Francis Berenbaum
email: francis.berenbaum@aphp.fr

Thomas Schnitzer

Alan Kivitz

Lars Viktrup

Elizabeth Johnston

Ruoyong Yang

Ed Whalen

Leslie Tive

David Semel

Gender, age, disease severity, body mass index and diabetes may not affect response to subcutaneous tanezumab in patients with osteoarthritis after 16 weeks of treatment. A subgroup analysis of placebocontrolled trials. RUNNING TITLE: Tanezumab subgroup analysis

Keywords: NCT02709486, NCT01089725 osteoarthritis, tanezumab, treatment response, patient characteristics, pooled analysis

clinical-studys/study-data-and-results for more information), Pfizer will provide access to individual de-identified participant data from Pfizer-sponsored global interventional clinical studies conducted for medicines, vaccines and medical devices (1) for indications that have been approved in the US and/or EU or (2) in programs that have been terminated (ie, development for all indications has been discontinued). Pfizer and Lilly will also consider requests for the protocol, data dictionary and statistical analysis plan. Data may be requested from Pfizer studies 24 months after study completion. The de-identified participant data will be made available to researchers whose proposals meet the research criteria and other conditions, and for which an exception does not apply, via a secure portal. To gain access, data requestors must enter into a data access agreement with Pfizer. TITLE: Gender, age, disease severity, body mass index and diabetes do not affect response to subcutaneous tanezumab in patients with osteoarthritis. A subgroup analyses of placebo-controlled trials. ABSTRACT Aim: To assess impact of pre-specified patient characteristics on efficacy and safety of subcutaneous tanezumab in patients with osteoarthritis (OA). Methods: Data were pooled from two (efficacy; N = 1545) or three (safety; N = 1754) phase 3 placebo-controlled trials. Change from baseline to week 16 in Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) Pain, WOMAC Physical Function, and patient global assessment of OA (PGA-OA) scores was compared between tanezumab (2.5 and 5 mg) and placebo groups via analysis of covariance. Treatment-emergent adverse events (TEAEs) were summarized descriptively. Analyses were done in patient subgroups (male or female; age <65, ≥65, or ≥75 years; body mass index [BMI] <25, 25 to <30, 30 to <35, or ≥35 kg/m 2 ; diabetes or no diabetes; baseline WOMAC Pain score <7 or ≥7; and Kellgren-Lawrence [KL] grade 2, 3, or 4 in the index joint) and the overall population).

Results: In all subgroups, improvements in WOMAC Pain were numerically greater and often statistically significant (P < .05) for both tanezumab groups compared with placebo. Results were similar for WOMAC Function and PGA-OA. TEAE profiles were generally consistent across subgroups and similar to the overall population (i.e., slightly higher rates of TEAEs, serious TEAEs, and severe TEAEs with tanezumab relative to placebo) with a few exceptions. Exceptions included females reporting slightly more TEAEs with tanezumab than males, and patients with diabetes reporting slightly more severe TEAEs with tanezumab than patients without diabetes. Additionally, TEAEs were more frequent with tanezumab than placebo in the age ≥65 and ≥75 years, but not the age <65 years, subgroups.

Conclusions: Efficacy and safety/tolerability of tanezumab may not be meaningfully impacted by gender, age, BMI, diabetes status, baseline pain severity or KL grade in the index joint. Conclusions are limited by low patient number in some subgroups. Clinicaltrials.gov: NCT02697773,

INTRODUCTION

Osteoarthritis (OA) represents a substantial global burden and is often associated with significant levels of pain and impairment of physical function. [START_REF] Cross | The global burden of hip and knee osteoarthritis: estimates from the global burden of disease 2010 study[END_REF] Current OA management concentrates on mitigating these symptoms through a combination of non-pharmacologic (eg, weight loss, exercise, education, cognitive behavioral therapy) and pharmacologic (eg, acetaminophen, nonsteroidal anti-inflammatory drugs [NSAIDs], opioids, duloxetine, intra-articular corticosteroids or hyaluronic acid) approaches. [START_REF] Kolasinski | American College of Rheumatology/Arthritis Foundation guideline for the management of osteoarthritis of the hand, hip, and knee[END_REF][START_REF] Geenen | EULAR recommendations for the health professional's approach to pain management in inflammatory arthritis and osteoarthritis[END_REF][START_REF] Grässel | Recent advances in the treatment of osteoarthritis [version 1; peer review: 3 approved[END_REF][START_REF] Bannuru | OARSI guidelines for the non-surgical management of knee, hip, and polyarticular osteoarthritis[END_REF] Managing OA-related pain, however, is difficult and many patients express dissatisfaction with treatment due to inadequate efficacy or tolerability issues, highlighting an unmet need for new safe and effective therapies. [START_REF] Ueda | Treatment satisfaction with pharmaceutical interventions in Japanese adults with osteoarthritis and chronic knee pain: an analysis of a web-based survey[END_REF][START_REF] Jackson | The burden of pain associated with osteoarthritis in the hip or knee from the patient's perspective: a multinational cross-sectional study[END_REF][START_REF] Gore | Therapy switching, augmentation, and discontinuation in patients with osteoarthritis and chronic low back pain[END_REF] Treatment response (efficacy and safety/tolerability) often varies between patients, even in highly controlled clinical trial settings, and may be attributed to differences in disease severity; patient characteristics such as age, weight, and gender; and the presence of certain comorbidities. [START_REF] Edwards | Patient phenotyping in clinical trials of chronic pain treatments: IMMPACT recommendations[END_REF][START_REF] Bierma-Zeinstra | Osteoarthritis subpopulations and implications for clinical trial design[END_REF][START_REF] Bruyère | Can we identify patients with high risk of osteoarthritis progression who will respond to treatment? A focus on epidemiology and phenotype of osteoarthritis[END_REF] Therefore, examining the efficacy and safety of a particular therapy in subgroups of patients, based on baseline demographic or clinical characteristics, could help identify subgroups of patients that may or may not respond favorably to treatment.

Tanezumab is a monoclonal antibody against nerve growth factor that is in clinical development for the treatment of the signs and symptoms of moderate-to-severe OA in patients with inadequate treatment response or intolerability to standard OA analgesics (eg, acetaminophen, NSAIDs, opioids). Three randomized, placebo-controlled, phase 3 studies have assessed the efficacy and safety of subcutaneous (SC) tanezumab in such patients. [START_REF] Birbara | Safety and efficacy of subcutaneous tanezumab in patients with knee or hip osteoarthritis[END_REF][START_REF] Schnitzer | Effect of Tanezumab on Joint Pain, Physical Function, and Patient Global Assessment of Osteoarthritis Among Patients With Osteoarthritis of the Hip or Knee: A Randomized Clinical Trial[END_REF][START_REF] Berenbaum | Subcutaneous tanezumab for osteoarthritis of the hip or knee: efficacy and safety results from a 24-week randomised phase III study with a 24-week follow-up period[END_REF] The current analysis pooled data from these trials to determine whether response (efficacy and safety/tolerability) to SC tanezumab is affected by patient characteristics including gender, age, body mass index (BMI), diabetes status, baseline pain severity, and Kellgren-Lawrence (KL) grade in the index joint.

