Andrew Atkinson
email: andrew.atkinson@insel.ch

Marcel Zwahlen

Diana Barger

Antonella D'

Arminio Monforte

Stephane De Wit

Jade Ghosn

Enrico Girardi

Veronica Svedhem

Philippe Morlat

Cristina Mussini

Antoni Noguera-Julian

Christoph Stephan

Giota Touloumi

Ole Kirk

Amanda Mocroft

Peter Reiss

Jose M Miro

James R Carpenter

Hansjakob Furrer

Withholding Primary Pneumocystis Pneumonia Prophylaxis in Virologically Suppressed Patients With Human Immunodeficiency Virus: An Emulation of a Pragmatic Trial in COHERE

Keywords: pneumocystis pneumonia, prophylaxis, HIV-RNA

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

and potentially increase the risk of antibacterial resistance due to prolonged usage.

The Collaboration of Observational HIV Epidemiological Research Europe (COHERE) in EuroCoord was a project-based collaboration that comprised 40 adult, pediatric, and motherchild HIV cohorts across Europe. The collaboration, which was active from 2005 to 2015, allowed for annual coordinated data collection via a centrally developed standardized operating procedure. The COHERE collaboration addressed novel research questions that could not be studied adequately in individual cohorts (http://www.cohere.org).

Previous analyses conducted on COHERE suggested that primary PcP prophylaxis can be safely withdrawn in patients with CD4 counts of 100-200 cells/µL if HIV RNA is suppressed [START_REF] Mocroft | Is it safe to discontinue primary pneumocystis jiroveci penumonia prophylaxis in patients with vorologically suppressed HIV infection and a CD4 cell count <200 cells/μL?[END_REF]. A more recent study added new findings, indicating that PcP incidence off prophylaxis was below 1 per 100 person-years (py) for virologically suppressed individuals with a CD4 count above 100 cells/µL, concluding that primary (and secondary) prophylaxis might not be needed in such cases [START_REF] Furrer | behalf of the Opportunistic Infections Project Team of the Collaboration of Observational HIV Epidemiological Research in Europe (COHERE) in EuroCoord. HIV replication is a major predictor of primary and recurrent pneumocystis pneumonia-implications for prophylaxis recommendations[END_REF]. However, it remains to be determined if PcP prophylaxis might be fully withdrawn for patients with consistently suppressed HIV viral load (VL), irrespective of CD4 count.

The current European AIDS Clinical Society (EACS) guidelines that are, at least partially, based on the results from these studies recommend the following rules for stopping primary PcP prophylaxis (page 105 of [START_REF][END_REF]): "Stop: if CD4 count >100 cells/μL and HIV-VL undetectable for over 3 months, " whereas the US National Institutes of Health (NIH) guidelines state [START_REF]Guidelines for the prevention and treatment of opportunistic infections in adults and adolescents with HIV[END_REF]: "Primary pneumocystis prophylaxis should be discontinued in adult and adolescent patients who have responded to ART with an increase in CD4 counts from <200 cells/mm 3 to >200 cells/ mm 3 for >3 months. "

The gold standard for estimating the risk of PcP would be to conduct a randomized trial. However, due to the low levels of PcP diagnoses for patients on ART, a randomized trial would be prohibitive both in terms of time and cost. Given the wealth of new data available in COHERE since initial studies focusing on PcP were carried out, the goal of our study was to investigate whether PcP prophylaxis might be withheld in all patients on ART with suppressed plasma HIV RNA (<400 copies/mL). We use the data to compare the risk of 2 PcP prophylaxis stopping strategies: (1) the existing guidelines with a CD4 count of 200 as threshold versus (2) a new strategy based solely on confirmed viral suppression. We estimate the risk of primary PcP in patients on combined antiretroviral therapy (cART) by applying the established causal inference approach in which observational data are used to emulate a hypothetical randomized trial comparing the 2 prophylaxis stopping strategies (eg, [START_REF] Danaei | Observational data for comparative effectiveness research: an emulation of randomised trials of statins and primary prevention of coronary heart disease[END_REF][START_REF] Cain | When to start treatment? A systematic approach to the comparison of dynamic regimes using observational data[END_REF][START_REF] Canigilia | Emulating a trial of joint dynamic strategies: an application to monitoring and treatment in HIV-positive individuals[END_REF][START_REF] Hernán | Using big data to emulate a target trial when a randomized trial is not available[END_REF]). This approach was pursued since the incidence of PcP for patients on cART is very low, and therefore a randomized trial would be prohibitive both in terms of time and cost. An "emulated trial" using observational data offers a viable alternative to estimate the risk of a proposed new treatment strategy.

METHODS

Hypothetical Target Trial

We emulated a pragmatic randomized controlled trial using observational data, and the natural starting point for this approach is to first define the hypothetical target trial to investigate the hypothesis. Since there is some degree of inconsistency between the current guidelines, in the interests of greater applicability for our study results we chose a least common denominator of both the EACS and NIH guidelines with just the CD4 threshold of 200 cells/μL as a criterion for stopping PcP prophylaxis.

The target trial is defined as a 2-arm, open-label study comparing the risk of 2 different strategies for taking and stopping PcP prophylaxis. Individuals with HIV are eligible to enter the hypothetical target trial if (1) they began follow-up in their cohort after 1998, (2) they started ART on or after this date (defined as any combination of 3 or more antiretrovirals of any type), (3) are 16 years or older, (4) have no history of previous PcP, and finally, (5) they are taking PcP prophylaxis in line with existing recommendations (ie, they have a CD4 count <200 cells/µL).

If eligible, patients are randomized to 1 of the 2 PcP prophylaxis strategies:

1. Strategy 1 (current guidelines): Continue taking PcP prophylaxis if CD4 count is <200 cells/µL and stop if CD4 increases from <200 cells/µL to >200 cells/µL for >3 months. Patients re-start prophylaxis if CD4 count is <200 cells/µL. 2. Strategy 2 (new): Continue taking PcP prophylaxis if HIV RNA ≥400 copies/mL and stop if the patient has confirmed viral suppression, defined as 2 consecutive HIV RNA measurements <400 copies/mL in an approximately 3-month period. (This lower limit of quantification was implemented to account for earlier follow-up visits in which detection thresholds were higher than the current 20 copies/mL.) Patients re-start when they are no longer virologically suppressed, defined as having 2 consecutive HIV RNA measurements ≥400 copies/mL.

Patients continue taking prophylaxis on the respective strategy until the above stopping conditions for their randomized arm have been met, and then they stop taking prophylaxis. They may re-start prophylaxis also according to the rules for the respective strategy. Individuals not complying with the stopping and re-starting rules of their randomized arm are considered to be deviating from the trial protocol. This process is summarized graphically in Supplementary Figure 1 noncompliance or adverse effects from treatment), die, or the administrative end of follow-up is reached (5 years).

The next section summarizes the steps taken to emulate the hypothetical target trial using observational data.

Emulated Trial

Study Population

To emulate the target trial, we included data from the 2015 merger of the COHERE database from 23 HIV cohorts for the period 2009 up to the first quarter of 2015. All patients were therefore treated when the guidelines for PcP prophylaxis were based only on the CD4 count threshold of 200 cells/μL. Information on patient characteristics (age, gender, geographical origin, and transmission category), use of ART (type of regimes and dates of start and discontinuation), CD4 cell counts and plasma HIV RNA over time and their dates of measurement, AIDS-defining conditions, and recorded dropouts and deaths was recorded. We selected patients in COHERE compliant with the same eligibility criteria as in the target trial defined in the previous section. Specifically, a patient was deemed eligible at the first visit at which the CD4 count was less than 200 cells/µL and he or she was taking PcP prophylaxis (in line with current guidelines). Baseline patient characteristics were defined as recorded at this visit (Table 1). All subsequent visits for such patients were included as the follow-up for that specific patient. It is assumed that patients continue ART treatment once started, irrespective of any intermittent periods of nonadherence.

A total of 4813 patients with 94 825 follow-up visits were eligible for the emulated trial (Table 1; Supplementary Appendix B "Data set definition" and Supplementary Figure 2). We defined the point of randomization for the emulated trial to be the first time point at which the eligibility criteria were met, defining this as "time 0" for the particular patient, and measuring the time in months from this starting point.

Randomization and Artificial Censoring

At the point of randomization, all patients are eligible for both of the stopping strategies. Therefore, we adopted the approach set out, for example, in Cain et al [START_REF] Cain | When to start treatment? A systematic approach to the comparison of dynamic regimes using observational data[END_REF], and replicate all patients, so that each patient is on both arms at the point of randomization (time 0). This cloning process means that at time 0 there are no differences between the patients assigned to the strategies. (However, this does mean that we have to compensate in the analysis for cloning the patients in this way; see "Statistical Methods".)

As in the target trial, follow-up visits from patients are included in the emulated trial until they are diagnosed with PcP or the administrative end of follow-up is reached (5 years). Visits are included for patients up to the point they drop out (for any reason) or die, after which they are censored as usual in a time-to-event analysis.

