Alexis Casanova
email: acasanova@amplitudesystemes.com

Quentin D'acremont

Giorgio Santarelli

Stefan Dilhaire

Antoine Courjaud

Ultrafast amplifier additive timing jitter characterization and control

We report on the characterization and long-term compensation of additive timing jitter introduced by a femtosecond ytterbium regenerative amplifier with a 100 kHz repetition rate. A balanced optical cross-correlation technique is used to generate a jitter error signal. This approach is well suited to characterize the additive timing jitter of Yb amplifiers seeded by narrow spectrum Yb oscillators. The balanced optical cross-correlator is in a noncollinear configuration allowing a background free coindence detection. This setup enables the measurement of additive timing jitter from the amplifier, with a noise floor of 300 as integrated from 10 Hz to 10 kHz. The measured additive timing jitter level is about 5 fs, integrated from 0.1 Hz to 10 kHz. The amplifier timing drift characterization and control are performed for more than an hour. Ultrafast amplifiers seeded by femtosecond oscillators are widely used in scientific experiments requiring high energy laser pulses such as free electron lasers (FELs), petawatt lasers, high harmon ics generation, plasma physics, terahertz (THz) generation, and optical parametric amplifiers (OPA) applications [1 4]. These experiments aim at reaching wavelengths ranging from x rays to THz waves by converting NIR laser pulses in highly nonlinear media. The input pulse energy required ranges from 100 μJ to 100 mJ to overcome low efficiency conversion processes. Despite significant improvements of other laser architectures in the last few years, especially concerning diode pumped solid state lasers [5,6], the regenerative amplifier remains the most suitable approach to deliver multi millijoule output energy.

However, the inherent principle of regenerative amplifiers implies a pulse buildup time on the order of a few microseconds which is equivalent to an optical path length of hundreds of meters. Thus, the regenerative amplification architecture is potentially sensitive to external perturbations such as mechanical vibrations, temperature fluctuations, and pressure variations. This becomes critical when the laser pulses must be synchron ized at a sub picosecond precision with other events, for in stance, in a synchrotron. In such conditions, the regenerative amplifier may become a dominant source of timing jitter in the synchronization budget.

Beyond particle accelerators, timing jitter is a crucial param eter for several applications. For example, femtosecond OPA efficiency depends on the temporal overlap of pump and seed pulses in a nonlinear crystal, the pump pulse being often pro vided by a regenerative amplifier to reach high output energies [3,7]. Relative pump to seed timing fluctuations result in amplitude variations of the OPA output. These thematics can be extended to a general issue concerning drift from free space optical links [8,9].

Several approaches have been pursued to reduce the timing jitter and drift from a laser amplifier. One method consists of using sum frequency generation (SFG) or an OPA to convert the seed pulse arrival time into a wavelength information [10,11]. Another method is to use an OPA technique instead of second harmonic generation to build a balanced optical cross correlator (BOC) [7]. To be effective, both methods require large seed spectra and pump pulse energies of several microjoules. Moreover, these setups are built for specific experi ments; therefore, it may be difficult or impractical to adapt them to a simpler seeder to amplifier configuration.

Our scope is to investigate short and long term timing jitter from an ytterbium regenerative amplifier with a 100 kHz rep etition rate, seeded by an ytterbium oscillator. We evaluate the excess timing jitter introduced by the amplifier by comparing its output timing with the seeding oscillator signal, using a balanced optical cross correlator [12]. This scheme gives a noise floor as low as 300 as integrated from 10 Hz to 10 kHz and a measured timing jitter of 5 fs, integrated from 0.1 Hz to 10 kHz. The stabilization of the amplifier repetition rate for more than an hour is also demonstrated.

The experimental setup for amplifier stabilization and characterization is described in Fig. 1. A commercial S pulse Ytterbium regenerative amplifier is seeded by a home built ytterbium oscillator with an average output power of 1 W at a 50 MHz repetition rate and a pulse duration of 250 fs. The integrated timing jitter of the oscillator is <15 fs from 1 kHz to 10 MHz. The regenerative amplifier yields an average output power of 1 W at a 100 kHz repetition rate and a pulse duration of 700 fs. The amplifier buildup time is 1 μs which corresponds to a free space travel distance of 300 m. The amplifier output beam travels across a tunable optical delay line (ODL). About 1% of the amplifier output power is injected into the balanced cross correlator. The tunable ODL range is 60 mm, which is equivalent to a timing compensation of about 200 ps. 30% of the oscillator output power is dedicated to seed the amplifier. The remaining 70% is sent to the BOC.

The balanced optical cross correlator (BOC) seen in Fig. 1 is based on a noncollinear SFG configuration similar to refer ence [13]. This approach allows one to spatially separate both the oscillator and amplifier beams and their undesired frequency doubled pulse trains from the cross correlation signal. The amplifier large beam diameter facilitates the spatial overlap of the beams in the noncollinear configuration. A 50:50 beam split ter and two 100 μm glass plates are included to introduce a time delay and allow the balanced detection operation. The SFG signals are then detected by two GaP photodetectors. The photo diode signal difference and filtering operation are made by custom low noise electronics.

The tunable ODL travel range allows us to evaluate the BOC transfer function over a span of 10 ps and to measure its slope in volts per femtosecond. To perform long and short term timing jitter measurement, we first acquire the BOC output signal with a data logger. The signal is then processed with a computer which returns a control signal to the transla tion stage, with a control bandwidth of about 1 Hz. Oscillator repetition rate fluctuations are converted in timing jitter and drift due to the large unbalance between the optical paths before the BOC. To limit this parasitic effect, the oscillator repetition rate is phase locked with the tenth harmonic detected by a fast photodiode to a stable 500 MHz radio frequency oscillator with a 1 kHz bandwidth. This allows us to set the optical path length between the oscillator and the BOC as the remainder of the division of the amplifier total path length by the oscillator path length.

Figure 2(a) shows the power spectral density (PSD) of the additive timing jitter introduced by the amplifier when the syn chronization is on. The PSD is measured at the BOC output from 10 Hz to 10 kHz with a fast Fourier transform (FFT) analyzer and with a data logger (NI USB 6210) to access the frequencies from 0.1 Hz to 10 Hz. The additive noise from the amplifier is below the typical jitter level of our oscillator. Figure 2(b) is the timing noise floor of the BOC. We evaluate the noise floor of our system by replacing the oscillator optical pulse train with an attenuated version of the amplifier output. The noise floor PSD we obtain is well below the measured additive timing jitter. This noise floor may be dominated by residual AM to PM conversion and/or mechanical instabilities of the BOC setup. The integrated timing jitter (0.1 Hz to 10 kHz) is also plotted in Fig. 2(c). The amplifier additive integrated jitter amounts to about 5 fs RMS from 0.1 Hz to 10 kHz. This value is dominated by acoustic like peaks in the kHz region. We attribute these peaks to mechanical reso nances in the laser amplifier enclosure. The integrated timing jitter measurement floor seen in Fig. 2(c) is about 300 as RMS from 10 Hz to 10 kHz. These values are consistent with previous published results for utrafast regenerative amplifiers [7,10,13]. The electronic noise floor is measured to be 10 -7 fs 2 ∕Hz, which corresponds to an integrated noise floor of 40 as RMS from 10 Hz to 10 kHz.

It is worth noting that the system has not been optimized for vibration sensitivity. One possibility to improve the design would be to change the delay line architecture between the oscillator and the BOC with a configuration that is less sensitive to mechanical perturbations. The BOC common mode rejec tion ratio has to be investigated as well. The SFG intensity can be improved by pulse picking and amplifying the seed oscillator prior to beam splitting and regenerative amplification. As a consequence, we can expect an even better resolution of our system over the observed bandwidth if needed. Note that our approach allows low measurement floor, even with a narrow spectrum seed oscillator.

Figure 3 shows long term measurement of the in loop error signal of (a) the closed loop system, (b) the ODL control signal, and (c) the measured temperature inside the laser enclosure. Note that the laser enclosure is temperature stabilized. We acquire the in loop error signal with a data logger, with a sampling rate f s 100 Hz. We can observe RMS and peak to peak fluctuations of 10 and 60 fs over 2 h correspondingly. The system has been routinely locked for more than 10 h with equivalent results. Figure 3(b) exhibits the ODL control signal with a peak to peak variation of about 1.2 ps over 1 h, which is larger than the amplifier pulse duration. Figure 3(c) shows the amplifier baseplate temperature variation which is strongly correlated to the ODL control signal. The sensitivity of the amplifier to the temperature is about 20 ps/K, which is very close to the aluminum baseplate linear thermal expansion co efficient. The long term timing fluctuations from the amplifier are dominated by residual temperature fluctuations. The addi tive amplifier short term timing jitter between 0.1 Hz and 10 kHz is below the jitter from the seeding laser oscillator, phase locked on a very low phase noise radio frequency refer ence. However, for a demanding experiment where long term synchronization is required, the additive amplifier timing drift still requires active control. Considering these results, we expect the additive timing fluctuations from the amplifier to increase with the total number of round trips inside the regenerative cavity, as the amplifier would become more and more sensitive to temperature and mechanical vibrations.

To summarize, we demonstrated a new BOC configuration allowing the characterization and control of short and long term timing fluctuations of an ytterbium regenerative amplifier. Our setup achieved a resolution of 300, as integrated from 10 Hz to 10 kHz, well below the integrated amplifier additive timing jitter of 5 fs (from 0.1 Hz to 10 kHz). The main sources of the amplifier additive timing jitter were mechanical vibra tions for short term timing jitter (10 Hz to 10 kHz) and the amplifier baseplate temperature fluctuations for the long term drift. Thus, this system is ready to be used in high precision synchronized experiments, such as swissFEL. In principle, this detection method enables timing jitter measurement up to Nyquist frequency. Therefore, this approach may be applied to any type of ultrafast amplifier system. In the future, we plan to use this setup for the characterization and control of a ytterbium regenerative amplifier with higher output energy and longer optical path length. We also intend to apply the same approach to ultrafast ytterbium fiber amplifiers to investigate their timing fluctuations.

  OCIS codes: (140.0140) Lasers and laser optics; (120.0120) Instrumentation, measurement, and metrology; (320.0320) Ultrafast optics.

Fig. 1 .

 1 Fig. 1. Experimental setup for the characterization and stabilization of the regenerative amplifier repetition rate, with a buildup time of 1 μs, corresponding to a 300 m optical path length. BOC, balanced optical cross correlator; GP, glass plate; HWP, half wave plate; LBO, lithium triborate; M, mirror; PBS, polarizing beam splitter; PhD, photodetector.

