Timothée Boyer Chammard

Elvis Temfack

Olivier Lortholary

Alexandre Alanio
email: alexandre.alanio@pasteur.fr

Timothée Boyer Chammard

Diagnostic and therapeutic strategies in cryptococcosis: impact on outcome

Keywords:

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.

DIAGNOSTIC STRATEGIES

The diagnosis of cryptococcal meningitis (CM) and more broadly, disseminated cryptococcosis, is easily made by direct examination of a pelleted biological fluid using India ink with the typical observation of rounded cells measuring about 3-10 μm in diameter surrounded a thick capsule represented as a clear halo. Sensitivity of this technique is around 86% [START_REF] Kambugu | Outcomes of cryptococcal meningitis in Uganda before and after the availability of highly active antiretroviral therapy[END_REF], Boulware et al. 2014b) but can be lower in HIV-negative patients in association with a low fungal burden [START_REF] Dromer | Determinants of disease presentation and outcome during cryptococcosis: the CryptoA/D study[END_REF]. The definitive diagnosis relies on cultures of biological samples in Sabouraud dextrose agar incubated at 30ºC [START_REF] Arendrup | ECIL-3 classical diagnostic procedures for the diagnosis of invasive fungal diseases in patients with leukaemia[END_REF]. Culture might be negative in case of exposure to antifungal therapy or may need longer incubation periods up to three weeks. Another easy way to diagnose cryptococcosis is the detection of cryptococcal antigen (CrAg). CrAg is composed of polysaccharide of the cryptococcal capsule which is shed in biological fluids and can be detected as a marker of the presence of the fungus. Detection of CrAg relies usually on specific monoclonal antibodies using different methods among which are latex agglutination (LA), enzyme-linked immunosorbent assays (ELISA) and most recently a lateral flow assay (LFA). CrAg detection in serum is presumptive of active cryptococcosis, especially in HIV patients and detection in cerebrospinal fluid (CSF) is diagnostic of CM [START_REF] Temstet | Evaluation of a monoclonal antibody-based latex agglutination test for diagnosis of cryptococcosis: comparison with two tests using polyclonal antibodies[END_REF].

However, cryptococcosis is a sub-acute infection. Consequently, CrAg can be detected in blood many weeks before the onset of symptoms [START_REF] French | Cryptococcal infection in a cohort of HIV-1-infected Ugandan adults[END_REF].

Serum CrAg positive patients are at 25% higher risk of developing CM within the first year of ART [START_REF] Letang | Cryptococcal antigenemia in immunocompromised human immunodeficiency virus patients in rural Tanzania: a preventable cause of early mortality[END_REF][START_REF] Meya | Preventing cryptococcosis -Shifting the paradigm in the era of highly active antiretroviral therapy[END_REF]. CrAg screening of patients before initiation of ART associated with preemptive fluconazole treatment once CrAg is positive was associated with low number of cases of CM [START_REF] Longley | Cryptococcal antigen screening in patients initiating ART in South Africa: a prospective cohort study[END_REF]. This combined strategy is associated with a significantly reduced mortality [START_REF] Mfinanga | Cryptococcal meningitis screening and community-based early adherence support in people with advanced HIV infection starting antiretroviral therapy in Tanzania and Zambia: an open-label, randomised controlled trial[END_REF].

As such, systematic pre-ART CrAg screening and preemptive fluconazole is recommended for those presenting with < 100 CD4 cell counts (WHO 2017). Though promising as a strategy to decrease the incidence of cryptococcal disease, especially when associated with timely ART initiation, its implementation still lags in many countries.

CrAg detection and diagnosis of cryptococcal disease

-Commercially available CrAg detection tests are highly sensitive and specific in serum and CSF in predicting the presence of meningitis in HIV positive patients presenting with evocative symptoms. CrAg detection in serum and CSF with symptoms suggestive of CM strongly correlates with culture-and/or India ink-confirmed CM. CSF CrAg is a rapid and reliable diagnostic test for confirmed HIV-associated CM [START_REF] Temfack | Cryptococcal antigen screening in asymptomatic HIV-infected antiretroviral Naïve patients in Cameroon and evaluation of the new semi-quantitative Biosynex CryptoPS Test[END_REF]. In a recently published study, the new semi-quantitative BioSynex/ Bio-Rad CryptoPS® lateral flow assay at point of screening, clearly classified patients with or without CM [START_REF] Temfack | Cryptococcal antigen screening in asymptomatic HIV-infected antiretroviral Naïve patients in Cameroon and evaluation of the new semi-quantitative Biosynex CryptoPS Test[END_REF]. Of note, serum and CSF CrAg should be preferentially considered at the time of diagnosis, as there is no clear correlation between changes in antigen titre and CSF cryptococcal culture [START_REF] Brouwer | Baseline correlation and comparative kinetics of cerebrospinal fluid colony-forming unit counts and antigen titers in cryptococcal meningitis[END_REF]. Persistence of detectable CrAg is frequent despite clinical improvement and complete recovery of the patient. The correlation between the evolution of CrAg levels during treatment and clinical improvement remains unclear and in case of early recurrence of symptoms, CSF CrAg should also be interpreted with caution.

Pre-ART CrAg screening and pre-emptive therapy -In HIV-infected patients with advanced AIDS disease, the prevalence of CrAg positivity ranged from 2-21%, with a median of 6% (IQR: 5-7) (Temfack et al., unpublished observations). CrAg-positive patients who were not offered fluconazole pre-emptive therapy are at higher risk of CM and death [START_REF] Desmet | The value of cryptococcal serum antigen screening among HIV-positive/AIDS patients in Kinshasa, Zaire[END_REF][START_REF] Jarvis | Screening for cryptococcal antigenemia in patients accessing an antiretroviral treatment program in South Africa[END_REF]). More so, about a third of CrAg-positive asymptomatic patients have evidence of CM [START_REF] Wake | High cryptococcal antigen titers in blood are predictive of subclinical cryptococcal meningitis among HIV-infected patients[END_REF]. In CrAg positive patients who received preemptive fluconazole initiated at 800 mg/day, there was a significant decrease in the relative risk of developing CM (from 21.4-6%) and of incident mortality (Temfack et al., unpublished observations).

THERAPEUTIC STRATEGIES

The main challenge for cryptococcosis management is for disseminated disease and meningoencephalitis, knowing that the printciple of therapy is similar for both. For both clinical presentations, recommended first-line antifungal therapy is based on a three-steps strategy including induction, consolidation and maintenance phases, that was first used in a landmark Mycoses Study Group trial [START_REF] Van Der Horst | Treatment of cryptococcal meningitis associated with the acquired immunodeficiency syndrome. National Institute of Allergy and Infectious Diseases Mycoses Study Group and AIDS Clinical Trials Group[END_REF]. The 2010 IDSA guidelines [START_REF] Perfect | Clinical Practice guidelines for the management of cryptococcal disease: 2010 update by the Infectious Diseases Society of America[END_REF] for management of disseminated disease placed amphotericin B and flucytosine as first choices for induction phase, with amphotericin B deoxycholate (0.7-1.0 mg/kg/day) given intravenously in combination with oral flucytosine 100 mg/ kg/day (at least two weeks for HIV-infected patients and organ transplant recipients, and four to six weeks for all other patients). When available, the use of liposomal formulation of amphotericin B is preferred, as it has been showed to induce less toxicity. The preferred therapy for pulmonary-limited disease, or non-disseminated disease is based on fluconazole therapy alone.