In addition, a patient can be "artificially censored" for 2 reasons: first, if they stop taking prophylaxis before meeting the defined stopping criteria for the assigned strategy, and second, if they keep taking prophylaxis when they should have stopped according to their strategy. So, for example, a patient on strategy 1 who does not stop prophylaxis when his/her CD4 count is more than 200 cells/µL is artificially censored. Analogously, a patient on the new strategy 2 is artificially censored if he/she is virologically suppressed and does not stop taking prophylaxis. As in the target trial, patients can have multiple periods of being on and off prophylaxis as long as they are compliant with their assigned strategy. A comparison of the target and emulated trials, and their differences, is presented in Supplementary Table S1 in Supplementary Appendix A.

Statistical Methods

The longitudinal dataset was expanded to have patient follow-up on a monthly basis (Supplementary Appendix B). We fitted a pooled logistic regression model to this expanded dataset to estimate the hazard ratio (HR) comparing the risk of the 2 treatment strategies. This approach provides a reasonable approximation to the Cox proportional hazards model when the risk of an event is small in any particular time window [START_REF] Efron | Logistic regression, survival analysis, and the Kaplan-Meier curve[END_REF][START_REF] Greene | Econometric analysis[END_REF].

To model the baseline hazard, we included "time" (measured in months from time 0 for each patient), along with its square and cubic terms. The model included an indicator variable for the strategy, along with an interaction term between stopping strategy and time to allow for nonproportional hazards, and the following baseline variables; gender, baseline age, geographical origin (Europe [reference], Africa, Asia, Latin America, North Africa, and Middle East), transmission mode (heterosexual [reference], men who have sex with men, injection drug user, other), baseline CD4 (and its square), cohort, baseline HIV RNA (and its square), calendar year at time 0 for this patient (to take into account changes in guidelines), indicator variable for censoring due to death or drop-out, and a variable defining the percentage of postbaseline (ie, postrandomization) follow-up time on ART. Where CD4 counts, HIV RNA measurements, and details of prophylaxis were not available for a patient in a particular month, we used the last observation carried forward (LOCF) method to impute the missing values. Due to the relatively low (<5%) number of missing records for baseline covariates, only the complete records were analyzed. Furthermore, we included inverse probability weights in the model to compensate for potential selection bias from artificial censoring. Details of the modelling approach, along with a subgroup analysis investigating "grace periods" for stopping prophylaxis, are defined in Supplementary Appendix C.

All analyses were carried out in R version 3.2.4 [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF], using the function svyglm in the package "survey" to calculate robust sandwich errors from logistic models. Throughout, we used a level of 0.05 as statistically significant.

Patient and Public Involvement

There was no patient or public involvement with regard to the design, conduct, reporting, or dissemination of the research.

Ethical Approval

Ethical approval was applied for and granted for the research from the appropriate body in the host country of the cohort contributing the data to COHERE.

RESULTS

There were 4813 patients included in the emulated trial, with 52 (1.1%) PcP diagnoses (Table 1). The median time between HIV RNA measurements was 2.8 months (interquartile range [IQR], 1.5, 3.7). The total follow-up time was 10 324 py on strategy 1 (existing prophylaxis guidelines; median, 4.3 py per patient; IQR, 1.3, 5.1) and 10 324 py on strategy 2 (based on viral suppression only; 2.9 py; IQR, 0.9, 5.1).

A crude rate comparison considering those patients still in follow-up after 60 months implied treatment strategy 2 had a lower rate of PcP diagnosis than did strategy 1 (2.1% vs 1.3%; P = .03). However, this difference was not mirrored in the unadjusted incidence rates (strategy 1: 4.2 events per 1000 py; 95% confidence interval [CI], 3.1, 5.3] vs strategy 2: 4.9 events per 1000 py; 95% CI, 3.6, 6.3; P = .4).

After fitting the pooled logistic regression model including all person-months, adjusting for baseline factors and including the inverse probability weights, the HR for the first 5 years of follow-up was .8 (95% CI, .6, 1.1; P = .2), indicating a marginal, but not statistically significant, lower risk on the stopping strategy 2 (Table 2). In the adjusted model, none of the covariates were significant at the 5% level, except for the variable defining the postbaseline ART adherence (P = .02). With this latter point in mind, we performed a further analysis limited to patients with postbaseline visits exclusively on ART, censoring patients at the first visit that they were no longer on ART. Fitting the analysis model to this smaller dataset of 4089 patients, the adjusted HR was attenuated slightly (HR, .9; 95% CI, [.6, 1.3], P = .6) (Figure 1).

Using the fitted parametric model we were able to estimate the survival probability over the course of the hypothetical trial period of 5 years (Supplementary Figure 3) and to estimate the difference in absolute risk between the 2 treatment strategies (ie, strategy 1-strategy 2) after 5 years (risk difference, 0.00; 95% CI, -.01, .01).

DISCUSSION

Comparison of the PcP prophylaxis-stopping strategies using a suitably adjusted model indicated that the risk using only confirmed and maintained plasma HIV RNA viral suppression on ART as the criterion for stopping PcP prophylaxis is the same as that for the current NIH guidelines using a CD4 count threshold of 200 cells/µL. We defined viral suppression to be at least 2 consecutive measurements over an approximately 3-month period. The newest EACS guidelines are less conservative than the prophylaxis-stopping rules we used as the comparator in our study, and therefore the study results presented here would tend to underestimate the potential benefit of a stopping strategy based solely on viral suppression.

A previous study using the COHERE data indicated that discontinuing or withholding primary prophylaxis in patients with CD4 counts above 100 cells/µL, suppressed viral load on ART, and without other immunodeficiencies is safe [START_REF] Mocroft | Is it safe to discontinue primary pneumocystis jiroveci penumonia prophylaxis in patients with vorologically suppressed HIV infection and a CD4 cell count <200 cells/μL?[END_REF]. To our knowledge, the present study involves the largest cohort of patients comparing the effects of stopping primary PCP prophylaxis in virologically suppressed patients irrespective of CD4 counts.

Our study extends results from smaller cohorts [START_REF] D'egidio | Pneumocystis jirovecci penumonia prophylaxis is not required with a CD4+ T-cell count <200 cells/mirol when viral replication is suppressed[END_REF][START_REF] Cheng | Risk of pneumocystis after early discontinuation of prophylaxis among HIV-infected patients receiving highly active antiretroviral therapy[END_REF][START_REF] Soriano | Discontinuation of secondary prophylaxis for opportunistic infections in HIVinfected patients receiving highly active antiretroviral therapy[END_REF], a randomized trial [START_REF] Chaiwarith | Discontinuation of primary and secondary prophylaxis for opportunistic infections in HIV-infected patients who had CD4+ cell count <200 cells/mm(3) but undetectable plasma HIV-1 RNA: an open-label randomized controlled trial[END_REF], and 2 reviews [START_REF] Constiniuk | Discontinuation of Pneumocystis jirovecci pneumonia prophylaxis with CD4 count <200 cells/ microL and virologic suppression: a systematic review[END_REF][START_REF] Sidhu | Discontinuing pneumocystis jirovecii pneumonia prophylaxis in HIV-infected patients with a CD4 cell count <200 cells/ mm3[END_REF]. In recent years, many physicians have stopped prescribing PcP prophylaxis in patients with suppressed viremia on ART, even with low CD4 counts [START_REF] Lin | Prescription of pneumocystis jirevecci pneumonia prophylaxis in HIV-infected patients[END_REF], and our results highlight an acceptably low risk associated with such an approach. Previous studies have used the trial emulation approach [START_REF] Danaei | Observational data for comparative effectiveness research: an emulation of randomised trials of statins and primary prevention of coronary heart disease[END_REF][START_REF] Cain | When to start treatment? A systematic approach to the comparison of dynamic regimes using observational data[END_REF][START_REF] Canigilia | Emulating a trial of joint dynamic strategies: an application to monitoring and treatment in HIV-positive individuals[END_REF], and our study highlights the generalizability of such methods. While using observational data in this way remains rather novel, the American Society of Clinical Oncology recently provided a cautious endorsement explaining "observational studies can also answer or inform questions that either have not been or cannot be answered by RCTs" ( [START_REF] Petito | Estimates of overall survival in patients with cancer receiving different treatment regimens: emulating hypothetical target trials in the Surveillance, Epidemiology, and End Results (SEER)-Medicare linked database[END_REF]; quoting from [START_REF] Visvanathan | Untapped potential of observational research to inform clinical decision making: American Society of Clinical Oncology Research statement[END_REF]). Our emulated trial aimed to mimic the design of a randomized trial as closely as possible, and thereafter to be precise and open about the limitations of the adopted approach. We make the assumption of no unmeasured confounding throughout-unfortunately, there is no definitive way of determining if this assumption is justified.

