M Conforti
email: matteo.conforti@ing.unibs.it

F Baronio

C De Angelis

M Marangoni

G Cerullo

Theory and experiments on multistep parametric processes in nonlinear optics

We present a comprehensive model for the description of different types of parametric interactions, associated with simultaneous phase-matching of several optical processes: the so-called multistep parametric interactions. Our approach is based on a recently derived single wave broadband equation that is able to describe general quadratic nonlinear optical interactions and can be solved with modest computational effort. We compare theoretical results with experiments on simultaneous second and third harmonic generation performed in periodically poled lithium tantalate crystals.

Introduction

Energy transfer between waves oscillating at different frequencies is one of the fundamental concepts in nonlinear optics and has been studied intensively since the early 1960s [START_REF] Franken | Generation of optical harmonics[END_REF]. Parametric processes naturally take place in nonlinear materials where the lowest-order nonlinear contribution to the polarization of the medium is quadratic in the electric field (P ∝ χ (2) E 2 ) and involve several waves at different frequencies [START_REF] Boyd | Nonlinear Optics[END_REF]. In order to be efficient the parametric coupling requires special relations between the phase velocities of the interacting waves to be satisfied, the so called phase matching conditions. Phase matching is either realized exactly by birefringence, or by periodically reversing the sign of the nonlinear coefficient χ (2) by quasi phase matching (QPM). The QPM technology permits also to engineer the nonlinear response of the medium and to obtain "nonlinear metamaterials" where the quadratic properties can be designed at will [START_REF] Fejer | Quasi-phasematched second harmonic generation: tuning and tolerances[END_REF][START_REF] Conforti | From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion[END_REF].

Multistep parametric interactions [START_REF] Saltiel | Multistep parametric processes in nonlinear optics[END_REF] represent a special type of second-order parametric process that involves several different second-order nonlinear interactions and is characterized by at least two different phase matching parameters. For example it is possible to obtain cascaded third harmonic generation (THG) from two concurrent processes: second harmonic generation (SHG) ω + ω = 2ω and sum frequency mixing (SFM) ω + 2ω = 3ω.

There exists a wide variety of multistep parametric processes, such as third harmonic generation [START_REF] Luo | Simultaneously efficient blue and red light generations in a periodically poled L i T aO 3[END_REF], two-color parametric interaction [START_REF] Couderc | Trapping of a weak probe through coupling with a two-color quadratic spatial soliton[END_REF], fourth harmonic generation [START_REF] Sukhorukov | Multistep cascading and fourth harmonic generation[END_REF], wavelength conversion [START_REF] Cardakli | Tunable all-optical time-slot-interchange and wavelength conversion using differencefrequency-generation and optical buffers[END_REF] and many others [START_REF] Saltiel | Multistep parametric processes in nonlinear optics[END_REF]. In general it is hard to phase match different processes by traditional methods such as birefringence phase matching, but QPM structures with chirped and aperiodic gratings supply the requested flexibility. This variety of phenomena requires a variety of mathematical models, that are usually derived in the frame of coupled slowly varying field envelopes. These models consist of quadratic nonlinear terms such as A 2 n , A m A n or A m A * n (in Ref. [START_REF] Saltiel | Multistep parametric processes in nonlinear optics[END_REF] a non exhaustive list of the most frequent cases is reported). It is clear that a deep knowledge of the physical setting is required in order to select the correct model, and all the matched processes have to be considered, in order to obtain consistent results: for example in a periodically poled lithium niobate (PPLN) sample several spatial harmonics of the grating can match several different processes. Sometimes the interacting waves have bandwidths that can be comparable with their frequency separation, making the division into harmonics somewhat arbitrary . For example one can ask if in a SFM process ω 1 + ω 2 = ω 3 , where ω 1 and ω 2 have a frequency separation comparable with their bandwidth, is more correct to model the process as a two or three wave process.

A different approach to the study of multistep quadratic phenomena is to exploit a single nonlinear envelope equation (NEE) for the entire frequency interval involved in the process.

Such a nonlinear envelope equation was derived recently and was shown to be accurate to model several quadratic processes [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF][START_REF] Conforti | Ultra-broadband optical phenomena in quadratic nonlinear media[END_REF]. This model is particularly suited for the purpose of numerically simulating the optical pulse propagation: in fact the computational resources required for the solution of this model are of the order of magnitude of those required for standard coupled wave equations.

In this article we show how to exploit the NEE for the description of multistep parametric processes. We show that a single wave equation can describe with great accuracy phenomena spanning several optical octaves. We also show that the relevant coupled wave models can be derived directly from the NEE with minimal assumptions.

The rest of the paper is organized as follows: in Section 2 we describe the single wave model and we outline the derivation of the the three wave equation describing cascading and third harmonic generation in a periodically poled lithium tantalate crystal (PPLT).

Finally we present out conclusions in section 4.

