MD Louis Lenfant
email: louis.lenfant@aphp.fr

MD Ugo Pinar

MD Morgan Roupret

PhD Pierre Mozer

PhD Emmanuel Chartier-Kastler

PhD Thomas Seisen

Role of Antimuscarinics Combined with Alpha-blockers in the Management of Urinary Storage Symptoms in Patients with Benign Prostatic Hyperplasia: An Updated Systematic Review and Meta-analysis

Keywords: Muscarinic Antagonists, Lower Urinary Tract Symptoms, Prostatic Hyperplasia, Systematic Review

Purpose: To evaluate the efficacy and safety of combining antimuscarinics with alpha-blockers to treat storage symptoms in men with benign prostatic hyperplasia.

Materials and Methods:

Searches were carried out on PubMed, MEDLINE, EMBASE, and Cochrane databases to identify randomized, placebo-controlled trials published before February 15, 2022, assessing the efficacy or safety of antimuscarinics in men with benign prostatic hyperplasia treated with alpha-blockers. Further meta-analyses were performed using standardized mean difference (SMD) and risk ratio (RR).

Results: A total of 12 randomized trials were included in the systematic review. The metaanalysis showed no impact of antimuscarinics on the number of urgencies per day (SMD= -0.23 [95%CI: -0.64-0.17]; p=0.21). However, the use of antimuscarinics was associated with a small reduction of micturition episodes per day (SMD= -0.19 [95%CI: -0.37; -0.01]; p=0.045). With regards to side-effects, post-void residual increased slightly in patients treated with antimuscarinics (SMD=0.26 [95%CI: 0.15; 0.37]; p<0.01). In addition, there was a higher risk of acute urinary retention (RR=3.26 [95%CI: 1.35; 7.86]; p=0.02), dry mouth (RR=3.43 [95%CI: 1.86; 6.32]; p<0.001), and constipation (RR=2.92 [95%CI: 1.48; 5.73]; p<0.001) with the use of antimuscarinics. Finally, the risk of treatment interruption due to adverse events was higher for the patients treated with antimuscarinics (RR=1.74 [95%CI: 1.27; 2.38]; p<0.01).

Conclusion:

The addition of antimuscarinics to alpha-blockers was not associated with a substantial reduction in urgencies and micturition episodes in BPH patients with storage symptoms. In addition, the toxicity profile was not in favor of antimuscarinic use in these patients.

INTRODUCTION

Benign prostatic hyperplasia (BPH) results from active proliferation of the transitional zone of the prostate gland. While the prevalence of this condition among men over 65-years of age is high, [START_REF] Mcvary | BPH: epidemiology and comorbidities[END_REF] the severity of related lower urinary tract symptoms and their impact on quality of life (QoL) vary from one patient to another. [START_REF] Irwin | Population-based survey of urinary incontinence, overactive bladder, and other lower urinary tract symptoms in five countries: results of the EPIC study[END_REF] Bladder outlet obstruction (BOO) is frequently associated with voiding symptoms, but storage symptoms including urgency and frequency are also observed in patients with BPH. [START_REF] D'ancona | The International Continence Society (ICS) report on the terminology for adult male lower urinary tract and pelvic floor symptoms and dysfunction[END_REF] Although BOO has long been considered the predominant etiology for storage symptoms in BPH patients, many recent data suggest that other bladder-related mechanisms may also be involved. [START_REF] Chapple | Lower urinary tract symptoms revisited: a broader clinical perspective[END_REF][START_REF] Roosen | A refocus on the bladder as the originator of storage lower urinary tract symptoms: a systematic review of the latest literature[END_REF] For example, a large population-based survey showed that the prevalence of storage symptoms increased with age without any difference between men and women. [START_REF] Irwin | Population-based survey of urinary incontinence, overactive bladder, and other lower urinary tract symptoms in five countries: results of the EPIC study[END_REF] In addition, several urodynamic studies reported that detrusor over-activity was more frequent in aging men, regardless of the presence of BOO. [START_REF] Andersen | The correlation between symptoms, cystometric and urodynamic findings[END_REF][START_REF] Christensen | Clinical manifestations of benign prostatic hyperplasia and indications for therapeutic intervention[END_REF] As a consequence, storage symptoms may be primarily age-related rather than prostate-centered.

Based on these data, the historically contra-intuitive use of antimuscarinics has been proposed to treat storage symptoms in BPH patients. Given the risk of acute urinary retention related to the inhibition of bladder contractions, it has been suggested that antimuscarinics should only be used in patients with a post-void residual (PVR) of less than 200 ml. [START_REF] Hao | Antimuscarinics and alpha-blockers or alphablockers monotherapy on lower urinary tract symptoms--a meta-analysis[END_REF] However, there is contradictory evidence to support the efficacy and safety of antimuscarinics in BPH patients. The last meta-analyses specifically assessing the efficacy of antimuscarinics on storage symptoms in men with BPH were published in 2013 [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF] and 2014, [START_REF] Hao | Antimuscarinics and alpha-blockers or alphablockers monotherapy on lower urinary tract symptoms--a meta-analysis[END_REF] and several large, randomized, controlled trials (RCTs) have been published since then. [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Solifenacin plus tamsulosin combination treatment in men with lower urinary tract symptoms and bladder outlet obstruction: a randomized controlled trial[END_REF][START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Van Kerrebroeck | Combination therapy with solifenacin and tamsulosin oral controlled absorption system in a single tablet for lower urinary tract symptoms in men: efficacy and safety results from the randomised controlled NEPTUNE trial[END_REF] The meta-analysis published in 2017 by Dahm et al. was not specifically dedicated to the evaluation of antimuscarinics in the setting of BPH and the main endpoint was the evaluation of the global International Prostate Symptoms Score (IPSS), which does not specifically assess storage symptoms. 13 Against this backdrop, our aim was to provide an updated systematic review and metaanalysis of available data from RCTs evaluating the role of antimuscarinics combined with alpha-blockers to treat BPH patients with storage symptoms using more specific instruments.

MATERIALS AND METHODS

Search strategy

A computerized, bibliographic search of PubMed, MEDLINE, EMBASE, and the Cochrane database was conducted in February 2022 using a free text protocol applying only "Humans" and "English" filters. Different combinations of the following keywords were used:

Inclusion and exclusion criteria

Based on the Preferred Reporting Items for Systematic Reviews and Meta-analyses (PRISMA) statement [START_REF] Page | The PRISMA 2020 statement: an updated guideline for reporting systematic reviews[END_REF] the Patient-Intervention-Comparison-Outcome-Study design (PICOS) approach was used to further select only double-blinded, placebo-controlled RCTs comparing the efficacy and/or safety of antimuscarinics plus alpha-blockers versus alpha-blockers and placebo to treat BPH patients with storage symptoms. Thus, open-label and retrospective studies as well as case reports were excluded from this systematic review. RCTs evaluating antimuscarinics in patients with storage symptoms related to idiopathic or neurogenic overactive bladder syndrome were also excluded. Diagnosis of storage symptoms had to be based on either patientreported symptoms, such as frequency, or urgency, or confirmed by urodynamic studies.

Intervention drugs had to be explicitly labeled as tolterodine, fesoterodine, propiverine, trospium, darifenacin, solifenacin, or oxybutynin, and studies involving non-oral administrations of antimuscarinics were excluded. Finally, reports had to evaluate the efficacy and/or safety of antimuscarinics, and those focusing on patient adherence or cost-effectiveness were excluded.