METHODS

Data sources

Patient-level data were pooled from two (efficacy: NCT02697773 and NCT02709486) or three (safety: NCT02697773, NCT02709486, and NCT01089725) randomized, placebo-controlled, phase 3 trials of SC tanezumab (Table 1). [START_REF] Birbara | Safety and efficacy of subcutaneous tanezumab in patients with knee or hip osteoarthritis[END_REF][START_REF] Schnitzer | Effect of Tanezumab on Joint Pain, Physical Function, and Patient Global Assessment of Osteoarthritis Among Patients With Osteoarthritis of the Hip or Knee: A Randomized Clinical Trial[END_REF][START_REF] Berenbaum | Subcutaneous tanezumab for osteoarthritis of the hip or knee: efficacy and safety results from a 24-week randomised phase III study with a 24-week follow-up period[END_REF] Study NCT01089725 was terminated early due to a class-wide partial clinical hold on anti-NGF therapies and 90.5% of treated patients received only 8 weeks of treatment (ie, one dose of study medication at baseline). Thus, this study was excluded from the current efficacy analyses (which were based on week 16 data). However, a majority (70.7%) of treated patients remained in study NCT01089725 for >16 weeks for safety evaluation. Therefore, study NCT01089725 is included in the current safety analyses, which were based on the full study (treatment + follow-up) periods and encompasses safety data from all phase 3, placebocontrolled OA studies of SC tanezumab conducted to date. In study NCT02697773, 87.5% and 79.9% of patients completed the treatment and full study (treatment + follow-up) periods, respectively. In study NCT02709486, 88.3% and 82.0% of patients completed the treatment and full study periods, respectively. All studies were conducted in compliance with the Declaration of Helsinki and the International Conference on Harmonisation Good Clinical Practice guidelines. Study protocols were approved by an institutional review board at each site and all patients provided written informed consent.

All studies enrolled patients aged ≥18 years with moderate-to-severe OA of the knee or hip and a history of inadequate response to other OA analgesics. Tanezumab was administered every 8 weeks for 16-24 weeks at doses ranging from 2.5-10 mg. Patients in the tanezumab 2.5/5 mg group of study NCT02697773 were included in the tanezumab 5 mg treatment arm in the current efficacy analyses.* The tanezumab 10 mg treatment arm from study NCT01089725 was not included in the current safety analysis since the 10 mg was not assessed in post-2015 trials NCT02697773 and NCT02709486. [START_REF] Hochberg | When Is osteonecrosis not osteonecrosis? Adjudication of reported serious adverse joint events in the tanezumab clinical development program[END_REF] Co-primary endpoints in each trial were change in Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC † ) Pain, WOMAC Physical Function, and patient global assessment of OA (PGA-OA) scores from baseline to end of the treatment period.

Subgroups of interest

In the current study, efficacy and safety analyses were done in the overall pooled patient populations and in prespecified subgroups of interest that included male or female; age <65, ≥65, or ≥75 years; BMI of <25, 25 to <30, 30 to <35, or ≥35 kg/m 2 ; diabetes or no diabetes; baseline WOMAC Pain score <7 or ≥7 (scores range from 0-10 with ≥7 representing severe pain); and KL grade 2, 3, or 4 in the index joint (KL grades of 0, 1, 2, 3, and 4 represent no, doubtful, minimal, moderate, and severe OA, respectively). Index joint was defined as the most painful joint at screening with a qualifying WOMAC Pain score and radiographic KL grade as confirmed by a central reader. Patients were included in the diabetes subgroup if they had a medical history of type 1 or type 2 diabetes mellitus, hyperglycemia, or had a baseline hemoglobin A1c ≥6.5%.

Efficacy analysis

Efficacy was based on change, from baseline to week 16, in WOMAC Pain, WOMAC Physical Function, and PGA-OA scores using patient-level data derived from studies NCT02697773 and NCT02709486. Week 16 was chosen since it was the longest treatment duration common to both studies. WOMAC Pain and Physical Function scores range from 0-10, with higher score indicating greater pain severity or function impairment, respectively. PGA-OA scores range from 1-5, with higher scores indicating worse disease status.

Least squares (LS) mean changes from baseline to week 16 in these measures were assessed for the placebo, tanezumab 2.5 mg, and tanezumab 5 mg treatment arms using an analysis of covariance (ANCOVA) model including terms for baseline score of the corresponding endpoint, baseline daily average pain score, index joint (hip or knee), treatment, and study. A multiple imputation approach was used for missing data, dependent on the reason for missing data. For patients with missing data due to discontinuation prior to week 16 for lack of efficacy, for an adverse event, or death, imputation was based on sampling from a normal distribution using a mean value equal to the patient's baseline efficacy value and the SD (over all treatment groups) of the observed efficacy data at week 16. For patients with missing data for any other reason, imputation was based on sampling from a normal distribution using a mean value of the patient's last observed efficacy value and SD (over all treatment groups) of the observed efficacy data at week 16. The proportion of patients achieving ≥50% (substantial) and ≥30% (moderate) improvement from baseline to week 16 in WOMAC Pain was assessed using a logistic regression method with terms for baseline WOMAC Pain subscale score and baseline daily average pain score, and classification variables of index joint, treatment, and study. A mixed last-observation carried forward/baseline-observation carried forward approach was used for missing data.

Treatment comparisons were based on LS mean differences from placebo (or odds ratios for 50% and 30% responder data), associated 95% confidence intervals, and P-values. For all comparisons, nominal significance was declared if the two-tailed test for the difference between treatment groups was significant at the 0.05 level.

Safety analysis

Safety was based on the occurrence of treatment-emergent adverse events (TEAEs) over the full study (treatment + safety follow-up combined) periods in studies NCT02697773 (16-week treatment + 24-week followup), NCT02709486 (24-week treatment + 24-week follow-up), and NCT01089725 (16-week treatment + 8-week follow-up). TEAEs were coded using Medical Dictionary for Regulatory Activities v22.0, with severity assessed by site investigators, and were summarized descriptively in the overall patient population and in the subgroups of interest for the placebo, tanezumab 2.5 mg, and tanezumab 5 mg treatment arms.

RESULTS

Efficacy

Patient demographics

The overall efficacy population included 1545 patients (placebo = 514, tanezumab 2.5 mg = 514, tanezumab 5 mg = 517). The population was predominantly female (67.3%), white (80.5%), and had an approximate mean age of 63 years (Table 2). OA disease duration ranged from 7.9 to 8.7 years across groups, with a majority (84.1%) of patients having knee as the index joint. Mean baseline WOMAC Pain, WOMAC Physical Function, and PGA-OA scores were 6.9, 7.0, and 3.5, respectively, in all treatment groups.

Change in WOMAC Pain, WOMAC Physical Function, and PGA-OA

Improvements in WOMAC Pain from baseline to week 16 were numerically greater and often statistically significant (P < .05) in both tanezumab groups compared with placebo, irrespective of gender, age, BMI, diabetes status, baseline WOMAC Pain score, or KL grade in the index joint (Figure 1; the number of patients in each subgroup can be seen within the figure). The only improvements that did not reach the level of significance for tanezumab vs placebo were the 2.5 mg dose in the male subgroup, the 2.5 mg dose in the age ≥75 years subgroup, the 5 mg dose in the <25 kg/m 2 BMI subgroup, both doses in the ≥35 kg/m 2 BMI subgroup, the 2.5 mg dose in the patients with diabetes subgroup, and both doses in the KL grade 2 in the index joint subgroup. In the overall patient population (all subgroups combined), improvements in WOMAC Pain from baseline to week 16 were significantly greater in both tanezumab groups compared with the placebo group. LS mean (standard error) change from baseline was -3.1 (0.12) for tanezumab 2.5 mg and -3.2 (0.12) for tanezumab 5 mg compared with -2.5 (0.12) for placebo (both tanezumab groups P < .0001 vs placebo). Improvements in WOMAC Physical Function (Supplemental Figure 1) and PGA-OA (Supplemental Figure 2) also favored tanezumab over placebo in all the subgroups of interest and often reached the level of significance.

