Mads Antoine Thiery-Vuillemin
email: a.thieryvuillemin@mac.com

Hvid Poulsen

Edouard Lagneau

Guillaume Ploussard

Alison Birtle

Louis-Marie Dourthe

Dominique Beal-Ardisson

Mads Hvid Poulsen

Elias Pintus

Redas Trepiakas

Florence Lefresne

Martin Lukac

Suzy Van Sanden

Genevieve Pissart

Alison Reid

Impact of Abiraterone Acetate plus Prednisone or Enzalutamide on Patient-reported Outcomes in Patients with Metastatic Castration-resistant Prostate Cancer: Final 12-mo Analysis from the Observational AQUARiUS Study

Keywords: Abiraterone, AQUARiUS, Enzalutamide, Metastatic castration-resistant prostate cancer, Patient-reported outcomes, Prednisone, Quality of life

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

Outcome measurements and statistical analysis: PROs were collected over 12 mo using Functional Assessment of Cancer Therapy-Cognitive Function (FACT-Cog), Brief Fatigue Inventory-Short Form (BFI-SF), Brief Pain Inventory-Short Form, and European Organisation for Research and Treatment of Cancer-Quality of Life Questionnaire (QLQ-C30) at baseline and routine visits. Outcomes included mean change in PROs, patients with clinically meaningful worsening (CMW) in PROs, and safety. Data were analysed using repeated measures linear and logistic models adjusted for baseline characteristics.

Results and limitations: Abiraterone-treated (N = 105) and enzalutamide-treated (N = 106) patients were included. Key PRO items (cognitive impairments and fatigue) were significantly (p < 0.05) in favour of abiraterone versus enzalutamide during the study.

"Perceived cognitive impairment" and "comments from others" (FACT-Cog); "fatigue right now", "usual level of fatigue", and "worst level of fatigue" (BFI-SF); and "cognitive functioning" and "fatigue" (QLQ-C30) were significantly in favour of abiraterone over enzalutamide for three or more consecutive periods up to month 12. From study initiation, significantly fewer patients receiving abiraterone experienced one or more CMW episode in cognition and fatigue. Fatigue and asthenia (adverse events) were lower with abiraterone than with enzalutamide (5% vs 15% and 10% vs 11%, respectively). There were no treatment-related deaths. Limitations included lack of randomisation.

Conclusions:

In a real-world setting, this 12-mo analysis suggests an advantage of abiraterone acetate plus prednisone over enzalutamide in reducing fatigue and cognitive function; this finding occurred early after treatment initiation. This difference should be considered when choosing treatment.

Patient summary:

This study looked at the effect of two treatments (abiraterone acetate plus prednisone and enzalutamide) for metastatic castration-resistant prostate cancer on Introduction Approximately 90% of patients with metastatic prostate cancer respond well to initial androgen deprivation therapy (ADT) [START_REF] Dueregger | The use of dietary supplements to alleviate androgen deprivation therapy side effects during prostate cancer treatment[END_REF]. However, 20% of patients progress to castrationresistant prostate cancer within 5 yr [START_REF] Crawford | Navigating the evolving therapeutic landscape in advanced prostate cancer[END_REF]. Metastatic castration-resistant prostate cancer (mCRPC) has a 5-yr survival rate of ~30% [START_REF]Prostate cancer: statistics[END_REF]. Metastatic CRPC is characterised by worsening symptoms, progressive decline in health-related quality of life (HRQoL), and increasing pain [START_REF] Kirby | Characterising the castration-resistant prostate cancer population: a systematic review[END_REF][START_REF] Nussbaum | Patient experience in the treatment of metastatic castration-resistant prostate cancer: state of the science[END_REF].

Until 2010, docetaxel was the standard treatment for mCRPC, but life-extending noncytotoxic therapies that may have positive impacts on patient-reported outcomes (PROs) are now available [START_REF] Nussbaum | Patient experience in the treatment of metastatic castration-resistant prostate cancer: state of the science[END_REF]. Abiraterone acetate plus prednisone (abiraterone) and enzalutamide target androgen signalling, delay radiographic progression, increase survival, and improve HRQoL and pain outcomes in chemotherapy-naïve patients with mCRPC compared with placebo ± prednisone [START_REF] Ryan | Abiraterone in metastatic prostate cancer without previous chemotherapy[END_REF][START_REF] Beer | Enzalutamide in metastatic prostate cancer before chemotherapy[END_REF][START_REF] Loriot | Effect of enzalutamide on health-related quality of life, pain, and skeletal-related events in asymptomatic and minimally symptomatic, chemotherapy-naive patients with metastatic castration-resistant prostate cancer (PREVAIL): results from a randomised, phase 3 trial[END_REF]. However, few studies have examined the impact of these treatments on HRQoL under real-world conditions.

AQUARiUS was a 12-mo phase IV study to evaluate the effect of abiraterone versus enzalutamide on PROs in patients with chemotherapy-naïve mCRPC in a real-world setting.

Results from interim 3-and 6-mo analyses of AQUARiUS showed that more favourable outcomes were achieved with abiraterone than with enzalutamide for PROs of cognition and fatigue [START_REF] Thiery-Vuillemin | Impact of abiraterone acetate plus prednisone or enzalutamide on fatigue and cognition in patients with metastatic castration-resistant prostate cancer: initial results from the observational AQUARiUS study[END_REF][START_REF] Thiery-Vuillemin | Six-month patient-reported outcome (PRO) results from AQUARiUS, a prospective, observational, multicenter phase 4 study in patients (Pts) with metastatic castration-resistant prostate cancer (mCRPC) receiving abiraterone acetate + prednisone (AAP) or enzalutamide (ENZ)[END_REF]. We report the final 12-mo results of AQUARiUS.

2.

Patients and methods

Study design

AQUARiUS (NCT02813408) was a 12-mo, two-cohort, prospective, observational, nonrandomised, multicentre, phase IV study conducted by office-or hospital-based urology and/or oncology specialists in Denmark (n = 3), France (n = 14), and the UK (n = 10). An independent ethics committee or institutional review board reviewed/received study notification, in all countries, as required by local regulations. The study was conducted in accordance with the Declaration of Helsinki. The design has been reported [START_REF] Thiery-Vuillemin | Impact of abiraterone acetate plus prednisone or enzalutamide on fatigue and cognition in patients with metastatic castration-resistant prostate cancer: initial results from the observational AQUARiUS study[END_REF][START_REF] Thiery-Vuillemin | Six-month patient-reported outcome (PRO) results from AQUARiUS, a prospective, observational, multicenter phase 4 study in patients (Pts) with metastatic castration-resistant prostate cancer (mCRPC) receiving abiraterone acetate + prednisone (AAP) or enzalutamide (ENZ)[END_REF].

Patients

Patients were eligible if they were ≥18 yr old with a histologically or cytologically confirmed diagnosis of adenocarcinoma of the prostate, and with documented metastatic disease and castration resistance, and if they were initiated on abiraterone or enzalutamide for asymptomatic or mildly symptomatic mCRPC after ADT failure.

Patients were excluded if they had prior chemotherapy to treat mCRPC or prior chemotherapy/cytotoxic agent to treat metastatic hormone-sensitive prostate cancer in the previous 12 mo. Inclusion and exclusion criteria have been described [START_REF] Thiery-Vuillemin | Impact of abiraterone acetate plus prednisone or enzalutamide on fatigue and cognition in patients with metastatic castration-resistant prostate cancer: initial results from the observational AQUARiUS study[END_REF]. All patients provided informed consent to participate. Patients meeting the study criteria were consecutively invited to participate in the study to minimise recruitment bias.

Intervention

The decision to prescribe abiraterone or enzalutamide was made by the treating physician in accordance with their usual practice.

Data collection

Medical records were used to capture clinical and demographic data. PRO data were collected prospectively using paper questionnaires completed by the patient at baseline and during routine visits to the clinic over 12 mo (the cut-off date was 21 March 2018).

Questionnaires included the Functional Assessment of Cancer Therapy-Cognitive Function (FACT-Cog) [START_REF] Cella | The Functional Assessment of Cancer Therapy scale: development and validation of the general measure[END_REF], Brief Fatigue Inventory-Short Form (BFI-SF) [START_REF] Mendoza | The rapid assessment of fatigue severity in cancer patients: use of the Brief Fatigue Inventory[END_REF], Brief Pain Inventory-Short Form (BPI-SF) [START_REF] Cleeland | The brief pain inventory user guide[END_REF], and European Organisation for Research and Treatment of Cancer-Quality of Life Questionnaire (EORTC QLQ-C30) [START_REF]EORTC QLQ-C30 scoring manual[END_REF] (see Supplementary Table 1 for further details on questionnaires). Questionnaire data were entered onto electronic case report forms (eCRFs) by an external vendor in accordance with quality control policies for data entry.

Outcomes

Two primary analyses were performed: for which odds ratios (ORs; 95% CI) were reported. An additional analysis was performed to determine the time (in months) to the first PRO item showing CMW of symptoms.

Secondary analyses included patient-reported adverse events (AEs), monitored from the 1st day of treatment to 30 d after last treatment exposure. Cause, severity, possible relationship to study drug, and outcome of AEs were recorded in eCRFs by the treating physician during routine visits. Medical resource use data were not source document verified and were inconsistent with visit dates reported in the eCRFs, so meaningful conclusions could not be made, and these data were not reported.

Statistical analyses

It was estimated that a sample size of 211 patients, balanced equally between cohorts, would be required to detect a CMW value of ≥0.5 SD between treatment cohorts, with 85%

power at the 5% level of significance (p < 0.05). CMW (≥0.5 SD) was based on published minimally important difference ranges (0.5-1.0 SD) for the questionnaires used in the study [START_REF] Mendoza | The rapid assessment of fatigue severity in cancer patients: use of the Brief Fatigue Inventory[END_REF][START_REF] Wong | Minimal clinically important differences in the brief pain inventory in patients with bone metastases[END_REF][START_REF] Cheung | Minimal clinically important difference (MCID) for the functional assessment of cancer therapy: cognitive function (FACT-Cog) in breast cancer patients[END_REF].

