MD João Pedro Ferreira
email: j.ferreira@chru-nancy.fr

MD Faiez Zannad

PhD Stuart J Pocock

MD Stefan D Anker

MD Javed Butler

MD Gerasimos Filippatos

MD Martina Brueckmann

MD Waheed Jamal

MD Dominik Steubl

Dipl.Math Elke Schueler

MD Milton Packer

Interplay of Mineralocorticoid Receptor Antagonists and Empagliflozin in Heart Failure

Keywords: heart failure, empagliflozin, mineralocorticoid receptor antagonists

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

6

Condensed Abstract

Both MRAs and SGLT2 inhibitors are beneficial in heart failure with a reduced ejection fraction. We studied the interplay of empagliflozin and MRAs in EMPEROR-Reduced. The effects of empagliflozin on most efficacy endpoints and on safety were similar in patients receiving or not MRAs. Among MRA nonusers, the empagliflozin group was 35% less likely than the placebo group to initiate treatment with an MRA following randomization. Among MRA users, the empagliflozin group were 22% less likely than the placebo group to discontinue treatment with an MRA following randomization. Severe hyperkalemia was less common in the empagliflozin group.

Introduction

Mineralocorticoid receptor antagonists (MRAs) reduce the morbidity and mortality of patients with heart failure and reduced ejection fraction. The Eplerenone in Mild Patients Hospitalization and Survival Study in Heart Failure (EMPHASIS-HF) and the Randomized Aldactone Evaluation Study (RALES) and trials demonstrated that both eplerenone and spironolactone decreased the risk of cardiovascular death and the risk of hospitalization for heart failure in patients with mild and moderate-to-severe symptoms, respectively. [START_REF] Zannad | Eplerenone in patients with systolic heart failure and mild symptoms[END_REF][START_REF] Pitt | The effect of spironolactone on morbidity and mortality in patients with severe heart failure. Randomized Aldactone Evaluation Study Investigators[END_REF] More recently, the Dapagliflozin and Prevention of Adverse Outcomes in Heart Failure (DAPA-HF) trial and the Cardiovascular and Empagliflozin Outcome Trial in Patients with Chronic Heart Failure and a Reduced Ejection Fraction (EMPEROR-Reduced) showed that the sodium glucose co-transporter 2 (SGLT2) inhibitors, dapagliflozin and empagliflozin, reduced the composite risk of cardiovascular death or hospitalization for heart failure in patients with heart failure and a reduced ejection fraction, with and without diabetes, with similar effects in patients receiving or not receiving an MRA before randomization. [START_REF] Mcmurray | Dapagliflozin in Patients with Heart Failure and Reduced Ejection Fraction[END_REF][START_REF] Packer | Cardiovascular and Renal Outcomes with Empagliflozin in Heart Failure[END_REF][START_REF] Zannad | SGLT2 inhibitors in patients with heart failure with reduced ejection fraction: a meta-analysis of the EMPEROR-Reduced and DAPA-HF trials[END_REF] However, it is not clear how MRAs influence other measures of efficacy (as well as the safety) of SGLT2 inhibitors and whether SGLT2 inhibition may influence the utilization of MRAs.

In this secondary analysis of the EMPEROR-Reduced trial, we examined the influence of MRA use at baseline on the efficacy and safety of empagliflozin and whether empagliflozin influenced the prescribing of MRAs following randomization.

Methods

The EMPEROR-Reduced trial was a randomized, double-blind, parallel-group, placebocontrolled and event-driven study, whose design has been described previously. [START_REF] Packer | Cardiovascular and Renal Outcomes with Empagliflozin in Heart Failure[END_REF] Participants were men or women with chronic heart failure (functional class II, III or IV) with a left ventricular ejection fraction ≤40%, who were receiving appropriate background treatment for heart failure. We preferentially enrolled patients with an ejection fraction ≤30% by requiring those with a higher ejection fraction to have been hospitalized for HF within 12 months or to have markedly increased levels of N-terminal prohormone B-type natriuretic peptide (NT-proBNP), i.e., 1000 pg/ml or 2500 pg/ml in those with an ejection fraction of 31-35% or 36-40%, respectively; these thresholds were doubled in patients with atrial fibrillation. The Ethics Committee of each of the 520 sites in 20 countries approved the protocol and all patients gave written informed consent.

Randomization

Patients were randomized double-blind (in a 1:1 ratio) to receive placebo or empagliflozin 10 mg daily, in addition to their usual therapy. The use of MRAs at baseline was not a stratification variable. Following entry into the trial, all appropriate treatments for heart failure or other medical conditions (including MRAs) could be initiated, discontinued or altered at the clinical discretion of the investigator. Patients were periodically assessed at study visits for major outcomes, symptoms and functional capacity related to heart failure, initiation or discontinuation of new treatments for heart failure (including MRAs), vital signs and biomarkers reflecting changes in the course of heart failure or the action of SGLT2 inhibitors, and adverse events. All randomized patients were followed for the occurrence of prespecified outcomes for the entire duration of the trial, regardless of whether the study participants were taking their study medications or adhered to the schedule of study visits.

Trial Endpoints

The primary endpoint was the composite of adjudicated cardiovascular death hospitalization for heart failure, analyzed as time to first event. The first secondary endpoint was the occurrence of all adjudicated hospitalizations for heart failure (including first and recurrent events). The second secondary end point was the analysis of the slope of the change in estimated glomerular filtration rate (eGFR) during double-blind treatment, which was supported by an analysis of a composite of serious adverse renal events, defined by the need for chronic dialysis or renal transplant or a sustained ≥40% drop in eGFR or a sustained eGFR <15 mL/min/1.73m 2 (if the baseline eGFR was ≥30 mL/min/1.73m 2 ) or <10 mL/min/1.73m 2 (if the baseline eGFR was <30 mL/min/1.73m 2 ). Additional analyses included the individual components of the primary endpoint as well as all-cause mortality, health status as assessed by the Kansas City Cardiomyopathy Questionnaire (KCCQ), systolic blood pressure, body weight and laboratory parameters.

The pre-specified safety analyses included serious adverse events, adverse events associated with discontinuation of a trial treatment, adverse events of special interest (i.e., volume depletion, renal events, and hypotension). The occurrence of a serum potassium concentration >5.5 mmol/L and >6.0 mmol/L as well as investigator-reported hyperkalemia were also analyzed.

Statistical Analysis

Baseline characteristics were compared in MRA users and nonusers using chi square test for categorical variables and t-test for continuous variables. For time-to-first-event analyses, differences between the placebo and empagliflozin groups were assessed using a Cox proportional hazards model, with prespecified baseline covariates of age, gender, geographical region, diabetes, left ventricular ejection fraction, and estimated glomerular filtration rate. For the analysis of total (first and repeated) events, the differences between the placebo and empagliflozin groups were assessed using a joint frailty model, with cardiovascular death (for recurrent heart failure hospitalizations) or all-cause mortality (for recurrent hospitalization for any reason) as competing risks. For the analysis of changes in vital signs and laboratory measurements, treatment effects were assessed based on changes from baseline using a mixed model for repeated measures. Between-group difference in the slope of change in eGFR were analyzed based on data on double-blind treatment and using a random intercept random slope model. All analyses used the same covariates as in the Cox model and included the baseline variable as an additional covariate, where applicable. For all efficacy measures, separate analyses were performed according to the use or nonuse of MRAs at baseline, and differences in the effect of empagliflozin in users and nonusers were assessed by interaction terms. Values for NT-pro BNP were log-transformed. Analyses of safety were performed based on all patients who had received at least one dose of the study medication. P values and 95% confidence intervals presented in this report have not been adjusted for multiplicity, and therefore inferences drawn from these statistics may not be reproducible. All analyses were performed using SAS, version 9.4 (SAS Institute).

Results

In the EMPEROR-Reduced trial, 71% of the patients were treated with an MRA at the time of randomization. In general, MRA users were younger with better renal function, but with a lower ejection fraction, systolic blood pressure and NT-pro BNP. Additionally, MRA users were more likely to have been hospitalized for heart failure within 12 months and be treated with a loop diuretic, but were less likely to have received a cardiac device for the treatment of heart failure. Other clinical features are shown in Table 1. MRA users were prescribed spironolactone in 2091 (78.6%) and eplerenone in 570 (21.4%) patients. Of the patients treated with an MRA, 1355 were randomized to placebo and 1306 to empagliflozin; of the patients not treated with an MRA, 512 were randomized to placebo and 557 to empagliflozin.

