Martin Lott

Philippe Roux

Léonard Seydoux

Benoit Tallon

Adrien Pelat

Sergey Skipetrov

Andrea Colombi

Localized modes on a metasurface through multi-wave interactions

In this paper, we describe the manifestation of localized states through coherent and incoherent analyses of a diffuse elastic wavefield inside a two-dimensional metamaterial made of a collection of vertical long beams glued to a thin plate. We demonstrate that localized states arise due to multi-wave interactions at the beamplate attachment when the beams acts as coupled resonators for both compressional and flexural resonances on the metasurface. Due to the low-quality factor compressional resonance of the beams, inside the main bandgap the modal density of the system drops to near a high-quality factor flexural resonance of the beams, and blocks the diffusion process of the wavefield intensity. This experiment physically highlights the tight-binding-like coupling in the localized regime for this two-dimensional metamaterial.

Introduction

In a strongly disordered scattering medium, it is possible to trap waves into energy loops 1 . At the sample size, the trapping induces spatial renormalization of the diffusion coefficient, which highlights the nonconventional transport property of the medium [2][3][4][5] . The diffusion process can then be stopped without the use of absorption, on a length known as the localization length. At a length scale that corresponds to a few scatterers inside the medium, the intensity spatially spreads in a deterministic way, due to interference effects. This phenomenon is known as modal localization, and it is the local elementary block for the global trapping effect of the wavefield intensity.

Recent two-dimensional studies have directly imaged microwave localized modes inside a disordered microwave cavity 6,[START_REF] Laurent | Mortessagne[END_REF] . In this case, the medium consists of a random distribution of resonant scatterers and has a large frequency bandgap. Around some isolated frequencies, the localized modes are clearly visible and spread according to distances roughly equal to the mean distance between the scatterers.

In this context, so-called locally resonant metamaterials [8][9][10][11][12] are good candidates for studying the transport phenomena of elastic and nonscalar guided waves. Indeed, the coupling between resonators to a homogeneous substrate degenerates the dispersion relationship and creates bandgaps (i.e., Fano-like resonances), in the same way as for electromagnetic waves. At the bandgap edges, the effective wavenumber is a) martin.lott@univ-grenoble-alpes.fr typically very high, and each resonator acts as a strong scatterer, and deviates the waves for lengths shorter than the wavelength.

In this paper, we revisit the experiment from Rupin et al. 13 , where a two-dimensional cluster of beams attached to a plate was used to create a metasurface. The beams showed both compressional and flexural resonances that were characterized by a low-quality and a high-quality factor, respectively (i.e., wide-spreading vs. narrow resonant peaks). These resonances strongly modify the A 0 flexural Lamb waves of the free plate [14][15][START_REF] Lott | Fundamentals and Applications of Acoustic Metamaterials[END_REF] . In these studies, they showed that large bandgaps are associated with compressional resonances that are contaminated by narrow leakage associated with each flexural resonance. As flexural resonances are coupled to the plate through the bending of the A 0 Lamb mode, changing to a thinner plate enhanced the flexural bandgap leakage [START_REF] Roux | New trends toward locally-resonant metamaterials at the mesoscopic scale[END_REF] . This narrow leakage is a potential candidate for localized modes, because it induces a sharp transition zone inside a frequency band that is dominated by a deficit of wave states (namely, the bandgap).

In addition to the mesoscopic measurements of the wavefield properties, as for the Ioffe-Regel criterion and the spatial intensity distribution function, we measure here the long-range coupling between the resonators themselves as a proxy for the modal density of the system [START_REF] Hildebrand | [END_REF] . The experimental set-up described below allows dense and three-dimensional sampling of the wavefield from the plate itself to the top of the beams, which is key for our analysis. Through this exhaustive dataset, we propose a novel matrix-based estimator of the modal density of such a system. Furthermore, we show direct agreement between the resonator motion and the wavefield properties inside the plate substrate, which casts new light on the strong coupling regime that can arise in locally resonant elastic metamaterials. In this twodimensional system, we obtain a clear physical meaning of the mesoscopic scattering regime based on the local coupling between coupled beam-like resonators.

A. Experimental set-up

The studied metasurface is a small portion of a thin aluminum plate (2 mm thick, 2 m long, 1 m wide). The experimental scheme is shown in Figure 1. The plate has a peculiar boundary shape, to break its natural symmetries. A dense set of 98 beams that act as resonators are randomly glued onto the plate surface over an approximate area of 20 cm 2 , with a mean surface density of 0.5 beam/cm 2 . Each beam is 60-cm long and has a diameter of 6 mm. The frequency bandwidth considered in the experiment (1-8 kHz) limits the plate waves to the first symmetric and antisymmetric Lamb modes. Due to the plate thickness, most of the energy inside this frequency band propagates as an antisymmetric mode A 0 that is characterized by out-of-plane polarization. At around 2.5 kHz, the A 0 wavelength is 12 cm, with a wave speed of 300 m/s. The S 0 is a lot faster, with a wavelength of 2 m for a wave speed of 5000 m/s. The first antisymmetric A 0 Lamb wave mode is generated inside the plate using two piezoelectric 1-cmdiameter discs, one of which is located at the center of the beam cluster, and the other in the far-field of the metasurface area. A 3-s-long electrical chirp signal is applied to the piezo source with a frequency ramp starting at 1 kHz, and going up to 8 kHz. The wavefield is then recorded with a three-component laser vibrometer, and cross-correlated with the emitted signal, to provide a 500ms highly reverberated impulse response with an 80 dB signal-to-noise ratio.

The measurement platform that is used consists of three scanning laser vibrometers that can measure the three-component motion, which is mounted on a fiveaxis motorized robot. Through the motion of the robot, we define three scanning areas for which the wavefield is accurately sampled. The first zone concerns the plate in the metamaterial region, on the opposite side of the beam cluster (antisymmetric motion, with regard to the beam cluster side of the plate). The second zone is a free plate region distant from the metamaterial, and the last zone is the top of the beams. The final dataset gathers the exact position in the robot repository and the three components velocity field at each of the scanned points. Due to the expected drop in velocity, the metasurface region is spatially sampled with a 4-mm step size, while the base plate surface is scanned with an 8-mm step size, and the resonator motion with one point per beam. A control unit drives the robot and the laser, and generates and records the signals. Two dynamics can be clearly noted. On the one hand, the compressional motion with fast pressure waves traveling inside the beam is characterized by a low-qualityfactor resonance that spreads over a large frequency band (∼1.7-3.8 kHz). On the other hand, the flexural resonance has slow-traveling shear waves inside the beam, which is hallmark of a high-quality factor.

The first-order interpretation of the plate-plus-beam coupling has already been described in great detail (see 13,15,19,20 ). Briefly, the low-quality factor compressional motion (fig. 2 a, red curve) of the beams opens the hybridization bandgap, as measured through the intensity transmission of the A 0 Lamb mode that is emitted from the source outside the metamaterial region, and is measured inside the beam cluster (fig. 2 b) with the out-of-plane component. Inside the bandgap, the beam cluster clamps the plate and acts as a perfect reflector, which allows only the transmission of evanescent waves with a specific attenuation length. In the latter part of the reverberated signal, after a few tens of milliseconds of propagation, the high-quality factor flexural resonances induce narrowband transmitted intensity peaks (fig. 2 b, red arrows) that are mainly seen in the compressional resonance bandgap.

We then focus on the bandgap itself, between ∼1.7 kHz and 3.6 kHz. At each flexural resonance, waves can propagate inside the beam cluster. However, they do not have the same amplitude dynamics as propagative waves inside the passband. Indeed, as show in figure 2 c,d, when the pass-band dispersion effects are associated to the monopolar stress feedback from the beam to the plate, the bandgap leakage (fig. 2b, red arrows) at flexural frequencies is due to the dipolar bending moment at the beamplate attachment. In both cases, out-of-plane polarized A 0 Lamb waves are generated, and are thus detected with the out-of-plane component of the laser vibrometer. In the following, we focus on the complex dispersion curve of the out-of-plane A 0 waves.

C. Ioffe-Regel criterion of localization

The dispersion curve is experimentally obtained using a two-point correlation method. Through an averaging process performed over all of the set of receivers and a series of time windows in the coda of the reverberated signals, the two-point correlation function converges toward the effective Greens function of the propagation medium 20 . In the present case, the optimal model for the effective medium is that for the two-dimensional Greens function for A 0 waves in the plate, with the possibility that the effective wave vector will be complex.

