Panagiota Katsikouli

Aline C Viana

Marco Fiore

Alberto Tarable

L' étude de la fr équence d' échantillonnage des mouvements des humains

Keywords: Mobilité humaine, trajectoires spatio-temporelles, fréquence d'échantillonnage

Des études récentes ont profité des techniques de suivi basées sur les technologies de positionnement d'étudier afin la mobilité humaine. Ces recherches ont révélé, entre autres, une grande régularité spatio-temporelle des mouvements individuels. Sur la base de ces résultats, nous visons à répondre à la question "à quelle fréquence doit-on échantillonner les mouvements humains individuels afin qu'ils puissent être reconstruits à partir des échantillons recueillis avec un minimum de perte d'information ?". Notre quête d'une réponse a conduit à la découverte de (i) propriétés spectrales apparemment universelles de la mobilité humaine, et (ii) une loi de mise à l'échelle linéaire de l'erreur de localisation par rapport à l'intervalle d'échantillonnage. Nos résultats sont basés sur l'analyse des trajectoires GPS de 119 utilisateurs dans le monde entier. Les applications de nos résultats sont liées à un certain nombre de domaines pertinents pour l'informatique omniprésente, tels que l'informatique mobile économe en énergie, les opérations de service basées sur l'emplacement, le sondage actif des positions des abonnés dans les réseaux mobiles et la compression des données de trajectoire.

Introduction

Over the past few years, the pervasive usage of smart devices and location-tracking systems has made it possible to study and understand human mobility at unprecedented scales. Fig. 1, shows heatmaps and timeseries of the locations visited by two random users in our employed dataset. Although these plots convey three weeks of data, a small set of frequently visited places emerges along with systematic paths connecting them. We investigate whether the regularity of human mobility entails the possibility of sampling individual movements at reduced constant frequencies, while allowing for the reconstruction of trajectories that retain a vast portion -if not all-of their original level of detail. Identifying suitable frequencies for human mobility sampling would have applications in a number of fields, including but not limited to (i) mobile computing, where overly frequent GPS localization unnecessarily reduces the battery life of mobile devices, (ii) location-based service design, where unwarranted users' position data collection raises significant privacy concerns, (iii) cellular networks, where active probing of subscribers' positions is a costly task whose rate must be duly optimized, (iv) trajectory data compression, where information loss must be minimized.

Our problem is equivalent to asking "at what frequency should one periodically sample individual human movements so that they can be reconstructed from the collected samples with minimum loss of information ?". To respond, we consider mobility patterns as signals over time, and carry out a spectral analysis of human mobility. We find that the spectra of the movements of 119 individuals have very similar, flat shapes that suggest the absence of convenient sampling frequency thresholds. We then carry out a quantitative analysis of the user localization error in movements reconstructed from regular sampling at different periodicities. Our results unveil a linear scaling law of the error with respect to the span of the constant sampling interval. This law corroborates the outcome of the spectral analysis and has significant practical implications, as it controls the trade-off between accuracy and cost of measurements of human mobility.

Related Works. The objective in spatial data trajectory compression is to maintain the trajectory shape (see [START_REF] Katsikouli | Persistence based online signal and trajectory simplification for mobile devices[END_REF][START_REF] Muckell | Algorithms for compressing gps trajectory data : An empirical evaluation[END_REF] and references therein). We aim at also preserving the temporal dimension of movements. Our problem is also different from sampling to detect important locations [TCK13, CBT + 13]. Indeed, we are not interested in simplifying a pre-recorded GPS trajectory but to find convenient sampling frequencies for human trajectory data. To the best of our knowledge, this is the first work to thoroughly study the problem of finding a good constant sampling frequency at which to sample human mobility so that users' complete movements can be accurately reconstructed.

Reference dataset

Our study employs a dataset of real-world individual mobility data extracted from three different sources.

-MACACO data : collected as part of the EU CHIST-ERA MACACO project † -OpenStreetMap (OSM) data : collected by volunteers who recorded and uploaded their trajectories as a contribution to the OSM database ‡ -Geolife data : collected in Beijing by Microsoft Research Asia [ZLC + 08]. The GPS trajectories in our dataset represent human mobility with various means of transportation (e.g., walking, trams, airtravel, cycling etc.) and at various scales ; they cover sensibly different geographical and temporal spans with the quality of the data for a single user being typically very heterogeneous over time, with periods of days or weeks where GPS logs are completely absent. To build a consistent reference dataset, we segmented the mobility traces of all users into one-week trajectories, and analysed them separately. During each week, we bounded the mobility of each individual to the regions where the activity is concentrated. Then, we retained only the trajectories that contain complete GPS records in at least six out of seven distinct week days, resulting in 1,052 weeks of mobiliy. We finally argue that the sampling intervals in the one-week trajectories of our reference dataset, which are upto 15 minutes, are sufficient to capture human movements, and we consider them as our ground-truth in the remainder of the study.

Spectral analysis of human mobility

From a spectral analysis viewpoint, answering the question "at what frequency should one periodically sample individual human movements so that they can be reconstructed from the collected samples with minimum loss of information ?", means to consider human movements as a signal in time, and study its spectrum in frequency by applying Fast Fourier Transform.

First, we need to transform individual GPS trajectories into unidimensional time series. We opted for a parallel study of the two dimensions of the geographical space, by considering them in isolation. Instead of †. https ://macaco.inria.fr/macacoapp/ ‡. https ://www.openstreetmap.org using the absolute values of latitude and longitude, we replace them with the signed latitude and longitude displacements from the corresponding centre of mass of the one-week trajectory. Our transformations have the property of generating zero-mean signals whose frequency spectra have no DC components. We also analysed the correlation between the isolated latitude or longitude displacements and the actual travelled distance in the bidimensional space. The results we obtained let us conclude that both dimensions, when taken separately, still provide decent approximations of the overall mobility. Fig. 2 shows the spectra of the latitude (top) and longitude (bottom) displacement signals of a representative selection of one-week trajectories. The original spectra are in light blue, while a moving-average that better displays the overall trends is in dark blue. Vertical orange lines outline the frequencies that correspond to sampling intervals of 10 minutes (farthest from the central frequency), 1 hour and 12 hours (closest to the central frequency). We make two important remarks : (i) despite the diversity across the different one-week trajectories, all spectra have very similar shapes ; (ii) the shapes do not show evidence of a bandwidth threshold beyond which the spectra become clearly negligible, making it impossible to identify an operational point for effective sampling. We found that the observations above hold in the overwhelming majority of our user-base. We can explain both facts by considering that human mobility is a sequence of long periods where individuals are almost static and fast transitions between such important locations. While positions during stationary time intervals contribute to low-frequency spectral components and are hence easily captured by a sparse sampling, travelling causes discontinuities in the mobility signal and is much harder to sample. Thus, the spectra do not reveal whether, e.g., collecting samples at every 10 minutes is obviously more efficient than sampling at every hour.

A quantitative analysis of mobility sampling

We experimentally investigate the exact trade-off between the quality and cost of sampling in the context of human mobility. We create downsampled versions of the one-week trajectories in our reference dataset, using a wide range of sampling intervals, from 10 minutes to 12 hours. We then linearly interpolated the samples, and assess how such reconstructed trajectories compare to the original ones. We measure the error in retrieving a complete individual trajectory from sampled data by computing the average Haversine distance, which is the mean of all Haversine distances between the points of the reconstructed and original mobility recorded at the same time instant. Fig. 3 shows the evolution of the average Haversine error against the sampling interval, for a representative set of individuals in our reference dataset. Each plot presents results for all of the one-week trajectories of a specific user : as multiple one-week trajectories are aggregated in every plot, we outline the mean (dots), 25-75% quantiles (dark blue region) and 10-90% quantiles (light blue region) of the error measured over all trajectories of one user.

A surprisingly clear linear relationship characterizes all curves. Fittings on a simple linear model (solid lines in Fig. 3) show an excellent match for all users in our dataset. Fig. 4a portrays the CDF of the Root Mean Square Error (RMSE) between the linear fitting and the mean values (solid lines and dots respectively in Fig. 3) of the average Haversine distance, for all users, over sampling intervals that range from 10 minutes to 12 hours. The probability mass of the distribution is below 250 meters -a very reasonable RMSE for people travelling tens of km per day.

We also highlight that the slope α characterizes the ratio between the average Haversine distance and the sampling interval, or, equivalently, it explains the mean additional error of the reconstructed trajectory when increasing the time that intercurs between samples. Hence, it can be measured in meters per minute (m/min). When looking at the value of α, we remark that it is not identical across users. We study the heterogeneity of α in Fig. 4b, which portrays the CDF of the distance/interval ratio associated to all individuals in our reference dataset. Over 90% of users have slopes that are uniformly distributed between 1 and 4 m/min. Hence, for the vast majority of individuals, the inaccuracy of their recorded trajectory grows of 1 to 4 meters for each minute added to their movement sampling interval.

Discussion and Conclusions

Summarizing our findings, we assert that the average error incurred by trajectories reconstructed from periodic samples scales linearly with the constant sampling interval. This result is well aligned with the outcome of our spectral analysis in Sec. 3 : the linearity of the relationship between error and sampling interval explains the absence of an operational point for the effective sampling of human movements. In addition, we find that the linear scaling law is characterized by a comparable parameter, i.e., the errorto-interval ratio, across all our user base depending on the individual, the error typically grows 1 to 4 meters when adding one minute to the inter-sample time. We are currently investigating the physical reasons behind the diversity among different individuals as well as working on an adaptive algorithm that will tailor the sampling frequency to the user mobility over time. A full version of this work is published in IEEE Globecom 2017.

FIGURE 1 :

 1 FIGURE 1: Heatmaps and time-series of locations visited by two distinct users during three weeks : humans tend to commute between a limited set of specific locations and revisit locations in a periodic fashion.

FIGURE 2 :

 2 FIGURE 2: Frequency spectra of unidimensional movement signals, for a selection of one-week trajectories.

FIGURE 3 :

 3 FIGURE 3: Average Haversine distance between the original and the reconstructed trajectories of four users, versus sampling intervals between 10 minutes and 12 hours.

  FIGURE 4: (a) Distribution of the RMSE due to the linear approximation of the relation between the average Haversine distance and the sampling interval. (b) Distribution of the ratio between the average Haversine distance and the sampling interval. Plots are for all users.