Collection of PRO data began before baseline visit (abiraterone and enzalutamide initiation) and continued until termination of treatment or for ~12 mo, whichever occurred first. Questionnaire data were collected during routine visits and analysed by periods 1, 2, 3, 4-6, 7-9, and 10-12 (see tables and figures for day ranges used per period).

Primary analyses were based on all treated patients (intent-to-treat [ITT] population). Sensitivity analyses were performed using the censored population (ie, abiraterone patients who switched to enzalutamide were censored at the start of enzalutamide and vice versa). All statistical analyses were performed using SAS release 9.4 (SAS Institute Inc., Cary, NC, USA).

Results

Patients

Overall, 226 patients were examined for eligibility and 211 (ITT population) were included (N = 105 [abiraterone] and N = 106 [enzalutamide]). Patient disposition is shown in Figure 1.

Baseline characteristics were well balanced between abiraterone-and enzalutamide-treated patients (Table 1).

Supplementary Table 2 shows the completion rate for the different PRO items during the study. Only the lowest percentage per PRO item (worse-case scenario, all items must be answered to consider a PRO complete) was presented for all patients and for those "still in study". The overall median completion rate for the 12-mo period was 81% for patients still in the study (this rate was based on all 28 PRO questions and all periods, and for both treatments).

Treatment exposure

The mean (SD) overall treatment duration was 38.3 (17.2) wk in the abiraterone group and 38.7 (18.2) wk in the enzalutamide group.

The percentages of patients starting on the usual abiraterone dose (1000 mg) and enzalutamide dose (160 mg) were 97% and 95%, respectively. In both cohorts, 5% of patients had at least one dose reduction. Treatment was discontinued during the 12-mo period in 40% of abiraterone patients and 36% of enzalutamide patients. The main reasons for discontinuation were disease progression for 62% versus 53%, toxicity for 12% versus 21%, and others for 26% versus 26% for abiraterone versus enzalutamide (Fig. 1).

Treatment differences in PRO items from baseline to 12 mo

There were no statistically significant differences between cohorts in baseline PRO scores (absolute values) for each item apart from pain interference (BPI-SF); this difference was not clinically meaningful (Supplementary Table 3). After initiation of treatment, there were statistically significant differences (p < 0.05) in favour of abiraterone over enzalutamide for 18 PRO items, as highlighted in Supplementary Table 3. With a more conservative approach (at least three periods [≥50%] needed to be significant in consecutive periods), nine of these PRO items were statistically significantly in favour of abiraterone over enzalutamide; these were mainly related to cognition, fatigue, appetite loss, and nausea (Table 2, Supplementary Fig. 1, and Supplementary Table 3). There was also consistency across different questionnaires for items relating to cognition and fatigue. Statistically significant improvements in favour of abiraterone over enzalutamide were observed for "perceived cognitive impairment", "comments from others" (FACT-Cog), and "cognitive functioning"

(QLQ-C30), which were evident at period 1. Similar findings were observed for "worst-level of fatigue", "usual level of fatigue", "fatigue right now" (BFI-SF), and "fatigue" (QLQ-C30) (Table 2).

CMW in PRO items over 12 mo

We checked whether the nine significant items detailed above were also clinically meaningful-with cumulative CMW analysis over 12 mo. Only those with statistically significant results in consecutive periods in the "mean change from baseline" analysis were reported. Overall, significantly fewer patients receiving abiraterone versus enzalutamide experienced at least one episode of CMW in "perceived cognitive impairments" (49% vs 76%

[OR = 0.31; 95% CI: 0.14; 0.70; p = 0.005]), "comments from others" (32% vs 62% [OR = 0.14; 95% CI: 0.05; 0.39; p < 0.001]; FACT-Cog), "worst level of fatigue" (53% vs 79% [OR = 0.33; 95% CI: 0.15; 0.75; p = 0.008]; BFI-SF), "fatigue" (45% vs 74% [OR = 0.29; 95% CI: 0.14; 0.62; p = 0.001]), and "appetite loss" (36% vs 60% [OR = 0.38; 95% CI: 0.17; 0.88; p = 0.023]; QLQ-C30; Fig. 2). These differences could be observed early within the 1st month of treatment (see Supplementary Table 4). There were four other items with statistically significant CMW in favour of abiraterone (none for enzalutamide); these were "emotional functioning", "social functioning", "physical functioning", and "pain right now". These were not reported based on the conservative approach used.

Time to the first measured PRO item showing CMW of symptoms

Exploratory Kaplan-Meier analyses of the time to the first measured PRO item showing CMW suggested that early and sustained numerical differences favouring abiraterone over enzalutamide were observed for "perceived cognitive impairments" and "comments from others" (FACT-Cog), "worst level of fatigue" (BFI-SF), "fatigue", and "appetite loss" (QLQ-C30; Supplementary Fig. 2).

Sensitivity analyses

Results from the censoring analysis remained consistent with those presented for the ITT analyses (Supplementary Table 5). A model with a random centre effect was tested, and it gave very similar results (model not shown).

Safety outcomes

The percentages of patients with at least one AE over 12 mo were 69% (abiraterone) and 77% (enzalutamide; Supplementary Table 6). Both fatigue and asthenia were lower with abiraterone than with enzalutamide (5% vs 15% and 10% vs 11%, respectively). The percentages of patients with at least one serious AE were 23% (abiraterone) and 26%

(enzalutamide), and with AEs leading to treatment or study termination were 5%

(abiraterone) and 11% (enzalutamide). The most common serious AEs were infections and infestations (7% [abiraterone] and 9% [enzalutamide]), and renal and urinary disorders (5%

[abiraterone] and 7% [enzalutamide]). The percentages of patients with AEs leading to hospitalisation were 22% (abiraterone) and 24% (enzalutamide). There were five deaths in the abiraterone group and seven in the enzalutamide group, but none were treatment related.

Discussion

AQUARiUS evaluated the impact of abiraterone and enzalutamide on self-reported PROs in patients with mCRPC over 12 mo in a real-world setting. The study showed that abiraterone was consistently associated with less cognitive impairments and fatigue than enzalutamide over the 12-mo period, and these differences were observed early after treatment initiation. The proportion of patients with CMW in cognitive impairments and fatigue was also statistically significantly lower with abiraterone than with enzalutamide over the 12-mo study period. The findings were consistent across specific (FACT-Cog and BFI-SF) and general (QLQ-C30) instruments. Findings relating to pain (BPI-SF), however, were inconclusive, as results were not consistent over consecutive periods and not confirmed by different analyses. Overall, these findings confirm those reported in the 3-and 6-mo analyses of AQUARiUS [START_REF] Thiery-Vuillemin | Impact of abiraterone acetate plus prednisone or enzalutamide on fatigue and cognition in patients with metastatic castration-resistant prostate cancer: initial results from the observational AQUARiUS study[END_REF][START_REF] Thiery-Vuillemin | Six-month patient-reported outcome (PRO) results from AQUARiUS, a prospective, observational, multicenter phase 4 study in patients (Pts) with metastatic castration-resistant prostate cancer (mCRPC) receiving abiraterone acetate + prednisone (AAP) or enzalutamide (ENZ)[END_REF]. The AQUARiUS outcomes are also comparable with those observed in randomised studies [START_REF] Basch | Abiraterone acetate plus prednisone versus prednisone alone in chemotherapy-naive men with metastatic castration-resistant prostate cancer: patient-reported outcome results of a randomised phase 3 trial[END_REF][START_REF] Cella | Relationship between patient-reported outcomes and clinical outcomes in metastatic castration-resistant prostate cancer: post hoc analysis of COU-AA-301 and COU-AA-302[END_REF][START_REF] Beer | The association between health-related qualityof-life scores and clinical outcomes in metastatic castration-resistant prostate cancer patients: Exploratory analyses of AFFIRM and PREVAIL studies[END_REF]. Results of a randomised phase II study also showed that total Functional Assessment of Cancer Therapy-Prostate scores favoured abiraterone over enzalutamide [START_REF] Khalaf | Health-related quality of life for abiraterone plus prednisone versus enzalutamide in patients with metastatic castration-resistant prostate cancer: results from a phase II randomized trial[END_REF].

The importance of determining the impact of treatment on PROs was shown in post hoc analyses of the pivotal studies for abiraterone and enzalutamide [START_REF] Cella | Relationship between patient-reported outcomes and clinical outcomes in metastatic castration-resistant prostate cancer: post hoc analysis of COU-AA-301 and COU-AA-302[END_REF][START_REF] Beer | The association between health-related qualityof-life scores and clinical outcomes in metastatic castration-resistant prostate cancer patients: Exploratory analyses of AFFIRM and PREVAIL studies[END_REF]. PRO improvements (eg, pain, functional well-being, and physical well-being) with abiraterone or enzalutamide were significantly associated with longer overall survival and radiographic progression-free survival. Worsening in PROs was also associated with a higher likelihood of radiographic progression compared with improvement/no change for patients treated with abiraterone.

However, AQUARiUS was not designed to assess this association.

This is the first study to report safety outcomes for abiraterone and enzalutamide in the same study according to a real-world setting. Rate of comorbidities, old age of patients, and opioid and sedative consumption are closer to routine practice than to registration trials [START_REF] Ryan | Abiraterone in metastatic prostate cancer without previous chemotherapy[END_REF][START_REF] Beer | Enzalutamide in metastatic prostate cancer before chemotherapy[END_REF]. However, the safety profiles were consistent with those reported in randomised studies [START_REF] Moreira | Differential side effects profile in patients with mCRPC treated with abiraterone or enzalutamide: a meta-analysis of randomized controlled trials[END_REF]. Abiraterone was associated with less asthenia and fatigue than enzalutamide, which is comparable with the findings from meta-analyses of randomised studies (COU-AA-301, COU-AA-302, AFFIRM, and PREVAIL) [START_REF] Moreira | Differential side effects profile in patients with mCRPC treated with abiraterone or enzalutamide: a meta-analysis of randomized controlled trials[END_REF]. The number of neurological AEs was higher in the enzalutamide cohort. The incidence of cardiac and vascular disorders was low and comparable between groups. However, it is important to note that there was potential for under-reporting of safety data due to the observational design.

Other limitations of AQUARiUS include lack of randomisation, although baseline characteristics were well balanced and a modelling approach correcting for all relevant baseline characteristics was used to minimise the confounding bias. There was no formal correction for multiple testing, but significant results were interpreted with extreme caution. Kaplan-Meyer plots of the time to the first measured PRO item showing CMW should also be interpreted with caution as these were post hoc; limitations of this analysis are listed in Supplementary Figure 2.