Systematic review process

After the removal of duplicates, 496 reports were independently reviewed by two authors (L.L. and U.P.) based on title and/or abstract screening to further select 52 studies for full-text assessment. Any disagreement regarding study inclusion was resolved by discussion between the two reviewers (L.L. and U.P.) and a third reviewer (T.S.) was asked to decide if no consensus could be reached. Based on the aforementioned inclusion and exclusion criteria, a final cross-checked selection was made of 12 RCTs published between 2005 and 2016. [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Solifenacin plus tamsulosin combination treatment in men with lower urinary tract symptoms and bladder outlet obstruction: a randomized controlled trial[END_REF][START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Van Kerrebroeck | Combination therapy with solifenacin and tamsulosin oral controlled absorption system in a single tablet for lower urinary tract symptoms in men: efficacy and safety results from the randomised controlled NEPTUNE trial[END_REF][START_REF] Abrams | Safety and tolerability of tolterodine for the treatment of overactive bladder in men with bladder outlet obstruction[END_REF][START_REF] Chapple | Tolterodine treatment improves storage symptoms suggestive of overactive bladder in men treated with alpha-blockers[END_REF][START_REF] Kaplan | Add-on fesoterodine for residual storage symptoms suggestive of overactive bladder in men receiving alpha-blocker treatment for lower urinary tract symptoms[END_REF][START_REF] Kaplan | Tolterodine and tamsulosin for treatment of men with lower urinary tract symptoms and overactive bladder: a randomized controlled trial[END_REF][START_REF] Lee | Combination treatment with propiverine hydrochloride plus doxazosin controlled release gastrointestinal therapeutic system formulation for overactive bladder and coexisting benign prostatic obstruction: a prospective, randomized, controlled multicenter study[END_REF][START_REF] Lee | Initial combined treatment with anticholinergics and alpha-blockers for men with lower urinary tract symptoms related to BPH and overactive bladder: a prospective, randomized, multi-center, double-blind, placebo-controlled study[END_REF][START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF][START_REF] Sener | Efficacy and safety of propiverine and terazosine combination for one year in male patients with luts and detrusor overactivity[END_REF][START_REF] Yamaguchi | Solifenacin as add-on therapy for overactive bladder symptoms in men treated for lower urinary tract symptoms--ASSIST, randomized controlled study[END_REF] Covidence systematic review software (Veritas Health Innovation, Melbourne, Australia) was used to select the studies during the systematic review process, as presented in the PRISMA flowchart (Figure 1).

Data extraction

Data were independently extracted from all included studies by two authors (L.L. and U.P.) with subsequent cross-checks to ensure their accuracy. Covidence systematic review software was used to create a standardized data extraction form that was created a priori. First, studylevel data included the reference, study design, study period, country, overall number of patients included, as well as inclusion and exclusion criteria. Second, patient-level data included mean IPSS and age at inclusion. Third, treatment-level data included the drug and dosage administered and the sample size of each intervention group. It should be noted that when several antimuscarinic dosages were evaluated in the included RCTs, only data from the highest dosage group were extracted. Fourth, efficacy outcomes included the number of urgency and micturitions per day as well as IPSS QoL scores. Finally, safety outcomes included the PVR, and the occurrence of adverse events, such as acute urinary retention (AUR), dry mouth, constipation, and dizziness.

Risk of bias assessment

Two reviewers (L.L. and U.P.) independently assessed the risk of bias (RoB) in each included study using the Cochrane Collaboration RoB tool, which includes seven domains for RCTs: random sequence generation, allocation concealment, participant and staff blinding, outcome assessment blinding, incomplete outcome data, selective reporting, and other sources of bias. [START_REF] Higgins | The Cochrane Collaboration's tool for assessing risk of bias in randomised trials[END_REF] Any disagreement with regards RoB assessment was resolved by discussion between the two reviewers (L.L. and U.P.) and a third reviewer (T.S.) was asked to decide if no consensus could be reached.

Data and statistical analyses

A narrative synthesis of included RCTs was first performed. Descriptive statistics were used to summarize extracted baseline study-, patient-and treatment-level data. Continuous outcomes were described using raw mean differences and standard deviation, or alternatively, median and interquartile ranges, whereas frequencies and proportions were used for categorical outcomes.

A similar methodology was used to report subjective outcomes for efficacy including QoL data.

In addition, meta-analyses were also performed of objective efficacy and safety data.

Our primary endpoint was the efficacy of antimuscarinics using mean changes in urgency and frequency. Raw effect size data in the form of means and standard deviations of the two groups were pooled to calculate standardized mean differences (SMDs). The secondary endpoint was the safety of antimuscarinics including mean changes in PVR calculated using similar a SMD methodology, but also the risk of AUR, dry mouth, constipation, and dizziness calculated using risk ratios (RRs) with the Mantel-Haenszel method to evaluate the weights of studies for binary outcome data. A random effect model was used to calculate these pooled estimates of treatment effects and their 95% confidence intervals [CIs]. Between-study heterogeneity was assessed using the Higgins and Thompson's I 2 statistic, which quantifies inconsistency across trials to assess the impact of heterogeneity on the meta-analyses. If substantial heterogeneity was observed (I 2 >50%), we attempted to determine possible reasons by examining individual trials and subgroup characteristics to check for outlier studies. Sensitivity analyses were further conducted by excluding outlier studies to evaluate the robustness of the pooled effect estimate.

Meta-regression analyses were performed for clinically relevant covariates to further explore potential reasons for heterogeneity and to assess if differences in true effect sizes could be explained outcomes at the study level. Finally, Funnel plots were used to explore bias in the results of meta-analyses. [START_REF] Sterne | Recommendations for examining and interpreting funnel plot asymmetry in meta-analyses of randomised controlled trials[END_REF] Egger's test was used to assess funnel plot asymmetry for continuous outcomes with intervention effects measured as mean differences. 26 All meta-analyses were performed using open-source R statistical software v.4.0.4 (R Foundation for Statistical Computing, Vienna, Austria).

RESULTS

Overall characteristics of the included studies

Overall, 12 RCTs were included [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Solifenacin plus tamsulosin combination treatment in men with lower urinary tract symptoms and bladder outlet obstruction: a randomized controlled trial[END_REF][START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Van Kerrebroeck | Combination therapy with solifenacin and tamsulosin oral controlled absorption system in a single tablet for lower urinary tract symptoms in men: efficacy and safety results from the randomised controlled NEPTUNE trial[END_REF][START_REF] Abrams | Safety and tolerability of tolterodine for the treatment of overactive bladder in men with bladder outlet obstruction[END_REF][START_REF] Chapple | Tolterodine treatment improves storage symptoms suggestive of overactive bladder in men treated with alpha-blockers[END_REF][START_REF] Kaplan | Add-on fesoterodine for residual storage symptoms suggestive of overactive bladder in men receiving alpha-blocker treatment for lower urinary tract symptoms[END_REF][START_REF] Kaplan | Tolterodine and tamsulosin for treatment of men with lower urinary tract symptoms and overactive bladder: a randomized controlled trial[END_REF][START_REF] Lee | Combination treatment with propiverine hydrochloride plus doxazosin controlled release gastrointestinal therapeutic system formulation for overactive bladder and coexisting benign prostatic obstruction: a prospective, randomized, controlled multicenter study[END_REF][START_REF] Lee | Initial combined treatment with anticholinergics and alpha-blockers for men with lower urinary tract symptoms related to BPH and overactive bladder: a prospective, randomized, multi-center, double-blind, placebo-controlled study[END_REF][START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF][START_REF] Sener | Efficacy and safety of propiverine and terazosine combination for one year in male patients with luts and detrusor overactivity[END_REF][START_REF] Yamaguchi | Solifenacin as add-on therapy for overactive bladder symptoms in men treated for lower urinary tract symptoms--ASSIST, randomized controlled study[END_REF] among which two types of patient populations were identified. In six studies, patients without prior BPH treatment were randomized to alphablockers alone or in combination with antimuscarinics. [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Van Kerrebroeck | Combination therapy with solifenacin and tamsulosin oral controlled absorption system in a single tablet for lower urinary tract symptoms in men: efficacy and safety results from the randomised controlled NEPTUNE trial[END_REF][START_REF] Kaplan | Tolterodine and tamsulosin for treatment of men with lower urinary tract symptoms and overactive bladder: a randomized controlled trial[END_REF][START_REF] Lee | Combination treatment with propiverine hydrochloride plus doxazosin controlled release gastrointestinal therapeutic system formulation for overactive bladder and coexisting benign prostatic obstruction: a prospective, randomized, controlled multicenter study[END_REF][START_REF] Lee | Initial combined treatment with anticholinergics and alpha-blockers for men with lower urinary tract symptoms related to BPH and overactive bladder: a prospective, randomized, multi-center, double-blind, placebo-controlled study[END_REF] Of the remaining studies, five included patients with persistent storage symptoms despite prior treatment with alphablockers [START_REF] Kaplan | Solifenacin plus tamsulosin combination treatment in men with lower urinary tract symptoms and bladder outlet obstruction: a randomized controlled trial[END_REF][START_REF] Chapple | Tolterodine treatment improves storage symptoms suggestive of overactive bladder in men treated with alpha-blockers[END_REF][START_REF] Kaplan | Add-on fesoterodine for residual storage symptoms suggestive of overactive bladder in men receiving alpha-blocker treatment for lower urinary tract symptoms[END_REF][START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF][START_REF] Yamaguchi | Solifenacin as add-on therapy for overactive bladder symptoms in men treated for lower urinary tract symptoms--ASSIST, randomized controlled study[END_REF] and one study did not report whether or not alpha-blockers were used before randomization. [START_REF] Sener | Efficacy and safety of propiverine and terazosine combination for one year in male patients with luts and detrusor overactivity[END_REF] Several RCTs reported the results for each of the antimuscarinics tolterodine, solifenacin, and propiverine, while two single trials reported the results for oxybutynin and fesoterodine. The sample size in the overall population was 4634 patients of whom 2273 received tamsulosin alone and 2361 received alpha-blockers in combination with antimuscarinics. The overall characteristics of the included studies are shown in Table 1.