Caroline M Moore
email: caroline.moore@ucl.ac.uk

Nicola L Robertson

Fatima Jichi

Adebiyi Damola

Gareth Ambler

Francesco Giganti

Ashley J Ridout

Simon R J Bott

Mathias Winkler

Hashim U Ahmed

Simon Rj Bott

Manit Arya

Anita V Mitra

Neil Mccartan

Alex Freeman

Charles Jameson

Ramiro Castro

Giulio Gambarota

Brandon J Whitcher

Clare Allen

Alex Kirkham

Mark Emberton

The Effect of Dutasteride on Magnetic Resonance Imaging Defined Prostate Cancer: MAPPED-A Randomized, Placebo Controlled, Double-Blind Clinical Trial

Keywords: Prostate cancer, active surveillance, MRI, targeted biopsy, dutasteride

Materials and methods

In this randomized, double-blind, placebo-controlled trial, men with biopsyproven low-intermediate risk prostate cancer (up to Gleason 3+4 and PSA up to 15 ng/ml) who had an MR visible lesion of >/= 0.2ml on T2-weighted sequences were randomized to daily dutasteride 0.5mg or placebo for 6 months. Lesion volume was assessed at baseline, 3 and 6 months, with an image-guided biopsy to the lesion at study exit. The primary endpoint was percentage reduction in lesion volume over 6 months. This trial was registered with the European Clinical Trials register (EudraCT 2009-102405-18).

Results

Forty-two men were recruited between June 2010 and January 2012. In the dutasteride group, the average volumes at baseline and 6 months were 0.55ml and 0.38ml respectively, and the average percentage reduction was 36%. In the placebo group, the average volumes at baseline and 6 months were 0.65ml and 0.76ml respectively, and the average percentage reduction was -12%. The difference in percentage reductions between groups was 48% (95% CI 27.4-68.3%. p< 0.0001). The most common adverse event was deterioration in

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erectile function (25% in men randomized to dutasteride, 16% in men randomized to placebo).

Conclusions

Dutasteride was associated with a significant reduction in prostate cancer volume on T2 weighted MRI images compared to placebo.

Introduction

Dutasteride is licensed for men with lower urinary tract symptoms (LUTS) associated with benign prostatic enlargement [START_REF] Roehrborn | Efficacy and safety of a dual inhibitor of 5-alpha-reductase types 1 and 2 (dutasteride) in men with benign prostatic hyperplasia[END_REF] . When assessed in placebo controlled prostate cancer studies, dutasteride reduced the proportion of men diagnosed with prostate cancer by 22.8% over 4 years 2, and was associated with lower rates of disease progression over 3 years in men on active surveillance [START_REF] Fleshner | Dutasteride in localised prostate cancer management: the REDEEM randomised, double-blind, placebocontrolled trial[END_REF] . Magnetic resonance imaging (MRI) of the prostate allows assessment of prostate cancer volume over time [START_REF] Kirkham | How good is MRI at detecting and characterising cancer within the prostate?[END_REF][START_REF] Villers | Dynamic contrast enhanced, pelvic phased array magnetic resonance imaging of localized prostate cancer for predicting tumor volume: correlation with radical prostatectomy findings[END_REF] . We report the use of MRI to assess the effect of dutasteride on prostate cancer volume in men on active surveillance.

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Materials and methods

This

Primary and secondary objectives

The primary objective was to evaluate the percentage reduction in tumour volume, assessed by T2-weighted MRI (T2W-MRI), following exposure to dutasteride 0.5 mg daily for six months, compared to placebo.

Secondary objectives included percentage reduction in tumour volume at 3 months, and on functional MR sequences, namely dynamic contrast enhanced (DCE) imaging and diffusion weighted imaging (DWI). Prostate cancer typically shows rapid wash-in and wash-out on DCE imaging due to the increased vascularity of its' highly permeable neo-vessels. DWI of prostate cancer typically shows restricted diffusion due to disorganized tissue structure, seen as high signal (bright areas) on high b-value imaging [START_REF] Barentsz | ESUR prostate MR guidelines 2012[END_REF] .

Patient population

The full study protocol has been published [START_REF] Robertson | MAPPED study design: A 6month randomised controlled study to evaluate the effect of dutasteride on prostate cancer volume using magnetic resonance imaging[END_REF] . Eligible men met the UK NICE 2008 active surveillance criteria [START_REF]Prostate cancer: diagnosis and treatment[END_REF] (up to Gleason 3+4 disease, PSA up to 15 ng/ml), based on biopsy within the preceeding two years. In addition, men were required to have a >0.2 ml lesion on the standard of care T2-weighted MRI.

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All eligible consented men had a study specific standardized multi-parametric 3 Tesla MRI. The images were reviewed by one study radiologist (CA, AK) to assess for the presence of a T2W lesion >0.2 ml. When confirmed, the man was randomized to placebo or dutasteride 0.5mg daily for 6 months. MRI was repeated at 3 and 6 months. Each MRI scan was reported separately by each of the study radiologists, who were blinded to treatment allocation.

An exit biopsy was offered to all men, with 10 standard cores and additional cores targeted to the MRI lesion using visual registration [START_REF] Moore | Standards of Reporting for MRI-targeted Biopsy Studies (START) of the Prostate: Recommendations From An International Working Group[END_REF] . Men were not rebiopsied for discordant histology at baseline, in order to avoid post biopsy artefact on the study MRI scans. The concordance between baseline histology, lesion on MRI and exit histology was assessed.

PSA, renal and liver function and adverse events were assessed at baseline, 3 and 6 months.

Randomization and masking

Seventeen patients were required to complete each arm to detect a 20% reduction in tumor volume, assuming a common standard deviation of 20.5%, power of 80% and 5% statistical significance. To account for failure to complete, we aimed to recruit 21 men per arm. Men were individually randomized using block randomization with varying block sizes. The statistical team at University College London (UCL) supplied UCLH pharmacy with a randomization list to allocate study medication. The patients, study doctors, radiologists and

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histopathologists were blind to treatment allocation. The radiologists and histopathologists were also blind to PSA results.

Outcomes and statistical analysis

The primary outcome was percentage reduction in tumor volume between baseline and 6 months on T2-weighted imaging. This was calculated using (baseline volume -6-month volume)/baseline volume 100%, where baseline volume and 6 months volume were the average of the measurements from the two raters. This percentage reduction was compared between the dutasteride and placebo groups using a t-test, with statistical significance set at a p-value of 0.05.

The secondary outcomes of percentage reduction in tumor volume as measured by DCE and DWI at six months, were compared between the dutasteride and placebo groups using a t-test. Each analysis was repeated for percentage reduction in tumour volume at three months.

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Results

Fifty six men consented to the study after meeting eligibility criteria based on standard of care MRI, histology and blood tests. Seven men were excluded as the study PSA or renal function did not meet the eligibility criteria. Forty-nine men had a study MRI scan. Seven of these men were excluded on MRI criteria.

Forty-two men were randomized between June 2010 and January 2012 (CONSORT [START_REF] Begg | Improving the quality of reporting of randomized controlled trials. The CONSORT statement[END_REF] diagram, figure 1).

Twenty-one men were randomized to each group (Table 1). One man exited the dutasteride group prior to taking any study medication as he wished to conceive a child. Another man exited the placebo group to have active treatment in response to upgrading from Gleason 3 + 3 to Gleason 3 + 4 at UCLH pathology review. The primary endpoint was derived in 20 men within each group.

When the baseline scans were formally reported, it was noted that a total of 11/42 randomized men had lesions which measured as less than 0.2ml on T2weighted images by one (n = 5) or both (n=6) study radiologists. All of these men had lesions volume of 0.2ml on other sequences (DCE and diffusion weighted imaging) by at least one of the radiologists. One of the men was deemed to have a visible lesion by one radiologist but not by the other. Eight men had a mean lesion volume of <0.2ml for the lesion on T2 weighted imaging. These men were included in the analysis, as they had met the original inclusion criteria (one study radiologist assessing the MRI scan as showing a 0.2ml lesion on MRI). The agreement in volume assessment between the 2 raters was formally assessed.

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Primary outcome

Change in total tumor volume on T2W-MRI over 6 months (Table 2)

In the dutasteride group, the average volumes at baseline and 6 months were 0.55ml and 0.38ml respectively, and the average percentage reduction was 36%. In the placebo group, the average volumes at baseline and 6 months were 0.65ml and 0.76ml respectively, and the average percentage reduction was -12% 9ie 12% growth). The difference in percentage reductions between groups was 48% (95% CI 27.4 -68.3, p <0.0001) (figure 2).

Fifteen of twenty men (75%) randomized to dutasteride had a reduction in tumor volume; 5/20 (25%) had stability (defined as < 20% volume change). No man randomized to dutasteride had an increase in tumor volume. In contrast, 2 men (10%) in the placebo group had a reduction in volume, 13 (65%) were stable and 5 (25%) exhibited an increase of > 20% (figure 3).

Secondary outcomes

Change in tumor volume on functional imaging over 6 months (Table 2)

On DCE imaging, men randomized to dutasteride had a 42.3% reduction in tumor volume at 6 months compared to 4.2% reduction in men men randomized to placebo (38% mean difference between groups, 95% CI: 16.59-59.64, p= 0.001).

On diffusion weighted imaging, men randomized to dutasteride had a 33% reduction in tumor volume at 6 months compared to a 7% increase in the men

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randomized to placebo, conferring a mean difference between groups of 40% (95% CI 21.31 -59.63, p = 0.0001).

Change in tumor volume over 3 months (T2-weighted imaging)

At 3 months there was a 34% reduction in tumor volume in the dutasteride group compared to 0.21% in the placebo group (mean difference between groups 34% (95% CI 21 -47%, p = < 0.0001).

Change in prostate volume

The average percentage reduction in prostate volume at 6 months was 15% in the dutasteride group, compared to an increase of 3.3% in the placebo group (mean difference between groups 18.8% (95% CI 14.9 -22.8%, p<0.0001).

Agreement between raters

The agreement between the raters was investigated using a Bland-Altman plot based on the percentage reduction in tumor volume calculated separately for each individual rater (supplementary figure 1).

In addition, the analysis of the primary outcome was repeated for each rater separately (supplementary table 1). One patient was deemed by one rater to have an increase in volume and the other rater to have a reduction in volume, and this had a significant effect on the results of rater 1. The table shows the analysis with and without this outlier. The results are consistent across the raters although rater 1 has recorded a larger volume reduction than rater 2.

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Histological data (Supplementary table 2)

All men were offered an end of study biopsy, although 12 of 40 men declined.

Histological upgrading to Gleason > 4+ 3 was seen in 3/15 (20%) in the dutasteride group, and in 6/13 (46%) in the placebo group.

Concordance of MRI lesions and histology (supplementary table 3)

Thirty seven men of forty men had histological confirmation of the MRI lesion at baseline or exit. The remaining three men had discordant baseline histology, of whom one had a negative exit biopsy (placebo group) and two declined the exit biopsy (one placebo, one dutasteride).