2. Nonlinear χ (2) envelope equation

The starting point of our analysis is the following equation [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF][START_REF] Conforti | Ultra-broadband optical phenomena in quadratic nonlinear media[END_REF]:

∂A ∂z + iDA = -iG(z) χ (2) ω 0 4n 0 c 1 - i ω 0 ∂ ∂t A 2 e iω 0 t-iβ 0 z +2|A| 2 e -iω 0 t+iβ 0 z , ( 1
)
where c is the velocity of light in vacuum, ε(ω) = 1 + χ(ω), χ(ω) is the linear electric susceptibility and k(ω) = (ω/c) ε(ω) is the propagation constant, A = A(z, t) is the electric field envelope as defined in [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF], ω 0 is a reference frequency, 2) is the nonlinear coefficient and G(z) describes the spatial modulation of the nonlinear material. The dispersive operator is defined as

n 0 = n(ω 0 ) is the reference refractive index, β 0 = k(ω 0 ) is the reference wavevector, χ (
D = ∞ m=1 1 m! k m (-i ∂ ∂t ) m where k m = ∂ m k ∂ω m (ω 0 ). It is possible, if needed, to
consider other and more general expressions of the dispersive operator [START_REF] Sh | Pade approximant for the refractive index and nonlocal envelope equations[END_REF].

The nonlinear envelope equation ( 1), first order in propagation coordinate, provides a powerful means of describing light pulse propagation in dispersive quadratically nonlinear media. Using an approach exploited in [START_REF] Brabec | Nonlinear optical pulse propagation in the single-cycle regime[END_REF] for cubic nonlinear media, equation

(1) was first reported in [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF], then derived rigorously extending its range of validity in [START_REF] Conforti | Ultra-broadband optical phenomena in quadratic nonlinear media[END_REF] . Equation ( 1) can be solved easily by split-step Fourier method exploiting the exact frequency dependence of the propagation constant for the linear step, and a fourth order Runge-Kutta scheme for the nonlinear step.

It has been pointed out that evolution equations like (1) describe unidirectional elec-tromagnetic propagation with a surprising degree of accuracy, almost without restrictions on the frequency extent of the signals [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF][START_REF] Brabec | Nonlinear optical pulse propagation in the single-cycle regime[END_REF][START_REF] Kolesik | Nonlinear optical pulse propagation simulation: From Maxwell's to unidirectional equations[END_REF][START_REF] Kinsler | Optical pulse propagation with minimal approximations[END_REF]: in this sense they can be considered as forward Maxwell equations [START_REF] Housakou | Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers[END_REF].

In order to show the validity of Eq. ( 1) over a bandwidth of several optical octaves and its ability to model multistep parametric processes, we derive from (1) the three-wave model for the description of cascading third harmonic generation [START_REF] Saltiel | Multistep parametric processes in nonlinear optics[END_REF][START_REF] Qin | Theoretical investigations of efficient cascaded third-harmonic generation in quasi-phase-matched and -mismatched configurations[END_REF]. We consider an electric field envelope composed of three waves:

A(z, t) = A 1 (z, t)e i(Ω-ω 0 )t-i(q 1 -β 0 )z + A 2 (z, t)e i(2Ω-ω 0 )t-i(q 2 -β 0 )z + A 3 (z, t)e i(3Ω-ω 0 )t-i(q 3 -β 0 )z , ( 2
)
where Ω is the fundamental frequency and q 1 , q 2 , q 3 are the propagation constants at fundamental, second and third harmonic.

In order to handle the dispersive term in a straightforward manner we first Fourier transform the Eq. ( 1):

∂ Â(ω) ∂z + i[k(ω + ω 0 ) -k 0 ] Â(ω) = N L (3)
where N L stands for the Fourier transform of the nonlinear terms. The difference in the frequency dependence between Â(.) and k(.) is given by the fact that A is the envelope and is centered around zero in frequency domain, whereas the wavenumber k has the same dependence of the total electric field, that is centered around ω 0 . It must be kept in mind that this frequency offset is totally arbitrary and it is not assumed that the band of the envelope must be narrow with respect to ω 0 [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF].

Now inserting (Fourier transform of) ansatz ( 2) in (3) we obtain:

3 m=1 ∂ Âm ∂z (ω -(mΩ -ω 0 )) + (4) i[k(ω + ω 0 ) -q m ] Âm (ω -(mΩ -ω 0 )) e -i(qm-β 0 )z = N L
that after inverse Fourier transform reads:

3 m=1 ∂A m ∂z + iD m A m e i(mΩ-ω 0 )-i(qm-β 0 )z = -iG(z) χ (2) ω 0 4n 0 c 1 - i ω 0 ∂ ∂t A 2 1 e i(2Ω-ω 0 )t-i(2q 1 -β 0 )z +2A 1 A 2 e i(3Ω-ω 0 )t-i(q 1 +q 2 -β 0 )z +2A 2 A * 1 e i(Ω-ω 0 )t-i(q 2 -q 1 -β 0 )z +2A 3 A * 2 e i(Ω-ω 0 )t-i(q 3 -q 1 -β 0 )z +2A 3 A * 1 e i(2Ω-ω 0 )t-i(q 3 -q 1 -β 0 )z + R , ( 5
)
where R is the remainder including terms oscillating at other frequencies. The dispersive terms D n represent inverse Fourier transforms of the real propagation constant k(ω + ω 0 ) expanded in power series around the reference wavenumbers q n .