Fig. 2 .

 2 Fig. 2. (a) Excess timing jitter spectral density from the regenerative amplifier with synchronization. (b) Measurement noise floor acquired by performing the amplifier pulses auto correlation with the BOC. (c) Corresponding integrated timing jitters.

Fig. 3 .

 3 Fig. 3. Amplifier timing drift measurement over 100 min. (a) Timing error signal. (b) Tunable ODL control signal. (c) Regenerative amplifier baseplate temperature.

Acknowledgment. The authors want to thank Albert Romann and Marta Divall for fruitful discussions. They also acknowledge Sébastien Courtois, Benoît Trophème, and Damien Sangla for their support.

Funding. French National Research and Technology Agency (ANRT) (2013/1184, 2013/1196); French National Research Agency (ANR) (ANR 10 NANO 0014).


Caroline Collange
email: caroline.collange@inria.fr

Mioara Joldes
email: joldes@laas.fr

Jean-Michel Muller
email: jean-michel.muller@ens-lyon.fr

Valentina Popescu
email: valentina.popescu@ens-lyon.fr

Parallel floating-point expansions

Keywords: floating-point arithmetic, floating-point expansions, high precision arithmetic, multiple-precision arithmetic, graphics processing unit, parallel computations

published or not. The documents may come

I. INTRODUCTION

In numerical computing, we sometimes encounter calculations that require more precision than the one offered by current processors. A way of handling such higher-precision calculations is to represent real numbers by floating-point (FP) expansions, i.e., by unevaluated sums of FP numbers. Several slightly different definitions of what a FP expansion is have been introduced in the literature, together with different arithmetic algorithms for manipulating them [START_REF] Priest | Algorithms for arbitrary precision floating point arithmetic[END_REF], [START_REF] Shewchuk | Adaptive precision floating-point arithmetic and fast robust geometric predicates[END_REF], [START_REF] Hida | Algorithms for quad-double precision floating-point arithmetic[END_REF]. Choosing between these algorithms frequently depends on a compromise between accuracy, speed, and safety (because some of the more complex algorithms are just heuristic: they do not come with a proof). With Graphics Processing Units (GPUs) oriented implementation in mind, other important aspects are parallelism and locality. Should we try to parallelize the arithmetic algorithms that manipulate FP expansions (or, rather, build variants that are suitable for parallelization), or should we keep them sequential and try to parallelize at a higher level? In a previous study [START_REF] Joldes | Searching for sinks for the hénon map using a multipleprecision gpu arithmetic library[END_REF], some of us have dealt with an "embarrassingly parallel" problem with compact intermediate data. However, many applications do not provide as much parallelism. Even for those that do, locality can be a problem. Increasing the precision of sequential arithmetic operations requires a corresponding increase in the amount of intermediate data to keep. Thus, parallel arithmetic algorithms are attractive not just by the extra parallelism they provide, but also by the locality improvements they enable. Here, with different numerical problems in mind, we parallelize the addition and multiplication of FP expansions on GPUs.

Section II recalls the basic notions related to FP expansions. In Sections III and IV we present data-parallel algorithms for addition and multiplication, respectively, of FP expansions. The implementation details are given in Section V followed by some performance assessment in Section VI. We finish by concluding our work in Section VII.

II. FLOATING-POINT EXPANSIONS

A normal binary precision-p floating-point (FP) number is a number of the form

x = M x • 2 ex-p+1 , with 2 p-1 ≤ |M x | ≤ 2 p -1.
The integer e x is called the exponent of x, and M x • 2 -p+1 is called the significand of x. Using Golberg's definition we denote ulp(x) = 2 ex-p+1 [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]Chap. 2].

A natural generalization of the notion of double-double or quad-double is the notion of floating-point expansion.

Definition II.1. A FP expansion u with n terms is the unevaluated sum of n FP numbers u 0 , u 1 , . . . , u n-1 , in which all nonzero terms are ordered by magnitude (i.e., if v is the sequence obtained by removing all zeros in the sequence u, and if sequence v contains m terms, |v i | ≥ |v i+1 |, for all 0 ≤ i < m -1).

Arithmetics on FP expansions have been introduced by

Priest [START_REF] Priest | Algorithms for arbitrary precision floating point arithmetic[END_REF], and later on by Shewchuk [START_REF] Shewchuk | Adaptive precision floating-point arithmetic and fast robust geometric predicates[END_REF].

To make sure that such an expansion carries significantly more information than only one FP number, it is required that the u i 's do not "overlap". The notion of (non-)overlapping varies depending on the authors. Intuitively, the stronger the sense of the nonoverlapping definition, the more difficult it is to obtain, implying extra manipulation of the FP expansions. In what follows we give the definition that we chose to use, referred to as ulp-nonoverlapping, in an attempt to allow for a more relaxed manipulation of the FP expansions. We specify first that even if a FP expansion may contain interleaving zeros, the definition that follows applies only to the non-zero terms of the expansion (i.e., the array v in Definition II.1).

Definition II.2. An expansion u

0 , u 1 , . . . , u n-1 is ulp- nonoverlapping if for all 0 < i < n, we have |u i | ≤ ulp(u i-1 ).
In other words, the components are either P-nonoverlapping (that is, nonoverlapping according to Priest's definition [START_REF] Priest | On properties of floating-point arithmetics: Numerical stability and the cost of accurate computations[END_REF]) or they overlap by one bit, in which case the second component is a power of two.

Depending on the nonoverlapping type of an expansion, when using standard FP formats for representation, the exponent range forces a constraint on the number of terms. The largest expansion can be obtained when the largest term is close to overflow and the smallest is close to underflow. We remark that, when using ulp-nonoverlapping expansions, for the two most common FP formats, the constraints are:

• for double-precision (exponent range [-1022, 1023]) the maximum expansion size is 39; • for single-precision (exponent range [-126, 127]) the maximum is 12.

In this article we deal with so called parallel FP expansions, i.e., the expansion is stored on parallel execution threads, with one term/thread. This implies that the user has to launch as many threads as the expansion size.

The majority of algorithms performing arithmetic operations on expansions are based on the so-called error-free transforms (EFT) (such as the algorithms 2Sum, Fast2Sum, Dekker's product and 2MultFMA presented for instance in [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]), that make it possible to compute both the result and the rounding error of a FP addition or multiplication. This implies that in general, each such EFT applied to two FP numbers, also returns two FP numbers. More precisely, the sum of two FP numbers can be represented exactly as a FP number which is the correct rounding of the sum, plus a second FP number corresponding to the rounding error. Under certain assumptions, this decomposition can be computed at a very low cost by a simple sequence of standard precision FP operations. For instance, assuming that a and b are two FP numbers, a simple algorithm (called 2Sum and due to Knuth [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]) computes S and E, the decomposition of a + b using only 6 FP operations. Similarly, if a FMA operator1 is available, 2ProdFMA [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF] returns P and the error E (namely ab-P ) in 2 FP operations.

Algorithms like these can be extended to be used with arbitrary precision computations by chaining, resulting into the so-called distillation algorithms [START_REF] Rump | Accurate floating-point summation part I: Faithful rounding[END_REF], for summing several FP numbers. From these we make use of the VecSum algorithm [START_REF] Shewchuk | Adaptive precision floating-point arithmetic and fast robust geometric predicates[END_REF], [START_REF] Ogita | Accurate sum and dot product[END_REF], which is simply a chain of 2Sum that performs an EFT on n FP numbers.

A potential problem appears when subsequent computations are done using these results; the size of the exact result is going to increase more and more. To avoid this, some "truncation" methods (both on-the-fly or a-posteriori) may be used to compute only an approximation of the exact result. Also, so-called (re-)normalization algorithms are used to render the result non-overlapping, which implies also a potential reduction in the number of components. III. DATA-PARALLEL ADDITION ALGORITHM FOR FP EXPANSIONS An algorithm that performs addition of two FP expansions x and y with n and m terms, respectively, will return a FP expansion with at most n + m terms as the exact result. This implies a continuous increase in the number of terms as subsequent computations are done using the obtained result. This is why, in practice, the results are "truncated" to obtain only an approximation of x + y.

Many variants of algorithms that compute the sum of two FP expansions have been presented in the literature [START_REF] Priest | Algorithms for arbitrary precision floating point arithmetic[END_REF], [START_REF] Shewchuk | Adaptive precision floating-point arithmetic and fast robust geometric predicates[END_REF], [START_REF] Hida | Algorithms for quad-double precision floating-point arithmetic[END_REF], [START_REF] Rump | Accurate floating-point summation part I: Faithful rounding[END_REF], each using different methods and having different complexities, but, from our knowledge, none of these algorithms are implemented in parallel and, even more, some of them are highly sequential, making them unsuitable for parallel architectures.

In what follows we will present a safe data-parallel algorithm for adding FP expansions that offers a tight error bound on the result and it allows to prove a clear constraint between the terms of the result. We also present a fast version with more relaxed error bounds based on the same scheme that may be used if computations are not cancellation-prone.

The safe data-parallel summation algorithm is presented in Fig. 1 and Algorithm 1. All arithmetic operations including EFTs like 2Sum are performed in parallel element-wise on R-element vectors. We assume vectors can be merged and elements inside a vector can be shuffled. These assumptions make the algorithms applicable to most SIMD units, including Intel SSE/AVX instruction set extensions [START_REF] Firasta | Intel AVX: New frontiers in performance improvements and energy efficiency[END_REF] and recent Nvidia GPUs [START_REF]Kepler GK110 architecture[END_REF].

Algorithm 1 Safe data-parallel algorithm for adding FP expansions.

Input: FP expansion vectors x = (x 0 , x 1 , . . . , x R-1 ) and y = (y 0 , y 1 , . . . , y R-1 ). Output: FP expansion vector s = (s 0 , s 1 , . . . , s R-1 ).

1: a ← (x 0 , 0, 0, . . . , 0) 2: b ← (y 0 , 0, 0, . . . , 0)

3: (s, e) ← 2Sum(a, b) 4: for i ← 1 to R do 5: e ← (x i , e 0 , e 1 , . . . , e R-2 ) //Shift right & insert xi 6:
(s, e) ← 2Sum(s, e )

7:

e ← (y i , e 0 , e 1 , . . . , e R-2 ) //Shift right & insert yi 8:

(s, e) ← 2Sum(s, e ) 9: end for 10: for i ← 1 to R -2 do (s, e) ← 2Sum(s, e ) 13: end for 14: e ← (0, e 0 , e 1 , . . . , e R-2 )//Shift right 15: s ← s + e 16: return s.