Figure 3a shows the real (blue) and imaginary (red) parts of the effective wavenumber that were extracted using the two-point correlation method. The real part (k) corresponds to the propagating wavenumber that is connected to the effective wavelength λ = 2π/ (k) in the medium. The imaginary part (k) corresponds to an attenuation length = 1/2 (k) that is classically described as the elastic mean free path in the framework of mesoscopic physics, where multiple scattering has important roles 11 . The first bandgap starts between ∼1.5 kHz and 2 kHz and ends just before 4 kHz. On the edge of the pass-band (i.e. ≥ 1.5 kHz), the wavelength becomes so small that the beams start to act as strong point-like scatterers. This results in an increase in the imaginary part of the wave vector in the effective Greens function. Inside the stopband, the real and imaginary parts are of the same order of magnitude, as predicted by Williams et al. 19 . The waves are highly attenuated. When the frequency reaches a flexural resonance of the beams, a strong variation in the dispersion curve emerges that reflects the possibility for flexural-type beam excitation to propagate as out-of-plane waves out of the metamaterial area.

The product between the effective propagating wavenumber and the mean free path represents the Ioffe-Regle criterion of localization k ∼ 1, and it describes the relative weight between propagation and damping inside the metamaterial area (fig. 3b). We observe that k 1 in the passband where attenuation is negligible. On the other hand, k strongly decreases in the bandgap, which indicates the strong damping of out-of-plane waves in the metamaterial. We observe that k can reach values of ≤ 1 within the frequency bandgap, which means that the wave is scattered on a distance less than or equal to one wavelength. In particular, at the flexural resonances, k 0.5 is the first sign of potential localized states in the metamaterial region.

Like in Laurent [START_REF] Laurent | Mortessagne[END_REF] , we can evaluate the localization length ζ from the real and imaginary parts of the wavenumber. We obtain 1 cm at 2205 Hz, 1.3 cm at 2750 Hz, and 1.6 cm at 3350 Hz, which correspond to the lowest distances between the beams. These three distances are also much smaller than the size of the sample (L∼20 cm), which means that localized modes can appear clearly at the frequencies within the random beam configuration.

D. Intensity distribution

Localized states are associated with exotic spatial field patterns at the scatterer scale. According to what has been seen before, the waves are scattered at the wavelength scale at the flexural resonances of the beams, and the randomness of the beam distribution starts then to dictate how the intensity of the wavefield spreads spatially.

Looking at the space and time distribution of the intensity is another way to detect localized states. For transmission through a disordered waveguide, the normalized intensity distribution can be described by Equations ( 1) and (2) 2,5,21 :

i∞ -i∞ dx iπ K 0 (2 √ -ux) exp(-φ(x)) (1)
with

φ(x) = g log 2 1 + x g + x g ( 2
)
where g is the dimensionless conductance that describes the strength of interference effects, and u is the dimensionless intensity ratio, whereby u = I/ I . If g 1, the waves are diffuse over all of the sample. If g ∼ 1 the waves can be localized.

In our set-up, we investigate the distribution of the speckle intensity inside and outside the metamaterial region, and far from the plate boundary. We make the assumption of an ergodic dynamical system. For a 15ms sliding time window, we compute the ratio I/ I at each frequency, and we gather the intensity ratio over the whole set of time windows in the coda and receiver positions inside the metasurface. The intensity distribution is plotted in figure 4b,c, both outside and inside the metasurface, for two frequencies that belong to the first bandgap. Deviation in the curve shape from the Rayleigh-like free-plate behavior is clearly seen close to the flexural resonance, at around 2200 Hz.

The intensity distribution can be fitted to a theory according to the conductance g (Eqs. ( 1), ( 2)), as shown in figure 4b,c. Instead of the conductance g, we prefer to plot the resistivity of 1/g in figure 4a for the speckle intensity patterns captured both inside and outside the beam cluster. The color bar represents a normalized version of the in-plane average motion at the top of the beams, which is expected to reach a maximum at each flexural resonance. This confirms that the resistivity gets stronger when the beams are excited in their transverse motion at flexural resonances inside the bandgap.

Indeed, the out-of-plane A 0 -like plate motion inside the bandgap is prevented by the beam anti-resonance compressional motion that blocks the propagating plate wave outside the metamaterial. However, this condition is no longer true for local bending motions at each beamplate attachment. As such, blocking the out-of-plane motion does not prevent a bending moment at the bottom of the beams (fig. 2b), which becomes overriding at a flexural resonance. We note that this local coupling appears to decrease along the bandgap, with the resistivity parameter 1/g slowly decreasing from one flexural motion to the other (e.g., see fig. 4a, for the three peaks of the resistivity parameter between 1.7 kHz and 3.7 kHz). Indeed, as shown in figure 2b, the coupling with the flexural motion of the beams is through the bending moment of the plate, which is proportional to the square of the effective wavenumber, and which slowly decreases from 2 kHz to 4 kHz along the bandgap (fig. 3a).

Finally, we observe that the flexural resonance inside the passband (fig. 4a, at ∼ 1.3 kHz) does not reveal such behavior. Indeed, whereas A 0 waves in the pass-band can still travel inside the metamaterial area, the beams only communicate through tight-binding mechanisms associated with evanescent waves in the bandgap 22 .

E. Visualization and robustness of the localized modes

Localized states should be robust in terms of the boundary conditions. Here, rather than changing the size of the metamaterial region, as did Mortessagne et al. (2007), we first visually check the robustness of the mode with the source located inside the metamaterial. If the waves are trapped by a localized state, the spatial representation of the intensity in the late part of the reverberating coda should be independent of the recorded time window. Moreover, the energy should be located around the source position (which is located in the middle of the sample), and spread over a small distance (i.e., localization length of ∼1.2 cm, as estimated above). Figure 5 shows the intensity inside the metamaterial as one color map that has the beam position and their corresponding flexural intensity motion superimposed as another color map. We used a 0.4-s time window, and started after 0.1 s of propagation for this analysis. Around the three frequencies that correspond to the minimum conductance inside the bandgap (fig. 4a, red crosses), the intensity is strictly located inside the metamaterial region without any interaction with its boundary, and thus with no leakage into the plate, as shown in figure 5a-c. For a frequency next to the first flexural resonance inside the bandgap (fig. 5d), we observe leakage outside the metamaterial. Moreover, as expected through the resistivity 1/g (fig. 4), the spatial spreading of the three localized modes (fig. 5a-c) visually increases along the bandgap, in agreement with the effective wavelength (see fig. 3a).

Another way to quantify the existence of localized modes associated with tight-binding coupling is to investigate the correlation between the beam group motion inside the bandgap. As indicated above, if the beams are distant from each other, they cannot transmit waves one to the other, as the plate is clamped by the collective beam behavior in the bandgap over the metamaterial re- gion. However, when the beams get close to each other, they start to exchange their energy through evanescent waves. One way to evaluate this short-distance correlation is through a cross-spectral matrix, which is computed with the three components of the 98 beams that form the metamaterial. This correlation is defined as given in Equation (3):

C ij (ω) = q i (ω)q j (ω) (3)
where q i is the three components (v x , v y , v z ) of the beams, and indicates the complex conjugate. Such a correlations matrix is Hermitian, with 98 × 3 = 294 lines and columns. We built the average cross-spectral density matrix (CSDM) as for the two-point correlation method, on an overall average over several realizations of the wavefield, as associated with different time windows of the highly reverberated signals. The selected time windows need to be sufficiently long to include waves that travel over the whole set of beams, from bottom to top, and from top to bottom. As the flexural waves in the beams are much slower than the compressional waves, a 75-ms time window is chosen, which leads to an ensemble average of over 20 realizations.

If the beam motions are completely independent one from the other, the maximum number of accessible states is reached for the system. If coupling between beams appears, the eigenvalue distribution of the CSDM is affected according to 4,23 . The eigenvalue distribution, which is also known as Shannon entropy, reflects the quantity of information a system can hold; namely, the modal density. This can be quantified at each frequency with the Shannon entropy S defined as in Equation ( 4):

S = -Σ λi log( λi ) ( 4
)
where λi is the i-th trace-normalized eigenvalue of the CSDM 24 . When S drops, the waves lose some degrees of freedom in terms of their propagation. Here, because of the wavefield equipartition, a frequency variation in the eigenvalue distribution is due to a variation in the modal density of the system itself 23 . Figure 6a shows the frequency-dependent Shannon entropy for our metamaterial. Once the studied frequency reaches the bandgap (i.e., fig. 6a, after 1.7 kHz), we only measure the noise at the top of the beams. Indeed, there is no energy on average inside the metamaterial, as depicted in figure 6d. The beams are static and the residual noise is fully incoherent, which corresponds to the maximum available entropy for the given number of realizations. At the flexural resonances, the curve of figure 6a can be seen to have gaps, which reach values below those in the passband (i.e., before 1.7 kHz). The modal density is then lower near the flexural resonant frequencies than in the free plate, which confirms the presence of localized modes. Note that the three entropy drops related to the three flexural resonances inside the bandgap (fig. 6, red arrows 1-3) correspond to the field patterns in figure 5a-c (respectively).