Proportion of patients with ≥50% or ≥30% improvement in WOMAC Pain

The proportion of patients achieving ≥50% improvement in WOMAC Pain from baseline to week 16 was numerically greater and often statistically significant (P < .05) in both tanezumab groups compared with placebo, irrespective of gender, age, BMI, diabetes status, baseline WOMAC Pain score, or KL grade in the index joint (Figure 2). The only proportion of 50% responders that did not reach the level of significance for tanezumab vs placebo were the 2.5 mg dose in the age ≥75 years subgroup, the 5 mg dose in the <25 kg/m 2 BMI subgroup, both doses in the ≥35 kg/m 2 BMI subgroup, both doses in the patients with diabetes subgroup, and both doses in the KL grade 2 in the index joint subgroup. In the overall patient population (all subgroups combined), the proportion of patients achieving ≥50% improvement in WOMAC Pain was significantly greater in both tanezumab (2.5 mg = 51.9%, 5 mg = 51.8%; both P < .0001) groups compared with the placebo (36.8%) group in the overall population. Similar to the 50% responder threshold, the proportion of patients achieving ≥30% improvement in WOMAC Pain in the tanezumab (2.5 mg = 68.0%, 5 mg = 69.4%; both P < .0001) groups was significantly greater than the placebo (55.6%) group in the overall population and in many subgroups of interest (Supplemental Figure 3).

Safety

Patient demographics

The overall safety population included 1754 patients (placebo = 586, tanezumab 2.5 mg = 602, and tanezumab 5 mg = 566). Demographics of the overall safety population (Table 3) were similar to those of the overall efficacy population.

Adverse events (AEs)

TEAEs in the overall safety population are summarized in Table 4. TEAE rates were largely similar in the tanezumab groups (2.5 mg = 62.8%, 5 mg = 62.0%) relative to the placebo group (60.9%). Rates of serious (placebo = 3.6%, tanezumab 2.5 mg = 5.3%, tanezumab 5 mg = 5.5%) and severe (placebo = 3.9%, tanezumab 2.5 mg = 4.7%, tanezumab 5 mg = 6.4%) TEAEs were slightly higher with tanezumab relative to placebo. Rates of treatment discontinuations due to TEAEs, however, were lower in the tanezumab groups (2.5 mg = 1.3%, 5 mg = 0.9%) than the placebo group (2.0%). Among common TEAEs (TEAEs occurring in ≥3% of patients in any treatment group), rates of peripheral edema (placebo = 0.3%, tanezumab 2.5 mg = 1.2%, tanezumab 5 mg = 3.2%) and paresthesia (placebo = 1.2%, tanezumab 2.5 mg = 2.5%, tanezumab 5 mg = 3.0%) were at least twice as high in both tanezumab groups than in the placebo group.

A summary of AEs in the subgroups of interest can be found in Supplemental Tables 1-6. In general, the profile of TEAEs (proportion of overall TEAEs, serious TEAEs, severe TEAEs, discontinuations due to TEAEs, and common TEAEs) in most subgroups of interest was broadly similar to the profile in the overall patient population, and only a few differences were noted. For example, females reported numerically more TEAEs with tanezumab (2.5 mg = 64.8%, 5 mg = 66.0%) than males (2.5 mg = 58.8%, 5 mg = 54.5%). In addition, overall TEAEs were more frequent in both tanezumab groups than the placebo group in the age ≥65 and age ≥75 years, but not the age <65 years, subgroups. Finally, patients with diabetes reported more severe TEAEs with tanezumab (placebo = 2.2%, tanezumab 2.5 mg = 7.8%, tanezumab 5 mg = 12.0%) than patients without diabetes (placebo = 4.3%, tanezumab 2.5 mg = 4.0%, tanezumab 5 mg = 5.2%), though overall rates of discontinuations due to TEAEs were similar (1.1%-2.2%).

DISCUSSION

In this analysis of over 1500 patients with moderate-to-severe OA of the knee and hip and a history of inadequate response to other standard OA analgesics, improvements in pain, function, and PGA-OA favored SC tanezumab (2.5 mg and 5 mg) over placebo irrespective of gender, age, BMI, diabetes status, baseline pain severity, or KL grade in the index joint. The TEAE profile of tanezumab was mostly consistent among the subgroups and broadly similar to the profile in the overall patient population.

The subgroups in these analyses were selected to determine the consistency of treatment response across a number of demographic and clinical variables. Evidence suggests that males and females experience pain, and respond to some treatments, differently. [START_REF] Gordon | Enhancement of morphine analgesia by the GABAB agonist baclofen[END_REF][START_REF] Gear | The kappa opioid nalbuphine produces gender-and dose-dependent analgesia and antianalgesia in patients with postoperative pain[END_REF][START_REF] Pud | Can personality traits and gender predict the response to morphine? An experimental cold pain study[END_REF] Though the underlying mechanisms are unknown, this may be due to inherent differences in drug metabolism, levels of drug receptors, psychological factors, the endocrine system, or processing pathways in males and females. [START_REF] Paller | Sex-based differences in pain perception and treatment[END_REF][START_REF] Fillingim | Sex differences in analgesic responses: evidence from experimental pain models[END_REF][START_REF] Templeton | Sex and gender issues in pain management[END_REF] Advanced age is also known to affect response to small molecule analgesics due to physiological changes associated with aging, including increased drug absorption due to slowing of the gastrointestinal tract, decreased drug metabolism due to decreased hepatic function, and reduced renal excretion. [START_REF] Mangoni | Age-related changes in pharmacokinetics and pharmacodynamics: basic principles and practical applications[END_REF][START_REF] Klotz | Pharmacokinetics and drug metabolism in the elderly[END_REF] In addition to age, body weight and BMI can also affect the pharmacokinetics of drug therapies by affecting absorption, distribution, and clearance. Likewise, diabetes can potentially affect treatment response via changes in drug pharmacokinetics due to altered drug absorption (via changes in blood flow in subcutaneous adipose tissue or muscle tissue) or excretion (via decreased renal function). [START_REF] Dostalek | Effect of diabetes mellitus on pharmacokinetic and pharmacodynamic properties of drugs[END_REF] The presence of diabetic peripheral neuropathy, which alters nociceptive signaling, could also affect analgesic responses in patients with diabetes. [START_REF] Rosenberger | Challenges of neuropathic pain: focus on diabetic neuropathy[END_REF] Finally, disease severity may also affect response to treatment.

There is some evidence in knee OA, for example, that radiologic severity of disease (assessed by KL grading) may be a possible predictor of efficacy response to intra-articular corticosteroid injection. [START_REF] Maricar | Predictors of response to intra-articular steroid injections in knee osteoarthritis-a systematic review[END_REF][START_REF] Maricar | Structural predictors of response to intra-articular steroid injection in symptomatic knee osteoarthritis[END_REF] It should be noted, however, that OA severity as assessed by radiographic KL grading does not necessarily correlate with OA-related pain severity. [START_REF] Bedson | The discordance between clinical and radiographic knee osteoarthritis: a systematic search and summary of the literature[END_REF][START_REF] Son | Absence of pain in subjects with advanced radiographic knee osteoarthritis[END_REF] Thus, we assessed both objective (KL grade) and subjective (baseline pain scores in the index joint) measures of OA severity.