The study also has several strengths; it has a large sample size, despite a decline in questionnaire response over 12 mo. Data are based on validated questionnaires, with consistency across cognitive and fatigue items (FACT-Cog, BFI-SF, and QLQ-C30). The findings are confirmed by several different analyses (mean change from baseline, CMW

[overall and at individual periods], and time to worsening of symptoms). The results are also supported by sensitivity analyses. The treatment periods were preplanned with monthly intervals at the beginning of the study to capture early treatment differences.

Conclusions

This 12-mo study showed that patients with mCRPC who were treated with abiraterone experienced significantly less fatigue and cognitive impairments than enzalutamide-treated patients. This difference occurred early after treatment initiation. In a real-world setting, it suggests an advantage of abiraterone over enzalutamide in reducing fatigue and cognitive function. This difference should be considered when choosing treatment. These results are also in line with other published data [START_REF] Khalaf | Health-related quality of life for abiraterone plus prednisone versus enzalutamide in patients with metastatic castration-resistant prostate cancer: results from a phase II randomized trial[END_REF]. The safety outcomes were consistent with the known safety profile of each drug, but abiraterone was associated with fewer fatigue, asthenia, and neurological AEs than enzalutamide. Overall, these data confirm previously published interim analyses from this study [START_REF] Thiery-Vuillemin | Impact of abiraterone acetate plus prednisone or enzalutamide on fatigue and cognition in patients with metastatic castration-resistant prostate cancer: initial results from the observational AQUARiUS study[END_REF][START_REF] Thiery-Vuillemin | Six-month patient-reported outcome (PRO) results from AQUARiUS, a prospective, observational, multicenter phase 4 study in patients (Pts) with metastatic castration-resistant prostate cancer (mCRPC) receiving abiraterone acetate + prednisone (AAP) or enzalutamide (ENZ)[END_REF], and support the positive impact of abiraterone and enzalutamide treatment on HRQoL under real-world conditions. AQUARiUS also shows that the results achieved in the pivotal studies for each drug can be translated

Funding/Support and role of the sponsor:

This study is funded by Janssen EMEA. The sponsor was involved in data management.

Acknowledgements:

The authors would like to acknowledge the dedicated efforts of the investigational sites that contributed to the study and the patients who allowed collection of their data. A list of all investigators can be found in the Supplementary material. Writing

( 1 )

 1 mean change from baseline in PRO item scores (continuous outcome), with treatment differences summarised by the mean (95% confidence intervals [CIs]); and (2) the percentage of patients who experienced at least one clinically meaningful (ie, 0.5 × standard deviation [SD] of the baseline PRO) worsening (CMW) versus improvement/no change in PRO items (binary outcome) in the 12-mo period,

Fig. 1 -Fig. 2 -

 12 Fig. 1 -Patient flow chart during the study. Abiraterone = abiraterone acetate plus prednisone; ITT

  into clinical practice. matter or materials discussed in the manuscript (eg, employment/affiliation, grants or funding, consultancies, honoraria, stock ownership or options, expert testimony, royalties, or patents filed, received, or pending), are the following: Antoine Thiery-Vuillemin has received grants and nonfinancial support from Janssen, personal fees from Astellas, grants from Sanofi, and grants and personal fees from Ipsen, Roche, BMS, and Pfizer. Guillaume Ploussard has served on advisory boards, and has received honoraria from Astellas Ferring, Bouchara, and Janssen. Alison Birtle has received personal fees from Janssen, Sanofi Aventis, Astellas, and Roche. Elias Pintus has received personal fees from Astellas, BMS, and Clovis Oncology, and nonfinancial support from Astellas, Clovis Oncology, and Janssen.

Alison Reid has received honoraria and travel/meeting assistance from Janssen and travel/meeting assistance from Astellas. Mads Hvid Poulsen, Edouard Lagneau, Louis-Marie Dourthe, Dominique Beal-Ardisson, and Redas Trepiakas have nothing to disclose. Florence Lefresne, Genevieve Pissart, and Suzy Van Sanden are employees of Janssen Pharmaceutica N.V. and hold stock in Johnson & Johnson. Martin Lukac is an employee of PAREXEL International Czech Republic s.r.o, on behalf of Janssen Pharmaceutica N.V., Beerse, Belgium.

Table 1 -Baseline characteristics of abiraterone-and enzalutamide-treated patients

 1 Abiraterone = abiraterone acetate plus prednisone; ECOG = Eastern Cooperative Oncology Group; IQR = interquartile range; mCRPC = metastatic castration-resistant prostate cancer; PSA = prostate-specific antigen.

	Abiraterone (N = 105)	Enzalutamide (N =
		106)

assistance was provided by Samantha Phillips from PAREXEL and was funded by Janssen.

The authors also acknowledge Nathalie Allietta, Hervé Besson, Laurent Antoni, Marieke Buitenhuis, Paul Robinson, Lindsay Dearden, Martin Kluska, Emma Lee, and Divyagiri Seshagiri, employees of Janssen EMEA, who contributed to the study and analysis. The data sharing policy of Janssen Pharmaceutical Companies of Johnson & Johnson is available at https://www.janssen.com/clinical-trials/transparency. As noted on this site, requests for access to the study data can be submitted through Yale Open Data Access (YODA) Project site at http://yoda.yale.edu.

Study concept and design: Thiery-Vuillemin, Reid, Van Sanden, Pissart. Acquisition of data: Thiery-Vuillemin, Poulsen, Lagneau, Ploussard, Birtle, Dourthe, Beal-Ardisson, Pintus, Trepiakas, Reid. Analysis and interpretation of data: Thiery-Vuillemin, Reid, Lefresne, Lukac, Van Sanden, Pissart. Drafting of the manuscript: Thiery-Vuillemin, Poulsen, Lagneau, Ploussard, Birtle, Dourthe, Beal-Ardisson, Pintus, Trepiakas, Lefresne, Lukac, Van Sanden, Pissart, Reid. Critical revision of the manuscript for important intellectual content: Thiery-Vuillemin, Poulsen, Lagneau, Ploussard, Birtle, Dourthe, Beal-Ardisson, Pintus, Trepiakas, Lefresne, Lukac, Van Sanden, Pissart, Reid. Statistical analysis: Van Sanden. Obtaining funding: None. Administrative, technical, or material support: None. Supervision: None. Other: None. Financial disclosures: Antoine Thiery-Vuillemin certifies that all conflicts of interest, including specific financial interests and relationships and affiliations relevant to the subject

Author contributions: Antoine Thiery-Vuillemin had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis.


Alia Mansar

Hélène Serier-Brault

Microwave-assisted synthesis to prepare metalorganic framework for luminescence thermometry

The lanthanide metal-organic framework, La1.23Na0.30(1,3-BDC)2(H2O)•4H2O (1,3-BDC: 1,3benzenedicarboxylate), has been for the first time synthesized under microwave-assisted hydrothermal conditions with a synthesis duration of 5 minutes. The substitution of La 3+ by Eu 3+ and Tb 3+ was possible until a maximal content of 10%.mol and the investigation of luminescence properties reveals that emitting cations can be located in the chains formed by the LnO10 polyhedra but also into the polar channels proposed by the structure, where some La 3+ crystallographic sites are also available. Finally, when codoped by Eu 3+ and Tb 3+ , the materials exhibit thermometric properties in the cryogenic range with a maximal relative thermal sensitivity around 1% K -1 at 80K.