Baseline patient characteristics

At inclusion, the lower age limit ranged between 40-and 50-years, daily urgency and frequency had to be >1-3 and 8, respectively, and the upper limit of PVR varied from 50-200 ml. Mean IPSS at inclusion ranged from 13.3-29.2 and the primary endpoint was assessed at 12 weeks in most studies. The baseline study characteristics are presented in Table 1.

Risk of bias assessment

Despite the high level of evidence of these placebo-controlled RCTs, more than half of the studies had a high or uncertain RoB regarding either random sequence generation, allocation concealment, blinding of participant and personnel, or blinding of outcomes. In these reports, the method of randomization was often not described precisely leading to uncertainty regarding the RoB. Figure 2 shows the RoB assessment for the included studies.

Efficacy outcomes

Subjective measures

IPSS QoL scores were reported in five trials including 1656 patients. The raw mean reduction of IPSS QoL ranged between -1.08 and -0.25. However, the meta-analysis showed no significant reduction in the IPSS QoL score with the use of antimuscarinics (SMD= -0.48 [95%CI: -1.05; 0.09]; I 2 =86% p=0.08; Figure 3A). The substantial between-study heterogeneity was caused by the RCTs of Lee et al. and Cai et al. reporting extreme effects. [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Lee | Initial combined treatment with anticholinergics and alpha-blockers for men with lower urinary tract symptoms related to BPH and overactive bladder: a prospective, randomized, multi-center, double-blind, placebo-controlled study[END_REF] Meta-regression analysis showed that previous treatment with alpha-blockers did not influence the study effect size (p=0.7) and that 94% of the data variability could be attributed to the remaining between-study heterogeneity.

Objective efficacy outcomes

Urgency

Overall, eight RCTs including 3602 patients evaluated the impact of antimuscarinics on urgency. 10-12, 16-18, 20, 23 The raw mean reduction in urgency episodes ranged from 0 to -1.4 per 24 h. However, the meta-analysis showed no significant reduction in the number of urgency episodes per 24 h with the use of antimuscarinics (SMD= -0.25 [95%CI: -0.68; 0.19]; I 2 =88% p=0.2; Figure 3B). The substantial between-study heterogeneity was caused by the RCT of Lee et al. reporting an extreme effect. [START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF] After excluding this study, the sensitivity meta-analysis showed a SMD= -0.07 [95%CI: -0.12; -0.02]; p=0.01, I 2 =0%.

Meta-regression analysis showed that previous treatment with alpha-blockers did not influence the study effect size (p=0.35) and that 95% of the data variability could be attributed to the remaining between-study heterogeneity.

Frequency

Overall, eight RCTs including 3398 patients evaluated the impact of antimuscarinics on frequency. 10-12, 16, 17, 19, 20, 23 The raw mean reduction in frequency episodes ranged between -0.2 and -1.1 per 24 h. The meta-analysis confirmed the significant reduction in number of frequency episodes per 24 h with the use of antimuscarinics (SMD= -0.19 [95%CI: -0.37; -0.01]; p=0.045, I 2 =63%; Figure 3C).

The substantial between-study heterogeneity was caused by the RCT of Lee et al.

reporting an extreme effect. [START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF] After excluding this study, the sensitivity meta-analysis showed a SMD= -0.10 [95%CI: -0.19; -0.02]; p=0.01, I 2 =0%.

Meta-regression analysis showed that previous treatment with alpha-blockers did not influence the study effect size (p=0.20) and that 67% of the data variability in could be attributed to the remaining between-study heterogeneity.

Risk of publication bias for urgency and frequency

Funnel plots for urgency and frequency including all studies from each meta-analyses showed asymmetry (p=0.07 and p<0.01 for the Egger's test for funnel plot asymmetry, respectively; Figure 4).

Objective safety outcomes

Post-void residual and acute urinary retention

Overall, seven RCTs including 2003 patients evaluated the impact of antimuscarinics on PVR. [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Solifenacin plus tamsulosin combination treatment in men with lower urinary tract symptoms and bladder outlet obstruction: a randomized controlled trial[END_REF][START_REF] Chapple | Tolterodine treatment improves storage symptoms suggestive of overactive bladder in men treated with alpha-blockers[END_REF][START_REF] Lee | Combination treatment with propiverine hydrochloride plus doxazosin controlled release gastrointestinal therapeutic system formulation for overactive bladder and coexisting benign prostatic obstruction: a prospective, randomized, controlled multicenter study[END_REF][START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF][START_REF] Sener | Efficacy and safety of propiverine and terazosine combination for one year in male patients with luts and detrusor overactivity[END_REF][START_REF] Yamaguchi | Solifenacin as add-on therapy for overactive bladder symptoms in men treated for lower urinary tract symptoms--ASSIST, randomized controlled study[END_REF] The meta-analysis showed a statistically significant increase in PVR with the use of antimuscarinics (SMD=0.26 [95%CI: 0.13; 0.39]; p<0.01, I 2 =26%; Figure 5A). However, the raw mean increase in PVR ranged from 1-25.5 mL which was not considered clinically significant. Nevertheless, the impact of antimuscarinics on the risk of AUR was evaluated in nine RCTs, 9, 11, 12, 16-19, 21, 23 which showed an increased relative risk ranging from 0-14.5. The meta-analysis confirmed the significantly increased risk of AUR with the use of antimuscarinics (RR=3.26 [95%CI: 1.35; 7.86]; p=0.02, I 2 =0%; Figure 5B).