For the sake of simplicity, we only present here the "input-R-output-R" version of the algorithm, even though the generalized version allows for different input sizes. The algorithm is based on a pipelined error propagation. We start by adding the first elements of each expansions, x 0 and y 0 on the first vector component. We continue to add the rest of the elements on the first component one by one and propagating the error upwards, to the other vector components. When we run out of elements to add we continue to propagate the errors for another R -1 steps by injecting 0s on the first component. In the last step of the algorithm there is no need to use EFT, since we are not going to propagate the errors anymore; this is why we use only simple addition.

By using this scheme to add the two expansions we ensure that the most significant term of the output, s 0 , is the sum of the inputs rounded to nearest. Moreover, the terms of the output are arranged in terms of magnitude in decreasing order, with some constraints. Let us prove this in what follows.

It is easily seen that the parallel scheme presented in Fig. 1 can be reduced to the sequential one presented in Fig. 2.

If there is a "Sterbenz relation" between x 0 and y 0 (i.e., if they are of opposite signs and x0 2 ≤ |y 0 | ≤ |2x 0 |) then e 01 = 0 and s 01 = x 0 + y 0 . This implies that s 12 = e 02 and e 12 = 0, and so on. In this case we end up propagating a 0, to the end of the result expansion, and we are left with the same scheme as we began with (illustrated in Fig. 3). This means that we can eliminate the case in which we have a "Sterbenz relation". Proposition III.1. Let us assume a = a 0 , a 1 , . . . a n-1 , an array of FP numbers that satisfy |a i | ≤ 2 -i(p-1)+δ |a 0 |, for some (presumably small) integer δ and |t| ≤ 2 -p+ |a 0 |, for some (presumably small) integer . If s = s 0 , s 1 , . . . , s n is the sum obtained by adding t to a as shown in Fig. 4 (from left to right, propagating the error), then all the terms in s satisfy:

|s i | ≤ 2 -i(p-1)+δ+1 , for all 0 < i ≤ n.
Proof: From the proof of the algorithm 2Sum we know that

|e i | ≤ 1 2 ulp(s i ) = 2 -p |s i |. From |s 0 |(1 -2 -p ) ≤ |a 0 + t| ≤ |s 0 |(1 + 2 -p ) and |a 0 |(1 -2 -p+ ) ≤ |a 0 + t| ≤ |a 0 |(1 + 2 -p+ ), we get |s 0 | 1 -2 -p 1 + 2 -p+ ≤ |a 0 | ≤ |s 0 | 1 + 2 -p 1 -2 -p+ . (1)
It follows that

|a 1 | ≤ 2 -(p-1)+δ |a 0 | ≤ 2 -(p-1)+δ • 1 + 2 -p 1 -2 -p+ • |s 0 |. This gives |e 0 + a 1 | ≤ 2 -p 1 + 2 δ+1 (1 + 2 -p ) 1 -2 -p+ |s 0 | .
From which we deduce

|s 1 | ≤ 2 -p (1 + 2 -p ) 1 + 2 δ+1 (1 + 2 -p ) 1 -2 -p+ |s 0 | . (2)
We

|s i | < 2 -ip θ i |s 0 |, (3)
with

θ i = (1 + 2 -p ) i + 1 1 -2 -p+ i j=1 2 j+δ (1 + 2 -p ) i-j+2 . (4)
One easily finds

θ i = (1 + 2 -p ) i + 2 δ+1 (1 + 2 -p ) 2 1 -2 -p+ 2 i -(1 + 2 -p ) i 1 -2 -p , hence, θ i = 2 i+δ+1 H i ,
with

H i = (1 + 2 -p ) i 2 i+δ+1 + (1 + 2 -p ) 2 1 -2 -p+ 1 -(1+2 -p ) i 2 i 1 -2 -p .
Denote u = 2 -p . In all practical cases ≥ 2 and δ ≥ 0, so that H i ≤ G i , with

G i = 1 2 1 + u 2 i + (1 + u) 2 1 -4u 1 -1+u 2 i 1 -u .
We have,

G i = 1 -1 2 1+u 2 i + u(u+6) 1-4u - -1 2 1+u 2 i • 2u(7-3u) (1-4u)(1-u)
The only positive term (after the initial "1") in that sum is u(u+6) 1-4u , which is less than 7u for all pertinent values of u = 2 -p . Hence G i < 1 as soon as 2 i+1 ≤ 2 p /7, which occurs in all practical cases. This gives

|s i | < 2 -i(p-1)+δ |s 0 |, with δ = δ + 1.
This concludes our proof. Hence, in the array of Fig. 1, δ is increased by 1 at each line. For instance, if we add two order-n expansions (i.e., we use 2n -1 lines in the Array of Fig. 1, then the terms s 0 , s 1 , . . . of the result satisfy

|s i | ≤ 2 -(p-1)i+2n-1 |s 0 |,
and the expansion obtained by keeping the first n terms only represents the sum with an error less than

(2 -np+3n-1 + 2 -(n+1)p+3n + 2 -(n+2)p+3n+1 + • • • ) • |a 0 |,
i.e., with a relative error bounded by a value slightly larger than 2 -np+3n-1 .

We would like to also mention here a faster, relaxed version of the above algorithm, that requires at most R -1 steps in order to compute the result (the last step using only simple addition). This algorithm (Fig. 5) offers a worse error bound and it does not ensure the correct result when cancellation occurs, if no re-normalization algorithm is applied on the result. It performs best when FP expansions of the same sign and close magnitudes are added. We recommend using it for fast computations that can be validated a-posteriori.

IV. DATA-PARALLEL MULTIPLICATION ALGORITHM FOR FP EXPANSIONS

In general, an algorithm that performs the multiplication of two expansions x and y with n and m terms, respectively, will return a FP expansion with at most 2nm terms [START_REF] Priest | Algorithms for arbitrary precision floating point arithmetic[END_REF], which, as in the case of addition, implies an increase in the number of terms.

The algorithm that we present (Algorithm 2) computes an approximation of x • y, where x and y are two parallel expansion. Here we also present just the "input-R-output-R" version.

We consider two parallel FP expansions x and y, each with R terms and we compute the R most significant FP components of the product π = x • y. We use the following intuition: let ε = 1 2 ulp(π 0 ), then roughly speaking, if π is of of O(Λ), then e 0 is of order O(εΛ). This means that for each product (p, e) = 2MultFMA(x i , y j ), p is of order O(ε i+j Λ) and e of order O(ε i+j+1 Λ). We truncate the result on-the-fly, by considering only the terms for which 0 ≤ i + j ≤ R -1, since the smaller terms have an order of magnitude much smaller and usually they will not influence the result.

Algorithm 2 Data-parallel algorithm for multiplying FP expansions.

Input: FP expansion vectors x = (x 0 , x 1 , . . . , x R-1 ) and y = (y 0 , y 1 , . . . , y R-1 ). Output: FP expansion vector π = (π 0 , π 1 , . . . , π R-1 ).

1: s ← (0, . . . , 0) 2: π ← (0, . . . , 0)

3: for i ← 0 to R -2 do 4:
y ← (y i , y i , . . . , y i ) //Broadcast 5:

(p, e) ← 2MultFMA(x, y )

(s, e ) ← 2Sum(s, p)

7: while e = 0 do 10:

π i ← s 0 //Insert
(s, e) ← 2Sum(s, e)

11:

e ← (0, e 0 , e 1 , . . . , e R-2 ) //Shift right 12:

end while 13:

while e = 0 do 14:

(s, e ) ← 2Sum(s, e )

15:

e ← (0, e 0 , e 1 , . . . , e R-2 ) //Shift right 16:

end while 17: end for 18: p ← x • y 19: s ← s + p 20: π R-1 ← s 0 //Insert into vector 21: return π.

The multiplication algorithm runs as follows: at each iteration i of the for loop (lines 3-16) we compute p+e = x•y; we add p to the result of the same order, using an EFT, which also generates an error, e . After that, using the two while loops (lines 8 -11 and 12 -15) we propagate the two generated errors, e and e to the lower order results. In the last step of the algorithm, we do not use any EFT, because the errors that are supposed to be computed are going to be of order O(ε R Λ), and we do not need to propagate them anymore. This algorithm has the same behavior as the sequential algorithm presented in Fig. 6, but a graphical representation of the parallel one would be too difficult to read.

While the exact product xy of two FP expansions x and y is computed as

R-1;R-1 i=0;j=0 x i y j = 2R-2 k=0 i+j=k
x i y j , in this algorithm we "truncate" it on-the-fly by computing and adding only the relevant part of the scalar products (the first R k=1 k individual products) and after that outputting the R most significant components in the result. In what follows we compute the error bound in two steps: first we compute the error generated by "truncating" the partial products, and second we compute the error given by the discarded errors.

Proposition IV.1. (Error bound on the truncated products) Let x and y be two ulp-nonoverlapping FP expansions, with R terms. If, when computing the product xy we "truncate" the operations by computing and adding only the most significant individual products, the first R k=1 k products, then the maximum error that we can obtain is:

xy - R-1 k=0 i+j=k x i y j = 2R-2 k=R i+j=k x i y j ≤ |x 0 y 0 | 2 -(p-1)R R-1 1-2 -(p-1)
.

Proof: The maximum error given by the discarded products satisfies:

2R-2 k=R i+j=k x i y j ≤ 2R-2 k=R i+j=k 2 -p(i+j)+i+j |x 0 y 0 | ; ≤ |x 0 y 0 | 2R-2 k=R (2R -1 -k)2 -(p-1)k ; ≤ |x 0 y 0 | 2 -(p-1)R R-2 k =0 (R -1 -k )2 -(p-1)k . We now consider the function φ(α) = ∞ k=0 (R -1 -k)α k
and we get:

φ(α) = ∞ k=0 -kα k + ∞ k=0 (R -1)α k ; = -α d dα ∞ k=1 α k + (R -1) 1 1 -α ; = -α (1 -α) 2 + R -1 1 -α .
Using φ(2 -(p-1) ), we obtain: 1) , and since -2 -(p-1)

2R-2 k=R i+j=k x i y j ≤ |x 0 y 0 | 2 -(p-1)R • • -2 -(p-1) (1 -2 -(p-1) ) 2 + R -1 1 -2 -(p-
(1-2 -(p-1) ) 2 is negative and very small we conclude our proof by:

2R-2 k=R i+j=k x i y j ≤ |x 0 y 0 | 2 -(p-1)R R -1 1 -2 -(p-1) . ( 5
)
Proposition IV.2. (Final error bound) Let x and y, two ulpnonoverlapping FP expansions, with R terms. When computing π, an approximation of xy to R terms, as shown in Algorithm 2 and Fig. 6, the result satisfies:

|xy -π| ≤ |x 0 y 0 |2 -(p-1)R (R -1) • 1 + 2 p-2 (1 + 2 -p ) + +(R 3 -R)((R -1)!) 2 . (6)
Proof: From the definition we know that

|x i | ≤ 2 -i(p-1) |x 0 | and |y j | ≤ 2 -j(p-1) |y 0 |, so we can deduce |x i y j | ≤ 2 -(p-1)(i+j) |x 0 y 0 | .
For computing π 0 we use only 2MultFMA(x 0 , y 0 ), and we get |π 0 | ≤ |x 0 y 0 | (1 + 2 -p ).