To be sure that these drops come with specific and robust modal shapes, we use a correlation criterion [25][26][27] . Around each flexural resonant frequency, we look at the modal shape correlation between neighboring frequencies. In the frequency domain, this is equivalent to a scalar product on the eigenvectors that are obtained through the singular value decomposition of the CSDM. In figure 6a, we represent the acos value of the scalar product (which indicates the angle between the two vectors i and j; θ ij ) between the first eigenvector V 1 (ω) computed at each frequency from 1.5 kHz to 4.5 kHz. We can see that around each flexural resonance, the off-diagonal terms of the computed matrix confirm the correlation between specific modes at different frequencies, which is clearly highlighted in the magnification of the first flexural resonance from figure 6b that is shown in figure 6c. Other groups of modes appear, but not all of them are related to localized states inside the metamaterial. For example, in figure 6c, the black arrow indicated as 4 refers to the intensity field of figure 5d, where leakage to the free-plate medium was clearly observed.

F. Discussion

Using the computed matrix θ ij , we estimate the modal density of the system as the mode separation ∆ω (i.e., distance between elements along the diagonal) and their respective spectral width δω (i.e., thickness of the diagonal). The Thouless criterion is expressed as δω/∆ω, and correspond to the conductance of the system. In figure 6c, five modes appear between 2100 Hz and 2300 Hz, with a distance between them that is twice that of their respective spectral widths. This means that as expected from the conductance analysis performed from plate measurements (fig. 4), we find a conductance g ∼ 0.5 in this frequency band that is based on the resonator modal coupling analysis.

In theory, there is no mobility edge for two-dimensional systems 3 , and to get a clear signature of a localized state, we only need a localization length that is a lot shorter than L of the sample. It might be asked why this appears here in particular, at the flexural resonances inside the bandgap, and not somewhere else. Two reasons might be invoked here. First, the modal density of a randomly distributed locally resonant metamaterial shows strong fluctuations near the passband edges, which correspond here to the transmitted bands (due to the flexural resonances) inside the main bandgap 28 . Secondly, as specified by 29 , the effects of randomness are perceived differently according to the wave polarization. In particular, the in-plane motion in the study thereof highlights more sensitivity to randomness (i.e., equivalent to the beam flexural motion here) than to the out-of-plane motion (i.e., equivalent to the beam compressional motion). This makes sense with the present study: on the band edge of each flexural resonance inside the bandgap, the randomness of the metamaterial distribution is strongly enhanced, which produces the appearance of a localized state inside the metamaterial.

At some point, the mesoscopic response of the wavefield around these localized modes is only driven by the random spatial distribution of the beams attached to the plate. The Ioffe-Regel criterion value of 0.5 for the three localized modes highlighted in this experiment suggests a diffusion regime where a propagating wave is scattered on half of a wavelength. This makes sense with the dipolar diffusion process of one beam around its flexural resonance (fig. 2c). Inside the bandgap, the beam bases that are attached to the plate cannot move vertically, but can rotate, which is indeed possible every half wavelength.

Conclusion

In conclusion, we have linked the appearance of localized states to the multi-wave interactions of nearby beams that act as coupled resonators for the plate-plusbeam metasurface. Inside the large main bandgap that is due to the low-quality factor compressional motion of the beams, high-quality factor flexural resonances of the beams make localized modes appear. For frequencies that correspond to the three different flexural resonances inside the main bandgap, the value k ∼1 of the effective medium is in agreement with low conductance values (g ¡1) computed from the wavefield intensity pattern. At these frequencies, the flexural beam motion matches the out-of-plane plate wavefield, which confirms that the neighboring beams transmit energy through evanescent waves. The localization length then appears to be directly related to the effective wavelength inside the main bandgap. We also analyzed the modal density of the system through eigenvalue decomposition of the CSDM computed from the beam motion, which shows that localized modes are robust with respect to frequency. This study highlights the tight-binding coupling prevailing between the beams through the evanescent field that exists in a bandgap with a deficit of wave states.

FIG. 1 .

 1 FIG. 1. Scheme of the experiment. The cluster of beams (a) occupies a small area (20 cm 20 cm) of the 2-mm-thick plate. The three laser heads (b) extract the motions at the top of the beams, as well as on the opposite side of the plate surface through the positioning of the robot arm, depicted with another scanning position in (c). A central computer performs the signal generation and recording, as well as the driving the arm motion. Two piezoelectric sources, one inside and one outside the beam cluster (d), excite the A0 Lamb mode into the plate. The multi-reverberated recorded wavefield (e) includes the plate and beams three-component motion (f).

FIG. 2 .

 2 FIG. 2. Metasurface dynamics. (a) Average motion recorded at the top of the beams for compressional (red) and flexural (blue) motions for the source located on the free plate. (b) Transmitted intensity (out-of-plane component) inside the metasurface from the source located outside of the beam cluster. (c, d) Modal representations of the beam motion under flexural (c) and compressional (d) resonance. The resulting stress applied to the plate is either bipolar (bending) or monopolar (compression).

FIG. 3 .

 3 FIG. 3. Complex dispersion curves for the effective medium. (a) Real (blue) and imaginary (red) parts of the wavenumber extracted through the two-point correlation method. (b) Ioffe-Regel k parameter over frequency, interpreted as the ratio between the real and imaginary parts of the wavenumber.

FIG. 4 .

 4 FIG. 4. Intensity distribution: experimental results (b, c) and theoretical fit (a) for the source located on the free plate. (a) The resistivity 1/g (g is defined as the Thouless conductance) is plotted with respect to the frequency. The three red crosses indicate the flexural resonance locations, and the color bar indicates the average horizontal motion amplitude of the beams. The two arrows refer to two nearby frequencies with low and high resistivity for which the intensity distribution I/ I is plotted in (b) and (c). (b, c) Blue, curves correspond to Rayleigh-like intensity distribution measured outside of the metamaterial area; red, curves refer to the intensity distribution measured inside the metamaterial. The conductivity g is extracted from a theoretical fit from Equations (1) and (2). (b) is obtained at 2045 Hz, inside the bandgap, and (c) is obtained for a flexural resonance at 2205 Hz.

FIG. 5 .

 5 FIG. 5. Normalized intensity maps of the late coda of the signals measured for the source located inside the metamaterial. The horizontal (x-y) beam motion Ib is plotted with the white-to-black color map, and the out-of-plane (z) plate motion Ip is plotted with the green-to-yellow color map. (a-c) The three maxima of the resistivity 1/g (shown in Fig. 4a as red crosses, and in Fig. 6a as red arrows). and panel (d) The close frequency of the first maximum where the intensity is leaking outside of the metamaterial area (Fig. 6a, black arrow).

FIG. 6 .

 6 FIG. 6. Singular value decomposition of the cross-spectral density matrix (CSDM) of the three-component motion of the 98 beams. The Shannon entropy (a) is computed from the distribution of the CSDM eigenvalue at each frequency. The three red arrows (1-3) indicate the three minima of the entropy that are associated with the three flexural resonances, which correspond to Figure 5a-c. (b) The scalar product between the first eigenvector extracted between neighboring frequencies shows the robustness of the modal excitation, with magnification around the first flexural resonance (2200Hz) shown in (c). The blue and red shaded areas highlight the flexural resonances of the beams inside the bandgap (see Fig. 2a). (c) The black arrow (4) indicates another mode for the beam cluster, which corresponds to (d) in Figure 5. (d) The average three-component intensity It measured at the top of the beam.

Acknowledgments

The authors are grateful to Mathieu Scail and Julien Nicolas from Le Mans Universit, and to Pascal Audrain from IRT Jules Verne, for their strong contribution to the experiments using the so-called 3Dvib platform located at the ENSIM School of Engineering in Le Mans.