1.INTRODUCTION

Lanthanide-based MOFs (LnMOFs) are considered as promising functional materials for optical properties, especially due to their luminescent properties that can be used for applications such as chemical or temperature sensing, lightings, displays [START_REF] Lustig | Metal-organic frameworks: functional luminescent and photonic materials for sensing applications[END_REF][START_REF] Artizzu | Tailoring functionality through synthetic strategy in heterolanthanide[END_REF][START_REF] Zhao | A Robust Luminescent Tb(III)-MOF with Lewis Basic Pyridyl Sites for the Highly Sensitive Detection of Metal Ions and Small Molecules[END_REF][START_REF] Qin | A water-stable Tb(III)-based metal-organic gel (MOG) for detection of antibiotics and explosives[END_REF][START_REF] Rocha | Lanthanide Organic Framework Luminescent Thermometers[END_REF][START_REF] Hu | Luminescent metal-organic frameworks for chemical sensing and explosive detection[END_REF][START_REF] Busschaert | Applications of Supramolecular Anion Recognition[END_REF][START_REF] Cui | Metal-organic frameworks for luminescence thermometry[END_REF][START_REF] Tian | A luminescent metal-organic framework demonstrating ideal detection ability for nitroaromatic explosives[END_REF][START_REF] Zhao | A chiral lanthanide metal-organic framework for selective sensing of Fe(III) ions[END_REF][START_REF] Cui | Lanthanide-tricarboxylate frameworks: Synthesis, structure and photoluminescence property[END_REF][START_REF] Zhao | Surfactants as promising media in the field of metal-organic frameworks[END_REF]… In our group, we have been working for several years on LnMOFs for luminescence thermometry [START_REF] N'dala-Louika | Ratiometric mixed Eu-Tb metal-organic framework as a new cryogenic luminescent thermometer[END_REF][START_REF] Salomon | A Multifunctional Dual-Luminescent Polyoxometalate@Metal-Organic Framework EuW10@UiO-67 Composite as Chemical Probe and Temperature Sensor[END_REF][START_REF] Viravaux | Temperature sensors based on europium polyoxometalate and mesoporous terbium metal-organic framework[END_REF] which, compared to conventional contact thermometry, has unique and distinct advantages of fast response, high accuracy, being non-invasive and presenting high spatial resolution (typically submicron scale) where traditional methods are ineffective. [START_REF] Brites | Thermometry at the nanoscale[END_REF][START_REF] Savchuk | Novel low-cost, compact and fast signal processing sensor for ratiometric luminescent nanothermometry[END_REF][START_REF] Quintanilla | Guiding Rules for Selecting a Nanothermometer[END_REF] Among the diverse aspects to Ln 3+ -based luminescent thermometry, one of the most robust methodologies relies on the measurement of the intensity of two transitions of distinct Ln 3+ emitting centers. For mixed-metal LnMOFs, the temperature is most often determined from the ratio between the intensity of the 5 D4→ 7 F5 and of the 5 D0→ 7 F2 transitions of Tb 3+ and Eu 3+ , respectively. [START_REF] Rocha | Lanthanide Organic Framework Luminescent Thermometers[END_REF][START_REF] Cui | A Luminescent Mixed-Lanthanide Metal-Organic Framework Thermometer[END_REF][START_REF] Rao | A highly sensitive mixed lanthanide metal-organic framework self-calibrated luminescent thermometer[END_REF] The application of microwave heating in synthetic chemistry starts during the late 1980s. [START_REF] Zhu | Microwave-assisted synthesis of various ZnO hierarchical nanostructures: Effects of heating parameters of microwave oven[END_REF] This technique has been proposed as an alternative to conventional hydro-or solvothermal reactions due to several advantages: (i) energy efficiency, (ii) fast crystallization (increment in number of reaction sites), (iii) phase selectivity, (iv) high yields, (v) variety of morphologies, (vi) particle size control, (vii) lower temperatures and reaction times. Although microwave heating was first used for the synthesis of inorganic materials, this technique has been also developed for the synthesis of MOFs. [START_REF] Klinowski | Microwave-assisted synthesis of metalorganic frameworks[END_REF][START_REF] Głowniak | Advances in Microwave Synthesis of Nanoporous Materials[END_REF][START_REF] Liu | Microwave-assisted synthesis of Zr-based metal-organic framework (Zr-fum-fcu-MOF) for gas adsorption separation[END_REF][START_REF] Firmino | Microwave Synthesis of a photoluminescent Metal-Organic Framework based on a rigid tetraphosphonate linker[END_REF][START_REF] Behrens | Microwave-Assisted Synthesis of Defects Metal-Imidazolate-Amide-Imidate Frameworks and Improved CO2 Capture[END_REF][START_REF] Khan | Synthesis of metal-organic frameworks (MOFs) with microwave or ultrasound: Rapid reaction, phase-selectivity, and size reduction[END_REF] In our group, we have worked, during the last couple of years, on different MOFs [START_REF] N'dala-Louika | Ratiometric mixed Eu-Tb metal-organic framework as a new cryogenic luminescent thermometer[END_REF][START_REF] Trannoy | Engineering of Mixed Eu 3+ /Tb 3+ Metal-Organic Frameworks Luminescent Thermometers with Tunable Sensitivity[END_REF] built upon the same organic ligand, namely 1,3-BDC (1,3-benzene dicarboxylic acid or isophthalic acid), and with Eu 3+ and Tb 3+ as cations in order to elaborate luminescent thermometers. Our first compound [START_REF] N'dala-Louika | Ratiometric mixed Eu-Tb metal-organic framework as a new cryogenic luminescent thermometer[END_REF], [Tb0.87Eu0.13(1,3-BDC)3(H2O)2]H2O, exhibits a crystal structure where Ln 3+ polyhedral form dimers connected by organic linkers to, subsequently form helix chains. The material was performant as a cryogenic luminescent thermometer with a relative thermal sensitivity of 3.26% K -1 at 35.5 K. In a second investigation, we were focused on a series of mixed compounds, [Tb1-xEux(CH3COO)(1,3-BDC)(H2O)] [START_REF] Trannoy | Engineering of Mixed Eu 3+ /Tb 3+ Metal-Organic Frameworks Luminescent Thermometers with Tunable Sensitivity[END_REF], where the Ln 3+ polyhedra form also dimers but in a tridimensional network. In this study, we highlighted the role of the Eu/Tb ratio to tune the thermal sensitivity of the thermometers which were, in that case, sensitive in the physiological range (around 300K). To pursue our investigation of the role of inorganic network in LnMOFs on the thermometric properties while maintaining the same organic ligand, we herein investigate a new LnMOF that we recently discovered, namely Ln1.14Na0.57(BDC)2(H2O)4H2O (Ln = La, or Ce). [START_REF] Trannoy | Lanthanide Isophthalate Metal-Organic Frameworks: Crystal Structure, Thermal Behavior, and White Luminescence[END_REF] In this material, the inorganic network is composed by LnO10 polyhedra connected to form infinite chains linked by the organic linkers. Moreover, the crystal structure exhibits non-polar and polar channels, the latter one being filled by Ln 3+ and Na + cations.

Consequently, we report in this work the microwave synthesis of this new MOF in order to reduce the synthesis time and we have investigated the doping of the La counterpart by Eu 3+ and Tb 3+ to elaborate new luminescent thermometer.

EXPERIMENTAL SECTION

Materials and synthesis

Reagents and Chemicals: La(NO3)3, 6H2O (99.99%), Tb(NO3)36H2O (99.99%), Eu(NO3)36H2O (99.99%), and isophtalic acid (99%) were purchased from Alfa Aesar. All chemicals were used without further purification. Synthesis of LaMOF. 1,3-H2BDC (86 mg, 0.52 mmol) was dissolved in 0.6 mL of an aqueous NaOH solution (1.5 M) and in 10 mL H2O while La(NO3)36H2O (113 mg, 0.26 mmol) was dissolved in 2 mL of dezionized water. The cationic solution was added dropwise in the ligand solution under stirring and a white precipitate was formed. The pH of the mixture was measured at equal to 4.7. Then, the above mixture was placed in a sealed [START_REF] Liu | Mixed-Lanthanoid Metal-Organic Framework for Ratiometric Cryogenic Temperature Sensing[END_REF] Synthesis of LaMOF-5Eu-5Tb and LaMOF-7Eu-3Tb. The procedure was the same as that for NaLaMOF except that the cationic solution was composed by La(NO3)36H2O (101 mg, 0.234 mmol), Eu(NO3)36H2O (5.56 mg, 0.013 mmol), and Tb(NO3)36H2O (5.88 mg, 0.013 mmol) for LaMOF-5Eu-5Tb and by La(NO3)36H2O (101 mg, 0.234 mmol), Eu(NO3)36H2O (6.79 mg, 0.023 mmol), and Tb(NO3)36H2O (3.53 mg, 0.0078 mmol) for LaMOF-7Eu-3Tb. Both samples were synthesized at 170°C/15 min in a Discover SP CEM microwave. Anal. Calcd for LaMOF-5Eu-5Tb, La1.107Eu0.06Tb0.06Na0.30C16O13H18 (%): C, 32.12; H, 3.01. Found: C, 32.19; H, 3.52.

Anal. Calcd for LaMOF-7Eu-3Tb, La1.107Eu0.086Tb0.037Na0.30C16O13H18 (%): C, 32.13; H, 3.01. Found: C, 32.77; H, 3.31. The Eu and Tb contents (in percentage) were determined by ICP-AES to be equal to 48.7/51.3 and 70.1/29.9 LaMOF-5Eu-5Tb and LaMOF-7Eu-3Tb, respectively.

Characterization

Powder X-ray Diffraction spectra were monitored using a D8 Bruker diffractometer in the Bragg-Brentano geometry, equipped with a front germanium monochromator, a copper anode (CuK-L3 radiation λ=1.540598 Å) and a LynxEye PSD detector. The simulated pattern was obtained by the PowderCell 2.4 software from the cif file of the LaMOF compound (CCDC 2013538).

Thermogravimetric analyses (TGA) were performed by flowing dry air with a heating and cooling rate of 5°C/min on a SETARAM TG-DSC 111 between 20 and 800°C. Fourier transform infrared (FT-IR) spectra were recorded in the 4000-400 cm -1 range on a Bruker Vertex FTIR spectrometer equipped with a computer control using the OPUS software. For ICP-AES analyses, samples were dissolved in a 10% HNO3 solution and analyzed by ICP-AES. The calibration curve was established from the analysis of five standard solutions containing Na, La, Eu and Tb in various contents to cover a range in accordance with targeted concentrations. The standard solutions were prepared with high purity monoelemental solutions in acidic solution. Photoluminescence spectra were recorded on a Jobin-Yvon Fluorolog 3 fluorometer equipped with a photomultiplier (excitation source: 450 W Xe arc lamp) using the front face acquisition mode. The emission spectra were corrected for detection and optical spectral response of the spectrofluorimeter and the excitation spectra were weighed for the spectral distribution of the lamp intensity using a photodiode reference detector. The temperature-dependent photoluminescence measurements were recorded on the same spectrometer controlling the temperature by a cryostat coupled with the Fluorolog cooled by liquid nitrogen. Lifetimes measurements were measured with the same equipment but the excitation was performed by a UV Xenon flash tube while the time-dependence of the emission was recorded with the photomultiplier. The morphology of powder was investigated with a field-emission gun scanning electron microscope JEOL JSM7600F operating at 7 kV.

RESULTS AND DISCUSSION

Microwave synthesis

The compound LaMOF crystallizes in the tetragonal apolar space group P42212. [START_REF] Trannoy | Lanthanide Isophthalate Metal-Organic Frameworks: Crystal Structure, Thermal Behavior, and White Luminescence[END_REF] In the network, each La 3+ cation is coordinated by ten oxygen atoms, eight oxygen atoms arising from the two different carboxylate groups and two oxygen atoms from water molecules in a distorted bicapped square prismatic environment. The LaO10 coordination polyhedra are connected by a triangular face along the c axis to form infinite chains. Subsequently, each chain is linked to four others chains by the BDC 2-ligand with a distance between metal centers equal to 11.231(1) Å along the a axis (to form a 3D MOF network with channels running along the c axis (Figure 1a). The synthesis of LaMOF by microwave-assisted hydrothermal synthesis was performed for a concentration of [La 3+ ] equal to 0.02 mol/L and a Metal/Ligand ratio of 1:2. In a first step, the isophthalic ligand was dissolved by a stoichiometric quantity of a NaOH aqueous solution to reach a pH varying from 5 to 5.3 after the whole dissolution. Then the cationic solution was added dropwise to form a white powder, and the pH decreases to around 4.8. Finally, the mixture is placed the microwave oven. The system was investigated for three different temperatures, 150, 160 and 170°C, and for four synthesis durations (5, 15, 30 and 60 min) (Figure 2). For a temperature of 150°C and whatever the synthesis time, the obtained phase corresponds to the precursor already reported in our previous work, [START_REF] Trannoy | Lanthanide Isophthalate Metal-Organic Frameworks: Crystal Structure, Thermal Behavior, and White Luminescence[END_REF] which actually is the initial white precipitate that one formed by adding the cationic solution to the basic solution of ligand. At 160°C, the LaMOF material is obtained only for a synthesis time of 60 min, for lower duration a mixture with precursor is obtained. Finally, at 170°C the LaMOF compound is obtained only after 5 minutes (figure 2b). Le Bail refinement of this compound was performed to confirm the purity of the phase (Figure 1b). In our previous work where samples were synthesized by conventional hydrothermal route (180°C/5 days), we highlighted the presence of Na + cations in the structure originated from the use of sodium hydroxide solution to dissolve the organic ligand. Then, we concluded to a final chemical composition of La1.14Na0.57(1,3-BDC)2(H2O)4H2O, where polar channels exhibited by the crystal structure are occupied by disordered Ln 3+ and Na + cations surrounded by water molecules. Here, we decided to add NaOH solution in a stoichiometric way to dissolve the ligand, which leads to a concentration ten times smaller (9.10 -3 M) and a final chemical composition (determined by ICP analyses) of La1.23Na0.30(1,3-BDC)2(H2O)4H2O. Consequently, the La/Na in the final compound can be modulated by varying the amount of Na + during the synthesis.