Effect of Empagliflozin on Measures of Efficacy According to MRA Use

With respect to heart failure and renal outcomes, when compared with placebo, the effect of empagliflozin was similar regardless of the use of an MRA at baseline, Table 2 and Figure 1.

The hazard ratios for the effect of empagliflozin on the primary composite outcome of cardiovascular death or hospitalization for heart failure were 0.75 (95% CI, 0.63-0.88) and 0.76 (95% CI, 0.59-0.97) in MRA users and MRA nonusers, respectively, interaction P=0.93, Figure 1. The hazard ratios for the effect of empagliflozin on total hospitalizations for heart failure were 0.71 (95% CI, 0.56, 0.89) and 0.69 (95% CI, 0.48, 0.97), respectively, interaction P=0.88. Additionally, baseline use of an MRA did not influence the effect of empagliflozin on health status (assessed by KCCQ score), NYHA functional class and the severity of heart failure hospitalizations (i.e., hospitalizations requiring intensive care or receiving treatment with intravenous positive inotropic drugs, vasopressor agents or mechanical interventions), Table 2 and Supplemental Table 1.

The use of MRAs also did not influence the effect of empagliflozin to retard the worsening of renal function during double-blind therapy. As compared with placebo, empagliflozin slowed the rate of decline in eGFR in both MRA users and nonusers, +1.55 ± 0.39 in MRA users and +2.19 ± 0.59 in MRA nonusers, interaction P=0.36, Table 2. The use of MRAs did not influence the effect of empagliflozin on the composite of serious adverse renal outcomes (interaction P =0.18), although the number of events in MRA nonusers was small (<25 events), Table 2.

In contrast, for cardiovascular death and all-cause mortality, the hazard ratios for the effect of empagliflozin versus placebo appeared to be directionally different for MRA users and nonusers. The hazard ratio for the effect of empagliflozin was 0.82 (95% CI 0.65-1.05) in MRA users and 1.19 (95% CI 0.82-1.71) in MRA nonusers, interaction P=0.10. Similarly, for all-cause mortality, the hazard ratios for the effect of empagliflozin versus placebo were 0.84 (95% CI 0.68-1.03) in MRA users and 1.15 (0.84-1.59) in MRA nonusers, interaction P =0.098, Figure 2 and Table 2.

Effect of Empagliflozin on Utilization of MRAs Following Randomization

Among the 1069 MRA nonusers at baseline (512 on placebo and 557 on empagliflozin), those randomized to empagliflozin were 35% less likely to be initiated on treatment with a MRA during the follow-up (122 on placebo and 90 on empagliflozin), hazard ratio 0.65 (95% CI 0.49-0.85), P=0.0019. This differential post-randomization utilization was apparent shortly following initiation of double-blind treatment, Table 3 and Central Illustration.

Conversely, among the 2661 MRA users at baseline (1355 in placebo and 1306 in the empagliflozin group), those randomized to empagliflozin were less likely to discontinue or interrupt treatment with MRAs during double-blind therapy following randomization (210 in the placebo and 164 in the empagliflozin group), hazard ratio 0.78 (95% CI 0.64-0.96), P=0.018.

Influence of MRA on the Effect of Empagliflozin on Vital Signs, Biomarkers and Safety

The effects of empagliflozin on systolic blood pressure, body weight, glycated hemoglobin, uric acid, NT-pro BNP, and hematocrit were not influenced by use of MRA at baseline, Supplemental Table 2.

When MRA users and nonusers were combined, the risk of hyperkalemia was lower in the empagliflozin than in the placebo group (Table 4). The effect of empagliflozin was particularly noteworthy in preventing the occurrence of severe hyperkalemia (hazard ratio 0.70 (95% CI 0.47-1.04); however, the effect of empagliflozin on the risk of severe hyperkalemia did not differ in MRA users and non-users at baseline, interaction P=0.56. For all other assessments, the use of MRA at baseline did not influence the safety profile of empagliflozin, Supplemental Table 3.

Discussion

In large-scale randomized controlled trials in patients with type 2 diabetes, with chronic kidney disease and with heart failure and a reduced ejection fraction, SGLT2 inhibitors have had a remarkably consistent effect to reduce the risk of hospitalizations for heart failure and to slow the progression of renal disease. [START_REF] Mcmurray | Dapagliflozin in Patients with Heart Failure and Reduced Ejection Fraction[END_REF][START_REF] Packer | Cardiovascular and Renal Outcomes with Empagliflozin in Heart Failure[END_REF][START_REF] Zannad | SGLT2 inhibitors in patients with heart failure with reduced ejection fraction: a meta-analysis of the EMPEROR-Reduced and DAPA-HF trials[END_REF][START_REF] Wanner | Empagliflozin and Progression of Kidney Disease in Type 2 Diabetes[END_REF][START_REF] Fitchett | Heart failure outcomes with empagliflozin in patients with type 2 diabetes at high cardiovascular risk: results of the EMPA-REG OUTCOME(R) trial[END_REF][START_REF] Heerspink | Dapagliflozin in Patients with Chronic Kidney Disease[END_REF][START_REF] Perkovic | Canagliflozin and Renal Outcomes in Type 2 Diabetes and Nephropathy[END_REF] Most patients in the cardiovascular outcomes trials in type 2 diabetes were not treated with MRAs.

In the original publications of both the DAPA-HF and EMPEROR-Reduced trials, background use of MRAs at baseline did not influence the effect of dapagliflozin and empagliflozin on the primary composite of cardiovascular death or hospitalization for heart failure. [START_REF] Mcmurray | Dapagliflozin in Patients with Heart Failure and Reduced Ejection Fraction[END_REF][START_REF] Packer | Cardiovascular and Renal Outcomes with Empagliflozin in Heart Failure[END_REF] However, more data are needed to assess whether the use of spironolactone or eplerenone might influence the effects of SGLT2 inhibitors on other measures of efficacy or on safety. The current study demonstrates that the use of MRAs at randomization did not influence the benefits of empagliflozin to reduce the number or severity of heart failure hospitalizations and to improve health status (as assessed by the KCCQ) or functional capacity (as assessed by NYHA class). Furthermore, empagliflozin slowed the decline in glomerular filtration rate and reduced the risk of serious adverse renal outcomes similarly in patients who did or did not receive spironolactone or eplerenone, although the number of adverse renal events in MRA nonusers was small. Finally, the use of MRAs did not modify the effect of empagliflozin on physiological measures or biomarkers that reflect the actions of SGLT2 inhibitors, and MRA use did not adversely affect the safety of the drug when given to patients with heart failure and a reduced ejection fraction.

Nevertheless, empagliflozin had a meaningful influence on the utilization of MRAs following randomization in two important ways. First, patients in the empagliflozin group who had not been treated with spironolactone or eplerenone at baseline were 35% less likely to be initiated on an MRA following randomization, i.e., among nonusers of an MRA at baseline, 23.8% of placebo-treated patients but only 16.2% of empagliflozin-treated patients were initiated on therapy with an MRA during double-blind follow-up. This differential utilization may have been related to the benefits of empagliflozin on health status and NYHA functional class, which may have diminished clinical pressures (on the part of both the patient and physician) to intensify treatments with drugs known to be effective for the treatment of heart failure. An initial increase in serum creatinine (related to an effect on intrarenal hemodynamics) in empagliflozin-treated patients may have also discouraged initiation of treatment with an MRA. [START_REF] Kraus | Characterization and implications of the initial estimated glomerular filtration rate 'dip' upon sodium-glucose co-transporter-2 inhibition with empagliflozin in the EMPA-REG OUTCOME trial[END_REF] Second, patients in the empagliflozin group who had been treated with spironolactone or eplerenone were 22% less likely to discontinue treatment with these drugs following randomization, i.e., among users of an MRA at baseline, 15.5% of placebo-treated patients but only 12.6% of empagliflozin-treated patients stopped or interrupted treatment with an MRA. This differential utilization may have been related to favorable effects of empagliflozin to mitigate the occurrence of hyperkalemia, and to prevent worsening renal function during long-term treatment, since increases in serum creatinine or potassium can cause physicians to withhold treatment with MRAs in patients who are receiving these drugs. [START_REF] Masoudi | Adoption of spironolactone therapy for older patients with heart failure and left ventricular systolic dysfunction in the United States, 1998-2001[END_REF][START_REF] Ferreira | Mineralocorticoid receptor antagonist pattern of use in heart failure with reduced ejection fraction: findings from BIOSTAT-CHF[END_REF][START_REF] Cooper | Use of Mineralocorticoid Receptor Antagonists in Patients With Heart Failure and Comorbid Diabetes Mellitus or Chronic Kidney Disease[END_REF] Additionally, since hospitalizations are known to trigger changes in background therapy, it is possible that the lower rates of hospitalizations for heart failure in patients randomized to empagliflozin might have contributed to our finding of lower rates of initiation or discontinuation of MRAs in the empagliflozin group."