The aim of CM therapy is to improve survival thanks to rapid yeast clearance from the CSF. Quantitative clearance from CSF, called "early fungicidal activity" (EFA), is measured as the rate of yeast clearance per millilitre of CSF per day based on CFU counting. In a combined cohort of 262 patients, a slow rate of cryptococcal clearance was shown to be independently associated with increased mortality at two weeks and at 10 weeks (Bicanic et al. 2009b). Though, in a recently published meta-analysis, combining 2854 HIV-infected or non HIV-infected patients in 27 randomised CM treatment trials, the authors concluded that all-cause mortality remains the essential primary outcome to be used to assess therapy efficacy in clinical trials [START_REF] Montezuma-Rusca | Early fungicidal activity as a candidate surrogate endpoint for all-cause mortality in cryptococcal meningitis: a systematic review of the evidence[END_REF].

Thereafter, we summarise the main studies and clinical trials which have helped to improve cryptococcosis outcome and survival in the last decades.

COMBINATION ANTIFUNGAL THERAPY

Amphotericin B with flucytosine -The use of flucytosine combined with amphotericin B for CM therapy was already observed in a study in 1979, before the HIV pandemic, where authors concluded that the combination therapy allowed more rapid sterilisation of the cerebrospinal fluid (p < 0.001) and less nephrotoxicity (p < 0.05) than did amphotericin B alone, although results were obtained at different dosages than those currently recommended [START_REF] Bennett | A comparison of amphotericin B alone and combined with flucytosine in the treatment of cryptococcal meningitis[END_REF].

In 1997, the Mycoses Study Group randomised, double-blind, placebo-controlled, trial involving 408 HIV-infected patients evidenced that the use of a combination therapy with amphotericin B plus flucytosine for induction therapy was associated with an increased rate of CSF sterilisation and a better outcome than amphotericin B alone, in HIV-associated CM, although without improved survival [START_REF] Van Der Horst | Treatment of cryptococcal meningitis associated with the acquired immunodeficiency syndrome. National Institute of Allergy and Infectious Diseases Mycoses Study Group and AIDS Clinical Trials Group[END_REF].

Aiming to confirm these results, a randomised trial conducted in Thailand, also found that the efficacy of therapy (as measured by EFA) was shown to be significantly faster in the arm in which patients were treated with the combination of amphotericin B plus flucytosine than in any of the three other arms (amphotericin B alone, amphotericin B plus fluconazole, or a triple therapy with amphotericin B, fluconazole and flucytosine), in 64 HIV-infected patients with CM [START_REF] Brouwer | Combination antifungal therapies for HIV-associated cryptococcal meningitis: a randomised trial[END_REF].

This strategy was then also shown to be superior in the French prospective CryptoA/D study in HIV-positive or HIV-negative patients with cryptococcal infection with a mycological failure at week two of 26% in the amphotericin B and flucytosine group compared to a treatment failure of 56% with any other treatments (p < 0.001) [START_REF] Dromer | Determinants of disease presentation and outcome during cryptococcosis: the CryptoA/D study[END_REF], Dromer et al. 2008).

Moreover, a three-arm, open-label, randomised trial including about 300 HIV-infected patients enrolled, demonstrated a survival benefit with this combination compared to amphotericin alone, with a reduction of about 40% in the relative risk of death at 10 weeks with addition of flucytosine [START_REF] Day | Combination antifungal therapy for cryptococcal meningitis[END_REF].

Finally, the recently completed "Advancing Cryptococcal Meningitis Treatment for Africa" (ACTA) trial, in which 721 HIV-infected patients with first episode of CM were randomised over nine sites in four different countries in Sub-Saharan Africa, definitively evidenced that the combination of amphotericin B and flucytosine was significantly superior to Amphotericin B and fluconazole, leading to a substantial mortality reduction [hazard ratio for death at 10 weeks with flucytosine vs. fluconazole, 0.62; 95% confidence interval (CI), 0.45 to 0.84; p = 0.002] ( [START_REF] Molloy | Antifungal combinations for treatment of cryptococcal meningitis in Africa[END_REF].

Amphotericin B with fluconazole -When flucytosine is unavailable, the combination of amphotericin B with fluconazole is still recommended [START_REF] Perfect | Clinical Practice guidelines for the management of cryptococcal disease: 2010 update by the Infectious Diseases Society of America[END_REF]. In an open-label, three-arm, phase II trial including 143 HIVinfected patients, combination of amphotericin B with fluconazole 800 mg/day was found to have significantly better long-term outcomes than amphotericin B and fluconazole 400 mg/day or amphotericin alone [START_REF] Pappas | A phase II randomized trial of amphotericin B alone or combined with fluconazole in the treatment of HIV-associated cryptococcal meningitis[END_REF]). However, this have not been observed in the study conducted by [START_REF] Day | Combination antifungal therapy for cryptococcal meningitis[END_REF], with no statistically survival benefit found for patients receiving amphotericin with fluconazole 800 mg/day compared to those with amphotericin B alone. More so, in the ACTA trial, the treatment arm combining Amphotericin B and fluconazole had the worse outcome, clearly demonstrating that the combination of Amphotericin B and fluconazole is a not recommended for induction treatment [START_REF] Molloy | Antifungal combinations for treatment of cryptococcal meningitis in Africa[END_REF].

Amphotericin B with voriconazole -The use of voriconazole at the dosage of 300 mg twice daily, instead of highdose fluconazole (800 mg or 1200 mg/day) in induction therapy associated with amphotericin B was tested in a randomised 4-arms trial and found to have similar EFA, and could therefore be a theoretical option for therapy, despite its higher cost and potential interactions with other drugs including rifampicin [START_REF] Loyse | Comparison of the early fungicidal activity of high-dose fluconazole, voriconazole, and flucytosine as second-line drugs given in combination with amphotericin B for the treatment of HIV-associated cryptococcal meningitis[END_REF]). As voriconazole is not available in most of the countries endemic for cryptococcosis, this combination would be marginally used.

Short courses therapy -The amphotericin B and flucytosine combination regimen remains unavailable in most parts of the world harbouring the highest burden of disease (Africa and Asia), furthermore several days of amphotericin B deoxycholate is well known to cause important adverse events, such as anaemia, renal impairment, hypokalaemia, hypomagnesemia, or phlebitis, and its administration requires inpatient hospitalisation for intravenous administration and electrolytes monitoring [START_REF] Bicanic | Toxicity of amphotericin B deoxycholate-based induction therapy in patients with HIV-associated cryptococcal meningitis[END_REF].

Researcher in lower middle-income settings are now working to improve induction therapy phase with combination therapy, using oral combination or reducing duration of intravenous drugs, to reduce incidence of adverse events, and try to improve survival.

In a study in Uganda, based on 30 HIV-infected patients, a short 5-day course of amphotericin with highdose fluconazole 1200 mg/day was found to have a better EFA than findings with fluconazole alone in previous studies, suggesting that shorter courses of amphotericin may be used [START_REF] Muzoora | Short course amphotericin B with high dose fluconazole for HIV-associated cryptococcal meningitis[END_REF].

The multi-centre ACTA trial, testing short courses amphotericin B deoxycholate based therapies with either flucytosine or fluconazole, also showed a better survival with a 7-day course of amphotericin B and flucytosine compared with 14-day course (HR 0.56 (0.35-0.91)) [START_REF] Molloy | Antifungal combinations for treatment of cryptococcal meningitis in Africa[END_REF]. This prompted WHO experts to now recommend a 7-days amphotericin B IV therapy with flucytosine as first line induction therapy (followed by seven days of high-dose fluconazole) in low and middle-income countries (WHO 2018).

Fluconazole and flucytosine -Before the highly active antiretroviral therapy era, an open-label, single-arm, prospective trial studied the fluconazole (400 mg daily) and flucytosine (150 mg/kg daily) combination therapy for a 10-week antifungal therapy in 32 patients with AIDS, and found a clinical survival at 10 weeks which appeared to be better than previously reported with amphotericin B alone or fluconazole alone [START_REF] Larsen | Fluconazole combined with flucytosine for treatment of cryptococcal meningitis in patients with AIDS[END_REF].

