is metabolized to thioinosinic and thioguanylic acid by the hypoxanthine phosphoribosyltransferase. These metabolites affect ribonucleotide synthesis. [START_REF] El-Yazigi | Pharmacokinetics of azathioprine after repeated oral and single intravenous administration[END_REF] Therefore, azathioprine appears to be an appropriate candidate drug for uveitis in JIA. Azathioprine has also been introduced for the treatment of uveitis associated with JIA. [START_REF] Kvien | Azathioprine versus placebo in patients with juvenile rheumatoid arthritis: a single center double blind comparative study[END_REF][START_REF] Savolainen | Azathioprine in patients with juvenile chronic arthritis: a longterm followup study[END_REF] Savolainen et al. [START_REF] Savolainen | Azathioprine in patients with juvenile chronic arthritis: a longterm followup study[END_REF] reported that in the long term azathioprine could effectively treat arthritis in 129 patients with JIA. In this study, a total of 19 patients had chronic iridocyclitis in some phase of the disease. Three children had episodes before treatment with azathioprine and 14 patients had active uveitis when included in study. In the latter cases activity persisted during therapy. Iridocyclitis occurred for the first time in 2 patients during low-dose therapy. In contrast to our study, no specific time points were analyzed, the cell grade in the anterior chamber as a measure for activity was not specified as currently agreed, nor was the adjustment of the topical steroid treatment considered. [START_REF] Jabs | Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] It may be speculated that additional topical treatment was not adjusted properly to the inflammatory activity.

Hemady et al. examined the role of immunosuppressive drugs in the management of progressive, corticosteroid-resistant uveitis associated with JIA. [START_REF] Hemady | Immunosuppressive drugs in the management of progressive, corticosteroid-resistant uveitis associated with juvenile rheumatoid arthritis[END_REF] The mean duration of immunosuppressive therapy was 6.4 months.

Only four of 26 patients were treated with azathioprine. In 2 patients inflammation was controlled. The other two patients -one receiving azathioprine in combination with methotrexate-did not respond to the therapy.

In contrast to the rare published data, our observations in a larger and homogenous group of JIA patients suggest that azathioprine may be effective in the treatment of JIA-associated chronic anterior uveitis (). During the mean follow-up of 26 months under azathioprine monotherapy, uveitis inactivity was maintained in 16 of 26 children (61.5%). The sparing effect of antiinflammatory medication, however, differed markedly among the individuals.The effectiveness of other immunosuppressive drugs in the treatment of JIA-associated uveitis has also been investigated only in retrospective uncontrolled studies. In one study including 23 patients, remission was described in 417 of 661 months with MTX treatment [START_REF] Foeldvari | Methotrexate is an effective treatment for chronic uveitis associated with juvenile idiopathic arthritis[END_REF]. In another study, quiescence of uveitis was obtained with MTX in 32 of 35 patients [START_REF] Heiligenhaus | Methotrexate for uveitis associated with juvenile idiopathic arthritis: Value and requirement for additional anti-inflammatory medication[END_REF]. Inactivity of uveitis was noted with cyclosporine A in only 6 of 25 patients [START_REF] Tappeiner | Limited value of cyclosporine A for the treatment of patients with uveitis associated with juvenile idiopathic arthritis[END_REF].

The impact of azathioprine on the visual outcome could not be determined in this study, as the documented visual acuities were frequently unreliable due to children's incompliance or because of opacities in the visual axis. Although this case series did not include a control group, the data reveal that new uveitis complications occurred in some patients during the treatment period. Indeed, new complications were detected in 3 of the 26 patients (11.5%) included, and in 2 of the patients the first uveitis manifestation developed in the formerly uninvolved eye although they were being treated with azathioprine.

Currently, the first-choice immunosuppressive drug is MTX, which is quite effective in controlling uveitis in JIA patients. [START_REF] Foeldvari | Methotrexate is an effective treatment for chronic uveitis associated with juvenile idiopathic arthritis[END_REF][START_REF] Heiligenhaus | Methotrexate for uveitis associated with juvenile idiopathic arthritis: Value and requirement for additional anti-inflammatory medication[END_REF] The management of JIA children with uveitis who do not respond to MTX represents an important issue, as this is the most frequently used immunosuppressant in these patients.

It is, therefore, noteworthy that our data show that azathioprine was not less effective when it was added to an immunosuppressive drug already being given. When azathioprine was added to MTX in the children with active uveitis in our case series, inactivity was achieved in 71.4%. However, our observations show that the sparing effect of the other systemic immunosuppressive drugs and of topical steroids can vary. For the entire patient series, systemic immunosuppressive drugs and steroids could be reduced only in a minority of patients. It is possible that the similar outcomes with azathioprine monotherapy and combination therapy are due to the differences in severity between the groups. However, uveitis activity, duration of uveitis, and complication rates did not differ between the groups (data not shown)."

Although we could speculate that higher azathioprine dosages would have given better results in this study, the azathioprine dosages did not differ significantly between the responders and the nonresponders (monotherapy p=0.3 and combined therapy p=1.0). According to safety recommendations, the dosage for this patient group should be under 3 mg/kg body weight. It has been shown previously that azathioprine treatment at this dosage is safe and well-tolerated in children. [START_REF] Fuentes | High-dose azathioprine in children with inflammatory bowel disease[END_REF] Apart from the anti-inflammatory value of immunosuppressive agents, drug safety is of the outmost importance in JIA children with uveitis, especially as long-term treatment is usually required. Thus, long-term safety data are needed, which is a good reason for preferentially using particular drugs, e.g., MTX, Cyclosporine A and also azathioprine. Therefore, the use of azathioprine may be favored in patients who do not respond to MTX, and may be an alternative to some recently introduced drugs, e.g., TNF-alpha inhibitors, leflunomide, or mycofenolate mofetil.

In summary, the present data suggest that azathioprine may be reconsidered in the step-ladder approach for the treatment of children with JIA-associated uveitis. Although this study is somewhat limited by its retrospective design and short follow-up in some patients, the results suggest that the value of azathioprine monotherapy is higher than often believed. The addition of azathioprine may also be beneficial for patients not responding properly to other immunosuppressive drugs. Prospective randomized studies and long-

Table 3

 3

	, fig 1). While azathioprine

Table 1

 1 Patient chracteristics before treatment with azathioprine. Children with uveitis associated with juvenile idiopathic arthritis (JIA). JIA subgroups are classified according to the ILAR

	classification (n=41). Antinuclear antibodies (ANA), Aza = azathioprine.
	Juvenile idiopathic arthritis	41 (100%)
	Oligoarthritis	35 (85.4%)
	Polyarthritis	2 (4.9%)
	Enthesitis related arthritis	1 (2.4%)
	Other arthritis	3 (7.3%)
	ANA positivity	39 (95.1%)
	Active chronic anterior uveitis	41 (100%)
	Uni-/bilateral	13 (31.7%) / 28 (68.3%)
	Any uveitis complication	20 (48.8%)
	Female / male	28 (68.3%) / 13 (31.7%)
	Mean age at onset of arthritis (yrs)	3.6 (range 1 -15)
	Mean age at onset of uveitis (yrs)	4.38 (range 1 -15)
	Mean duration of uveitis before study (yrs)	2,82 (range 0-19)
	Previous topical corticosteroid treatment	41 (100%)
	Previous systemic DMARDs
	No others	29 (70.7%)
	Methotrexate	10 (24.4%)
	Cyclosporine A	6 (14.6%)
	Etanercept	1 (2.4%)
	Prednisolone	0 (0%)
	Mean follow-up time on Aza (months)	25 (range 3-107)
	DMARDs = disease-modifying antirheumatic drugs

Table 2

 2 Azathioprine monotherapy as initial immunosuppressive treatment. (n=26). Uveitis in patients with juvenile idiopathic arthritis (JIA). Inactivity of anterior uveitis was defined as < 1+ cells in the anterior chamber.[START_REF] Jabs | Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] *Sparing of second-line immunosuppressive drugs and topical steroids according to Saurenmann et al..[START_REF] Saurenmann | Tumour necrosis factor alpha inhibitors in the treatment of childhood uveitis[END_REF]. During entire follow-up = at all study visits.

	During entire follow-up

Table 3

 3 Azathioprine when added to treatment of patients already on any immunosuppressive drugs (n=15). Uveitis in patients with juvenile idiopathic arthritis (JIA). Inactivity of anterior uveitis

	was defined as < 1+ cells in the anterior chamber.[11] *Sparing of second-line
	immunosuppression and topical steroids according to Saurenmann et al..[12] During entire
	follow-up = at all study visits.
			During entire follow-up
		At 1 year n (%)	(mean 2.2 years)
	Inactivity of uveitis	5 / 9 (56.6%)	10 / 15 (66.7%)
	New uveitis complications	2 (22.2%)	3 (20.0%)
	Sparing of systemic immunosuppression*
	Good	nd	6 (40.0%)
	Moderate	nd	4 (26.7%)
	Poor	nd	5 (33.3%)
	Sparing of topical steroids
	Taper off	1 (11.1%)	6 (40.0%)
	Reduce to ≤ 2 drops / d	2 (22.2%)	0 (0%)
	Unchanged	3 (33.3%)	9 (60.0%)
	Increased	3 (33.3%)	0 (0%)

Table 4

 4 Azathioprine when added to treatment of patients already on methotrexate (n=7). Uveitis in patients with juvenile idiopathic arthritis (JIA). Inactivity of anterior uveitis was defined as < 1+ cells in the anterior chamber.[START_REF] Jabs | Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] *Sparing of second-line immunosuppression and topical steroids according to Saurenmann et al..[START_REF] Saurenmann | Tumour necrosis factor alpha inhibitors in the treatment of childhood uveitis[END_REF]. During entire follow-up = at all study visits.