Improvements in WOMAC Pain, WOMAC Function, and PGA-OA favored tanezumab over placebo in all subgroups and many, but not all, comparisons reached the level of statistical significance. Due to the low number of patients in many of the subgroups, however, it is not surprising that not all comparisons vs placebo were significant. Low patient numbers were particularly evident in the age ≥75 years (56-64), patients with diabetes (86-92), and <25 kg/m 2 BMI (64-72) subgroups. LS mean differences versus placebo in these subgroups with low patient numbers were similar to (and often greater than) the observed LS mean differences versus placebo observed in subgroups with greater patient numbers, but there was greater variability (i.e. 95% CIs) in the low patient number subgroups that contributed to the lack of statistical significance. In addition, a higher placebo response in the KL grade 2 subgroup (LS mean change from baseline in WOMAC Pain = -2.9) relative to the KL grade 3 subgroup (LS mean change from baseline in WOMAC Pain = -2.6) and, particularly, the KL grade 4 subgroup (LS mean change from baseline in WOMAC Pain = -1.8) may have contributed to the lack of significant effect for both doses of tanezumab in the KL grade 2 subgroup. This observation also suggests that there may be an inverse correlation between the magnitude of the placebo effect and the degree of structural OA severity. The LS mean change for the tanezumab groups was more similar across all KL grades, ranging from -2.9 to -3.4.

Overall, conclusions on the clinical significance of treatment effect in the various subgroups are difficult to make due to the limited number of patients in some subgroups and the fact that subgroups were not directly compared to each other. It is notable, however, that statistically greater proportions of tanezumab-treated patients achieved ≥30% and ≥50% improvement in WOMAC Pain compared to placebo-treated patients in most subgroups . These 30% and 50% thresholds represent moderate and substantial, respectively, improvements in pain for patients with chronic pain conditions and suggests that the benefits observed in this analysis were clinically meaningful in many of the subgroups. [START_REF] Dworkin | Interpreting the clinical importance of treatment outcomes in chronic pain clinical trials: IMMPACT recommendations[END_REF] Though the TEAE profile of tanezumab among the subgroups of interest was broadly consistent with the overall population, there were a few instances where the profile appeared somewhat different across subgroups. Females reported more TEAEs with tanezumab than males. It is possible that this may represent a modest, but real, difference between the genders as patient numbers were high in each subgroup and previous studies suggest that women experience (or report) adverse drug reactions at a higher frequency than men across all drug classes. [START_REF] Zucker | Sex differences in pharmacokinetics predict adverse drug reactions in women[END_REF] TEAE rates were higher among tanezumab-treated patients than placebo-treated patients in older (aged ≥65 and ≥75 years) patients, but not in younger (aged <65 years) patients. This could represent a difference in tanezumab's safety profile among older patients since the risk of AEs to drug treatments in general, as well as the risk for drug-drug interactions, is increased with advanced age. [START_REF] Davies | Adverse drug reactions in special populations -the elderly[END_REF] A more likely reason for this discrepancy, however, may be due to the higher TEAE rates among placebo-treated patients in the age <65 years subgroup (65.9%) relative to the age ≥65 years (50.7%) and age ≥75 years (54.7%) subgroups. Further, TEAE rates among tanezumab-treated patients were more comparable across the three age subgroups (58.6-65.7%) and similar to the TEAE rate among all tanezumab-treated patients in the overall safety population (62.0-62.8%).

Joint safety events, particularly rapidly progressive OA, are associated with nerve growth factor antibodies such as tanezumab in some patients. [START_REF] Schnitzer | Effect of Tanezumab on Joint Pain, Physical Function, and Patient Global Assessment of Osteoarthritis Among Patients With Osteoarthritis of the Hip or Knee: A Randomized Clinical Trial[END_REF][START_REF] Berenbaum | Subcutaneous tanezumab for osteoarthritis of the hip or knee: efficacy and safety results from a 24-week randomised phase III study with a 24-week follow-up period[END_REF][START_REF] Hochberg | When Is osteonecrosis not osteonecrosis? Adjudication of reported serious adverse joint events in the tanezumab clinical development program[END_REF] Though a detailed analysis of joint safety is beyond the scope of this paper, we note that OA was reported as an AE (representing new [non index joint] or worsening [index joint] cases of OA) more often among tanezumab-treated patients than among placebo-treated patients in the overall safety population and in many subgroups of interest.

Our findings regarding the use of SC tanezumab in patients with moderate-to-severe OA and a history of inadequate response to other analgesics agree with, and build upon, a previous analysis of intravenous (IV) tanezumab in patients with OA. 33 Like the current SC analysis, the previous IV analysis demonstrated that tanezumab provided significant improvement of pain, function, and global disease status in subpopulations of patients based on age, baseline pain severity, diabetes status, and BMI without identifying new safety risks in those subpopulations. 33 Our findings are also in broad agreement with a recent pharmacokinetic analysis of IV and SC data suggesting that tanezumab dosing does not need to be adjusted based on factors such as gender, age, or BMI. [submitted manuscript] Limitations of this study include the low number of patients in many subgroups and the post-hoc nature of our analyses. In addition, comparisons were with placebo, and different subgroups were not directly compared with each other (eg, male vs female). Likewise, the two doses of tanezumab were not directly compared with each other and conclusions on relative efficacy are limited. Efficacy was based on changes in pain after 16 weeks (two doses) of treatment and findings should not be generalized to longer treatment durations. Finally, it should be noted that the overall efficacy and safety populations were largely female (approximately two-thirds of all patients) and white (over 80% of patients). This may limit the ability to generalize our findings to other OA populations, particularly those of predominantly non-white racial groups.

However, strengths of the study include overall large patient population from which the subgroups were derived and similarity in trial design for the studies included in the analysis.

CONCLUSIONS

In summary, treatment with SC tanezumab (at doses of 2.5 or 5 mg) every 8 weeks provided improvements over placebo (often reaching the level of statistical significance) in pain, function, and overall OA disease status in all patient subgroups, which were based on gender, age, BMI, diabetes status, baseline pain severity, or KL grade in the index joint. The overall TEAE profile of tanezumab was mostly consistent among the subgroups, broadly similar to the profile in the overall patient population, and no new safety risks were identified in the subgroups.

These findings suggest that tanezumab efficacy is maintained regardless of comorbidity, disease severity, and selected clinical criteria in patients with moderate-to-severe OA and a history of inadequate response to standard OA analgesics. Conclusions, however, are limited by the small number of patients in some subgroups.

(2.2)

Abbreviations: TEAE, treatment-emergent adverse event. † This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Reported in ≥3% of patients in any treatment group. Bolding indicated the event was reported at a higher frequency in both tanezumab groups relative to the placebo group.