PSA changes (supplementary table 4)

In men on placebo, the direction of change of PSA mirrored the direction of change in tumour volume. All men on dutasteride showed a PSA reduction, with no clear correlation between the change in lesion volume and PSA.

Medication compliance

All men were compliant with medication according to the returned tablet count at the end of each 3 month prescription period.

Adverse events

No serious adverse events occurred, and no subject discontinued study medication due to adverse events. Deterioration of sexual function (at least a 4 point reduction in IIEF-15 [START_REF] Rosen | The International Index of Erectile Function (IIEF): A Multidimensional Scale for Assessment of Erectile Dysfunction[END_REF] ) occurred in 8 of 40 men (5/20 (25%) on dutasteride and 3/19 (16%) on placebo.

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Discussion

Dutasteride 0.5mg daily was associated with an average reduction in prostate cancer volume of 36%, compared to an average increase of 12% in men on placebo, as assessed by T2W-MRI over six months.

This volume change was associated with less histological upgrading at targeted biopsy on study exit when randomized to dutasteride than to placebo (20% versus 46%). However, the lack of targeted biopsies at baseline to match to targeted exit biopsies and the small sample size limit the ability to definitively ascribe this to the effect of dutasteride. The rate of upgrading in the placebo group was high compared to that seen in men undergoing scheduled repeat standard 10-12 core systematic biopsy on active surveillance protocols [START_REF] Dall'era | Active Surveillance for Early-stage Prostate Cancer: Review of the Current Literature[END_REF] .

However, this was expected due to the use of image-guided targeting as well as standard biopsy exit, compared to standard biopsy alone at entry, in the majority [START_REF] Siddiqui | Comparison of MR/ultrasound Fusion-guided Biopsy with Ultrasound-guided Biopsy for the Diagnosis of Prostate Cancer[END_REF] .

In 3 of 40 men there was a lack of concordance between biopsy and entry histology, not resolved on targeted biopsy at exit, which may represent a false positive MRI lesion, although 2 of the men declined the exit biopsy, and so this could not be confirmed.

There have been concerns that dutasteride may act selectively on low grade tumors, and may be associated with a greater likelihood of developing highgrade cancers [START_REF] Andriole | Modeling and analysis of Gleason score 8-10 prostate cancers in the REDUCE study[END_REF] , although radical prostatectomy data from randomized studies have not confirmed this [START_REF] Iczkowski | The dual 5-alpha-reductase inhibitor dutasteride induces atrophic changes and decreases relative cancer volume in human prostate[END_REF] .

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In this study, two men randomized to dutasteride had 1mm and 2mm respectively of Gleason 4 + 4 on targeted biopsy at exit, with an overall Gleason grade of 3 + 4. At entry, the first had 9mm Gleason 3 + 3 on standard transrectal biopsy, with a lesion volume of 0.97ml, which reduced to 0.64ml at 6 months.

The second had 4mm 3+ 4 on targeted biopsy at entry, with a lesion volume of 0.20ml, which reduced to 0.13ml at 6 months. This grade increase on targeted biopsy at exit would be compatible with the grade shift seen with a targeted biopsy approach [START_REF] Robertson | Prostate Cancer Risk Inflation as a Consequence of Image-targeted Biopsy of the Prostate: A Computer Simulation Study[END_REF] .

Dutasteride-associated reduction in prostate cancer volume was seen at 3 months, with marginal additional reduction by 6 months, with a similar timescale to prostate volume reduction in LUTS studies. We noted greater proportional reduction in tumor volume compared to whole prostate volume which suggests that prostate cancer may be more sensitive to androgen depletion than noncancerous prostate tissue.

The functional MRI sequences (DCE and DWI) assess perfusion and cell density, respectively. The percentage volume reduction in functional sequences was similar to that on T2Wi, although DWI tumor volumes were lower than on other sequences.

The adverse event profile was compatible with the known side effects of dutasteride, with significantly more men taking dutasteride (25% vs 16%)

showing deterioration in sexual function.

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The study is limited in both its sample size and duration. The 20% threshold for a reduction in tumor volume change was chosen, as it was the least change that might be considered both clinically meaningful and assessable by our independent observers. Sufficient men were recruited to achieve a power of 80% to detect a 20% volume change. A six-month interval of assessment is common in many active surveillance protocols [START_REF] Dall'era | Active Surveillance for Early-stage Prostate Cancer: Review of the Current Literature[END_REF] , and is known to show response to dutasteride in studies of men with LUTS. The fact that a study of this size and duration could meet its primary endpoint is promising for future exploratory studies of this type.

Another limitation of the study relates to the external validity of the study population. The eligibility criteria consisted of the UK NICE 2008 criteria for active surveillance in men with low or intermediate risk disease (Gleason sum 7 or less, PSA < 15ng/ml), clinical T2b disease). Of note, these criteria do not include any estimation of burden of disease such as the number of cores positive, maximum cancer core length or percentage cancer core involvement.

In addition to the UK NICE criteria, men were required to have a lesion visible on T2W-MRI.

Men with lesions seen exclusively on functional imaging were excluded. We do not know whether radiological phenotype predicts prognosis or responsiveness, although men with no visible lesion on MRI are at lower risk of progression than men with a visible lesion [START_REF] Park | Role of Multiparametric 3.0 Tesla Magnetic Resonance Imaging in Prostate Cancer Patients Eligible for Active Surveillance[END_REF] .

We assessed for an imbalance of tumor size in each group which might lead to a spurious result, if larger tumors are assumed to have a greater growth rate.

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Four men had tumor volumes >1 ml, 3 of whom were randomized to placebo.

We repeated the analysis with these men removed and found very similar results. In addition, a formal analysis that adjusted for baseline volume was performed with little change in the outcome.

Our findings may have important clinical implications. First, if MRI reliably detects prostate cancer volume change then it could be useful for reassessment during surveillance [START_REF] Graham | Prostate cancer: a summary of updated NICE guidance[END_REF] . Further work is needed to determine the imaging parameters (rate of change or an absolute threshold) at which transition to active treatment should be initiated, as recommended in the recent PRECISE guidelines [START_REF] Moore | Reporting MRI in men on active surveurveillance for prostate cancer. The PRECISE recommendations: A report of a European School of Oncology Task Force[END_REF] Second, this novel design might be useful in assessing well tolerated agents that might modify the natural history of prostate cancer (e.g. aspirin, vitamin D, and dietary modifications including cruciferous vegetables [START_REF] Veitonmäki | Use of Aspirin, but Not Other Non-steroidal Anti-inflammatory Drugs Is Associated with Decreased Prostate Cancer Risk at the Population Level[END_REF][START_REF] Marshall | Vitamin D3 supplementation at 4000 international units per day for one year results in a decrease of positive cores at repeat biopsy in subjects with low-risk prostate cancer under active surveillance[END_REF][START_REF] Liu | Cruciferous vegetables intake and risk of prostate cancer: a meta-analysis[END_REF] ). To date, large and expensive epidemiological studies of long duration 20 using repeat random biopsies in men on active surveillance 21 have been used. Our novel study design could allow initial screening of potential agents for further study.

Conclusions

Dutasteride is associated with a reduction in prostate cancer volume as assessed by T2W-MRI in men with biopsy proven low or intermediate risk prostate cancer suitable for active surveillance, over a six-month period. This The lesion is seen as high signal on diffusion weighted imaging at baseline (D), and stable at 3 months (E) and 6 months (F).

The lesion shows enhancement on dynamic contrast enhanced images at baseline (G), and whilst less conspicuous at 3 months (H) these are stable at 6 months (I ).

Figure S1

Bland Altman plot showing the relationship between the percentage reduction in tumor volume between raters.

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  investigator led study, sponsored by University College London, was approved by the Hammersmith & Queen Charlotte's & Chelsea Research Ethics Committee (UK) (09/H0707/84), the Medicines & Health Regulatory Agency and registered on the European Clinical Trials register (EudraCT 2009-102405-18).

2 .Figure 1 Figure 2

 212 Figure 1 CONSORT diagram showing men in study

Figure 3B :

 3B Figure 3B:A 71 year-old man from the placebo group with a right peripheral zone lesion at baseline on T2-weighted imaging (A). This is stable at 3 months (B) and 6 months (C).

Table 1 : Baseline characteristics in each group. Continuous variables summarized as mean (sd, range) and categorical variables summarized as n (%).

 1

	Dutasteride (n = 21)	Placebo (n = 21)

Table 2 : Change in total tumor volume between baseline and six months on T2-weighted imaging, dynamic contrast enhancement and diffusion weighted imaging

 2

		Dutasteride (20)	Placebo (20)	Mean
		Mean (sd, range)	Mean (sd, range)	difference
				(95% CI) p-
				value
	Total tumor volume (ml) at	0.55 (0.5,	0.65 (0.8,
	baseline (T2)	0.12 to 2.36)	0 to 3.27)
	Total tumor volume (ml) at	0.38 (0.4,	0.76 (0.9,
	6 months (T2)	0.02 to 1.59)	0.09 to 3.96)
	% change in total tumor			47.88
	volume from 0-6 months	-35.73 (25.7,	12.15 (37.2,	(27.42,
	(T2)	-82.61 to 13.21)	-48.48 to 119.16)	68.34), p
				<0.0001
	Total tumor volume at	0.58 (0.5,	0.67 (0.8,
	baseline (gadolinium/DCE)	0.055 to 2.342)	0.09 to 3.72)
	Total tumor volume at 6	0.35 (0.3,	0.66 (0.8,
	months (gadolinium/DCE)	0.025 to 1.328)	0.045 to 3.68)
	% change in total tumor			38.11
	volume from 0-6 months	-42.32 (25.6,	-4.20 (38.5,	(16.59,
	(gadolinium/DCE)	-88.37 to 3.76)	-64.41 to 102.22)	59.64)
				p=0.0010
	Total tumor volume at	0.48 (0.6,	0.42(0.3,
	baseline (diffusion)	0.07 to 2.695)	0 to 1.18)
	Total tumor volume at 6	0.30 (0.3,	0.45 (0.4,
	months (diffusion)	0.05 to 1.284)	0 to 1.165)
	% change in total tumor	-33.46 (21.8,	7.01 (35.9,	40.47

Acknowledgements

We would like to thank the men who gave their consent to participate in this study. Mark Emberton receives research support from the United Kingdom's National Institute of Health Research UCLH/UCL Biomedical Research Centre. Mark Emberton holds Senior Investigator status with the United Kingdom's National Institute of Health Research.

Disclosures

The study was investigator-led and sponsored by University College London.