Lorenza N C Dezanet

Sarah Maylin

Audrey Gabassi

Hayette Rougier

Patrick Miailhes

Caroline Lascoux-Combe

Julie Chas

Pierre-Marie Girard

Constance Delaugerre

Karine Lacombe
email: karine.lacombe2@aphp.fr

Dr Anders Boyd
email: anders.boyd@iplesp.upmc.fr

Prof Karine Lacombe

Kinetics of Hepatitis B Core-Related Antigen and Anti-Hepatitis B Core Antibody and Their Association With Serological Response in Human Immunodeficiency Virus-Hepatitis B Coinfection

Keywords: hepatitis B, HIV, seroclearance, hepatitis B core-related antigen, anti-hepatitis B core antibody

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

Background

In HIV-positive individuals, it was recently estimated that 7.4% worldwide were co-infected with chronic hepatitis B virus (HBV) [1]. HIV-HBV co-infection has been associated with increased risk of liver cirrhosis, hepatocellular carcinoma, and hepatic decompensation particularly when HBV replication is left uncontrolled [START_REF] Ioannou | The prevalence of cirrhosis and hepatocellular carcinoma in patients with human immunodeficiency virus infection[END_REF][START_REF] Hosaka | Long-term entecavir treatment reduces hepatocellular carcinoma incidence in patients with hepatitis B virus infection[END_REF]. Tenofovir (TDF)-containing antiretroviral therapy (ART) effectively provides dual activity against HIV and HBV in coinfected patients, allowing suppression of HBV replication [START_REF] Lampertico | EASL 2017 Clinical Practice Guidelines on the management of hepatitis B virus infection[END_REF]. Wide-spread TDF use in coinfected patients has led to substantial decreases in HBV-DNA detection, with almost 85% of patients able to achieve undetectable HBV-DNA [START_REF] Boyd | Rates and determinants of hepatitis B 'e' antigen and hepatitis B surface antigen seroclearance during long-term follow-up of patients coinfected with HIV and hepatitis B virus[END_REF].

In HBV mono-infected patients, the viral markers indicating improved prognosis are suppression of HBV-DNA, hepatitis B "e" antigen (HBeAg) seroclearance (for those with HBeAgpositive serology), and more importantly hepatitis B surface antigen (HBsAg) seroclearance [START_REF] Lampertico | EASL 2017 Clinical Practice Guidelines on the management of hepatitis B virus infection[END_REF].

During TDF use in HIV-HBV co-infected patients, almost half of those who are HBeAg-positive exhibit HBeAg-seroclearance and few overall have HBsAg-seroclearance [START_REF] Boyd | Hepatitis B surface antigen quantification as a predictor of seroclearance during treatment in HIV-hepatitis B virus coinfected patients from Sub-Saharan Africa: Seroclearance during treatment[END_REF][START_REF] Boyd | Use of hepatitis B surface and "e" antigen quantification during extensive treatment with tenofovir in patients co-infected with HIV-HBV[END_REF]. There are, however, exceptions in which higher rates of seroclearance are observed in co-infected patients with more severe immunosuppression [START_REF] Singh | HIV-hepatitis B virus coinfection: epidemiology, pathogenesis, and treatment[END_REF][START_REF] Chihota | High Rates of Hepatitis B Virus (HBV) Functional Cure Among Human Immunodeficiency Virus-HBV Coinfected Patients on Antiretroviral Therapy in Zambia[END_REF][START_REF] Matthews | Quantitative HBsAg and HBeAg Predict Hepatitis B Seroconversion after Initiation of HAART in HIV-HBV Coinfected Individuals[END_REF]. As antiviral therapy inhibits viral replication of HBV-DNA, its levels do not accurately reflect intrahepatic HBV activity in patients undergoing antiviral treatment [START_REF] Boyd | Decay of ccc-DNA marks persistence of intrahepatic viral DNA synthesis under tenofovir in HIV-HBV co-infected patients[END_REF]. Other markers of treatment efficacy, such as quantification of HBeAg (qHBeAg) and HBsAg (qHBsAg), have been useful in predicting seroclearance events, but their stability during long-term treatment makes their use in routine clinical practice debatable [START_REF] Boyd | Hepatitis B surface antigen quantification as a predictor of seroclearance during treatment in HIV-hepatitis B virus coinfected patients from Sub-Saharan Africa: Seroclearance during treatment[END_REF][START_REF] Martinot-Peignoux | The role of HBsAg quantification for monitoring natural history and treatment outcome[END_REF].

Recently, quantitative hepatitis B core-related antigen (qHBcrAg) and anti-hepatitis B core antibody (qAnti-HBc) have been gaining attention in HBV mono-infection. qHBcrAg consists of three proteins, hepatitis B core antigen (HBcAg), HBeAg and a 22KDa precore protein (p22cr), which are transcribed from the precore/core gene and share an identical 149 amino acid sequence [START_REF] Mak | Review article: hepatitis B core-related antigen (HBcrAg): an emerging marker for chronic hepatitis B virus infection[END_REF]. This surrogate marker strongly correlates with covalently-closed circular (ccc)DNA and total intrahepatic HBV-DNA during antiviral-induced HBV suppression [START_REF] Mak | Review article: hepatitis B core-related antigen (HBcrAg): an emerging marker for chronic hepatitis B virus infection[END_REF][START_REF] Wong | Hepatitis B Virus Core-Related Antigens as Markers for Monitoring Chronic Hepatitis B Infection[END_REF][START_REF] Tanaka | Hepatitis B Core-Related Antigen Assay Is Useful for Monitoring the Antiviral Effects of Nucleoside Analogue Therapy[END_REF][START_REF] Chen | Clinical evaluation of hepatitis B core-related antigen in chronic hepatitis B and hepatocellular carcinoma patients[END_REF][START_REF] Rokuhara | Clinical evaluation of a new enzyme immunoassay for hepatitis B virus core-related antigen; a marker distinct from viral DNA for monitoring lamivudine treatment[END_REF][START_REF] Chuaypen | Serum hepatitis B core-related antigen as a treatment predictor of pegylated interferon in patients with HBeAg-positive chronic hepatitis B[END_REF]. qAnti-HBc involves quantification of the standard marker anti-HBc antibodies and is thought to reflect HBV-specific adaptive immunity. Baseline levels of this marker have been shown to bear reliable prediction of HBeAg seroconversion in HBV mono-infected patients undergoing antiviral therapy [START_REF] Fan | Baseline quantitative hepatitis B core antibody titre alone strongly predicts HBeAg seroconversion across chronic hepatitis B patients treated with peginterferon or nucleos(t)ide analogues[END_REF].

Nevertheless, current research on these markers has mainly focused on HBV mono-infected patients, while no study to date has examined its relevance in HIV-HBV co-infection. This is particularly concerning as qAnti-HBc levels are linked to host immunity [START_REF] Lee | Interaction of the Hepatitis B Core Antigen and the Innate Immune System[END_REF], which could be impaired during HIV infection. The aim of the present study was to describe qHBcrAg and qAnti-HBc kinetics during long-term TDF treatment in a large, prospective cohort of ARTexperienced HIV-HBV co-infected patients. We further aimed to evaluate the association between these markers and HBeAg or HBsAg-seroclearance and their predictive capacity for these events.

Methods

Patients and data collection

Patients were selected from the French HIV-HBV Cohort Study [START_REF] Lacombe | Major role of hepatitis B genotypes in liver fibrosis during coinfection with HIV[END_REF]. Briefly, this prospective study recruited 308 HIV-infected patients with chronic HBV infection from seven clinical centers located in Paris and Lyon, France during May 2002-May 2003. Patients were included if they had HIV-positive serology confirmed by western blot and HBsAg-positive serology for more than 6 months. Participants were followed prospectively every six to twelve months until 2010-2011.

For this analysis, we included patients who initiated TDF-containing ART, had a minimum of two consecutive visits undergoing TDF-containing ART (lasting more than 6 months), with an available sample at TDF-initiation and at least once during follow-up. Patients with detectable HCV-RNA or HDV-RNA or those undergoing intensification with standard or pegylatedinterferon (peg-IFN) were excluded from analysis (Supplementary figure 1). All patients gave written informed consent to participate in the cohort and the study received ethical approval in accordance with the Helsinki declaration.

Baseline was defined as the study visit at or directly prior to TDF-initiation. Follow-up began at TDF-initiation and continued until last study visit, TDF discontinuation, meeting any one of the exclusion criteria, or death; whichever occurred first. Demographic information were collected at study inclusion. HIV-and HBV-related variables were collected before TDF initiation and at each follow-up visit.