More recently, a randomised clinical trial compared combination therapy with high dose fluconazole (1200 mg daily) and flucytosine (100 mg/kg daily) to high dose fluconazole alone for the 2-week induction therapy, in 41 HIV-infected patients with CM in Malawi, and found better EFA and survival for the combination therapy arm [START_REF] Nussbaum | Combination flucytosine and high-dose fluconazole compared with fluconazole monotherapy for the treatment of cryptococcal meningitis: a randomized trial in malawi[END_REF].

Finally, the ACTA trial including 721 randomised HIV-infected patients also showed that a combination oral therapy with high dose fluconazole (1200 mg/day) and flucytosine (100 mg/kg/day) for the 2-week induction therapy was non-inferior to the standard 2-week courses of amphotericin B based therapy with flucytosine or fluconazole [START_REF] Molloy | Antifungal combinations for treatment of cryptococcal meningitis in Africa[END_REF]. That made authors conclude that this regimen is an acceptable and effective option for CM therapy in Africa if amphotericin B is not available or not recommended.

FLUCONAZOLE MONOTHERAPY

Both 2010 IDSA and 2011 WHO guidelines recommend high-dose fluconazole monotherapy (1200 mg/ day) for 10-12 weeks if amphotericin and flucytosine are not available [START_REF] Perfect | Clinical Practice guidelines for the management of cryptococcal disease: 2010 update by the Infectious Diseases Society of America[END_REF], WHO 2011).

In a cohort study carried out in Malawi using high dose fluconazole (1200 mg) as induction therapy [START_REF] Gaskell | A prospective study of mortality from cryptococcal meningitis following treatment induction with 1200 mg oral fluconazole in Blantyre, Malawi[END_REF], mortality was unacceptably high, as it was in a study using 800 mg [START_REF] Rothe | A prospective longitudinal study of the clinical outcomes from cryptococcal meningitis following treatment induction with 800 mg oral fluconazole in Blantyre, Malawi[END_REF], showing that fluconazole monotherapy is not suitable as induction therapy.

A recently published CrAg screening study in Ethiopia in HIV-infected patients, highlighted that the use of high dose fluconazole monotherapy for induction phase is inadequate, leading to a 68% mortality rate at three months, in patients with CM [START_REF] Beyene | Inadequacy of high-dose fluconazole monotherapy among cerebrospinal fluid cryptococcal antigen (CrAg)-positive human immunodeficiency virus-infected persons in an Ethiopian CrAg screening program[END_REF].

USE OF LIPOSOMAL AMPHOTERICIN B

At the beginning of highly active antiretroviral therapy era, in 1997, a randomised trial comparing liposomal amphotericin B (L-AmB) and conventional deoxycholate amphotericin B in 28 HIV-infected patients with CM, observed that a 3-week course of 4 mg/kg L-AmB had a significantly earlier CSF culture conversion than a 3-week course standard amphotericin B, with similar clinical outcome and that L-AmB had significantly less renal adverse events [START_REF] Leenders | Liposomal amphotericin B (AmBisome) compared with amphotericin B both followed by oral fluconazole in the treatment of AIDS-associated cryptococcal meningitis[END_REF].

In a larger multi-centre trial, in which 267 patients were randomised, liposomal amphotericin B -therapy was similar in term of efficacy and overall mortality at 10 weeks, compared to conventional amphotericin B deoxycholate for HIV-associated CM. L-AmB at a dosage of 3 mg/kg/day was associated with significantly fewer adverse effects than at a dosage of 6 mg/kg/day, and compared to conventional amphotericin B [START_REF] Hamill | Comparison of 2 doses of liposomal amphotericin B and conventional amphotericin B deoxycholate for treatment of AIDS-associated acute cryptococcal meningitis: a randomized, double-blind clinical trial of efficacy and safety[END_REF].

More recently, 80 participants were enrolled in the AMBITION preliminary phase II trial comparing different short courses of L-AmB with high dose fluconazole. The single dose arm was non-inferior to the standard 14days arm in term of early fungicidal activity, compared to two-doses or three-doses arms [START_REF] Jarvis | Ambition-cm: high-dose liposomal amphotericin for HIVrelated cryptococcal meningitis[END_REF].

Based on these results, the AMBITION phase III (IS-RCTN 72509687) ongoing trial is currently taking place to evaluate whether a single high dose (10 mg/kg) of L-AmB given with high dose fluconazole and flucytosine is as effective as the newly recommended (based on ACTA trial) one-week daily-dosed amphotericin B deoxycholate based induction therapy with flucytosine as an induction therapy [START_REF] Molefi | AMBITION-cm: intermittent high dose AmBisome on a high dose fluconazole backbone for cryptococcal meningitis induction therapy in sub-Saharan Africa: study protocol for a randomized controlled trial[END_REF]. This trial will be the largest therapeutic trial on CM with 850 patients to be enrolled over a 3-year period in five different African countries.

ADJUVANT THERAPIES

Steroids -Adjuvant glucocorticoid therapy is successfully used in patients with non-fungal meningitis such as bacterial or tuberculous meningitis, and reduces mortality in some sub-groups. In CM IDSA guidelines [START_REF] Perfect | Clinical Practice guidelines for the management of cryptococcal disease: 2010 update by the Infectious Diseases Society of America[END_REF], experts recommend to consider the use of dexamethasone at high doses for patients having immune reconstitution inflammatory syndrome with severe central nervous system signs and symptoms, using a two-six-week course.

The CRYPTODEX double-blind, randomised, placebo-controlled trial, conducted in 6 countries over Africa and Asia, aimed to demonstrate the efficacy of adjunctive dexamethasone but was stopped after the enrolment of 451 patients (on an overall of 880 planned) for safety reasons [START_REF] Beardsley | Adjunctive dexamethasone in HIV-associated cryptococcal meningitis[END_REF]. All patients received the locally-available and standard-of-care combination antifungal therapy with amphotericin B and fluconazole, and were randomised to receive either dexamethasone (intravenous for the first two weeks, and then orally, tapered doses until the 6th week) or placebo for six weeks. The primary outcome was survival until 10 weeks after randomisation. There was no difference between groups -47% in the dexamethasone group and 41% in the placebo group died. EFA was measured and during the first two weeks of treatment, dexamethasone was significantly associated with slower rates of decline in the EFA in CSF than was placebo. Even if dexamethasone was significantly associated with a better reduction in CSF opening pressure during the first two weeks than was placebo, it was also associated with more adverse events and disability.

Interferon-gamma -In a prospective trial, with 90 HIV-infected patients randomised in three different arms [standard therapy with amphotericin B plus flucytosine, standard therapy with two doses of interferon-gamma (INF-γ), and standard therapy with six doses of INF-γ], an increased rate of clearance with two doses of adjunctive INF-γ than with standard therapy was shown, but there was no significant difference in mortality between groups [START_REF] Jarvis | Adjunctive interferon-γ immunotherapy for the treatment of HIV-associated cryptococcal meningitis[END_REF].

Sertraline -Previous studies have suggested that sertraline, a selective serotonin reuptake inhibitor (SSRI), frequently used as an anti-depressant, has a real in vitro and in vivo fungicidal activity against Cryptococcus neoformans, as its action has a synergistic effect with fluconazole in reducing the fungal burden in brain, kidney, and spleen [START_REF] Zhai | The antidepressant sertraline provides a promising therapeutic option for neurotropic cryptococcal infections[END_REF]. The first clinical open-label and dose-finding study was conducted in Uganda, recruiting 172 patients with HIV-associated CM, to assess the efficacy of adjunctive sertraline, with standard amphotericin B and high-dose fluconazole antifungal therapies [START_REF] Rhein | Efficacy of adjunctive sertraline for the treatment of HIV-associated cryptococcal meningitis: an open-label dose-ranging study[END_REF]. Authors concluded that patients had faster cryptococcal CSF clearance and a lower incidence of immune reconstitution inflammatory syndrome and relapse than that reported in the past.