For computing π 1 we use VecSum#1, with 3 entries of order O(εΛ): the error from the previous step, and two partial products, which are less than 2 -(p-1) |x 0 y 0 | (1 + 2 -p ). It is easily seen that all these entries are bounded by the same value. We define the following notation:

Ω 1 = 2 -(p-1) |x 0 y 0 | (1 + 2 -p ). (7)
It follows that 3 and the outputted errors are less than

π 1 < 3 • 2 -(p-1) |x 0 y 0 | (1 + 2 -p )
Ω 2 = 3•2 -2p+1 |x 0 y 0 | (1+2 -p ) 3 .
For computing π 2 we use VecSum#2 which is going to have 7 entries of order O(ε 2 Λ): 2 errors outputted by VecSum#1, bounded by Ω 2 ; 2 errors from the previous step's partial products, which are less than 2 -2p+1 |x 0 y 0 | (1 + 2 -p ); and 3 partial products, less than 2 -2(p-1) |x 0 y 0 | (1 + 2 -p ). We observe once more that all the entries are less than Ω 2 .

For the induction step we consider VecSum#i -1. For computing π i-1 we have (i-1) 2 +i entries, which we assume are all less than Ω i-1 . It follows:

π i-1 < (2Ω i-1 (1 + 2 -p ) + Ω i-1 )(1 + 2 -p ) + . . . < (i 2 -i + 1)Ω i-1 (1 + 2 -p ) i 2 -i
and also, the largest error term outputted and implicitly all the others are less than 1 2 ulp(π i-1 ).

This implies that all error terms are less than:

= (i 2 -i + 1)Ω i-1 (1 + 2 -p ) i 2 -i • 2 -p ; = 2 -(p-1) |x 0 y 0 | (1 + 2 -p ) • • i n=1 (n 2 -n + 1)(1 + 2 -p ) n 2 -n 2 -p ; = 2 -(p-1)-ip |x 0 y 0 | <2 (1 + 2 -p ) 1+2+...+(i 2 -i) • i n=1 (n 2 -n + 1); < 2 -(i+1)p+2 |x 0 y 0 | (i!) 2 . ( 8
)
and this last value will define Ω i . This implies that, since we use only simple summation for computing π R-1 , in the last step we neglect R 2 + R terms, all less than Ω R .

We also have to account for the errors that occur when computing the last partial products using only simple multiplication. This means R -1 terms less than

2 -R(p-1)+p-2 |x 0 y 0 | (1 + 2 -p )
When adding all these errors with the one given in ( 5) we get the following bound:

|xy -π| ≤ |x 0 y 0 |2 -(p-1)R (R -1) • 1 1 -2 -(p-1) <1 +2 p-2 (1 + 2 -p ) + + 2 -p-R+2 <1 (R 2 + R)(R -1)((R -1)!) 2 ≤ |x 0 y 0 |2 -(p-1)R (R -1) • 1 + 2 p-2 (1 + 2 -p ) + +(R 3 -R)((R -1)!) 2 . (9)

And this concludes our proof

Unfortunately, for the multiplication algorithm we are unable to prove any constraints on the terms of the result. Even though cancellation cannot happen when multiplying two FP numbers, it may happen during the summation process, in which case we can get |π i | < |π j |, with i < j. If this happens we can apply a re-normalization algorithm, like the one presented in [START_REF] Joldes | Arithmetic algorithms for extended precision using floating-point expansions[END_REF], in order to render the result nonoverlapping. Since this implies adding a sequential step at the end of Algorithm 2, slowing it's performance, we recommend using this algorithm only if computations are known not to be cancellation-prone or if the result can be verified a-posteriori.

V. WARP-SYNCHRONOUS GPU IMPLEMENTATION GPUs are highly multi-threaded SIMD architectures [START_REF] Garland | Understanding throughput-oriented architectures[END_REF]. They are programmed in languages like CUDA and OpenCL, that expose a pure multi-thread programming model. Programmers describe compute kernels as a single program run by many fine-grained threads. The compiler and hardware group these threads into so-called warps, containing 32 threads on current Nvidia architectures. Threads inside a warp run in lockstep and share a single control flow, and their instructions are executed on SIMD units, with one thread per lane.

This implicit SIMD model is equivalent to explicit SIMD: a GPU program can be also understood as computations on vectors from the point of view of a warp. This enables direct implementation of the data-parallel algorithms we propose. Recent additions to the available hardware primitives make this warp-synchronous programming style particularly efficient [START_REF]Kepler GK110 architecture[END_REF]. Our implementation targets GPUs with compute capability 3.0 or above, such as Kepler and Maxwell architectures, that support warp vote and shuffle instructions. The code was written in CUDA C, using double-precision numbers (i.e. p = 53).

Warp vote instructions perform boolean reductions across all threads within a warp. For instance, they can check whether a condition holds for all the threads, or for any of the threads of the warp. The __any function computes the logical OR of a warp-sized vector of predicates and broadcast it to all elements. We use it to implement the loop exit conditions of the FP expansion multiplication in Algorithm 2.

Warp shuffle instructions allow arbitrary communication between threads in a warp, without having to go through memory. They are analogous to shuffle or permute instruction in explicit SIMD instruction sets [START_REF] Firasta | Intel AVX: New frontiers in performance improvements and energy efficiency[END_REF]. We use them to shift vector components to propagate the errors across expansion terms, and to insert and extract scalar values inside vectors.

• shfl_up(x, n, R) and shfl_down shift components respectively upward or downward by n positions within each R-element vector; • shfl reads an arbitrary vector component within each vector lane. As mentioned in Section III, similar shuffle and permute operations are available in explicit SIMD instruction sets such as Intel AVX. GPUs that do not support the shuffle instruction can also exchange data through the OpenCL local memory at a cost in performance.

We illustrate the warp-synchronous implementation of Algorithm 2 for FP expansion multiplication in Fig. 7. The code appears from a single thread's perspective, but it runs in parallel and it takes decisions based on the vector lane within an expansion (i.e. threadIdx.x). Although the hardware only support shuffling 32-bit data, we implemented shuffle instructions on the double type by shuffling each half separately.

Although we present here a version of the code that is parameterized by only one parameter, R, our actual implementation uses different template parameters for inputs and output, meaning that we allow static generation of any inputoutput precision combinations.

We exploit both the parallelism that exist between expansion terms and across different expansions. To benefit from the SIMD execution and intra-warp communication primitives, all terms in a given expansion have to be computed by threads of the same warp. As warps have 32 threads on Nvidia architectures, the maximal supported expansion size is 32. Smaller expansions are packed together inside warps. Although this approach works with any expansion size R between 1 and 32 using appropriate padding, we recommend using power of two sizes, which allow filling the whole warp.

VI. COMPARISON AND DISCUSSION

In this section we present performance measurements obtained on a Tesla K40 GPU (Kepler architecture), using the CUDA 7.5 software architecture. The code also runs on the newer Maxwell architecture, although the performance of all algorithms becomes limited by the lower double-precision performance of current Maxwell-based GPUs. We measure throughput on embarrassingly-parallel computations. The values are given in million of operations per second (Mop/s). By one op we understand one operation using extended precision. The tests were done using random generated examples, running on 1024 blocks each with 1024, 512 or 256 execution threads, depending on the expansion size and the required resources to run the algorithms.

To analyze the effect of parallelism on memory footprint, we consider two different shared memory usage scenarios: one best case that assumes the application uses no intermediate data outside of the registers used for the computation, and one worst case where the application uses 256 bytes of CUDA shared memory for each term of the expansion.

For comparison, we report to Bailey's GQD library [START_REF] Hida | Algorithms for quad-double precision floating-point arithmetic[END_REF] and CAMPARY [START_REF] Joldes | Arithmetic algorithms for extended precision using floating-point expansions[END_REF], which offer extended precision using FP expansions on GPU. The arithmetic algorithms they employ are not by themselves parallel. Moreover, the GQD library is limited to double-double and quad-double precisions, and the algorithms used in the implementation are not provided with any correctness proof and there is no specific error bound given. CAMPARY is a recent software, that comes with correctness proofs and results guaranteed within a certain error bound.

In Table I we show the addition algorithm's performance for the best-case, no-memory configuration, followed by the performance obtained in the memory-constrained configuration, in Tables II. Even in the worst-case embarrassingly-parallel setup, the performance of data-parallel addition algorithm is competitive with the best known sequential algorithms for smaller expansions: the parallelism comes at little cost in number of operations per expansion.

For the multiplications algorithm we asses performance in Tables III and IV, for the best case setting and for the memory constrained one, respectively. From the data obtained in the best case scenario we observe that addition on larger expansions and multiplications suffer from parallelization overhead. The benefits of exploiting the parallelism available within each expansion are fully realized when parallelism is constrained by internal memory usage. The performance of data-parallel algorithms remains stable in this setup, while the performance of sequential algorithm decreases sharply with memory usage. Although the QD library remains faster on expansions of size 2 (double-double), the dataparallel algorithms significantly outperform their sequential counterparts for all larger expansions. The performance gap increases with the expansion size, eventually reaching an order of magnitude for 32-term expansions.

VII. CONCLUSION

We presented data-parallel algorithms for addition and multiplication of floating-point expansions. By taking advantage of data parallelism within FP expansions, they are suitable for SIMD architectures. We present fast addition and multiplication algorithms, as well as a safe addition algorithm with rigorous error bounds. A GPU implementation of the algorithms using warp-synchronous programming in CUDA is already competitive with state-of-the-art sequential algorithm in an idealistic embarrassingly parallel setup. However, dataparallel algorithms really shine in the more realistic case of applications that manipulate a sizable amount of intermediate data, significantly outperforming sequential algorithms for expansions of size 4 and greater.