The morphology of the LaMOF obtained at 170°C/5 min was assessed by SEM analyses to identify the effect of the time synthesis reduction on the grain size. Therefore, SEM analyses evidence the formation of micro-rods with an average width of 1 µm and an average length of 10 µm (Figure 3). Consequently, the microwave-assisted hydrothermal route enables to get the phase LaMOF in very short synthesis time (5 min) but the use of relatively high temperature, i.e. 170°C, do not allow to obtain of nanopowder in these conditions. The doped LaMOF material, namely LaMOF-5Eu-5Tb and LaMOF-7Eu-3Tb, where also prepared by microwave-assisted hydrothermal route at 170°C but the time duration was increased to 15 min to get pure phase. Both PXRD analysis (Figures S1 andS2) and FT-IR spectroscopy (Figure S3) confirmed the mixed Eu-Tb compounds are isostructural to undoped LaMOF [START_REF] Trannoy | Lanthanide Isophthalate Metal-Organic Frameworks: Crystal Structure, Thermal Behavior, and White Luminescence[END_REF].

Thermogravimetric (Figure S4) and elemental analyses confirmed the chemical compositions.

Moreover, the Tb:Eu molar ratio was determined to high precision by taking ICP-AES measurements (Table S1).

Luminescence properties

The photophysical properties at room temperature of both LaMOF-5Eu-5Tb and LaMOF-7Eu-3Tb were thoroughly investigated. As reported in our previous work [START_REF] Trannoy | Lanthanide Isophthalate Metal-Organic Frameworks: Crystal Structure, Thermal Behavior, and White Luminescence[END_REF], the LaMOF host lattice can be easily doped by small content of the optically active ions Eu 3+ and Tb 3+ to provide white luminescence. Herein, the total amount of luminescent ions was fixed at 10 mol. %, which 10 corresponds to the thermodynamic solubility limit in the framework, and two Eu/Tb ratio were tested, namely 1:1 and 1:0.42. Excitation spectra were monitored at room-temperature within the 5 D0  7 F2 Eu 3+ transition (λem = 614 nm), and the 5 D4  7 F5 Tb 3+ (λem = 542 nm) but also within the ligand broad emission at λem = 394 nm (Figures S5 andS6). The excitation band of the organic ligand, which is centered at 342 nm, is present on excitation spectra monitored at 614 and 542 nm but with a moderate intensity, the main excitation peak being centered at 297 nm for λem = 614 nm (Eu 3+ emission) and at 322 nm for λem = 542 nm (Tb 3+ emission). This second excitation band, located at higher energy than the ligand one, can be assigned to a O-Ln 3+ charge transfer, indicating that emitting lanthanides are also placed into MOF channels, surrounded by the water molecules. [START_REF] Song | Europium-based infinite coordination polymer nanospheres as an effective fluorescence probe for phosphate sensing[END_REF][START_REF] Cunha-Silva | Multi-functional rare-earth hybrid layered networks: Photoluminescence and catalysis studies[END_REF] The presence of this charge transfer band indicates the possibility to excite, in a separate way, the emitters located in the LnO10 chains of the network and those placed in the MOF channels (Figure 1).

The phosphorescence spectrum at 77 K (Figure S7) of LaMOF was recorded to identify the lowest-lying triplet energy state positioning of the organic ligands. Contrary to what expected, the phosphorescence spectrum is composed by two broad bands peaking at 394 nm and 477 nm, respectively. The peak centered at high energy was assigned to the S1  S0 transition as confirmed by the fluorescence spectrum of LaMOF recorded at 300K, while the peak centered at 477 nm corresponds to the T1  S0 transition. Time-resolved spectroscopy was used to precisely determine the triplet energy (Figure S8), and consequently, the triplet level can be estimated to be 22200 cm -1 from the shortest-wavelength, which is in a suitable energy range to sensitize both Eu 3+ and Tb 3+ according to the Latva's empirical rule. [START_REF] Latva | Correlation between the lowest triplet state energy level of the ligand and lanthanide(III) luminescence quantum yield[END_REF] The solid-state luminescence of LaMOF-5Eu-5Tb and LaMOF-7Eu-3Tb was investigated at room temperature for the three different excitation wavelengths. Under UV irradiation, the emission spectra of LaMOF-5Eu-5Tb (Figure S9) and LaMOF-7Eu-3Tb (Figure S10) display the characteristic green and red luminescence of Tb 3+ and Eu 3+ ions, respectively, which exhibit typical lines at 490, 545, 581, and 619 nm and at 587, 611, 652, and 695 nm, attributed to the Tb 3+ 5 D4  7 F6-3 transitions and the Eu 3+ 5 D0  7 F0-4 transitions, respectively. The main effect of excitation wavelength changes concerns the ligand emission, which is present at λexc = 322 or 342 nm whereas it is totally absent et λexc = 297 nm, in agreement with excitation spectra. Thus, under excitation at 322 or 342 nm, both materials exhibit almost white luminescence with CIE parameters close to (0.33; 0.33) (Figure S11, and Table S2) as already reported while at λexc =297 nm, the blue contribution brought by the ligand disappeared to generate orange and red emitters. This shift in the color coordinates is due to the Tb 3+ -to-Eu 3+ energy transfer, which was evidenced by the presence of 7 F6→ 5 D4 Tb 3+ transition (at 483 nm) within the 5 D0→ 7 F2 Eu 3+ transition in the excitation spectra of a mixed compound (inset in Figures S5 andS6). Nevertheless, the excitation band is very weak, assuming that the Tb 3+ -to-Eu 3+ energy transfer is not efficient at room-temperature.

The application of both Eu-Tb-NaLaMOF materials as ratiometric luminescent thermometers was characterized using the emission spectra in the 80-250 K range. All thermometry characterizations were performed at λexc = 297 nm where only the Eu 3+ and Tb 3+ emissions are present. As an illustrative example, Figure 4a presents the temperature dependent emission spectra of LaMOF-7Eu-3Tb in the 77-250 K range (similar data for the LaMOF-5Eu-5Tb material is reported in Figure S12). The temperature dependence of the ITb (green) and IEu (red) parameters is represented in Figure 4b. The integrated areas of the 5 D4→ 7 F5 (ITb) and of the 5 D0→ 7 F2 (IEu) emissions were used to define the thermometric parameter as ∆ = ITb/IEu, ITb and IEu being obtained using the emission spectra in the 530-560 nm and 603-633 nm intervals, respectively. In the 80-300K, ITb and IEu continuously decrease but the thermal quenching of ITb is more significant, with a decrease of 75% while IEu decrease of 30%. The thermometric parameter Δ of both samples (Figure 5a and Figure S12) is depicted according to the temperature in the 77-250 K range. It may be described by the phenomenological Mott-Seitz expression involving a single deactivation channel assuming a temperature dependence of the Eu 3+ emission is smaller than that of the Tb 3+ emission (Figure 4b and Figure S12b) according to the equation ( 2): [START_REF] Rocha | Lanthanide Organic Framework Luminescent Thermometers[END_REF][START_REF] N'dala-Louika | Ratiometric mixed Eu-Tb metal-organic framework as a new cryogenic luminescent thermometer[END_REF][START_REF] Trannoy | Engineering of Mixed Eu 3+ /Tb 3+ Metal-Organic Frameworks Luminescent Thermometers with Tunable Sensitivity[END_REF]]

∆(𝑇𝑇) = ∆ 0 1+𝛼𝛼 exp (-∆𝐸𝐸 𝑘𝑘 𝐵𝐵 𝑇𝑇) ⁄ (2) a) b)
where Δ0 is the Δ parameter at T  0 K, α = W0/WR is the ratio between the non-radiative and radiative decay rates in the limit of T 0 K and ΔE is the energy difference between the emitting level ( 5 D4) and the level responsible for its non-radiative deactivation. Fitting Equation (2) to the experimental Δ(T) curves, we find excellent correlation coefficients attesting the adequacy of this simple model to get the calibration curve of the hybrid materials. Finally, the ∆E value was found equal to 161 ± 4 cm -1 for LaMOF-7Eu-3Tb and 203 ± 3 cm -1 for LaMOF-5Eu-5Tb.

Temperature-dependence of lifetimes where characterized for LaMOF-7Eu-3Tb (Figure S14)

and it evidences a rapid decrease of the 5 D4 decay time between 80 and 300 K while the 5 D0 decay time is almost constant. Consequently, the Tb 3+ -to-Eu 3+ energy transfer does not govern the thermometric properties. Considering the phosphorescence emission of the ligand (Figures S7 andS8) which overlaps the 5 D4 level of Tb 3+ (487 nm), we can tentatively ascribe the deactivation channel to Tb 3+ -to-ligand back energy transfer. Then, in order to evaluate the thermal performances of mixed compounds, we calculated the relative thermal sensitivity defined as 𝑆𝑆 𝑟𝑟= |𝜕𝜕∆/𝜕𝜕𝑇𝑇| ∆ ⁄ which is a usual figure of merit to compare the performance of distinct systems. [START_REF] Rocha | Lanthanide Organic Framework Luminescent Thermometers[END_REF][START_REF] Brites | Lanthanides in Luminescent Thermometry[END_REF][START_REF] Dramicanin | Trends in luminescence thermometry[END_REF] Both materials exhibit thermal sensitivity in the cryogenic range (< 100 K) with a maximum of relative thermal sensitivity of 0.97 %K -1 at 75 K for LaMOF-7Eu-3Tb (Figure 5a), and 0.89 %K -1 at 100 K for LaMOF-5Eu-5Tb (Figure S13b).