By grouping together terms oscillating at the same frequency we obtain:

∂A 1 ∂z + iD 1 A 1 = -iG(z) χ (2) Ω 2n 0 c A 2 A * 1 e -i∆k 1 z + A 3 A * 2 e -i∆k 2 z , ∂A 2 ∂z + iD 2 A 2 = -iG(z) χ (2) Ω 2n 0 c A 2 1 e i∆k 1 z + 2A 3 A * 1 e -i∆k 2 z , ∂A 3 ∂z + iD 3 A 3 = -iG(z)3 χ (2) Ω 2n 0 c A 1 A * 2 e i∆k 2 z , (6)
where ∆k 1 = q 2 -2q 1 and ∆k 2 = q 3 -q 2 -q 1 . Dispersive operators are usually truncated at second order , so we take D n ≈ Dn = -ik 1n ∂ t -1/2k 2n ∂ tt . We can now expand the nonlinear coefficient modulation G(z) in Fourier series truncated at the third spatial harmonic: G(z) ≈ 2 π e i2π/Λz + 2 3π e i2π/Λz•3 where Λ is the QPM period. Usually in ferroelectric materials such as lithium niobate and lithium tantalate, the dispersion is such that first harmonic matches SHG and third harmonic matches THG, so it is possible to write:

∂A 1 ∂z + i D1 A 1 = -iσ 1 A 2 A * 1 e -iδk 1 z -iσ 3 A 3 A * 2 e -iδk 2 z ∂A 2 ∂z + i D2 A 2 = -iσ 2 A 2 1 e iδk 1 z -iσ 4 A 3 A * 1 e -iδk 2 z , ∂A 3 ∂z + i D3 A 3 = -iσ 5 A 1 A * 2 e iδk 2 z , (7)
where

δk 1 = ∆k 1 -2π/Λ and δk 2 = ∆k 2 -2π/Λ • 3 are the residual mismatches, σ 1,2 = d ef f,I Ω n 0 c , σ j = (j -2)d ef f,II Ω n 0 c [j = 3, 4, 5] and d ef f,I = χ (2) 2 2 π , d ef f,II = χ (2) 2 2 3π .
This system is the usual coupled wave equations describing cascading THG, as can be obtained by Maxwell's equations [START_REF] Saltiel | Multistep parametric processes in nonlinear optics[END_REF][START_REF] Qin | Theoretical investigations of efficient cascaded third-harmonic generation in quasi-phase-matched and -mismatched configurations[END_REF]. It is worth noting that the derived equations do not depend on ω 0 .

In order to show the validity of our approach, we simulated the propagation of a femtosecond pulse in a L = 5mm long periodically poled stoichiometric lithium tantalate sample (PPSLT). To model the refractive index dispersion we employed a Sellmeier model fitted from experimental data [START_REF] Bruner | Temperature[END_REF] (temperature T = 25 o C) and nonlinear coefficient

d 33 = χ (2)
/2 = 10.6pm/V . In the numerical solution of Eq. ( 1) we inserted the exact dispersion relation k(ω). We assumed a first order quasi phase matching (QPM) grating, with a period Λ = 17.4µm. We injected a T = 120f s FWHM long gaussian pulse, centered around 1400nm, with I = 20GW/cm 2 peak intensity. The corresponding residual phase mismatch is δk 1 = 10000m -1 and δk 2 = 8002m -1 . In the simulation we set the reference frequency ω 0 to be equal to the second harmonic of the input pulse: in this way the second harmonic is stationary in the reference frame (z ′ , τ ). The change of reference frame is introduced by the change of variables z = z and τ = t -k 1 z [START_REF] Conforti | Nonlinear envelope equation for broadband optical pulses in quadratic media[END_REF].

Figure 1 shows the amplitude of the electric field envelope. The two concurrent processes of second and third harmonic generation appear clearly. The crystal dispersion makes the fundamental frequency (FF), second harmonic (SH) and TH (third harmonic) pulses to separate each other during propagation. The FF pump depletion is significant with a conversion efficiency of 37% to second harmonic and 9.5% to third harmonic.