Our study has a number of other limitations. We emulate a target trial and it being unblinded has drawbacks; we cannot rule out potential behavioral changes associated with a patient knowing that he/she is on prophylaxis. The presence of undiagnosed PcP at the time the trial is started is a potential risk in both a hypothetical target and an emulated trial. In our observational data, certain physicians may be more, or less, cautious about prescribing prophylaxis, perhaps depending on unrecorded characteristics that may influence the outcome. We restricted follow-up to 5 years to mirror a realistic trial, but this means our risk analysis is accordingly limited to this time period. In terms of the general application of our results, it is important to note that, although data from 23 European cohorts were included in the analysis, 2 of the large European countries (France, United Kingdom) were potentially underrepresented in the analysis. In addition, our study does not include participants under 16 years of age, and therefore the conclusions are not generalizable to children living with HIV. Finally, and perhaps most importantly, 36% of patients (n = 1752) had a CD4 count of 100 cells/ μL or less and 16% of patients (n = 787) had a CD4 count of 50 cells/μL or less at baseline in the analysis. However, these patients contributed overproportionally to the number of PcP diagnoses with 23 of 52 (44%) and 12 of 52 (23%), respectively. Notwithstanding the results presented from this study, clinicians may require further reassurance of our findings before choosing to stop prophylaxis for these higher-risk groups.

From a methodological standpoint, we used a single imputation method (LOCF) to estimate the trajectory of the CD4 and RNA measurements over time. This has the same potential drawbacks of other single imputation methods in terms of variance estimation and potential bias. An alternative would be to multiply impute the time-varying covariates [START_REF] Keogh | Multiple imputation in Cox regression when there are time-varying effects of covariates[END_REF], and this is an area for potential further study. Furthermore, since inverse probability weighting inherently assumes patients are censored at random, a sensitivity analysis might be considered to investigate potentially noninformative censoring. In conclusion, HIV replication measured as plasma HIV RNA is a major contributor to the risk of developing primary PcP. In virologically suppressed patients on ART, irrespective of CD4 levels, the risk of PcP is marginally lower using viral suppression alone, compared with when prophylaxis is taken based on the CD4 count threshold according to current guidelines. The study suggests that primary PcP prophylaxis might be safely withheld in patients on ART with confirmed plasma viral suppression, regardless of their CD4 count.

Figure 1 .

 1 Figure 1. Hazard ratio estimates and 95% confidence intervals from the fitted unadjusted and adjusted models with and without IPW; hazard ratios <1 indicate that the new strategy using viral suppression as a criterion reduces risk compared with the existing strategy based on CD4 count. Abbreviation: IPW, inverse probability weighting. Downloaded from https://academic.oup.com/cid/article/73/2/195/5843639 by Universite de Bordeaux user on 29 September 2021

cid 2021:73 (15 July) • 197

	Downloaded from https://academic.oup.com/cid/article/73/2/195/5843639 by Universite de Bordeaux user on 29 September 2021
	in Supplementary
	Appendix A.
	Participants continue in the trial until they are diag-
	nosed with PcP (the endpoint), drop out (eg, due to protocol
	PcP Prophylaxis Stop Using HIV-RNA •

Table 1 . Baseline Characteristics of Patients in the Emulated Trial

 1

		Overall	No PcP Diagnosis	PcP Diagnosis	P
	Number of patients	4813	4761	52 (1.1%)
	Female, n (%)	1195 (24.8)	1182 (24.8)	13 (25.5)	.99
	Age, median [IQR], years	40 [35, 47]	40 [25, 47]	40 [33, 46]	.58
	Geographical orgin, n (%)				.54
	Europe	3938 (81.8)	3'895 (81.8)	43 (84.3)
	Africa	482 (10.0)	478 (10.0)	4 (7.8)
	Asia	83 (1.7)	82 (1.7)	1 (2.0)
	Latin America	236 (4.9)	235 (4.9)	1 (2.0)
	North Africa and Middle East	74 (1.5)	72 (1.5)	2 (3.9)
	HIV transmission mode, n (%)				.44
	MSM	1606 (33.4)	1586 (33.3)	20 (39.2)
	Heterosexual	1833 (38.1)	1815 (38.1)	18 (35.3)
	IDU	1155 (24.0)	1146 (24.1)	9 (17.6)
	Other	219 (4.6)	215 (4.5)	4 (7.8)
	CD4, median [IQR], cells/μL	130 [77, 169]	130 [77, 169]	120 [53, 159]	.11
	HIV RNA, median [IQR], copies/mL	1460 [107, 65 000]	1402 [102, 63 816]	46 700 [540, 227 600]	<.001
	Calendar year at start of emulated trial, median (IQR) 2003 [1999, 2008]	2003 [1999, 2008]	2002 [1998, 2006]	.18
	Follow-up on ART, median [IQR], %	84 [41, 100]	84 [41, 100]	100 [84, 100]	<.001

Table 2 . Patient Numbers at Baseline and at the End of Follow-up (60 months) Along With Crude Rate and Incidence Comparisons of Strategies and Hazard Ratio Estimates From the Pooled Logistic Model

 2 Unadjusted model has PcP diagnosis as a dependent variable and as independent variables, indicator variables for the strategy along with time, time-squared, and time-cubed. Interactions between time (and its square and cube) and the strategy were not significant at the 5% level. b Adjusted model contains the same terms as the unadjusted model, along with the baseline covariates age, age-squared, gender, mode of transmission, geographical origin, cohort, CD4, CD4-squared, log 10 HIV RNA, log 10 HIV RNA-squared, calendar year at time 0 for each patient, indicator variables for death and dropout, and the percentage postbaseline follow-up time on ART.

	Strategy 1: Existing
	Prophylaxis Guidelines	Strategy 2: New	P

a c Refer to subgroup analysis at the end of Supplementary Appendix C.

Abbreviations: ART, antiretroviral therapy; HIV, human immunodeficiency virus; IDU, injection drug user; IQR, interquartile range; MSM, men who have sex with men; PcP, pneumocystis pneumonia.

Acknowledgments. The authors thank Professor Miguel Hernan of Harvard University for numerous discussions and advice during the course of the project and preparation of the manuscript.

Notes COHERE Project Writing Committee Working Group. Anastasia Antoniadou, Stephane De Wit, Christoph Stephan, Katja Doerholt, Hansjakob Furrer, Enrico Girardi, Jade Ghosn, Ole Kirk, Jose Maria Miro Meda, Amanda Mocroft, Philippe Morlat, Cristina Mussini, Peter Reiss, Marcel Zwahlen. Opportunistic Infections Project Working Group. A. Antoniadou, A. Castagna, K. Doerholt, G. Fätkenheuer, D. Raben, R. Teira, R. Zangerle, in addition to those in the author list of the document. Steering Committee-Contributing Cohorts. Ali Judd (AALPHI), Robert Zangerle (AHIVCOS), Giota Touloumi (AMACS), Josiane Warszawski (ANRS CO1 EPF/ANRS CO11 Observatoire EPF), Laurence Meyer (ANRS CO2 SEROCO), François Dabis (ANRS CO3 Aquitaine), Murielle Mary Krause (ANRS CO4 FHDH), Jade Ghosn (ANRS CO6 PRIMO), Catherine Leport (ANRS CO8 Copilote), Linda Wittkop (ANRS CO13 HEPAVIH), Peter Reiss (ATHENA), Ferdinand Wit (ATHENA), Maria Prins (CASCADE), Heiner Bucher (CASCADE), Diana Gibb (CHIPS), Gerd Fätkenheuer (Cologne-Bonn), Julia Del Amo (CoRIS), Niels Obel (Danish HIV Cohort), Claire Thorne (ECS, NSHPC), Amanda Mocroft (EuroSIDA), Ole Kirk (EuroSIDA), Christoph Stephan (Frankfurt), Santiago Pérez-Hoyos (GEMES-Haemo), Osamah Hamouda (German ClinSurv), Barbara Bartmeyer (German ClinSurv), Nikoloz Chkhartishvili (Georgian National HIV/AIDS), Antoni Noguera-Julian (CORISPE-cat), Andrea Antinori (ICC), Antonella d' Arminio Monforte (ICONA), Norbert Brockmeyer (KOMPNET), Luis Prieto (Madrid PMTCT Cohort), Pablo Rojo Conejo (CORISPES-Madrid), Antoni Soriano-Arandes (NENEXP), Manuel Battegay (SHCS), Roger Kouyos (SHCS), Cristina Mussini (Modena Cohort), Jordi Casabona (PISCIS), Jose M. Miró (PISCIS), Antonella Castagna (San Raffaele), Deborah Konopnick (St.Pierre Cohort), Tessa Goetghebuer (St Pierre Paediatric Cohort), Anders Sönnerborg (Swedish InfCare), Carlo Torti (The Italian Master Cohort), Caroline Sabin (UK CHIC), Ramon Teira (VACH), Myriam Garrido (VACH), David Haerry (European AIDS Treatment Group). Executive Committee. Stéphane de Wit (Chair, St. Pierre University Hospital), Jose M. Miró (PISCIS), Dominique Costagliola (FHDH), Antonella d' Arminio-Monforte (ICONA), Antonella Castagna (San Raffaele), Julia del Amo (CoRIS), Amanda Mocroft (EuroSida), Dorthe Raben (Head, Copenhagen Regional Coordinating Centre), Geneviève Chêne (Head, Bordeaux Regional Coordinating Centre). Pediatric Cohort Representatives. Ali Judd, Pablo Rojo Conejo. Regional Coordinating Centres: Bordeaux RCC: Diana Barger, Christine Schwimmer, Monique Termote, Linda Wittkop; Copenhagen RCC: Casper M. Frederiksen, Dorthe Raben, Rikke Salbøl Brandt. Project Leads and Statisticians. Juan Berenguer, Julia Bohlius, Vincent Bouteloup, Heiner Bucher, Alessandro Cozzi-Lepri, François Dabis, Antonella d' Arminio Monforte, Mary-Anne Davies, Julia del Amo, Maria Dorrucci, David Dunn, Matthias Egger, Hansjakob Furrer, Marguerite Guiguet, Sophie Grabar, Ali Judd, Ole Kirk, Olivier Lambotte, Valériane Leroy, Sara Lodi, Sophie Matheron, Laurence Meyer, Jose M. Miró, Amanda Mocroft, Susana Monge, Fumiyo Nakagawa, Roger Paredes, Andrew Phillips, Massimo Puoti, Eliane Rohner, Michael Schomaker, Colette Smit, Jonathan Sterne, Rodolphe Thiebaut, Claire Thorne, Carlo Torti, Marc van der Valk, Linda Wittkop.