	During entire follow-up

Table 5

 5 Azathioprine when added to treatment of patients already on cyclosporine A (n=5). Uveitis in patients with juvenile idiopathic arthritis (JIA). Inactivity of anterior uveitis was defined as < 1+ cells in the anterior chamber.[START_REF] Jabs | Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] *Sparing of second-line immunosuppression and topical steroids according to Saurenmann et al..[START_REF] Saurenmann | Tumour necrosis factor alpha inhibitors in the treatment of childhood uveitis[END_REF] During entire follow-up = at all study visits.

	During entire follow-up

None of the authors has a financial or proprietary interest in any of the material or the method presented.

Licence for Publication

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in BJO and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence (http://group.bmj.com/products/journals/instructions-for-authors/licence-forms).

term registries are necessary to define the most effective and safest treatment regimen.

Figure 1

Azathioprine for uveitis in 41 children with juvenile idiopathic arthritis.

Competing Interest

Please list Competing Interests if they exist if not please include the following statement;

Competing Interest: None declared.


MD Andreas Christoph Lipski
email: andreas.lipski@uni-duisburg-essen.de

Anja Eckstein

Joachim Esser

Christian Loesch

Klaus Mann

Christoph Mohr

Bernhard Jurklies

Course of pattern-reversed visual evoked cortical potentials in 30 eyes after bony orbital decompression in dysthyroid optic neuropathy

Keywords: pattern VEP, VECP, Graves´ disease, DON

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In up to 8.6% of patients with Graves´ orbitopathy (GO), the clinical course is complicated by dysthyroid optic neuropathy (DON). [START_REF] Werner | The thyroid. A fundamental and clinical text[END_REF][START_REF] Neigel | Dysthyroid optic neuropathy. The crowded orbital apex syndrome[END_REF] Although exact mechanisms underlying DON still require illumination, the optic nerve with its vascular supply is vulnerable to compression especially in the orbital apex. [START_REF] Kennerdell | Apical optic nerve compression of dysthyroid optic neuropathy on computed tomography[END_REF] Secondary to inflammative swelling of extraocular muscles and orbital soft tissue, apical orbital crowding may thus lead to altered blood supply and impairment of axonal transport resulting in functional impairment of the optic nerve. [START_REF] Trobe | Optic nerve involvement in dysthyroidism[END_REF][START_REF] Forbes | Ophthalmopathy of Graves' disease: computerized volume measurements of the orbital fat and muscle[END_REF] As DON can be better correlated to periorbital swelling and impairment of motility than to lid retraction or proptosis, the diagnosis may be obscured in the absence of these clinical entities. [START_REF] Trobe | Optic nerve involvement in dysthyroidism[END_REF][START_REF] Feldon | Clinical classification of Graves' ophthalmopathy. Identification of risk factors for optic neuropathy[END_REF] Clinical signs for DON -decreased vision, an afferent pupillary defect or funduscopic abnormalities such as papilledema -should promptly result in therapeutic intervention. To enhance diagnostic precision, conduction of visual evoked cortical potentials (VECP) may be accompanied by orbital imaging. [START_REF] Neigel | Dysthyroid optic neuropathy. The crowded orbital apex syndrome[END_REF][START_REF] Kennerdell | Apical optic nerve compression of dysthyroid optic neuropathy on computed tomography[END_REF][START_REF] Trobe | Optic nerve involvement in dysthyroidism[END_REF][START_REF] Forbes | Ophthalmopathy of Graves' disease: computerized volume measurements of the orbital fat and muscle[END_REF][START_REF] Feldon | Clinical classification of Graves' ophthalmopathy. Identification of risk factors for optic neuropathy[END_REF][START_REF] Monteiro | Diagnostic ability of Barrett's index to detect dysthyroid optic neuropathy using multidetector computed tomography[END_REF][START_REF] Wijngaarde | Pattern EPS in endocrine orbitopathy[END_REF] Pattern-reversal VECP retain a valuable tool in the diagnosis and follow-up of DON as the disease is significantly associated with P100 amplitude loss and delayed deflection. [START_REF] Neigel | Dysthyroid optic neuropathy. The crowded orbital apex syndrome[END_REF][START_REF] Wijngaarde | Pattern EPS in endocrine orbitopathy[END_REF][START_REF] Esser | Therapy related follow-up studies of visually evoked cortical potentials in patients with endocrine ophthalmopathy (stage 6)[END_REF][START_REF] Setälä | The value of visual evoked potentials in optic neuropathy of Graves' disease[END_REF][START_REF] Tsaloumas | Flash and pattern visual evoked potentials in the diagnosis and monitoring of dysthyroid optic neuropathy[END_REF][START_REF] Salvi | The study of visual evoked potentials in patients with thyroid-associated ophthalmopathy identifies asymptomatic optic nerve involvement[END_REF][START_REF] Rutecka-Debniak | Visual evoked potentials in diagnosis and monitoring of optic neuropathy in the course of thyroid ophthalmopathy[END_REF][START_REF] Van Lith | Pattern evoked cortical potentials and compressive lesions along the visual pathways[END_REF] Pattern-VECP are furthermore capable to identify even subclinical courses of DON with high sensitivity. [START_REF] Salvi | The study of visual evoked potentials in patients with thyroid-associated ophthalmopathy identifies asymptomatic optic nerve involvement[END_REF] Established treatment for DON includes intravenous administration of corticosteroids and bony decompression of the orbit. Several surgical approaches with one-to fourwall orbital delamination have been considered with variable efficacy to reverse congestive proptosis and optic nerve impairment. [START_REF] Jorissen | Dysthyroid orbitopathy and orbital decompression. A review[END_REF] Previous studies using VECP to monitor DON during courses of steroid or radiation treatment concluded partial remission of DON, but only few patient reports exist on VECP development after orbital bony decompression.

With intention to selectively focus on the restoration of optic nerve function as reflected by the pattern-reversal VECP, we retrospectively analyzed clinical courses from 30 eyes of 15 patients with DON using pattern-VECP before and after bilateral bony orbital 3-wall decompression. [START_REF] Tveten | Orbital decompression in treatment of endocrine orbitopathy[END_REF][START_REF] Eckstein | Thyroid-associated ophthalmopathy.Pathogenesis, clinical presentation and examination, stage-dependent therapy[END_REF]

Material and methods

We retrospectively analyzed clinical outcomes, the results and courses of VECP in 15 non-consecutive GO patients (n= 30 eyes) with DON undergoing orbital decompression. Inclusion criteria were symptoms of DON (at least one of the following signs): an otherwise unexplained visual loss in at least one eye, presence of a (relative) afferent pupillary deficit, colour desaturation and a pathologic VECP.

Another prerequisite was a normalised thyroid hormone level as assessed by serologic quantification of free triiodothyronine and tetraiodothyronine (fT3, fT4).

The study´s intervention consisted of bilateral bony 3-wall orbital decompression using a combined coronar and anterior approach with osteotomy and resection of the lateral, medial and inferior orbital walls followed by dissection of the medial and posterior ethmoidal cells. Furthermore, orbital fat resection was performed after slitting of the periorbita. [START_REF] Tveten | Orbital decompression in treatment of endocrine orbitopathy[END_REF] Postoperative systemic steroid treatment with fluocortolone was administered at a cumulative dose of 1.55 g over 6 weeks starting with 1 mg/kg body weight per day orally.

Visual acuity assessment was performed on the same day before VECP conductance within one week before decompression surgery using standardized tests of Landolt optotypes at a distance of 5 m according to DIN 58220. Values were converted to logMAR for statistical analyses. Vision was compared both between values at VECP examinations as well as between preoperative and last examination results. Significant vision improvement was assumed when vision increased by at least 2 lines. VECP conductance was followed by mydriatic ophthalmoscopy. At the same date, Hertel´s exophthalmometry and biomicroscopic slit lamp examination with ophthalmoscopy were performed. Accordingly, clinical activity-and modified NOSPECS-scores were recorded as described elsewhere. [START_REF] Eckstein | Thyroid-associated ophthalmopathy.Pathogenesis, clinical presentation and examination, stage-dependent therapy[END_REF] Presence of an RAPD was assayed using the swinging-flashlight test at symmetrical test-and background luminances. Patients with documented corneal pathology (stippled or eroded corneal epithelium) were excluded from the study. VECP assessment was in full accordance with the ISCEV standards. [START_REF] Odom | Visual evoked potentials standard (2004)[END_REF] Cortical responses to reversing full-field checkerboard stimulations were measured using gold cup electrodes (Oz-Fz) and earlobe clip electrodes (Schuler GmbH, Freiburg, Germany) accessing a Jaeger/Toennies multiliner vision unit (Hoechberg, Germany).

Ten20™ EEG paste (D.O. Weaver & Co, Aurora, US) served as conductive gel after alcoholic skin scouring to achieve impedances below 5 Ω. Checkerboard squares of four different sizes were consecutively presented in a darkened testing chamber on a 25 x 20 cm rectangular monitor screen with a fixation point in the center at a distance of 100 cm. Monocular stimulation was adopted in all investigations. Rectangular stimulation square sizes were 13.7, 27.5, 60 and 120 minutes of angle (arcmin) at each conduction, respectively. Mean monitor luminance was 50 cd/m² and the checkerboard-reversal rate was set to 2 Hz thereby achieving a contrast of 80% between white and black checks. Signals were filtered at 1-100 Hz, amplified and averaged from 200 cycles with 1% artefact tolerance. Latencies and amplitudes of the first positive through (N75 to P100) as well as of the second negative through (P100 to N140) were used for evaluation. Values of recordings prior to and after orbital decompression were analyzed and compared to results obtained from an age-matched control group. P100 latency deterioration was defined as either prolongation of previously pathologic values or a newly pathologic latency in any of the four different checkerboard pattern sizes not registered during preoperative testing. Upper latency limits were calculated by adding two standard deviations to mean values from healthy age-matched subjects in our laboratory for each checkerboard pattern size.

Upper latency limits were 129 ms at 13.7 arcmin, 125.5 ms at 27.5 arcmin, 124 ms at 60 arcmin and 126 ms at 120 arcmin.