11 :

 11 e ← (0, e 0 , e 1 , . . . , e R-2 )12:

Fig. 1 :

 1 Fig. 1: Safe data-parallel FP expansion addition algorithm, illustrated for the case when R = 3 terms.

Fig. 2 :

 2 Fig. 2: Sequential scheme of the FP expansion addition algorithm.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Reduction of the sequential scheme in Fig. 2 based on the "Sterbenz relation". Now let us prove the following proposition that refers to only one horizontal line of the scheme, as presented in Fig. 4.

Fig. 5 :

 5 Fig. 5: Fast data-parallel FP expansion addition algorithm, illustrated for the case when R = 4 terms.

Fig. 6 :

 6 Fig. 6: Graphical representation of the sequential algorithm that behaves like Algorithm 2.

Fig. 7 :

 7 Fig. 7: Multiplication Algorithm 2 implemented in CUDA C, simplified for the case in which the inputs and the output have the same power-of-two size, R.

  can continue, by noticing that |e 1 | is bounded by 2 -p |s 1 |, and bounding |a 2 | by 2 -2(p-1)+δ 1+2 -p 1-2 -p+ • |s 0 |. This gives a bound on |e 1 + a 2 |, and a bound on s 2 is obtained by multiplying that last bound by (1 + 2 -p ). An easy induction finally gives

TABLE I :

 I Performance in Mop/s for adding two FP expansions on the GPU in the best case with no internal memory usage; R represent the number of terms in both input and output expansions. * precision not supported

	R	Safe (Alg. 1)	Fast	CAMPARY	QD
	2	6,294	11,914	3,188	41,549
	4	1,141	4,637	1,448	5,310
	8	314.4	2,234	493.6	*
	16	73.89	830.3	158.6	*
	32	13.57	131.4	41.77	*

TABLE II :

 II Performance in Mop/s for FP expansion addition al-

	gorithms in the memory-constrained case with 256B shared memory
	per expansion term
	R	Safe (Alg. 1)	Fast	CAMPARY	QD
	2	1,774	1,424	971.9	4,830
	4	337	528	131.3	391.4
	8	70.52	229.4	28.2	*
	16	16.67	103.9	6.52	*
	32	3.82	16.03	1.04	*

TABLE III :

 III Performance in Mop/s for multiplying two FP expansions on the GPU in the best case with no internal memory usage

	R	Algorithm 2	CAMPARY	QD
	2	6,484	11,100	27,390
	4	756.2	1,363	2,726
	8	135.2	47.19	*
	16	25.07	12.10	*
	32	2.63	2.62	*

TABLE IV :

 IV Performance in Mop/s for FP expansion multiplication algorithms in the memory-constrained case with 256B shared memory per expansion term

	R	Algorithm 2	CAMPARY	QD
	2	882.87	2,093	3,501
	4	99.37	180.8	250.9
	8	18.05	7.06	*
	16	4.04	0.84	*
	32	0.23	0.096	*

A FMA (Fused Multiply-Add) operator evaluates an expression of the form xy + t with one final rounding only.


Christel-Loic Tisse

Lionel Martin

Lionel Torres

Michel Robert

Person Identification Technique Using Human Iris Recognition

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


Anne-Laurence Tomi

Jérémie Sellam

MD Karine Lacombe

MD Soraya Fellahi

Manuela Sebire

MD Caroline Rey-Jouvin

MD Anne Miquel

MD Jean-Philippe Bastard

MD Emmanuel Maheu

MD Ida K Haugen

MD David T Felson

Jacqueline Capeau

PhD Pierre-Marie Girard

Francis Berenbaum
email: francis.berenbaum@aphp.fr

Jean-Luc Meynard

Prof Francis Berenbaum

Increased prevalence and severity of radiographic hand osteoarthritis in HIV-1infected patients associated with metabolic syndrome: data from the cross-sectional METAFIB-OA study

Keywords: osteoarthritis, HIV-1, hand, aging, metabolic syndrome, metabolic osteoarthritis

Objective To determine radiographic hand OA (HOA) prevalence in HIV-1-infected patients in comparison with the general population and to address whether MetS may increase the risk of HOA during HIV-1 infection.

Patients. HIV-1-infected patients with MetS (IDF criteria) aged 45-65 years were matched by age and gender to HIV-1-infected subjects without MetS and underwent hand radiographs.

Framingham OA cohort was used as general population cohort.

Methods. Radiographic HOA was defined as Kellgren-Lawrence score (KL) ≥ 2 on more than 1 joint. Radiographic severity was assessed by global KL score and number of OA joints. HOA prevalence was compared to that found in the Framingham study, stratified by age and sex. Logistic and linear regression models were used to determine risk factors of HOA in HIV-1-infected patients.

Results. 301 patients (88% male, mean age 53.45.0 years), were included, 152 with MetS and 149 without it. Overall HOA prevalence was 55.6% and was higher for those with MetS than those without it (64.7% vs 46.3%, p=0.002). When considering men within each age group, HOA frequency was greater in HIV-1-infected patients than the general population (all ages: 55.8% vs 38.7%; p<0.0001), due to the subgroup with MetS (64.9%; p<0.0001), as well as the subgroup without MetS, although not significantly (46.6%; p=0.09).

Risk of HOA was increased with MetS (odds ratio [OR] 2.23, 95% confidence interval [95%CI] 1.26-3.96) and age (OR 1.18,. HOA severity was greater for patients with MetS than those without. HOA was not associated with previous or current exposure to protease inhibitors or HIV infection-related markers.

Conclusions HOA frequency is greater in HIV-1-infected patients, especially those with MetS, than the general population.

NCT02353767

Human immunodeficiency virus (HIV)-associated mortality is greatly reduced because of the widespread use of efficient antiretroviral therapy [START_REF] Puhan | Excess mortality in patients with AIDS in the era of highly active antiretroviral therapy: temporal changes and risk factors[END_REF]. Consequently, in the United States and Europe, HIV-1-infected patients older than 50 years represent more than 50% of the followup population [START_REF] Collaboration | Life expectancy of individuals on combination antiretroviral therapy in high-income countries: a collaborative analysis of 14 cohort studies[END_REF][START_REF] Cfdca | HIV/AIDS surveillance report among persons aged 50 or over[END_REF][START_REF] Pharris | Ten years after Dublin: principal trends in HIV surveillance in the EU/EEA, 2004 to 2013[END_REF]. Likewise, age-related health problems such as cardiovascular diseases have been increasing with increasing age in patients with HIV because of chronic inflammation, immune activation and immunosenescence, and lifelong antiretroviral therapy [START_REF] Aberg | Aging, inflammation, and HIV infection[END_REF][START_REF] Capeau | Premature aging in human immunodeficiency virus (HIV) infected patients: detection, pathophysiological mechanisms and management[END_REF][START_REF] Brothers | Biologic aging, frailty, and age-related disease in chronic HIV infection[END_REF]. These comorbidities, called non-AIDS-related comorbidities, are more prevalent in HIV-infected patients than people of the same age without HIV, which suggests an extra "hit" of aging related to HIV-1 infection and/or antiretroviral therapy, which can lead to geriatric syndromes with impairment and frailty [START_REF] Deeks | HIV infection, antiretroviral treatment, ageing, and non-AIDS related morbidity[END_REF][START_REF] Effros | Workshop on HIV infection and aging: What is known and future research directions[END_REF][START_REF] Franceschi | Inflammaging and anti-inflammaging: A systemic perspective on aging and longevity emerged from studies in humans[END_REF][START_REF] High | HIV and Aging: State of Knowledge and Areas of Critical Need for Research. A Report to the NIH Office of AIDS Research by the HIV and Aging Working Group[END_REF].

Osteoarthritis (OA) is the most common rheumatologic disease due to aging, affecting about 6 million people with hip and/or knee OA in France [START_REF] Guillemin | Prevalence of symptomatic hip and knee osteoarthritis: a two-phase populationbased survey[END_REF] and 27 million in the United States [START_REF] Lawrence | Estimates of the prevalence of arthritis and other rheumatic conditions in the United States. Part II[END_REF]. The disease has never been studied in the setting of HIV comorbidities. Among all localizations, hand OA (HOA) is associated with pain, disability and deteriorated quality of life to the same extent as rheumatoid arthritis [START_REF] Dahaghin | Prevalence and pattern of radiographic hand osteoarthritis and association with pain and disability (the Rotterdam study)[END_REF][START_REF] Slatkowsky-Christensen | Health-related quality of life in women with symptomatic hand osteoarthritis: a comparison with rheumatoid arthritis patients, healthy controls, and normative data[END_REF][START_REF] Michon | Assessing health-related quality of life in hand osteoarthritis: a literature review[END_REF]. However, to what extent HIV-1positive patients have OA and HOA is unknown.

OA is a heterogeneous group of diseases that can be differentiated by the risk factors (i.e., aging, obesity, trauma). Each factor has specific pathophysiological pathways, all leading to joint destruction [START_REF] Bijlsma | Osteoarthritis: an update with relevance for clinical practice[END_REF][START_REF] Grotle | Obesity and osteoarthritis in knee, hip and/or hand: An epidemiological study in the general population with 10 years followup[END_REF]. Obesity-associated OA is one of the most-studied phenotypes characterized by the association of obesity or overweight with OA on weightbearing joints due to mechanical overload [START_REF] Courties | Metabolic stress-induced joint inflammation and osteoarthritis[END_REF]. However, recently, the demonstration of an association between obesity and HOA has shed light on a potential role of systemic metabolic disturbances in the pathophysiology of OA [START_REF] Yusuf | Association between weight or body mass index and hand osteoarthritis: a systematic review[END_REF]. Indeed, several studies have raised the possibility of an association of metabolic syndrome (MetS) and OA, but which did not persist after adjustment on body mass index or weight in some of them [START_REF] Puenpatom | Increased prevalence of metabolic syndrome in individuals with osteoarthritis: an analysis of NHANES III data[END_REF][START_REF] Visser | The relative contribution of mechanical stress and systemic processes in different types of osteoarthritis: the NEO study[END_REF][START_REF] Shin | Association between metabolic syndrome, radiographic knee osteoarthritis, and intensity of knee pain: results of a national survey[END_REF]. In addition to accelerated aging, HIV-infected patients frequently have MetS because HIV infection, via chronic inflammation and immune activation, which contributes to dyslipidemia and insulin resistance [START_REF] Alencastro | Metabolic syndrome and population attributable risk among HIV/AIDS patients: comparison between NCEP-ATPIII, IDF and AHA/NHLBI definitions[END_REF]. Moreover, antiretroviral therapy, especially protease inhibitors, can induce a lipodystrophic syndrome characterized by altered body fat composition, dyslipidemia and insulin resistance [START_REF] Wand | Metabolic syndrome, cardiovascular disease and type 2 diabetes mellitus after initiation of antiretroviral therapy in HIV infection[END_REF] and subsequent MetS development [START_REF] Mutimura | Insulin resistance change and antiretroviral therapy exposure in HIV-infected and uninfected Rwandan women: a longitudinal analysis[END_REF].