The potential for an effect of empagliflozin to reduce the risk of hyperkalemia is noteworthy, since a similar finding has been reported with the use of dapagliflozin in the patients with heart failure and a reduced ejection fraction who were enrolled in the DAPA-HF trial. [START_REF] Kristensen | Dapagliflozin reduces the risk of hyperkalaemia in patients with heart failure and reduced ejection fraction: a secondary analysis DAPA-HF[END_REF] In that trial, the effect of SGLT2 inhibition to reduce the risk of serum potassium >6.0 mmol/L was particularly notable in patients who were treated with an MRA at baseline, with a 50% relative reduction, an effect size that was similar to the 36% reduction in this risk seen in the EMPEROR-Reduced trial, although the latter was not statistically significant. Of note, SGLT2 inhibitors appeared to have little effect on serum potassium in patients with type 2 diabetes without heart failure, possibly because few patients in the diabetes trials were receiving MRAs. [START_REF] Yavin | Effect of the SGLT2 Inhibitor Dapagliflozin on Potassium Levels in Patients with Type 2 Diabetes Mellitus: A Pooled Analysis[END_REF] The mechanism by which SGLT2 inhibitors may selectively interfere with MRA-induced hyperkalemia is not known. Interestingly, a similar effect has been reported with the use of sacubitril/valsartan in patients with heart failure and a reduced ejection fraction. [START_REF] Desai | Reduced Risk of Hyperkalemia During Treatment of Heart Failure With Mineralocorticoid Receptor Antagonists by Use of Sacubitril/Valsartan Compared With Enalapril: A Secondary Analysis of the PARADIGM-HF Trial[END_REF] Might the effect of empagliflozin to promote differential utilization of MRAs following randomization have confounded the analyses of survival in the EMPEROR-Reduced trial?

As seen in Table 3, when compared with the empagliflozin group, 32 additional patients in the placebo group who were nonusers of an MRA at baseline were initiated on treatment with an MRA, and the differential enrichment was apparent almost immediately after the start of double-blind therapy. Since MRAs reduce mortality in heart failure with a reduced ejection fraction, [START_REF] Zannad | Eplerenone in patients with systolic heart failure and mild symptoms[END_REF][START_REF] Pitt | The effect of spironolactone on morbidity and mortality in patients with severe heart failure. Randomized Aldactone Evaluation Study Investigators[END_REF] it is likely that lives might have been saved preferentially in the placebo group, contributing to the apparent excess estimate of hazard in empagliflozin-treated patients who were MRA nonusers at randomization. Conversely, when compared with the placebo group, 46 additional patients in the empagliflozin group who were users of an MRA at randomization were maintained on treatment with an MRA, potentially contributing to the lower hazard of death observed in empagliflozin-treated patients who were MRA users at randomization. Therefore, differential post-randomization utilization of MRAs in both MRA users and nonusers may help to explain our finding of a possible qualitative interaction between baseline MRA use and the effect of empagliflozin on survival (interaction P=0.10, unadjusted for multiple comparisons). It is important to note that post-randomization differential enrichment is not methodologically accounted for in a conventional intention-totreat analysis. [START_REF] Wanis | Adjusting for adherence in randomized trials when adherence is measured as a continuous variable: An application to the Lipid Research Clinics Coronary Primary Prevention Trial[END_REF][START_REF] Berger | A review of methods for ensuring the comparability of comparison groups in randomized clinical trials[END_REF] If the differential utilization of MRAs following randomization exerted an influence on estimates of a treatment effect on mortality, it is possible that differences that occurred by chance in the use of MRAs at the time of randomization might also have influenced the estimates of the effect of empagliflozin. The EMPEROR-Reduced trial did not stratify randomization according to the use of MRAs, and by the play of chance, patients prescribed an MRA at baseline were more likely to be randomized to placebo, a difference that involved 49 patients. Conversely, in DAPA-HF, the use of MRAs was (by chance) more frequent in the patients randomized to dapagliflozin than placebo (by 22 patients). [START_REF] Mcmurray | Dapagliflozin in Patients with Heart Failure and Reduced Ejection Fraction[END_REF] Because randomization does not ensure balance of all baseline factors that can influence outcomes, imbalances in treatments that can affect outcomes can impact reported estimates of the size of treatment effects. Accordingly, it is important to adjust for prognostically relevant baseline covariates, and such adjustment (even when performed post hoc) can help to reconcile conflicting reports of the efficacy of similar treatments. [START_REF] Ferreira | Covariate adjusted reanalysis of the I-Preserve trial[END_REF] The current study should be interpreted in light of its strengths and limitations. The use of a MRA at baseline was included as one of 12 prespecified subgroups, but many of the analyses reported herein are post hoc, as is common in secondary manuscripts following the publication of the primary results of a large-scale trial. Given the exploratory nature of this report we did not perform correction for multiplicity of tests. Notwithstanding, our finding that the use of an MRA did not influence the effects of empagliflozin on most variables is consistent with similar findings in the DAPA-HF trial [START_REF] Docherty | Effects of dapagliflozin in DAPA-HF according to background heart failure therapy[END_REF] . However, it is not known if differential enrichment of MRA use following randomization was seen in the DAPA-HF trial.

Finally, the clinical impact of the differential use of MRAs at baseline and following randomization is dependent on assumptions about the magnitude of the survival advantage of MRAs and the duration of differential use, and these complexities undermine our ability to estimate the potential for post-randomization bias with a reasonable degree of precision.

In conclusion, with respect to adverse heart failure and renal outcomes, the use of an MRA did not influence the effects of empagliflozin in patients with heart failure and a reduced ejection fraction. This finding indicates that MRAs and SGLT2 inhibitors can be usefully combined to reduce morbidity and mortality and improve symptoms and health status without concerns of an adverse interaction. The observation that treatment with empagliflozin may influence physician decision-making regarding the utilization of MRAs may have important implications for the design and analysis of large-scale trials where background therapy with life-prolonging drugs is not tightly controlled and may confound estimates of a treatment effect of the study medication. In the clinical setting, a favorable symptomatic response to SGLT2 inhibition should not obviate the use of MRAs by practitioners. Furthermore, the possibility that the use of SGLT2 inhibitors may enhance the ability of patients to be maintained on treatment with an MRA provides further support for the value of combining these two classes of drugs to improve outcomes in patients with heart failure.

Clinical Perspectives

Clinical competencies

In EMPEROR-Reduced, the use of MRAs did not modify the effect of empagliflozin on heart failure and renal outcomes, and the concomitant use of empagliflozin with MRAs was welltolerated. Patients randomized to empagliflozin had less frequent MRA discontinuation than patients randomized to placebo.

Translational outlook

Empagliflozin may enable the use of MRA. The mechanisms by which this benefit may occur requires further study. Cardiac resynchronization therapy ‡ 149 (13.9) 293 (11.0) 0.012 Data are n (%) and mean ± SD unless otherwise stated. † Includes all the patients with an implantable cardioverter-defibrillator regardless of the presence or absence of cardiac resynchronization therapy. ‡ Includes all the patients who were receiving cardiac resynchronization therapy regardless of the presence or absence of a defibrillator. ACE=angiotensin-converting-enzyme; ARB=angiotensin-receptor-blocker; CABG=coronary artery bypass graft; eGFR=estimated glomerular filtration rate; HbA1c= glycated hemoglobin A1c; KCCQ=Kansas City Cardiomyopathy Questionnaire; NT-pro BNP=N-terminal pro hormone B-type natriuretic peptide; NYHA=New York Heart Association; PCI=percutaneous coronary intervention. P values and 95% confidence intervals presented in this report have not been adjusted for multiplicity, and therefore inferences drawn from these statistics may not be reproducible. Rate is expressed in events per 100 person-years. * Analysis performed in patients with potassium <5.5 mmol/L at baseline only; ** Analysis performed in patients with potassium <6.0 mmol/L at baseline only. Shown are adverse events up to 7 days and serum potassium levels up to 3 days following discontinuation of study medication. P values and 95% confidence intervals presented in this report have not been adjusted for multiplicity, and therefore inferences drawn from these statistics may not be reproducible.