We then simulated the same set up with coupled wave model [START_REF] Couderc | Trapping of a weak probe through coupling with a two-color quadratic spatial soliton[END_REF]. We approximated dispersion up to second order, calculating the relevant parameters from the Sellmeier relation. For inverse group velocities we used the following values (in s/m) k 11 = 7.2136 • 10 -9 , k 12 = 7.538 • 10 -9 , k 13 = 8.293e • 10 -9 ; whereas for group velocity dispersion we used (in

s 2 /m) k 21 = 1.0408 • 10 -25 , k 22 = 3.7514 • 10 -25 , k 23 = 7.9607 • 10 -25 .
To compare the results we filtered the envelope A around FF, SH and TH: the outcome of the comparison is shown in Fig. 2. The agreement is almost perfect for fundamental and second harmonic, whereas small differences can be noted on third harmonic. This can be ascribed to the higher dispersion at short wavelength, that makes the truncation at second order in Eqs. [START_REF] Couderc | Trapping of a weak probe through coupling with a two-color quadratic spatial soliton[END_REF] sightly inaccurate.

Experimental results

For the experimental observation of the cascading SHG and THG we chose a PPSLT crystal. focused in the middle of the PPSLT crystal in order to obtain an effective interaction length of 2mm. This mild focusing enables to neglect both linear and nonlinear spatial effects. At the output of the sample the frequency spectra of the pulses are detected by a spectrometer. Figure 3a) shows the outcome of the experiments performed with a FF pulse centered around 1390 nm, with a peak intensity I=15GW/cm 2 . The three periods 17.7µm, 18.1µm and 18.5µm phase match SH waves at 723.5nm, 730.9nm and 738nm, respectively. A discrete amount of non phase-matched second harmonic is visible between 650nm and 720nm. We can also see the generation of third harmonic at 475nm, 478.5nm and 482nm, respectively.

The amplitude of the TH peaks is around 5% of SH peaks.

Figure 3b) shows the results of the numerical solution of Eq. ( 1): we can see a good agreement with the experimental data. The two insets show the comparison between experiments (solid curves) and numerics (dashed curves) around the second and third harmonics.

At the SH numerical and experimental curves are perfectly superimposed: peak positions and linewidths are perfectly captured by the theory. For what concerns third harmonic, we can see a perfect reproduction of the peak positions, but with a larger experimental linewidth. This discrepancy can be ascribed to the insufficient resolution of the spectrum analyzer used in the experiments.

Conclusion

We presented a model for the description of different types of parametric interactions in nonlinear optics which are associated with simultaneous phase-matching of several optical 10 processes, based on a recently derived single wave boradband equation. This model is able to describe general quadratic nonlinear optical interactions and can be solved with modest computational efforts. We compared theoretical results with experiment on simultaneous second and third harmonic generation performed in periodically poled lithium tantalate crystals.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Propagation of a femtosecond pulse in a PPLT crystal. a) Evolution of the field

Fig. 3 .

 3 Fig. 3. Experimental (a) and numerical (b) SH and TH spectra at the crystal output


MD, MSc Jean Devignes
email: j.devignes@chru-nancy.fr

PhD Malika Smaïl-Tabbone

Alex Hervé

Geoffroy Cagninacci

PhD Marie-Dominique Devignes

MD, MSc Thomas Lecompte

MD Stéphane Zuily

MD Denis Wahl

Extended aPL persistence beyond 12 weeks 1 11 pages, 2 tables, 1 figure Extended persistence of antiphospholipid antibodies beyond the twelve-week time interval: association with baseline antiphospholipid antibodies titres

Keywords: Antiphospholipid antibodies, Antiphospholipid syndrome, Lupus anticoagulant, Anticardiolipin antibodies

Introduction

The confirmation time interval for presence of antiphospholipid antibodies (aPL) has been extended to 12 weeks since epiphenomenal antibodies may disappear after six weeks. Our aim was to analyse extended persistence of aPL positivity beyond the 12-week interval.

Methods

We retrospectively analysed our database of 23,856 aPL test samples collected between 2005 and 2017 from 17,367 consecutive patients. Two groups of patients were identified among aPLpositive patients, confirmed at 12 weeks: with or without extended persistence beyond confirmatory testing. Percentages of extended persistence are given according to the initial aPLpositivity profiles and baseline laboratory variables are compared between the two groups.

Results

Three hundred and twenty-seven patients confirmed aPL-positive had subsequent testing. The vast majority of them displayed extended persistence on the long term: 89.6% and up to 97.9% for patients with initial triple positivity. In extended persistent positive patients, there were more LA positive initial samples, and baseline LA test values and IgG aCL titres were higher than in non-persistent positive patients.

Conclusion

Data from a large database of an aPL referral laboratory showed that the time interval of 12 weeks defining persistence of aPL positivity was appropriate for the majority of patients. Furthermore, we found baseline features associated with extended persistence.