Statistical analysis

Statistical evaluation was performed using a two-sided Student t-test (level of significance = 5%) after verification of parametric distribution using the Kolmogorow-Smirnoff index. Values in the text represent mean ± sd unless otherwise indicated.

Results

Patients

Thirthy eyes of 15 patients with GO were included into the study of which 93% were female. All patients were previously treated either by radioiodine therapy (n=4) or by thyroidectomy (n=11) and had progressive GO despite previous administration of at least one cycle of systemic steroid therapy. Furthermore, bilateral orbital irradiation was performed in 12 patients in the past without achieving sufficient remission.

Because in most cases pretreatment was not coordinated by our center, both orbital irradiation and steroid therapy were administered at variable treatment and dosing regimes. [START_REF] Johnson | A retrospective study on the efficacy of total absorbed orbital doses of 12, 16 and 20 Gy combined with systemic steroid treatment in patients with Graves´ orbitopathy[END_REF] History of smoking was present in 87% of patients. One patient received octreotide immunosuppression because of psoriasis comorbidity. Additional patient characteristics are depicted in Table 1.

The mean interval of initial diagnosis of GO and orbital decompression was 1.1 ± 1.0 years. CAS before decompression was 7 ± 2 and NOSPECS was 9 ± 2 (mean ± sd).

Visual acuity

During the postoperative period, mean logMAR visual acuity (VA) increased, statistically significant, by 2.4 lines (from 0.38 ± 0.25 right before surgery to 0.14 ± 0.1 after 30 ± 13 months). Mean vision of patients at follow-up VECP examinations 10 ± 7 months after decompression already improved significantly to logMAR VA=0.18 ± 0.16, p=0.014 (p<0.02). At last examinations 30 ± 13 months after decompression, further mean vision improvement to 0.14 ± 0.12 logMAR VA was denoted (p=0.0001; Fig. 1).

After decompression surgery, all eyes maintained or improvied vision during followup. 50% of patients significantly gained vision by 2 or more lines. Figure 2 further itemizes vision responses to orbital decompression in 3 baseline decimal vision groups. Clearly, patients with low baseline vision preferentially profited from surgery.

Vision gain of up to 6 lines in two eyes and a mean gain of 4.5 lines were evident in this subgroup of 8 eyes. Furthermore was stabilization or improvement evident both in the intermediate (0.5-0.32) as well as in the good baseline vision groups (Fig. 2).

Exophthalmometry

Hertel´s exophthalmometry revealed statistically significant reductions of 6.4 ± 3 mm 30 ± 13 months after decompression, from 22.4 ± 3.3 mm before surgery to 16.1 ± 2.6 mm, p<0.0001).

VECP amplitudes

During the VECP observation period of 12 ± 11 months, mean VECP-P100 amplitudes of 30 eyes increased, statistically significant, by 4.4, 2.6, 2.6 and 1.9 µV at 13.7, 27.5, 60 and 120 arcmin stimulation checkerboard size, according to increases of 92%, 43%, 44% and 31%, respectively (p=0.002, 0.019, 0.012 and 0.064). P100 amplitudes generated by large patterns of 120 arcmin did not change after decompression. Figure 3 illustrates median courses of amplitudes in the respective check-size groups over the observation period and Table 2 depicts P100 amplitude and latencies before and after decompression in detail. VECP latencies P100 latencies after decompression surgery were not different in a statistically significant manner compared to the recordings prior decompression. Total incidences of pathologic VECP P100 latency values decreased after decompression surgery.

Median latency outcome after 12 ± 11 months depended on the stimulation pattern size (figure 4). While decreased latencies were seen at 13.7 arcmin pattern size, prolongations were evident at 27.5 and 120 arcmin. Median P100 latencies at 60 arcmin remained stable but offered a leaning to increase during follow-up.

Discussion

Here we report on the beneficial results of bony orbital decompression to prevent irreversible vision loss due to Efficient reduction of proptosis was associated with gain of visual acuity and improvement of pattern-evoked VECP amplitudes after 12 months in our group of patients showing persistent active disease despite previous systemic steroid therapy. To our knowledge, this is the largest study implementing VECP conductance to monitor optic nerve function before and after bony orbital decompression.

All patients were previously treated either by radioiodine or thyroidectomy and revealed a normalisation of fT3 and fT4 serum values at the timepoint of decompression surgery as well as during follow-up. All patients previously received systemic steroids and 13 patients underwent additional orbital irradiation without sufficient or definite remission of GO prior decompression. Pretreatment was not always coordinated by our clinic and variations in prior treatment regimes were thus unavoidable. However the circumstances leading to DON in these patients were not subject of the study.

Bilateral surgical intervention was performed by the same surgeon with a standardized dosing regimen of postoperative steroids. The bilateral approach was choosen even in cases with asymmetrical optic nerve function because of relatively symmetric proptosis on both eyes in all patients.

Monitoring of DON in response to therapy has been achieved using different techniques of pattern-and flash-evoked VECP.[9, 10] Tsaloumas compared both during radiation or decompression therapy and concluded not to have found substantial differences between the techniques. [START_REF] Tsaloumas | Flash and pattern visual evoked potentials in the diagnosis and monitoring of dysthyroid optic neuropathy[END_REF] However pattern-evoked VECP may be affected by patient compliance, media opacities or blurred vision primarily impairing amplitude potentials while latencies remain unaffected or may be compromised to a much lesser extent as far as retinal luminance remains unaffected. [START_REF] Tsaloumas | Flash and pattern visual evoked potentials in the diagnosis and monitoring of dysthyroid optic neuropathy[END_REF][START_REF] Adachi-Usami | Principles and practice of clinical electrophysiology of vision[END_REF] Although our patients underwent systemic steroid treatment, significant cataract formation during the VECP follow-up period was not observed.

Ongoing visual rehabilitation over the longer clinical follow-up period further underlines that cataract formation is unlikely to have affected VECP results in our study. Potential changes in retinal luminance caused by altered pupil diameters following orbital surgery secondary to intraoperative trauma to pupillomotoric nerve fibres remain unlikely and should not appear on a regular basis. Frequently occurring postoperative complications rather included strabism with diplopia and hypesthesia of the face skin. [START_REF] Tveten | Orbital decompression in treatment of endocrine orbitopathy[END_REF] While Tsaloumas and co-workers reported on immediate mean latency recovery 2 weeks after orbital decompression in 7 eyes with DON using pattern-reversed stimuli, Setälä and associates observed indiscriminate mean flash-evoked cortical response latencies after 7 years of follow-up in 7 patients. [START_REF] Setälä | The value of visual evoked potentials in optic neuropathy of Graves' disease[END_REF][START_REF] Tsaloumas | Flash and pattern visual evoked potentials in the diagnosis and monitoring of dysthyroid optic neuropathy[END_REF] This is in line with our findings after a mean follow-up of ten months -latencies did not reveal significant regeneration and even deteriorated in some patients after bony orbital

Figure 5

 5 Figure5further depicts latency courses assigned to baseline vision groups before and after decompression. Here, all vision groups developed latency prolongation when stimulated by large pattern sizes (120 arcmin). In the intermediate vision group, patients initially and after decompression surgery had prolonged latencies during testing with small check-sizes with a tendency to normalise after therapy. New postoperative latency deterioration was punctually evident in all vision groups, always restricted to the smallest pattern size not affected in the preoperative examination (data not shown).Worsening of latencies assigned to stimulation pattern size was evident in 4 patients of the low and intermediate vision groups, respectively, as well as in 3 patients in the high baseline vision group. Here, one patient developed latency deterioration in all of the four already preoperatively pathologic check-sizes despite maintaining unchanged vision during follow-up.

Fig. 2 :Fig. 3 :Fig. 4 :

 234 Fig.2: Decimal vision responses to decompression surgery assigned to baseline vision groups. All patients gained vision (arrows) or remained stable during follow-up of 30 ± 13 months after therapy (triangles). Patients with lower baseline vision experienced better visionincreases.

Fig. 5 :Tab. 1 :

 51 Fig.5: VECP P100 latency changes after decompression surgery assigned to baseline visionafter stimulation with four different pattern sizes(13.7-120 arcmin). For deatails: see text.

Table 1 :

 1 Patient characteristics

		age	GD*	GO*	follow-up (months)	VA pre/post	Hertel pre/post	pretreatment	comorbidity	smoking
	pt.	(years) (years) (months)	Clinical	VECP	OD	OS	OD	OS
	1	60	1.2	5	35	11	0.2/0.7	0.4/0.6	22/13.5	23/15	2St, R, I	none	Y (20)
	2	55	0.9	7	24	25	0.6/1.0	0.6/1.0	27/17	27/17	2St, R, S	none	Y (10)
	3	51	8.6	30	39	7	0.3/0.4	0.5/0.7	22/15.5 22/15.5	2St, 2R, S	psoriasis	Y (30)
	4	62	42.6	5	26	28	0.3/1.0	0.3/1.0	23/17	23/17	St, R, S	none	N
	5	68	0.8	10	28	4	0.6/0.6	0.6/0.7	21/16	21/17	St, S, I	HT	Y (6)
	6	46	0.9	13	24	17	0.9/1.0	0.9/1.0 21.5/17.5 22.5/18.5	St, R, S	none	Y (20)
	7	55	0.5	9	54	3	0.3/0.8 0.16/0.63 27/22	27/20	2St, R, S	HT	Y (20)
	8	43	1.6	23	28	35	0.8/0.8	0.6/1.0	20/14	21/14	St, R; S	none	Y (10)
	9	67	51.7	8	55	79	0.6/0.6	0.5/0.7	16/14	18/14	St, R, S, 2I	MI; FH	N
	10	46	0.9	8	27	3	0.9/0.9	0.9/1.0	22/13	23/14	2St, S	none	Y (20)
	11	48	1.7	20	61	5	0.8/1.0	0.8/1.0	22/17	23/16.5	St, R, S	rheumatism Y (20)
	12	73	0.5	7	17	11	0.5/0.5	0.2/0.6	16/12	16/12	St, R, I	AA	Y (10)
	13	44	1.0	0.7	0.6	1	0.8/1.0	0.5/0.8	23/17	24/16	2St, R, S	none	Y (10)
	14	76	51.8	46	16	27	0.08/0.5 0.6/0.6	25/20	23/20	St, 2S	HT, AC	Y (30)
	15	49	0.6	7	19	22	0.4/0.5 0.32/0.6	26/17	26/17	St, R, S	none	Y (10)