Other adverse events

Overall, the risk of dry mouth, constipation, and dizziness was evaluated in 11, 9-12, 15-19, 21-23 10, 9-12, 16-19, 22, 23 and four [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Kaplan | Tolterodine and tamsulosin for treatment of men with lower urinary tract symptoms and overactive bladder: a randomized controlled trial[END_REF][START_REF] Lee | Combination treatment with propiverine hydrochloride plus doxazosin controlled release gastrointestinal therapeutic system formulation for overactive bladder and coexisting benign prostatic obstruction: a prospective, randomized, controlled multicenter study[END_REF] RCTs, respectively. Contradictory findings were observed for all these side-effects with the risk of dry mouth, constipation, and dizziness ranging from 0.94- The substantial between-study heterogeneity regarding the side-effect dry mouth was caused by the VICTOR trial reporting extreme effects. After excluding this study, the sensitivity metaanalysis showed RR=3.03 [95%CI: 2.30; 4.00]; p<0.001, I 2 =0%. Meta-regression analysis for dry mouth showed that previous treatment with alpha-blockers did not influence the study effect size (p=0.22) and that 73% of the variability in our data could be attributed to the remaining between-study heterogeneity. Finally, the risk of treatment interruption due to adverse events assessed in nine RCTs including 3810 patients was higher for those treated with antimuscarinics (RR=1.74 [95%CI: 1.27; 2.38]; p<0.01, I 2 =0%, Figure 5F).

DISCUSSION

This systematic review and meta-analysis evaluated the efficacy and safety of antimuscarinics to treat storage symptoms in men with BPH. Our principal finding was that treatment with antimuscarinics was not associated with any substantial benefit on urgency and frequency over established treatment with alpha-blockers. Furthermore, there were more side-effects with antimuscarinics compared to alpha-blockers when the data were sufficient to have a pooled comparison.

The main strength of this systematic review is that it addresses a clinically relevant issue that most urologists frequently encounter in their daily practice. It is the first review to specifically evaluate the efficacy of antimuscarinics on storage symptoms in patients with BPH.

Indeed, most reviews have evaluated the efficacy of antimuscarinics on total IPSS or Qmax, 13,[START_REF] Fullhase | Systematic review of combination drug therapy for non-neurogenic male lower urinary tract symptoms[END_REF] but these parameters do not specifically assess storage symptoms. It was deemed necessary to fill this gap by evaluating the effect of antimuscarinics on frequency and urgency in a systematic review with a stringent methodology following a pre-established protocol and PRISMA criteria, including randomized, placebo-controlled studies with a high level of evidence. Most previous systematic reviews included open-label studies and did not include a rigorous assessment of the quality of evidence beyond the study design, which despite randomization was often subject to measurement bias.

Tolterodine and solifenacin were the most frequently studied agents in this review and were evaluated in five and four RCTs, respectively. Their efficacy at reducing urgency and micturition was limited. The maximum raw mean reduction of urgency and frequency were -1.4 and -1.1 episodes per day, respectively. As shown previously in the setting of idiopathic overactive bladder, [START_REF] Lee | Comparison of doxazosin with or without tolterodine in men with symptomatic bladder outlet obstruction and an overactive bladder[END_REF] the differences in efficacy between the types of drugs were minimal and the overall efficacy was limited, despite statistically significant but small differences when comparing the efficacy of the drugs with placebo.

A meta-analysis of urgency and frequency showed substantial between-study heterogeneity. The examination of individual trials and subgroup characteristics identified Lee et al. 2011 [START_REF] Sener | Efficacy and safety of propiverine and terazosine combination for one year in male patients with luts and detrusor overactivity[END_REF] as an outlier for both outcomes, mainly because of its wide confidence interval.

Further sensitivity analyses excluding outliers reported statistically significant results showing a non-clinically relevant reduction of urgency (SMD=0.07). Six studies included patients without prior BPH treatment [START_REF] Cai | Efficacy and Safety of Medium-to-long-term Use of Tolterodine Extended Release with or without Tamsulosin in Patients with Benign Prostate Hyperplasia and Larger Prostate Size: A Double-blind, Placebocontrolled, Randomized Clinical Trial[END_REF][START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Van Kerrebroeck | Combination therapy with solifenacin and tamsulosin oral controlled absorption system in a single tablet for lower urinary tract symptoms in men: efficacy and safety results from the randomised controlled NEPTUNE trial[END_REF][START_REF] Kaplan | Tolterodine and tamsulosin for treatment of men with lower urinary tract symptoms and overactive bladder: a randomized controlled trial[END_REF][START_REF] Lee | Combination treatment with propiverine hydrochloride plus doxazosin controlled release gastrointestinal therapeutic system formulation for overactive bladder and coexisting benign prostatic obstruction: a prospective, randomized, controlled multicenter study[END_REF][START_REF] Lee | Initial combined treatment with anticholinergics and alpha-blockers for men with lower urinary tract symptoms related to BPH and overactive bladder: a prospective, randomized, multi-center, double-blind, placebo-controlled study[END_REF] while five studies included patients with persistent storage symptoms despite prior treatment with alpha-blockers. [START_REF] Kaplan | Solifenacin plus tamsulosin combination treatment in men with lower urinary tract symptoms and bladder outlet obstruction: a randomized controlled trial[END_REF][START_REF] Chapple | Tolterodine treatment improves storage symptoms suggestive of overactive bladder in men treated with alpha-blockers[END_REF][START_REF] Kaplan | Add-on fesoterodine for residual storage symptoms suggestive of overactive bladder in men receiving alpha-blocker treatment for lower urinary tract symptoms[END_REF][START_REF] Macdiarmid | Efficacy and safety of extendedrelease oxybutynin in combination with tamsulosin for treatment of lower urinary tract symptoms in men: randomized, double-blind, placebo-controlled study[END_REF][START_REF] Yamaguchi | Solifenacin as add-on therapy for overactive bladder symptoms in men treated for lower urinary tract symptoms--ASSIST, randomized controlled study[END_REF] However, metaregression analyses showed that previous treatment with alpha-blockers did not influence the study effect size with regard to urgency and frequency.

The risk of publication bias estimation through funnel plots and Eger's test showed significant asymmetry. The asymmetry was particularly observed for frequency (p<0.01) with studies seeming to be missing in areas of non-significance, which could be due to reporting studies [START_REF] Kaplan | Add-on fesoterodine for residual storage symptoms suggestive of overactive bladder in men receiving alpha-blocker treatment for lower urinary tract symptoms[END_REF][START_REF] Kaplan | Tolterodine and tamsulosin for treatment of men with lower urinary tract symptoms and overactive bladder: a randomized controlled trial[END_REF][START_REF] Sener | Efficacy and safety of propiverine and terazosine combination for one year in male patients with luts and detrusor overactivity[END_REF] required ≥3 urgency episodes per 24 h at baseline. Second, differences between alpha-blocker treatments may also be accountable for the differences in efficacy between studies. Tamsulosin was the alpha-blocker prescribed in the majority of studies, but other studies included patients with other alpha-blockers prescribed before study initiation. Third, variability in the prescribed dose of antimuscarinic and, in particular, the difference in protocol with flexible-dose escalation (VICTOR and Kaplan et al. 2012 studies [START_REF] Kaplan | Safety and tolerability of solifenacin add-on therapy to alpha-blocker treated men with residual urgency and frequency[END_REF][START_REF] Kaplan | Add-on fesoterodine for residual storage symptoms suggestive of overactive bladder in men receiving alpha-blocker treatment for lower urinary tract symptoms[END_REF] or fixed-dose studies may also account for the variation in treatment efficacy. The variability in efficacy associated with the higher rate of side-effects could lead to a rethink of the prescribing model by moving towards on-demand treatment, which could be useful to maximize patient compliance and have a reasonable ratio between benefit and adverse effects.

The pathophysiological mechanism leading the patient to have urgency may vary from patient to patient and may also explain the differences in efficacy of symptomatic treatment with antimuscarinics. When urgency is related to an irritative spike in the bladder mucosa triggering uninhibited detrusor contractions, the expected benefit of antimuscarinics may be high. In other patients, sphincter insufficiency due to fatigue and age-related amyotrophy can lead to urine passage through the urethra and trigger an urge to urinate through a uretrovesical reflex mechanism without real bladder hyperactivity. For these patients, the expected efficacy of antimuscarinics would be quite limited. Identifying patients with muscarinic receptor dysregulation resulting in increased sensitivity to acetylcholine due to chronic obstruction warrants further research. Understanding which patients are likely to respond, or not, could significantly shift the outcomes of future similar studies.