Introduction

According to the revised Sapporo classification for definite antiphospholipid syndrome (APS) [START_REF] Miyakis | International consensus statement on an update of the classification criteria for definite antiphospholipid syndrome (APS)[END_REF], the classification criteria are met when at least one clinical criterion is present together with at least one antiphospholipid antibodies (aPL) positive test persisting over time. Current laboratory criteria are the lupus anticoagulant (LA), IgG or IgM anti-cardiolipin antibodies (aCL), or IgG or IgM anti-β2-glycoprotein I antibodies (aβ2GPI). The time interval defining persistence was extended from six [START_REF] Wilson | International consensus statement on preliminary classification criteria for definite antiphospholipid syndrome: report of an international workshop[END_REF] to 12 weeks [START_REF] Miyakis | International consensus statement on an update of the classification criteria for definite antiphospholipid syndrome (APS)[END_REF] in 2006 because of concerns that epiphenomenal aPL -as in infections [START_REF] Abdel-Wahab | Systematic review of case reports of antiphospholipid syndrome following infection[END_REF] may disappear in some instances after more than six weeks. However, the 12-week interval was mostly based on expert opinion and little data are available about persistence of aPL positivity beyond this time frame. Indeed, some studies have questioned the relevance of using the 12-week instead of the six-week interval, but they included a limited number of subjects and lacked clear statistical significance [START_REF] Park | Clinical application of revised laboratory classification criteria for antiphospholipid antibody syndrome: is the follow-up interval of 12 weeks instead of 6 weeks significantly useful?[END_REF][START_REF] Solano | Comparison of the 1999 Sapporo and 2006 revised criteria for the classification of the antiphospholipid syndrome[END_REF]. Published data on confirmation of aPL positivity at 12 weeks were conducted in patients with APS. In particular, it was shown that most (53 out of 54) subjects with triple positivity at initial testing had their aPL profile confirmed after 12 weeks, whereas single-positive and double-positive groups had their positivity confirmed only for 21 out of 53 and 42 out of 50 subjects, respectively [START_REF] Pengo | Confirmation of initial antiphospholipid antibody positivity depends on the antiphospholipid antibody profile[END_REF][START_REF] Pengo | Diagnostics and treatment of thrombotic antiphopholipid syndrome (APS): a personal perspective[END_REF]. However, no systematic study did analyse further testing beyond the confirmation at 12 weeks to assess persistence of aPL on the long term.

We retrospectively analysed our database derived from aPL laboratory testing in a real world setting of an aPL referral laboratory, with the main objective to evaluate the course of aPL positivity beyond confirmatory testing at 12 weeks; the secondary objective was to identify baseline features associated with such an extended persistence.

Patients, material and methods

Laboratory tests were performed according to established guidelines with the same tests and reagents for the LA and aCL tests throughout the study period (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017), whereas the aβ2GPI test was performed routinely from 2011 onwards. For LA testing, two screening coagulation tests were performed: dilute Russell's viper venom time (dRVVT LAC Screen ® , Siemens) and Activated Partial Thromboplastin Test (PTTLA ® Diagnostica Stago) [START_REF] Pengo | Update of the guidelines for lupus anticoagulant detection. Subcommittee on lupus anticoagulant/antiphospholipid antibody of the scientific and standardisation committee of the international society on thrombosis and haemostasis[END_REF]. All samples from patients anticoagulated with vitamin K antagonists (VKAs) that could not be tested with DRVVT have been tested with PTTLA, except for those on heparin treatment. For the two tests, positivity was defined by a LA ratio (clotting time of the patient sample relative to that of a locally prepared pool of plasmas obtained from healthy blood donors) greater than 1.2. In case of positive screening, confirmation tests were performed after 1:1 mixing with pooled plasma from healthy blood donors, using the same threshold ratio of 1.2. The LA test was deemed positive when at least one of the two LA ratios (PTTLA or dRVVT) was greater than the threshold, with mixing and confirmation tests alike.

Regarding aCL antibodies, IgG and IgM ELISAs (Enzyme-Linked ImmunoSorbent Assays) were performed according to guidelines applicable at the time of the testing [START_REF] Lakos | International consensus guidelines on anticardiolipin and anti-β2-glycoprotein I testing: report from the 13th International Congress on Antiphospholipid Antibodies[END_REF]. IgG and IgM aβ2GPI

ELISAs were performed with a commercially available kit (Orgentec ® ). For both ELISAs, the calibration was performed according to Harris's standards and the thresholds for positivity were defined as the 99 th percentile of the titre measured in a population of more than 170 healthy blood donors [START_REF] Wahl | Definition and significance of high positivity aCL ELISA[END_REF]. The cut-off values for aCL positivity were equal to 14 GPL and 5 MPL for IgG and IgM respectively, and those for aβ2GPI positivity were equal to 8 GPL and 8 MPL for both IgG and IgM.

Samples were deemed aPL-positive when at least one laboratory test was positive (above thresholds) and aPL-negative when all performed tests were negative (below thresholds).