Following these results, the "Adjunctive Sertraline for the Treatment of HIV-associated Cryptococcal Meningitis (ASTRO-CM)" phase III randomised placebo-controlled clinical trial had been evaluating if sertraline dosed initially at 400 mg/day for two weeks, followed by 200 mg/day for an additional 10 weeks prior to tapering, could have an 18-week survival benefit, compared with placebo when receiving standard available induction therapy of amphotericin B deoxycholate and fluconazole 800 mg/day. The trial was stopped after enrolling 460 of the 550 patients planned for futility, showing a 18-week mortality of 52% in sertraline group and 46% in placebo group (p = 0.15) [START_REF] Rhein | Adjunctive sertraline in HIV-associated cryptococcal meningitis[END_REF].

Tamoxifen -Tamoxifen, the oestrogen receptor antagonist drug usually used for breast cancer, has been shown to be fungicidal and synergistic with fluconazole and amphotericin B in vitro and in vivo in a mouse model of disseminated cryptococcosis [START_REF] Butts | Estrogen receptor antagonists are anti-cryptococcal agents that directly bind EF hand proteins and synergize with fluconazole in vivo[END_REF]. At acceptably concentrations for humans, tamoxifen combined with fluconazole decreased brain fungal burden, and has demonstrated to inhibit the growth of C. neoformans within macrophages, which are not accessible by classical antifungal drugs. It has also the benefit to have a good oral bioavailability.

Therefore, an open-label, phase II, randomised trial ("A randomised trial of tamoxifen combined with amphotericin B and fluconazole for cryptococcal meningitis", NCT03112031) will enrol 50 patients in Vietnam, comparing EFA according to the strategy between (amphotericin B plus fluconazole ± tamoxifen during the first two weeks).

ELEVATED OPENING CEREBROSPINAL FLUID PRESSURE

Elevated intracranial pressure (CSF pressure ≥ 25 cm H2O) is a usual complication of CM, secondary to a failure of CSF resorption because of obstruction by cryptococcal polysaccharide capsule and yeasts.

In several large randomised trials (van der [START_REF] Van Der Horst | Treatment of cryptococcal meningitis associated with the acquired immunodeficiency syndrome. National Institute of Allergy and Infectious Diseases Mycoses Study Group and AIDS Clinical Trials Group[END_REF][START_REF] Brouwer | Combination antifungal therapies for HIV-associated cryptococcal meningitis: a randomised trial[END_REF][START_REF] Bicanic | Fungal burden, early fungicidal activity, and outcome in cryptococcal meningitis in antiretroviral-naive or antiretroviral-experienced patients treated with amphotericin B or fluconazole[END_REF]) of amphotericin B-based therapy for CM, the effect of baseline CSF opening pressure was evaluated [START_REF] Graybill | Diagnosis and management of increased intracranial pressure in patients with AIDS and cryptococcal meningitis. The NIAID Mycoses Study Group and AIDS Cooperative Treatment Groups[END_REF], Bicanic et al. 2009a) and led to recommend to relieve by large-volume CSF drainage, performing daily therapeutic LPs if the CSF pressure is ≥ 25 cm of CSF with symptoms of increased intracranial pressure during induction therapy.

Another randomised trial comparing acetazolamide or placebo, with amphotericin B alone was prematurely stopped after recruitment of 24 patients in Thailand with CM, presenting with headache and an elevated opening cerebrospinal fluid pressure ≥ 20 cm H20, after having observed that patients who received acetazolamide developed significantly lower venous bicarbonate levels and higher chloride levels and had more frequent serious adverse events than patients who received placebo [START_REF] Newton | A randomized, double-blind, placebo-controlled trial of acetazolamide for the treatment of elevated intracranial pressure in cryptococcal meningitis[END_REF].

In the COAT trial, the effect of therapeutic lumbar puncture (LP) was evaluated in HIV-infected patients treated for CM, and investigators found that therapeutic LPs were associated with a 69% relative improvement in survival, regardless of initial intracranial pressure (Boulware et al. 2014a, Rolfes et al. 2014).

ANTIRETROVIRAL THERAPY FOR HIV-INFECTED PATIENTS

Antiretroviral therapy (ART) is the corner stone of treatment in HIV-infected patients with advanced AIDS disease. In patients with CM, it is now recommended delaying ART for four-six weeks after starting antifungal therapy (WHO 2017). This deferring strategy has indeed shown to significantly improve survival, compared to early ART initiation strategy (Boulware et al. 2014a).

NON-HIV-INFECTED PATIENTS

In solid organ transplant recipients, cryptococcal disease has different presentation, with less disseminated disease, and more lung-limited presentation [START_REF] Singh | Cryptococcosis in solid organ transplant recipients: current state of the science[END_REF]). This could be caused by the use of calcineurin inhibitor immunosuppressive agents -such as cyclosporine or tacrolimus. These agents are synergistic with antifungals, and their use had shown better outcomes [START_REF] Kontoyiannis | Calcineurin inhibitor agents interact synergistically with antifungal agents in vitro against Cryptococcus neoformans isolates: correlation with outcome in solid organ transplant recipients with cryptococcosis[END_REF]. The optimal way to manage immunosuppressive regimen is to do sequential or stepwise reduction of immunosuppressive agents, and to consider decreasing corticosteroid therapy first [START_REF] Perfect | Clinical Practice guidelines for the management of cryptococcal disease: 2010 update by the Infectious Diseases Society of America[END_REF]. For disseminated presentations, lipid formulations of amphotericin B are preferred to standard amphotericin B deoxycholate based therapy, given the potential risk of renal deterioration and drug interactions in these patients.

IN CONCLUSION

Cryptococcal disease is a death-related disease in immunosuppressed patients, especially in advanced AIDS disease patients. Over the past 15 years, many studies and clinical trials have led to improve prevention, diagnosis, therapy and outcome of patients with cryptococcal meningitis. Efforts are now needed to implement those strategies in middle and low-income settings using affordable and efficient diagnostic strategies and drug regimen. These new strategies are being evaluated and will be implemented progressively to improve patient management.

AUTHORS' CONTRIBUTION

TBC, ET and AA wrote the paper; OL and ET -major criticism of the manuscript.


Vincent Savaux

Yves Louët

PAPR Analysis as a Ratio of Two Random Variables: Application to Multicarrier Systems with Low Subcarriers Number

Keywords: PAPR, CCDF, Multicarrier, OFDM

The distribution of the peak to average power ratio (PAPR) of multicarrier systems is studied in the context of signals featuring low subcarriers number. Considering the mean power of the signal as a random variable, a general cumulative distribution function of the PAPR is suggested. We show that the new formulation is valid not only for low subcarriers number, but also for larger subcarriers number. Thus, an asymptotic analysis using a large subcarriers number proves a perfect concordance between the suggested approach and the state-ofthe-art derivations only valid for large subcarriers number. All the theoretical developments are supported through simulations, and we discuss the case of oversampled signals as well.

I. INTRODUCTION

Multicarrier signals are well known to be prone to high power fluctuations due to the inherent summation of independent information carried on different tones. The most common way to quantify these power fluctuations is to define a metric taking into account its maximum value relatively to its mean. This gave birth to the peak factor definition namely the peak to average power ratio (PAPR) parameter highly discussed for decades especially in the context of multicarrier systems, such as orthogonal frequency division multiplexing (OFDM) (see [START_REF] Wunder | The PAPR Problem in OFDM Transmission: New Directions for a Long-Lasting Problem[END_REF] and references therein). PAPR is defined as the ratio of the maximum power and the mean power of a signal. An accurate PAPR derivation of multicarrier signal may be difficult given the initial statistic of the signal to be considered. A straightforward way to circumvent this issue is to directly upper bound the PAPR value regardless of any statistical hypothesis. As a result, a PAPR upper bound is found to be equal to N.f (M ), N being the useful subcarriers number of the OFDM systems and f (M ) a real value depending on the digital modulation constellation size M (it is easily shown that f (M ) equals 3 M -1 M +1 ). However this upper bound appears to be very far from reality and almost never reached due to the random behavior of the signal. As a consequence, the only way to provide a thorough PAPR model is to derive its distribution function or similarly its complementary cumulative distribution function (CCDF) viewing PAPR as a random variable due to the random character of the signal itself.