Central Illustration. Effect of Empagliflozin on the Use of Mineralocorticoid Receptor Antagonists Following Randomization

Caption: Among MRA nonusers, the empagliflozin group was 35% less likely than the placebo group to initiate treatment with an MRA following randomization. Among MRA users, the empagliflozin group were 22% less likely than the placebo group to discontinue treatment with an MRA following

Figure 1 .

 1 Figure 1. Effect of Empagliflozin on Major Heart Failure Endpoints, According to Use of Mineralocorticoid-Receptor Antagonist at Baseline

Figure 2 .

 2 Figure 2. Effect of Empagliflozin on Cardiovascular Death and All-Cause Mortality, According to Use of Mineralocorticoid-Receptor Antagonist at Baseline

  randomization. Left panel (a): time to initiation of MRA treatment in patients not using an MRA at baseline. Right panel (b): time to discontinuation or temporary interruption of MRA in patients using MRA at baseline. HR= hazard ratio; CI=confidence interval; MRA=mineralocorticoid receptor antagonist *Model includes age, baseline estimated glomerular filtration rate, region, baseline diabetes status, sex, baseline left ventricular ejection fraction, treatment, baseline use of MRA and treatment by baseline use of MRA interaction.

Table 1 . Clinical Characteristics, According to the Use of a Mineralocorticoid Receptor Antagonist at Baseline

 1

	No MRA use at	MRA use at
	randomization (n=1069)	randomization (n=2661)	P value

Table 2 . Effect of Empagliflozin on Prespecified Measures of Efficacy, According to the Use of Mineralocorticoid Receptor Antagonists at Baseline Mineralocorticoid receptor antagonist nonusers (n=1069)

 2 Patients who died prior 52 weeks are considered as having not improved or deteriorated. Missing scores are imputed for surviving patients. Ceiling effects were managed as follows: if a patient had a baseline value  5 points, they were defined as having a 5-point deterioration if they had a value 5 points at 52 weeks; conversely, if a patient had a baseline value 95 points, they were defined as having a 5 point improvement if they had a value 95 points at 52 weeks. ‡ Logistic regression analysis adjusted for age, baseline eGFR, treatment arm, region, baseline diabetes status, sex, baseline left ventricular ejection fraction, baseline NYHA class, baseline use of MRA and baseline use of MRA by treatment interaction. No improvement/ deterioration was imputed for patients who died or had missing data due to being lost to follow-up or withdrawal of consent at 52 weeks.The proportion of responders were compared between treatment groups using a logistic regression model including for age, baseline eGFR, baseline KCCQ value. region, baseline diabetes status, sex, baseline LVEF, treatment arm, baseline use of MRA and treatment arm by baseline use of MRA interaction. Multiple imputations were used to impute missing data. The observed number and proportion of responders, odds ratio between treatment groups, 95%

					Mineralocorticoid receptor
		Placebo (n=512)	Empaglifloz in (n=557)	Treatment effect	antagonist users (n=2661) Placebo (n=1355) Empaglifloz Treatment in (n=1306) effect	Interactio n P value
		n/N (%)	n/N (%)	-	n/N (%)	n/N (%)	-	-
	Cardiovascular
	death or heart failure	132 (25.8)	118 (21.2)	0.76 (0.59, 0.97)	330 (24.4)	243 (18.6)	0.75 (0.63, 0.88)	0.93
	hospitalization
	Total number
	of heart failure hospitalization	165	126	0.69 (0.48, 0.97)	388	262	0.71 (0.56, 0.89)	0.88
	s
	Slope of
	decline in eGFR (ml/min/1.73m	-2.74 ± 0.43	-0.55 ± 0.40	2.19 (1.04, 3.35)	-2.09 ± 0.27	-0.55 ± 0.28 1.55 (0.79, 2.30)	0.36
	2 /yr)
	Composite kidney endpoint	12 (2.3)	11 (2.0)	0.80 (0.35, 1.82)	46 (3.4)	19 (1.5)	0.41 (0.24, 0.71)	0.18
	Time to first
	hospitalization for heart	106 (20.7)	82 (14.7)	0.65 (0.49, 0.87)	236 (17.4)	164 (12.6)	0.71 (0.58, 0.87)	0.66
	failure
	Cardiovascular death	51 (10.0)	67 (12.0)	1.19 (0.82, 1.71)	151 (11.1)	120 (9.2)	0.82 (0.65, 1.05)	0.10
	All-cause mortality	68 (13.3)	87 (15.6)	1.15 (0.84, 1.59)	198 (14.6)	162 (12.4)	0.84 (0.68, 1.03)	0.098
	Change in KCCQ clinical summary score at 52 weeks
	Increase >5 ¥	227 (44.3)	292 (52.5)	1.39 (1.06, 1.83)	650 (48.0)	665 (50.9)	1.16 (0.97, 1.37)	0.26
	Decrease >5 ¥ 175 (34.2)	159 (28.6)	0.77 (0.58, 1.03)	432 (31.9)	385 (29.5)	0.87 (0.73, 1.05)	0.48
	Change in NYHA functional class at 52 weeks
	Odds ratio for improvement ‡	81/425 (19.1)	109/479 (22.8)	1.25 (0.88, 1.78)	222/1110 (20.0)	251/1057 (23.7)	1.32 (1.06, 1.66)	0.81
	Odds ratio for worsening ‡	71/425 (16.7)	78/479 (16.3)	0.97 (0.68, 1.38)	208/1110 (18.7)	164/1057 (15.5)	0.78 (0.62, 0.98)	0.31
	Treatment effect is displayed as hazard ratio and 95% confidence intervals for all variables, except for KCCQ and
	NYHA class (which are displayed as odds ratios and 95% confidence intervals and the slope of eGFR (which is
	displayed as absolute difference ± standard deviation.

¥ confidence interval and 2-sided p-value estimated from each imputed dataset were combined using Rubin's rule, and the combined results are presented.

Table 3 . Post-randomization Initiation and Discontinuation of Mineralocorticoid Receptor Antagonists in the Placebo and Empagliflozin Groups

 3

		Placebo	Empagliflozin
		n/N (%)	Rate per 100 patient-	n/N (%)	Rate per 100 patient-	Hazard ratio (95%CI)
			years		years
	Initiation of MRA in patients not receiving an MRA at baseline	122/512 (23.8)	21.2	90/557 (16.2)	13.2	0.65 (0.49, 0.85) P=0.0019
	Discontinuation or interruption of MRA in patients receiving an MRA at baseline	210/1355 (15.5)	13.2	164/1306 (12.6)	10.7	0.78 (0.64, 0.96) P=0.018
	CI = confidence interval; MRA = mineralocorticoid receptor antagonist.

Table 4 . Effect of empagliflozin on the occurrence of hyperkalemia All patients No use of mineralocorticoid receptor antagonist at baseline Use of mineralocorticoid receptor antagonist at baseline

 4

		Placebo	Empagliflozin		Placebo	Empagliflozi n		Placebo	Empagliflozin
		n/N (%)	Rat e	n/N (%)	Rat e	HR (95%CI)	n/N (%)	Rate	n/N (%)	Rate	HR (95%CI)	n/N (%) Rate n/N (%)	Rat e	HR (95%CI)	Interactio n P value
	Serum potassium >5.5 mmol/L*	179/175 0 (10.2)	10.3	164/1752 (9.4)	9.2	0.89 (0.72, 1.09)	53/ 484 (11.0)	11.0	47/ 527 (8.9)	8.4	0.79 (0.53, 1.17)	126/1266 (10.0)	10.0	117/ (9.6) 1225	9.5	0.93 1.20) (0.72,	0.48
	Serum potassium >6.0 mmol/L* *	57/1824 (3.1)	3.0	42/1811 (2.3)	2.2	0.70 (0.47, 1.04)	18/501 (3.6)	3.4	17/54 3 (3.1)	2.8	0.81 (0.42, 1.58)	39/1323 (2.9)	2.8	25/1268 (2.0)	1.9	0.64 1.05) (0.38,	0.56
	Hyperkalemia as adverse event reported by investigator	127/186 3 (6.8)	5.9	109/1863 (5.9)	5.0	0.82 (0.64, 1.06)	32/512 (6.3)	5.3	31/55 7 (5.6)	4.6	0.87 (0.53, 1.42)	95/1351 (7.0)	6.2	78/1306 (6.0)	5.2	0.81 1.09) (0.60,	0.82

Acknowledgments

Graphical assistance was provided by Matthew Smith of Elevate Scientific Solution and supported financially by Boehringer Ingelheim.