The study was supported financially by GSK who also provided supplies of both drug and placebo. GSK had no input into the design, conduct and analysis of the study. The manuscript has been reviewed by GSK but final editorial control rests with the principal investigator (ME), who serves as guarantor of the study.


Anson Ma

Francisco Chinestat

Malcolm Mackley

The Rheology and Microstructure of Carbon N anotube ( CNT) Suspensions

Keywords: Carbon Nanotube suspension, Processing, Rheo-optics, Shear-induced assembly PACS: 6L46,Fg, 6L48,+c, 8Ll6,qVc, 8Ll6,Fg

This conference paper is concerned with the rheology and microstructure of Carbon Nanotube suspensions, Carbon Nanotubes (CNTs) are cylinders of rolled graphene sheets of high aspect ratio and they belong to a relatively new class of fibrous material that can potentially be used for high-performance nano-composites and nano-devices, For these applications, understanding the rheology and microstructure of CNT suspensions is of key importance as this would allow for processing CNTs into macroscopically usable form with controlled properties, This paper reports the rheology and optical microstructure of surface-treated and untreated CNT suspensions, In terms of rheology, both types of CNT suspensions exhibited a shear-thinning characteristic in steady simple shear flows and their rheological responses were subsequently modelled using a Fokker-Planck (FP) based orientation model that incorporates aggregation effect in the case of untreated CNT suspension, In terms of unusual mesostructures, it was observed that by applying appropriate flow conditions to an untreated CNT suspension, a novel type of mesoscopic filament structure could be formed, where the CNT filaments aligned perpendicular to the flow direction and in the vorticity direction, The formation of these anisotropic structures has similarities to the hand-spinning of cotton into fibres and these flow-induced structures might have strategic benefits for some potential applications of CNT

INTRODUCTION

Carbon Nanotubes (CNTs) are rolled cylinders of graphene sheets having high aspect ratio, low density, high mechanical strength and high thermal conductivity [START_REF] Iijima | Helical microtubules of graphitic carbon[END_REF] [2], Because of these intrinsic properties, they have been researched in a number of application areas such as high-performance nanocomposites and nanodevices [START_REF] Calvert | Nanotube composites: A recipe for strength[END_REF] [START_REF] De Heer | A Carbon Nanotube field-emission electron source[END_REF], For most of these applications, understanding the rheology and microstructure of CNT suspensions is important in relation to processing this type of nanomaterials into some macroscopically usable forms with predictable properties, The objectives of this paper are to highlight several key experimental findings, both in terms of rheological and optical studies of CNT suspensions, and secondly, to discuss briefly the basis on which CNT suspensions can be modelled using a Fokker-Planck (FP) based diffusion-advection modeL Finally, flow-induced formation of some unusual CNT mesostructures is also described,

CARBON NANOTUBE SUSPENSIONS

At present, enhancing the bulk conductivity of a polymer film is identified as one of the most promising potential applications for CNTs [START_REF] Sandler | Development of a dispersion process for Carbon Nanotubes in an epoxy matrix and the resulting electrical properties[END_REF] [START_REF] Shaffer | Dispersion and packing of Carbon Nanotubes[END_REF] and for such an application, it is thought desirable to have a well-dispersed system of CNT. However, this can be a laborious and challenging task because in general, a mixing time of several hours is required in order to establish a stable rheology for CNT suspensions, and even then dispersion at the nano-scale is not guaranteed [START_REF] Huang | Dispersion rheology of Carbon Nanotubes in a polymer matrix[END_REF]. In this regard, introducing functional groups onto the surface of CNTs, also known as functionalization, has proven to be one possible way of enhancing the dispersion of CNT in a processing matrix [START_REF] Banerjee | Rational chemical strategies for Carbon Nanotube functionalization[END_REF]. The actual chemistry involved in surface-treating CNTs may differ, but functionalization generally works on the principle of increasing steric hindrance or introducing electrostatic repulsion between CNTs, thereby preventing effective 7r -7r stacking and subsequent aggregation of CNTs (see for example [START_REF] Dyke | Unbundled and highly functionalized Carbon Nanotubes from aqueous reactions[END_REF]). Figure 1 shows typical optical micrographs of surface-treated and untreated CNTs suspended in epoxy resin1 [START_REF] Ma | Rheological modelling of suspensions involving surfacetreated Carbon Nanotubes[END_REF]. It is clear from the micrographs that the untreated CNT suspension exhibited an optical microstructure (Fig. l(a)), whilst by contrast, there is no discernable microstructure for the treated CNT suspension (Fig. l(b)), indicating that the mixture was well-dispersed at the micron-level.