Manuscript submitted in March 2018

Vincent Savaux is with IRT b<>com, Rennes, France (e-mails: vincent.savaux@b-com.com, Phone: +33 256 35 82 16.)

Yves Louët is with CentraleSupélec and IETR, Rennes, FR (e-mail: yves.louet@centralesupelec.fr) Considering a large subcarriers number N , a conventional analysis assumes that each time sample of the OFDM signal follows a complex Gaussian distribution. The overall signal can then be modeled as a random vector of N independent Gaussian samples. This has been first considered in [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], [START_REF] Van Nee | OFDM for wireless multimedia communications[END_REF] where the CCDF of the PAPR is given in a very simple form and remains one of the most popular derivation. But the latter derivation ((2) in the paper) remains only valid for an oversampling factor equal to 1, which makes the formula inaccurate for oversampled signals simulating continuous phenomenons. The reason is that oversampling leads to correlated samples. Still in [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], [START_REF] Van Nee | OFDM for wireless multimedia communications[END_REF], the authors extended their work to provide an approximation of the PAPR CCDF valid for large oversampling factors, but this approximation is empirical and yields some discrepancies with simulations for low subcarriers number (N < 128).

To extend [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], authors suggested more thorough derivations valid for lower oversampling values (4 or 8) [4]- [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF] and different subcarriers numbers. These works are based either on the level crossing rate analysis of the peak distribution (since the envelop of the OFDM signal can be approximated as a Gaussian process), or theoretical links between peaks of sampled and continuous signals, called extreme value theory. However, these suggested expressions of PAPR CCDF are close to the simulation results only for N ≥ 64.

Nevertheless the recent interest in OFDM systems with low subcarriers number (e.g. narrowband internet of things (NB-IoT) signal with 12 subcarriers [START_REF]3GPP TS 36.211[END_REF]) should push us to revisit the aforementioned PAPR derivations. The reason lies in the PAPR denominator expression: it could not be approximated as the signal power expectation as previously, since the ergodic condition is not valid anymore. Even though the central limit theorem remains valid (down to 10 carriers [START_REF] Rudziński | Normalized Gaussian Approach to Statistical Modeling of OFDM Signals[END_REF]), the denominator itself has to be reconsidered as a random variable and no more as a constant value (the mean power).

Thus, the main contributions of this paper are summarized as follows.

1) We derive a general PAPR cumulative function valid for large and low subcarriers numbers, down to 12 subcarriers. Our results are supported by original theoretical developments. The main point is that the denominator of PAPR CCDF is viewed as a random variable. 2) We show that an asymptotic analysis for large N leads to existing results [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], [START_REF] Van Nee | OFDM for wireless multimedia communications[END_REF], which confirms the validity of our approach. Moreover, the first and second order moments of PAPR are derived from our suggested CCDF, and perfectly match those found in the state of the art [START_REF] Ochiai | On the Distribution of the Peak to Average Power Ratio in OFDM Signals[END_REF]. In addition, simulations results validate the theoretical developments, which shows the interest to reconsider PAPR expression for low subcarriers number. Furthermore, we discuss the extension of PAPR CCDF for low N value and large oversampling rate. Fig. 1 positions the suggested work according to the number of subcarriers N and the oversampling factor L. Such as highlighted in Fig. 1, the suggested CCDF is relevant for low N value, as it was a pending issue, but it remains valid for larger N value as well.

The remainder of the paper is organized as follows. Section II presents the problem statement. Section III provides the new PAPR CCDF, first validated in Section IV to match the state-of-the-art results in an asymptotic analysis, and second in Section V through simulation results, which cover both signals at Nyquist rate and oversampled signals. Several concluding remarks are made in Section VI. Main proofs are given in Appendix to maintain the flow of the paper.

II. PROBLEM STATEMENT

Let us consider a signal x(t) sampled at Nyquist rate 1/t s over an observation window of duration N t s , with t s the sampling time. It is assumed that the samples x n , n = 0, .., N -1 are independent and identically distributed (iid) and obey a complex white Gaussian distribution CN (0, σ 2 ), where σ 2 is the variance of x n , i.e. σ 2 = E{|x n | 2 }. This hypothesis fits the multicarrier signal such as OFDM or filter bank multicarrier (FBMC) [START_REF] Farhang-Boroujeny | OFDM Versus Filter Bank Multicarrier[END_REF], according to the central limit theorem for N > 10 [9]. Note that in that case, N is usually equal to the number of samples per symbol, which corresponds to the number of subcarriers of the signal at Nyquist rate. According to these assumptions, the PAPR of the signal can be expressed as

λ = max n∈ 0,N -1 |x n | 2 1 N ∑ N -1 n=0 |x n | 2 . ( 1
)
It is worth mentioning that, in the literature, the mean square in the denominator expression is almost systematically replaced by the expectation as

1 N ∑ N -1 n=0 |x n | 2 ≈ E{|x n | 2 } = σ 2
, assuming large N . This approximation allows for a simplification of the analytical developments, as the CCDF of the PAPR is derived from the numerator of (1). Since the elements |x n | 2 are iid and obey a Chi-squared (χ 2 ) distribution with two degrees of freedom, the CCDF is written as:

CCDF (λ) = 1 - N -1 ∏ n=0 P( 1 σ 2 |x n | 2 ≤ λ) = 1 - ( 1 -e -λ ) N , (2)
where P(.) means "probability of an event". However (2) holds when N is large enough. In fact, as shown in Fig. 2, the CCDF in (2) holds for N ≥ 128. Therefore, a new expression of the CCDF should be derived for lower N values, i.e. when the mean 1

N ∑ N -1 n=0 |x n | 2 cannot
be approximated by the expectation. We hereby propose a more general expression of the PAPR fitting low N values. Then, we present an asymptotic analysis of the PAPR when N tends to infinity, which shows that the known results of the literature can be obtained from the suggested general expression.

III. SUGGESTED CCDF

For clarity purpose, we denote by n * the index corresponding to the maximum value of |x n | 2 , i.e.

n * = arg max n∈ 0,N -1 |x n | 2 .
(3)

Accordingly, we define the set of indexes Ξ = {0, 1, .., N -1}, and the subset Ξ * which does not contain n * , i.e. Ξ * ∪{n * } = Ξ and Ξ * ∩ {n * } = ∅.

When N is low, the numerator and denominator of the PAPR in (1) cannot be considered as independent variables, since the maximum is included in the sum. Therefore we should rewrite the denominator of PAPR by using Ξ * subset as

1 N N -1 ∑ n=0 |x n | 2 = max n∈ 0,N -1 |x n | 2 N + 1 N ∑ n∈Ξ * |x n | 2 , ( 4
)
where the sum in the right side of (4) does not contain the maximum value. Thus, the general expression of the CCDF is obtained by substituting ( 4) into (1):

CCDF (λ) =P ( max n∈ 0,N -1 |x n | 2 1 N ∑ N -1 n=0 |x n | 2 ≥ λ ) =P ( max n∈ 0,N -1 |x n | 2 ≥ λ ( max n∈ 0,N -1 |x n | 2 N + 1 N ∑ n∈Ξ * |x n | 2 ) ) =P ( max n∈ 0,N -1 |x n | 2 X ≥ λ 1 -λ N λ 1 N ∑ n∈Ξ * |x n | 2 Y ) , ( 5
)
where λ, X, and Y have been defined for clarity purpose. Note that (5) holds only if λ < N , which is a reasonable condition since we assume that N , although low-valued, is large enough to make the central limit theorem valid, in case of multicarrier signal. In order to definitely address this issue in the rest of the developments, we arbitrarily assume that N ≥ 12, which is consistent with the considered CCDF range (see Fig. 2), as well as with the condition λ < N imposed by λ in [START_REF] Sharif | On the Peak to Average of OFDM Signals Based on Oversampling[END_REF].