Funding

The EMPEROR-Reduced trial was funded by Boehringer Ingelheim and Eli Lilly (EMPEROR-Reduced ClinicalTrials.gov number, NCT03057977).

Disclosures

FZ reports personal fees from Boehringer Ingelheim during the conduct of the study; personal fees from Janssen, Novartis, Boston Scientific, Amgen, CVRx, AstraZeneca, Vifor Fresenius, Cardior, Cereno pharmacuetical, Applied Therapeutics, Merck, Bayer, and Cellprothera

Data sharing

Data will be made available upon request in adherence with transparency conventions in medical research and through requests to the corresponding author. The executive committee of EMPEROR has developed a comprehensive analysis plan and numerous prespecified analyses, which will be presented in future scientific meetings and publications. At a later time-point, the full database will be made available in adherence with the transparency policy of the sponsor (available at https://trials.boehringeringelheim.com/transparency_policy.html).


Gabriel Antoniu
email: gabriel.antoniu@inria.fr.

Loïc Cudennec

Majd Ghareeb

Osamu Tatebe

Building Hierarchical Grid Storage Using the GFARM Global File System and the JUXMEM Grid Data-Sharing Service ⋆

As more and more large-scale applications need to generate and process very large volumes of data, the need for adequate storage facilities is growing. It becomes crucial to efficiently and reliably store and retrieve large sets of data that may be shared at the global scale. Based on previous systems for global data sharing (global file systems, grid data-sharing services), this paper proposes a hierarchical approach for grid storage, which combines the access efficiency of RAM storage with the scalability and persistence of the global file system approach. Our proposal has been validated through a prototype that couples the GFARM file system with the JUXMEM data-sharing service. Experiments on the Grid'5000 testbed confirm the advantages of our approach.

Introduction

An increasing number of applications in various fields (such as genetics, nuclear physics, health, environment, cosmology, etc.) are nowadays exploiting large-scale, distributed computing infrastructures for simulation or information processing. This leads to the generation of very large volumes of data. The need to store, manage and process these data in a proper way leads to several important requirements. First, a large storage capacity is needed. Second, as these large volumes of data may be produced by (or used as an input for) long and costly computations, data persistence is essential. To address these requirements, file-based secondary storage has usually been favored in most grid storage systems. On the other hand, data need to be efficiently accessed in a distributed way at a large scale. As the cost of disk read/write operations may significantly limit the performance of data accesses, the use of faster-access RAM storage appears as a promising approach. The concept of grid-data sharing service [START_REF] Antoniu | JuxMem: An adaptive supportive platform for data sharing on the grid[END_REF][START_REF] Antoniu | How to bring together fault tolerance and data consistency to enable grid data sharing[END_REF] explores this idea by providing the abstraction of a globally shared memory space, built by aggregating the RAM storage made available by thousands of grid nodes. However, the overall storage capacity is limited to the aggregated RAM storage available.

This paper proposes an architecture for large-scale, distributed grid storage whose goal is to leverage at the same time the efficiency of RAM accesses and the largercapacity, persistent disk storage available on a grid. Our architecture implements a gridscale memory hierarchy by interconnecting a grid data-sharing service (acting as a gridscale RAM) and a grid file system.

The remaining of the paper is organized as follows. Section 2 discusses related work. Section 3 briefly describes the two systems on which we rely: the JUXMEM grid data-sharing service and the GFARM grid file system; then, it introduces our hybrid architecture. Section 4 gives details on the interaction between the two systems. An experimental evaluation of our approach is presented in Section 5. Section 6 concludes the paper and discusses future research directions.

Related Work

One of the major goals of grid infrastructures is to transparently provide access to computational and storage resources, by hiding the details about which resources are used and where they are located, as much as possible. Regarding data storage and management, this goal is still far from being achieved, as most current grid data management systems require explicit data transfers before and after the computations. GridFTP [START_REF] Allcock | Data management and transfer in high-performance computational grid environments[END_REF], Chirp [START_REF]Chirp protocol specification[END_REF] are typical examples of two file transfer tools adapted to grid infrastructures, providing for instance support for parallel streams, authentication, checkpoint/restart in case of failures, etc. Based on such tools, catalogue-based data localization and management services have been built, such as RLS [START_REF] Dunno | DataGrid prototype 1. EU-DataGrid collaboration[END_REF], Reptor [START_REF] Kunszt | File-based replica management[END_REF], Optor [START_REF] Kunszt | File-based replica management[END_REF], LDR [START_REF]Lightweight data replicator[END_REF]. Such catalogues allow the user to manually register and characterize data copies, but do not provide any support for transparent access, nor for automatic consistency maintenance.

In contrast, the concept of grid file system provides a familiar, file-oriented API allowing to transparently access physically distributed data through globally unique, logical file paths. The applications simply open and access such files as if they were stored on a local file system. A very large distributed storage space is thus made available to existing applications that usually use file storage, with no need for modifications. This approach has been taken by a few projects like GFARM [START_REF] Tatebe | Grid datafarm architecture for petascale data intensive computing[END_REF], GridNFS [START_REF] Honeyman | GridNFS: global storage for global collaborations[END_REF], LegionFS [START_REF] White | LegionFS: a secure and scalable file system supporting cross-domain high-performance applications[END_REF], etc.

The transparent data access model is equally defended by the concept of grid datasharing service, illustrated by the JUXMEM platform (described in detail in the next section). This service provides the grid applications with the abstraction of a globally shared memory in which data can be easily stored and accessed through global identifiers. Compared to the grid file system approach, this approach improves access efficiency by totally relying on RAM storage. Besides the fact that a RAM access is more efficient than a disk access, the system can leverage locality-optimization schemes developed within the Distributed Shared Memory (DSM) consistency protocols that serve as a basis for the system's design. However, the system's storage capacity is limited by the overall RAM available on the infrastructure.

Combining RAM and disk storage to achieve scalability, persistence and efficiency

As previously shown, grid file systems provide a convenient way to persistently store very large volumes of data into distributed files, whereas memory-based grid datasharing services provide a more efficient data access. To address all these issues at the same time, we propose a hierarchical storage system that combines a data sharing service with a grid file system.

The JUXMEM data sharing service

Providing a transparent data access model in an efficient way has been one of the major motivations of research efforts on distributed shared memory (DSM) systems [START_REF] Protic | Distributed shared memory: concepts and systems[END_REF]. However, the efficiency of traditional DSM consistency protocols has not proved scalable. As the grids brought forward new hypotheses (a larger scale, a dynamic infrastructure with increased failure probability), a new approach to transparent memorybased data sharing was needed. To address this challenge, the concept of grid data sharing service has been proposed [START_REF] Antoniu | JuxMem: An adaptive supportive platform for data sharing on the grid[END_REF]. The idea is to rely on results from several areas: location-transparent access and consistency protocols of DSM systems; algorithms for fault-tolerant distributed systems; scalability and techniques to support volatility in peer-to-peer (P2P) systems.

From the user's point of view, JUXMEM provides an API inspired by DSM systems allowing to perform memory allocation, data access through global IDs (e.g. pointers), and lock-based synchronization. Users can dynamically allocate shared memory using the juxmem_malloc primitive, which returns a global data ID. This ID can be used by other nodes in order to access existing data, through the use of the juxmem_mmap function. It is the responsibility of the implementation of the grid data-sharing service to localize the data and perform the necessary data transfers based on this ID. This is how a grid data-sharing service provides a transparent access to data. Both juxmem_malloc and juxmem_mmap primitives provide a local pointer that can be used to directly access the data. Note that, at the implementation level, the juxmem_mmap primitive does not rely on the mmap call. It will provide the calling client with a full copy of the data, whatever the grain of the subsequent accesses made by that client.