RHEOLOGY

The rheology of both treated and untreated CNTs suspended in epoxy resin were studied and modelled for simple rheometric flows. It was observed experimentally that both types of suspensions exhibited a shear-thinning characteristic when subject to steady simple shear and the apparent viscosity ( 1Ja) of the suspension decreased asymptotically to the matrix viscosity as shear rate increased (Fig. 2). Given the same weight concentration of CNTs, the untreated CNT suspension exhibited a significantly larger viscosity enhancement effect at low shear rates compared with the treated CNT suspension, as also shown in Fig. 2. For treated CNT suspensions, the observed shear-thinning characteristic can be modelled using a simple orientation model that in the past has been used for short-fibre suspensions [START_REF] Petrie | The rheology of fibre suspensions[END_REF]. The model basically assumes that CNTs behave essentially as rigid rods and that they can orient and align in a simple shear flow. There are, however, also randomizing events due to Brownian motion and tube interaction (for semi-dilute or concentrated suspensions), which can be represented using a rotary diffusion coefficient (Dr) [START_REF] Folgar | Orientation behaviour offibers in concentrated suspensions[END_REF] and the time evolution of CNT orientation distribution can be described using a Fokker-Planck based diffusion-advection equation [START_REF] Hinch | The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles[END_REF]. Dr and an adjustable parameter NP that depends on the concentration and aspect ratio are the key parameters in the model and such model can successfully describe the shear-thinning characteristics of treated CNT suspensions as shown in Fig. 3. Besides simple shear flows, the model was also capable of describing some kinetic responses of CNT suspensions in smallamplitude oscillatory shear flows, thereby supporting the basis for the model itself [START_REF] Ma | Rheological modelling of suspensions involving surfacetreated Carbon Nanotubes[END_REF].

~~~•••••••••••••• ==•••• ~ 0 0 0 V V~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~ ~ $ $ ""' (
However, in the case of untreated CNT suspensions where aggregates of CNTs were optically observed, the simple orientation model failed to describe the shear-thinning characteristic observed experimentally, and therefore a more sophisticated model is needed. An Aggregation/Orientation (AO) model has been proposed in this context by considering extra contribution from CNT aggregates [START_REF] Ma | Rheological modelling of suspensions involving untreated Carbon Nanotubes in shear flow[END_REF]. The model assumes that in an aggregating suspension of CNTs, CNTs can exist in different entanglement states and they can have different mobility to orient in a simple shear flow. The Fokker-Planck equation was then modified to accommodate for a hierarchy of microstructures. In this model, transformation between microstructures is possible depending on the shear rate applied to the system, and thus matching the phenomena of CNT aggregation and disaggregation as observed experimentally [15] [16].

UNUSUAL FLOW MICROSTRUCTURE

The flow microstructure of untreated CNT suspensions was followed using the Cambridge Shear System (Linkam Instruments Inc.), which essentially consists of a shear cell coupled with an optical microscope. Although shear can induce alignment of CNTs, it was also reported that low shear could induce aggregation of CNTs [15] [16]. In most cases, a homogeneous dispersion of CNT is desired, however there are other situations where a highly heterogeneous structure can be of strategic benefit. It was discovered that some unusual filament mesostructures could be formed by applying appropriate flow conditions to an untreated CNT suspension [START_REF] Ma | Experimental observation on the flow-induced assembly of Carbon Nanotube suspensions to form Mesoscopic Helical Filaments[END_REF]. Figure 4 shows the time evolution of optical texture as low shear was applied to some untreated CNTs (0.03%) suspended in a low-viscosity epoxy matrix2 • Initially, there were only isotropic aggregates of CNTs (Fig. 4(a)), but as they rotated within a simple shear flow, they drew in nearby CNT aggregates to form long Mesoscopic Helical Filaments (MHFs) that aligned perpendicular to the flow and in the vorticity direction. The way in which CNT aggregates are helically wound to form this kind of filament structure is intriguing

and not yet fully understood. In terms of applications, although the advantage of having a MHF structure of CNTs has not yet been explored, it is expected that the anisotropic nature of these mesostructures will produce preferential conductivity along the length of the filaments and thereby providing unusual anisotropic electrical properties to the system as a whole. 1 ) was applied to 0.03% untreated CNTs suspended in a low-viscosity epoxy matrix (gap size= 130 J-Lm) [START_REF] Ma | Experimental observation on the flow-induced assembly of Carbon Nanotube suspensions to form Mesoscopic Helical Filaments[END_REF].

CONCLUSIONS

In conclusion, this paper reported recent findings on the rheo-optical studies of Carbon Nanotube (CNT) suspensions which are of general relevance to the processing of CNTs.

It was found that for surface-treated CNT suspensions that exhibit no discernable optical microstructure, CNTs can essentially be modelled as rigid fibres that can orient in a shear flow. The rheology of flowing CNTs can be successfully described using a simple orientation model and essential information on the orientation distribution of CNTs can be obtained from a Fokker-Planck (FP) description. The FP based orientation model was capable of predicting both steady and transient responses of treated CNT suspensions, but failed in the case of untreated CNT suspensions where CNT aggregates were optically observed. This has led to the development of a more appropriate Aggregation/ Orientation (AO) model, in which a hierarchy of CNT aggregate structures was considered. Finally, in terms of optical microstructure, although low shear can induce aggregation and should be avoided in cases where a uniformly dispersed system of CNTs is desired, it can also be used strategically to produce some unusual Mesoscopic Helical Filament (MHF) structures that are highly anisotropic.

FIGURE 1 .

 1 FIGURE 1. Optical micrographs of 0.3% (a) surface-treated and (b) untreated CNT suspensions [1 0].

  0.3% Untreated CNT . • ... / . . . . . . . .. . . . . . . . . . ..

FIGURE 2 .

 2 FIGURE 2. Apparent viscosity (1Ja) as a function of shear rate (i) for (a) surface-treated and (b) untreated CNT suspensions [1 0].

FIGURE 3 .

 3 FIGURE 3. Simple orientation model fitting of 0.2% and 0.3% treated CNT suspensions. Dr and Np are the key fitting parameters in the model. (1)a =apparent viscosity and y =shear rate.) [l 0]

FIGURE 4 .

 4 FIGURE 4. Time evolution of optical texture as low shear (y = 0.5 s-1 ) was applied to 0.03% untreated CNTs suspended in a low-viscosity epoxy matrix (gap size= 130 J-Lm)[START_REF] Ma | Experimental observation on the flow-induced assembly of Carbon Nanotube suspensions to form Mesoscopic Helical Filaments[END_REF].

The surface-treated CNTs were single-walled and were supplied by Nanocomposites Inc., whereas the untreated CNTs were multi-walled and were provided by the Department of Materials Science and Metallurgy, University of Cambridge. The epoxy resin used was Araldite LY556 from Huntsman Inc.

The low-viscosity epoxy has a viscosity of0.4 Pa.s and was supplied by Epoxies Inc. (UV 60-7155).

ACKNOWLEDGMENTS

We would like to thank Nanocomposites Inc. and Prof. A. H. Windle from the Department of Materials Science and Metallurgy at the University of Cambridge for providing the base Carbon Nanotubes. Anson Ma would also like to to acknowledge the Croucher Foundation Scholarship (Hong Kong) and the Overseas Research Students Awards Scheme (ORSAS) (United Kingdom) for providing financial support.


Yang Gao

Suenne Kim

Si Zhou

Hsiang-Chih Chiu

Daniel Nélias

Claire Berger

Walt De Heer

Laura Polloni

Roman Sordan

Angelo Bongiorno

Elisa Riedo

Elastic coupling between layers in two-dimensional materials

Two-dimensional materials, such as graphene and MoS 2 , are films of a few atomic layers in thickness with strong in-plane bonds and weak interactions between the layers. The in-plane elasticity has been widely studied in bending experiments where a suspended film is deformed substantially; however, little is known about the films' elastic modulus perpendicular to the planes, as the measurement of the out-of-plane elasticity of supported 2D films requires indentation depths smaller than the films' interlayer distance. Here, we report on sub-ångström-resolution indentation measurements of the perpendicularto-the-plane elasticity of 2D materials. Our indentation data, combined with semi-analytical models and density functional theory, are then used to study the perpendicular elasticity of few-layer-thick graphene and graphene oxide films. We find that the perpendicular Young's modulus of graphene oxide films reaches a maximum when one complete water layer is intercalated between the graphitic planes. This non-destructive methodology can map interlayer coupling and intercalation in 2D films.

most studied 2D material is graphene, existing as a single layer of graphite [START_REF] Novoselov | Electric field effect in atomically thin carbon films[END_REF] or a few-layer-thick epitaxial graphene (EG) film [START_REF] De Heer | Epitaxial graphene[END_REF] . Graphene possesses a large in-plane Young's modulus 10 (∼1 TPa) as well as high intrinsic carrier mobility [START_REF] Baringhaus | Exceptional ballistic transport in epitaxial graphene nanoribbons[END_REF][START_REF] Bolotin | Ultrahigh electron mobility in suspended graphene[END_REF] and high in-plane thermal conductivity [START_REF] Balandin | Superior thermal conductivity of single-layer graphene[END_REF] . Besides graphene, also 2D films of graphene oxide 14- 17 (GO), hexagonal boron nitride [START_REF] Ci | Atomic layers of hybridized boron nitride and graphene domains[END_REF] and transition metal dichalcogenides [START_REF] Ramakrishna Matte | MoS 2 and WS 2 analogues of graphene[END_REF][START_REF] Castellanos-Gomez | Mechanical properties of freely suspended semiconducting graphene-like layers based on MoS 2[END_REF][START_REF] Radisavljevic | Single-layer MoS 2 transistors[END_REF] such as MoS 2 exhibit unique and excellent properties and hold great promise for nanotechnology applications. One of the main characteristics of 2D materials is the high anisotropy between the in-plane and perpendicular-to-the-plane properties. For example, in graphite, owing to the strong covalent bonds between atoms in the plane, and the weak Van der Waals interlayer interaction, the in-plane Young's modulus is E = 1 TPa (ref. 22), whereas the interlayer perpendicular-to-the-plane Young's modulus is only E ⊥ =36 GPa (ref. 22). Recent studies have suggested that the mechanical properties of 2D materials are strongly correlated to their structure and properties [START_REF] Park | Nanomechanical and charge transport properties of two-dimensional atomic sheets[END_REF][START_REF] Lee | Measurement of the elastic properties and intrinsic strength of monolayer graphene[END_REF][START_REF] Kwon | Probing nanoscale conductance of monolayer graphene under pressure[END_REF][START_REF] Lee | Frictional characteristics of atomically thin sheets[END_REF][START_REF] Lee | Elastic and frictional properties of graphene[END_REF] . The inplane Young's modulus [START_REF] Park | Nanomechanical and charge transport properties of two-dimensional atomic sheets[END_REF][START_REF] Lee | Measurement of the elastic properties and intrinsic strength of monolayer graphene[END_REF][START_REF] Lee | Elastic and frictional properties of graphene[END_REF][START_REF] Stewart | Atomistic simulations of nanoindentation on the basal plane of crystalline molybdenum disulfide (MoS 2 )[END_REF][START_REF] Suk | Mechanical properties of monolayer graphene oxide[END_REF] of exfoliated graphene and MoS 2 has been widely studied in bending experiments where a film is suspended on trenches or holes, and an atomic force microscopy (AFM) tip is used to bend the suspended film with deformations of tens and hundreds of nanometres. On the other hand, very little is known about the elasticity perpendicular to the planes, hereafter called perpendicular or interlayer elasticity, of 2D materials composed of very few atomic layers. Recent calculations have investigated the out-of-plane shear and Young's modulus of carbon nanotubes and graphene [START_REF] Hajgató | Out-of-plane shear and out-of plane Young's modulus of double-layer graphene[END_REF] . Experimentally, resonance ultrasound spectroscopy was used to study the elastic constants and the anisotropy between the in-plane and perpendicular-to-the-plane directions of thin films [START_REF] Nakamura | Resonance ultrasound spectroscopy with laser-Doppler interferometry for studying elastic properties of thin films[END_REF] . Investigations of the perpendicular-to-the-plane elasticity of fewlayer-thick 2D films have not been reported to the best of our knowledge, and remain an experimental challenge because they require indentations on supported-as opposed to suspended-2D films, where the indentation should remain smaller than the films' interlayer distance, that is, less than a few ångströms. Nevertheless, the interlayer elastic coupling is particularly important because it is related to the thermal [START_REF] Chiritescu | Ultralow thermal conductivity in disordered, layered WSe 2 crystals[END_REF] , electronic [START_REF] Riedl | Structural and electronic properties of epitaxial graphene on SiC(0 0 0 1): A review of growth, characterization, transfer doping and hydrogen intercalation[END_REF] , tribological [START_REF] Lee | Frictional characteristics of atomically thin sheets[END_REF][START_REF] Lucas | Hindered rolling and friction anisotropy in supported carbon nanotubes[END_REF] and optical [START_REF] Tan | The shear mode of multilayer graphene[END_REF] properties of 2D films. The perpendicular elasticity is expected to be affected by the structure and chemistry of the layers, the presence of stacking and intrinsic defects, and intercalation, which is a critical process for doping and tuning mechanical and electronic properties in 2D films. Mapping the interlayer elastic coupling in 2D films is therefore an important technological and scientific advancement.