Let us now investigate the distributions of X and Y , denoted by f X and f Y , respectively. From (2), it can be deduced that the distribution of X is the derivative of the CDF

N -1 ∏ n=0 P(|X| 2 ≤ x) = ( 1 -e -x σ 2 ) N , (6)
where it can be noted that, unlike (2), |X| 2 is not normalized by σ 2 , since the denominator is not considered as constant anymore. Thus, we obtain the corresponding distribution:

f X (x) = N σ 2 (1 -e -x σ 2 ) N -1 e -x σ 2 . ( 7
)
The derivation of the distribution of Y requires more developments. On one hand, it can be noted that Y is the sum of N -1 independent elements |x n | 2 , which are not normalized by σ 2 . On the other hand, each element in the sum can be rewritten as

|x n | 2 = |Re(x n )| 2 + |Im(x n )| 2 ,
where Re and Im stand for the real and the imaginary parts, respectively. Since Re(x n ) (resp. Im(x n )) is a zero-mean real Gaussian variable, then we can deduce that f Y is similar to a χ 2 distribution with 2N -2 degrees of freedom. In Appendix A, it is proved that the distribution of Y is expressed as

f Y (y) = (N -1) N -1 σ 2(N -1) y,N Γ(N -1) y N -2 e - (N -1)y σ 2 y,N , (8)
where Γ(.) is the Gamma function [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], and the parameter σ 2 y,N is equal to

σ 2 y,N = σ 2 ( 1 - N -1 ∑ k=0 (-1) k ( N -1 k ) (k + 1) 2 ) Θ N . ( 9
)
Θ N in (9) has been defined for clarity purpose.

It is worth mentioning that X and Y in (5) are uncorrelated, but not independent, since we know that X ≥ N N -1 Y . Therefore, the joint distribution f X,Y may be hard to derive, then we make the following approximation:

CCDF (λ) = P(X ≥ λY, λY ≥ 0) ≈ ∫ +∞ 0 f Y (y) ∫ +∞ λy f X (x)dxdy = 1 - ∫ +∞ 0 ( 1 -e -λy σ 2 ) N f Y (y)dy. ( 10
)
Given the binomial theorem

( 1 -e -λy σ 2 ) N = ∑ N k=0 ( N k ) (-1) k e -k λy σ 2
to develop [START_REF] Farhang-Boroujeny | OFDM Versus Filter Bank Multicarrier[END_REF] with [START_REF]3GPP TS 36.211[END_REF], we obtain:

CCDF (λ) ≈ 1 - N ∑ k=0 (N -1) N -1 ( N k ) (-1) k (Θ N σ 2 ) (N -1) Γ(N -1) × ∫ +∞ 0 y N -2 e - (N -1)y Θ N σ 2 -k λy σ 2 dy. ( 11
)
By substituting = y

( k λ σ 2 + N -1 Θ N σ 2
) in the exponential, we recognize the Gamma function [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], then [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] yields

CCDF (λ) ≈ 1 - N ∑ k=0 ( N k ) (-1) k (N -1) N -1 Θ N -1 N ( N -1 Θ N + k λ) N -1 = - N ∑ k=1 ( N k ) (-1) k (1 + kΘ N λ N -1 ) N -1 . ( 12
)
It must be emphasized that ( 12) can be considered as a general CCDF expression of PAPR, which holds for low subcarriers number (or equivalently, for low samples number). In order to show that this expression still holds for large N value as well, we derive in Section IV an asymptotic analysis of PAPR when N tends to infinity.

IV. ASYMPTOTIC PAPR ANALYSIS

For clarity purpose, we remove the notation lim N →+∞ throughout this section, as we assume that N is large in all developments. Also, we note ∼ instead of ∼ N →+∞ . First, we show that the CCDF (12) asymptotically tends to the usual expression (2), and second, we derive the mean and variance of PAPR.

A. Asymptotic CCDF

As preliminary results, it is straightforward to show that λ ∼ λ, and Θ N ∼ 1. This equivalence can be proved by reminding that σ 2 y,N = σ 2 Θ N on one hand, and from (27),

σ 2 y,N = σ 2 -E { |Xi,m| 2 N }
∼ σ 2 on the other hand. Then, for any x ∈ R we have (1 + x N ) N ∼ e x . The substitution of the previous equivalences into [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], in addition to the binomial theorem, lead to

CCDF (λ) ∼ - N ∑ k=1 ( N k ) (-1) k e kλ = 1 - N ∑ k=0 ( N k ) (-1) k e kλ = 1 - ( 1 -e -λ ) N . ( 13
)
We recognize [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], which shows the relevance of the general expression of the CCDF in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], which holds for large N values.

B. Asymptotic Mean and Variance of PAPR 1) Mean of PAPR:

It is must be noticed that the mean and variance of PAPR cannot be directly derived from [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] since it involves integrals with respect to λ from 0 to infinity. However, it has been stated that (5) holds only if λ < N . Therefore, we make the following approximation: we suppose that N is large enough so that the numerator and the denominator of the PAPR are uncorrelated, but the denominator is still considered as a random variable. Then, we can rewrite [START_REF] Sharif | On the Peak to Average of OFDM Signals Based on Oversampling[END_REF] and redefine the variable Y as

CCDF (λ) = P ( max n∈ 0,N -1 |x n | 2 X ≥ λ 1 N N -1 ∑ n=0 |x n | 2 Y ) (14)
where X has the same distribution as previously (see ( 7)), and Y obeys a χ 2 distribution with 2N degrees of freedom and parameter σ 2 . By following the same reasoning as in ( 10)-( 12), the CCDF from ( 14) can be derived as follows:

CCDF (λ) = - N ∑ k=1 ( N k ) (-1) k (1 + kλ N ) N . ( 15
)
We then deduce the distribution of PAPR, denoted f λ as

f λ (λ) = ∂ ∂λ ( 1 -CCDF (λ) ) = - N ∑ k=1 ( N k ) (-1) k k (1 + kλ N ) N +1 . ( 16
)
The mean, denoted by µ P AP R , whose derivation is proved in Appendix B, is expressed as

µ P AP R = ∫ +∞ 0 λf λ (λ)dλ = - N ∑ k=1 ( N k ) (-1) k N 2 k(N -1)(N -2) ∼ γ + ln(N ), ( 17
)
where γ is the Euler-Mascheroni constant. It must be emphasized that (17) matches the results presented in [START_REF] Ochiai | Peak-Power Reduction Schemes in OFDM Systems : A Review[END_REF], where the usual PAPR CCDF (2) is used.

2) Variance of PAPR: The variance of the PAPR, denoted by ν P AP R , is obtained from ( 15)-(17) as

ν P AP R = ∫ +∞ 0 λ 2 f λ (λ)dλ νP AP R -µ 2 P AP R . ( 18
)
In the following, we focus on the development of νP AP R .

Similarly to (17), the asymptotic νP AP R value is expressed as

νP AP R = - N ∑ k=1 ( N k ) (-1) k 2N 3 k 2 (N -1)(N -2)(N -3) ∼ π 2 6 + (γ + ln(N )) 2 , ( 19
)
from which we finally deduce:

ν P AP R ∼ π 2 6 . ( 20
)
We prove (19) in Appendix C. Once again, this result matches those in [START_REF] Ochiai | Peak-Power Reduction Schemes in OFDM Systems : A Review[END_REF], which validates the suggested PAPR form.