JUXMEM's architecture mirrors a grid consisting of a federation of distributed clusters and it is therefore expressed in terms of hierarchical groups of nodes. In order to cope with possible failures that may threaten data persistence, JUXMEM includes automatic replication mechanisms. When allocating a memory block, the client may specify: 1) on how many clusters the data should be replicated; 2) on how many providers in each cluster the data should be replicated; 3) the consistency protocol that should be used to manage the access to this data. As a result of the allocation procedure, a set of distributed replicas are created, called data group. This group has a fault-tolerant, self-organizing behavior: failures of its members are automatically detected, and failed nodes are transparently replaced, in order to maintain the replication degrees specified by the user. To do this transparently, while guaranteeing the correctness of the data accesses that may take place during the failures, we rely on fault-tolerant algorithms for group membership and atomic multicast. To favor scalability and take into account the latency hierarchy previously discussed, hierarchical adaptations of these algorithms are implemented (see [START_REF] Antoniu | How to bring together fault tolerance and data consistency to enable grid data sharing[END_REF] for details).

From an implementation point of view, the fault-tolerant algorithms used are leaderbased.

The main consistency model provided by JUXMEM is entry consistency, first introduced in [START_REF] Bershad | The Midway distributed shared memory system[END_REF]. In this model, which we consider as well-adapted to grid data sharing, processes that need to access data need to properly synchronize by acquiring a lock associated to that data. This is done by calling juxmem_acquire_read (prior to a read access) or juxmem_acquire (prior to a write access). Note that juxmem_acquire_read allows multiple readers to simultaneously access the same data. The juxmem_release primitive must be called after the access, to release the lock. These synchronization primitives allow the implementation to provide consistency guarantees according to the consistency protocol specified by the user at the allocation time.

The GFARM distributed file system

GFARM FS is a distributed file system federating local file systems. Typically, it federates local file systems on compute nodes in several clusters. A node that provides a local file system is called a file system node.

Physically, files can be replicated and stored on any file system node, but can be accessed transparently by a client. File replicas can be created by a GFARM replicate command from a file system node to another file system node. Consistency among file replicas is maintained by a close-to-open consistency like AFS [START_REF] Kazar | Synchronization and caching issues in the andrew file system[END_REF] when a file is updated. Every other out-of-date file replicas except an up-to-date replica will be deleted when the updated file is closed.

GFARM FS is designed to achieve scalable file I/O performance in a distributed environment by putting priority to a local file system, and file-affinity scheduling. When a client is also a file system node, the local file system is a part of a GFARM FS. The data transfer from file system nodes to a client can be reduced by exploiting local file access. When a new file is created, the file is created on the local file system if there is enough disk space. When one of file replicas is stored on a local file system, the local file replica is chosen to be accessed. A file-affinity scheduling is a scheduling policy of a process allocation such that a file system node that has a specified file replica has priority to be scheduled. This increases a chance of a local file access.

GFARM FS consists of a file system metadata server and multiple file system nodes. A file system daemon called gfsd runs on every file system node. Client nodes access a GFARM FS by mounting it. It is developed in open source at http://sourceforge.net/ projects/gfarm/.

Our proposal: a hybrid grid memory hierarchy

In order to take advantage of the more persistent, larger-capacity storage provided by the GFARM grid file system, while keeping data access efficient (i.e. not impacted by disk access delays), our approach consists in using GFARM as secondary storage for the JUXMEM data sharing service. The main idea is to allow applications to use JUXMEM's more efficient memory-oriented API, while letting JUXMEM to persistently store data on disk files by making calls to GFARM in the background. These calls are internally issued by JUXMEM, so they are totally transparent to the user, as illustrated on Figure 1. Besides, as explained in Section 4, their cost is also generally transparent, as the disk accesses are performed asynchronously in most cases, in order to avoid them to impact the efficiency of the application's data accesses. With respect to persistence, as seen in the previous section, JUXMEM already enhances data availability in the presence of failures by using data replication strategies. However, every piece of data is stored in physical memory, making this system prone to hard failures of nodes. JUXMEM's consensus algorithm used to implement atomic multicast and group membership operations, supports multiple simultaneous failures within each any group of replicas, as long as a majority of nodes remain correct in the group. However, in a grid, failures of whole clusters may happen (e.g. due to air conditioning problems). This leads to the loss of all data stored exclusively in RAM, if all replicas are stored within the failing cluster. In the context of heavy, long-duration scientific applications, it may be costly (if ever possible) to regenerate the data, e.g. by restarting the computation that produced it. A more efficient approach to ensure data persistence is to use secondary, disk-based storage. Another scenario where disk storage is more appropriate than RAM storage corresponds to situation where data is read and processed a long time after it has been produced (e.g. after several weeks).

Finally, thanks to the use of GFARM, the storage space made available to applications that use JUXMEM's memory-oriented API is significantly increased: JUXMEM can act as a shared cache for actively accessed data, while GFARM ensures a large capacity for long term storage.

Implementing the JUXMEM-GFARM interaction

In the approach presented here, we have chosen to exhibit the memory-oriented API provided by the JUXMEM data-sharing to the grid applications: the user can dynamically allocate memory in the grid storage space, map it to its own address space and access it through local pointers. On the other side, we internally use GFARM's global IDs (i.e. globally shared file names) to persistently store JUXMEM's shared data on physical files, but these IDs, as well as the usage of this file-oriented API are hidden from the user.

We therefore need to define which JUXMEM entities should interact with GFARM, when JUXMEM should flush data to GFARM and when this data should be restored to JUXMEM. In the first version of our hybrid architecture, we decide that, for each data, a single JUXMEM provider should interact with GFARM, as a client: the data group leader. Note that JUXMEM and GFARM storage systems may share (or not!) the same physical nodes for data storage. Setting up physically distinct topologies may however be justified by the different requirements of JUXMEM (size of physical memory) and GFARM (available disk space), but also by the need to enhance fault tolerance. For the sake of simplicity, we have chosen to flush data to GFARM whenever a client updates the corresponding JUXMEM memory providers at the end of a critical section during which the data is modified. According to the entry consistency protocol currently implemented within JUXMEM, this happens each time a client releases the lock associated to the data. Using this particular moment enforces the atomicity of the write operation in GFARM: no JUXMEM client can access the data until both shared memory and file system versions are synchronized. (This basic setting can further be refined by tuning the flush frequency as a more complex function of the data modification frequency.)

Flushing data from JUXMEM to GFARM

Figure 2 describes the interactions needed between the main entities involved in a flush operation, for a basic scenario where the user performs an acquire-write-release sequence. Once the user acquires the lock, it gets the data from the corresponding data group; it can then modify the data in its local memory, as exclusive access is guaranteed. When the client releases the lock, the modified data is sent to the data group. All members of the data group update their copy and an acknowledgment is sent to the client to let it continue his computation, while the data group leader flushes the data into a logical GFARM file uniquely identified by a file name identical to the corresponding JuxMem data id. To further improve fault tolerance, this file containing raw data may subsequently be replicated on several GFARM file storage nodes using the dedicated GFARM replicate command. This GFARM replication degree can be specified when allocating the memory in JUXMEM for that data.

Note that the acknowledgement to the client is sent before the completion of the flush to GFARM. This asynchronous strategy improves performance on the client side, as it does not need to wait the data to be written in GFARM before proceeding to the next computation. However, in order to maintain data consistency, the associated lock can not be re-assigned until the data has been written in both systems. Any node which subsequently tries to acquire the associated lock in exclusive mode will be blocked until the data flush to GFARM is complete. However, read operations can proceed in parallel with the flush operation. The benefits of our asynchronous scheme are therefore real for write-once data, or if the frequency of writes to the same data does not fall under a threshold where clients have to wait for JUXMEM and GFARM updates. This threshold depends on the data size and on network communication performance.

Data flushing from JUXMEM to GFARM introduces an overhead discussed in section 5. If during a computation step the write frequency to some data is too high, the time needed to flush into GFARM may degrade the user application performances. In that case, it may be assumed that this computation step is short enough to decide a lower frequency for data flushes to GFARM (e.g. one flush every ten modifications instead of one flush after each modification). Flush frequency to GFARM should thus be seen as a finely tuned parameter that makes a trade-off between reliability and performance.

Restoring data from GFARM to JUXMEM

Restoring data consists in reading a data from the file system in order to store it in the data sharing service. This operation occurs in two cases: 1) a JUXMEM client accesses a data that is no longer in the service, because of failures or because it was produced a long time before; and 2) the user would like to restore a given version of the data within a rollback procedure. As for flushing, one of the simplest way to achieve the restoring operation is to use the data group leader for reading the data from the file system.