The operating temperature of LaMOF when codoped by Eu 3+ and Tb 3+ is perfectly in agreement with the chain topology exhibited by the Ln 3+ polyhedra. Indeed, in the literature, many cryogenic luminescent thermometers based on mixed Eu-Tb MOFs exhibits a chain topology of the inorganic network. [START_REF] Liu | Mixed-Lanthanoid Metal-Organic Framework for Ratiometric Cryogenic Temperature Sensing[END_REF][START_REF] Ananias | Cryogenic Nanothermometer Based on the MIL-103(Tb,Eu) Metal-Organic Framework[END_REF][START_REF] Abdelhameed | Luminescent Nanothermometers Obtained by Post-Synthetic Modification of Metal-Organic Framework MIL-68[END_REF] Thus, the distance Ln-Ln is very short favoring the occurrence of energy transfer between two Ln 3+ or energy migration.

Finally, the temperature uncertainty 𝛿𝛿𝑇𝑇 = 1 𝑆𝑆 𝑟𝑟 × 𝛿𝛿∆ ∆ ⁄ ⁄ , (𝛿𝛿∆ ∆ ⁄ corresponds to the average error imposed by the photomultiplier detection and was equal to 0.2%) which defines the smallest temperature change that may be ascertained in a given measurement, was evaluated as amounting to 0.02 K for both samples at 80K (Figure 5b and S13b). To evaluate whether the material could be used as a reliable thermometer, cycle tests were performed on LaMOF-7Eu-3Tb, and the repeatability was calculated according the equation ( 3): [START_REF] N'dala-Louika | Ratiometric mixed Eu-Tb metal-organic framework as a new cryogenic luminescent thermometer[END_REF][START_REF] Rocha | Near-Infrared Ratiometric Luminescent Thermometer Based on a New Lanthanide Silicate[END_REF] 𝑅𝑅 = 1 -

𝑚𝑚𝑚𝑚𝑚𝑚|∆ 𝑐𝑐 -∆ 𝑖𝑖 | ∆ 𝑐𝑐 (3)
Where ∆C is the mean thermometric parameter, and ∆i is the value of each measurement of the thermometric parameter. Then, four cycles between 80 and 150K were carried out (Figure S15) to †If an author's address is different than the one given in the affiliation line, this information may be included here.

Figure 1 .

 1 Figure 1. a) projection along the c axis of the LaMOF structure (blue: LaO10 polyhedra, red:

Figure 2 .

 2 Figure 2. a) Schematic diagram of obtained phases according the synthesis parameters (yellow

Figure 3 .

 3 Figure 3. SEM photograph of LaMOF synthesized at 170°C/5 min (C = 0.02 M, M/L = 2) by

Figure 4 .

 4 Figure 4. (a) Emission spectra of LaMOF-7Eu-3Tb in the 80-300K range with the excitation

Figure 5 .

 5 Figure 5. (a) Temperature dependence of ∆ in the 80-300K range for LaMOF-7Eu-3Tb with

ACKNOWLEDGMENT

HB thanks Nicolas Stephant, engineer at the Institute of Matériaux Jean Rouxel of Nantes, for SEM images.

get a repeatability R > 97%, which confirms the robustness of the materials for an application as luminescent thermometer.

CONCLUSIONS

The LaMOF compound can be synthesized by microwave-assisted hydrothermal route with a very short synthesis time (170°C/5 min). The temperature synthesis cannot be decreased below 170°C, which limits the obtaining of nanoparticles. When co-doped by Eu 3+ and Tb 3+ , the material can be used as cryogenic luminescent thermometer with a maximum of relative thermal sensitivity of 0.97 %K -1 at 75 K for the LaMOF-7Eu-3Tb. As already evidenced in our previous work, [START_REF] Trannoy | Engineering of Mixed Eu 3+ /Tb 3+ Metal-Organic Frameworks Luminescent Thermometers with Tunable Sensitivity[END_REF] an increase of the Eu content has an impact on the thermal sensitivity which is improved.

ASSOCIATED CONTENT Supporting Information. The following files are available free of charge.

PXRD, Thermal analyses, FTIR, CIE parameters, Luminescence properties (PDF) AUTHOR INFORMATION

Corresponding Author * Helene.brault@cnrs-imn.fr

Present Addresses

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

SYNOPSIS.

La-MOF built upon 1,3-benzenedicarboxylic acid has been synthesized by microwave-assisted hydrothermal route, reducing the synthesis time to a few minutes. The codoping of the MOF by small amounts of Eu 3+ and Tb 3+ leads to new luminescent thermometers operating in the cryogenic range.


Patient education in the context of Parkinson disease cared for with deep brain stimulation (DBS): teaching models, goals and actual implications

Marie Gaille, Mathilde

Several key factors and motivations with important national variations: health education tradition and policies, chronic disease care, patients' participation and patient's movements, patient empowerment, institutionalization both at a national and an international, health budget costs. A history in the making (Bartlett, 1986).

Tensions and semantic disputes: to distinguish between three main visions of patient education:

• To transmit an information/knowledge to the patient in order to help her/him to better deal with her/his condition (vertical or patient centered educational model) • A knowledge co-elaborated with the patient in order to help her/him to live better with her/his condition.

• Patient education as a space of co-elaboration/negotiation of a shared knowledge.

Results:

❖ A clarification of the multiplicity goals and teaching models of patient education is needed. ❖ A specific attention must be paid to patient education in the context of technological care. ❖ An attention must be paid to the long-term patient education effectiveness.

Methodology: review of international literature, conceptual and historical analysis, fieldwork : observation of patient education in the context of Parkinson' disease and interviews with clinicians and patients.

A patient education "in the making" with new issues concerning the patients' use of and learn from a technological care both from short-and long-term perspectives: How far can/will learning go?

The educational model recommended by the HAS not only reproduces the traditional medical approach, but also refers in its implementation to school-based or management strategies. (…) However, both the arrival and the duration of the chronic disease have a critical impact not only on the physiological level but also on the psychological balance. (Lacroix 2009)

  Patient education in the context of Parkinson disease cared for with deep brain stimulation (DBS): teaching models, goals and actual implications. Marie GAILLE, philosopher, Senior Researcher SPHERE (UMR 7219) CNRS Université de Paris Mathilde LANCELOT, philosopher, PhD, Research Assistant -SPHERE (UMR 7219) CNRS Université de Paris Patient Education for Parkinson disease cared with DBS: What is at stake? Starting points: • A recent history of patient education in the context of Parkinson' disease in France (2007) and a history in making for patient education for Parkinson' patient cared with DBS in France (2013). • From an international perspective, patient education models in the context of Parkinson's' disease could be traced back to 1994 (Montgomery et al.) • Several development of patient education for Parkinson' disease cared with or without DBS are in progress in France and elsewhere. Practices differ in terms of contents, places (clinical centers, hospitals, education center, associations) and funding (National and public health funds and/or pharmaceutical industries). Questions: What do these education programs bring to patients? Which goal ? What meaning do they bring to the management of chronic diseases and technological care? Do they meet the needs of ageing people, the need to support longer and more technological lives? Analysis: Several aims and motivations and tensions between different aims: Patient autonomy; "Managing" and better understanding of the disease and treatments; lessen consult's number and frequency; reduction of public health costs in the context of chronic and long-term disease; sharing medical knowledge and disease' experiences; transform the patient-physician relationship, etc. « Patient Education » : What is it in a nutshell ? Starting observations: • From the 1960 on, Western societies have developed patient education, a flourishing and diverse practice in several countries since then. OMS recommendations since the 1990. • In France, patient education as an informal practice and as a practice regulated by Law since 2009. • A lack of reflexivity both in epistemological and ethical terms on this practice. Questions to be investigated: Which knowledge? Which educational relationship? Which goals? Analysis: Several words and meanings: « éducation thérapeutique », « patient education », patient-manager, health manager, health education, etc. Several actors & disciplines. Several aims: knowledge, know-how, being and behavior / « Savoirs-savoir faire et savoir être ».


Charles-Gérard Lucas

Patrice Abry

Herwig Wendt

Gustavo Didier

COUNTING THE NUMBER OF DIFFERENT SCALING EXPONENTS IN MULTIVARIATE SCALE-FREE DYNAMICS: CLUSTERING BY BOOTSTRAP IN THE WAVELET DOMAIN

Keywords: Multivariate selfsimilarity, multivariate scaling exponents, multivariate wavelet transform, bootstrap

Multivariate selfsimilarity has become a classical tool to analyze collections of time series recorded jointly on one same system. Often, it amounts to estimating as many scaling exponents as time series. However, this leaves open the important question how many such scaling exponents are actually different. Elaborating on earlier work aiming to test the hypothesis that all exponents are equal, we intend here to count the number of different scaling exponents from a single finite size multivariate time series. To this end, we devise an original clustering procedure that combines a wavelet domain block multivariate bootstrap scheme with a test strategy for a reduced set of multiple hypotheses on the pairwise equality of scaling exponents that are relevant to clustering. Monte Carlo simulations, making use of synthetic multivariate selfsimilar processes, assess the relevance and performance of the proposed procedure under different scenarios and demonstrate that the proposed method yields practically satisfactory cluster number and size estimations.

INTRODUCTION

Context.