Table 2 :

 2 VECP P100 latencies and amplitudes

	VECP P100 latencies (ms, OD pre/post//OS pre/post)


MD Benjamin Guigui

MD Nicolas Leveziel

PhD Virginie Martinet

MD Nathalie Massamba

MD Margaret Sterkers

MD Gabriel Coscas

MD Eric H Souied
email: esouied@hotmail.com

Angiography features of early onset drusen

Keywords: basal laminar drusen, Malattia Leventinese, fluorescein angiography, indocyanine green angiography, large colloid drusen

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Drusen are extra cellular deposits that accumulate between the retinal pigment epithelium (RPE) and the inner collagenous layer of Bruch's membrane [START_REF] Gass | Drusen and disciform macular detachment and degeneration[END_REF]. They are commonly considered as the initial pathologic stage of age-related macular degeneration (AMD) [START_REF] Pauleikhoff | Drusen as risk factors in agerelated macular disease[END_REF]. Drusen usually appear after the age of 50 years old but they are not only associated with AMD; they have also been observed earlier. The commonly described types of early onset drusen (EOD) are basal laminar drusen (BLD) and Malattia Leventinese (ML).

BLD were described in 1977 by Gass as small round yellow lesions randomly scattered in the macula and in the mid periphery of the retina [START_REF] Gass | Stereoscopic atlas of macular diseases[END_REF]. Commonly appearing in the early adulthood, they are easier visualized by fluorescein angiography (FA), harbouring a typical "stars in the sky" appearance on early phases.

Vitelliform macular detachment can frequently be associated. This type of EOD has also been related with membranoproliferative glomerulonephritis type II (MPGN II) [START_REF] Leys | Fundus changes in membranoproliferative glomerulonephritis type II. A fluorescein angiographic study of 23 patients[END_REF]. ML, also called dominant drusen, has been described in 1925 by Vogt as a form of familial drusen with autosomal dominant inheritance, initially observed in patients living in the Leventine valley in the Ticino canton of southern Switzerland. This unusual inherited macular degeneration is caused by a mutation in the EFEMP1 (EGF containing fribrillin-like extracellular matrix protein 1) gene [START_REF] Stone | A single EFEMP1 mutation associated with both Malattia Leventinese and Doyne honeycomb retinal dystrophy[END_REF]. The diagnosis is based on fundus examination showing small radial drusen [START_REF] Piguet | Dominantly inherited drusen represent more than one disorder: a historical review[END_REF].

Our purpose was to analyze and classify phenotypes of EOD on the basis of FA and indocyanine green angiography (ICGA) features.

Methods

A prospective study was realized between 2006 and 2009 at the Creteil University

Eye Clinic, France. A complete ophthalmologic examination was performed for all patients including best corrected visual acuity (BCVA), fundus examination, color fundus photographs, red free, infrared, fundus autofluorescence frames (Heidelberg Retina Angiograph Heidelberg Engineering, Heidelberg, Germany), FA (Zeiss FF450IR, Carl Zeiss Inc, San Leandro, CA, USA or HRA, Heidelberg Engineering, Heidelberg, Germany), infrared and confocal ICGA (HRA, Heidelberg). This study was performed in agreement with the Declaration of Helsinki for research involving human subjects, and in agreement with our local ethics committee.

Criteria for inclusion were presence of drusen on fundus examination diagnosed before the age of 50 years old. Criteria for exclusion were Age Related Macular Degeneration (AMD), Stargardt disease, vitelliform macular dystrophy, severe diabetic retinopathy, media opacities that could not allow a precise examination of the retina. Diagnostic criteria for BLD were based on fundus examination and FA (stars in the sky pattern). Diagnostic criteria for ML were mainly based on fundus examination (small radial drusen associated with large and round drusen located around the optic nerve in the macular area).

Results

A total of 96 eyes of 48 patients (33 women and 15 men) were included in this prospective study. Mean age at presentation was 45. showed numerous variable sized small yellow white subretinal deposits with located in the macula and the peripheral retina (figure 1A). In 30/64 eyes (47%), BLD were arranged with clustered groups scattered throughout the retina. In 18/64 eyes (28%), an area without drusen was observed in the temporal part of the perifoveolar region (figure 1A). Vitelliform macular detachment was associated with BLD in 30/62 eyes (48%), bilateral in 90% of cases. BLD were complicated by choroidal neovascularization (CNV) in 6/64 eyes (9%) of 6 patients. Large central atrophy area was observed in 13/64 eyes (20%) of 8 patients. Red free and infrared frames showed pigmentary changes with focal atrophy of the retinal pigment epithelium (RPE) in 44/64 eyes (69%). Fundus autofluorescence (AF) showed an ill-defined autofluorescence of the drusen (figure 1B). On the early phases of FA, drusen are easily visualized as intense hyperfluorescent spots. A typical "star in the sky" pattern was observed in all cases (figure 1C). On late phases, fluorescence decreased slowly. On the early phases of confocal ICGA, BLD were faintly visualized as hyperfluorescent spots. On the middle and late phases, BLD appeared hyperfluorescent and confluent. However, some drusen appeared hypofluorescent on the late phases of ICG (figure 1F). ML was diagnosed in 5/48 patients (10%, 4 women and 1 man; table 1). Two patients were sisters. All patients had autosomal dominant inheritance. Mean age at diagnosis was 32 years old and mean age at inclusion was 34 years old (from 27 to 44). Mean BCVA was 20/50 (range from 20/25 to 20/200). In 2/10 eyes (20%) of 2 patients, the poor visual acuity was explained by a central chorioretinal atrophy.

Color fundus photographs showed whitish drusen of different size associated with pigmentary changes (atrophy, pigment migration and hyperplasia). The largest drusen were round, confluent, localized around the macula and around the optic disc.

The smallest drusen, with a characteristic radial distribution were mostly localized temporally to the macular area (figure 2). On red free frames, the bright round drusen were easily distinguished from the small radial drusen. AF frames revealed an intense autofluorescence of large drusen whereas small drusen were faintly visible (figure 2B). On the early phases of FA, large central drusen were hypofluorescent and small peripheral radial drusen were highly hyperfluorescent (figure 2C). On the late phases of FA, fluorescence of large drusen increased, harboring a fuzzy aspect.

They surrounded a central hypofluorescent area, corresponding to a relative central sparing (figure 2D). On the opposite, hyperfluorescence of small radial drusen decreased on the late phases of FA. No CNV was observed. On the early phases of ICGA, large drusen were hypofluorescent and small peripheral radial drusen were hyperfluorescent. On the late phases of ICGA, large drusen presented as hyperfluorescent spots surrounded by a hypofluorescent halo in all eyes whereas small drusen were mildly hyperfluorescent or not clearly seen (figure 2F).

All the remaining patients (11/48; 23%), 9 women and 2 men presented similar angiographic features of their EOD (table 1). No medical history or familial retinal disease was noted. Mean age was 35 years old at diagnosis and 40 years old at presentation (from 26 to 53). Mean BCVA was 20/25 (range from 20/20 to 20/40).

Color fundus photographs showed an aspect of large colloid drusen (LCD). Size of drusen was about 200 to 300 µm (figure 3). No pigmentary changes or vitelliform macular detachment was observed. In 6/22 eyes (27%) of 3 patients, these LCD were preferentially concentrated on the temporal perifoveolar region, in 4/22 eyes (18%) of 2 patients, they were preferentially concentrated on the macular region and in the remaining 12/22 eyes, they were evenly distributed on the posterior pole of the retina. In 10/22 eyes (45%), these LCD were confluent. Some small drusen could be associated and were more peripheral. On red free frames, large bright round drusen were easily distinguished and surrounded by a thin dark ring (figure 3B). AF frames revealed autofluorescence of the LCD (figure 3E). On the early phases of FA, LCD were mildly hyperfluorescent. Their hyperfluorescence was not homogenous and began on the middle periphery of the drusen, sparing a small dark center and a peripheral thin hypofluorescent ring (figure 3C). During the sequence, fluorescence of the drusen increased and on the late phases of FA, drusen were homogenously hyperfluorescent (figure 3D). No leakage was observed. No choroidal neovascularization or atrophy was noted in the 22 eyes. The small drusen, inconstantly associated with LCD, were hyperfluorescent during all the sequence. On the early phases of ICGA, these LCD were hypofluorescent, in all patients (figure 3F). In 2/22 eyes (9%), some drusen were also isofluorescent. On the intermediary phases of ICGA, most of drusen were hypofluorescent. On the late phases of ICGA, different pattern of staining were observed and different features of LCD could be detected in a same eye (figure 3G and3H). In 4/22 eyes (18%), they were mainly hyperfluorescent and in 18/22 eyes (82%), they were mainly hypofluorescent. However, hypofluorescent drusen were not homogeneous on the intermediary and late phases, appearing as a small dark center surrounded by a discreetly more fluorescent large halo, bordered with a peripheral thin hypofluorescent dark annulus (figure 3). This unusual feature was observed in all eyes. Furthermore, some hyperfluorescent pin points were noted in 12/22 eyes (55%).

Discussion

In our study, our purpose was to analyze and classify the different types of EOD on the basis of their clinical and angiographic characteristics.