Considering the increased life expectancy, accentuated aging and high prevalence of MetS in HIV-infected patients, we hypothesized that HOA could represent a novel HIVassociated non-AIDS-related comorbidity. Likewise, we aimed to determine whether MetS might be associated with risk of radiographic HOA in HIV-infected patients and whether the prevalence of HOA in HIV-infected patients is greater than in the general population.

Methods

Patients

The present study, called METAFIB-OA, is an ancillary study of the Metabolic Syndrome and Fibrosis (METAFIB) study (NCT02353767). METAFIB is a cross-sectional single-center study that recruited 458 HIV-infected patients >18 years old from January 2011 to December 2012 from outpatient clinics. HIV-positive status for at least 5 years was confirmed by western blot analysis or ELISA in patients without chronic viral hepatitis co-infection to investigate the impact of MetS on liver fibrosis. All patients were HIV-1-positive and were separated in 2 subgroups, with (HIV-1+MetS+) and without MetS (HIV-1+MetS-). MetS was defined by the International Diabetes Federation (IDF) criteria (Supplementary file Table 1) [START_REF] Alberti | Harmonizing the Metabolic Syndrome A Joint Interim Statement of the International Diabetes Federation Task Force on Epidemiology and Prevention; National Heart, Lung, and Blood Institute; American Heart Association; World Heart Federation; International Atherosclerosis Society; and International Association for the Study of Obesity[END_REF]. Patients with MetS+ were matched to those without MetS+ by sex and age (± 5 years), with a ratio of 1:1.

For all patients, metabolic and HIV-1 infection clinical characteristics were recorded at the time of participation to METAFIB-OA cross-sectional study. A fasting blood sample was obtained from all subjects for assessing blood cell count, CD4+ and CD8+ T-cell count, ultrasensitive HIV-1 viral (usHIV-1) load, lipid levels, glycemia, and insulinemia for homeostasis model assessment insulin resistance (HOMA-IR) calculation by the standard procedure of the hospital. The lowest count (nadir) for the CD4+ and CD8+ T cells was extracted from medical files for patients.

Between September 2011 and April 2012, all patients between 45 and 64 years from the METAFIB were contacted for the ancillary METAFIB-OA study. Exclusion criteria were pregnancy, inflammatory rheumatic disease or Dupuytren's disease. To compare the prevalence of HOA between METAFIB-OA patients and the general population, we used data for the Framingham cohort, a population-based cohort consisting of the Offspring and the Community cohorts, in which HOA prevalence had been previously examined [START_REF] Feinleib | The Framingham Offspring Study. Design and preliminary data[END_REF][START_REF] Haugen | Prevalence, incidence and progression of hand osteoarthritis in the general population: the Framingham Osteoarthritis Study[END_REF].

The study was approved by the Institutional Review Board (Comité de Protection des Personnes, Paris Ile de France V, Paris). All patients gave informed consent.

Radiographic definition of HOA and severity

For METAFIB-OA study (i.e., between September 2011 and April 2012), all patients underwent bilateral postero-anterior hand radiography at 100% at APHP Saint-Antoine Hospital, and two trained assessors (A-LT and CR-J) scored an equal number of radiographs. The readers were blinded to clinical data and subgroup (MetS+ or MetS-). Hand radiographs were graded by the Kellgren-Lawrence grading scale (KL) [START_REF] Kellgren | Radiological Assessment of Osteo-arthrosis[END_REF], which assesses the distal interphalangeal (DIP), proximal interphalangeal (PIP), interphalangeal thumb (IP-1), metacarpal (MCP) and first carpometacarpal (CMC-1) joints with a grading system from 0 to 4 (0, no OA; 1, doubtful OA; 2, definite minimal OA; 3, moderate OA; 4, severe OA). Any patient with at least 1 finger joint scored at KL grade ≥ 2 was considered to have radiographic HOA. Thumb-base OA was also considered separately as unilateral or bilateral with KL score ≥ 2 on 1 or 2 CMC-1 joints (31) because it is more likely related to loading than the other three joints (DIP, PIP, or MCP) [START_REF] Chaisson | Grip strength and the risk of developing radiographic hand osteoarthritis: results from the Framingham Study[END_REF][START_REF] De Monsabert | Quantification of finger joint loadings using musculoskeletal modelling clarifies mechanical risk factors of hand osteoarthritis[END_REF][START_REF] Hunter | Trapeziometacarpal subluxation predisposes to incident trapeziometacarpal osteoarthritis (OA): the Framingham Study[END_REF][START_REF] Jonsson | High hand joint mobility is associated with radiological CMC1 osteoarthritis: the AGES-Reykjavik study[END_REF].

Radiographic severity was assessed by 1) the global KL score (sum of the scores for all joints; range 0-128), 2) number of OA joints (KL ≥2; range 1-32), 3) number of patients with erosive HOA according to the Verbruggen-Veys anatomical phase score (VV), consisting of five phases with a numerical value representing the evolution of HOA (N, normal joint; S, stationary OA with osteophytes and joint-space narrowing; J, complete loss of joint space in the whole or part of the joint; E, subchondral erosion; R, remodeling of subchondral plate) [START_REF] Verbruggen | Numerical scoring systems for the anatomic evolution of osteoarthritis of the finger joints[END_REF]. The DIP, PIP, IP-1 and MCP joints were assessed. Erosive HOA diagnosis was based by the presence of at least 1 erosive joint on bilateral HOA radiographs (E or R phase).

Before this scoring, both readers (A-LT and CR-J) underwent a training session to assess the inter-and intra-observer reproducibility with 20 hand radiographs from routine practice. Reproducibility between the readers was estimated first by the Cohen kappa coefficient for inter-reader concordance and then by the intraclass correlation coefficient (ICC) for intra-reader concordance. For the total KL scale, the scores were 0.64 (95% confidence interval [95% CI] 0.61-0.67) and 0.95 (95% CI 0.80-0.98), respectively. For the VV score, the ICCs were 0.9 (95% CI 0.91-0.99) and 0.98 (95% CI 0.91-0.99), respectively.

HOA symptoms

Patients from METAFIB who participated to the ancillary METAFIB-OA study completed a brief standardized questionnaire at the time of radiography about their dominant hand, menopausal status for women, and history of psoriasis. For joint pain assessment, the following binary question was asked: "On most days, do you have any pain, aching or stiffness in any of your joints?"

Biomarker assessment

A blood sample was performed at the time of the inclusion in the main METAFIB study, so some months before METAFIB-OA. High-sensitivity C-reactive protein (hsCRP) was measured by nephelometry on an IMMAGE analyzer (Beckman-Coulter, Villepinte, France).

We measured plasma levels of high-sensitive interleukin 6 (IL-6) level, reflecting global inflammation and aging, soluble CD14 (sCD14) and soluble CD163 (sCD163) (Quantikine ELISA Kit, R&D Systems, Oxford, UK), 2 markers of monocyte/macrophage activation involved in HIV-related chronic immune activation [START_REF] Leeansyah | Soluble biomarkers of HIV transmission, disease progression and comorbidities[END_REF], and leptin (Quantikine; R&D Systems, Oxford, UK) and total adiponectin (ALPCO, EUROBIO, Les Ulis, France), 2 adipokines involved in MetS [START_REF] Ma | Inflammatory mediators involved in the progression of the metabolic syndrome[END_REF][START_REF] Maury | Adipokine dysregulation, adipose tissue inflammation and metabolic syndrome[END_REF] and OA [START_REF] Abella | Adipokines, metabolic syndrome and rheumatic diseases[END_REF].

Statistical analysis

Descriptive data are presented as mean ± SD, median (interquartile range [IQR]) or number (%). HIV+MetS+ and HIV+MetS-patients and subjects with and without HOA were compared by chi-square test for categorical variables and Wilcoxon rank-sum test for continuous variables.

We compared HOA prevalence from the METAFIB-OA with the community-based cohort from the Framingham study that estimated HOA prevalence by a slightly modified version of the KL scale (i.e., HOA diagnosis with ≥ 1 joint radiographic OA by a modified KL scale) by chi-square test by gender and age group.

To determine the factors associated with HOA or thumb-base OA, we calculated crude odds ratios (ORs) with 95% confidence intervals (CIs) by univariate modeling in the entire population. We explored variables related to demographic characteristics, HIV-1 infection features and metabolic disturbances. Variables associated with HOA diagnosis on univariate analysis with p < 0.

Results

Population characteristics

The main METAFIB study included 222 HIV1+MetS+ and 222 HIV1+MetS-subjects; 173 patients with MetS+ and 166 without MetS-45 to 64 years old were screened for the METAFIB-OA study (Figure 1). Characteristics of the study population are in Table 1, with stratification by MetS status. The 2 groups did not differ except for all MetS characteristics (p<0.0001), as expected. HOA symptoms were reported by 27% of the population.

HOA prevalence and structural severity in the METAFIB-OA group

Radiographic HOA prevalence and severity are stratified by gender in Table 2. Overall radiographic HOA was significantly more frequent in patients with MetS than those without it (64.5% vs 46.3%; p=0.002). Stratification by gender yielded similar results (64.9% in male with MetS vs 46.5% in male without MetS, p=0.002 and 61.1% in female with MetS vs 44.4% in without MetS, p=0.002). The same difference was observed for thumb-base OA.

Radiographic structural severity based on KL total score and number of OA joints was significantly more pronounced in MetS+ patients than those without it (p=0.002). Few cases of erosive HOA were observed in MetS+ subgroup (n=5, 3.3%) as well as in MetS-subgroup (n=2, 1.3%), with no significant difference between them.

Comparison of METAFIB-OA group with aged-matched general population from the

Framingham cohort

For patients 45 to 64 years old, mean HOA prevalence was greater in the METAFIB group than the Framingham population (55.5% vs 39.3%, p<0.0001) (Table 3). This difference was mainly due to MetS+ patients in the METAFIB-OA cohort (64.5% in HIV+MetS+ vs 39.3% in Framingham, p<0.0001, for all ages), but there was also a numeric difference of HOA prevalence between Framingham cohort and HIV+MetS-patients (for all ages: 46.3% in HIV+MetS-vs 39.3% in Framingham p=0.09).