Modern real-world applications often entail the joint analysis of multivariate times series, collected from one same system by a (possibly large) number of sensors. Scale-free dynamics has proven a fruitful paradigm to monitor the status of the system for many applications very different in nature (cf. e.g., [START_REF] Abry | Multivariate scale-free temporal dynamics: From spectral (fourier) to fractal (wavelet) analysis[END_REF]). Multivariate scale-free analysis amounts to estimating the scaling exponents of the data, and it is common practice to estimate as many exponents as there are components in the collected data [START_REF] Abry | Wavelet estimation for operator fractional Brownian motion[END_REF][START_REF] Abry | Wavelet eigenvalue regression for nvariate operator fractional Brownian motion[END_REF]. However, to correctly interpret the status of a system, knowledge of how many of the estimated scaling exponents actually correspond to different values, and estimating how many components are controlled by each of these distinct scaling exponent, is crucial. This is e.g., the case in neurosciences with the analysis of infraslow brain activity (cf. [START_REF] La Rocca | Revisiting functional connectivity for infraslow scale-free brain dynamics using complex wavelets[END_REF]). These two issues have been rarely addressed in the context of multivariate scale-free dynamics and form the core of this work.

Related works.

Selfsimilarity and fractional Brownian motion have long been recognized as relevant models for the scale-free temporal dynamics of single time series [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF][START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF][START_REF] Pipiras | Long-Range Dependence and Self-Similarity[END_REF]. In practice, the estimation of the selfsimilarity exponent is thus central and can be efficiently and robustly performed by means of wavelet transforms [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF][START_REF] Veitch | A wavelet-based joint estimator of the parameters of long-range dependence[END_REF]. The multivariate time series encountered in many modern applications call both for multivariate selfsimilarity models, such as the recently proposed operator fractional Brownian motion (ofBm) [START_REF] Maejima | Operator-self-similar stable processes[END_REF][START_REF] Mason | Sample path properties of operatorself-similar Gaussian random fields[END_REF][START_REF] Didier | Integral representations and properties of operator fractional Brownian motions[END_REF][START_REF] Didier | Exponents, symmetry groups and classification of operator fractional Brownian motions[END_REF], and for multivariate wavelet representation based estimation procedures, such as those developed in [START_REF] Abry | Wavelet estimation for operator fractional Brownian motion[END_REF][START_REF] Abry | Wavelet eigenvalue regression for nvariate operator fractional Brownian motion[END_REF][START_REF] Wendt | Bootstrap-based bias reduction for the estimation of the self-similarity exponents of multivariate time series[END_REF][START_REF] Lucas | Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series[END_REF]. These estimation procedures output as many selfsimilarity parameter estimates as there are time series in the data. The actual and relevant use of such collections of estimates for understanding the system under study requires, as a first step, determining how many of the estimates are actually different. In earlier works [START_REF] Lucas | Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series[END_REF][START_REF] Wendt | Wavelet domain bootstrap for testing the equality of bivariate self-similarity exponents[END_REF], we proposed a waveletdomain block-bootstrap strategy for deciding whether all selfsimilarity exponents are equal or not. Yet, this leaves untouched the critical issue of counting the number of equal selfsimilarity exponents. Goals, contributions and outline. The present work aims to devise a clustering procedure that, from a single observation of finite size multivariate data, is able to count the number of actually distinct selfsimilarity exponents. To that end, the definition of multivariate selfsimilarity is briefly recalled in Section 2 together with the eigenwavelet-based selfsimilarity parameter estimation procedure. The proposed clustering procedure, the key contribution of the present work, is then detailed in Section 3: Building up on [START_REF] Lucas | Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series[END_REF][START_REF] Wendt | Wavelet domain bootstrap for testing the equality of bivariate self-similarity exponents[END_REF], the clustering strategy consists in an original multiple hypothesis test procedure for the equality of the reduced set of pairs of selfsimilarity exponents which, after ordering from small to large values, are neighbors. By construction, the so-obtained test decisions translate directly into clustering results. To estimate the null distributions of the test statistics, we also propose an original wavelet domain time-scale-component-block bootstrap scheme. Monte Carlo experiments using ofBm, reported in Section 4, are used to assess the relevance and performance of the proposed clustering procedure. Results demonstrate that the proposed bootstrap procedure accurately approximates the null distributions, and that our procedure yields satisfactory clustering performance under several scenarios, with different sample sizes or number of components with identical selfsimilarity exponents, both in terms of number of exponents that are actually different, and in terms of the number of components each of them control. OfBm synthesis, selfsimilarity exponent estimation and test procedures are implemented by the authors and will be made available upon publication.

MULTIVARIATE SELFSIMILARITY

Model: Operator fractional Brownian motion

Fractional Brownian motion (fBm) is the only Gaussian, self-similar stochastic process with stationary increments. It is classically used as a paradigm for (univariate) selfsimilarity [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises and applications[END_REF][START_REF] Samorodnitsky | Stable non-Gaussian random processes[END_REF][START_REF] Pipiras | Long-Range Dependence and Self-Similarity[END_REF]. In the present work, we use a collection, X {XH 1 (t), . . . , XH M (t)} t∈R , of M fBm, each with possibly different selfsimilarity exponents H = (H1, . . . , HM ), 0 < H1 ≤ . . . ≤ HM < 1, made dependent via a M ×M covariance matrix ΣX . To model M -variate selfsimilarity, these components are linearly mixed via a M ×M real-valued and

Y {Y H,Σ X ,W 1 (t), . . . , Y H,Σ X ,W M (t)} t∈R W {XH 1 (t), . . . , XH M (t)} t∈R = W X. (1)
M -fBm is a specific case of the more general and inspiring operator fractional Brownian motion, constructed in [START_REF] Maejima | Operator-self-similar stable processes[END_REF][START_REF] Mason | Sample path properties of operatorself-similar Gaussian random fields[END_REF][START_REF] Didier | Integral representations and properties of operator fractional Brownian motions[END_REF][START_REF] Didier | Exponents, symmetry groups and classification of operator fractional Brownian motions[END_REF], which indicates that H and ΣX are dependent parameters that cannot be chosen independently [START_REF] Didier | Integral representations and properties of operator fractional Brownian motions[END_REF].

Estimation: Wavelet analysis

The key step in selfsimilarity analysis lies in the estimation of the H = (H1, . . . , HM ) from the M -variate time series Y . An estimation procedure based on the multivariate discrete wavelet transform (DWT) was proposed in [START_REF] Abry | Wavelet estimation for operator fractional Brownian motion[END_REF][START_REF] Abry | Wavelet eigenvalue regression for nvariate operator fractional Brownian motion[END_REF]

. Let DY m (2 j , k) = 2 -j/2 ψ0(2 -j t- k)|Ym(t) , ∀k ∈ Z, ∀j ∈ {j1, .
. . , j2} denote the univariate DWT coefficients, with ψ0 the mother wavelet [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. Multivariate DWT is defined by stacking of the univariate DWT, DY (2 j , k) = (DY 1 (2 j , k), . . . , DY M (2 j , k)), ∀k ∈ Z, ∀j ∈ {j1, . . . , j2}, with ∀m ∈ {1, . . . , M }. The behaviors along scales 2 j of the eigenvalues of the M × M multivariate wavelet spectrum

S(2 j ) 1 nj n j k=1 DY (2 j , k)DY (2 j , k) * (2)
computed at scale 2 j , provide relevant estimation of H. Yet, this approach was shown to suffer from the so-called repulsion effect in computing equal eigenvalues, thus inducing biased estimates for H [START_REF] Lucas | Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series[END_REF]. To limit the resulting bias, it was proposed to compute modified wavelet spectra with equal number of wavelet coefficients contributing at each scale:

S (w) (2 j ) 1 nj 2 wn j 2 k=1+(w-1)n j 2 DY (2 j , k)DY (2 j , k) * , (3)
for w = 1, . . . , 2 j-j 2 . The eigenvalues {λ

(w) 1 (2 j ), . . . , λ (w)
M (2 j )} are computed independently at each scale 2 j and for each nonoverlapping window w, hence affected by a repulsion effect of same intensity across all scales. Then, estimators Ĥ1, . . . , ĤM are defined as linear regressions, against scales 2 j 1 , . . . , 2 j 2 , of the logarithms λm(2 j ) 2 j 2 -j 2 j-j 2 w=1 log 2 (λ (w) m (2 j )) averaged across windows w:

Ĥm = j 2 j=j 1 vj λm(2 j ) 2 - 1 2 , ∀m = 1, . . . , M, (4)
with vj classical regression weights verifying j jvj = 1 and j vj = 0 (cf. [START_REF] Veitch | A wavelet-based joint estimator of the parameters of long-range dependence[END_REF]). The relevance of this estimation procedure applied to M -fBm was assessed in [START_REF] Lucas | Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series[END_REF].

MULTIVARIATE TIME SCALE BLOCK-BOOTSTRAP BASED CLUSTERING

Testing multiple ordered pairwise hypotheses

For notational simplicity, assume that the vector H is sorted, ∀m = 1, . . . , M -1, Hm+1 ≥ Hm. Our strategy for estimating the groups of equal exponents among H consists in testing the equality of the M -1 pairs of components that are neighbors in H, with correction for multiple test decisions. The positions of the detected nonequalities among these M -1 pairs of ordered exponents, if any, then defines the boundaries for the clusters for the distinct exponents.

The null hypotheses for these M -1 pairwise tests are thus defined as

H (m) 0 : Hm+1 = Hm, m = 1, . . . , M -1. (5)
To construct a test for each hypothesis, we first compute, from a single observation of finite size M -variate data, the vector of M estimates Ĥ as described in Section 2.2, and sort its entries in ascending order, Ĥτ = ( Ĥτ(1) , . . . , Ĥτ(M) ) with Ĥτ(m+1) ≥ Ĥτ(m) , m = 1, . . . , M -1. We then compute the M -1 test statistics defined as

δm = Ĥτ(m+1) -Ĥτ(m) . (6)
It can been shown that the entries of Ĥ are asymptotically Gaussian and weakly dependent, which naturally leads to postulate that under

H (m) 0 , δm follows a half-normal distribution, { √ 2/σm √ π} exp(-δ 2 m /2σ 2
m ). Our numerical simulations validate that this is a reasonable approximation, see Section 4.2 below. The tests for (5) can thus be computed as reject

H (m) 0 if δm > γm, m = 1, . . . , M -1, (7)
where each γm is a rejection threshold. However, since the scale parameters σm are unknown and a priori depend on H, ΣX and W , the values for γm are also unknown. In order to perform the tests, we therefore propose to estimate σm using a bootstrap procedure [START_REF] Lahiri | Resampling Methods for Dependent Data[END_REF][START_REF] Zoubir | Bootstrap techniques for signal processing[END_REF] that is described next.