Modulated nanoindentation experiments

Here, an unconventional AFM-based method allowing for subångström-resolution indentations coupled with a semi-analytical method (SAM) is used to measure the elasticity of 2D films in the direction perpendicular to the layers, for a number of layers as small as two. This AFM-based method is capable of maintaining an indentation depth smaller than the interlayer distance, with a resolution of 0.1 Å. We report on AFM-SAM investigations of the perpendicular elastic modulus of EG, epitaxial graphene oxide (EGO), and conventional GO films at varying ambient humidity. These studies, together with density functional theory (DFT) calculations, highlight how the interlayer elastic coupling in graphene and GO films is affected by the films' chemistry, structure, water intercalation, and number of layers. Interestingly, the perpendicular Young's modulus of GO increases by increasing the amount of water trapped in between the layers until a full monolayer is produced. Then, when the second water layer is forming, the modulus decreases. Finally, our studies show that when using sub-interlayer-distance indentations in 2D films thicker than a few layers, the AFM indentation curves are very sensitive to the elastic modulus perpendicular to the layers, E ⊥ , and almost independent of the value of the in-plane Young's modulus, E (see Fig. 1a). As a consequence, the indentation force curves can be fitted with a simple modified Hertz model as if the film is an isotropic material, where Young's modulus of the film is indeed the perpendicular modulus E ⊥ . This work is therefore offering a new experimental and theoretical framework to investigate the interlayer elastic coupling in 2D films.

The experimental measurement of the perpendicular-to-theplane elasticity of 2D films is here achieved by using an unconventional AFM-based method that we call modulated nanoindentation [START_REF] Chiu | Morphology dependence of radial elasticity in multiwalled boron nitride nanotubes[END_REF][START_REF] Lucas | Aspect ratio dependence of the elastic properties of ZnO nanobelts[END_REF]35 (MoNI). In particular, we have investigated highly oriented pyrolytic graphite (HOPG), 4H-SiC, EG films grown on a SiC substrate [START_REF] Berger | Electronic confinement and coherence in patterned epitaxial graphene[END_REF][START_REF] De Heer | Epitaxial graphene[END_REF] , EGO films grown on a SiC substrate [START_REF] Kim | Room-temperature metastability of multilayer graphene oxide films[END_REF] , and regular GO films deposited with the conventional Hummers' method [START_REF] Hummers | Preparation of graphitic oxide[END_REF] on a Si substrate. More details about indentation experiments, sample preparation and properties of the films are provided in Methods and Supplementary Information (see also Supplementary Figs 12345). During a typical MoNI experiment, an AFM tip, which is vertically oscillated at a fixed frequency with a ∼0.1 Å amplitude, applies an increasing pressure to a 2D film surface, in the z-direction perpendicular to the film surface (see Fig. 1a). The oscillations are applied and controlled by a lock-in amplifier, while a constant normal force F z between the tip and the 2D film is maintained constant by the feedback loop of the AFM. By working with a constant force, any thermal drift is avoided. Furthermore, using a lock-in detection system together with a differential measurement allows us to measure very shallow indentations, usually smaller than 1-3 Å, with a resolution of 0.1 Å. Indeed, instead of measuring directly the normal force as a function of the indentation, z indent , we measure the slope of the force versus indentation curve at each constant normal force, namely k cont (F z ). Force versus indentation curves are then obtained by integrating the equation dF z =k cont (F z )•dz indent as follows:

z indent (F z ) = Fz 0 dF z k cont (F z ) (1)
with the sample and can be easily determined experimentally (Supplementary Figs 2 and 3 for more details) [START_REF] Lantz | Atomic-force-microscope study of contact area and friction on NbSe 2[END_REF][START_REF] Carpick | Lateral stiffness: A new nanomechanical measurement for the determination of shear strengths with friction force microscopy[END_REF][START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF] . Furthermore, for SiC and the 2D materials investigated here, when the load is equal to F po the contact area is indeed zero and z indent (F po )=0 (see Supplementary Information). As shown in Supplementary Fig. 3, k cont (F z )-which is proportional to the contact radius and the squared root of the indentation depth-drops to zero when the load reaches F po . We underline that the F z versus z indent curves could in principle be shifted along the z indent axis in very soft and adhesive materials depending on the value of z indent (F po ); however, the shape of these curves does not depend on the type of adhesive contact. We also note that in Supplementary Figs 2 and3 the pull-off force can change significantly from curve to curve; however, F z remains consistent. This shift in the absolute value of the normal force is due to humidity and the drifting of the laser position on the photo-detector during the different individual experiments (we take hundreds of curves for each sample and condition). More detailed discussions can be found in the Supplementary Information and Supplementary Figs 4 and5. However, the clear determination of the pull-off force during each indentation experiment allows one to always determine the corrected 'absolute' normal load as F z = (F z -F po ), and to compare different curves independently of the laser position shifting and presence of adhesion forces. In the Supplementary Information we discuss in detail why for the materials investigated here we can account for the adhesion force by simply adding to the load the pull-off force. This adhesive contact mechanics model is called the Derjaguin-Muller-Toporov (DMT) model [START_REF] Grierson | Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy[END_REF][START_REF] Derjaguin | Effect of contact deformations on the adhesion of particles[END_REF] and it describes well stiff materials and small tip radii. Figure 1b shows the measured (and offset-corrected) F z versus z indent at ambient humidity for different 2D films, namely EG and EGO. We remark that both EG and EGO in Fig. 1b are 10 layers thick. Figure 1b also shows the force versus indentation curve for a single crystal of silicon carbide, the substrate over which EG and EGO are grown. The indentation curves, all performed with the same AFM tip, clearly indicate a larger stiffness for 10-layer EG than 10-layer EGO. As expected from previous studies, SiC shows a very large stiffness. 4H-SiC has a hexagonal crystal structure; however, its elastic behaviour is very isotropic, and therefore a simple Hertzian offset-corrected contact model can be used to extract from the force versus indentation curves Young's modulus of the material. For the configuration of a sphere (the AFM tip) pressing on a flat surface of an isotropic half-space with a normal force F z the Hertz model gives:

F z = 4 3 E * (R) 1/2 z 3/2 indent ( 2
)
where to fit the experimental measurements of F z versus z indent , as shown in Fig. 1b for a quasi-isotropic sample such as SiC. Indeed, the fitting procedure gives E S H iC ertz = 400 GPa, for a tip radius R=114 nm (the details of tip radius measurement can be found in the Supplementary Information and Supplementary Fig. 6), which is in excellent agreement with the literature. Similar experiments have also been performed on other standard samples such as ZnO single crystals to ensure the ability of MoNI to obtain reliable results (see details in the Supplementary Information and Supplementary Fig. 2). However, the Hertz relationship in (2) is in principle not well suited to model the contact mechanics between an AFM spherical tip and an anisotropic material such as graphite and other 2D materials because the Hertz model was originally valid only for isotropic half-spaces [START_REF] Johnson | Contact Mechanics[END_REF][START_REF] Turner | Contact on a transversely isotropic half-space, or between two transversely isotropic bodies[END_REF] . This experimental set-up is therefore uniquely suited to investigate the perpendicular elastic modulus of 2D films, which are only a few layers thick. We note that the lower limit of the integral in equation ( 1) is zero only when there are no adhesion forces. In the case of an adhesive contact, the lower limit is the pull-off force, F po , defined as the negative load at which the AFM tip loses the contact

E * = ((1 -(υ sample Hertz ) 2 /E sample Hertz ) + (1 -(υ tip

Semi-analytical methods

A better insight into the contact-pressure distribution as a function of the material elastic constants can be obtained using simulations with SAMs, which have proved their efficiency in describing the contact mechanics of anisotropic materials [START_REF] Bagault | Contact analyses for anisotropic half-space coated with an anisotropic layer: Effect of the anisotropy on the pressure distribution and contact area[END_REF][START_REF] Bagault | Contact analyses for anisotropic half space: Effect of the anisotropy on the pressure distribution and contact area[END_REF][START_REF] Jacq | Development of a three-dimensional semi-analytical elastic-plastic contact code[END_REF] . Here, we use SAM to simulate the force versus indentation curves in graphite. We use graphite elastic constants found in the literature [START_REF] Kelly | Physics of Graphite[END_REF] (see Methods), and we model the indentation of an AFM silicon tip (R= 100 nm) in a graphite sample deposited on SiC. We use this configuration because the 2D materials studied here have been deposited either on SiC (EG and EGO) or Si (conventional GO). Figure 2a shows the results from the SAM simulations on graphite along with the experimental curves obtained by MoNI on a bulk sample of HOPG. In Fig. 2a, the SAM-simulated curve agrees extremely well with the experiments on HOPG; it is important to note that in the SAM simulations for graphite we use, according to the literature, as inplane Young's modulus E =1.046 TPa, and as z-axis (perpendicular to the planes) Young's modulus E ⊥ =(36.4 ±1) GPa (more details are given in the Methods). Interestingly, when the Hertz model in ( 2) is applied to fit the experimental indentation curves measured on graphite, as if graphite was an isotropic material, the result of the fitting procedure gives as the single isotropic modulus E H H er O tz PG =(33 ±3) GPa, for R = 100 nm like the AFM tip radius. The Hertz model fitting curve is also reported in Fig. 2a to show the perfect agreement with experiments and SAM simulations. The Hertz model is therefore able with a simple fitting procedure to obtain a value of Young's modulus that is equal, within an error of 10%, to the most accepted value for the perpendicularto-the-plane Young's modulus of graphite, that is, E ⊥ =36 GPa.

⊥

The excellent consistency between experiment, simulation and Hertz model when studying the indentation of 2D films with extremely small indentation depths is a direct consequence of the following SAM observations. If we consider sub-interlayer distance indentations in a transversally isotropic (orthotropic) material having E ⊥ = E Hertz and E varying up to one order of magnitude compared with E Hertz , we find that the contact pressure and contact area for a given pressure remain almost the same as in an isotropic material having E =E Hertz . On the other hand, the contact pressure changes markedly compared with the isotropic case when varying E in the same range and maintaining E = E Hertz (ref. 44). Overall these results indicate that for sub-nanometre AFM indentations, much smaller than the film's thickness, the force versus indentation curves are very sensitive to E ⊥ , and almost independent of the value of E . This result can be understood with a back-of-the-envelope calculation of the in-plane and perpendicular-to-the-plane stress distribution in a layered material when indenting the material perpendicular to the planes (for details see Supplementary Information and Supplementary Fig. 8). From these calculations it seems clear that the key parameter controlling sub-nanometre indentations is the ratio a 2 /l 0 z indent , where a is the contact radius, and l 0 is the length of an sp 2 bond. For example, when indenting graphite with an AFM tip with z indent < 0.3 nm, this simple model shows that we are sensing only E ⊥ . A more precise analysis can be developed by using SAM calculations. In particular, the contact pressure distribution profiles in the case of sub-nanometre indentations for the Hertz model in the case of an isotropic material with E Hertz = 33 GPa (the value of E ⊥ in graphite), and in the case of E Hertz = 1.046 TPa (the value of E in graphite), are plotted in Fig. 2b along with the SAM simulations for bulk graphite and stiffer than those ones measured on the buffer layer, and on 10layer EG, as clear in Fig. 2c and Supplementary Fig. 7. This result can be understood by the following considerations. For a film of 10-layer graphene, we are probing only E ⊥ , which is ∼36 GPa, because the 0.1 nm indentation is easily distributed among the 0.01 nm displacements per layer, supposing an infinitely rigid substrate. In contrast, when the AFM tip is indenting (z indent ∼ 0.1 nm) 1layer EG on an infinitely rigid substrate, the whole indentation is carried on by that single layer sitting at an interatomic distance of 0.3 nm from the substrate, and strongly resisting such a huge deformation. Therefore, for 1-layer graphene on SiC we are now probing a complex convolution of effects and properties, which include E (∼1 TPa), E ⊥ , and the elasticity of the SiC substrate (E ∼ 400 GPa). In conclusion, these experiments show that MoNI measurements can be sensitive to the substrate interaction and to the number of layers, although more calculations will be required to fully explain the results for 1-layer graphene.