Considering the differences in sex ratios between the 2 cohorts and the impact of aging on OA development, we compared HOA prevalence stratified on sex and age. The prevalence of HOA was significantly higher for men in the METAFIB cohort than men in the Framingham study within each age group. Furthermore, a comparison of age-matched men from the Framingham and the METAFIB cohort by presence and absence of MetS showed that this difference was due to a higher prevalence of HOA in the METAFIB cases (for all ages: 64.9%) than in Framingham subjects (38.7%; p<0.0001). However, we also found greater prevalence of HOA in METAFIB male controls than men from the Framingham study, although not significantly (for all ages, 46.6%, vs 38.7%; p=0.09).

The METAFIB cohort contained few females (n=36, but prevalence of HOA was greater for the METAFIB women than Framingham women, although not significantly (Table 3). Conversely, unilateral or bilateral thumb-base OA frequency did not differ between the METAFIB group and Framingham cohort (data not shown).

Determinants of HOA

In univariate analysis, age, CD4 T-cell count, detectable hsHIV-1 viral load, HOMA-IR, triglycerides level and presence of MetS were associated with HOA (Table 4).

In multivariate analysis, only presence of MetS (adjusted OR=2.23, 95% CI 1.26-3.96; p=0.002) and age (adjusted OR per year=1.18, 95% CI 1.12-1.25; p=0.00001) remained independently associated with HOA (see Figure 1 in the supplement). These 2 factors were associated with HOA severity (See Table 2

in the supplement).

Considering each metabolic component separately, only insulin resistance assessed by HOMA-IR and triglycerides level were associated but not significantly with HOA on univariate analysis (p=0.06 and p=0.07, respectively) but not after adjustment (Table 4).

For thumb-base OA, only age remained significantly associated with HOA (OR=1.10, 95% CI (1.04-1.17); p=0.001), with a nonsignificant association with MetS (OR=1.86, 95% CI 0.98-3.45; p=0.06) (See Table 3

in the supplement).

Of note, HOA diagnosis and severity were not associated with HIV-1 infection characteristics or HIV infection-related markers (Table 4, Supplementary tables 2 and 3).

Association between biological markers and HOA

To further elucidate the mechanism, MetS or HIV-related characteristics, that may favor HOA, we tested a set of metabolic or HIV-related biomarkers. Plasma sCD14 level was significantly higher in patients with than without HOA (2203.8 vs 2010.8 pg/mL; p=0.02) (See Table 4 in the supplement). This finding was corroborated by univariate analysis finding log(sCD14) level associated with HOA in the whole study population (OR=4.9, 95% CI 1.1-21.6; p=0.03). However, after adjustment for MetS, this association became nonsignificant (OR=3.9, 95% CI 0.9-17.2; p=0.07). Plasma levels of adipokines, hsCRP and sCD163 did not differ by HOA diagnosis.

Discussion

In this study, radiographic HOA prevalence was determined for the first time in HIV-infected patients and found to be higher than in the general population from the Framingham cohort. This frequency was further increased in HIV-infected patients with MetS. Age and MetS were associated with HOA during HIV-1 infection. Furthermore, these 2 factors were also associated with HOA radiographic severity.

Several diseases such as atherosclerosis complications and cancer have emerged as key points in the global therapeutic management of HIV-infected patients. Surprisingly, although OA represents the most common age-related joint disease and with the growing interest in the role of systemic cardiometabolic disturbances in OA pathophysiology, only one preliminary study showed that in 35 HIV-infected men, total body and android fat mass were inversely related to knee cartilage volume measured by MRI [START_REF] Fillipas | The relationship between body composition and knee structure in patients with human immunodeficiency virus[END_REF]. Here, taking advantage of the unique METAFIB study including HIV-1-infected patients with or without MetS, we observed a higher prevalence of radiographic HOA in HIV-infected patients as compared with the Framingham cohort, representing the general population. Differences between the 2 cohorts were obvious in men, but the small sample of women limits the power of the statistical analysis. In men, although the prevalence of HOA increased with age in both cohorts, HOA occurred more frequently in HIV-1-infected patients, especially those with MetS. Interestingly, men from the METAFIB cohort without MetS showed a higher prevalence of HOA, although not significantly, than men from the Framingham cohort in each age group. Such a result emphasizes the effect of aging during HIV infection beyond that of MetS.

Several studies have suggested an association between obesity and HOA [START_REF] Yusuf | Association between weight or body mass index and hand osteoarthritis: a systematic review[END_REF]. With obesity, the conditions hypertension, dyslipidemia, and glucose intolerance alone or together could increase the risk of OA. In the Rotterdam cohort (42), prevalence of HOA was higher in overweight patients with hypertension and diabetes than patients with only overweight. Here, the model of HIV-1 infection further supports the "metabolic" OA phenotype because the prevalence of HOA was greater in patients with than without MetS [START_REF] Bijlsma | Osteoarthritis: an update with relevance for clinical practice[END_REF][START_REF] Grotle | Obesity and osteoarthritis in knee, hip and/or hand: An epidemiological study in the general population with 10 years followup[END_REF][START_REF] Courties | Metabolic stress-induced joint inflammation and osteoarthritis[END_REF]. Of note, obesity was not associated with HOA diagnosis in the METAFIB-OA study: such a result agrees with those from the population-based Netherlands Epidemiology of Obesity cohort, finding HOA linked more to metabolic systemic factors than weight itself [START_REF] Visser | The relative contribution of mechanical stress and systemic processes in different types of osteoarthritis: the NEO study[END_REF]. Radiographic severity was more severe in patients with MetS, so MetS is a risk factor and also an aggravating factor of radiographic HOA during HIV infection. Radiographic HOA could progress quickly in these patients because radiographic severity is associated with radiographic progression [START_REF] Bijsterbosch | Clinical and radiographic disease course of hand osteoarthritis and determinants of outcome after 6 years[END_REF].

The association between HIV and HOA complicated by MetS and aging emphasizes the involvement of systemic metabolic inflammatory mediators. Interestingly, plasma sCD14 level was associated with HOA but less so after adjustment for MetS, which suggests that sCD14 and thus macrophage activation induced by MetS could be a link between MetS and HOA and its severity. Of note, plasma sCD14 level is associated with symptoms and radiographic progression in knee OA, and synovial fluid sCD14 level is associated with activated macrophages infiltrating knee synovium [START_REF] Daghestani | Soluble macrophage biomarkers indicate inflammatory phenotypes in patients with knee osteoarthritis[END_REF]. HOA in patients with MetS may have increased synovial inflammation, as is found in patients with knee OA and type 2 diabetes [START_REF] Schett | Diabetes is an independent predictor for severe osteoarthritis: results from a longitudinal cohort study[END_REF].

Separate analyses of thumb-base OA showed that this localization was not more prevalent in HIV patients than the general population. Such a result was expected because mechanical factors and the morphology of the trapezo-metacarpal joint are crucial for this localization. However, thumb-base OA was more prevalent in HIV patients with than without MetS. So, metabolic disturbances may represent an additional risk factor, focal mechanical injury being a precipitating event [START_REF] Anakwe | Osteoarthritis at the base of the thumb[END_REF][START_REF] Radin | Role of mechanical factors in pathogenesis of primary osteoarthritis[END_REF].

This study has several limitations. First, hand radiography was performed 1 year after assessment of MetS and serum samples taken for examining exploratory markers. However, OA progression is a very slow process, so this bias may have minimal effect on our findings.

Second, the Framingham study used the modified version of the KL scoring system, with HOA definition based on the presence of osteophytes and/or definite joint-space narrowing [START_REF] Haugen | Prevalence, incidence and progression of hand osteoarthritis in the general population: the Framingham Osteoarthritis Study[END_REF], whereas the original KL scale, used in METAFIB-OA, defined HOA exclusively by presence of osteophytes. Furthermore, in the Framingham study, the evaluation of OA in the thumb base included assessment of both the CMC-1 and triscaphoid joints, whereas only the CMC-1 joints were assessed in METAFIB-OA. Hence, the scoring system of the Framingham study would classify more joints with definite OA. Consequently, the differences between the 2 cohorts may have been underestimated. Finally, only 25% of patients reported hand pain. However, this frequency agrees with data from other cohorts and OA pain may fluctuate through time [START_REF] Dahaghin | Prevalence and pattern of radiographic hand osteoarthritis and association with pain and disability (the Rotterdam study)[END_REF][START_REF] Haugen | Prevalence, incidence and progression of hand osteoarthritis in the general population: the Framingham Osteoarthritis Study[END_REF]. Third, to compare HOA between the French METAFIB-OA group and the general population, we have used the Framingham cohort although characteristics of general population in United States and France are certainly different. However, no HOA assessment in a population-based cohort is available in France until now.

In conclusion, HIV-1 infection represents a special setting in which the risk of radiographic HOA and its severity is increased, due to accelerated aging and MetS. Beyond classical OA phenotypes, the "HIV-related OA" subtype could be thus individualized.

  2 were entered in a backward stepwise multivariate conditional logistic regression model. Multivariate linear regression was used to determine factors associated with structural radiographic severity, as defined above (3 definitions). MetS was entered in the logistic model along with other variables and kept in the final model, whatever the level of significance. Results are presented as β regression coefficients. All measurements were log-transformed to remove positive skewness and were compared between patients with and without HOA by the Wilcoxon rank-sum test. Univariate and multivariate analyses with a logistic model were adjusted for MetS. All analyses involved use of STATA v12.1 (StataCorp, College Station, TX, USA) and p <0.05 was considered statistically significant.

Figure 1 .

 1 Figure 1. Flow chart of METAFIB-OA Legends: MetS+: presence of metabolic syndrome, MetS-: absence of metabolic syndrome; OA: osteoarthritis.

able 1. Baseline characteristics of the METAFIB-OA cohort

  Hypertension was defined as systolic pressure ≥140 mm Hg and/or diastolic pressure ≥90 mm Hg. Diabetes was defined as glycemia >6 mmol/L. OA: osteoarthritis, MetS: metabolic syndrome, usHIV: ultra-sensitive HIV, CDC: Centers for Disease Control and Prevention, BMI: body mass index; HDL: high-density lipoprotein; LDL: lowdensity lipoprotein, cho: cholesterol, HOMA-IR score: Homeostasis Model Assessment of Insulin Resistance, MetS+: presence of metabolic syndrome, MetS-: absence of metabolic syndrome.