Supplementary Data

Supplementary materials are available at Clinical Infectious Diseases online. Consisting of data provided by the authors to benefit the reader, the posted materials are not copyedited and are the sole responsibility of the authors, so questions or comments should be addressed to the corresponding author. Author contributions. A. A., M. Z., J. R. C., and H. F. were responsible for the concept and methodology. A. A. performed data curation and analysis and prepared the first draft of the manuscript. H. F., J. R. C., and M. Z. were responsible for supervision of the work. All authors were responsible for reviewing and editing the manuscript.

Dissemination declaration. We do not intend to disseminate the results to study participants and/or patient organizations. The results from the study may filter into the appropriate international guidelines.

Data sharing. Data used for the analysis will generally not be publicly available but can be made available based on the approval by the chair of the executive committee of COHERE (Stephane De Wit; refer to author list).

Disclaimer. The funders had no role in the study design, data collection, data analysis, data interpretation, or writing of the report. The corresponding author had full access to all the data in the study and had final responsibility for the decision to submit for publication.

Financial support. The COHERE study group has received unrestricted funding from Agence Nationale de Recherches sur le SIDA et les Hépatites Virales (ANRS), France; HIV Monitoring Foundation, The Netherlands; and the Augustinus Foundation, Denmark. The research leading to these results received funding from the European Union Seventh Framework Program (FP7/2007-2013) under EuroCoord grant agreement number 260694. A list of the funders of the participating cohorts can be found at www.COHERE.org. Research reported in this publication was supported by the Swiss National Science Foundation (grant number 324730_149792, principal investigator: H. F.), specifically for H. F. and A. A.

Potential conflicts of interest. A. A. is also employed by the University Children's Hospital in Basel. D. B. has been part of an advisory board and received speaking fees from Gilead. S. D. W. received grants from Gilead Sciences, Janssen, Merck, Sharp & Dohme (MSD), and ViiV Healthcare, all paid to his institution. A. N.-J. was supported by "Subvencions per a la Intensificacio de Facultatius Especialistes" (Departament de Salut de la Generalitat de Catalunya, Programa PERIS 2016-2020) (SLT008/18/00193). C. M. has participated on advisory boards and received study grants and/or speaker honoraria from AbbVie, Gilead Sciences, ViiV Healthcare, Janssen, Angelini, BMS, and MSD. C. S. received honoraria and travel support from Gilead Sciences, MSD, and Janssen-Cilag, outside of the submitted work. E. G. received unrestricted research grants from Gilead Sciences and Mylan, a travel grant from Gilead Sciences, and honoraria from Gilead Sciences, ViiV Healthcare, Janssen, Mylan, and Angelini, outside the submitted work. O. K. has received consulting honoraria and travel support for conferences from Gilead Sciences, Janssen, MSD, and ViiV Healthcare, outside the submitted work. A. M. has received honoraria from Gilead Sciences, ViiV Healthcare, and A. Craig Eiland PC, outside of the submitted work. P. R. reports grants from Gilead Sciences, ViiV Healthcare, MSD, and Janssen, and other honoraria from Gilead Sciences, ViiV Healthcare, MSD, and Teva, outside the submitted work. J. M. M. received a personal 80:20 research grant from Institut d'Investigacions Biomèdiques August Pi i Sunyer (IDIBAPS), Barcelona, Spain, during 2017-2021, and has received consulting honoraria and/or research grants from AbbVie, Angelini, Contrafect, Cubist, Genentech, Gilead Sciences, Janssen, Medtronic, MSD, Novartis, Pfizer, and ViiV Healthcare, outside of the submitted work. J. R. C. is supported by the UK Medical Research Council grant MC_UU_12023/21 and has received a grant to his department from Astra Zeneca and consulting honoraria from Pfizer and GSK, outside of the submitted work. H. F. reports grants to the institution from ViiV Healthcare, Gilead Sciences, MSD, AbbVie, and Sandoz, outside the submitted work. P. M. reports personal fees from Gilead Sciences and MSD and nonfinancial support from Gilead Sciences, MSD, and ViiV Healthcare, outside the submitted work. All other authors report no potential conflicts. All authors have submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest. Conflicts that the editors consider relevant to the content of the manuscript have been disclosed.


Zhangjie Peng

Tianshu Li

Member, IEEE Cunhua Pan

Member, IEEE Hong Ren

Fellow, IEEE Marco Di Renzo

Analysis and Optimization for RIS-Aided Multi-Pair Communications Relying on Statistical CSI

Keywords: Reconfigurable intelligent surface (RIS), intelligent reflecting surface (IRS), statistical channel state information (CSI), multi-pair communication, genetic algorithm

In this paper, we investigate a reconfigurable intelligent surface (RIS) aided multi-pair communication system, in which multi-pair users exchange information via an RIS. We derive an approximate expression for the achievable rate by assuming that statistical channel state information (CSI) is available. A genetic algorithm (GA) to solve the rate maximization problem is proposed as well. In particular, we consider implementations of RISs with continuous phase shifts (CPSs) and discrete phase shifts (DPSs). Simulation results verify the obtained results and show that the proposed GA method has almost the same performance as the globally optimal solution. In addition, numerical results show that three quantization bits can achieve a large portion of the achievable rate for the CPSs setup.

I. INTRODUCTION

Reconfigurable intelligent surfaces (RISs) are an emerging transmission technology that is capable of configuring the wireless channel into desirable forms by appropriately optimizing the response of their individual scattering elements [1]. Due to their appealing features of low cost and low power consumption, RISs have attracted extensive research attention [START_REF] Huang | Reconfigurable intelligent surfaces for energy efficiency in wireless communication[END_REF]- [START_REF] Renzo | Smart radio environments empowered by recongurable AI meta-surfaces: An idea whose time has come[END_REF]. Some initial efforts have been devoted to the study of various RIS-aided communication systems and applications, such as physical layer security in [START_REF] Shen | Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications[END_REF], [START_REF] Yu | Robust and secure wireless communications via intelligent reflecting surfaces[END_REF], multicell networks in [START_REF] Pan | Multicell MIMO communications relying on intelligent reflecting surfaces[END_REF], full duplex systems in [START_REF] Peng | Multiuser full-duplex two-way communications via intelligent reflecting surface[END_REF], mobile edge computing in [START_REF] Bai | Latency minimization for intelligent reflecting surface aided mobile edge computing[END_REF], and wireless power transfer in [START_REF] Pan | Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF]. However, a paucity of contribution studied RIS-aided multi-pair communication systems, which is a typical communication scenario due to the rapid increase of the number of machine-type devices in future networks.

(Corresponding authors: Cunhua Pan.) Z. Peng is with the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai 200234, China, and also with the Institute of Artificial Intelligence on Education, Shanghai Normal University, Shanghai 200234, China (e-mail: pengzhangjie@shnu.edu.cn).

T. Li is with the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai 200234, China (e-mail: 1000479056@smail.shnu.edu.cn).

C. Pan and H. Ren are with the School of Electronic Engineering and Computer Science at Queen Mary University of London, London E1 4NS, U.K. (e-mail: c.pan,h.ren@qmul.ac.uk).

W. Xu is with the National Mobile Communications Research Lab, Southeast University, Nanjing 210096, China, and also with Henan Joint International Research Laboratory of Intelligent Networking and Data Analysis, Zhengzhou University, Zhengzhou, 450001 China (wxu@seu.edu.cn).