V. SIMULATIONS AND DISCUSSION

A. Simulations Results 1) Nyquist Rate Signal: Simulations have been carried out to show the relevance of the proposed theoretical developments regarding PAPR of multicarrier signal with low subcarriers number (N ∈ {16, 32, 64, 128}). In all simulations, a 16quadratic amplitude modulation (QAM) constellation has been used. Note that higher order constellations leads to very similar results. The simulation results have been averaged over 10000 independent runs. Fig. 3 depicts the CCDFs of PAPR versus λ (in dB), for (a) N =16, 32 subcarriers, and (b) N =64, 128 subcarriers. For each subcarriers number configuration, three curves are compared: one obtained through simulations, and two obtained from theory, referred as "usual" in (2), and "proposed" in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. It can be generally observed that the suggested theoretical PAPR CCDF is closer to simulations than the usual one, for any N ∈ {16, 32, 64, 128}. In particular, Fig. 3 shows that at CCDF (λ) = 10 -4 , the difference between proposed CCDF and simulation is less than 0.1 dB, whereas compared to (2), it is equal to 1, 0.5, 0.3, and 0.2 for N =16, 32, 64, and 128, respectively. This shows the relevance of the proposed PAPR analysis for N ≤ 128 compared to the usual one. Furthermore, it shows that the larger N , the closer to simulation the CCDFs. However, it can be noted that the proposed CCDF does not perfectly match the simulations, in particular in small λ ranges, where it is even slightly worse than CCDF in [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF]. This is mainly due to the approximation made in [START_REF] Farhang-Boroujeny | OFDM Versus Filter Bank Multicarrier[END_REF].

Fig. 4 shows the mean and variance of PAPR versus N from 16 to 2048. Theoretical results in ( 17) and ( 20) are compared with those obtained through simulations. It can be clearly observed that µ P AP R tends to γ + ln(N ), and ν P AP R tends to π 2 6 as N increases. This shows the relevance of the asymptotic mean and variance analysis of PAPR, and then confirms the results presented in [START_REF] Ochiai | Peak-Power Reduction Schemes in OFDM Systems : A Review[END_REF].

2) Oversampled Signal: It must be emphasized that previous developments and results have been presented for multicarrier signals sampled at Nyquist rate. However, for more practical scenarios, the PAPR of oversampled signal should be considered. In order to fit oversampled signals with low subcarriers number, we propose two approximations of CCDF (λ):

• Van Nee based approximation [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], i.e. N is substituted by ⌊αN ⌉ in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], where ⌊.⌉ is the nearest integer function. In that case, the coefficient α is empirically set in order to fit the curves obtained by simulation. In the following, this approximation is referred as "adapted", since it is adapted from Van Nee approach in [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF].

• Proposed approximation, where λ is substituted by λ = βλ 1-βλ N in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], where β is a parameter which is empirically set. In the following, this approximation is referred as "proposed". Furthermore, we compared the suggested approximation with CCDF expressions derived in the literature [START_REF] Ochiai | On the Distribution of the Peak to Average Power Ratio in OFDM Signals[END_REF], [START_REF] Jiang | Derivation of PAPR Distribution for OFDM Wireless Systems Based on Extreme Value Theory[END_REF], [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF]. In [START_REF] Ochiai | On the Distribution of the Peak to Average Power Ratio in OFDM Signals[END_REF], the authors expressed the CCDF of PAPR as:

CCDF (λ) = 1 -exp ( -N e -λ √ πλ 3 ) . ( 21
)
In [START_REF] Jiang | Derivation of PAPR Distribution for OFDM Wireless Systems Based on Extreme Value Theory[END_REF], [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF], the authors based their developments on extreme value theory. The CCDF in [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF] is expressed as

CCDF (λ) = 1 -exp ( -N e -λ √ π log(N ) 3
) , ( 22
)
and this result has been extended in [START_REF] Jiang | Derivation of PAPR Distribution for OFDM Wireless Systems Based on Extreme Value Theory[END_REF] to more a realistic model where subcarriers may not be active, and with different power distribution. In that more general case, the CCDF of PAPR can be written as

CCDF (λ) = 1 -exp ( -2Ke -λ √ 2ρπλK 3N ) , ( 23
)
where K is half the number of activated subcarriers, and ρ is related to the power allocation.

In order to validate the suggested approximations, Figs. 5 and6 show the CCDFs of PAPR versus λ for adapted and proposed approximations, respectively. The oversampling rate is equal to 20, then the PAPR of the signal is very close to that of the continuous signal. In Fig. 5, Van Nee based adaptation has been carried out with α = 1.5. Furthermore, the curve referred as "theory, original" corresponds to the multicarrier signal with N = 16, and sampled at Nyquist rate such as proposed in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. It can be seen that the CCDF in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] underestimates the actual CCDF of oversampled signal of about 0.4 dB. In addition, the Van Nee approximations do not perfectly match the curves obtained through simulations. Thus, a specific α value should be set for every N value. In Fig. 6, the proposed CCDF approximation has been carried out with β = 0.92. In that case, it can be observed that the proposed CCDFs match well the curves obtained through simulations for CCDF (λ) ≤ 0.1. However, it must be noted that further simulations (not shown here) revealed that this approximation does not hold anymore for N > 128 with β = 0.92 (β should then be adapted), whereas Van Nee's does with α = 2.8 such as suggested in [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF]. Thus, we can conclude that Van Nee's approximation can still be used for oversampled multicarrier signals with large N number (N ≥ 128), whereas the proposed one fits signals with low N number (N < 128), even though it can also be used for larger N by adapting β value. Fig. 7 compares the CCDF of PAPR of the proposed approximation with those of the aforementioned state-of-theart [START_REF] Ochiai | On the Distribution of the Peak to Average Power Ratio in OFDM Signals[END_REF], [START_REF] Jiang | Derivation of PAPR Distribution for OFDM Wireless Systems Based on Extreme Value Theory[END_REF], [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF], using a 16-QAM and the same oversampling rate as previously. The CCDF obtained through simulations has been plotted as a reference. Two subcarriers numbers are considered: N =16, in Fig. 7-(a), and N = 64 in Fig. 7-(b). The curve of CCDF corresponding to (23) has been obtained by considering that all subcarriers are active, i.e. K = N 2 . Note that in the case where ρ = 1, then (23) reduces to (21). Therefore, we arbitrarily set ρ = 0.6 to distinguish (23) and (21). It can observed in Fig. 7-(a) that the proposed approximation almost match the simulations, whereas the three other approximations deviate from simulation for λ ≥ 7 dB. However, in Fig. 7-(b), all the approximations are closer to the simulation. In particular, it is highlighted that, at CCDF (λ) = 0.0001, the proposed approximation, (22), and (23) are less than 0.2 dB far from the simulation. This shows the relevance of the suggested analysis and the general CCDF of PAPR expression [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] for low subcarriers number N < 64.

B. Discussion

The various simulations results show the relevance of the suggested general expression of CCDF of PAPR, in particular for low subcarriers numbers 12 ≤ N ≤ 64. However, it can be noted that [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] involves the derivation of two binomial coefficients per element of the sum. Therefore, the calculus of (12) may be not tractable in practice. In fact, if N is higher than 128, the derivation of the binomial coefficients exceeds the computing capacity of most of the present computers. However, it can be noted that Θ N in (9) quickly converges towards 1 N (γ + ln(N )) (this is not proved in this paper), which reduces the computation cost of the CCDF for N > 32. Otherwise, computation tricks should be used to obtain the binomial coefficient in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], e.g. by considering logarithmic additions instead of multiplications.