		Total	HIV-1+	HIV-1+	P value
		(n=301)	MetS+	MetS-
			(n=152)	(n=149)
	Demographic features
	Male gender, n (%)	265 (88.0)	134 (88.2)	131 (87.9)	0.9
	Age (years), mean (SD)	53.4 (5.0)	53.5 (4.9)	53.4 (5.1)	0.7
	Post-menopausal status (n=37) n (%)	26 (70.3)	14 (77.8)	12 (63.2)	0.5
	Predominant side, n (%)
	Right-handed	250 (83.1)	133 (87.5)	117 (78.5)
	Left-handed	32 (10.6)	13 (8.6)	19 (12.8)
	Mixed	19 (6.3)	6 (3.9)	13 (8.7)	0.1
	Hand OA features
	Hand trauma history, n (%)	62 (20.6)	26 (%)	36 (%)	0.2
	Psoriasis, n (%)	13 (4.3)	4 (2.6)	9 (6.0)	0.2
	Pain, n (%)	80 (26.6)	42 (27.6)	38 (25.5)	0.7
	HIV features
	Duration of HIV infection (years), mean (SD) 17.7 (7.3)	17.0 (7.1)	18.3 (7.5)	0.2
	CDC-C stage, n (%)	81 (26.9)	43 (28.3)	38 (25.5)	0.8
	CD4 level (/mm 3 ), mean (SD)	623 (265)	625 (257)	621 (274)	0.9
	CD4/CD8 ratio, mean (SD)	0.87 (0.37) 0.86 (0.4)	0.87 (0.3)	0.4
	Undetectable usHIV viral load, n (%)	240 (79.7)	82 (53.9)	90 (60.8)	0.2
	Duration of exposure to protease inhibitors	47.7 (29.2) 26.9 (31.1)	27.4 (33.6)	0.8
	(months), mean (SD)
	Metabolic syndrome components
	Waist circumference (cm), mean (SD)	92.2 (10.8) 98.2 (10.1)	86.3 (8.1)	<0.0001
	BMI (kg/m2), mean (SD)	24.9 (5.8)	26.4 (4.8)	23.3 (6.3)	<0.0001
	Obesity (BMI≥30), n (%)	32 (10.6)	26 (17.1)	6 (4.0)	<0.0001
	Hypertension, n (%)	56 (18.6)	40 (26.3)	16 (10.7)	<0.0001
	Triglycerides (mmol/l), mean (SD)	1.95 (1.9)	2.47 (1.7)	1.42 (2.0)	<0.0001
	HDL-cho (mmol/l), mean (SD)	1.21 (0.4)	1.05 (0.3)	1.36 (0.4)	<0.0001
	LDL-cho (mmol/l), mean (SD)	2.9 (0.9)	3.1 (0.8)	2.8 (0.9)	0.0003
	Glycemia (mmol/L), mean (SD)	5.45 (1.08) 5.85 (1.3)	5.05 (0.64)	<0.0001
	HOMA-IR score, mean (SD)	2.42 (2.79) 3.51 (3.50)	1.31 (0.92)	<0.0001
	Type 2 diabetes, n (%)	65 (21.6)	57 (37.5)	8 (5.4)	<0.0001
	Data are number (%) or mean (SD) for the total population, and number (%) and median (interquartile range
	[IQR]) for cases and controls.

Table 2 . Radiographic hand OA and severity in the whole METAFIB-OA cohort and in cases and controls

 2

		Total (n=301)	MetS+ HIV-1+	MetS-HIV-1+	p value
			(n=152)	(n=149)
	OA
	HAND OA ≥1 joint KL ≥2
	Men	148 (49.2)	87 (57.2)	61 (40.9)
	Women	19 (6.3)	11 (7.2)	8 (5.3)
	Total cohort	167 (55.5)	98 ()	69 (46.3)	0.002
	Thumb-base OA
	KL ≥2 on 1 or 2 sides
	Men	55 (18.3)	37 (24.3)	18 (12.1)
	Women	6 (2.0)	3 (2.0)	3 (2.0)
	Total cohort	61 (20.3)	40 (26.1)	21 (14.1)	0.01
	Erosive OA
	Men	7 (2.3)	5 (3.3)	2 (1.3)
	Women	0	0	0
	Total cohort	7 (2.3)	5 (3.3)	2(1.3)	0.5
	SEVERITY CRITERIA
	Sum of KL scores	5.2 ± 8.8	6.7 ± 0.9	3.7 ± 0.5	0.002
	No. of joints with KL ≥2	2.5 ±4.1	3.2 ± 0.4	1.8 ± 0.2	0.002
	Data are number (%) or mean  SD
	HOA: hand OA, KL: Kellgren-Lawrence score, MetS+: presence of metabolic syndrome,
	MetS-: absence of metabolic syndrome

Table 3 . Prevalence of radiographic hand OA in the general population (Framingham cohort study) and in the METAFIB-OA study by age.

 3

	Age groups	Framingham	METAFIB	METAFIB	METAFIB	METAFIB vs	METAFIB	METAFIB
	(years)	total population	cohort	HIV-1+	HIV-1+	Framingham	Cases vs	Controls vs
		(n=1508)	(n=301)	MetS+	MetS-		Framingham	Framingham
				(n=152)	(n=149)
	Men + women (45-64) 39.3	55.5	64.5	46.3	<0.0001	<0.0001	0.09
	MEN	n=641	n=265	n=134	n=131
	45-49	10.3	28.6	32.1	25.7	0.004	0.006	0.03
	50-54	31.9	50.0	57.1	41.5	0.003	0.001	0.23
	55-59	43.9	68.6	80.6	55.9	0.0004	<0.0001	0.20
	60-64	56.4	88.1	100	76.2	0.0002	0.0001	0.08
	All ages (45-64)	38.7	55.8	64.9	46.6	<0.0001	<0.0001	0.09
	WOMEN	n=867	n=36	n=18	n=18
	45-49	13.5	38.5	50.0	20.0	0.02	0.006	0.68
	50-54	26.0	46.7	57.1	37.5	0.08	0.07	0.47
	55-59	45.8	100	100	100	0.005	0.06	0.03
	60-64	63.2	0	0	0	0.19	NA	0.19
	All ages (45-64)	39.7	52.8	61.1	44.4	0.12	0.07	0.68
	Radiographic hand OA definition was presence of ≥ 1 affected joint with KL score ≥2.
	MetS+: presence of metabolic syndrome, MetS-: absence of metabolic syndrome, KL: Kellgren-Lawrence score

Table 4 . Univariate and multivariate analysis of associations between sociodemographic factors, HIV infection characteristics and metabolic variables and radiographic hand OA in the METAFIB-OA study population

 4 Radiographic hand OA definition was presence of ≥ 1 affected joint with a KL score ≥2. Hypertension was defined as systolic pressure ≥140 mm Hg and/or diastolic pressure ≥90 mm Hg. Diabetes was defined as glycemia >7 mmol/L. HOA+: hand OA presence, HOA-: hand OA absence. OR, odds ratio; 95% CI, 95% confidence interval. MetS: metabolic syndrome, BMI: body mass index; HDL: high density lipoprotein; HOMA-IR score (Homeostasis Model Assessment of Insulin Resistance)

	Variable	HAND OA Diagnosis			Univariate analysis		Multivariate analysis
		HAND OA+	HAND OA-	P	OR	95% CI	P value OR	95% CI	P value
		(n=167)	(n=134)	value
	Socio-demographic variables
	Age (years), mean (SD)	55.1(5.0)	51.4 (4.3)	10 -5	1.18	1.12-1.25	10 -5	1.18	1.11-1.25	10 -5
	Male gender, n (%)	148	117	0.5	0.88	0.44-1.78	0.5	-
	Previous hand trauma, n (%)
	HIV characteristics
	Duration of HIV infection (years),	18.0 (7.4)	17.3 (7.1)	0.5	1.01	0.98-1.04	0.5	-
	mean (SD)
	CD4 level (/mm3), mean (SD)	600 (251)	652 (281)	0.1	0.99	0.99-1.00	0.1	0.99	0.99-1.00	0.2
	Undectable hsHIV viral load, n (%)	87/161 (29.0) 85/129 (28.2) 0.03	1.64	1.02-2.65	0.05	1.37	0.80-2.34	0.3
	(n=290)
	Duration of exposure to protease							-
	inhibitors (months), mean (SD)	27.5 (32.6)	26.6 (31.8)	0.7	1.00	0.99-1.01	0.8
	Metabolic variables
	Waist circumference (cm), mean	92.8 (10.5)	92.0 (11.2)	0.4	1.01	0.99 -1.03	0.6	-
	(SD)
	Obesity (BMI≥30), n (%)	14 (4.7)	18 (6.0)	0.1	0.98	0.94-1.02	0.3	-
	Hypertension, n (%)	35	21	0.2	1.43	0.79-2.59	0.3	-
	Triglycerides (mmol/l), mean (SD)	2.14 (2.36)	1.72 (1.1)	0.07	1.19	0.99-1.43	0.07	1.06	0.9-1.25	0.5
	HDL-chol (mmol/l), mean (SD)	1.2 (0.4)	1.2 (0.4)	0.7	0.93	0.51-1.70	0.8	-
	HOMA-IR score, mean (SD) (n=297) 2.48 (2.78)	2.36 (2.81)	0.06	1.02	0.93-1.10	0.7	-
	Diabetes, n (%)	36 (%)	29 (%)	0.6	0.99	0.57-1.73	0.9	-
	MetS, n (%)	98 (32.6)	54 (%)	10 -3	2.10	1.32-3.34	0.002	2.18	1.26-3.96	0.005

Acknowledgments

Hayette Rougier (Infectious diseases department, AP-HP Saint-Antoine HospitalParis, France) for logistics concerns in the METAFIB-OA substudy; Raphaèle Séror (Rheumatology department, AP-HP Bicêtre Hospital, Le Kremlin Bicêtre, France) for advice concerning reproducibility in the reading of hand radiographs; and Laura Smales (BioMedEditing, Toronto, Canada).

Funding sources

NIH AR47785 for Framingham cohort access and Bristol Myer Squibb (BMS) that financially supported the assessment of biomarkers but was not involved in the analysis of the study or interpretation of the results.

All co-authors are independent of these funding sources.

Conflicts of interest

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive license (or non-exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in BMJ editions and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence.

All co-authors had full access to all of the data (including statistical reports and tables) in the study and take the responsibility for the integrity of the data and the accuracy of the data analysis.

Authorship

All authors fulfill the 4 following criteria:

1) Substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; AND 2) Drafting the work or revising it critically for important intellectual content; AND 3) Final approval of the version to be published; AND 4) Agreement to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.