M. Di Renzo is with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 3 Rue Joliot-Curie, 91192 Gif-sur-Yvette, France. (marco.di-renzo@universite-paris-saclay.fr) In multi-pair communication systems, the direct communication links may be blocked in both indoor and outdoor scenarios. In particular, the direct signals may be blocked by trees and large buildings in outdoor scenarios. In indoor scenarios, the signals may be blocked by thick walls, especially in high-frequency millimeter-wave communication systems. To counteract these issues, we consider the deployment of an RIS in order to assist the transmission between pairs of devices. Compared with existing research works on relayassisted multi-pair communications in [START_REF] Xu | On uplink performance of multiuser massive MIMO relay network with limited RF chains[END_REF], [START_REF] Xu | Spectral and energy efficiency of multi-pair massive MIMO relay network with hybrid processing[END_REF], RISs have some appealing advantages. In particular, an RIS has low power consumption and does not need power amplifiers and signal processing units after configuration [1], [START_REF] Renzo | Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison[END_REF].

RIS 1 U A U Ai U AK 1 a g ai g aK g 1 b g bi g bK g 1 U B U Bi U BK
Against this background, we study the transmission design for an RIS-aided multi-pair communication system. Unlike most of the existing papers [START_REF] Shen | Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications[END_REF]- [START_REF] Pan | Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF] where instantaneous channel state information (CSI) is assumed to be known, we consider the availability of statistical CSI [START_REF] Guo | Outage probability analysis and minimization in intelligent reflecting surface-assisted MISO systems[END_REF] that is easier to obtain since it varies more slowly. Specifically, our contributions are as follows: 1) We derive the achievable rate of the considered system model; 2) we propose a genetic algorithm (GA) method to solve the phase shifts optimization problem, by considering the setups of continuous phase shifts (CPSs) and discrete phase shifts (DPSs); 3) we provide extensive simulation results to demonstrate the correctness of our derived results, and to show that three bits are enough to discretize the phase shifts, which provides useful engineering insights for designing RIS-aided systems.

The rest of the paper is organized as follows. In Section II, we introduce the RIS-aided multi-pair communication system model. We derive the achievable rate in Section III and optimize the phase shifts in Section IV. Numerical results are provided to demonstrate the correctness of our analysis in Section V. Conclusions are drawn in Section VI.

II. SYSTEM MODEL

We consider an RIS-aided multi-pair communication system, where K pairs of users exchange information via an RIS, as shown in Fig. 1. The RIS consists of L reflective elements that are capable of customizing the channel by optimizing their phase shifts. The phase shift matrix Θ is denoted by Θ = diag(e jθ1 , . . . , e jθ ℓ , . . . , e jθL ), where θ ℓ is the phase shift of the ℓth reflective element. We denote the ith singleantenna transmitter as U Ai and the ith single-antenna receiver as U Bi , for i = 1, ..., K.

The channel between U Ai and the RIS and the channel between the RIS and U Bi can be written as

g ai = √ α ai h ai , (1)
g bi = √ α bi h bi , (2)
where α ai and α bi denote the large-scale fading coefficients, and g i ∈ C L×1 and h i ∈ C L×1 denote the fast fading vectors. Rician fading is assumed for all channels, thus the vectors h ai and h bi can be expressed as

h ai = ε i ε i + 1 h ai + 1 ε i + 1 hai , (3)
h bi = β i β i + 1 h bi + 1 β i + 1 hbi , (4)
where ε i and β i denote the Rician factors, hai ∈ C L×1 and hbi ∈ C L×1 denote the non-line-of-sight channel vectors, whose entries are independent and identically distributed standard Gaussian random variables, i.e., CN (0, 1), and h ai ∈ C L×1 and h bi ∈ C L×1 denote the line-of-sight channel vectors. In particular, h ai and h bi can be expressed as

h ai = [1, e j2π d λ sinςi , . . . , e j2π d λ (L-1)sinςi ] T , (5)
h bi = [1, e j2π d λ sinϕi , . . . , e j2π d λ (L-1)sinϕi ] T , (6)
where ϕ i and ς i represent the ith pair of users' angle of arrival (AoA) and angle of departure (AoD), respectively. In the present paper, we consider RISs that are made of discrete elements that are spaced half of the wavelength apart [1]. Therefore, we assume d = λ 2 in the rest of the paper. We assume the availability of statistical CSI at U Ai , for i = 1, . . . , K. The statistical CSI can be readily obtained since it varies much slowly than the instantaneous CSI. The signal received at U Bi is given by

y i = g T bi Θ K j=1 √ p j g aj x j + n i = √ p i g T bi Θg ai x i Desired signal + K j=1,j =i √ p j g T bi Θg aj x j
Inter-user interference

+ n i Noise , (7)
where x j ∼ CN (0, 1) represents the signal transmitted by U Aj , p j denotes the transmit power of U Aj , and n i ∼ CN (0, σ 2 i ) is the additive white Gaussian noise (AWGN) at U Bi , for i = 1, . . . , K.

From ( 7), we observe that y i consists of three parts: the desired signal that U Bi wants to receive, the interference produced by other multi-pair users, and the noise. Furthermore, the signal-to-interference plus noise ratio (SINR) at U Bi is given by

γ i = p i α bi α ai h T bi Θh ai 2 K j=1,j =i p j α bi α aj h T bi Θh aj 2 + σ 2 i . (8)
Hence, the ergodic achievable rate for U Bi can be expressed as

R i = E{log 2 (1 + γ i )}, (9)
and the sum achievable rate can be written as

C = K i=1 R i . (10)

III. ACHIEVABLE RATE ANALYSIS

In this section, we analyze the system performance. We derive an approximate expression for the achievable rate in the following theorem.

Theorem 1. In an RIS-aided multi-pair communication system, the average achievable rate of U Bi can be approximated as

R i ≈ Ri log 2      1 + p i α bi α ai εiβiΩi,i+L(εi+βi)+L (εi+1)(βi+1) K j=1,j =i p j α bi α aj εiβjΩi,j +L(εi+βj )+L (εi+1)(βj+1) + σ 2 i      , (11)
where Ω i,i and Ω i,j are, respectively, defined as

Ω i,i = L + 2 1≤m<n≤L cos[θ n -θ m + (n -m)π(sinϕ i +sinς i )], (12)
Ω i,j = L + 2 1≤m<n≤L cos[θ n -θ m + (n -m)π(sinϕ i +sinς j )]. (13)
Proof: Using Lemma 1 in [START_REF] Zhang | Power scaling of uplink massive MIMO systems with arbitrary-rank channel means[END_REF], R i in ( 9) can be approximated as

R i ≈ log 2      1 + p i α bi α ai E h T bi Θh ai 2 K j=1,j =i p j α bi α bj E h T bi Θh aj 2 + σ 2 i      . ( 14
)
The ℓth element of h ai and the ℓth element of h bi can be written as

[h ai ] ℓ = ε i ε i + 1 e j(ℓ-1)πsinϕi + 1 ε i + 1 (s ℓi + jt ℓi ), (15)
[h bi ] ℓ = β i β i + 1 e j(ℓ-1)πsinςi + 1 β i + 1 (u ℓi + jv ℓi ), (16)
where s ℓi ∼ N (0, 1/2) and t ℓi ∼ N (0, 1/2) denote the independent real and imaginary parts of [ hai ] ℓ , respectively, and u ℓj ∼ N (0, 1/2) and v ℓj ∼ N (0, 1/2) denote the independent real and imaginary parts of [ hbi ] ℓ , respectively. Let us first consider the term E h T bi Θh aj 2 in [START_REF] Renzo | Reconfigurable intelligent surfaces vs. relaying: Differences, similarities, and performance comparison[END_REF]. By substituting [START_REF] Guo | Outage probability analysis and minimization in intelligent reflecting surface-assisted MISO systems[END_REF] and ( 16) into E h T bi Θh aj 2 , it can be rewritten as

E h T bi Θh aj 2 = E R h T bi Θh aj 2 + I h T bi Θg aj 2 , ( 17
)
where the terms R h T bi Θh aj and I h T bi Θg aj are the real and imaginary parts of h T bi Θh aj . They are respectively given by ( 18) and (19) (at the bottom of this page), where λ ij is given by

λ ij = 1 (ε i + 1)(β j + 1) . ( 20
)
With the aid of some algebraic calculations, we obtain

E h T bi Θh aj 2 = ε i β j Ω i,j + L(ε i + β j ) + L (ε i + 1)(β j + 1) . (21)
A similar method can be used to calculate the term E h T bi Θh ai 2 in ( 14), which yields

E h T bi Θh ai 2 = ε i β i Ω i,i + L(ε i + β i ) + L (ε i + 1)(β i + 1) . ( 22
)
By substituting ( 21) and ( 22) into ( 14), we obtain the final result in [START_REF] Pan | Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF]. This completes the proof.