In our series, drusen were more commonly observed in women than in men (ratio 2/1) as described in previous series of patients with BLD [START_REF] Gass | Adult vitelliform macular detachment occurring in patients with basal laminar drusen[END_REF][START_REF] Cohen | Visual function and course of basal laminar drusen combined with vitelliform macular detachment[END_REF]. BLD were the most frequent type of drusen observed in young people (67%) and ML accounted for 10% of patients (table 2). It is notable that the remaining EOD patients (23%) shared similar angiographic features, distinct from BLD and ML. We named them large colloid drusen (LCD) throughout this manuscript. Patients with BLD or LCD had no familial retinal history while all patients suffering from ML had a parent affected. Mean visual acuity was better in LCD eyes (20/25) than BLD eyes (20/40) or ML eyes (20/50), although the difference was not significant, probably because of a lack of power related to a relatively small number of patients in each subgroup. The better visual acuity in the group of LCD could be explained by the younger age and the lack of follow-up. Moreover, LCD were not associated with atrophy or CNV in our series, as sometimes observed in ML or BLD. Our observational study was not designed to analyze the progression of LCD. It remains possible that long term complications such as retinal pigment atrophy and visual loss could be observed with a long term follow up. Patients affected with BLD were significantly older than patients affected with ML (p=0,02) and than patients affected with LCD (not significant; p>0,05). In our series, BLD were complicated by CNV in 10% eyes. In other studies, the occurrence of CNV secondary to BLD varies from 4 to 36%. [START_REF] Leys | Fundus changes in membranoproliferative glomerulonephritis type II. A fluorescein angiographic study of 23 patients[END_REF][START_REF] Gass | Adult vitelliform macular detachment occurring in patients with basal laminar drusen[END_REF][START_REF] Cohen | Visual function and course of basal laminar drusen combined with vitelliform macular detachment[END_REF]. Although CNV have not been associated with ML in our series, few cases of CNV complicating ML have been described in the literature [START_REF] Pager | Malattia leventinese presenting with subretinal neovascular membrane and hemorrhage[END_REF]. However, we are aware that the lack of follow-up could be a limit of our study. There is possibility that LCD are complicated by atrophy or CNV in a longer follow-up period.

Main findings of our study are ICGA features showing 3 distinct patterns. Two types of BLD are distinguished with ICG (hyperfluorescent drusen with some hypofluorescent drusen). Moreover, vitelliform macular detachment observed in 48% of our BLD patients is also easily differentiated from CNV by ICG [START_REF] Quaranta | Indocyanine green angiography of basal laminar drusen in the retinal pigment epithelium associated with vitelliform macular degeneration[END_REF]. Vitelliform macular detachment appears progressively and can be associated with poor visual outcome when it evolutes to a geographic atrophy. In ML, ICGA differentiates two types of drusen [START_REF] Souied | Indocyanine green angiography features of Malattia leventinese[END_REF]. On late phases of ICGA, large drusen of ML appear as hyperfluorescent spots surrounded by halos of hypofluorescence. The significance of this peculiar "pustuliform-like" feature in ML is not clear. It could be explained by the physical or substructural characteristics of the drusen. In LCD, the drusen appear on intermediary and late phases of ICGA like a hypofluorescent center surrounded by a hyperfluorescent halo. This halo is bordered with a peripheral thin hypofluorescent ring, harboring a "donut shape" (figure 3). This unusual feature appears as the opposite image of ICGA features of large drusen of ML. This "donut-shape" can also be distinguished on infrared frames and on early phases of FA. In age related soft drusen, an original autofluorescence pattern characterized by a central area of decreased autofluorescence surrounded by an annulus of increased autofluorescence has been described, suggesting changes in retinal pigment epithelium. [START_REF] Delori | Autofluorescence distribution associated with drusen in age-related macular degeneration[END_REF] The difference in lipid composition between the core and the edge of the drusen could be responsible of these patterns of staining. Curcio et al. demonstrated that AMD drusen possess a central distinct structural core domain, surrounded by a cholesterol ester shell [START_REF] Curcio | Accumulation of cholesterol with age in human Bruch's membrane[END_REF][START_REF] Mullins | Human ocular drusen possess novel core domains with a distinct carbohydrate composition[END_REF]. A core-like structure was also described for BLD [START_REF] Mullins | Structure and composition of drusen associated with glomerulonephritis: implications for the role of complement activation in drusen biogenesis[END_REF][START_REF] Russell | Comparison of color to fluorescein angiographic images from patients with early-adult onset grouped drusen suggests drusen substructure[END_REF]. Thus, a difference of feature of fluorescence between the center and the periphery in a same drusen could also be explained by structual differences. Other examinations like high definition spectral-domain optical coherence tomography or adaptative optics could be useful to analyse these unusual features and to explain structural differences. Further studies are needed to analyze and compare SD OCT features in EOD. This is a phenotypic report. An another limit of the study is the lack of genetic testing in our series. No genotypic study or molecular analysis has been not done. Here, we observed three types of EOD but there are probably other types of drusen. Several macular diseases have been described in young people and our findings could be helpful to establish a continuum. Strong association between AMD and Tyr402His variant of the complement factor H gene (CFH) has been established [START_REF] Klein | Complement factor H polymorphism in agerelated macular degeneration[END_REF]. This polymorphism is also associated with BLD [START_REF] Grassi | Complement factor H polymorphism p.Tyr402His and cuticular Drusen[END_REF][START_REF] Boon | Basal laminar drusen caused by compound heterozygous variants in the CFH gene[END_REF] and with MPGN II [START_REF] Appel | Membranoproliferative glomerulonephritis type II (dense deposit disease): an update[END_REF] . In both diseases, a basal activation of the alternative pathway of the complement cascade leads to immune deposits in the lamina densa of the glomerular basement membrane (MPGN II) or in Bruch's membrane (BLD). X linked Alport syndrome is characterised by a central and peripheral dot-and-fleck retinopathy associated with renal failure. It can look like BLD and a retinal "lozenge" or "dull macular reflex" has been described in some patients. The pathogenesis of the retinal dots and flecks is independent of CFH-dependent mechanisms and, may depend on the nature of the underlying COL4A5 mutations [START_REF] Liu | The dot-and-fleck retinopathy of X linked Alport syndrome is independent of complement factor H (CFH) gene polymorphisms[END_REF]. On the other hand, the Arg345Trp mutation of the EFEMPI gene is associated to ML and to Doyne honeycomb retinal dystrophy [START_REF] Stone | A single EFEMP1 mutation associated with both Malattia Leventinese and Doyne honeycomb retinal dystrophy[END_REF]. This gene encodes fibuline 3, an extracellular matrix protein [START_REF] Marmorstein | Association of EFEMP1 with malattia leventinese and age-related macular degeneration: a mini-review[END_REF]. An abnormal accumulation of the Arg345Trp variant of EFEMP1 into RPE cells and in Bruch's membrane may cause drusen formation [START_REF] Fu | The R345W mutation in EFEMP1 is pathogenic and causes AMD-like deposits in mice[END_REF]. Most young adults with drusen associated with chromosome 6 have fine macular drusen and good vision whereas affected infants and children may have congenital atrophic maculopathy. In a family with autosomal dominant drusen and atrophic macular degeneration, the gene maps on chromosome 6q14. This locus is distinct from, but adjacent to, the loci for cone-rod dystrophy 7 (CORD7) and North Carolina macular dystrophy (MCDR1) [START_REF] Kniazeva | A new locus for dominant drusen and macular degeneration maps to chromosome 6q14[END_REF]. Sorsby fundus dystrophy is a rare autosomal dominant early-onset maculopathy. Drusen and atrophy or metaplasia of the retinal pigment epithelium can be observed. Furthermore CNV can occur. An extensive macular atrophy with pseudodrusen occurring in the middle-aged adult has been recently described [START_REF] Hamel | Extensive macular atrophy with pseudodrusen-like appearance: a new clinical entity[END_REF].

Further studies are needed to investigate the genetics of LCD.

In conclusion, in this study we describe a third type of EOD, harboring particular features on FA and ICG, distinct from BLD and ML features. This third type, called LCD, account for 23% of EOD and seems to be associated to good visual outcomes. FA and ICG features are useful to distinguish the 3 subtypes of EOD (table 2).

Figure 1 Patient # 7 ,

 17 Figure 1

Figure 2 Patient

 2 Figure 2

Figure 3 Patient

 3 Figure 3


Lazhar Fekih-Ahmed
email: lazhar.fekihahmed@enit.rnu.tn

ON THE ZEROS OF THE RIEMANN ZETA FUNCTION

Keywords: Mathematics Subject Classification. Primary 11M26; Secondary 11M06, 11M45 Number Theory, Riemann Zeta function, Riemann Hypothesis

published or not. The documents may come

PART I 1. Introduction

It is well known that the Riemaan zeta function defined by the Dirichlet series

(1.1) ζ(s) = 1 1 s + 1 2 s + • • • + 1 k s + • • • = ∞ n=1 n -s
converges for ℜ(s) > 1, and can be analytically continued to the whole complex plane with one singularity, a simple pole with residue 1 at s = 1. It is also well known that ζ(s) satisfies the functional equation:

(1.2)

χ(s)ζ(s) = ζ(1 -s)χ(1 -s) with χ(s) = π -s 2 Γ( s 2 ),
and that the zeros of ζ(s) come into two types. The trivial zeros which occur at all negative even integers s = -2, -4, • • • , and the nontrivial zeros which occur at certain values of s ∈ C, 0 < ℜ(s) < 1.

The Riemann hypothesis states that the nontrivial zeros of ζ(s) all have real part ℜ(s) = 1 2 . From the functional equation (1.2), the Riemann hypothesis is equivalent to ζ(s) not having any zeros in the strip 0 < ℜ(s) < 1 2 .

2. An Analytic Continuation of ζ(s)

Let (2.1) S n (s) = 1 - n -1 1 2 -s + n -1 2 3 -s -• • • + (-1) n-1 (n) -s , n ≥ 2
with S 1 (s) = 1.