A relational database was built to store, organize and query the data related to 23,856 samples from 17,367 consecutive patients visiting the Region Lorraine aPL referral centre at the Nancy University Hospital between 2005 and 2017. Data were directly imported from an extraction of records from our laboratory information system (GLIMS™; MIPS) into the database (MySQL system), thanks to a customized wrapper. SQL (Standard Query Language) procedures were developed on the database firstly to classify each sample as aPL-positive or not, depending on the results of the laboratory tests, and secondly to identify different subsets of patients according to the course of their aPL positivity. The underlying algorithm is displayed in Figure 1. The patient subgroup of interest was identified using five questions as (i) displaying an initial aPL-positive test with subsequent testing (questions 1&2 = yes), (ii) having their aPL positivity confirmed at an interval of at least 12 weeks (question 3 = yes; median time window: 26 weeks), and (iii) undergoing aPL testing thereafter (question 4 = yes). Finally (question 5), these patients were divided into two groups according to the persistence of positive aPL tests until the last studied sample (Extended Persistent Positive: EPP group), or not (Non-Persistently Positive: NPP group).

EPP and NPP groups were compared regarding to qualitative data (counts with percentages) using the exact Fisher test. Quantitative data, calculated for initial positive tests only, were presented as median and interquartile range (IQR=Q3-Q1) because of the skewed distribution of the values and were compared using a non-parametric Wilcoxon test. A p-value < 0.05 was considered significant.

Results

Course of aPL positivity beyond the 12-week time interval

To group patients according to their aPL positivity trajectories, the 23,856 sample results related to 17,367 consecutive patients (studied as per routine between 2005 and 2017) were computationally processed as described in Figure 1. We found 82 % of the patients negative for aPL (n=14,617), and 2074 patients with a positive result that was not controlled at least 12 weeks apart. Out of the 676 remaining patients, 115 (17%) were not found positive at the confirmatory testing and out of the 561 confirmed ones, 234 were not further tested. Thus, there were 327 aPLpositive patients, with confirmation at least 12 weeks apart and with subsequent testing. These patients have been followed up beyond confirmatory testing for a median duration of 94 weeks (IQR: 173, max: 393). Negative testing occurred for 34 out of the 327 patients during the followup period, yielding a percentage of extended persistence beyond confirmatory testing of 89.6%.

Extended persistence and initial aPL profiles

Extended persistence was analysed with regard to initial aPL profile for a subgroup of 124 patients assayed with the three laboratory tests from 2011 onwards. Results are detailed in Table 1. In this group, the total percentage of extended persistence was 96.0% for a median follow-up duration of 56 weeks (IQR: 85, max: 278). Interestingly, this percentage was around 97% for patients displaying double and triple positivity and slightly (but not significantly) lower (93.3%) for those with single positivity. Among patients with initial single positivity, those with category IIa (LA only, 14 patients) were all extended persistent positive (EPP). In the double-positive group, the only non-persistent positive (NPP) patient had both LA and IgM aCL positivity at initial testing, and in the triple-positive group, the only NPP patient had the IgG isotype of aCL and combined IgG and IgM isotypes of a2GPI at initial testing.

Baseline features and extended persistence

Comparisons of demographic and laboratory data between the 293 EPP and 34 NPP patients are presented in Table 2. The number of samples per patient for aPL testing was similar in the EPP and NPP groups (5 [IQR: 3] vs. 5 [IQR: 4] respectively). Laboratory results were analysed according to initial positivity. In comparison to the NPP group, there were significantly more LA positive samples in the EPP group, and baseline LA ratios and IgG aCL titres were significantly higher.

Discussion

In the revised Sapporo criteria [START_REF] Miyakis | International consensus statement on an update of the classification criteria for definite antiphospholipid syndrome (APS)[END_REF], the time-interval for aPL confirmation was extended from six to 12 weeks on an empirical basis and expert opinion. Our retrospective study, conducted in a real world setting of an aPL referral laboratory, shows for the first time, that beyond the confirmation of aPL positivity at least 12-week apart, patients remain positive on subsequent testing in more than 89% of the cases up to 393 weeks of follow-up (median value: 94 weeks). Thus, these results fully support the choice that was made for the revised criteria. Whether a time-interval beyond six weeks but shorter than 12 is suitable remains to be studied in a prospective manner.

The subgroup of 124 patients tested with the three aPL tests enabled us to analyse extended persistence of aPL in relation with the initial aPL-positivity profile. Over their relatively short follow-up period (about one year on average), we observed very high percentages of extended persistence of aPL (> 93%) regardless of initial single, double or triple positivity. Moreover, in the EPP group, we detected all three possible types of initial double positivity (aCL and aβ2GPI, LA negative), (LA and aCL, aβ2GPI negative) and (LA and aβ2GPI, aCL negative). The first one was the most frequent (19 out of 31) and corresponds to the only double positive combination described by Pengo and Denas in their recent review [START_REF] Pengo | Diagnostics and treatment of thrombotic antiphopholipid syndrome (APS): a personal perspective[END_REF].