Assessing markers of viral activity

Serum HBV-DNA was quantified at baseline and every 6 months by a real-time PCR assay (COBAS® AmpliPrep/COBAS TaqMan®, detection limit: 12 IU/mL; or COBAS® Amplicor HBV Monitor, detection limit: 60 IU/mL; Roche Diagnostics, Meylan, France). HIV-RNA viral load was measured at cohort inclusion and every six months using either a branched-DNA (b-DNA Quantiplex 3.0, detection limit: 50 copies/mL, Bayer Diagnostics, Cergy Pontoise, France) or real-time PCR technique (COBAS® AmpliPrep/COBAS TaqMan® HIV-1 test, detection limit: 40 copies/mL, Roche Molecular Systems, Meylan, France).

Qualitative HBsAg, HBeAg, and antibodies were detected yearly using commercial enzymelinked immunoassays (EIAs) (Diasorin, Antony, France). All other markers were collected at baseline and every 6 months. Serum qHBsAg levels (IU/mL) were quantified using the ARCHITECT HBsAg assay (Abbott Laboratories, Rungis, France) [START_REF] Boyd | Use of hepatitis B surface and "e" antigen quantification during extensive treatment with tenofovir in patients co-infected with HIV-HBV[END_REF][START_REF] Deguchi | Quantitation of hepatitis B surface antigen by an automated chemiluminescent microparticle immunoassay[END_REF]. Serum qHBeAg levels were quantified using either ARCHITECT or Elecsys HBeAg assay (with Modular E170 analyzer; Roche Diagnostics). qHBeAg levels were expressed in Paul Ehrlich Institute units (PEIU)/mL [START_REF] Maylin | Quantification of hepatitis B e antigen between Elecsys HBeAg and Architect HBeAg assays among patients infected with hepatitis B virus[END_REF]. qHBcrAg (U/mL) was measured using a commercially-available, automated HBcrAg chemiluminescence EIA (Lumipulse® G System, FujiRebio Europe, Gent, Belgium) [START_REF] Kimura | Sensitive Enzyme Immunoassay for Hepatitis B Virus Core-Related Antigens and Their Correlation to Virus Load[END_REF]. When initial qHBcrAg levels were above 7 log10 U/mL, a 1/100 dilution was performed to obtain results within the range of quantification. Finally, IgG and IgM anti-HBc antibodies were quantified using ARCHITECT Anti-HBc II assay (Abbott Laboratories, Rungis, France) with an automated ARCHITECT i4000 system. Anti-HBc antibody levels were reported in PEIU/mL [START_REF] Yuan | Quantitative hepatitis B core antibody level may help predict treatment response in chronic hepatitis B patients[END_REF].

Statistical analysis

All qHBcrAg and qAnti-HBc units were log10 transformed. Linear regression was used to estimate univariable differences in baseline qHBcrAg or qAnti-HBc, and their 95% confidence intervals (CI), across levels of determinants. A multivariable model was constructed by adding all covariables with a p<0.2 in univariable analysis and removing non-significant variables in backwards-stepwise fashion.

In longitudinal analysis, on-treatment kinetics of the changes of qHBcrAg (ΔqHBcrAg) and qAnti-HBc (ΔqAnti-HBc) were estimated using mixed-effect linear regression models with a random-intercept to account for between-patient variability at baseline [START_REF] Rabe-Hesketh | Multilevel and longitudinal modeling using stata[END_REF]. Models were stratified by HBeAg-status. Given the bi-phasic nature of ΔqHBcrAg kinetics among HBeAgpositive patients in preliminary analysis, we modeled qHBcrAg kinetics according to <18 and >18 months of TDF treatment. All of the models were adjusted a priori for factors influencing liver fibrosis or HBV replication: body mass index (BMI), age, concomitant lamivudine (LAM) treatment, cumulative treatment duration with LAM, undetectable HBV-DNA, CD4+ cell count >350 cells/mm 3 (if not already stratified), and baseline qHBcrAg or qAnti-HBc levels. For specific genetic and immunological determinants, we also included a cross-product term between TDF duration and presence of determinant along with its individual components (separately for each determinant), from which stratum-specific estimates could be calculated and differences in ΔHBcrAg and ΔqAnti-HBc could be tested.

We then evaluated the relationship between qHBcrAg or qAnti-HBc and HBeAg-seroclearance.

First, univariable hazards ratios (HR) and 95%CI comparing hazards of HBeAg-seroclearance between continuous baseline levels of markers (qHBcrAg and qAnti-HBc separately) were estimated using Cox proportional hazards models. A multivariable model was constructed using the backwards-selection approach as described above. Second, we performed several analyses to explore the most adequate thresholds for both markers, individually or combined.

We selected the following criteria to assess the predictive capacity of HBeAg-seroclearance: qHBcrAg log10 U/mL <7.5 at baseline and <6.5 at months 12, 24 and 36; qAnti-HBc >4.1 log10 PEIU/mL at baseline; qHBcrAg <7.5 log10 U/mL and qAnti-HBc >4.1 log10 PEIU/mL at baseline; qHBcrAg <7.5 log10 U/mL or qAnti-HBc >4.1 log10 PEIU/mL at baseline; and HBV-DNA <60 IU/mL at months 12, 24 and 36. Time-dependent receiver operating characteristic (ROC) curves were used to evaluate the sensitivity (Se) and specificity (Sp) of each criterion as predictor of HBeAgseroclearance until specific time-points of TDF duration [START_REF] Heagerty | Time-dependent ROC curves for censored survival data and a diagnostic marker[END_REF].

All statistical analysis was performed using STATA (v15.1, College Station, TX, USA) or RStudio (v1.1.453, Vienna, Austria) and significance was defined as a p-value <0.05.

Results

Baseline characteristics of the study population

Of the 158 patients, 95 (60%) were HBeAg-positive and 63 (40%) HBeAg-negative. At TDFinitiation, median age was 41.1 years (IQR=35.5-48.0) and the majority was male (84.2%). All patients were ART-experienced with median CD4+ cell count at 404/mm 3 (IQR=295-552) and 57.3% (n=90) having undetectable HIV-RNA. A total of 123 (78.3%) had detectable HBV-DNA with a median viral load of 5.2 log10 IU/mL (IQR=3.0-7.2). For those with available data on HBV genotypes, most patients harbored genotype A (68%), followed by E (12%), G (12%) and D (8%). The G to A nucleotide substitution at position 1896 of the precore region was identified in 21.8% of patients (n=22/101).

As shown in Table 1, HBeAg-positive patients were more often male (p<0.001), born in zone of low/moderate HBV-prevalence (p<0.001), diagnosed with an AIDS-defining event (p=0.04), had a longer duration of known HIV-infection (p=0.007), higher levels of HIV-RNA (p=0.03) and qHBsAg (p<0.001) than HBeAg-negative patients (Table 1). As expected, HBV-DNA and ALT/AST levels were higher among HBeAg-positive versus HBeAg-negative patients (both p<0.001).

Baseline determinants of qHBcrAg and qAnti-HBc levels

At TDF-initiation, median levels of unadjusted qHBcrAg were significantly higher in HBeAgpositive (7.8 log10 U/mL; IQR=7.0-8.2; p<0.001) versus HBeAg-negative patients (3.0 log10 U/mL; IQR=2.5-4.0), while median levels of unadjusted qAnti-HBc were greater in HBeAg-negative (3.4 log10 PEIU/mL; IQR=3.0-3.9; p=0.008) versus HBeAg-positive patients (2.8 log10 PEIU/mL; IQR= 1.1-4.0) (Table 1). In multivariable analysis (Table 2), baseline qHBcrAg levels (adjusted mean log10 U/mL) were higher in those with an AIDS-defining event (6.3 versus without advanced HIV disease, 5.7; p=0.06), ALT >70 IU/L (6.7 versus ALT ≤70 IU/L, 5.5; p<0.001) and higher serum HBV-DNA (p<0.001). Baseline qAnti-HBc levels (adjusted mean log10 PEIU/mL) were higher in older individuals (p=0.09), those without an AIDS-defining illness (3.0 versus with AIDS-defining illness, 2.3; p=0.047), with higher CD4+ cell count (p<0.001) and with HBeAg-negative status at baseline (3.4 versus HBeAg-negative, 2.5; p<0.001).

On-treatment kinetics of qHBcrAg and qAnti-HBc levels

Median follow-up of TDF treatment was 4.6 years (IQR=2.9-7.6). Adjusted ΔHBcrAg was faster in HBeAg-positive versus HBeAg-negative patients, overall and during the first 18-months of treatment (Table 3; figure 1A). In HBeAg-positive patients, adjusted ΔqHBcrAg during TDF treatment was biphasic, with a significantly more rapid decline during the first 18 months (-0.051 log10 U/mL/month) compared to thereafter (-0.011 log10 U/mL/month; p=0.007). In HBeAg-negative patients, adjusted ΔqHBcrAg was linear during TDF treatment (-0.003 log10 U/mL/month). ΔqHBcrAg was faster in HBeAg-negative patients with HBV genotype D infection patients (p<0.001) and slower with genotype A infection (p=0.01) compared to other genotypes, while faster ΔqHBcrAg was observed in HBeAg-negative patients with precore mutations compared to without (p=0.009).