✷ Substituting ( 11) into [START_REF] Bai | Latency minimization for intelligent reflecting surface aided mobile edge computing[END_REF], we obtain the sum achievable rate. According to Theorem 1, if α bi , α ai , AoA, AoD, ε i , and σ i are kept fixed, the sum achievable rate is determined by the number of user pairs K, the transmit power p i , the phase shift matrix Θ and the number of reflective elements L.

IV. PHASE SHIFTS OPTIMIZATION

In this section, we formulate and solve the phase shifts optimization problem in order to maximize the sum achievable rate. Both case studies of CPSs and DPSs are considered.

A. Optimal CPSs Design

To begin with, we consider the scenario adopting CPSs. The optimization problem is formulated as To tackle this issue, we propose a GA method in Algorithm 1. We associate each individual to a 1 × L phase shift vector θ = [θ 1 , . . . , θ ℓ , . . . , θ L ] and θ ℓ corresponds to its ℓth gene.

max Θ K i=1 Ri (23a) s.t. θ ℓ ∈ [0, 2π) ∀ℓ = 1, . . . , L. (23b)
The initial population, the evaluation and sort operations, the selection function, the crossover function and the mutation function of the proposed GA method are described as follows.

1) Initial population: N t individuals are generated, by generating each gene as a variable that is randomly distributed within [0, 2π). This is referred to as the initial population.

2) Evaluation and sort: Each individual is evaluated by using the fitness function

f (θ) = 1 K i=1 Ri . ( 24
)
In particular, the lower the fitness function is, the higher the evaluation of the individual and the position in the corresponding sorted list are.

R h T bi Θh aj = λ ij L ℓ=1 ( ε i β j (cos(θ ℓ + (ℓ -1)π(sinϕ i + sinς j ))) + cosθ ℓ (s ℓi u ℓj -t ℓi v ℓj ) -sinθ ℓ (s ℓi v ℓj + t ℓi u ℓj ) ) + λ ij L ℓ=1 ( √ ε i ((cosθ ℓ -sinθ ℓ )cos((ℓ -1)πsinϕ i )u ℓj -(cosθ ℓ + sinθ ℓ )sin((ℓ -1)πsinϕ i )v ℓj ) + β j ((cosθ ℓ -sinθ ℓ )(cos((ℓ -1)πsinς j )s ℓi -(cosθ ℓ + sinθ ℓ )sin((ℓ -1)πsinς j )t ℓi )) ) (18)
I h T bi Θh aj = λ ij L ℓ=1 ( ε i β j (sin(θ ℓ + (ℓ -1)π(sinϕ i + sinς j ))) + sinθ ℓ (s ℓi u ℓj -t ℓi v ℓj ) + cosθ ℓ (s ℓi v ℓj + t ℓi u ℓj ) ) + λ ij L ℓ=1 ( √ ε i ((cosθ ℓ + sinθ ℓ )cos((ℓ -1)πsinϕ i )u ℓj + (cosθ ℓ -sinθ ℓ )sin((ℓ -1)πsinϕ i )v ℓj ) + β i ((cosθ ℓ + sinθ ℓ )cos((ℓ -1)πsinς j )s ℓj + (cosθ ℓ -sinθ ℓ )sin((ℓ -1)πsinς j )t ℓj ) ) (19)
3) Selection function: The selection function is used to obtain two candidates, θ 1 and θ 2 , from N t individuals. The individuals with larger values of the fitness function have a lower probability to be selected. θ 1 and θ 2 are obtained by using Algorithm 2.

Algorithm 2: Selection Algorithm

1 Generate a random number c between 0 and 1; 2 Define a cumulative-probabilities vector r = [1/N t , . . . , r/N t , . . . , 1], for r = 1, ..., N t .; 3 Denote r ′ /N t as the nearest maximum number in r to c; 4 Take the r ′ th individual in the list that is sorted by using (24).

Algorithm 3: Single-Point Crossing Algorithm

1 Get θ 1 = [θ (1) 1 , . . . , θ (1)
ℓ , . . . , θ

L ] and

θ 2 = [θ (2) 1 , . . . , θ (2)
ℓ , . . . , θ

L ] selected by Algorithm 2; 2 Identify a crossover point ℓ ′ ∈ [1, L] randomly; 3 Crossover θ 1 and θ 2 at crossover point ℓ ′ ; 4 Obtain two children

θ c1 = [θ (1) 1 , . . . , θ (1)
ℓ ′ , θ (2)
ℓ ′ +1 , . . . , θ

L ] and

θ c2 = [θ (2) 1 , . . . , θ (2)
ℓ ′ , θ (1)
ℓ ′ +1 , . . . , θ (1)
L ].

4) Crossover function:

The crossover function operates on θ 1 and θ 2 , and generates two crossover children θ c1 and θ c2 . A single-point crossing algorithm is employed, which is described in Algorithm 3.

5) Mutation function: The mutation function operates on (N t -N e )L genes from (N t -N e ) individuals (except for the elites). Each gene is capable of mutating to a random number between 0 and 2π with mutation rate.

Similar as [START_REF] Ye | Tradeoff caching strategy of the[END_REF], the complexity of the proposed GA algorithm is nN t , where N t is the population size, and n is the number of generations evaluated. Moreover, n is determined by the convergence behavior of the GA.

B. Optimal DPSs Design

In practice, only a limited number of phase shifts can be used. We assume that each reflective element is encoded with B bits, and thus 2 B phase shifts can be chosen to enhance the signal reflected by the RIS. We denote the DPS matrix as Θ = diag(e j θ1 , . . . , e j θℓ , . . . , e j θL ), where θℓ is the DPS of the ℓth reflective element. Replacing the CPSs in Problem (23) by the DPSs, the optimization problem for DPSs can be formulated as

max Θ K i=1 Ri (25a) s.t. θℓ ∈ {0, 2π/2 B , . . . , 2π(2 B -1)/2 B } ∀ℓ = 1, . . . , L. (25b)
It is observed that Problem (25) is similar to Problem (23). Accordingly, the GA method proposed for solving the CPSs optimization problem can be applied to solve the DPSs optimization problem as well.

V. NUMERICAL RESULTS

We evaluate the impact of different parameters on the sum achievable rate. We assume that the Rician factor is ε i = 10, the noise power is σ 2 i = 1, and the transmission power is SNR = p i , for i = 1, ..., K. The main parameters for the GA are: N t = 100, N s = 50, N c = 50, N e = 1, n max = 10000, P m = 0.1, and ε = 10 -6 . The other parameters are summarized in Table I, where the AOA and the AOD are randomly distributed within [0, 2π), and the large-scale fading coefficients α ai and α bi are set according to [START_REF] Zhang | Power scaling of uplink massive MIMO systems with arbitrary-rank channel means[END_REF].

In Fig. 2, we illustrate the sum achievable rate (10) versus the SNR obtained from the analytical expression in [START_REF] Pan | Intelligent reflecting surface aided MIMO broadcasting for simultaneous wireless information and power transfer[END_REF] and Monte-Carlo simulations from [START_REF] Peng | Multiuser full-duplex two-way communications via intelligent reflecting surface[END_REF]. Two quantization bits are assumed. A good analytical agreement with Monte-Carlo simulation results is obtained, which verifies the derivations. We observe that the sum achievable rate increases with the number of reflecting elements L, as expected.

Fig. 3 depicts the sum achievable rate versus the SNR, by assuming two quantization bits and comparing different optimization schemes. Compared with the scheme based on randomly chosen phase shifts, the proposed GA and exhaustive search methods can achieve higher sum achievable rate. It is interesting to observe that the proposed GA method has almost the same performance as the globally optimal solution obtained by using the exhaustive search method.

Fig. 4 shows the sum achievable rate versus B (the number of quantization) for the scenarios adopting DPSs. The sum achievable rate for the DPSs setup increases rapidly when B is small, while the curve gradually saturates when B is large. However, it is known that using a large number of quantization bits to control the phase shifts incurs high hardware cost and power consumption. The figure shows that three quantization bits can achieve a large portion of the sum achievable rate for the CPSs setup, which provides useful engineering design insights for RIS-aided systems. Fig. 5 depicts the sum achievable rate versus the Rician factor, with two quantization bits and SNR = 20 dB. The figure shows that the sum achievable rate increases with the Rician factor. With the increase of the Rician factor, we can observe that the gaps of the sum rates between the random scheme and the proposed GA method converge to the fixed values. It is because with a large Rician factor, the channels are dominated by the LOS component.

VI. CONCLUSION

In this paper, we investigated RIS-aided communications for multiple pairs of users. We derived an approximate expression for the achievable rate. Based on the derived analytical framework, we developed a GA-based algorithm for optimizing the achievable sum achievable rate, which can be applied to CPS-based and DPS-based implementations. Simulation results verified the effectiveness of the proposed GA-based method. outage probability and fronthaul usage in a cloud-RAN," IEEE Trans. Veh. Technol., vol. 67, no. 7, pp. 6383-6397, Jan. 2018.

Fig. 1 .

 1 Fig. 1. System model for RIS-aided multi-pair communications.

Fig. 2 .