Multivariate wavelet domain bootstrap for δm

Following schemes proposed in earlier works [START_REF] Lucas | Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series[END_REF][START_REF] Wendt | Wavelet domain bootstrap for testing the equality of bivariate self-similarity exponents[END_REF], collections of bootstrap resamples of the multivariate wavelet coefficients D(2 j , k), k = 1, . . . , nj are computed by a multivariate waveletdomain block-bootstrap procedure that preserves their scale-time multivariate (cross-)dependence structure (while component-wise resampling would, to the contrary, destroy the cross-component dependencies [START_REF] Wendt | Wavelet domain bootstrap for testing the equality of bivariate self-similarity exponents[END_REF][START_REF] Lahiri | Resampling Methods for Dependent Data[END_REF]). Technically, at each scale 2 j , R block bootstrap resamples D * (r) j = (D * (r) (2 j , 1), . . . , D * (r) (2 j , nj)), r = 1, . . . , R are drawn with replacement from (D(2 j , k), . . . , D(2 j , k + LB -1)), k = 1, . . . , nj. The overlapping blocks have size LB in time, and range across all scales and all components jointly. From each bootstrap sample D * (r) j

, bootstrap estimates S * (r,w) (2 j ) and Ĥ * (r) m are computed using Eqs. [START_REF] Abry | Wavelet eigenvalue regression for nvariate operator fractional Brownian motion[END_REF][START_REF] La Rocca | Revisiting functional connectivity for infraslow scale-free brain dynamics using complex wavelets[END_REF].

In order to obtain an approximation of the null distribution of δm, under any possible hypothesis (that is, for H (m) 0 true or not true), the following further steps are required. First, the average value is subtracted from each of the bootstrap resample components, H * (r)

σ * 2 m = Var * ( δ * m )/ 1 - 2 π . (8)
Finally, the bootstrap estimated p-values associated with the rejection of H (m) 0 , m = 1, . . . , M -1, are computed as

p * m = 1 -F δm σ * m , ( 9
)
where F is the cumulative density function of the standardized halfnormal distribution.

Multiple hypotheses test decisions and clustering

The bootstrap p-values ( 9) can now be used to perform the tests [START_REF] Pipiras | Long-Range Dependence and Self-Similarity[END_REF]. Specifically, to correct for multiple hypotheses, a decision d (m) to reject or not independently each of the null hypothesis H (m) 0 at a preset level of false discovery rate (FDR) α is taken using the Benjamini-Hochberg correction procedure [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF],

d (m) α = 1 : p * m < α M -1 π(m), (10)
where π(•) denotes the permutation that orders the p-values, p π(1) < . . . < p π(M -1) . Finally, the vector of M -1 decisions is an indicator vector for the cluster boundaries for H: if there are P rejected hypotheses H (m) 0 , there are P + 1 distinct selfsimilarity exponents and clusters. The detected cluster labels for Hm are defined as

Cα(m) = m m =1 Dα(m ) (11)
where Dα = (1, d

α , . . . , d

(M -1) α
). The whole clustering strategy described in Section 3 is summarized in Fig. 1. The covariance matrix ΣX is chosen such that all diagonal entries are set to 1 and all non-diagonal entries are set to r = 0.5. The M × M invertible matrix W is randomly selected and kept fixed for all experiments. Wavelet analysis is performed with the least asymmetric Daubechies3 wavelet across analysis scales 2 j 1 = 2 8 to 2 j 2 = 2 11 . R = 500 block-bootstrap resamples are drawn from overlapping blocks of size LB = 6 (corresponding to the size of the Daubechies3 mother-wavelet time support). Clustering performance assessment. The adjusted random index (ARI) and Normalized Mutual Information (NMI) [START_REF] Vinh | Information theoretic measures for clusterings comparison: Variants, properties, normalization and correction for chance[END_REF] are used to quantify clustering performance, i.e., how well components with same H are grouped together and components with different H are separated: ARI measures the number of pairs of elements that are correctly grouped or separated, whereas NMI measures the joint entropy of the distributions of estimated and correct clustering.

PERFORMANCE ASSESSMENT

Performance of the clustering strategy

We finally assess the clustering performance for the four scenarios defined above. Fig. 4 reports the histograms of the estimated number of clusters for the different scenarios and three different levels of false discovery rate α = (0.01, 0.05, 0.10). For Scenario1, for which all M -1 = 5 null hypotheses are true, we obtain actual false discovery rates of (0.02, 0.06, 0.11), respectively, which is in good agreement with the preset values. This further corroborates the above numerical analysis on the relevance of our test and detection procedure for null hypotheses. For Scenario2, Scenario3 and Scenario4, for which more than one cluster are to be detected, our proposed procedure detects the correct number of clusters in a majority of the cases. This demonstrates, first, that our proposed test has practically reasonable power (aka detection probability), and second, that our clustering strategy leads to satisfactory results concerning the number of detected groups of equal Hm. As expected, the rate of cluster number over-estimation (under-estimation) increases (decreases) with increasing preset false discovery rate α, respectively.

Table 2 further quantifies this clustering performance analysis and shows average values for NMI and ARI with 95% confidence interval (obtained as averages across realizations) for all four scenarios and false discovery rate α = 0.05. The results indicate that the proposed procedure overall leads to practically satisfactory performance, with NMI values of up to 0.87, and ARI values of up to 0.94. It is interesting to note that Scenario4 is more difficult to cluster than Scenario3, which contains the same number of clusters and values for Hm but equi-sized clusters.

Overall, these results unambiguously validate the relevance practically satisfactory performance of the proposed procedure. Table 2: Clustering performance with 95% confidence interval for significance level ↵ = 0.05.

Scenario1 Scenario2 Scenario3 Scenario4 NMI 0.66 ± 0.02 0.87 ± 0.01 0.79 ± 0.01 ARI 0.94 ± 0.02 0.60 ± 0.03 0.68 ± 0.02 0.59 ± 0.02

CONCLUSION

This work devises a clustering method to group identical selfsimilarity exponents of multivariate time series from an estimation procedure. The proposed approached is based on a pairwise bootstrap testing procedure which has been numerically assessed. The clustering strategy displays good performances.

ACKNOWLEDGMENT

Supported by PhD Grant DGA/AID (no 01D20019023), ANR-16-CE33-0020 MultiFracs, ANR-18-CE45-0007 MU-TATION. Table 2: Clustering performance with 95% confidence interval for significance level α = 0.05.

Scenario1

Scenario2 Scenario3 Scenario4 NMI n/a 0.66 ± 0.02 0.87 ± 0.01 0.79 ± 0.01 ARI 0.94 ± 0.02 0.60 ± 0.03 0.68 ± 0.02 0.59 ± 0.02

CONCLUSION AND PERSPECTIVES

The present work constitutes a first attempt to count the number of scaling exponents, estimated from multivariate data, that are actually different. It relies on combining a wavelet eigenvalues-based estimation procedure, with a multivariate wavelet domain block bootstrap scheme, and false discovery rate corrections for multiple hypotheses testing. Monte Carlo experiments show that the bootstrap procedure satisfactorily reproduces the real distributions of the test statistics and that the overall procedure yields very satisfactory performance in estimating the actual number of scaling exponents and in grouping identical ones together, under several scenarios with one, two or three clusters, with clusters of possibly unbalanced size.

Matlab routines permitting the estimation of the multivariate selfsimilarity exponents as well as their clustering are publicly available 1 as part of our reproducible research effort.

Future investigations will include exploring whether the use of M (M -1)/2 non-sorted hypotheses further improves performance compared to using M -1 sorted pair hypotheses. Large dimension issues (i.e., situations where the number of components grows at fixed rate with the sample size) will also be investigated.

ACKNOWLEDGMENT

Supported by PhD Grant DGA/AID (no 01D20019023), ANR-16-CE33-0020 MultiFracs, ANR-18-CE45-0007 MUTATION. G. Didier's long term visits to ENS Lyon were sponsored by ENS Lyon, the CNRS and the Simons Foundation collaboration grant #714014.

1 https://github.com/charlesglucas/ofbm_tools

Fig. 1 :

 1 Fig. 1: Flowchart of the clustering strategy.

  m , m = 1, . . . , M, r = 1, . . . , R. τ * (r,1) , . . . , H * (r) τ * (r,M ) ), with H * (r) τ * (r,m+1) ≥ H * (r) τ * (r,m) , m = 1, . . . , M -1, r = 1, . . . , R. Then, the bootstrap test statistics δ * (r) m = H * (r) τ * (r,m+1) -H * (r) τ * (r,m) , r = 1, . . . , R, are computed and their bootstrap variance Var * can be used to obtain estimates for the scale parameters σm under the null hypotheses

4. 1 .Fig.

 1 Fig. Distributions of the test statistics δm. Standardized Monte Carlo statistics δm/σm against a standardized half-normal distribution under H1 = . . . = H6 = 0.8 (Scenario1).

4. 2 . 1 .Fig. 3 :

 213 Fig. 3: Distributions of bootstrap statistics δ * m . Quantile-quantileplots of bootstrap standardized δ * m /σ * m for an arbitrarily chosen realization against a standardized half-normal distribution under Sce-nario2 (H (m) 0 is not true for m = 3 and is true for m = 3, and M = 6); black solid lines correspond with identical quantiles, dashed red lines join the first and third quantiles of the distributions.

Table 1 :

 1 Monte Carlo estimates of scale parameters σm and bootstrap scale parameter estimates σ * m (Monte Carlo average and standard deviation) for H1 = . . . = HM = 0.8 (Scenario1). m = 1 m = 2 m = 3 m = 4 m = 5 σm × 10 2

Fig. 4 :

 4 Fig. 4: Estimated number of clusters. Histograms of the estimated numbers of clusters for (a) 1 cluster, (b) 2 clusters and (c-d) 3 clusters with M = 6.

Fig. 4 :

 4 Fig. 4: Estimated number of clusters. Histograms of the estimated numbers of clusters for (a) 1 cluster, (b) 2 clusters and (c-d) 3 clusters with M = 6.