Supplemental Table 2. Summary of

Discontinued study due to TEAE Abbreviations: TEAE, treatment-emergent adverse event. † This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Reported in ≥3% of patients in any treatment group. Bolding indicated the event was reported at a higher frequency in both tanezumab groups relative to the placebo group. † This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Reported in ≥2% of patients in any treatment group. Bolding indicates the event was reported at a higher frequency in both tanezumab groups relative to the placebo group. Abbreviations: TEAE, treatment-emergent adverse event. † This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Patients were included in the yes diabetes group if they had a medical history of type 1 or type 2 diabetes mellitus, hyperglycemia, or who had a baseline hemoglobin A1c ≥6.5%. § Reported in ≥3% of patients in any treatment group. Bolding indicated the event was reported at a higher frequency in both tanezumab groups relative to the placebo group. Abbreviations: TEAE, treatment-emergent adverse event. † This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Reported in ≥3% of patients in any treatment group. Bolding indicated the event was reported at a higher frequency in both tanezumab groups relative to the placebo group. -Abbreviations: KL, Kellgren-Lawrence; TEAE, treatment-emergent adverse event. † This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Reported in ≥3% of patients in any treatment group. Bolding indicated the event was reported at a higher frequency in both tanezumab groups relative to the placebo group.

Supplemental Table 3. Summary of

Supplemental Table 4. Summary of

Supplemental Table 5. Summary of

Supplemental Table 6. Summary of

 24 

 24 Abbreviations: IV, intravenous; NSAID, nonsteroidal anti-inflammatory drug; OA, osteoarthritis; PGA-OA, patient global assessment of osteoarthritis; SC, subcutaneous; WOMAC, Western Ontario and McMaster Universities Osteoarthritis Index. † Diagnosed per American College of Rheumatology criteria with radiographic confirmation (Kellgren-Lawrence grade ≥2). ‡ Patients received 2.5 mg at baseline and 5 mg at week 8. These patients were included in the 5 mg group in the current analyses. § Due to early termination, 90.5% of patients received only 8 weeks of treatment (ie, one dose of study medication at baseline). As a result, study NCT01089725 was included in the current safety analysis but excluded from the 16-week efficacy analysis.

FIGURESFigure 1 .

 1 FIGURESFigure 1. Change in WOMAC Pain † from baseline to week 16 (NCT02697773 and NCT02709486).

Figure 2 .Supplemental Table 1 .

 21 Figure 2. Proportion of patients achieving ≥50% improvement in WOMAC Pain † from baseline to week 16 (NCT02697773 and NCT02709486).

  TEAEs over the full study (treatment + follow-up) period † in age subgroups of the safety population (

  TEAEs over the full study (treatment + follow-up) period † in BMI subgroups of the safety population (

  Abbreviations: BMI, body mass index; TEAE, treatment-emergent adverse event.

  TEAEs over the full study (treatment + follow-up) period † in diabetes subgroups of the safety population (NCT02697773, NCT02709486, and NCT01089725)

  TEAEs over the full study (treatment + follow-up) period † in baseline pain subgroups of the safety population (NCT02697773, NCT02709486, and NCT01089725)

  TEAEs over the full study (treatment + follow-up) period † in KL subgroups of the safety population (

TABLE 2 . Demographics of the overall efficacy population (NCT02697773 and NCT02709486)

 2 Western Ontario and McMaster Universities Osteoarthritis Index. † Scores range from 0-10, with higher scores indicating greater pain severity. ‡ Scores range from 0-10, with higher scores indicating greater functional impairment. § Scores range from 1 = very good to 5 = very poor.

			Tanezumab	Tanezumab
		Placebo	2.5 mg	5 mg
	Characteristic	(n = 514)	(n = 514)	(n = 517)
	Gender, n (%)
	Male	161 (31.3)	171 (33.3)	173 (33.5)
	Female	353 (68.7)	343 (66.7)	344 (66.5)
	Mean (SD) age, years	62.5 (9.8)	63.2 (9.4)	63.4 (9.9)
	Race, n (%)
	White	403 (78.4)	423 (82.3)	418 (80.9)
	Black or African American	60 (11.7)	43 (8.4)	50 (9.7)
	Asian	47 (9.1)	43 (8.4)	42 (8.1)
	Other	4 (0.8)	5 (1.0)	7 (1.4)
	Mean (SD) disease duration, years	8.7 (8.1)	7.9 (7.8)	8.3 (7.2)
	Mean (SD) baseline WOMAC Pain †	6.9 (1.1)	6.9 (1.1)	6.9 (1.1)
	Mean (SD) baseline WOMAC Physical Function ‡	7.0 (1.1)	7.0 (1.0)	7.0 (1.1)
	Mean (SD) baseline PGA-OA §	3.5 (0.6)	3.5 (0.6)	3.5 (0.6)
	Index joint, n (%) ¶
	Hip	80 (15.6)	83 (16.1)	83 (16.1)

¶ Index joint was defined as the most painful joint at baseline with a qualifying WOMAC Pain score and Kellgren-Lawrence grade as confirmed by a central reader.

TABLE 3 . Demographics of the overall safety population (NCT02697773, NCT02709486 and NCT01089725)

 3 Western Ontario and McMaster Universities Osteoarthritis Index. † Scores range from 0-10, with higher scores indicating greater pain severity. ‡ Scores range from 0-10, with higher scores indicating greater functional impairment. § Scores range from 1 = very good to 5 = very poor.

			Tanezumab	Tanezumab
		Placebo	2.5 mg	5 mg
	Characteristic	(n = 586)	(n = 602)	= 566)
	Gender, n (%)
	Male	186 (31.7)	199 (33.1)	198 (35.0)
	Female	400 (68.3)	403 (66.9)	368 (65.0)
	Mean (SD) age, years	62.3 (10.2)	62.9 (9.5)	63.2 (10.1)
	Race, n (%)
	White	463 (79.0)	494 (82.1)	458 (80.9)
	Black or African American	70 (11.9)	54 (9.0)	56 (9.9)
	Asian	49 (8.4)	47 (7.8)	44 (7.8)
	Other	4 (0.7)	7 (1.2)	8 (1.4)
	Mean (SD) disease duration, years	8.9 (8.4)	8.0 (7.9)	8.4 (7.5)
	Mean (SD) baseline WOMAC Pain †	7.0 (1.1)	7.0 (1.2)	7.0 (1.2)
	Mean (SD) baseline WOMAC Physical Function ‡	7.0 (1.1)	7.0 (1.1)	7.0 (1.1)
	Mean (SD) baseline PGA-OA §	3.5 (0.6)	3.5 (0.6)	3.5 (0.6)
	Index joint, n (%) ¶
	Hip	80 (13.7)	88 (14.6)	78 (13.8)

¶ Index joint was defined as the most painful joint at baseline with a qualifying WOMAC Pain score and Kellgren-Lawrence grade as confirmed by a central reader.