One of the main concerns about prescribing antimuscarinics in patients with BPH is the risk of AUR and increased PVR. In this meta-analysis, we found a significant, yet not clinically relevant, PVR mean difference for patients treated with antimuscarinics (SMD=13.3 [95%CI: 5.2; 21.3]; p<0.01). However, most of these studies excluded patients with high PVR before treatment. Therefore, these results are not transferable to BPH patients with significant PVR.

Notwithstanding the exclusion of patients with high PVR and the small difference in mean PVR after treatment, the risk of AUR was increased more than threefold in patients treated with antimuscarinics (RR=3.26 [95%CI: 1.35; 7.86]; p=0.02, I 2 =0%), but the overall incidence was less than 3% in all trials.

Our results are consistent with those of a previous meta-analysis. Füllhase et al. reported that add-on combination therapy, compared with alpha-blockers alone, showed a 9.2% reduction in 24-h voiding frequency and a reduction of 1.1 urgency episodes per day. [START_REF] Fullhase | Systematic review of combination drug therapy for non-neurogenic male lower urinary tract symptoms[END_REF] In a more recent meta-analysis, Dahm et al. evaluated the effect of add-on therapy with an antimuscarinic on the mean change in IPSS score and IPSS QoL, but did not report any findings regarding urgency or frequency. In this study, darifenacin, fesoterodine, solifenacin, oxybutynin, tolterodine, and trospium were not associated with a significant improvement in the IPSS score compared to the control group. Finally, a systematic review by Kaplan et al. [START_REF] Kaplan | Antimuscarinics for treatment of storage lower urinary tract symptoms in men: a systematic review[END_REF] reported the results of the randomized, double-blind TIMES, ADAM, VICTOR, MacDiarmid et al. studies and several other prospective studies, [START_REF] Athanasopoulos | Safety of anticholinergics in patients with benign prostatic hyperplasia[END_REF][START_REF] Lee | Comparison of doxazosin with or without tolterodine in men with symptomatic bladder outlet obstruction and an overactive bladder[END_REF][START_REF] Yang | Efficacy and safety of combined therapy with terazosin and tolteradine for patients with lower urinary tract symptoms associated with benign prostatic hyperplasia: a prospective study[END_REF] and concluded that antimuscarinic plus alpha-blocker therapy provided a significant benefit to men with BPH and storage symptoms.

However, this was a systematic review without any meta-analysis and the authors did not provide detailed results on mean change in micturition or urgency. The meta-analysis conducted in our study found a significant but small reduction in urgency and frequency. Although our results are in line with current EAU and AUA guidelines, 32, 33 which advocate cautious use of antimuscarinics in patients with high PVR, we also relativize the observed clinical effectiveness of antimuscarinics on storage symptoms. The level of evidence on which the recommendations were based was intermediate (i.e., Level 2) because it was partly derived from open-label studies and post-hoc analysis. Our meta-analysis of high-level evidence studies should be useful in tempering these recommendations and in guiding patients towards on-demand treatment while warning of the risk of side-effects and variable efficacy.

Despite the aforementioned strengths and the add-on value to the existing literature, this study is not devoid of limitations. First, the placebo group cannot be considered a no-treatment group in these studies, especially because patients were required to complete a voiding diary.

The latter provides insight into behavioral adjustments that can improve symptoms by educating patients about their voiding habits. Second, the primary endpoint was measured at 12 weeks for most studies. Thus, there is no clear evidence that the efficacy of treatment, however limited, persists beyond this period. Similarly, there is no evidence that the safety profile and side-effects remain the same after 3 months of treatment in patients with BPH. However, since bladder alterations due to chronic obstruction may take months to completely resolve, studies with longer follow-up may yield different results. Third, the current study did not identify any predictive factors of efficacy of antimuscarinic therapy in patients with BPH. Previous posthoc analyses showed that a combination therapy with tolterodine and tamsulosin was effective in subjects regardless of whether their baseline prostate size or serum prostate-specific antigen level was above or below the respective study median. [START_REF] Roehrborn | Tolterodine extended release with or without tamsulosin in men with lower urinary tract symptoms including overactive bladder symptoms: effects of prostate size[END_REF][START_REF] Roehrborn | Effects of serum PSA on efficacy of tolterodine extended release with or without tamsulosin in men with LUTS, including OAB[END_REF] Fourth, some studies were not designed specifically to measure the improvement in frequency and urgency as main endpoints and this might lead to possible bias. Finally, the variability between inclusion and exclusion criteria in all the studies included leads to heterogeneity in the severity of baseline symptoms, which complicates the comparison of raw mean reduction in the main outcomes of interest.

CONCLUSION

Treatment with antimuscarinics was not associated with a short-term substantial reduction of urgency and frequency over established treatment with alpha-blockers. Furthermore, there were also more adverse events with antimuscarinics than with alpha-blockers when the data were sufficient for a pooled comparison. Clinicians should be aware of these findings in order to explain to their patients the expected limited short-term benefit of antimuscarinic therapy on storage symptoms and the potential side-effects of this treatment. A negative SMD means that the experimental group has a lower mean score than the control group (i.e., a reduction in the assessed outcome for the antimuscarinic group)

(

  Muscarinic Antagonists[MeSH] OR Tolterodine[All Fields] OR Fesoterodine[All Fields] OR Propiverine[All Fields] OR Trospium[All Fields] OR Darifenacin[All Fields] OR Solifenacin[All Fields] OR Oxybutynin[All Fields]) AND (Prostatic Hyperplasia[All Fields] OR 'Lower Urinary Tract symptoms'[All Fields]). Studies including the following keywords were excluded from the search: (Review[Publication Type] OR Surgery[MeSH] OR Compliance[MeSH] OR Cost Analysis[MeSH] OR Validation Study[Publication Type]). This systematic review was registered on PROSPERO (registration number CRD42022310941).

34. 2

 2 , 0.97-16.1, and 0.48-8.3, respectively. The meta-analyses showed a significantly increased risk of dry mouth (RR=3.10 [95%CI: 1.71; 5.62]; p<0.001, I 2 =76%; Figure 5C), constipation (RR=3.52 [95%CI: 2.23; 5.54]; p<0.001, I 2 =0%; Figure 5D), and dizziness (RR=1.02 [95%CI: 0.22; 4.69]; p=0.9, I 2 =34%; Figure 5E) with the use of antimuscarinics.

  bias. Several factors might explain the limited observed efficacy of antimuscarinics: first, the heterogeneity between studies regarding the selection criteria, and particularly the difference in severity of storage symptoms at baseline, complicates the comparison. Whereas the ADAM, VICTOR, Lee at al. 2011, and Lee et al. 2005 studies 11, 16, 20 required that eligible subjects had ≥1 urgency episodes per 24 h at baseline, the TIMES, Sener et al. 2013, and 2012 Kaplan et al.

Figure 1 .

 1 Figure 1. Preferred Reporting Items for Systematic Reviews and Meta-Analyses

Figure 2 .

 2 Figure 2. Risk of bias assessment.

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Comparison of alpha-blockers vs. alpha-blockers plus antimuscarinics on (A)

Data availability Statement

The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.


Antoine Seguin

Yann Bertho

Baptiste Darbois Texier

Penetrating a granular medium by successive impacts

We consider the penetration dynamics of a vertical cylinder into a dry granular medium subjected to successive impacts. The depth of the impactor below the free surface zN first evolves linearly with the impact number N and then follows a power-law evolution zN ∝ N 1/3 . The depth reached by the cylinder after a given number of impacts is observed to increase with the impact energy but to decrease with its diameter and the density of the granular medium. We develop a model that accounts for the quasi-static and inertial granular forces applying on the cylinder to rationalize our observations. This approach reveals the existence of two intrusion regimes for large and small impact numbers, allowing all data to be rescaled on a master curve. Then, we extend the study to the effect of sidewalls on the dynamics of the impactor. We show that lateral confinement changes the dependence of the impactor depth on the impact number zN (N ). This effect is accounted for by considering the increase of the granular drag with the lateral confinement.