The ability of the Hertz relationship in equation ( 2) to model sub-nanometre indentations in transversally isotropic 2D materials is further confirmed by Fig. 2d. We have fitted all (except for 1-layer EG) the measured F z (z indent ) curves with Hertz equation ( 2) leaving as a free fitting parameter the exponent, b, of z indent , that is, by using the function

F z = C • z b
indent , where C is a constant. The results are reported in Fig. 2d, and they show that b is on average equal to (1.4 ±0.1), a value in very close agreement with the exponent expected for a perfect Hertzian contact where b=1.5 (see equation ( 2)). By fitting with the Hertz equation (2) all of the exper-imental indentation curves for the different 2D materials films in ambient atmosphere (see Table 1), we find that as mentioned above the perpendicular elastic modulus is larger in 10layer EG films than in HOPG; this difference is probably related to the absence of defects in 10-layer EG compared with HOPG. We also find that 10-layer EGO is much softer than 10-layer EG; this result can be explained by the increase of the interlayer distance d, in EGO compared with EG, precisely from d = 3.4 Å to d =9.3 Å (ref. 48) as also ob-served in X-ray diffraction spectroscopy measurements [START_REF] Zhou | Film structure of epitaxial graphene oxide on SiC: Insight on the relationship between interlayer spacing, water content, and intralayer structure[END_REF] . The more surprising result is the difference in the perpendicular elasticity between EGO films and GO films produced with the conventional exfoliation/filtration/deposition method (see Methods). In ambient humidity we precisely obtain: E ⊥ =(35 ±10) GPa (for conventional GO), E ⊥ = (36 ± 3) GPa (for 10-layer EG), E ⊥ =(33 ±3) GPa (for HOPG), and E ⊥ =(23 ±4) GPa (for EGO). To understand the origin of the large interlayer perpendicular elasticity in conventional GO compared with EGO, we have performed DFT calculations on the elasticity of GO with a different amount of intercalated water. We have then compared these calculations with MoNI experiments on GO and EGO performed at different ambient humidities.

Density functional theory study of perpendicular elasticity

DFT calculations have been performed on model structures of GO consisting of periodic stacks of graphene layers fully oxidized by either hydroxyl or epoxide groups, including increasing concentrations of water molecules, and presenting AA stacking (see Supplementary Information and Table 1). For each model, we used a DFT-D2 (refs 49,50) scheme (technical details and the full list of calculations are discussed in Supplementary Information, and Supplementary Figs 9101112) to perform a full structural optimization and determine the zero-temperature interlayer spacing. Subsequently, we applied a pressure and used DFT-D2 to estimate the z-axis Young's modulus from the energy versus displacement curves. A summary of the DFT results reporting E and interlayer distance as a function of intercalated water percentage for GO structures fully oxidized with hydroxyls is reported in Table 2. Figure 3a shows the calculated F z versus displacement curves at varying intercalated water content for the case of graphene fully oxidized with hydroxyl groups, because 50-nm-thick graphite. It can be easily concluded that the pressure distributions for an isotropic material having E = 33 GPa, bulk graphite, and 50-nm-thick graphite are almost all the same. On the other hand, the pressure distribution changes markedly when considering an isotropic material with E = 1.046 TPa. We conclude that when studying 2D materials with sub-nanometre indentations, the Hertz model is an extremely simple and accurate model to fit the experimental AFM indentation curves and obtain the perpendicular Young's modulus E ⊥ of 2D films having a thickness larger than the indentation depth.

Perpendicular-to-the-plane elasticity

After having tested and gathered a better understanding of subnanometre indentations in orthotropic layered materials such as bulk graphite, we have then used MoNI to investigate the interlayer elasticity of very thin films of layered materials. Initially, MoNI is performed on a film of 10-layer-thick EG grown on SiC, as shown in Fig. 2c. We remark that 0.1 nm indentations in supported graphene films with a thickness of ∼10 layers are not very sensitive to the presence of the SiC substrate, as also shown by SAM calculations. Similarly to the fitting procedure used in Fig. 2a, we use the Hertz model to fit the indentations curves on 10-layer EG. We have repeated several measurements, and the Hertz fit provides for 10-layer EG a modulus perpendicular to the planes equal to E ⊥ =(36 ±3) GPa, the same value as ideal graphite, and larger than the value measured on HOPG (E ⊥ =33 ±3 GPa). It is worth noting that in Fig. 2a for the SAM simulations and the MoNI experiments on HOPG we use a tip radius of R = (100 ± 10) nm, whereas the MoNI experiment on 10-layer EG in Fig. 2c was performed with an AFM tip of R = (114 ± 10) nm; therefore, to compare these curves we need to rescale them by a factor of R 1/2 (see equation ( 2)).

To investigate the role of the substrate and number of layers, we have then focused our experiments on the first layers of EG grown on the Si face of SiC (0001). More details on these experiments are reported in the Methods, in the Supplementary Information and in Supplementary Fig. 7. Much attention in the scientific community is at present directed towards single-layer EG grown in the Si face of SiC (refs 31,47). This system is very interesting because between the SiC substrate and the first 'well-decoupled' 1-layer graphene, there is a honeycomb carbon layer called the buffer layer [START_REF] Riedl | Structural and electronic properties of epitaxial graphene on SiC(0 0 0 1): A review of growth, characterization, transfer doping and hydrogen intercalation[END_REF] that has different properties from graphene, and it has a strong interaction with the SiC substrate. This substrate interaction strongly modifies the electronic structure of the carbon buffer layer, which is in fact not conducting. Extensive research is underway about the properties of the decoupled first 1-layer graphene and the buffer layer on SiC. Here, we have performed MoNI experiments on the decoupled first 1-layer graphene and on the buffer layer grown on the Si face of SiC, as shown in Fig. 2c and Supplementary Fig. 7. We find that the buffer layer produces indentation curves very similar to the ones on bare SiC (also reported in Fig. 2c). This result is in agreement with the notion that C atoms in the buffer layer have a strong interaction with the SiC substrate; indeed this interaction is often described as covalent bonding [START_REF] Riedl | Structural and electronic properties of epitaxial graphene on SiC(0 0 0 1): A review of growth, characterization, transfer doping and hydrogen intercalation[END_REF] . On the other hand, we obtain stiffer indentation curves on the decoupled 1-layer graphene on SiC. These curves are ⊥ previous experiments have shown that conventional GO films are mainly composed of hydroxyls groups [START_REF] Kim | Room-temperature metastability of multilayer graphene oxide films[END_REF] . In comparison, Fig. 3b presents the experimental MoNI results on conventional GO at different relative humidities (RHs). Figure 3c,d shows the resulting E as a function of intercalated water percentage compared with carbon (for DFT calculations) and RH (for MoNI experiments), respectively. A summary of the experimental results is also reported in Table 1. The agreement between experiments and DFT calculations is striking. DFT calculations and experiments show that E ⊥ increases with the amount of intercalated H 2 O molecules (and RH), reaching a maximum value of about 31 GPa at 25% H 2 O, and 35 GPa at RH = 25%, for the DFT calculations and indentation experiments, respectively. The perpendicular elastic modulus then decreases down to 20 GPa at 50% water, and 23 GPa at 50% RH, for the DFT calculations and indentation experiments, respectively. We remark that the presence of a maximum in the perpendicular modulus found in both experiments and calculations is a key result; however, the excellent quantitative agreement between water content (from DFT) and RH (from experiments) might be only a coincidence. The DFT calculations clearly give an insight into the atomistic origin of the behaviour of E ⊥ as a function of intercalated water. DFT calculations show that the interlayer distance and perpendicular Young's modulus in GO change abruptly between the case of dry layers and a multilayer film including a small amount of H 2 O (<6%), whereas more gradual variations of these physical properties are obtained for increasing the water content between 6.25% and 25% (Fig. 3c, and Table 2). In particular, E ⊥ drops from about 35 GPa (close to EG) in dry films to about 11-14 GPa when the layered structure includes 6.25% H 2 O. This behaviour can be understood by considering that when the amount of water is only a few percentage of the carbon amount, H 2 O molecules swell the graphene structure, increasing the interlayer distance from 3.4 Å to about 6.2 Å, but leaving the interlayer space mainly empty and therefore producing a soft structure with a low perpendicular elastic modulus. This interlayer modulus increases with increasing amount of water, which fills the interlayer space without changing too much the interlayer distance (Table 2). However, at 25% water, H 2 O molecules have completely filled a water layer in between the layers, and this situation corresponds with the maximum in perpendicular elastic modulus. Above 25% water, the perpendicular elastic modulus decreases because a second water layer starts to form in between the layers, further swelling and softening the GO structure. Owing to this explanation, it is now possible to understand the different values of E ⊥ in EGO compared with conventional GO. EGO is not a porous structure, and water intercalation is minimal [START_REF] Zhou | Film structure of epitaxial graphene oxide on SiC: Insight on the relationship between interlayer spacing, water content, and intralayer structure[END_REF] and independent of humidity. For this reason we find that E ⊥ in EGO remains constant and ∼22 GPa for all RHs. On the other hand, conventional GO is a porous structure [START_REF] Zhu | Microwave assisted exfoliation and reduction of graphite oxide for ultracapacitors[END_REF] where the amount of intercalated water can change depending on the humidity; therefore, in agreement with the DFT calculations we observe a maximum in conventional GO when varying the RH.

Outlook

In conclusion, we have presented a new methodology, which combines sub-ångström-resolution indentation measurements and SAMs to study the elasticity perpendicular to the plane of few-layerthick 2D materials. The comparison between indentation experiments and SAMs has also demonstrated that a simpler approach to interpret experimental sub-nanometre indentation curves in fewlayer-thick 2D films is to use the Hertz model. We showed that the fitting of the experimental indentation curves with the Hertz model provides, to a good approximation, the value of Young's modulus perpendicular to the plane of 2D films. The experimental study of EG and different types of GO film, combined with DFT calculations, has demonstrated that the interlayer elasticity is extremely sensitive to the presence of intercalated molecules in between the planes. In particular, these studies show that intercalated water in GO can markedly change the interlayer elastic modulus, which at first decreases when a small amount of water is intercalated in between the layers and the structure is swelled, and then increases with increasing amount of water until H 2 O molecules have completely filled a water layer in between the layers. Above this point, the perpendicular elastic modulus decreases because a second water layer starts to form in between the layers, further swelling and softening the GO structure. This understanding can also explain the different behaviour of E ⊥ in conventional GO and EGO. Finally, the here proposed subångström-resolution indentation measurements applied to EG are shown to be sensitive to the substrate interaction and to the number of layers. The results reported here provide a new path to study the interlayer elastic coupling and the Van der Waals forces in few-layerthick 2D materials, and shed new light on the use of the Hertz model in investigating the perpendicular-to-the-plane Young's modulus of 2D films. This study will impact a variety of fields, from electronics to phononics, allowing new investigations and understanding of the relationship between molecular structure, thermal conductivity, electronic properties, and phonon propagation in layered materials. For example, through local measurements of the elastic modulus, MoNI could probe interlayer and substrate interaction, as well as the presence of dopants/intercalates, which are extremely important for modulating the electronic properties of 2D materials. Furthermore, the interlayer elasticity is strictly connected with the out-ofplane thermal properties of layered materials. The here discussed interlayer elasticity measurements could help in the understanding of the origin of the extremely low out-of-plane thermal conductivity found in thin films of transition metal dichalcogenides [START_REF] Chiritescu | Ultralow thermal conductivity in disordered, layered WSe 2 crystals[END_REF] .