TABLE 4 . Summary of TEAEs the overall safety population over the full study (treatment + follow-up) period † (NCT02697773, NCT02709486, and NCT01089725)

 4

			Tanezumab	Tanezumab
		Placebo	2.5 mg	5 mg
	Patients, n (%)	(n = 586)	(n = 602)	(n = 566)
	With any TEAE	357 (60.9)	378 (62.8)	351 (62.0)
	With any serious TEAE	21 (3.6)	32 (5.3)	31 (5.5)
	With any severe TEAE	23 (3.9)	28 (4.7)	36 (6.4)
	Discontinued treatment due to TEAE	12 (2.0)	8 (1.3)	5 (0.9)
	Discontinued study due to TEAE	5 (0.9)	10 (1.7)	2 (0.4)
	Common TEAEs ‡
	Arthralgia	95 (16.2)	91 (15.1)	83 (14.7)
	Nasopharyngitis	49 (8.4)	61 (10.1)	47 (8.3)
	Back pain	32 (5.5)	42 (7.0)	34 (6.0)
	Headache	33 (5.6)	34 (5.6)	24 (4.2)
	Osteoarthritis	19 (3.2)	22 (3.7)	24 (4.2)
	Pain in extremity	16 (2.7)	26 (4.3)	21 (3.7)
	Musculoskeletal pain	23 (3.9)	31 (5.1)	20 (3.5)
	Fall	21 (3.6)	35 (5.8)	19 (3.4)

† © 1996 Nicholas Bellamy. WOMAC® is a registered trademark of Nicholas Bellamy (CDN, EU, USA).

† This is a total of 40, 48, and 24 weeks for NCT02697773, NCT02709486, and NCT01089725, respectively. ‡ Reported in ≥3% of patients in any treatment group. Bolding indicated the event was reported at a higher frequency in both tanezumab groups relative to the placebo group.

Medical writing support was provided by Matt Soulsby, PhD, of Engage Scientific Solutions and was funded by Pfizer and Eli Lilly and Company. study support from Pfizer, Eli Lilly and Company, Regeneron, Galapagos, Taiwan Liposome Corporation, and Anika Therapeutics and has served as a consultant or on an advisory board for Pfizer, Eli Lilly and Company, Glaxo-Smith Kline, AstraZeneca, Noven, Galapagos, and Merck. FB has received grants through his institution from TRB Chemedica, MSD, and Pfizer; has worked as a consultant to Novartis, MSD, Pfizer, Eli Lilly and Company, UCB, AbbVie, Roche, Servier, Sanofi-Aventis, Flexion Therapeutics, Expanscience, GSK, Biogen, Nordic, Pfizer Inc. (manufacturer of tanezumab) and Eli Lilly and Company. Authors from Pfizer Inc. and Eli Lilly and Company contributed to study design; data collection (Pfizer), management (Pfizer), and interpretation of data; and preparation, review, and approval of the manuscript.


Ruggero Lot
email: rlot@laas.fr

Layla Martin-Samos

Stefano De Gironcoli

Anne Hémeryck
email: anne.hemeryck@laas.fr

Developing a Neural Network potential to investigate interface phenomena in solid-phase epitaxy

In this work, we develop a new neural network potential for silicon and perform accurate molecular dynamics simulations of the liquid, amorphous and diamond phases. The potential is tested against several physical properties and the solid phase epitaxy process is simulated.

I. INTRODUCTION

Silicon based modern technology is facing an ongoing miniaturization challenge because faster and less powerhungry devices are required for all sort of applications, from small and super portable sensors to the development of new supercomputer chips. Industrial fabrication has already moved towards the exploitation of the third dimension as a way to pack chips reaching the astonishing small scale of tens of nanometer per single transistor. At such a small scale a plethora of phenomena that drive the device formation need a more accurate theoretical understanding to allow for more reliable device production.

In the multi-scale modeling approach often adopted to explain macroscopic experimental observations [START_REF] Zographos | Multiscale modeling of doping processes in advanced semiconductor devices[END_REF], a critical step that drastically impacts the accuracy of the theoretical prediction is the choice of an empirical potential to describe the atomistic scale. Common empirical potentials used in studies of the liquid, amorphous and diamond phases -crucial for the investigation of any step in the formation of a silicon device -have more or less complex functional forms, inspired by intuition and physical insight, with few parameters fitted either to macroscopic properties or accurate calculations [START_REF] Tersoff | New empirical model for the structural properties of silicon[END_REF], [START_REF] Stillinger | Computer simulation of local order in condensed phases of silicon[END_REF]. One of the big problems of these potentials is their lack of flexibility, so that if they aim to be general they suffer limited accuracy and can fail to correctly describe the individual phases. On the other hand their great advantage is to be computationally very efficient as compared to more accurate first-principles calculations.

In recent years very flexible machine-learning (ML) potentials, trained on ab-initio data, are emerging as an attempt to get the best of the two worlds [START_REF] Lot | PANNA: Properties from artificial neural network architectures[END_REF]- [START_REF] Zuo | Performance and cost assessment of machine learning interatomic potentials[END_REF]: reaching the accuracy of first principle calculations at the computational cost of an empirical potential. Several different ML techniques are available, with advantages and disadvantages [START_REF] Zuo | Performance and cost assessment of machine learning interatomic potentials[END_REF]. After sketching the methodology in Sec. II, Sec. III and Sec. IV are dedicated to the creation of the dataset and the testing of the potential. Sec. V simulates the solid phase epitaxy (SPE).

II. METHODOLOGY

The construction of a neural network potential can follow several approaches [START_REF] Behler | Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations[END_REF]. We adopted the method originally proposed by Behler et al. [START_REF] Behler | Generalized neural-network representation of high-dimensional potential-energy surfaces[END_REF] for its generality, that allows the treatment of systems with a varying number of atoms, and gives a good compromise between speed and accuracy. Within this framework, as for most empirical potentials, one makes the assumption that the total energy E of a system can be expressed as a sum over atomic energy contributions:

E = i E i . (1)
where the index i labels the atoms in the unit cell. Each atomic energy term is expressed as a function of the local environment surrounding a central atom and is realized by a neural networks (NN). Different NNs are used to compute the contribution of different atomic species.

The complete definition of a NN potential requires the specification of i) an encoding for the local atomic environments in the form of a vector that can be used as input to the NN and ii) the specification of the NN architecture (shape) to process it. Here the input, or descriptor, is implemented adopting the Symmetry Function (SF) concept first proposed in [START_REF] Behler | Generalized neural-network representation of high-dimensional potential-energy surfaces[END_REF] and later extended in [START_REF] Lot | PANNA: Properties from artificial neural network architectures[END_REF], [START_REF] Smith | ANI-1: An extensible neural network potential with DFT accuracy at force field computational cost[END_REF]. SFs are continuous many-body functions G ({r i | |r i -r ref | < r c }) of the atomic coordinates r i within a cutoff distance, r c , from a reference atom (situated at r ref ) that encode the local information in a translationally, rotationally and permutationally invariant way. The adopted descriptor comprises purely radial SF components, sampling the distance between r min = 0.5 Å and r c = 4.6 Å over 32 bins, and combined angular-radial terms with 12 angular and 6 radial bins between 0 and π and 1.5 and 4.6 Å, respectively. Further hyperparameters were fixed to η = 16.0 in the radial terms and ξ = 50 and η = 6.0 in the angular-radial part. This results in SFs with a total of 104 (32 radial and 12 × 6 = 72 angular) components.

As for the specific shape of the NN we adopted an all-to-allconnected feed-forward (FFNN) architecture for its flexibility. Formally a L-layer FFNN can be defined as:

x (l) i = f (l)   nodes l-1 j w (l) i,j x (l-1) j + b (l) i   , l = 1, ..., L, (2)
where for each layer, l, and each of its nodes, i, the output, x is the single neuron of the output layer containing the predicted atomic energy contribution. A sketch of the SF generation and of the NN architecture used can be seen in the left panel of Fig. 1.