Using the well-known identity, valid for ℜ(s) > 0:

(2.2)

n -s = 1 Γ(s)
∞ 0 e -nt t s-1 dt, we can rewrite S n (s) in (2.1) as:

S n (s) = n-1 k=0 (-1) k n -1 k (k + 1) -s = 1 Γ(s) ∞ 0 n-1 k=0 (-1) k n -1 k e -(k+1)t t s-1 dt (2.3) = 1 Γ(s) ∞ 0 (1 -e -t ) n-1 e -t t s-1 dt, since n-1 k=0 (-1) k n -1 k e -(k+1)t = e -t (1 -e -t ) n-1 .
We have 2 -1 1 -e -t e -t t s-1 dt.

∞ n=1 S n (s) n + 1 = 1 Γ(s) ∞ n=1 ∞ 0 (1 -e -t ) n-1 n + 1 e -t t s-1 dt (2.4) = 1 Γ(s) ∞ 0 ∞ n=1 (1 -e -t ) n-1 n + 1 e -t t s-1 dt (2.5) = 1 Γ(s) ∞ 0 t (1 -e -t )
(2.6) Before we proceed further, some remarks are in order:

Remark 2.1. The interchange of the summation and integration in equation (2.5) is valid because the series

∞ n=1 ∞ 0
(1-e -t ) n-1 n+1 e -t t s-1 dt converges absolutely and uniformly for 0 < t < ∞. To see this, we show uniform convergence for the dominating series

∞ n=1 ∞ 0 (1-e -t ) n-1 n+1
e -t t σ-1 dt, σ = ℜ(s). Indeed, let K = max((1 -e -t ) n-1 e -t/2 ), 0 < t < ∞. A straightforward calculation of the derivative shows that K = (1 -1 2n-1 ) n-1 1 √ 2n-1 and is attained when e -t = 1 2n-1 . Now, for n ≥ 2 we have

1 n + 1 ∞ 0 (1 -e -t ) n-1 e -t t σ-1 dt = 1 n + 1 ∞ 0 (1 -e -t ) n-1 e -t/2 (e -t/2 t σ-1 ) dt ≤ K n + 1 ∞ 0 e -t/2 t σ-1 dt = 1 n + 1 (1 - 1 2n -1 ) n-1 2 σ Γ(σ) √ 2n -1 (2.7) ≤ 2 σ Γ(σ) (n + 1) √ 2n -1 .
The last inequality implies that the dominating series converges by the comparison test.

Remark 2.2. To get equation (2.6) we used the identity:

(2.8) ∞ n=1 (1 -e -t ) n-1 n + 1 = t (1 -e -t ) 2 - 1 1 -e -t ,
which can be obtained by putting

X = 1-e -t into - log(1 -X) X 2 - 1 X = ∞ n=1 X n-1 n + 1 . Now, since (2.9) d dt -te -t 1 -e -t = te -t (1 -e -t ) 2 - e -t 1 -e -t ,
an integration by parts in (2.6) yields (2.10)

∞ n=1 S n (s) n + 1 = s -1 Γ(s) ∞ 0 e -t t s-1 1 -e -t dt = (s -1)ζ(s),
which is valid when ℜ(s) > 1. And since the integral (2.6) is valid for ℜ(s) > 0, then we have proved

Theorem 2.3. Let φ(t) = t (1-e -t ) 2 -1 1-e -t , then for ℜ(s) > 0, (2.11) (s -1)ζ(s) = 1 Γ(s) ∞ 0 φ(t)e -t t s-1 dt.
Remark 2.4. Although we will not need it in the rest of the paper, we can also obtain an analytic continuation of (s -1)ζ(s) when ℜ(s) ≤ 0. We simply rewrite (2.11) as a contour integral

(2.12) Γ(1 -s) 2πi C φ(t)e -t t s-1 dt,
where C is the Hankel contour consisting of the three parts C = C -∪ C ǫ ∪ C + : a path which extends from (-∞, -ǫ), around the origin counter clockwise on a circle of center the origin and of radius ǫ and back to (-ǫ, -∞), where ǫ is an arbitrarily small positive number. The integral (2.12) now defines (s -1)ζ(s) for all s ∈ C.

Remark 2.5. In particular, when s = k is a positive integer, we have yet another formula for ζ(k):

(2.13) (k -1)ζ(k) = 1 (k -1)! ∞ 0 φ(t)e -t t k-1 dt.
Remark 2.6. The above integral formula for (s -1)ζ(s), although obtained by elementary means, does not seem to be found in the literature. As for the series formula, it has been obtained by a different method in [START_REF] Ser | Sur une expression de la fonction ζ(s) de Riemann[END_REF]. A series formula that is different but similar in form and often mentioned in the literature is that of Hasse [START_REF] Hasse | Ein Summierungsverfahren für die Riemannsche Reihe[END_REF].

A Series Expansion of (s -1)ζ(s)Γ(s)

The analytic function (s-1)ζ(s)Γ(s) can be represented by a Taylor series around any point s 0 = 1 + iy on the vertical line σ = 1. In particular, for s 0 = 1 we obtain the well-known power series [START_REF] Apostol | Formulas for Higher Derivatives of the Riemann Zeta Function[END_REF]:

(3.1) (s -1)ζ(s)Γ(s) = a 0 + a 1 (s -1) + a 2 (s -1) 2 + a 3 (s -1) 3 + • • •
where the coefficients a 0 = 1 and a n are defined by

a n = 1 n! lim s→1 d n ds n ∞ 0 φ(t)e -t t s-1 dt (3.2) = 1 n! ∞ 0 φ(t)e -t lim s→1 d n ds n t s-1 dt (3.3) = 1 n! ∞ 0 φ(t)e -t (log t) n dt, (3.4)
with φ(t) being the function defined in Theorem 2.3. The coefficients a n are very important in the evaluation of ζ (k) (0) as given by Apostol in [START_REF] Apostol | Formulas for Higher Derivatives of the Riemann Zeta Function[END_REF]. Up to now the a n are regarded as unknowns and as difficult to approximate as ζ (k) (0) itself as pointed out by Lehmer [START_REF] Lehmer | The Sum of Like Powers of the Zeros of the Riemann Zeta Function[END_REF]. The formula above solves the exact evaluation problem of the ζ (k) (0) and many other variant formulae.

The following proposition provides more information on the sequence {a n }. Proposition 3.1. For n large enough, the coefficients a n are given by

a n = (-1) n 1 2 - 1 6
1 2 n+1 + O( 1 4 n ).
Proof. The expression of a n can be split into the sum

a n = 1 n! 1 0 φ(t)e -t (log t) n dt + 1 n! ∞ 1 φ(t)e -t (log t) n dt = (-1) n 2 + (-1) n n! 1 0 φ(t)e -t - 1 2 log 1 t n dt (3.5) + 1 n! ∞ 1 φ(t)e -t (log t) n dt.
To obtain an estimate the first integral in (3.5), we use equation (2.9). A differentiation with respect to t of the following expansion which defines the Bernoulli numbers 1

(3.6) te -t 1 -e -t = ∞ n=0 B n n! t n , |t| < 2π gives (3.7) φ(t)e -t - 1 2 = ∞ n=2 -B n (n -1)! t n-1 = - 1 6 t + 1 180 t 3 - 1 5040 t 5 + • • • Now, since 1 B 0 = 1 B 1 = -1/2, B 2 = 1/6, B 3 = 0, B 4 = -1/30, B 5 = 0, B 6 = 1/42, B 7 = 0, B 8 = -1/30 etc. (3.8) 1 n! 1 0 t m log 1 t n dt = 1 (m + 1) n+1
for all n, m positive, the first integral in (3.5) without the factor (-1) n has an expansion

1 n! 1 0 φ(t)e -t - 1 2 log 1 t n dt = ∞ n=2 -B n (n -1)!(n + 1) n+1 = - 1 6 1 2 n+1 + 1 180 1 4 n+1 -• • • (3.9)
To estimate the second integral in (3.5), we use the bound

(3.10) (log t) n < e ǫt
which is valid for all t ≥ n 1+ǫ and for n large enough and where ǫ is any positive small number. We split the integral into two parts

1 n! ∞ 1 φ(t)e -t (log t) n dt = 1 n! n 1+ǫ 1 φ(t)e -t (log t) n dt + 1 n! ∞ n 1+ǫ φ(t)e -t (log t) n dt (3.11) ≤ C 0 n! (1 + ǫ) n log(n) n + 1 2n! e -(1-ǫ)n 1+ǫ 1 -ǫ , (3.12)
where

C 0 = ∞ 1 φ(t)e -t dt = 0.58. Clearly, the term 1 2n! e -(1-ǫ)n 1+ǫ
1-ǫ is extremely small for n large enough. In particular, for n ≥ 2, it is less than C a n where C is a positive constant and a is any positive constant greater than 2.

Using Stirling formula n! ∼ √ 2πn n e n , we can verify that the term C0 n! (1 + ǫ) n log(n) n can also be made less than C a n for n large enough. Taking a = 4 for example, we obtain

(3.13) 1 n! ∞ 1 φ(t)e -t (log t) n dt ≤ C 4 n .
By combining the above estimates, we obtain

(3.14) a n = (-1) n 1 2 - 1 6
1 2 n+1 + O( 1 4 n ).
For any s 0 of the form s 0 = 1 + iy the corresponding Taylor series is

(3.15) (s -1)ζ(s)Γ(s) = b 0 + b 1 (s -s 0 ) + b 2 (s -s 0 ) 2 + b 3 (s -s 0 ) 3 + • • •
where the b n is expressed as

(3.16) b n = 1 n! ∞ 0 φ(t)e -t (log t) n t iy dt.
An asymptotic estimate of the coefficients b n is given by the following proposition whose proof is the same as the proof of Proposition 3.1.

Proposition 3.2. The radius of convergence of the series (3.15) is 1 + y 2 and for n large enough, the coefficients b n are given by

b n = 1 2 (-1) n (1 + iy) n+1 - 1 6 (-1) n (2 + iy) n+1 + O( 1 ( 16 + y 2 ) n
).