Both HBeAg-positive and HBeAg-negative patients had linear ΔqAnti-HBc during TDF treatment (Figure 2A), while ΔqAnti-HBc was not different between HBeAg-positive versus HBeAgnegative patients (-0.011 log10 PEIU/mL/month for both; p=0.7) (Table 3). In HBeAg-positive patients, adjusted ΔqAnti-HBc was significantly faster in HBV genotype A versus non-A infection (p=0.02). In both HBeAg-positive and HBeAg-negative patients, those with CD4+ cell count >350/mm 3 versus ≤350/mm 3 and those with a nadir CD4+ cell count >200/mm 3 versus ≤200/mm 3 exhibited a significantly faster ΔqAnti-HBc during TDF treatment (Table 3).

Of note, baseline qHBsAg levels were not significantly correlated with individual ΔqHBcrAg in HBeAg-positive individuals (Bland-Altman rho=-0.06, p=0.11) and HBeAg-negative individuals (Bland-Altman rho=0.0004, p=0.9). Baseline qHBsAg levels were also not significantly correlated with individual ΔqAnti-HBc in HBeAg-positive individuals (Bland-Altman rho=0.06, p=0.12) and HBeAg-negative individuals (Bland-Altman rho=-0.08, p=0.12).

Association of qHBcrAg and qAnti-HBc levels with seroclearance

During TDF treatment, 26 HBeAg-positive patients achieved HBeAg-seroclearance (cumulative incidence=27.4%, 95%CI=18.7%-37.5%). Of the patients with HBeAg-seroclearance, HBeAg seroconversion occurred in 8 (cumulative incidence=8.4%, 95%CI=3.7%-15.9%). Patients with lower baseline qHBcrAg level had a higher rate of HBeAg-seroclearance (Figure 1B) after adjusting for change in CD4+ cell count from previous visit (adjusted-HR=0.48, 95%CI=0.33-0.70) (Table 4). Patients with higher baseline qAnti-HBc level had higher rates of HBeAgseroclearance (HR=1.49, 95%CI=1.08-2.07) (Figure 2B), while no other variable was below the p-value threshold considered for multivariable analysis (Table 4). In addition, ΔqAnti-HBc was also faster among patients achieving HBeAg-seroclearance during follow-up (-0.017 log10 PEIU/mL/month) compared to those without HBeAg-loss (-0.010 PEIU/mL/month; p<0.001).

HBsAg-seroclearance occurred in only 5 patients (cumulative incidence=3.2%, 95%CI=1.0%-7.2%). There were no statistically significant differences in their baseline characteristics compared to those without HBsAg-seroclearance (Supplementary table 1). Among those with HBsAg-seroclearance, median baseline qHBcrAg was 6.7 log10 U/mL (range HBeAg-positive=6.7-8.0, HBeAg-negative=3.7-4.0) and median baseline qAnti-HBc was 3.2 log10 PEIU/mL (range HBeAg-positive=1.2-4.6, HBeAg-negative=1.8-3.6). ΔqHBcrAg appeared steeper during the first 12 months of TDF treatment in these individuals (Figure 1C), while the change in qAnti-HBc was flat during follow-up (Figure 2C). Given the few numbers of patients with HBsAg-seroclearance, this endpoint was not considered further in analysis.

Predictive capacity of HBV serological markers on HBeAg-seroclearance

Table 5 gives the Se and Sp of various criteria in predicting HBeAg-seroclearance at specific time-points during TDF treatment. As expected, undetectable HBV-DNA for most time-points provided optimal Se, but consistently low Sp in predicting long-term HBeAg-seroclearance. qHBcrAg <6.5 log10 U/mL at months 12, 24 or 36 showed comparable Se to undetectable HBV-DNA, but higher Sp especially for predicting HBeAg-seroclearance in the 12 months following qHBcrAg measurement. Baseline qAnti-HBc >4.1 log10 PEIU/mL showed high Sp, but very low Se in predicting HBeAg-seroclearance. When combining qHBcrAg <7.5 log10 U/mL and qAnti-HBc >4.1 log10 PEIU/mL, both assessed at baseline, we found the highest levels of Sp in predicting HBeAg-seroclearance of the studied markers, but low Se.

Discussion

Despite the extensive research on novel biomarkers of HBV viral activity, none to date have examined their kinetics or association with seroclearance in HIV-HBV co-infected patients undergoing TDF treatment. We have demonstrated a gradual decrease of on-treatment qHBcrAg and qAnti-HBc levels, compatible with a cumulative therapeutic benefit of long-term nucleos(t)ide analogue (NA) therapy. Our findings indicated that lower baseline levels of qHBcrAg and higher qAnti-HBc antibodies were strong, independent predictors of HBeAg seroclearance, suggesting some clinical applicability in TDF-treated HIV-HBV co-infected patients. To our knowledge, this study is the first to report the clinical utility of qHBcrAg or qAnti-HBc levels as a predictor of HBeAg-seroclearance in the treated HIV-HBV co-infected population.

Despite the fact that all previous studies examining qHBcrAg and qAnti-HBc antibodies have been performed in HBV mono-infected patients, we did observe several similarities to our coinfected population [START_REF] Fan | Baseline quantitative hepatitis B core antibody titre alone strongly predicts HBeAg seroconversion across chronic hepatitis B patients treated with peginterferon or nucleos(t)ide analogues[END_REF][START_REF] Wang | HBsAg and HBcrAg as predictors of HBeAg seroconversion in HBeAg-positive patients treated with nucleos(t)ide analogues[END_REF]. A decrease in qHBcrAg during TDF treatment was indeed observed for both HBeAg-positive or HBeAg-negative patients [START_REF] Wang | HBsAg and HBcrAg as predictors of HBeAg seroconversion in HBeAg-positive patients treated with nucleos(t)ide analogues[END_REF][START_REF] Ma | HBcrAg Identifies Patients Failing to Achieve HBeAg Seroconversion Treated with Pegylated Interferon Alfa-2b[END_REF]. In addition, we demonstrated a bi-phasic decline in HBeAg-positive patients, which probably reflected the initial, effective inhibition of circulating virus in the blood associated with TDF followed by reductions of the cccDNA pool [START_REF] Lai | Reduction of covalently closed circular DNA with long-term nucleos(t)ide analogue treatment in chronic hepatitis B[END_REF]. For qAnti-HBc antibodies, its kinetics in TDF-treated HIV-HBV coinfected patients showed a steady decline, for both HBeAg-positive or HBeAg-negative patients, similar to HBV mono-infected individuals treated with peg-IFN or NAs [START_REF] Fan | Baseline quantitative hepatitis B core antibody titre alone strongly predicts HBeAg seroconversion across chronic hepatitis B patients treated with peginterferon or nucleos(t)ide analogues[END_REF][START_REF] Hu | Level of Hepatitis B (HB) Core Antibody Associates With Seroclearance of HBV DNA and HB Surface Antigen in HB e Antigen-Seronegative Patients[END_REF][START_REF] Hou | Quantitative Hepatitis B Core Antibody Level Is a New Predictor for Treatment Response In HBeAg-positive Chronic Hepatitis B Patients Receiving Peginterferon[END_REF]. Interestingly, the observation of higher median baseline qAnti-HBc levels in HBeAg-negative versus HBeAgpositive patients could be the result of increased immune activity in patients previously able to clear HBeAg [START_REF] Liao | Monitoring of serum HBV RNA, HBcrAg, HBsAg and anti-HBc levels in patients during long-term nucleoside/nucleotide analogue therapy[END_REF][START_REF] Song | Quantitative hepatitis B core antibody levels in the natural history of hepatitis B virus infection[END_REF].

We were able to identify other viral characteristics associated with qHBcrAg and qAnti-HBc kinetics. The presence of a precore mutation in HBeAg-negative patients appeared to accelerate declines in qHBcrAg. Indeed, as precore mutations can block the synthesis of HBeAg without adversely affecting HBV replication [START_REF] Carman | Mutation preventing formation of hepatitis B e antigen in patients with chronic hepatitis B infection[END_REF], the more rapid decrease in qHBcrAg might partly reflect viral suppression linked to TDF treatment. Whether this is due to reduction of the cccDNA pool in hepatocytes or from immunological clearance of virus in patients harboring precore mutant variants is unknown and cannot be confirmed with the data collected in our study. Nevertheless, it should be noted that the rates of declines observed in these patients were much slower than most of the average rates of declines in HBeAg-positive patients.