 2 Fig. 2. Sum achievable rate versus SNR with B = 2 and GA.

Fig. 3 .

 3 Fig. 3. Sum achievable rate versus SNR with B = 2 by various schemes.

Fig. 4 .

 4 Fig. 4. Sum achievable rate versus B with SNR = 20 dB.

[ 1 ]Fig. 5 .

 15 Fig. 5. Sum achievable rate versus Rician factor with B = 2 and SNR = 20 dB.


Daniel Lee

Hiroki Takahashi

Aany S L Thankamony

Jean-Philippe Dacquin

Michel Bardet

Olivier Lafon

Gaël De Paëpe

Enhanced solid-state NMR correlation spectroscopy of quadrupolar nuclei using dynamic nuclear polarization

Keywords: dynamic nuclear polarization, solid-state nuclear magnetic resonance, magic angle spinning, dipolar recoupling, quadrupolar nuclei, polarization transfer, structure determination, mesoporous alumina. Supporting Information

Owing to a true sensitivity enhancement, corresponding to 4 to 5 orders of magnitude of time-savings, for a solid-state NMR experiment performed under dynamic nuclear polarization conditions compared to a conventional solid-state NMR experiment, it is shown that it is possible to record interface-selective 27 Al-27 Al two-dimensional dipolar correlation spectra on mesoporous alumina, an advanced material with potential industrial applications. The low efficiency of cross-polarization and also of dipolar recoupling for quadrupolar nuclei has been completely negated using this technique. The important presence of penta-coordinated aluminium has not only been observed, but its role in bridging interfacial tetra-and hexa-coordinated aluminium has been determined. Such structural information, collected at low temperature (~103 K) and 9.4 Tesla with the use of dynamic nuclear polarization, would have been impossible to obtain under standard conditions, even using a higher magnetic field. However, here it is demonstrated that this information can be obtained in only 4 hours. This work clearly opens a new avenue for the application of solid-state NMR to quadrupolar nuclei and notably the atomic-scale structure determination of catalysis materials, such as mesoporous alumina.

Porous materials are a cornerstone of the chemical industry since they can be employed in heterogeneous catalysis, separation processes, sensors, photonics, biomaterials etc. However, their rational design is currently limited by the lack of applicable atomic-resolution characterization methods. Nuclear magnetic resonance (NMR) spectroscopy presents itself as a technique that is able to provide detailed information on the atomic-level structure and dynamics of these heterogeneous and often disordered systems. Unfortunately, the intrinsic insensitivity of NMR impedes the examination of interfaces; these regions being crucial to the properties of porous materials.

Owing to the pioneering work of Griffin and co-workers, 1 recent developments in the combination of solid-state NMR (SSNMR) under Magic-Angle Spinning (MAS) with dynamic nuclear polarization (DNP) at high magnetic field have facilitated the acquisition of high-resolution NMR spectra with greatly enhanced signal-to-noise ratios (S/N). 2 This heralds the imagination and implementation of previously unrealistic experiments. In this work, one such experiment is performed on technologically and catalytically important mesoporous alumina (Al2O3). 3 This thermally stable (800°C) and high surface area material (typically 200-400 m 2 g -1 ) possessing ordered mesopores (Figure 1 (a)) has a high concentration of research interests due to its catalytic significance in the production of biodiesel, [START_REF] Wilson | Book Chapter in Catalysis for Alternative Energy Generation[END_REF] along with many other applications. [START_REF] Bejenaru | [END_REF] Utilizing 'green' and renewable fuels is forever becoming increasingly important and the understanding of mesoporous alumina and its specific role in the catalysed biodiesel production process will lead to improved manufacture methods and therefore increased system viability.

A particular obstacle when knowledge of the local environment of 27 Al nuclei is desired is that conventional SSNMR radio-frequency (RF) irradiation sequences return very low efficiencies for quadrupolar nuclei. However, in spite of the difficulties arising from the quadrupolar nature of 27 Al, herein it is shown that DNP-enhanced SSNMR can be used to perform sophisticated yet fast, interface-selective characterization measurements. Previous studies have shown that when protons are only located at the surface/interface of much larger systems then cross-polarization (CP) from protons to the nuclei of interest can result in a surface/interface-selective technique. 6 This technique has been combined with DNP to study surfaces with greatly improved sensitivity 7 and also -alumina where CP is very inefficient. 8 In this work the DNP is dominated by the cross-effect (CE) 9 between two unpaired electrons in a biradical (TOTAPOL 10 ) and protons in a glass-forming solvent matrix or protons at the interface in the mesoporous alumina. The experiments were performed on a Bruker BioSpin solid-state DNP-NMR spectrometer. 11 This spectrometer is equipped with a gyrotron and connecting transmission line, able to provide microwave (w) irradiation at 263 GHz and approximately 5 W at the sample, in combination with an AVANCE™ III 400 MHz wide-bore NMR system and a low-temperature MAS probe capable of achieving spinning rates for 3.2 mm rotors of up to 17 kHz at sample temperatures of approximately 100 K.

The 'DNP-ready' sample was prepared by impregnation with an excess of solution. 12 The mesoporous alumina was first heated to 408 K for 6 hours to ensure that the sample, including the pores, was dry. 46 L of 20 mM TOTAPOL in a glass-forming mixture of [ 2 H6]-DMSO, H2O and 2 H2O (78 %, 8 %, 14 % according to weight, respectively) was added to cover 28 mg of the mesoporous alumina powder. This mixture was stirred vigorously and then left to rest at room temperature 24 hours to allow time for the impregnation of the pores by the DNP chemical matrix. A repetition of centrifuging the mixture followed by pipetting any supernatant was employed to remove the excess liquid. The remaining wet powder was added to a thin-walled zirconia 3.2 mm MAS rotor. It must be stated that, unlike other systems, since it is not necessary to dissolve/disperse 1 the impregnated mesoporous alumina in a DNP matrix then these systems benefit from the ability to use the maximum amount of the sample of interest possible without compromising the DNP-enhancement. The NMR signal enhancement as a result of DNP in this system can be seen in Figure 1 (b). With w irradiation the CE enhances the polarization of protons within the sample. This polarization is then transferred to the central transition (CT) of the 27 Al nuclei via low power CP under MAS (CPMAS) 13 and the resulting spectrum is recorded (red line). The experiment is repeated under the same conditions except without the w irradiation (blue line). Here, the enhancement, , is measured simply by comparing the intensities of the spectral peaks resulting from the CPMAS experiments with and without w irradiation. For this system  was 15. This was the same for each of the 27 Al peaks. For comparison, the direct excitation under MAS (DEMAS) spectrum of 27 Al is also shown in Figure 1 (b) (black line). As an aside, a similar DNP-ready sample that used an impregnation time of only 1 hour was also tested. For this sample  was 11. Thus, it appears that longer impregnation times are necessary to facilitate the penetration of the TOTAPOL solution into the pores.

In the spectra of Figure 1 8 the substantial presence of Al V resonances observed here indicates that the water-based DNP matrix does not significantly coordinate to these sites and thus this solution is applicable for the study of Al V sites using DNP-enhanced SSNMR measurements of similar systems. Furthermore, this abundance of Al V sites at the interface demonstrates the industrial potential for this mesoporous alumina. Not only does this system exhibit greater surface area than the more commonly employed -alumina but it appears that these Al V sites, which can act as binding sites for the active phase, 14 are much more plentiful.

Not only does the impregnation solution provide the biradical necessary for DNP and the glass-forming solvents to best utilize this biradical, but this matrix also provides extra protons that enhance the CP signal, irrespective of DNP. Furthermore, the requirement of low temperatures for efficient DNP has the added advantage of increasing nuclear relaxation times, with the usually fast relaxation of quadrupolar nuclei during spin-lock periods limiting the success of CP for measurements at room temperature. The combination of low temperatures and the addition of these extra, useful protons means that the time-savings using this system for DNP-enhanced CPMAS experiments, when compared to ubiquitous SSNMR systems, cannot be predicted by simply allowing for the DNP enhancement (), the low-temperature gain (Boltzman factor and reduced thermal noise) whilst taking into account sample 'bleaching' 15 and the change in relaxation times due to the paramagnetic effects of the added biradical. The only true comparison in this (and similar) system(s) is to record spectra under specific DNP conditions and under normal SSNMR conditions and compare the S/N per unit time. 16 Note that resolution is also an extremely important factor and apparent linewidths should also be taken into account. However, in this system the linewidths are dominated by disorder and there are negligible changes between the two experimental conditions. A similar CPMAS experiment was recorded under conventional SSNMR conditions (9.4 T, 298 K, 8 kHz MAS on a 3.2 mm rotor; not shown). Comparing the S/N per unit time between the DNP-enhanced CPMAS spectrum and the conventional CPMAS spectrum showed that using DNP for this system resulted in a true enhancement factor of 184 (time-saving factor of ~ 34000). This demonstrates the considerable advantage of using DNP in the study of this system.