When a client requires access to some data, the JUXMEM service is queried to check if the corresponding data group is still present in memory. If not, the corresponding file (if any) is searched for in the GFARM file system, using the data ID specified by the client. If the data is found in GFARM, a corresponding JUXMEM data group is created, then the data is sent to the client.

Feasibility study: evaluation

We have designed and implemented a prototype in which the JUXMEM data-sharing service uses GFARM according to the interaction scheme explained in Section 4. We use our prototype to run a synthetic application simulating producer-consumer access patterns. Each piece of data is written once and can be accessed independently. This scenario is inspired by real applications, such as climate (e.g. ocean-atmosphere) modeling applications based on code-coupling [START_REF] Valcke | OASIS 3 user's guide[END_REF]. We measure the average time to write (respectively read) a piece of data by performing 20 successive accesses. The goal of this preliminary evaluation is to show that thanks to our combined approach, the JUXMEM service is enhanced with persistence guarantees provided by GFARM, whereas the access cost remains the same (i.e. not impacted by disk access delays) in most cases.

Evaluations are performed using the Grid'5000 [START_REF]The Grid'5000 project[END_REF] testbed. We use 7 nodes of a Grid'5000 cluster made of Intel Xeon 5148 LV CPUs running at 2.3 GHz, outfitted with 4 GB of RAM each, SATA hard drives (57 MB/s peak throughput) and interconnected by a Gigabit Ethernet network. The theoretical maximum network bandwidth is thus 125 MB/s; however, if we consider the IP and TCP header overhead, this maximum becomes slightly lower: 117.5 MB/s when MTU = 1500 B. GFARM runs on 3 nodes (a metadata server, a cache metadata server, a file system node) and JUXMEM uses 4 nodes (a producer, a consumer, a manager and a memory provider also acting as a GFARM client).

In order to have a reference for our evaluation, we first ran the scenario described above using GFARM only, without JUXMEM. Read and write access times for GFARM for different data sizes, are provided on Figures 3(a) and 3(b). The average read throughput is 69 MB/s and the average write throughput is 42 MB/s. Note that GFARM's read throughput is higher than the peak hard drive throughput (57 MB/s), which indicates that GFARM benefits here from some cache effects. These results are given as reference values for comparison with the access times provided by our hierarchical grid storage system. To evaluate the cost of writing data using our hierarchical grid storage, we consider two scenarios: a common-case scenario and a worst-case scenario.

In the first scenario, we consider that write accesses to a same piece of data are infrequent. In such a case, thanks to our asynchronous scheme, the data flush to GFARM after a write session is complete before the next write to that data into JUXMEM. This is the case of all write accesses for our producer-consumer scenario, because all data are written only once. Figure 3(b) indicates a write throughput of 89 MB/s for our hierarchical storage system, equal to the pure JUXMEM throughput. This represents an improvement by 112% compared to the pure GFARM throughput (42 MB/s), essentially due to the relative costs of memory accesses compared to disk accesses. Although the data is written to GFARM in both schemes, the improvement is possible thanks to the fact that, once the data is written into JUXMEM, the producer can continue its computation while the data is flushed into GFARM. Note that the JUXMEM write cost includes the cost of synchronization (acquire/release).

In a second scenario, we consider that write accesses to a same piece of data are frequent. In this case, each write (but the first one) has to wait for the previous one to be flushed into GFARM. Basically, the access time is therefore the time to write into JUXMEM plus the time to write into GFARM. This leads to a 26 MB/s write throughput, which is slightly (7%) lower than the theoretical throughput (28.5 MB) that can be estimated based on the separate write throughputs of JUXMEM and GFARM. In this case, as explained at the end of Section 4.1, it may be reasonable to reduce the flush frequency, i.e. to allow several write sessions to JUXMEM to proceed before flushing data to GFARM.

We equally analyze the cost of reading accesses for our hierarchical grid storage using two scenarios. We first consider that the data is present in JUXMEM (i.e. in physical memory). Figure 3(a) shows the time to read a data in such a configuration. The read throughput for our hierarchical prototype is again equal to JUXMEM's read throughput and reaches 100 MB/s in this case, i.e. an improvement by 45% compared to an optimized GFARM throughput of 69 MB/s (which already benefits from some cache effects, as explained above). For reference, we remind the reader that, theoretically, the maximum network throughput is 117 MB/s. In this configuration, we can claim that our hierarchical grid storage provides GFARM's persistence guarantees, while the user only "pays" the cost of JUXMEM accesses (which includes, as previously, the synchronization cost, and does not benefit of any specific optimization).

We also consider a reading scenario where the data is not hosted by any JUXMEM provider in physical memory, but remains available in the GFARM file system. The time to read includes the time to retrieve the data from GFARM, to store the data in JUXMEM and to send it to the client. In this configuration, the throughput reaches 39 MB/s. This is a worst-case scenario that only occurs when the data is not present in JUXMEM. Note that this cost is incurred only once, as subsequent read accesses to the same piece of data benefit from JUXMEM's access cost (100 MB/s).

Conclusion and future work

While grid file systems provide an elegant solution for persistent storage of large volumes of data on physically distributed files, the concept of grid data-sharing service offers efficient access to globally shared data by relying on RAM storage. We propose a hierarchical grid storage system that simultaneously addresses these issues, based on the JUXMEM grid-data sharing service and on the GFARM grid file system. The main idea is to allow applications to use JUXMEM's efficient memory-oriented API, while letting JUXMEM persistently and transparently store data on GFARM disk files.

Our experiments performed on the Grid'5000 testbed confirm the advantages of our approach: in most cases the data access cost is not impacted by the cost of disk accesses (100 MB/s read throughput, 89 MB/s write throughput). At the same time, thanks to GFARM, the storage space made available to applications that use JUXMEM's memoryoriented API is significantly increased: JUXMEM can act as a shared cache for actively accessed data, while GFARM ensures a large capacity for long term, persistent storage.

As a future work, we plan to extend our experiments to more complex configurations where replication is used to improve fault tolerance. We plan to evaluate the induced overhead, and to address this issue using parallel data accesses between JUXMEM providers and GFARM storage nodes. We equally intend to develop and compare multiple cache strategies allowing JUXMEM to efficiently act as a cache for actively used data, while GFARM would serve for long-term storage.

Fig. 1 .

 1 Fig. 1. The JUXMEM-GFARM architecture.

Fig. 2 .

 2 Fig. 2. Sequence diagram of basic JUXMEM-GFARM interactions.

Fig. 3 .

 3 Fig. 3. Cost of read (a) and write (b) operations in JUXMEM and GFARM.

Acknowledgment This work has been supported by the Sakura programme for bilateral Japan-France collaborations, by the AIST (Tsukuba, Japan), by the French National Agency for Research project LEGO (ANR-05-CIGC-11) and by the NEGST France-Japan research collaboration programme. It has been also supported by a grant of Sun Microsystems and a grant from the Regional Council of Brittany, France.

The experiments presented in this paper were carried out using the Grid'5000 experimental testbed, an initiative from the French Ministry of Research through the ACI GRID incentive action, INRIA, CNRS and RENATER and other contributing partners (see http://www.grid5000.fr/).


Christophe Chesneau

Yogesh J Bagul
email: yjbagul@gmail.com

A NEW BOUNDING TECHNIQUE BASED ON INFINITE PRODUCT DECOMPOSITION

Keywords: 2010 Mathematics Subject Classification. 26A09, 26D07, 33B10 Infinite product, trigonometric and hyperbolic functions, exponential bounds

In this note, a new strategy is proposed to obtain bounds for functions having product decompositions. Applications are given for trigonometric and hyperbolic functions, thus improving some existing inequalities in the literature. Some graphics illustrate the findings.

A new result

As stated in [START_REF] Kostić | Generalized inequalities for ratio functions of trigonometric and hyperbolic functions[END_REF], infinite products are rarely used to establish analytical inequalities. The main aim of this note is to propose a new technique for obtaining bounds of some functions by the excellent use of their infinite product representations. The next theorem is the main result of the paper.