On-treatment qAnti-HBc kinetics were faster in HBV genotype A versus non-A genotypes, specifically among HBeAg-positive patients. In studies on the natural history of HBV monoinfection, HBeAg-positive patients with HBV genotype A have demonstrated higher rates of sustained transaminase normalization, HBV-DNA clearance, transition to inactive HBV carrier state after HBeAg-seroconversion, and HBsAg-seroclearance [START_REF] Sánchez-Tapias | Influence of hepatitis B virus genotype on the long-term outcome of chronic hepatitis B in western patients[END_REF][START_REF] Campenhout Mjh Van | Hepatitis B core-related antigen levels are associated with response to entecavir and peginterferon add-on therapy in hepatitis B e antigen-positive chronic hepatitis B patients[END_REF]. These data would suggest that a more robust antiviral immune response could be elicited against genotype A infection, yet the exact mechanisms remain unknown ( [START_REF] Lin | Hepatitis B Virus Genotypes and Variants[END_REF]). The faster decrease in qAnti-HBc levels builds on these findings, suggesting that anti-HBV-specific immunity could be more active in HBeAg-positive genotype A infected patients, even during TDF treatment.

Interestingly, given that HIV-related immunity also exerts an effect on anti-HBV-specific immunity [START_REF] Crane | No Increase in Hepatitis B Virus (HBV)-Specific CD8+ T Cells in Patients with HIV-1-HBV Coinfections following HBV-Active Highly Active Antiretroviral Therapy[END_REF], qAnti-HBc levels would assumedly be influenced by the degree of HIV-induced immunosuppression. This effect was already apparent at baseline, at which point qAnti-HBc levels were lower in patients with a previous AIDS-defining illness and lower CD4+ T cell counts. Additionally, more rapid declines of qAnti-HBc were observed in patients with higher baseline (>350 cells/mm 3 ) and nadir (>200 cells/mm 3 ) CD4+ cell counts, regardless of baseline HBeAg-status, supporting the concept that more pronounced immunosuppression lends to slower declines of qAnti-HBc [START_REF] Thio | Hepatitis B and human immunodeficiency virus coinfection[END_REF]. How this translates to HBeAg-seroclearance is unclear.

Faster slopes of qAnti-HBc were associated with HBeAg-seroclearance, suggesting that longer time spent at higher levels of qAnti-HBc levels (i.e. with more active anti-HBV immunity) does not lead to seroclearance. In addition, we observed no significant association with CD4+ cell counts and HBeAg-seroclearance.

More importantly, we found that higher levels of qHBcrAg at the time of initiating TDFcontaining ART were independently associated with lower rates of HBeAg-seroclearance in HBeAg-positive patients. Previous research linking baseline qHBcrAg levels with HBeAgseroclearance have mostly focused on treatment with peg-IFN. In 46 patients treated with peg-IFN, Chuaypen et al [START_REF] Chuaypen | Serum hepatitis B core-related antigen as a treatment predictor of pegylated interferon in patients with HBeAg-positive chronic hepatitis B[END_REF] reported that those with baseline qHBcrAg levels >8 log10 U/mL had a low risk of HBeAg seroclearance and suppression of HBV-DNA at 12 weeks post-treatment.

Others have found that qHBcrAg levels at week 12 of peg-IFN treatment were predictive of post-treatment HBeAg-seroclearance [START_REF] Ma | HBcrAg Identifies Patients Failing to Achieve HBeAg Seroconversion Treated with Pegylated Interferon Alfa-2b[END_REF], while another study investigating entecavir (ETV) with or without peg-IFN add-on therapy observed that higher qHBcrAg levels at both week 24 and week 36 were independently associated with a lower risk of HBeAg-seroclearance and HBV-DNA suppression 24-weeks post-treatment in both treatment arms [START_REF] Campenhout Mjh Van | Hepatitis B core-related antigen levels are associated with response to entecavir and peginterferon add-on therapy in hepatitis B e antigen-positive chronic hepatitis B patients[END_REF]. With regards to NA-based therapy, baseline HBcrAg levels >5.7 log10 U/mL have been also associated with lack of HBeAg-seroconversion at 6 and 12 months of treatment [START_REF] Wang | HBsAg and HBcrAg as predictors of HBeAg seroconversion in HBeAg-positive patients treated with nucleos(t)ide analogues[END_REF] and faster declines of qHBcrAg were observed in patients with HBeAg-seroclearance as compared to those without [START_REF] Liao | Monitoring of serum HBV RNA, HBcrAg, HBsAg and anti-HBc levels in patients during long-term nucleoside/nucleotide analogue therapy[END_REF]. Our findings in TDF-treated HIV-HBV co-infected patients fall in line with these studies.

Likewise, we found an association between anti-HBc antibody levels and HBeAg-seroclearance, indicating that patients with higher baseline total qAnti-HBc levels exhibited higher rates of HBeAg-seroclearance. Previous research has suggested that baseline qAnti-HBc strongly predicts HBeAg-seroconversion in populations of HBV mono-infected patients receiving either NA or peg-IFN therapy [START_REF] Fan | Baseline quantitative hepatitis B core antibody titre alone strongly predicts HBeAg seroconversion across chronic hepatitis B patients treated with peginterferon or nucleos(t)ide analogues[END_REF][START_REF] Yuan | Quantitative hepatitis B core antibody level may help predict treatment response in chronic hepatitis B patients[END_REF][START_REF] Hou | Quantitative Hepatitis B Core Antibody Level Is a New Predictor for Treatment Response In HBeAg-positive Chronic Hepatitis B Patients Receiving Peginterferon[END_REF]. The fact that an association between qAnti-HBc levels and HBeAg-seroclearance was observed in HIV-HBV co-infected patients, given the effect of HIVinduced immunosuppression on its quantification and sometimes higher rates of HBeAgseroclearance observed after anti-HBV containing ART [START_REF] Chihota | High Rates of Hepatitis B Virus (HBV) Functional Cure Among Human Immunodeficiency Virus-HBV Coinfected Patients on Antiretroviral Therapy in Zambia[END_REF][START_REF] Matthews | Quantitative HBsAg and HBeAg Predict Hepatitis B Seroconversion after Initiation of HAART in HIV-HBV Coinfected Individuals[END_REF], is noteworthy.

In terms of predictive capacity, we confirm the high sensitivity of qHBcrAg. We add to previous studies by showing that qHBcrAg levels have higher specificity in predicting HBeAgseroclearance within 12-months of being measured, with waning specificity in predicting HBeAg-seroclearance thereafter. This finding implies that repeated measurements of qHBcrAg would be needed in order to continuously and more accurately gauge the risk of HBeAgseroclearance. Indeed, other studies focusing on time-specific measurements of serum qHBcrAg levels have involved peg-IFN treatment in HBV mono-infected patients, with qHBcrAg measured at 12 weeks representing an important milestone for treatment response and stopping rules [START_REF] Chuaypen | Serum hepatitis B core-related antigen as a treatment predictor of pegylated interferon in patients with HBeAg-positive chronic hepatitis B[END_REF][START_REF] Ma | HBcrAg Identifies Patients Failing to Achieve HBeAg Seroconversion Treated with Pegylated Interferon Alfa-2b[END_REF][START_REF] Wang | Comparison of hepatitis B virus core-related antigen and hepatitis B surface antigen for predicting HBeAg seroconversion in chronic hepatitis B patients with pegylated interferon therapy[END_REF]. These studies have also demonstrated higher Se, but modest Sp when using qHBcrAg to predict HBeAg-seroconversion. On the other hand, serum anti-HBc antibodies had higher Sp at baseline than at any point during follow-up. Even when used as a complementary marker with qHBcrAg, predictive capacity did not improve. Nevertheless, the performance of these markers appears to be similar to qHBsAg and qHBeAg from previous studies in TDF-treated co-infected individuals [START_REF] Maylin | Kinetics of hepatitis B surface and envelope antigen and prediction of treatment response to tenofovir in antiretroviral-experienced HIVhepatitis B virus-infected patients[END_REF]. Coupled with the marginally higher Sp when compared to using undetectable HBV-DNA to predict HBeAg-seroclearance, it is debatable to what extent these markers should be used in routine care [START_REF] Lin | Hepatitis B Virus Genotypes and Variants[END_REF].