In our case we used two hidden layers (hence L = 3) of 256 and 128 nodes with hyperbolic tangent activation functions and a single output node with linear activation. This is at variance with the Gaussian activation previously adopted [START_REF] Lot | PANNA: Properties from artificial neural network architectures[END_REF]- [START_REF] Smith | ANI-1: An extensible neural network potential with DFT accuracy at force field computational cost[END_REF] and required a precondition of the input that must be transformed from the positive defined SF to a symmetrical distribution around the origin. To obtain this we added a linear, non trainable, layer to the network to perform a precomputed principal components analysis on the input lowering its dimension from 104 to 86 components. After this step we are left with a model with 59905 degrees of freedom.

Given a NN architecture the model is optimized (trained) minimizing the deviation of the predicted values from the ones computed from first-principles for a set of structures representative of the phenomenon of interest. The creation of such a dataset is detailed in Sec.III. The model error estimate (loss function) is minimized via stochastic gradient descent where mini-batches of the training data are randomly selected and used to compute a noisy estimate of the loss-function gradient and update the model parameters. A portion of the dataset is kept aside for validation on previously unseen data (a standard procedure in ML). Training is achieved when no further improvement is seen on the validation loss. The training procedure is sketched in the right panel of Fig. 1.

III. CREATION OF THE DATASET

For empirical potential such as the ones proposed by Tersoff [START_REF] Tersoff | New empirical model for the structural properties of silicon[END_REF] or Stillinger and Weber (SW) [START_REF] Stillinger | Computer simulation of local order in condensed phases of silicon[END_REF], one defines a parametric functional form and fixes its few parameters in such a way that a set of stable phases and macroscopic properties are correctly reproduced, assuming that this will allow him/her to perform studies of other related properties. If this is not the case there is not much one can do about. For NN potentials Fig. 1. On the left side: a pictorial view of a FFNN can be realized as an ensemble of nodes arranged in layers. Each node or neuron is a uniquely parametrized mathematical function that takes as input the vector that is passed to the layer it is part of and returns as output the value of the function for the given input. Layers are chained together from the "input" to the "output" in a hierarchical manner such that the output of the above layer is given as input to the next layer. The vector passed to the input layer is commonly defined as the features vector and the output layer is composed of a single neuron the output of which is the local contribution to the total energy. The extrapolation from one FFNN to a potential is trivial: each features vector is labeled with the species of the central atom and is processed accordingly by a different FFNN. The combination of all these networks (one per species) defines a full potential. On the right side: the main training/validation procedure is sketched.

the situation is quite different: once the (great) flexibility of the class of representable functionals is defined by the choice of a specific architecture, the many parameters defining the model are determined in order to reproduce accurate calculation performed on a number or configurations deemed representative. The training/validation dataset is crafted by sampling configurations that will likely be explored during the process of interest. As we are aiming at describing the complex crystal/amorphous interface dynamics in SPE, a number of different phase space regions have been included in the dataset: crystalline and amorphous bulk, surfaces, liquids, and more. The final accuracy in each region depends on the amount of sampling and on the variability of local configurations: regions with limited variability require little sampling while regions with more entropy require larger amounts of data points.

Different parts of the phase space were sampled with different details but all following a common strategy: molecular dynamics (MD) were performed using LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF]; from these MD trajectories a number of uncorrelated samples were extracted and their accurate energy and forces were obtained with the Quantum ESPRESSO [START_REF] Giannozzi | QuantumESPRESSO toward the exascale[END_REF] DFT package (with a cutoff of 50 Ry and a 46.10 Å k-points grid linear-density). These calculated values become part of the dataset. The exchange-correlation functional initially selected was PBE-sol, a PBE-type functional optimized for solid and surfaces [START_REF] Perdew | Restoring the density-gradient expansion for exchange in solids and surfaces[END_REF]. The dataset energies and forces were later recomputed with the original PBE functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] to address the unexpected poor description of the melting temperature obtained with PBE-sol (more about this in Results Sec. IV).

Diamond bulk was sampled around its harmonic minimum using Tersoff potential. For other phases, that show higher complexity, we proceeded with an iterative scheme similar to the one we proposed in [START_REF] Shaidu | A systematic approach to generating accurate neural network potentials: The case of carbon[END_REF]. A first round of MD sampling is performed with empirical or (rough) NN potentials. Points collected from these MDs are used to train new, more accurate, NNs. These potentials are iteratively used to perform more MDs exploring additional, more relevant, geometries to be added in the dataset to improve the quality of the potential. This procedure stops when no appreciable additional phase space sampling is observed in subsequent iterations. Some of the configurations from earlier iterations might then be removed from the dataset as not sampled anymore by the accurate self-consistent potential.

The sampling of the amorphous phase was conducted starting from Tersoff potential and annealing setup similar to one used in previous SPE studies [START_REF] Krzeminski | Molecular dynamics simulation of the recrystallization of amorphous Si layers: Comprehensive study of the dependence of the recrystallization velocity on the interatomic potential[END_REF]. The liquid was sampled in a similar way as the amorphous. We also added a few solid/liquid interface samples to include realistically intermediate local environments.

Sampling of crystalline surfaces was a bit tricky as none of the commonly used empirical potentials was able to describe correctly the asymmetric dimerization present at the (001) surface. We therefore generated a preliminary NN potential by using part of the database developed by Bartok et al. for their GAP potential [START_REF] Bartók | Machine learning a general-purpose interatomic potential for silicon[END_REF]. Despite not being optimal for NN application this database was sufficient to provide a fair starting point reducing the number of self-consistent iterations needed and was kept in the training set for all iterations.

During the evolution, the dataset variability was monitored using a t-sne bidimensional projection of the explored dataset as presented in Fig. 2. This showed how configurations generated by Tersoff and early generation NN potentials explore different geometries from those consistently found by newer generations of NN potentials.

Approximately 16 thousand points were collected, for a total of 890 thousand local environments. After removing the data from [START_REF] Bartók | Machine learning a general-purpose interatomic potential for silicon[END_REF], the structures that were no longer visited during the latest iterations of the self consistent procedure, and applying a random uniform sub-sampling on the remaining dataset we were left with approximately 5 thousand data points, or 250 thousand local environments. A standard splitting of the dataset in a training part (containing 80% of the data) and a validation part (containing 20% of the data) has been performed, paying attention that the same fraction of training and validation data were present in every subset so as to allow an homogeneous testing in different regions of the phase space.

IV. RESULTS

The overall validation accuracy of the NN potential trained on the previously described dataset, with energies evaluated according to the PBE exchange-correlation functional, is of 5.6 meV/atom for the energy per atom root mean square error (RMSE) and of 153 meV/Å mean error per forces component. This accuracy is comparable with the one obtained by others with similar approaches [START_REF] Artrith | High-dimensional neural network potentials for metal surfaces: A prototype study for copper[END_REF]. A more detailed analysis of the validation accuracy decomposed over the different sampled macro regions of the phase space described in the previous all amorphous bulk liquid surfaces Fig. 2. The figure shows the t-sne applied to the structures in the dataset using the fingerprint defined in Valle et al. [START_REF] Valle | Crystal fingerprint space -a novel paradigm for studying crystal-structure sets[END_REF] and the related distance. In gray all the available structures are presented while colored points refer to the final dataset. Different colors highlight macro-regions of the phase space. Small sparse gray islands are points from the variegate dataset in [START_REF] Bartók | Machine learning a general-purpose interatomic potential for silicon[END_REF]. Big dense gray islands instead are region sampled by older iterations of the potential excluded because no more visited. section is shown in Fig. 3 and shows that the accuracy is rather uniform across the phase space. In order to get a feeling of the quality of the potential associated to these reported numerical errors we tested it investigating a few physical properties that can be compared with experiments and/or DFT references. A comprehensive test has been carried out for the diamond structure where we checked the accuracy of the Energy-Volume curve and computed the bulk modulus obtaining a value of 98.0 GPa, close to the experimental value of 97.6 GPa [START_REF] Hopcroft | What is the young's modulus of silicon?[END_REF]. Full phonon dispersions were also investigated obtaining results very close to the DFT reference (see bottom panel of Fig. 4).