The complete dataset of 293 EPP and 34 NPP patients was large enough to study the baseline values of the LA and aCL tests associated with extended persistence. Patients found LA positive, and those with high IgG aCL titres or with high LA ratios (which can be considered as result of a high titre as well) at initial testing, were significantly more frequent in the EPP than in the NPP group. This extends existing reports concerning confirmation at 12 weeks, showing that presence of a LA is associated with aPL persistence at that time-point [START_REF] Pengo | Confirmation of initial antiphospholipid antibody positivity depends on the antiphospholipid antibody profile[END_REF] and that patients with transient aPL-positivity have lower aCL titres than patients with confirmed aPL-positivity [START_REF] Wahl | Definition and significance of high positivity aCL ELISA[END_REF].

One limitation of our study is that it was a retrospective evaluation due to the real-world setting.

However, such a setting ensured thorough exhaustiveness and reduced recruitment bias since all consecutive samples were analysed. Another limitation was that aβ2GPI antibodies were not tested routinely before 2011. Thus, aβ2GPI single-positive samples before 2011 were classified as aPL negative. This could potentially lead to ignoring positive patients from our study or underestimating the percentage of extended persistence. However, this uncertainty does not contradict the conclusion of the paper.

Conclusion

We report what is to the best of our knowledge the first real-world evaluation of the 12-week time interval required for APS laboratory classification criteria, performed with a large sized dataset.

Our data indicate that among aPL patients controlled after 12 weeks and confirmed positive, a vast majority remains persistently positive over time. Thus, the time frame of 12 weeks to control the initial aPL positivity seems appropriate to identify patients who will be persistently positive for aPL over the long run. Moreover, the database system and the analytical procedures developed for this study make it possible to expand our aPL patient registry in the future and consider further large-scale studies to improve the diagnosis, treatment and follow-up of these patients.

  The confirmatory testing was performed at least 12 weeks apart the initial one.Qualitative data are presented as counts and percentages of corresponding group: n (%). Except for age that is presented as mean ± standard deviation, quantitative data are presented as median value and interquartile range [IQR] of positive tests: m [IQR] because of their skewed distribution. In bold are highlighted the differences between the PP and NPP groups that are statistically significant (p<0.05). aCL: anticardiolipin antibodies, aCL positivity cut-offs were equal to 14 and 5 GPL and MPL for IgG and IgM respectively; LA: Lupus anticoagulant, LA positivity cut-off was a ratio ≥ 1.2 for dRVVT or PTTLA screening tests (ratio of clotting time of the patient sample relative to that of a locally prepared pool of plasmas obtained from healthy blood donors).

Figure 1 :

 1 Figure 1: Flowchart of patient features according to the results of aPL testing.

Table 1 : Extended persistence of aPL beyond confirmatory testing at least 12 weeks apart, according to initial aPL positivity profile. EPP: Extended Persistent aPL-positive, NPP: Non-Persistent aPL-positive.

 1 IgM only; 10 both IgG and IgM isotypes; one IgG aCL and IgM a2GPI, 8 IgG aCL and combined IgG and IgM isotypes for a2GPI (including the NPP sample), 6 IgM aCL and combined IgG and IgM isotypes for a2GPI, 3 combined IgG and IgM isotypes for aCL and IgG a2GPI, and one combined IgG and IgM isotypes for aCL and IgM a2GPI.

	Group of patients	Total number	EPP	NPP	% of extended persistence
	Subgroup tested for LA, aCL and a2GPI	124	119	5	96.0 %
	Single Positive	45	42	3	93.3%
	Class IIa (LA)	14	14	0
	Class IIb (aCL)	18	16	2
	Class IIc (a2GPI)	13	12	1
	Double Positive	31	30	1	96.8%
	aCL and a2GPI, LA negative	19	19	0
	LA and aCL, a2GPI negative	7	6	1
	LA and a2GPI, aCL negative	5	5	0
	Triple Positive	48*	47	1	97.9%
	*: Isotype distribution:
	similar for aCL and IgG a2GPI; 15 IgG only; 4

Table 2 : Characteristics of patients and baseline laboratory variables at the time of the initial positive testing according to extended persistence (EPP) or not (NPP) of aPL positivity beyond confirmatory testing.

 2

	Characteristics	Total (n=327)	EPP (n=293)	NPP (n=34)	P value
	Demographic variables
	Age (years)	45.3 ±16.4	45.2 ±15.2	46.2 ±17.8	0.64
	Women	242 (74.0%)	214 (73.0%)	28 (82.4%)	0.30
	Laboratory variables
	LA positive	208 (63.6%)	197 (67.2%)	11 (32.4%)	0.0001
	LA ratio (PTTLA)	2.03 [1.12]	2.04 [1.13]	1.24 [0.06]	0.001
	LA ratio (dRVVT)	1.51 [0.57]	1.53 [0.58]	1.35 [0.17]	0.02
	aCL positive	206 (63.0%)	180 (61.4%)	26 (76.5%)	0.09
	IgG titre	44 [94]	47 [92]	24 [10]	0.015
	IgM titre	13 [12]	13 [12]	10 [10]	0.17

Acknowledgements

Funding Support: We are grateful to Association des Chefs de Service for supporting AH. This work was funded by the Contrat de Plan Etat Région (IT2MP: Innovation Technologiques, Modélisation et Médecine Personnalisée) involving FEDER, Région Grand-Est, INSERM, Inria, CNRS and Université de Lorraine.