Certain limitations of our study must be acknowledged. We only included analysis on HBeAg seroclearance and not HBeAg seroconversion, which could be considered a more desired endpoint of treatment response [START_REF] Lampertico | EASL 2017 Clinical Practice Guidelines on the management of hepatitis B virus infection[END_REF]. Considering that few patients achieved HBsAg seroclearance, our analysis on this endpoint was wholly descriptive. Second, there was substantial heterogeneity of HBcrAg levels observed during follow-up, making interpretability of this marker somewhat difficult. Finally, data on HBV genetic variability were available for patients with HBV-DNA replication, representing a group with more active HBV infection.

In conclusion, we provide a thorough understanding of qHBcrAg and qAnti-HBc kinetics, and their clinical applicability, in the context of HIV-HBV co-infection. Serum levels of qHBcrAg and qAnti-HBc could be useful in predicting HBeAg-seroclearance in HIV-HBV co-infected patients undergoing long-term TDF-containing ART. This is based on the high sensitivity of qHBcrAg and high specificity of qAnti-HBc, when compared to undetectable HBV-DNA, in predicting HBeAgseroclearance. Nevertheless, whether they provide further clinical utility compared to qHBsAg and qHBeAg remains debatable.

Notes

Conflicts of interest statement. C.D. reports receiving grants outside the submitted work from

ViiV and Gilead, and K.L. reports receiving advisory board fees from Gilead. All other authors have no potential conflicts of interest to report.

Funding. This work was supported by SIDACTION (AO 19); and the Agence nationale de recherches sur le Sida et les hépatites virales (ANRS). Gilead Sciences, Inc. provided an unrestricted grant for the French HIV-HBV cohort and was not involved in any part of the data collection, analysis and manuscript writing.

Meetings. These findings have been presented at the Réunion Annuelle AC42 -Réseau

National Hépatites de l'ANRS 2019, in February 2019, held in Paris, France; and, at the EASL-AASLD HBV Endpoint Cure Conference, in March 2019, held in London, England (abstract numbers: P04-01YI and P04-02YI). a The following variables were not considered in multivariable analysis: both models -HBV and HIV-infection duration (unable to assess true infection duration), zone of HBV-prevalence, sex (collinear with other variables), AST < 70 IU/mL (preferred over other transaminase variables) and HIV RNA (no longer below p-value threshold); qHBcrAg -HBeAg status, precore mutation and qHBsAg (collinearity) ; qAnti-HBc -CD4+ nadir >250/mm 3 , concomitant LAM treatment, and precore mutation (no longer below p-value threshold). Decline in markers were determined using a mixed-effect linear model adjusted for baseline levels, body mass index, age, concomitant lamivudine treatment, cumulative treatment duration with lamivudine, HBV-DNA level and CD4 + cells count (if not stratified).

a Slopes were compared between determinant groups, p values derived from a Wald X 2 -test of an interaction term (calculated as the cross-product between duration of tenofovir treatment and presence of determinant), which is included in a mixed-effect linear model with its individual components. In multivariable analysis, qHBeAg and qHBsAg were not further considered as the intent was to study qHBcrAg and qAnti-HBc antibodies. Univariable HR are provided for comparison.

a Model 1: the following variables were no longer below the p-value threshold -age at baseline and CD4 cell count >350/mm 3

Table 1 . Baseline characteristics of patients treated with tenofovir HBeAg status p a

 1

	HBeAg-positive	HBeAg-negative
	(n = 95)	(n = 63)

Table 2 . Baseline determinants of hepatitis B core-related antigen and anti-hepatitis B core antibody

 2

	qHBcrAg (log10 U/mL)

Table 3 . Kinetics of hepatitis core-related antigen and anti-hepatitis B core antibody during follow-up Decline in qHBcrAg log10 U/mL per month a Decline in qAnti-HBc log10 PEIU/mL per month a

 3

			HBeAg+		HBeAg-		HBeAg+		HBeAg-
		<18 months	p for	>18 months	p for		p for		p for		p for
		Δ (95% CI)	intx a	Δ (95% CI)	intx	Δ (95% CI)	intx	Δ (95% CI)	intx	Δ (95% CI)	intx
	Overall	N =				N = 63		N=95		N=63
		-0.051 (-0.062, -0.040)		-0.011 (-0.024, 0.003)		-0.003 (-0.006, -0.001)		-0.011 (-0.013, -0.009)		-0.011 (-0.013, -0.009)
	HBV genotype	N =				N = 22		N=82		N=22
	A	-0.054 (-0.068, -0.040)	0.9	-0.068 (-0.090, -0.045)	0.29	0.001 (-0.006, 0.008)	0.01	-0.014 (-0.016, -0.012)	0.02	-0.005 (-0.011, 0.001)	0.6
	D	-0.024 (-0.064, 0.017)	0.12	-0.038 (-0.054, -0.021)	0.4	-0.018 (-0.029, -0.008)	<0.001	-0.009 (-0.014, -0.003)	0.08	-0.002 (-0.012, 0.008)	0.4
	E	-0.041 (-0.092, 0.011)	0.7	-0.041 (-0.060, -0.019)	0.9	-0.003 (-0.015, 0.008)	0.9	-0.013 (-0.020, -0.005)	0.9	-0.011 (-0.021, -0.002)	0.17
	G	-0.075 (-0.117, -0.034)	0.27	-0.049 (-0.065, -0.033)	0.24	-		-0.013 (-0.018, -0.008)	0.9	-	-
	YMDD mutation b	N =	0.17		0.07	N = 21		N=82	0.6	N=21
	Yes	-0.070 (-0.094, -0.046)		-0.044 (-0.060, -0.028)		-		-0.014 (-0.018, -0.011)		-	-
	No	-0.051 (-0.065, -0.036)		-0.011 (-0.015, -0.007)		-0.004 (-0.010, 0.002)		-0.014 (-0.016, -0.011)		-0.070 (-0.132, -0.008)
	Precore mutation c	N =	0.3		0.9	N = 19	0.009	N=82	0.5	N=19	0.3
	Yes	-0.048 (-0.083, -0.014)		-0.049 (-0.064, -0.033)		-0.007 (-0.014, -0.001)		-0.012 (-0.016, -0.007)		-0.004 (-0.010, 0.002)
	No	-0.053 (-0.067, -0.040)		-0.011 (-0.014, -0.007)		0.005 (-0.003, 0.014)		-0.013 (-0.016, -0.011)		-0.009 (-0.017, -0.001)

HBeAg, hepatitis B e antigen; HBV, hepatitis B virus; PEIU, Paul Ehrlich Institute units; qAnti-HBc, quantified anti-hepatitis B core antibody; qHBcrAg, quantified hepatitis B core-related antigen.

Table 4 -Factors associated to HBeAg-seroclearance during follow-up

 4 , alanine aminotransferase; ART, antiretroviral treatment; BMI, body mass index; HBeAg, hepatitis B e antigen; HBV, hepatitis B virus; HR, hazard ratio; LAM, lamivudine; PEIU, Paul Ehrlich Institute units; qAnti-HBc, quantified anti-hepatitis B core antibody; qHBcrAg, hepatitis B core-related antigen; qHBeAg, quantified hepatitis B e antigen; qHBsAg, quantified hepatitis B surface antigen. Analysis on the 95 HBeAg-positive patients, among whom 26 had HBeAg-loss.

	Risk-Factor

ALT

Table 5 . Quantifiable HBV markers in predicting HBeAg seroclearance Criteria N

 5

				Classification Probabilities a
	M24	M36	M48	M60	M72	M96
	n=78	n=63	n=40	n=34	n=27	n=13
	Se	Sp	Se	Sp	Se	Sp	Se	Sp	Se	Sp	Se	Sp

b Model 2: the following variables were no longer below the p-value threshold -age at baseline, CD4 cell count >350/mm 3 and per 250/mm 3 change from prior visit. c All HR are adjusted for the variables listed in the column.d Time-varying covariate

Acknowledgements. This study has been sponsored by the Institut de Médecine et d'Epidémiologie Appliquée (IMEA). L.D. was awarded a post-doctoral fellowship from the Agence nationale de recherches sur le Sida et les hépatites virales (ANRS).

Corresponding author. Dr a As sensitivity (Se) and specificity (Sp) are calculated from survival probabilities, all patients (N) are included in analysis.

b HBV-was defined as <60 IU/mL For information, the number of patients (n) considered (i.e. those not lost to follow-up or having HBeAg seroclearance) during each time-interval has been included. † † Not evaluated 1

Supplementary Table 1. Baseline characteristics of patients with and without