A comparison of the NMR signal intensity for each aluminium coordination state between the DNP-enhanced CPMAS experiment and the DEMAS experiment can only give a very crude indication of the influence of protons on each state. The 27 Al IV peak is 2.4 times larger in the DNPenhanced CPMAS spectrum, the 27 Al V peak is 4.6 times larger and the 27 Al VI peak is 6.1 times larger. Note that the linewidths (and thus the apparent transverse dephasing times, T2*) do not vary between the two spectra and, accordingly, proton couplings are not the limiting factor of these linewidths. A more complete analysis of the relative influence of protons can be performed using measurements of the time constant associated with the homogeneous contribution to the transverse dephasing which cannot be refocused with a -pulse, T2'. T2' values were measured using a pseudo-two-dimensional spin-echo experiment with varied, rotor-synchronized echo times, the pulse sequence for which is given in Figure S1. Figure 2 illustrates such measurements for each of the aluminium peaks in the mesoporous alumina sample, both with and without heteronuclear decoupling. This is used to demonstrate the effect protons have on each T2', and thus on each aluminium site. Note that the employed heteronuclear decoupling was deemed sufficient in that an increase in power did not affect the value for T2' (this was observed for optimized SPINAL-64 17 decoupling with powers above 90 kHz; data not shown). The ratio of the T2' value measured with decoupling to the value measured without decoupling was 2.3 for Al IV , 2.9 for Al V and 3.2 for Al VI . This is in agreement with the relative intensities between the CPMAS and DEMAS spectra. These corroborating observations indicate that the interfacial Al VI sites are more heavily influenced by protons than the interfacial Al V sites which, in turn, are more heavily influenced by protons than the interfacial Al IV sites. This is consistent with an increase in the number of hydroxo or aqua groups for an increasing coordination number.

Figure 2. Normalized intensity decay for each 27 Al peak with and without 100 kHz of SPINAL-64 heteronuclear decoupling during a spin-echo period as a function of the (rotor-synchronized) spin-echo time, used to determine T2'. This DNPenhanced experiment was performed with a MAS rate of 8 kHz, a sample temperature of ~103 K and the pulse sequence shown in Figure S1.

The lack of change in T2* for each spectral peak between the CPMAS and DEMAS experiments noted above could indicate that heteronuclear decoupling is not necessary. Nonetheless, as can clearly be seen from Figure 2, protons have a significant impact on the spin dynamics of 27 Al nuclei. Moreover, in this system the large influence of protons is not just limited to experiments that begin with a CP step. However, this realization is beyond the scope of this work but will be discussed in detail in a future publication.

Figure 2 illustrates the importance of the heteronuclear decoupling during the spin-echo period; these periods (and thus T2' values) being vital in most solid-state NMR experiments involving J-couplings. 18 A further demonstration of its importance is given in Figure S2 where the spectral intensities of each 27 Al coordination state after CP and then a spin-echo period of total duration 4 ms are compared with and without 1 H decoupling. Globally the intensities without decoupling are reduced by 60 % compared to those with decoupling. T2' values are also important in solid-state NMR experiments involving dipolar couplings (REDOR-type experiments etc. 19 ). Furthermore, other dipolar recoupling sequences will experience magnetization decay times on the order of T2' which can also be heavily affected by proton couplings. Thus, heteronuclear decoupling during these sequences can also be highly beneficial, as long as interference effects are avoided. Total acquisition time for each spectrum was ~4 hours. For illustrative purposes, the DNP-enhanced CPMAS spectrum (Figure 1 (b)), with the three different aluminium peaks labeled, is also shown.

The supercycled symmetry-based dipolar recoupling sequence BR22 1 has been shown to be of great utility when recording homonuclear correlation spectra. 20 However, the use of this, and other, 21 sequences for 27 Al has been limited to bulk studies by the low efficiency (approximately 5 %), which results from the intricate spin dynamics of quadrupolar nuclei in the presence of RF field and MAS. The entirety of the adapted pulse sequence used here for interface-selective, DNP-enhanced correlation spectroscopy of quadrupolar nuclei is given in Figure S1. This pulse sequence firstly transfers hyperpolarized proton magnetization to nearby half-integer spin quadrupolar nuclei via a CP step. Then, the sequence creates double-quantum (DQ) coherences between the CTs of dipolar-coupled quadrupolar nuclei whilst removing DQ coherence contributions involving satellite transitions of a single 27 Al nucleus. The spectra resulting from this sequence then give information about the spatial proximities between neighboring nuclei. The variables of this experiment can be optimized in a one-dimensional fashion (no indirect-dimension increments). Using this 1D experiment the importance of heteronuclear decoupling during the recoupling periods and also the whole sequence can be clearly demonstrated (Figure S2). Moreover, demanding experiments usually require careful calibrations of pulses and delays. Here, calibrations of the parameters required for maximum efficacy of the recoupling sequence could be performed on the sample of interest in less than an hour owing to the huge gain in S/N due to the DNP conditions. Under conventional SSNMR conditions these calibrations would require much longer experimental times even with model samples.

Once calibrated, the two-dimensional version of this pulse sequence was used to record the first correlation spectra of quadrupolar nuclei using DNP. Furthermore, since the sequence begins with a CP step, observed correlations would only be between interfacial 27 Al nuclei. Spectra resulting from the implementation of this sequence on the mesoporous alumina sample are shown in Figure 3. It has been shown that the BR22 1 recoupling sequence has a dependency on offset 19a so three spectra with different offsets were recorded to remove any uncertainty in the data. Possibly the most striking point is that each spectrum only took approximately four hours to acquire. Under conventional SSNMR conditions the acquisition of equivalent data would require over fifteen years for each spectrum! It can be seen that dipolar correlations between all sites are present (including same site correlations between different nuclei e.g. Al VI -Al VI correlations) except for correlations between Al IV and Al VI sites. Therefore, at the interface of this material Al V sites connect Al IV and Al VI sites. To summarize, DNP enhancements ( of about 15 have been observed on an industrially important sample of mesoporous alumina using a 9.4 T solid-state DNP-NMR spectrometer. More importantly, this translates into a true sensitivity gain per unit time of 184 when compared to equivalent room temperature measurements. Transcending the sensi-tivity limitations of cross-polarization and also dipolar recoupling for quadrupolar nuclei, the corresponding 4 to 5 orders of magnitude of time-savings allow not only the acquisition of very fast 1D CPMAS experiments and pseudo-2D experiments but also interface-selective 2D homonuclear dipolar correlation experiments. These 1D and 2D experiments clearly demonstrate that it is possible to probe spatial proximities involving quadrupolar nuclei with the use of DNP-enhanced SSNMR with greatly reduced timescales. Moreover, the absence of Al IV -Al VI correlations combined with the presence of Al IV -Al V and Al V -Al VI correlations shows that interfacial Al V bridges interfacial Al IV and Al VI sites. This structural data, together with the results of the influence of protons on the T2' of each interfacial site, can be combined with ab initio calculations to help elucidate the surface structure of this amorphous material. This work is planned for a future publication.

ASSOCIATED CONTENT

Supporting Information Placeholder Figures S1 andS2. Experimental details and pulse sequence diagrams along with spectra demonstrating the importance of heteronuclear decoupling for this study. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 .

 1 Figure 1. (a) TEM micrograph (left) and associated schematic (right) showing the presence of ordered hexagonal mesopores throughout the alumina sample studied in this work. (b) 27 Al NMR spectra of this sample recorded with a MAS rate of 8 kHz and a sample temperature of ~103 K.

  (b) three discrete 27 Al peaks are observed. These correspond to hexa-coordinated aluminium, Al VI , with a chemical shift ( 27 Al) = 5 ppm, penta-coordinated aluminium, Al V , (( 27 Al) = 36 ppm) and tetra-coordinated aluminium, Al IV , (( 27 Al) = 70 ppm) sites. Contrary to the literature,

Figure 3 .

 3 Figure 3. DNP-enhanced, interface-selective DQ-SQ 27 Al homonuclear dipolar correlation spectra of the mesoporous alumina recorded with a MAS rate of 8 kHz, a sample temperature of ~103 K and the pulse sequence shown in Figure S1.

ACKNOWLEDGMENTS

This work was supported by the ANR (ANR08-CEXC-003-01) and funding from the RTB. G.D.P. was supported by EU Marie Curie (PIEF-GA-2009-237646) for part of the work. D.L. was supported by CEA-EUROTALENTS (PCOFUND- GA-2008-228664). Dr. M. Giffard is acknowledged for the synthesis of the TOTAPOL biradical. The UCCS members are grateful for funding provided by Region Nord/Pas de Calais, Europe (FEDER), CNRS, French Minister of Science, FR-3050, USTL, ENSCL, and ANR-2010-JCJC-0811-01. O. L. and A.S.L.T. acknowledge X. Lu, J. Trébosc, H. Vezin and J.-P. Amoureux for helpful discussions.

AUTHOR INFORMATION

Corresponding Author *gael.depaepe@cea.fr


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