isotropic. One of the advantages of SAMs when applied to the contact problem compared with more widely used numerical techniques is the computing time, which is at least one or two orders of magnitude shorter. For more details on SAM for the numerical procedure used to solve the contact problem for an elastic substrate coated with an anisotropic layer, see refs 44,45. For transversely isotropic material, there are 5 independent elastic constants in the elasticity stiffness tensor: C 11 , C 12 , C 13 , C 33 and C 44 . These 5 elastic constants can be transformed to the well-known engineering notation in the following way. Note that G xy and v xy are correlated, which means that only one of them is independent. Graphite's elastic constants used for SAM have been previously reported in the literature [START_REF] Kelly | Physics of Graphite[END_REF] and they are C 11 = (1,060 ± 20) GPa, C 12 = (180 ± 20) GPa, C 13 = (15 ± 5) GPa, C 33 = (36.5 ± 1) GPa and C 44 = (4.5 ± 0.5) GPa. For graphite, this gives E = (1.06 ± 0.02) TPa, and E ⊥ = (36.4 ± 1) GPa.

The SiC substrate was approximated as cubic with E = 450 GPa, ν = 0.17 and G = E/2(1 + ν) = 192.31 GPa.

DFT calculation. Density functional theory (DFT) calculations are performed by using the PWscf code of the QUANTUM Espresso toolkit [START_REF] Giannozzi | QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materials[END_REF] . We use a plane-wave basis set with an energy cutoff of 120 Ry to represent the Kohn-Sham wavefunction, norm-conserving pseudo-potentials for all atomic species [START_REF] Troullier | Efficient pseudopotentials for plane-wave calculations[END_REF] , and the exchange-correlation functional proposed by Perdew, Burke and Ernzerhof 53 . To model hydrated GO, we consider two ordered structures-graphene sheets fully covered by either hydroxyl or epoxide groups, with various amounts of water molecules intercalated in between the oxide layers [START_REF] Kim | Room-temperature metastability of multilayer graphene oxide films[END_REF][START_REF] Zhou | Origin of the chemical and kinetic stability of graphene oxide[END_REF][START_REF] Zhou | Film structure of epitaxial graphene oxide on SiC: Insight on the relationship between interlayer spacing, water content, and intralayer structure[END_REF] . The GO sheets are arranged in either ' AA' or ' AB' stacking geometry. We use the primitive unit cell of each model as the supercell. We use 3 × 3 × 3 and 3 × 3 × 1 gamma-centred Monkhorst-Pack meshes to sample the Brillouin zone of the supercells including one and two oxide layers, respectively. To account for the London dispersion forces in multilayer GO, we adopt the semi-empirical DFT-D2 approach proposed previously [START_REF] Grimme | Semiempirical GGA-type density functional constructed with a long-range dispersion correction[END_REF] . The GO models are fully optimized for both the electronic and cell degrees of freedom using the DFT-D2 scheme. To calculate the perpendicular-to-the-plane Young's modulus, we started from the equilibrium structure of each model, and we vary the lattice parameter along the z direction with a strain step of ∼0.2% up to a 2% strain. The ionic positions of GO models are then gently relaxed to obtain the total energy and stress tensor under each strain. The E ⊥ value of each model is calculated by interpolating the pressure along the z direction with respect to the volume using the equation: E ⊥ = -V dP/dV , where P is the pressure along the z direction, and V is the volume of GO models. More details are reported in the Supplementary Information.

Figure 1 .

 1 Figure 1. Modulated nanoindentation experiments. a, Schematic diagram of the experimental set-up, where a spherical AFM tip vibrates while indenting a few-layer-thick film of graphene or GO. b, Experimentally measured indentation curves for single-crystal SiC, 10-layer-thick EG, and 10-layer-thick EGO. All three curves were obtained with the same AFM tip, R=114 nm.

Hertz ) 2

 2 s ratio and Young's moduli of, respectively, the investigated sample and the AFM tip, with R the AFM-tip radius[START_REF] Johnson | Contact Mechanics[END_REF] . For the silicon tip used in the MoNI measurements, E t H ip ertz = 169 GPa and υ t H ip ertz = 0.27. The elastic modulus can therefore be obtained by using the relationship(2)

Figure 2 .

 2 Figure 2. Experimental, SAM-simulated and Hertz indentation curves. a, Experimentally measured indentation curves in HOPG (filled circles), semi-analytical model simulations of indentation in graphite (open circles), and Hertzian fitting (continuum line) of the indentation curves on HOPG. The indenting tip radius was 100 nm. b, Contact-pressure distribution profiles for Hertz contacts and SAM simulations of indentation in graphite. Note that for bulk graphite and for a graphite film 50 nm thick, the SAM simulations and the contact distribution profiles almost overlap. c, Experimental indentation curves on 10-layer-thick EG, 1-layer-thick EG, bu er-layer EG, and SiC. d, Statistical analysis of exponent number b in the fitting function F z =C * z b indent . For EGO and GO, the RH is indicated. b average is 1.40.

Figure 3 .

 3 Figure 3. DFT and experimental results for conventional GO films. a, The DFT-calculated F z versus displacement curves for di erent water contents in graphene fully oxidized with hydroxyl groups. b, Experimental F z versus indentation depth curves at di erent RHs in conventional GO. All of the curves were obtained with the same AFM tip. c,d, Experimental and DFT results of E ⊥ of GO as a function of water content and RH, respectively. The insets are schematic diagrams of the corresponding atomistic structures showing how water molecules fill the interlayer spacing. Each experimental point of E ⊥ is an average value of more than 30 di erent measurements.

E

  = E x = E y = (C 11 -C 12 )(C 11 C 33 + C 12 C 33 -2C 13 C 13 )/ (C 11 C 33 -C 13 C 13 ) E ⊥ = E z = C 33 -2C 13 C 13 /(C 11 + C 12 ) v zx = C 13 /(C 11 + C 12 ) G xz = C 44 G xy = (C 11 -C 12 )/2 v xy = v yx = (E x /G xy ) -1

Table 1 . Summary of the experimental results of E ⊥ (GPa) at di erent RHs. Relative humidity 10 ± 2% 15 ± 3% 25 ± 3% 35 ± 3% 50 ± 3%

 1

	10-layer EGO	22 ± 3	-	23 ± 4	19 ± 3	22 ± 3
	Conventional GO 21 ± 6	26 ± 6	35 ± 10 -	23 ± 7
	10-layer EG	-	-	36 ± 3	-	-
	HOPG	-	-	33 ± 3	-	-

Table 2 . Summary of the DFT results reporting interlayer distance and E as a function of di erent fractions of intercalated w ⊥ ater for GO structures fully oxidized with hydroxyls.

 2

	Water fraction (%)	Interlayer distance (Å)	E ⊥ (GPa)
	0	6.2	34.9
	6.25	8.6	13.4
	12.5	8.7	18.2
	25	8.9	31.3
	50	11.3	20.7

Acknowledgements Y.G., S.K., H-C.C. and E.R., acknowledge the support of the Office of Basic Energy Sciences of the US Department of Energy (DE-FG02-06ER46293). S.Z. and A.B. acknowledge the support of the National Science Foundation (NSF) grant CMMI 1436375. S.Z., A.B., C.B. and W.d.H. acknowledge the support of the NSF grant DMR-0820382. C.B. acknowledges partial financial support from the European Flagship Graphene. A.B. acknowledges the support of the NSF grant CHE-0946869. R.S. acknowledges the support of the Italian Cariplo Foundation, project No. 2011-0373. We thank J-P. Turmaud for the EG on Si sample.

Methods

Modulated nanoindentation measurements. The oscillations are applied at 1 kHz to the atomic force microscopy (AFM) tip by a piezoelectric stage rigidly attached to the AFM cantilever-tip system, and controlled by a lock-in amplifier (Stanford Research Systems, SR830), while a constant normal force F z between the tip and the 2D films is maintained constant by the feedback loop of the AFM (see Supplementary Information for more details). To maintain the linear elastic regime, the piezo-stage oscillations are chosen to be only ∼0.1 Å. During the indentation, the driving fixed piezo-stage oscillation amplitude z piezo is equal to the sum of the cantilever bending and tip-2D film normal deformation. Under such circumstances, the AFM cantilever and the tip-film contact can be considered as two springs connected in series: the cantilever with stiffness k lev and the tip-film contact with stiffness k cont . The force required to stretch these two springs in series with a total displacement z piezo is equal to the normal force variation F z . This experimental configuration allows us to measure the total stiffness k tot at each normal load F z , fixed by the feedback loop of the AFM, from the following relation:

where F z is the variation of the normal force caused by the piezo-stage oscillation and is much smaller than F z , k lev is the spring constant of the cantilever, and k cont is the tip-film contact stiffness. As k lev can be measured independently, the measurement of F z / z piezo at different normal loads F z allows us to acquire the tip-film contact stiffness k cont as a function of F z . Force versus indentation curves are obtained by integrating dF z =k cont (F z )•dz indent . The data reported in Table 1 are averaged values obtained from 10, 210, 90 and 12 measurements on highly oriented pyrolytic graphite, graphene oxide (GO), epitaxial graphene oxide (EGO) and epitaxial graphene (EG) respectively. More details on the MoNI experiments are reported in the Supplementary Information.

2D materials. We have studied 10-layer graphene grown epitaxially on the C face of a 4H-SiC substrate 2 . We have also investigated the buffer layer and the first decoupled 1-layer graphene, both grown epitaxially on the Si face of 4H-SiC (0001) (refs 31,47; see Supplementary Fig. 7).

To prepare EGO samples, the 10-layer EG samples on SiC chips are then directly oxidized shortly after preparation by using a milder Hummers' method [START_REF] Zhou | Film structure of epitaxial graphene oxide on SiC: Insight on the relationship between interlayer spacing, water content, and intralayer structure[END_REF] , which avoids graphene exfoliation and dispersion in solution. Once the reactions are terminated, EGO films on the SiC chips are picked up from the solution and rinsed with deionized water for 1 min. The EGO films, 10 layers thick, are finally blow-dried by nitrogen gas.

GO films are prepared by drop-casting colloidal GO dispersion on a Si chip and leaving it to dry at 80 • C. Stable colloidal GO dispersions are produced by a modified Hummers' method [START_REF] Hummers | Preparation of graphitic oxide[END_REF] . Once the reaction is terminated, the oxidized portion is separated and cleaned from unoxidized graphite and other residual species. The cleaning is performed by centrifugation of the obtained suspension at 7,197g for 30 min to remove both the acidic content and ions. The solid content is collected and redispersed with deionized H 2 O. This operation is repeated in sequence until the pH of the supernatant is close to neutrality. At that point, the exfoliation of graphite oxide is performed by prolonged and vigorous shaking, forming a brownish colloidal suspension of GO flakes. The subsequent collection of the purified supernatant results in stable aqueous GO suspensions, which are drop-casted on Si to form a film with a thickness of 40-50 nm (ref. 51).

Semi-analytical method. The theoretical model used here to predict the indentation curves as well as the distribution of contact pressure and the contact area is based on what is known as semi-analytical methods. SAM is an alternative to the finite element method, quite close to the boundary element method, suitable for continuum mechanics problems. The main advantage of the boundary element method over SAM is that the first one is more versatile because it uses surface or volume integrals whatever the shape of the volume of interest. Conversely, SAM uses analytical solutions of these integrals for simple geometries (such as sphere or cuboid), which reduces the computation to a simple summation of elementary solutions. SAM consists of the numerical summation of elementary analytical solutions, such as the effect of a point load on a layered half-space, as far as they are known. The difficulty for anisotropic elastic coating and/or substrate is that the set of equations to be solved requires manipulating complex numbers with conjugate pairs, leading to double roots when the material tends to behave as elastically