I. INTRODUCTION

Probing the ground to extract samples or to dig up buried objects is at the heart of many civil engineering activities and exploration missions. Several applications, such as tunneling or underground scanning, would benefit from the development of subterranean robots capable of penetrating and moving below the surface [1,2]. Among the techniques used to penetrate a probe into the ground, several rely on the transfer of kinetic energy to the digging impactor as in drilling [3]. This transfer can also be done using a succession of steps like when someone drives a nail, i.e., hammering. In particular, the hammering mechanism underpins the operation of space probes that are designed to dive below the sandy surface of other planets and inspect their physical properties [4,5]. Despite its common use, the hammering technique as a means of penetrating a granular soil is little explored from a physical point of view.

The study of granular media has followed an intensive effort these last decades, with a particular focus on the stress state within these materials. The pioneering work in this field is by Janssen who considered the stress distribution in a confined granular medium [6]. He observed that the pressure exerted on the base of the container saturates at a value much smaller than the hydrostatic pressure that would be measured for a liquid column. Instead, the pressure saturates at a constant value that scales as the weight of grains over a vertical distance comparable to the width of the container. His continuum phenomenological model relies on the hypothesis that the frictional contact forces between the grains and the walls are at their Coulomb threshold and that the vertical stresses are redistributed proportionally to the horizontal ones. This effect, characteristic of granular media, has led to numerous works in static [7][8][9] or dynamical configurations [10,11].

Over the last few decades, many works were devoted to the dynamics of an object impacting a granular medium. A large part of this effort concerns spherical projectiles, well suited to the study of impact crater morphology [12][13][14][15]. This configuration has implications in geophysics for meteorite impacts [16,[START_REF] Katsuragi | Physics of Soft Impact and Cratering[END_REF], a problem in which the effective gravity must be taken into account [START_REF] Altshuler | [END_REF]. Different experiments and numerical simulations permitted to estimate the drag force applying on the sphere during its intrusion [19][20][21][22][23][24][25][26][27][28]. In parallel, theoretical models were proposed to rationalize these observations [29][30][31]. A widely used model to describe the drag force during the impact of a sphere is that which considers the sum of two contributions: a collisional term and a Coulomb friction term [21]. The collisional term evolves as v 2 , where v is the penetration velocity of the sphere, and is associated to the energy dissipation by collisions with the grains constituting the granular medium [32,33]. This expression for the inertial drag was shown to hold for objects of different shapes with a prefactor that accounts for their geometry [34]. The frictional term, by analogy with the hydrostatic pressure in a classical fluid, is described with an evolution as ρgz, where z is the penetration depth of the object [35]. This prediction holds for small depths and different models were proposed for the drag force at larger depths. In this limit, the quasi-static drag force on a penetrating object was described by a generic term in z α , with α > 1 and that depends on the object geometry [36][37][38]. More recently, the quasi-static granular drag experienced by a slender object was rationalized as the sum of a z term and a z 2 term [39]. This model is the result of two contributions: the pressure of the granular force at the tip of the slender object (∝ z) and the granular frictional force on the lateral side of the submerged part, integrated over the entire depth (∝ z 2 ). The previous expressions for the drag force permitted to capture finely the dynamics of an object that realizes a single impact in a granular medium. However, the predictions of these models regarding a succession of impacts remain to be explored.

In this paper, we study experimentally the penetration of a slender object (called impactor hereafter) into a dry granular material by successive impacts. First, we measure the evolution of the penetration depth of the impactor as a function of the number of impacts it receives, in a granular medium that can be considered as infinite. We highlight the existence of two regimes and we develop a continuous modeling of the penetration depth. Second, we explore the influence of a lateral confinement during the penetration and we show that friction to the walls introduces a modification of the scaling law of the penetration depth.

II. EXPERIMENTAL SETUP

The experimental setup consists of a cylindrical container of diameter D, height 300 mm, containing rigid beads of diameter d g and density ρ g [Fig. 1(a)]. The beads used are made of different materials, plastic, glass, and metal of density ρ g = 920, 2500, and 7740 kg m -3 , and diameters d g = 1, 0.45 and 2 mm, respectively. A controlled preparation of the granular medium is obtained by pouring the grains into the container and then tapping it at least ten times on the floor, ensuring an initial volume fraction φ = 0.62±0.01. We define the bulk density of the granular medium as ρ = φ ρ g . The impactor is a hollow cylinder of external diameter d (ranging from 16 to 50 mm) and mass m 0 (ranging from 60 to 460 g) ending with a 60 • angle conical head, as sketched in Fig. 1(a). Initially, the tip of the impactor is placed at a depth z 0 = 50 mm below the surface of the granular medium to ensure its stability. The impacts are generated by a cylindrical mass m i = 209 g dropped inside the hollow cylinder from a height h, without any initial velocity. When the mass reaches the bottom of the impactor, it transfers its momentum and the impactor moves deeper in the granular medium. The impacts are reproduced several times in a row, keeping constant the dropping height h and thus the impact energy E = m i gh. A picture of the impactor is taken after each impact and allows one to measure its final depth z N below the granular surface as a function of the impact number N . Particular care has been taken to systematically center the impacting mass with the axis of the impactor to ensure that the system remains vertical during successive impacts (deviation less than 2 • ). In addition, we checked that the time interval between impacts has no influence on the intrusion dynamics in a range between 10 s and 100 s. This interval time is much larger than the characteristic time required for a grain to fall from its own diameter d g /g 10 ms. In all the experiments, the ratio of the impactor diameter over the bead diameter d/d g is always larger than 10, allowing to describe the medium in the limit of continuum mechanics [39]. Note that we do not measure significant variations of the mean packing fraction during the successive impacts, as each impact reinforces the initial tapping protocol. Also, to avoid any interaction with the bottom of the container, the distance between the tip of the impactor and the bottom is at least ten times larger than the diameter of the grains [40].

III. PENETRATION IN UNBOUNDED GRANULAR MEDIUM A. Experimental results

We first consider the case where the diameter of the container is large compared to the impactor diameter (D > 8 d) and side walls have a negligible effect on the impactor penetration [23]. A typical penetration dynamics obtained under these conditions is shown in Fig. 1(b) for an impactor entering into glass beads and experiencing N = 125 successive impacts. We observe that the dynamics is fast for the first impacts then slows down with the number of impacts, but continues to progress without saturating with depth. The inset in Fig. 1(b) displays the logarithmic plot of the data and shows that the penetration dynamics is well approximated by the scaling law z N ∝ N 1/3 when N 10.

The inset in Fig. 2(a) presents the penetration dynamics z N for an impactor entering glass beads and submitted to different impact energies E. We observe that the intrusion dynamics keeps the same increasing shape whatever the impact energy, but that the depth after a given number of impacts increases with E. The same data are presented in Fig. 2(a) as a function of the total impact energy delivered to the impactor N E m i /m, where m = m 0 + m i . It turns out that experiments performed with different pairs of h and m i but the same impact energy E = m i gh leads to the same results, highlighting that the impact energy is the relevant parameter in this problem.

The effect of the impactor diameter d on the penetration depth is displayed in Fig. 2(b) as a function of the impact number N , by keeping constant the impact energy E. The intrusion dynamics is then observed to be slower for larger impactor diameters.

Finally, we explore the influence of the density of the granular medium on the penetration by successive impacts [Fig. 2(c)]. We realize experiments with grains of different materials by keeping unchanged the impactor characteristics (m 0 = 210 g, m i = 209 g, and d = 33 mm) as well as the impact energy (E = 0.52 J). We observe that increasing the bulk density ρ of the granular medium significantly slows down the intrusion dynamics.