In practice, we conclude from the different simulations results that (12) should be used for very low subcarriers number, typically N < 128 at Nyquist rate, and N < 64 for oversampled signals. For higher N values, other approximations [START_REF] Van Nee | Reducing the peak-to-average power ratio of OFDM[END_REF], [START_REF] Ochiai | On the Distribution of the Peak to Average Power Ratio in OFDM Signals[END_REF], [START_REF] Jiang | Derivation of PAPR Distribution for OFDM Wireless Systems Based on Extreme Value Theory[END_REF], [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF] can be considered for simplicity matter.

VI. CONCLUSION

In this paper, we presented a new and general expression of PAPR CCDF for multicarrier systems, which holds for large and low subcarriers number values, down to N ≥ 12. This formula has been derived by considering the denominator of PAPR as a random variable, whereas it is usually assumed to be a constant. This led us to derive an original distribution function for this denominator. In addition, we showed that an asymptotic analysis of the PAPR leads to the existing results in the literature, in terms of CCDF, mean, and variance of PAPR. These developments, supported by various simulations results, (a) CCDF of PAPR versus λ (dB), N =16 subcarriers.

APPENDIX

A. Proof of ( 8)-( 9)

Let Ω X = {X 0 , X 1 , .., X N -1 } be a set of N independent zero-mean complex white Gaussian variables with variance

σ 2 = E{|X n | 2 }. Then, the sum Z = 1 N ∑ N -1 n=0 |X n | 2 obeys a χ 2 distribution
with 2N degrees of freedom and parameter σ 2 Z (due to the fact that X n are not normalized), i.e. the distribution of Z is written as

f Z (z) = N N σ 2N Z Γ(N ) z N -1 e -N z σ 2 Z , ( 24
)
where the parameter σ 2 Z can be straightforwardly deduced from (24). In fact, on one hand, we have

E { Z } = 1 N N -1 ∑ n=0 E{|X n | 2 } = σ 2 , ( 25
)
and, on the other hand, by property of random variables featuring continuous density functions, it can be shown that

E { Z } = ∫ +∞ 0 zf Z (z) = σ 2 Z , (26)
which yields σ 2 Z = σ 2 . Note that the latter feature holds for any degree of χ 2 distribution. Thus, suppose that X i , i ∈ 0, N -1 , is randomly chosen in Ω X , and let Z ′ be defined as

Z ′ = Z-|Xi| 2 N . Then Z ′ obeys a χ 2 distribution with 2N -2 degrees of freedom with parameter σ 2 Z ′ = E { Z -|Xi| 2 N } = N -1 N σ 2 . Now, suppose that X i is chosen such that |X i | 2 = |X i,m | 2 , where |X i,m | 2 = max i∈ 0,N -1 |X i | 2 (the subscript m points out that |X i,m | 2 is
the maximum value). Then, the variable Y = Z -|Xi,m| 2 N still obey a χ 2 with 2N -2 degrees of freedom with new parameter σ 2 Y , which is derived below. It must be noted that the distribution of X = |X i,m | 2 is given in [START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF], therefore the value of σ 2 Y is derived as follows:

σ 2 Y = E { Z - |X i,m | 2 N } = σ 2 - 1 N ∫ +∞ 0 xf X (x)dx. (27)
Then, the binomial theorem applied to f X (x) leads to

f X (x) = N σ 2 (1 -e -x σ 2 ) N -1 e -x σ 2 = N -1 ∑ k=0 ( N -1 k ) (-1) k N σ 2 e -(k+1)x σ 2 . ( 28
)
Therefore, substituting (28) into (27) yields

σ 2 Y = σ 2 - 1 N ∫ +∞ 0 N -1 ∑ k=0 ( N -1 k ) (-1) k xN σ 2 e -(k+1)x σ 2
dx.

(29) Then we substitute w = (k+1)x σ 2 to recognize the Gamma function, which finally leads to [START_REF] Rudziński | Normalized Gaussian Approach to Statistical Modeling of OFDM Signals[END_REF], and then concludes the proof.

B. Proof of (17)

We use the following equivalence ( N k ) ∼ N k k! to simplify µ P AP R in (17):

µ P AP R ∼ - N ∑ k=1 (-1) k N k kk! . ( 30
) Let E 1 (z) = ∫ +∞ z e -t
t dt, z ∈ C\R -, be the exponential integral function, defined in [START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. The series representation of E 1 is

E 1 (z) = -γ -ln(z) - +∞ ∑ k=1 (-1) k z k kk! . ( 31
)
Since lim z→+∞ E 1 (z) = 0, and reminding that N → +∞, then the substitution of z by N in (31) leads to µ P AP R in (17), which concludes the proof.

C. Proof of (19)

Similarly to the reasoning in Appendix B, it should be first noticed that the first line in (19) becomes

νP AP R ∼ -2 N ∑ k=1 (-1) k N k k 2 k! . ( 32
)
Then, we use the series expansion of the so-called generalized integro-exponential function (see (2.3) and (2.10) in [START_REF] Milgram | The Generalized Integro-Exponential Function[END_REF]), denoted by E j s , such as hereby described:

E j s (z) = 1 Γ(j + 1)
where Ψ k,s is the polygamma function [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], whose analytical values are provided in [START_REF] Milgram | The Generalized Integro-Exponential Function[END_REF], Table 1, and in [START_REF] Abu-Shumays | Transcendental Functions Generalizing the Exponential Integrals[END_REF]. It can be noticed that, when s = j = 1:

E 1 1 (z) = +∞ ∑ k=1 (-1) k z k k 2 k! + 1 2 2 ∑ k=0 ( 2 k
) ln(z) 2-k Ψ k,1 , (

where

Ψ 0,1 = 1 Ψ 1,1 = γ Ψ 2,1 = γ 2 + π 2 6 . ( 35
)
From the integral form of E j s (z) in (33) we have lim

Fig. 1 .

 1 Fig. 1. Overview of the suggested PAPR analysis compared with state-ofthe-art, according to subcarriers N and oversampling factor L.

Fig. 2 .

 2 Fig. 2. CCDF of PAPR versus λ for N ∈ {16, 32, 128, 512}, comparison between theory in (2) and simulations.

  CCDF of PAPR versus λ (dB), N =16, 32 subcarriers.

  CCDF of PAPR versus λ (dB), N =64, 128 subcarriers.

Fig. 3 .

 3 Fig.3. CCDF of PAPR versus λ (dB), comparison of simulated (exact) CCDF, and usual theoretical CCDF (2) and proposed theoretical CCDF (12), using N =16, 32, 64, 128 subcarriers.

6 νFig. 4 .

 64 Fig. 4. Mean and variance of PAPR versus N for N = 16, 32, 64, 128, 256, 512, 1024, 2048, comparison between (17), (20), and simulations.

Fig. 5 .

 5 Fig.5. CCDF of PAPR versus λ (dB) of oversampled OFDM signal using Van Nee based approximation. The "original" corresponds to Nyquist rate model in[START_REF]Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. The results have been obtained with an oversampling rate of 20, α = 1.5.

Fig. 6 .

 6 Fig.6. CCDF of PAPR versus λ (dB) of oversampled OFDM signal using proposed approximation. The results have been obtained with an oversampling rate of 20, β = 0.92.

  et al.[START_REF] Ochiai | On the Distribution of the Peak to Average Power Ratio in OFDM Signals[END_REF] theory, Wei et al.[START_REF] Wei | A modem extreme value theory approach to calculating the distribution of the peak-to-average power ratio in OFDM systems[END_REF] theory, Jiang et al.[START_REF] Jiang | Derivation of PAPR Distribution for OFDM Wireless Systems Based on Extreme Value Theory[END_REF]

∫ +∞ 1 ln

 1 (t) j t -s e -zt dt = 1+j-k Ψ k,s ,

z→+∞ E 1 1

 1 (z) = 0. Then considering z = N and substituting (32) into (34) finally leads toνP AP R ∼ π 2 6 + (γ + ln(N )) 2 , (36)which concludes the proof.

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TCOMM.2018.2857460Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.