SPE dynamics critically depends on the temperature involved and commonly adopted empirical potentials either fail terribly in describing the thermal stability of the different phases or, if they succeed in it, fail in other aspects.

For example, the SW potential has a melting temperature of 1691 ± 20 K [START_REF] Broughton | Phase diagram of silicon by molecular dynamics[END_REF] but generates amorphous of poor quality. Tersoff potential, on the other hand, is believed to generate realistic amorphous geometries but it predicts a melting tem-perature of 2547 ± 22 K [START_REF] Cook | Comparison of semi-empirical potential functions for silicon and germanium[END_REF]. Errors in the predicted melting temperature are likely to impact defect formation and diffusion dynamics, relevant for SPE. We hence turned to the study of the bulk melting temperature, equilibrating samples where crystalline and liquid phases coexist [START_REF] Zhang | A comparison of methods for melting point calculation using molecular dynamics simulations[END_REF]- [START_REF] Morris | The melting lines of model systems calculated from coexistence simulations[END_REF].

We were disappointed to find out that the PBE-sol functional originally adopted gave a NN potential whose bulk melting temperature was as low as 1194 ± 28 K. This induced us, after several cross checks, to resort to the original PBE xcfunctional, obtaining a melting temperature of 1468 ± 11 K, in line with the literature value of 1540 ± 50 K [START_REF] Yoo | The melting temperature of bulk silicon from ab initio molecular dynamics simulations[END_REF], still slightly off w.r.t. the experimental value of 1685 ± 2 K. The unsuitability of PBE-sol functional to describe melting behavior of materials has been recently independently confirmed for GAP potentials [START_REF] Jinnouchi | On-the-fly machine learning force field generation: Application to melting points[END_REF].

Having obtained a reasonable description of the liquid/solid transition we proceeded to study the amorphous phase. We followed the procedure described in Ref. [START_REF] Laaziri | High-energy x-ray diffraction study of pure amorphous silicon[END_REF], [START_REF] Lampin | Molecular dynamics simulations of the solid phase epitaxy of si: Growth mechanism and orientation effects[END_REF] where liquid samples were quenched with a 10K/ps ramp. Amorphous samples of very good quality are thus obtained. In the top panel of Fig. 4 the structure factor from the largest simulated cell (4096 atoms) is compared with experimental results. Next we examined in detail the (001) surfaces that play a very important role on microelectronic devices [START_REF] Ramstad | Theoretical study of the si(100) surface reconstruction[END_REF]. Several reconstructions have been proposed in [START_REF] Ramstad | Theoretical study of the si(100) surface reconstruction[END_REF]: p(1x1) also known as Ideal, p(2x1) symmetric, p(2x1) asymmetric, p(2x2) and c(4x2). Of these four structures only the last three have proven to be stable with sufficiently accurate calculations. The energy differences among predicted structures can be seen in Fig. 5. Geometries were also compared and our NN potential has shown to be able to correctly predict the dimer tilting angle of 18 • for the p(2x1) asymmetric phase as well as the 19 • angle of the p(2x2) and c(4x2) ones, in accordance with DFT.

V. PRELIMINARY OBSERVATION OF THE SPE MECHANISMS

The overall broad accuracy shown by our NN potential for so many properties gives us confidence in its adequacy to explore the more complex phenomenon of SPE. Using a MD procedure similar to the one described in [START_REF] Lampin | Molecular dynamics simulations of the solid phase epitaxy of si: Growth mechanism and orientation effects[END_REF], we created an amorphous/crystalline diamond interface and performed a first SPE simulation at 1050 K. Already in this preliminary simulation we can see that 3D pyramidal motives, expected from the different reconstructions speed measured experimentally [START_REF] Olson | Kinetics of solid phase crystallization in amorphous silicon[END_REF], appear (presented in Fig. 6). A detailed study of SPE mechanism and speed is underway. VI. CONCLUSIONS AND OUTLOOK We have shown that NN potentials are a valid way to model physical phenomena at the atomistic level through the creation and testing of a potential for silicon from a diverse dataset, crafted from crystalline, amorphous, liquid and surface configurations. As a byproduct, we found that PBE-sol xc-functional is not suitable to study thermal related phenomena in silicon. With the developed potential we are investigating the SPE process, focusing on the underlying recrystallization mechanism and speed. We expect that the resulting insight will allow us to improve the prediction of TCAD codes, possibly solving the longstanding problem of the experimentally observed difference in recrystallization speed in different growth directions.

  formed as a linear combination of the previous layer information using layer dependent bias, b (l) i , and weights, w

  , and passing this intermediate result through a layer dependent activation function, f (l) . The special case x (0) j is the input vector that is in our case the collection of 104 SF components, while x (L) 1

Fig. 3 .

 3 Fig. 3. Distributions of the errors per dataset in energies (top panel, box corresponds to the first quartile) and Forces (bottom panel, different boxes represent different quartiles) are presented. It can be seen how phases with the highest variability (liquid and amorphous) show the highest errors. At this point different RMSE can be computed for energies and forces depending on the subset of points: 5.8 meV -170 meV/Å for the amorphous, 0.9 meV -66 meV/Å for the bulk, 7.5 meV -196 meV/Å for the liquid and 3.4 meV -103 meV/Å for the surfaces.

Fig. 4 .

 4 Fig.4. Top Panel: Amorphous silicon structure factor S(q). Dots are experimental points[START_REF] Laaziri | High resolution radial distribution function of pure amorphous silicon[END_REF], green line from NN potential. Bottom Panel: Diamond phonons of silicon. DFT prediction with PBE (black line) and NN (red line).

Fig. 5 .

 5 Fig.5. Surfaces energies. From left to right, PBE-Sol energies, PBE energies, NN energies for PBE geometries, NN energies after relaxation. The c(4x2) structure has always been taken as zero reference. DFT geometries have been obtained by relaxation with the BFGS algorithm, as a force threshold we opted for the default 25 meV/Å. Minimization with NN potentials were performed with the same threshold. Both DFT calculations predicted the p(2x1) symmetric structure as stable with a low k-points grid, the reported energy is a successive SCF calculation with the high accuracy grid. In the last column the s p(2x1) geometry has relaxed to the a p(2x1) structure. The p(2x2) and the c(4x2) are incorrectly ordered but their difference in energy is below our error threshold.

Fig. 6 .

 6 Fig. 6. 3D pyramidal motives appearing during the solid phase epitaxy simulation at 1050K. Only tetrahedrally coordinated atoms with angles compatible with the diamond structure are presented for simplicity.

VII. ACKNOWLEDGEMENTS We acknowledge funding from European Union Horizon 2020 research and innovation program under grant agreement No. 871813 MUNDFAB and No. 824143 MaX CoE.