B. Model

To model the evolution of the penetration depth z N with the number N of successive impacts, we consider the forces that apply on the impactor during the advancing phases. The impactor experiences its own weight, the force resulting from the impact of the internal mass, and the resistance of the granular medium. Here, the weight of the impactor mg is small compared to the granular drag and can be neglected in the force balance. The mechanical action exerted by the impacting mass is accounted through momentum conservation during the collision of the internal mass with the bottom of the cylinder. The impacting mass m i dropped from a height h without initial velocity has a mechanical energy m i gh and a velocity before impact √ 2gh. The collision of the internal mass m i with the penetrating object of mass m 0 is assumed to be perfectly inelastic. In these conditions, the conservation of the momentum of the system provides that the initial velocity of the impactor just after the impact is v i = √ 2gh m i /m. The resistive forces resulting from the interaction of the impactor with the granular material have both a collisional and a frictional origin. At each new impact, two temporal regimes can be distinguished [24]: a first phase corresponding to the earliest instants where the inertial resistance dominates and where the drag force strongly depends on the velocity of the impactor; a second phase where the quasi-static resistance dominates, independent of the impactor velocity. In the following, we will focus on the terms corresponding to these two regimes, one after the other.

By considering first the case where the velocity of the impactor is high and the inertial contribution of granular drag, proportional to the cross sectional area (∼ d 2 ) and to the density of the granular medium ρ, overcomes the quasi-static contribution, the equation of motion for the impactor of mass m is written as

m dv dt = -K v ρd 2 v 2 , (1)
where v = dz/dt and K v is a characteristic prefactor of the collision dissipation. As in the case of a sphere impact in a granular medium, K v is expected to depend on the packing fraction φ [12,24,31]. The model is developed in the limit of continuous depths z and then discretized to z N to be compared with experiments.

At the beginning of impact number N , the impactor is located at the depth z = z N -1 of the previous impact and its velocity is v = v i . By using the relation

dv/dt = (dv/dz)(dz/dt) = v dv/dz = (1/2) d(v 2 )/dz, Eq. (1) leads to v 2 (z) v 2 i = exp - z -z N -1 L 0 , (2)
where L 0 = m/2K v ρd 2 is a characteristic length, which scales as m/ρ d 2 . This expression is valid at the first instants of the penetration dynamics. The final depth z N is reached when the impactor velocity vanishes, i.e., v(z N ) = 0. We obtain a relation between two successive rest positions z N -z N -1 = L 0 , which, by iteration from the initial depth z 0 , leads to the following prediction:

z N -z 0 L 0 = N. (3)
This relationship describes the first asymptotic regime that is expected to occur during the first set of impacts at shallow depths. We now consider the opposite situation where the quasi-static contribution of the drag force overcomes the inertial term and which occurs in the limit of low velocities or high depths. In this limit, the resistance experienced by the impactor from the granular medium is proportional to z 2 [39,41]. We neglect here the resistive force term of the pressure at the tip of the impactor, which should evolve as z, assuming it is approximately cylindrical. Under these conditions, the equation of motion for the penetration dynamics of the impactor is written as

m dv dt = -K z ρgd z 2 , (4)
where K z is a characteristic prefactor of the friction force. This equation can be made non-dimensional by using L 0 and by introducing a characteristic time T 0 = (K v d/K z g) 1/2 , which scales as (d/g) 1/2 . For the impact number N , the initial conditions are z = z N -1 and v = v i . Using the same relation as before for dv/dt, we obtain the following solution:

v 2 (z) v 2 i = 1 - 1 3 L 0 v i T 0 2 z 3 -z 3 N -1 L 3 0 . (5)
The final depth z N is reached when v(z N ) = 0. We obtain a relation between two successive impacts, which, by iteration with respect to the initial depth, leads to the following prediction:

z 3 N -z 3 0 L 3 0 = 3N T 0 v i L 0 2 . (6)
In the limit of large depths (z N z 0 ), we can approximate that z 3 N -z 3 0 (z N -z 0 ) 3 and the previous equation leads to:

z N -z 0 L 0 = 3N T 0 v i L 0 2 1/3 . (7)
To summarize, two intrusion regimes should be considered, where z N ∼ mN/ρd 2 ∝ N in the limit of small depths, and z N ∼ (mv 2 i N/ρgd) 1/3 ∝ N 1/3 in the limit of large depths. Note that both predictions provide that the depth of the impactor after a given number of impacts decreases with the impactor diameter d and the density of the granular medium ρ, in qualitative agreement with experiments displayed in Figs. 2(b) and 2(c). At intermediate depths, we expect a crossover between these two regimes and we suggest to connect these two predictions by the arbitrary expression

z N -z 0 z c = N N c 1 + N N c 2/3 , (8)
where z c is the crossover depth and N c the number of impacts associated to this crossover. Note that shallow penetration depths corresponding to N/N c 1 and large penetration depths corresponding to N/N c 1, lead, respectively, to the limiting cases z N ∝ N and z N ∝ N 1/3 discussed before. We consider z c and N c as free parameters and we seek the best fit of the measured intrusion dynamics with Eq. (8). Figure 3(a) presents all the intrusion dynamics normalized by N c and z c estimated through this procedure. We conclude that the proposed modeling is compatible with the experimental data since all our measurements collapse on a master curve which corresponds to Eq. ( 8). The identification of Eqs. ( 3), ( 6), and ( 8) leads to the following expressions for the crossover parameters:

z c = √ 3 v i T 0 ∼ v i (d/g) 1/2 and N c = √ 3 T 0 v i /L 0 ∼ ρv i d 5/2 /(m g 1/2
). In the following, we compare the estimations provided by our fitting procedure to these predictions. Figure 3(b) shows the evolution of z c /d as a function of v i /(d g) 1/2 . We observe a good agreement between the experimental data and the scaling of z c predicted by this modeling. The fit of the data highlighted by the solid line in Fig. 3(b) provides an estimate of the ratio K v /K z = 0.4 ± 0.2. Figure 3(c) presents the evolution of N c as a function of ρv i d 5/2 /(m g 1/2 ). The fit of the data [solid line in Fig. 3(c)] provides an estimate of the ratio (K v 3 /K z ) 1/2 = 4 ± 3. The agreement between the measurements and predictions is still reasonable even though there is more dispersion on the data. This dispersion arises from the fact that N c is limited to a narrow range of values (between 1 and 10) while the uncertainty on this parameter can hardly be less than 1. Moreover the variations of prefactors in granular media are known to be highly fluctuating [35,39]. These two scaling laws allow to give an estimate of K v 6 and K z 15. Note that the values of K v and K z are known to depend on the shape of the impactor. Indeed, a collisional model permitted to propose a dependency for K v with the geometry of the object [34]. In the quasistatic limit, an Archimedean law was proposed for the granular drag at low depths, which permits to express the influence of the head geometry on the resistive force [35].

The crossover between the two intrusion regimes is described by the ratio between the collisional and the frictional forces. The transition between these two regimes can be rationalized with a non-dimensional number which corresponds in this case to the Bagnold number Ba defined as:

Ba = ρd 2 v 2 ρgdz 2 = d z v 2 gz = d z Fr 2 , (9)
where Fr = v/ √ gz is a dynamic Froude number.

This definition drops the constant relative to the ratio K v /K z which is of the order of unity. When Ba 1, the quasi-static contribution of the granular resistance predominates, whereas when Ba 1, the inertial term overcomes. In these experiments, this number is related to the Froude number which compares two characteristic velocities: the initial velocity due to the impact of the mass and the velocity linked to the hydrostatic pressure. The dynamic side comes from the fact that the more the object sinks, the smaller this number naturally becomes.

IV. PENETRATION IN A CONFINED GRANULAR MEDIUM

In this section, we consider the effect of a lateral confinement on the intrusion dynamics of the impactor over successive impacts. We follow the same experimental procedure as previously, but we now investigate larger ratio of the impactor diameter over reservoir diameter d/D where the influence of the lateral walls is not negligible anymore. Whereas previous experiments were conducted for d/D < 0.125, we now increase this ratio up to 0.4. Figure 4 shows the intrusion dynamics of an impactor penetrating into glass beads with different values of the ratio d/D. When d/D is smaller than 0.3, all the intrusion dynamics collapse on the prediction of Eq. ( 8) when rescaled by the crossover depth z c and the impact number N c . However, when d/D becomes larger than 0.3, the intrusion dynamics starts to deviate from the unbounded case. The lateral confinement not only reduces the depth of the impactor after a given number of impacts, but also changes the penetration law as highlighted by the semi-logarithmic scales of Fig. 4 that reveal a transition towards a logarithmic trend, z N ∝ ln N , when d/D 0.3. To characterize this change in behavior, we fit the intrusion dynamics by the following law:

z N -z 0 z c = N N c α+2/3 1 + N N c 2/3 , (10)
where α is a free parameter equals to α = 1/3 when the effect of the lateral confinement is negligible, recovering Eq. (8), while in the limit of large N , this relation scales as z N ∝ N α . Therefore, we look for the exponents α that provide the best fits of the intrusion dynamics and we plot them in the inset of Fig. 4 as a function of the ratio d/D. We observe that α 0.33 when d/D 0.3 and decreases towards zero for larger values of the ratio d/D. When the friction on the lateral walls is mobilized, the pressure in the material does not increase linearly with depth, but rather saturates exponentially towards a constant value [6]. This corresponds to a dynamical Janssen effect resulting from the mobilization of the friction and is already observed when moving a vertical wall upwards [10]. However, in practice, we do not observe any saturation of the force experienced by a rod plunging into a granular medium [37,39,42]. The moving wall is the side of the impactor that goes downwards and thus mobilizes friction in the opposite direction. In this case, the usual expression of the Janssen force adopts a different form since there is no saturation: we therefore consider that the pressure in the material increases exponentially with depth. This exponential dependence was already proposed to rationalize the force experienced by the bottom wall of a narrow granular column as it pushes up the grains [43]. This dependency of the pressure with depth is reflected in the drag force that applies on the impactor. Indeed, a previous study revealed that the presence of lateral walls increases the drag experienced by a sphere intruding at low speeds into a granular tank [44]. To account for this effect up to the limit of extreme confinements, we suggest an exponential variation of the force with depth

m dv dt = -K z ρgdλ 2 exp z λ , (11)
where λ is the characteristic depth of the evolution of the pressure profile. Following the same method as previously, the solution of this equation is:

v 2 (z) -v 2 i = - 2 K z ρgdλ 3 m exp z λ -exp z N -1 λ . (12)
Looking for z N where v(z N ) = 0, and using the characteristic length L 0 and the characteristic time T 0 introduced in the unbounded case, we obtain:

exp z N λ -exp z N -1 λ = v 2 i L 0 T 2 0 λ 3 . (13)
By iteration, we obtain a prediction for the intrusion dynamics in the presence of lateral confinement:

z N λ = ln exp z 0 λ + v 2 i L 0 T 2 0 λ 3 N . (14)
In the limit of large depths (z N z 0 ), this prediction scales as z N ∝ ln N and thus has the same scaling as the observations made for d/D 0.3 (Fig. 4). Since a power law of the form z N ∝ N α is always larger than ln N at large N , this explains why the fits of the intrusion dynamics with Eq. ( 10) have an exponent α that decreases towards zero when the effect of the confinement dominates.

V. CONCLUSION

In this study, we analyze the different intrusion regimes of a cylindrical impactor that penetrates a granular medium through N successive impacts. In an unbounded medium, the dynamics of the impactor during the first impacts is ruled by a balance between the force resulting from the impact of the internal mass and the inertial resistance of the granular medium due to collision dissipation (∝ v 2 ), and leads to an intrusion law of the form z N ∝ N . For larger impact numbers, as the impactor moves deeper in the grains, the input energy is balanced by the quasi-static force applied by the granular medium on its lateral surface (∝ z 2 ). Thus, the intrusion dynamics becomes sublinear as z N ∝ N 1/3 . We develop a theoretical framework that accounts for the forces applied by the grains on the impactor. The scaling laws predicted by the model are in agreement with the experimental measurements. The existence of these two regimes of penetration dynamics reveals a crossover depth z c ∼ v i (d/g) 1/2 which depends on the geometry of the object and the impact velocity, also well captured by the model.

In a confined granular medium, the penetration dynamics is modified as the granular forces are modulated by the friction at the walls. The more confined the environment, the more difficult is the penetration by successive impacts since the resistant force increases. Consequently, the intrusion dynamics transits towards a logarithmic trend, z N ∝ ln N in the case of extreme confinement. We extend the theoretical framework that accounts for the forces applied by the granular medium on the impactor in this case.

The agreement between the predictions of this model and our measurements proves that we now have a reliable understanding of the granular forces applying on an impactor to predict its dynamics when its submitted to complex solicitations such as a series of impacts. This understanding offers promising opportunities to optimize the design of autonomous digging systems based on this principle and dedicated to explore any granular soils. In future studies, we can explore the effect of cohesion of the granular medium on the intrusion dynamics of the impactor to get closer to practical situations.

Figure 1 .

 1 Figure 1. (a) Sketch of the experimental setup and notations used. (b) Depth of the impactor zN as a function of the impact number N , for an intrusion in glass beads (dg = 0.45 mm and ρ = 1550 kg m -3 ). The impactor has a diameter d = 50 mm and a mass m0 = 460 g. The impacting mass is mi = 209 g and is released from h = 25.5 cm corresponding to an energy E = 0.52 J. Inset: Same data in a logarithmic plot. The black line represents zN ∝ N 1/3 .

UFigure 2 .

 2 Figure 2. (a) Depth of the impactor zN as a function of the accumulated impacting energy N E mi/m for different impact energies E. Inset: zN as a function of the impact number N . For intrusions in glass beads (dg = 0.45 mm and ρ = 1550 kg m -3 ), the impactor has a diameter d = 33 mm, a mass m0 = 210 g and the impacting mass is mi = 209 g. (b) Depth zN as a function of the impact number N for different diameters d, with dg = 0.45 mm and ρ = 1550 kg m -3 , by keeping constant the impact energy E and the mass ratio mi/m. (c) Depth zN as a function of the impact number N for different densities of the granular medium ρ, with d = 33 mm and a mass m0 = 210 g, the impacting mass mi = 209 g is released from h = 255 mm.

Figure 3 .

 3 Figure 3. (a) Normalized impactor depth (zN -z0)/zc as a function of the normalized impact number N/Nc. The solid line represents the prediction of Eq. (8). (b) Crossover distance zc deduced from the fit and normalized by the impactor diameter d as function of the experimental parameter vi/(d g) 1/2 . The solid line corresponds to zc/d = A vi/(d g) 1/2 , with A = 1.1. (c) Crossover impact number Nc as a function of the experimental parameter ρvi d 5/2 /(m g 1/2 ). The solid line corresponds to Nc = Bρvi d 5/2 /(m g 1/2 ), with B = 15. Symbols have the same meaning as in Fig. 2.

Figure 4 .

 4 Figure 4. Normalized depth zN /zc as a function of the normalized impact number N/Nc for increasing ratio of impactor diameter over the reservoir diameter d/D. The upper solid line indicates the prediction for unbounded intrusion from Eq. (8) and the lower solid line represents the extreme case zN ∝ ln N when the lateral confinement matters. Inset: Estimated exponent α from the fit of the intrusion dynamics with Eq. (10) as a function of d/D. The gray region distinguishes confined cases from the unbounded case.

ACKNOWLEDGMENTS

We are grateful to J. Amarni, A. Aubertin, L. Auffray, C. Manquest, and R. Pidoux for their contribution to the development of the experimental setup. We thank Hugo Gerard for preliminary experiments. This work was supported by ANR PIA funding: Grant No. ANR-20-IDEES-0002.