Y Marrocchi

L G Vacher

M Piralla

THERMAL CONDITIONS OF FLUID CIRCULATION IN HYDRATED ASTEROIDS

Introduction: Asteroids are super-kilometer-sized bodies formed by gravitational instability in turbulent regions of the protoplanetary disk. Hydrated C-type asteroids -the most common bodies in the main asteroid belt experienced subsequent heating due to heat released by the decay of 26 Al. However, quantitative estimation of their thermal histories has not yet been possible, leading to considerable uncertainty on the physical characteristics of earlyformed primitive rocky bodies. Aqueously-formed carbonaceous chondritic minerals such as carbonates represent direct snapshots of the asteroidal fluids and can in theory be used to decipher their thermal evolution. However, determining carbonate precipitation temperatures requires knowing the O-isotopic compositions of their parent fluids, which itself requires to know the carbonate precipitation temperatures, leading to a seemingly endless problem.

Here we report in situ O-isotopic analyses of Ca-carbonates and propose a new isotopic alteration model that reconciles petrographic observations and formation temperatures of CM carbonates.

Methods: Detailed observations of calcite grains in polished sections of the CM2 Murchison and Mukundpura were made using a scanning electron microscope (SEM) JEOL JSM-6510 equipped with an Energy Dispersive Xray (EDX) Genesis detector. Their oxygen isotopic compositions were determined using a CAMECA ims 1280 HR2 ion microprobe at CRPG (Nancy, France).

Results: Only calcite grains surrounded by serpentine (hereafter T1 calcite) were found in the two polished sections of Murchison. O-isotopic measurements show that Murchison's T1 calcites have δ 18 O values between 25.5 and 41.5‰, δ 17 O values between 14.5 and 23.5‰ and Δ 17 O values between -1.1 and 1.8‰ (n = 24). In contrary, two types of carbonates were found in the polished section of Mukundpura: (i) T1 calcite grains and (ii) polycrystalline grains of calcite containing Fe-Ni sulfide inclusions (hereafter T2 calcite). O-isotopic analyses on Mukundpura's Ca-carbonates reveal that (i) T1 calcites have δ 18 O values between 27.4 and 38.2‰, δ 17 O values between 13.3 and 18.5‰ and Δ 17 O values between -1.4 and -0.2‰ (n = 8), whereas (ii) T2 calcites have δ 18 O values between 12.4 and 18.6‰, δ 17 O values between 4.2 and 7.3‰ and Δ 17 O values between -4 and -0.9‰ (n = 13).

Discussion: Our data define a mass-independent trend that is similar, within errors, from those commonly reported in different CM chondrites [1][2][3]. In first approximation, this trend suggest that the O-isotopic compositions of CM carbonates is essentially controlled by variable degrees of isotopic exchanges between 16 O-rich anhydrous silicates and a 17-18 O-rich fluid [4]. However, when considering only T1 calcites, their O-isotopic compositions define a distinct trend that is indistinguishable, within errors, from the TFL (i.e., average Δ 17 O = -0.4 ± 1.0). Conversely, T2 calcites exhibit clustered δ 17 Oδ 18 O values with average Δ 17 O of -2.6 ± 1.0, which artificially generate the aforementioned mass-independent trend. This demonstrates that petrographic observations of carbonates are essential for understanding the O-isotopic evolution of asteroidal fluids and the constraints they bear.

Direct estimations of the O-isotopic composition of asteroidal alteration fluids was possible thanks to CO 2 clumped-isotope thermometry (Δ 47 , [5]). Based on measurements performed on the T1 calcite-dominated CM chondrites Murchison and Murray, this led to alteration fluids ranging from δ 18 O = 2 ‰ and Δ 17 O = -0.55 ‰ to δ 18 O = 6.8 ‰ and Δ 17 O = -0.49 ‰. As these fluids experienced oxygen isotopic exchanges with the anhydrous CM chondrite protolith (δ 18 O = -3.8 ‰, δ 17 O = -6.5 ‰; [6]), they define two trends that delimit the possible O-isotopic compositions of the alteration fluids. Considering these extreme trends, the precipitation temperatures of each calcite and serpentine can be calculated according their respective isotopic fractionation factor α. Our isotopic approach allows a formation sequence of CM secondary phases to be proposed, characterized by a temperature increase from T1 calcites < serpentines < T2 calcites. This implies that fluid circulation in CM parent body-(ies) occurs during a prograde evolution of the temperature, where T1 calcites formed first at T = 0-30 °C, followed by the precipitation of (Fe,S)-rich serpentine and tochilinite at T = 30-100°C and then T2 calcites at T = 110-140°C.