Theorem 1. Let us consider a differentiable function f (x) on (0, +∞) such that we can express f (x) as either:

• Case I: f (x) = k∈K (1 + a k x c ), or

• Case II: f (x) = k∈K (1 -a k x c ),
for x ∈ C, where C is a subset of (0, +∞) and K is a subset of Z, c is a positive real number and (a k ) k∈K is a sequence of positive real numbers. Then,

• for f (x) as defined in Case I and any x such that 2 1/c x ∈ C, the following inequality holds:

f (x) f (x) < c 2x log f (2 1/c x) ,
• for f (x) as defined in Case II and any x such that 2 1/c x ∈ C and x c sup k∈K a k < 1, the following inequality holds:

f (x) f (x) > c 2x log f (2 1/c x) .
Proof. In order to prove the inequality of the first item, let us recall a result by [6] saying that, for any y > 0,

log(1 + y) > y 1 + y/2 . (1.1)
Now, we have

f (x) f (x) = {log[f (x)]} = k∈K log (1 + a k x c ) = k∈K ca k x c-1 1 + a k x c = c 2x k∈K 2a k x c 1 + a k x c < c 2x k∈K log (1 + 2a k x c ) = c 2x k∈K log 1 + a k (2 1/c x) c = c 2x log f (2 1/c x) . (1.2)
This ends the proof of the first inequality. The proof of the inequality in the second item is similar; it is enough to use the reverse of the result of [6] which holds for any y ∈ (0, 1), that is log(1 -y) < -y/(1 -y/2). By proceeding as in (1.2), we get the desired inequality. The proof of Theorem 1 is complete.

The rest of the study concerns several applications of Theorem 1, showing that it can produce sharp bounds for diverse trigonometric and hyperbolic functions of interest.

Applications

The inequalities

sin x x < exp log π 2 (x cot x -1) , x ∈ 0, π 2 , (2.1)
and

exp 1 2 (x coth x -1) < sinh x x , x > 0 (2.2)
were established by [START_REF] Bhayo | On certain old and new trigonometric and hyperbolic inequalities[END_REF] and [START_REF] Sándor | Two sharp inequalities for trigonometric and hyperbolic functions[END_REF], respectively. See also [START_REF] Bagul | On a result of Bhayo and Sándor[END_REF] for an alternative corrected proof of (2.2). We first refine inequality (2.1) for most of the values of x ∈ (0, π/2). Moreover, the obtained upper bound is valid in a larger interval (0, π).

Proposition 1. For any x ∈ (0, π/ √ 2), we have

cot x > 1 x 1 + log sin(x √ 2) x √ 2 .
Equivalently, for any x ∈ (0, π), we have

sin x x < exp x √ 2 cot x √ 2 -1 .
Proof. The proof follows from Theorem 1 applied with f (x) = (sin x)/x. Since

sin x x = +∞ k=1 1 - x 2 π 2 k , x ∈ (0, π),
we are in the Case II with

C = (0, π), K = N/{0}, a k = 1/(π 2 k 2 ) and c = 2.
The proof ends by remarking that

f (x)/f (x) = cot x -1/x.
The refinement of (2.1) in Proposition 1 is illustrated in Figure 1 where the curve of the difference function of the upper bounds is plotted, governed by the following equation: From Figure 1, it is clear that the new upper bound is sharper than the one established by [START_REF] Bhayo | On certain old and new trigonometric and hyperbolic inequalities[END_REF], except for a small range of values included into (λ, π/2) where λ ≈ 1.545. Also, we refine inequality (2.2) in the following proposition.

d(x) = exp log π 2 (x cot x -1) -exp x √ 2 cot x √ 2 -1 , x ∈ 0, π 2
Proposition 2. For any x > 0, we have

coth x < 1 x 1 + log sinh(x √ 2) x √ 2 .
Equivalently, for any x > 0, we have

exp x √ 2 coth x √ 2 -1 < sinh x x .
Proof. The proof follows from Theorem 1 applied with f (x) = (sinh x)/x. Since

f (x) = +∞ k=1 1 + x 2 π 2 k 2 , x > 0,
we are in the Case I with C = (0, +∞), K = N/{0}, a k = 1/(π 2 k 2 ) and c = 2. The proof ends by remarking that f (x)/f (x) = coth x -1/x.

The improvement of the new lower bound is illustrated in Figure 2, where the following difference function is plotted: From Figure 2, we see that the new lower bound is sharper than the one proposed by [START_REF] Sándor | Two sharp inequalities for trigonometric and hyperbolic functions[END_REF].

j(x) = exp 1 2 (x coth x -1) -exp x √ 2 coth x √ 2 -1 , x ∈ (0, 1.5).
In [START_REF] Chesneau | A note on some new bounds for trigonometric functions using infinite products[END_REF] and [START_REF] Bagul | On exponential bounds of hyperbolic cosine[END_REF], it is respectively proved that

cos x < exp - x 2 2 , x ∈ (0, π/2) (2.3) and exp θx 2 < cosh x, x ∈ (0, α), (2.4)
where α > 0 and θ = log(cosh α)/α 2 . Sharper exponential bounds for cos x and cosh x can be given by refining (2.3) and (2.4) as follows:

Proposition 3. For any x ∈ (0, π/(2 √ 2
)), we have

tan x < - 1 x log cos(x √ 2) .
Equivalently, for any x ∈ (0, π/2), we have

cos x < exp - x √ 2 tan x √ 2 .
Proof. The proof follows from Theorem 1 applied with f (x) = cos x. Since

cos x = +∞ k=0 1 - 4x 2 π 2 (2k + 1) 2 , x ∈ (0, π/2),
we are in the Case II with C = (0, π/2), K = N, a k = 4/[π 2 (2k + 1) 2 ] and c = 2. The proof ends by remarking that f (x)/f (x) = -tan x.

The upper bound in Proposition 3 refines (2.3) since tan x > x for x ∈ (0, π/2). Proposition 4. For any x > 0, we have

tanh x < 1 x log cosh(x √ 2) .
Equivalently, for any x > 0, we have

exp x √ 2 tanh x √ 2 < cosh x.
Proof. The proof follows from Theorem 1 applied with f (x) = cosh x. Since

f (x) = +∞ k=0 1 + 4x 2 π 2 (2k + 1) 2 , x > 0,
we are in the Case I with C = (0, +∞), K = N, a k = 4/[π 2 (2k + 1) 2 ] and c = 2. The proof ends by remarking that f (x)/f (x) = tanh x.

Applications of the previous result can be of various kinds, allowing the determination of simple bounds for various integral or series bounds having no closed-form. For instance, the integral x * 0 e y tanh(y) dy with x * > 0 is complex to treat mathematically, but thanks to Proposition 4, we have

x * 0 e y tanh(y) dy ≤ x * 0 cosh(y √ 2)dy = 1 √ 2 sinh(x * √ 2).
We also established the following consequence of Theorem 1.

Proposition 5. Let us adopt the setting and notations of Theorem 1. Let f 1 (x) and f 2 (x) be functions satisfying the Case I and Case II, respectively, with the same quantities involved. Then, for any x such that 2 1/(2c) x ∈ C and x c sup k∈K a k < 1, the following inequality holds:

f 1 (x) f 1 (x) + f 2 (x) f 2 (x) > c x log f 1 (2 1/(2c) x) + log f 2 (2 1/(2c) x) .
Proof. The proof follows from Theorem 1 applied with f (x) = f 1 (x)f 2 (x). Since

f 1 (x)f 2 (x) = k∈K (1 -a k x c ) k∈K (1 + a k x c ) = k∈K 1 -a 2 k x 2c ,
we are in the Case II with the sequence (a 2 k ) k∈K and the constant 2c. The proof ends by remarking that f

(x)/f (x) = [f 1 (x)f 2 (x) + f 1 (x)f 2 (x)]/[f 1 (x)f 2 (x)] = f 1 (x)/f 1 (x) + f 2 (x)/f 2 (x).
As an example of application of Proposition 5, by considering f 1 (x) = cosh x and f 2 (x) = cos x, then we can apply this result with K = N, a k = 4/[π 2 (2k + 1) 2 ] and c = 2, for x ∈ (0, π/2 5/4 ), we get tanh x -tan x > 2 x log cosh(2 1/4 x) + log cos(2 1/4 x) .

(2.5)

This inequality is shown in Figure 3 for x ∈ (1, 1.3), illustrating its sharpness.

Figure 1 .

 1 Figure 1. Curve for the function d(x) for x ∈ (0, π/2).

Difference of comparable old and new lower bounds for sinh x / x x Figure 2 .

 x2 Figure 2. Curve for the function j(x) for x ∈ (0, 1.5).

Figure 3 .

 3 Figure 3. Illustration of the lower bound of tanh x -tan x as formulated in (2.5).