The zeros of ζ(s)

The Taylor series expansion (s -1)ζ(s)Γ(s) provides us with a tool to study the zeros of ζ(s) is a neighborhood of s 0 = 1 + iy. We have

(4.1) (s -1)ζ(s)Γ(s) = b 0 + b 1 (s -s 0 ) + b 2 (s -s 0 ) 2 + b 3 (s -s 0 ) 3 + • • • with (4.2) b 0 = iyζ(1 + iy)Γ(1 + iy), (4.3
) b n = 1 n! ∞ 0 φ(t)e -t (log t) n t iy dt.
It is a well-know fact [START_REF] Hadamard | Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques[END_REF] that ζ(1 + iy) = 0 for all y. Therefore b 0 = 0 for all y and the inverse of (s -1)ζ(s)Γ(s) is well-defined and can be expanded into a power series of the form

(4.4) 1 (s -1)ζ(s)Γ(s) = c 0 + c 1 (s -s 0 ) + c 2 (s -s 0 ) 2 + c 3 (s -s 0 ) 3 + • • • ,
where the coefficients c n are given by (4.5)

c n = (-1) n ∆ n b n+1 0 , with (4.6) ∆ n = b 1 b 0 • • • • • • 0 b 2 b 1 b 0 • • • 0 . . . . . . . . . . . . . . . b n-1 b n-2 b n-3 • • • b 0 b n b n-1 b n-2 • • • b 1 . Let D(s 0 , 1 2
) be the open disk of center s 0 and radius 1 2 . The zeros of ζ(s) are the same as those of (s -1)ζ(s)Γ(s) in the right half plane; therefore, ζ(s) = 0 in D(s 0 , 1 2 ) for any y is equivalent to the radius of convergence of the series (4.4) being at least 1 2 for any y.

Now, the union of the strips S 1 = {s ∈ C : 1 2 < σ < 1} and S 2 = {s ∈ C : 0 < σ < 1 2 } form the critical strip minus the critical line σ = 1 2 . Moreover, the strip S = {s ∈ C : 1 2 < σ < 3 2 } can be written as

(4.7) S = y D(1 + iy, 1 2 )
We conclude from the above, that if ζ(s) doest not have a zero inside the strip S and a fortiori does not have a zero in the strip S 1 , then by the functional equation ζ(s) cannot have a zero inside S 2 neither. We thus have proved Theorem 4.1. The Riemann hypothesis is equivalent to either

(1) the series (4.1) does not have any zero in the disk D(s 0 , 1 2 ) for any y.

(2) the radius of convergence of the series (4.4) is at least 1 2 for any y. Remark 4.2. We have been able to prove that the series (4.1) does not have any zero in the disk D(1, 1 2 ) (i.e y = 0). The proof is trivial and uses the criterion of Petrovitch [START_REF] Landau | Sur quelques théorèmes de M. Pétrovitch relatifs aux zéros des analytiques[END_REF] for power series. We also have been able to prove the well-known result that D(1, 1) is a zero-free region. We have been unable to generalize the proof to any y because as y gets large the value of |b 0 | become very small compared to that of |b 1 |. The typical power series and polynomial non-zero regions criteria are inapplicable. More knowledge on the ratios |b n |/|b 0 | is needed.

Remark 4.3. Another criterion of the Riemann hypothesis can be formulated using the conformal mapping s = 1 1-z which maps the plane ℜ(s) > 1 2 onto the unit disk |z| < 1. We can write

( z 1 -z )ζ( 1 1 -z )Γ( 1 1 -z ) f (z) = ∞ 0 φ(t)e -t t z 1-z dt = ∞ n=0 ãn z n , (4.8)
where the coefficients ãn are given by (4.9) ãn = ∞ 0 φ(t)e -t L n (-ln t) dt, L n (x) = L 0 n (x) being the Laguerre polynomial of order 0. The Riemann hypothesis is equivalent to the function f (z) having no zeros in the unit disk.

Although the above formulations of the Riemann hypothesis seem to be promising since exact information on the coefficients is known, we will not pursue this approach. The new approach that we will adopt is presented next.

PART II

In this part we will pursue a completely different approach from the one presented in PART I. Using the new series representation of the zeta function of the first part, exact information on its zeros is provided based on Tauberian-like results.

The Series representation of (s -1)ζ(s)

In PART I, we showed that (s -1)ζ(s) process both an integral and a series representation valid for ℜ(s) > 0. In the remaining of the paper we will only consider the series representation. We recall the series representation valid when ℜ(s) > 1:

(5.1)

∞ n=1 S n (s) n + 1 = s -1 Γ(s) ∞ 0 e -t t s-1 1 -e -t dt = (s -1)ζ(s),
where S n (s) is given by

(5.2) S n (s) = n-1 k=0 (-1) k n -1 k (k + 1) -s .
First, we provide another proof of the validity of the series representation for ℜ(s) > 0. To prove the analytic continuation when ℜ(s) > 0, we need to evaluate the sum when ℜ(s) > 0. The next lemma, which will also be needed in the rest of the paper, provides such an estimation. It provides an estimate of the exact asymptotic order of growth of Sn(s) n+1 when n is large.

Lemma 5.1. S n (s) n + 1 ∼ 1 n(n + 1)(log n) 1-s Γ(s)
for n large enough and for all s =

σ + it, ℜ(s) > 0, s / ∈ {1, 2, • • • }. Proof. By putting k = m -1 in (5.
2), we have by definition

S n (s) = n m=1 n -1 m -1 (-1) m-1 m -s = n m=1 m n n m (-1) m-1 m -s = -1 n n m=1 n m (-1) m m 1-s = -1 n ∆ n (s -1), (5.3) where ∆ n (λ) n m=1 n m (-1) m m -λ .
The asymptotic expansion of sums of the form ∆ n (λ), with λ ∈ C being nonintegral has been given in Theorem 3 of Flajolet et al. [START_REF] Flajolet | Mellin Transforms and Asymptotics: Finite Differences and Rice's Integrals[END_REF]. With a slight modification of notation, the authors in [START_REF] Flajolet | Mellin Transforms and Asymptotics: Finite Differences and Rice's Integrals[END_REF] have shown that ∆ n (λ) has an asymptotic expansion in descending powers of log n of the form

(5.4) -∆ n (λ) ∼ (log n) λ ∞ j=0 (-1) j Γ (j) (1) j!Γ(1 + λ -j) 1 (log n) j
We apply the theorem to ∆ n (λ) with λ = s -1 to get

(5.5) ∆ n (s -1) ∼ -(log n) s-1 Γ(s) ,
which leads to the result

(5.6) S n (s) ∼ 1 n(log n) 1-s Γ(s)
.

The Lemma follows from dividing equation (5.6) by n + 1.

Now to obtain an analytic continuation when ℜ(s) > 0, we simply observe that the logarithmic test of series in combination with the asymptotic value of S n (s) provided by Lemma 5.1 imply that the absolute value of the series on the left hand side of (5.1) is dominated by a uniformly convergent series for all finite s whose real part is greater than 0.

Remark 5.2. By Weierstrass theorem, we can see that the function (s-1)ζ(s) can be extended outside of the domain ℜ(s) > 1 and that it does not have any singularity when ℜ(s) > 0. Moreover, by repeating the same process for ℜ(s) > -k, k ∈ N, it is clear that the series defines an analytic continuation of ζ(s) valid for all s ∈ C.

Preparation Lemmas

Throughout this section, we suppose that 0 < ℜ(s) < 1. For a fixed s = σ + it, we associate with ζ(s) the following power series:

(6.1) (s -1)ζ(s, x) S 1 (s) 2 x + S 2 (s) 3 x 2 + • • • + S n-1 (s) n x n-1 + S n (s) n + 1 x n + • • • x ∈ R.
Let's also further define the "comparison" power series by

(6.2) Φ(x) (1 -x) log(1 -x) -x s = φ 0 + φ 1 x + φ 2 x 2 + • • • + φ n x n + • • •
It is easy to verify that for σ > 0

(6.3) lim x→1 (1 -x) log(1 -x) -x s = 0.
Furthermore, direct calculation of Φ ′ (x) yields the expression

(6.4) Φ ′ (x) = log(1 -x) -x s -1 + s - s log(1 -x) - s x .
Clearly, Φ ′ (x) is well-defined for all x ∈ [0, 1) and satisfies (6.5) lim

x→1 |Φ ′ (x)| = ∞.
In other words, the function Φ ′ (x) is a continuous well-defined function of x, converges for all values of x ∈ [0, 1) and diverges when x → 1. Moreover, because Φ(x) is analytic at x = 0, Φ ′ (x) must possess the following power series expansion around x = 0:

(6.6) Φ ′ (x) = φ 1 + 2φ 2 x + • • • + nφ n x n-1 + • • •
Finally, we associate to the series (6.6) the following positive coefficients power series:

(6.7) Φ(x) |φ 1 | + 2|φ 2 |x + • • • + n|φ n |x n-1 + • • •
The proofs in the remaining of this section will be based on two theorems. The first theorem, which is due to Nörlund [START_REF] Nörlund | Sur les valeurs asymptotiques des nombres et des polynmes de Bernoulli[END_REF] and more recently generalized by Flajolet et al., estimates the asymptotic behavior of the coefficients of certain powers series: Theorem 6.1 ([6]). Let α be a positive integer and β be a real or complex number,

β / ∈ {0, 1, 2, • • • }. Define the function f (z) by (6.8) f (z) = (1 -z) α ( 1 z log 1 1 -z β .
Then, the Taylor coefficients f n of f (z) satisfy (6.9)

f n ∼ n -α-1 (log n) β e 1 1! -(β) (log n) + e 2 2! β(β -1) (log n) 2 + • • • , with (6.10) e k = d k ds k 1 Γ(-s) s=α .
The derivatives in (6.10) when s = α is a positive integer can be evaluated with the help of the identity:

(6.11) 1 Γ(-s) = - sin(πs) π Γ(1 + s).
For example, the value of e 1 when α is a positive integer is given by (6.12)

e 1 = - d ds sin(πs) π Γ(1 + s) s=α = -cos(πα)Γ(1 + α),
and in this case (6.13)

f n ∼ β cos(πα)Γ(1 + α) n 1+α (log n) 1-β .
The second theorem, due to Appell [START_REF] Appell | Sur certaines séries ordonnées par rapport aux puissances d'une variables[END_REF], is the counterpart of l'Hospital's rule for divergent positive coefficients power series: Theorem 6.2 ([3] p. 66). Let f (x), g(x) be two real power series of the form

(6.14) f (x) = ∞ n=1 a n x n , g(x) = ∞ n=1 b n x n , a n , b n > 0 for all n > N, 0 < x < 1.
We further suppose that

• the series ∞ n=1 a n , ∞ n=1 b n are both divergent so that x = 1 is a singular point of both f (x) and g(x). • lim n→∞ a n b n = l, then, (6.15) lim x→1 f (x) g(x) = l.
Our first result establishes an important property on the behavior of the derivative of the function Φ(x) when x is close to 1: Lemma 6.3. There exists an x 0 ∈ (0, 1) and a constant C independent of x such that for all x ∈ (x 0 , 1) we have

|Φ ′ (x)| Φ(x) > C > 0.
Proof. From (6.4), (6.16)

|Φ ′ (x)| = log(1 -x) -x σ -1 + s - s log(1 -x) - s x .
Let's suppose that the power series expansion of log(1 -x) -x σ Ψ(x) is given by (6.17)

Ψ(x) = ψ 0 + ψ 1 x + ψ 2 x 2 + • • • + ψ n x n + • • • ,
then applying Theorem 6.1 with α = 0 and β = σ, implies that for large values of n, the coefficients ψ n satisfy the following asymptotic value:

(6.18) ψ n ∼ σ n(log n) 1-σ . Similarly, for Φ ′ (x) = φ 1 + 2φ 2 x + • • • + nφ n x n-1 + • • • , Theorem 6.
1 with α = 1 and β = s, implies that for large values of n, the coefficients φ n satisfy the following asymptotic estimates:

(6.19) φ n ∼ -s n 2 (log n) 1-s .
The asymptotic value of n|φ n | imply by Abel's Theorem and the logarithmic test of series that, like Ψ(x), the series Φ(x) goes to infinity as x approaches 1.

We thus have (6.20)

|Φ ′ (x)| Φ(x) = Ψ(x)x ∞ n=1 n|φ n |x n -1 + s - s log(1 -x) - s x . Now as x approaches 1, -1 + s - s log(1-x) -s
x approaches 1 so that given any small ǫ > 0 we can find x 1 such that for x ∈ (x 1 , 1), -1 + s -

s log(1-x) -s x > 1 -ǫ. Moreover, since Ψ(x)x = ∞ n=1 ψ n-1
x n and ∞ n=1 n|φ n |x n both go to infinity as x approches 1, and since the asymptotic estimates (6.18)-(6.19) of ψ n-1 and φ n verify (6.21) lim

n→∞ ψ n-1 n|φ n | = σ |s| ,
then Theorem 6.2 gives

(6.22) lim x→1 ∞ n=1 ψ n-1 x n ∞ n=1 n|φ n |x n = σ |s| .
In other words, given any small ǫ > 0 we can find x 2 such that for x ∈ (x 2 , 1),

Ψ(x)x ∞ n=1 n|φn|x n > σ |s| -ǫ.
To complete the proof take for example C = σ 4|s| and x 0 = max{x 1 , x 2 }.

The second lemma that we need establishes a relationship between the derivative of ζ(s, x) and that of Φ(x): Lemma 6.4. Let Φ be defined as above and let ζ ′ (s, x) be d dx ζ(s, x), then

(6.23) lim x→1 (s -1)ζ ′ (s, x) Φ ′ (x) = -1 sΓ(s)
,

where the limit is taken from below.

Proof. We have (6.24)

(s -1)ζ ′ (s, x) = S 1 (s) 2 + 2S 2 (s) 3 x + • • • + (n -1)S n-1 (s) n x n-2 + nS n (s) n + 1 x n-1 + • • • x ∈ R. Lemma 5.1 gives S n (s) n + 1 ∼ 1 n(n + 1)(log n) 1-s Γ(s)
for n large enough and for all

s = σ + it, 0 < ℜ(s) < 1.
(s -1)ζ ′ (s, x) = -1 sΓ(s) Φ ′ (x) + ∞ n=1 ǫ n nφ n x n-1 .
Now by dividing all sides of (6.28) by Φ ′ (x), 0 < x < 1, and taking the limit as x → 1, we get (6.29) lim

x→1 (s -1)ζ ′ (s, x) Φ ′ (x) = -1 sΓ(s) + lim x→1 ∞ n=1 ǫ n nφ n x n-1 Φ ′ (x) .
To prove the lemma it suffices to show that (6.30) lim

x→1 ∞ n=1 ǫ n nφ n x n-1 Φ ′ (x) = 0.
Indeed, using our first preparation Lemma 6.3, simple calculations yield

lim x→1 | ∞ n=1 ǫ n nφ n x n-1 | |Φ ′ (x)| ≤ lim x→1 ∞ n=1 n|ǫ n ||φ n |x n-1 |Φ ′ (x)| ≤ 1 C lim x→1 ∞ n=1 n|ǫ n ||φ n |x n-1 Φ(x) , (6.31)
where Φ(x) is defined in (6.7). If the series in the numerator is convergent, the result is obvious. If not, the two series in the right hand side of the last inequality are both divergent positive coefficients power series. An application of Theorem 6.2 shows that the limit in (6.31) is equal to the limit of (6.32) lim

n→∞ n|ǫ n ||φ n | n|φ n | = 0,
and the lemma is proved. Now let s be a nontrivial zero of ζ(s). By Abel's theorem lim x→1 (s -1)ζ(s, x) = (s -1)ζ(s, 1) = 0. In addition, lim x→1 Φ(x) = 0. The next and last preparation lemma shows that because of the particular function Φ(x), l'Hospital's rule which usually does not apply to vector valued or complex valued function, does apply for this particular case:

(6.33) lim x→1 (s -1)ζ(s, x) Φ(x) = lim x→1 (s -1)ζ ′ (s, x) Φ ′ (x) = -1 sΓ(s) .
Lemma 6.5. Let Φ be defined as above and let s be a nontrivial zero of ζ(s), then

(6.34) lim x→1 (s -1)ζ(s, x) Φ(x) = -1 sΓ(s)
where the limit is taken from below.

Proof. From Lemma 6.

ζ(s, 1-ǫ)-ζ(s, x) + lim ǫ→0 Φ(1 -ǫ) -Φ(x) sΓ(s) = 1 x δ(y)Φ ′ (y) dy.
Now recalling that lim ǫ→0 Φ(1 -ǫ) = 0, and that s is a zero of ζ(s) so that lim ǫ→0 ζ(s, 1 -ǫ) = 0, and dividing both sides of (6.38) by Φ(x), we finally get

(6.39) (s -1)ζ(s, x) Φ(x) + 1 sΓ(s) = - 1 x δ(y)Φ ′ (y) dy Φ(x) ,
by which we obtain:

2 The integral is an improper integral, i.e. it is defined as lim ǫ→0 The last limit in (6.42) consists of a limit of the ratio of two real functions that satisfy the hypothesis of l'Hospital's rule. That is

(6.43) K lim x→1 1 x δ(y) Φ(y) ′ dy |Φ(x)| = K lim x→1 δ(x) Φ(x) ′ |Φ(x)| ′ = K lim x→1 δ(x) = 0.
Consequently, (6.44) lim

x→1 (s -1)ζ(s, x) Φ(x) + 1 sΓ(s) = 0,
and the lemma is proved.

Equation (6.34) in Lemma 6.5 is quite a remarquable identity. It says that if s is a nontrivial zero of ζ(s) and even though for this particular value of s, lim x→1 (s -1)ζ(s, x) = 0 and lim x→1 Φ(x) = 0, the limit lim x→1 (s -1)ζ(s, x) Φ(x) is welldefined and is equal to -1 sΓ(s) .

For a nontrivial zero s, Φ(x) is in some sense a measure of the rate of convergence of (s -1)ζ(s, x) to zero and hence of the manner (s -1)ζ(s) goes to zero. By using the above identity and comparing the rate of convergence of two symmetric zeros with respect to the critical line, we can deduce the Riemann hypothesis.

Proof of Riemann Hypothesis

We proceed by contradiction. Suppose that s = σ + it is a nontrivial zero of ζ(s) with 0 < σ < 1 2 . From the functional equation (1.2), 1 -s must also be a nontrivial zero of ζ(s).

We have from the previous analysis Similarly, for 1 -s, we define the comparison function Φ(x) which is analogous to Φ(x) but with 1 -s in place of s to get This contradicts equation (7.5) unless σ = 1 2 in which case the limit in (7.5) is equal to 1. So there cannot be a zero such that 0 < σ < 1 2 and therefore all the zeros must lie on the line σ = 1 2 . The proof is complete. References

1 x

 1 is always bounded by a suitable constant, say K, for y ∈ (x 0 , 1), x 0 close to 1, the limit in(6.40) is less than or equal to (6.42) K lim x→1 δ(y) Φ(y) ′ dy |Φ(x)| .

( 1 ( 1 Φ

 11 Taking absolute values and dividing equation (7.1) by (7.2) 3 , we must then have(7.3) lim x→1 -s)ζ(s, x) sζ(1 -s, x) -s)ζ(s, x) sζ(1 -s, x) lim x→1 x→1 | (1-s)ζ(s,x) sζ(1-s,x) | is equal to | (1-s)ζ(s)sζ(1-s) |. The latter quantity is a finite non-zero value since by continuity with respect to s the functional equation (1number. But exact calculation of the left hand side of (7.5) gives: