Benoît Blossier
email: blossier@ijclab.in2p3.fr

Jochen Heitger
email: heitger@wwu.de

Jan Neuendorf
email: jan.neuendorf@wwu.de

Teseo San

𝒂 J Heitger

𝒃 J Neuendorf 𝒃

José 𝒂

Direct access to hadronic decay parameters with twisted boundary conditions

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

Our exploratory study looks for direct access to the resonant hadronic transition amplitude without resorting to the Lüscher formalism. We study the decay 𝜓 (3770) → DD by applying partially twisted boundary conditions (P𝜃BCs) to the quenched charm quark, circumventing possible problems with final state interactions. If successful, we could compute the dependence of the transition amplitude on the charm-quark mass and test the predictions made by phenomenological quarkpair-creation models. Finally, we study if and to what extent an extraction of the excited state 𝜓 (3770) is necessary for this analysis.

The 39th International Symposium on Lattice Field Theory (LATTICE2022) 8th-13th August 2022

Rheinische Friedrich-Wilhelms-Universität Bonn, Germany

Introduction

Investigating hadronic decays from lattice QCD is notoriously challenging. Indeed, analytical properties of decay amplitudes, like the poles in the complex plane that can be identified as resonances, are not straightforward to recover from correlation functions in Euclidean space. However, Lüscher's formalism and its various generalisations have made a breakthrough in the study of processes like the 𝜌 meson decay, or scatterings of meson-meson, meson-baryon and baryon-baryon states. As those approaches relate the decay parameters to the spectrum of interacting particles in finite volume, they require computations in several physical volumes to identify singular points on phase shift curves. As a consequence, the numerical work can be costly.

McNeile et al. [START_REF] Ukqcd Collaboration | Mixing of scalar glueballs and flavor singlet scalar mesons[END_REF] proposed a different method, which gives direct access to the decay matrix elements under the condition that the kinematical configuration is close to the threshold. In our exploratory work, we combine the approach in [START_REF] Ukqcd Collaboration | Mixing of scalar glueballs and flavor singlet scalar mesons[END_REF] with the use of P𝜃BCs so that the threshold condition can be established off the momenta quantisation in finite volume.

Particle decay under investigation

To apply the proposal in [START_REF] Ukqcd Collaboration | Mixing of scalar glueballs and flavor singlet scalar mesons[END_REF], we study the charmonium state 𝜓 (3770), which is a vector particle with quantum numbers I G (J PC ) = 0 -(1 --) and mass 𝑀 𝜓 (3770) = 3773.7(4) MeV [START_REF] Workman | Review of Particle Physics[END_REF]. We consider its main decay channel, 𝜓 (3770) → DD, depicted in fig. 1, with (Γ D 0 + Γ D ± )/Γ total > 90%. Since we do not account for isospin breaking (IB), we cannot distinguish between the neutral and charged D mesons in the final state. Then, the braket that we aim to compute is 𝑥 ≡ DD 𝜓 (3770) .

(2.1)

This decay is ideal for probing [START_REF] Ukqcd Collaboration | Mixing of scalar glueballs and flavor singlet scalar mesons[END_REF] because it occurs nearly at the threshold in an experiment, 𝑀 𝜓 (3770) -2𝑀 D 0 ∼ 44 MeV [START_REF] Workman | Review of Particle Physics[END_REF]. The relevant experimental spectrum for our work appears in fig. 2. Of course, the unphysical quark masses used in the simulations and a finite lattice spacing can affect the particular values of the masses seen in fig. 2, and they might even make the DD state heavier than 𝜓 (3770). As the initial and final states, 𝜓 (3770) and DD, are very close in mass, the twist angle required to fix the kinematical configuration at the threshold is expected to be small. Moreover, twisted boundary conditions boost each meson of the DD system in the opposite direction, increasing the system energy while it stays at rest.

Since we study a process with final-state interactions between the D and D mesons, we may only apply P𝜃BC to the quenched charm quark [START_REF] Sachrajda | Twisted boundary conditions in lattice simulations[END_REF], keeping finite-volume effects under control. The decay 𝜓 (3770) → DD has been thoroughly studied in experiments, and it offers a nice playground to test the reliability of theoretical frameworks like early effective quark models. In this project, we concentrate on the 3 𝑃 0 quark pair creation model [START_REF] Le Yaouanc | Naive quark pair creation model and baryon decays[END_REF], which was fruitful for understanding the dynamics of hadronic meson decay in two mesons [START_REF] Kokoski | Meson Decays by Flux Tube Breaking[END_REF]. It postulates an effective charm-quark mass dependence only through the wave functions that describe each of the three hadrons entering the process, and introduces a universal coupling 𝛾. In turn, we compute the matrix element

𝐻 𝑞 𝐻 𝑞 hℎ(1 - , 𝑛 = 3) 𝐸 [𝐻 𝑞 𝐻 𝑞 ]=𝑚( hℎ (1 - , 𝑛=3))
on the lattice, and compare the dependence on the mass 𝑚 ℎ of the heavy quark ℎ.

Methodology

We consider correlation functions of operators 𝑂 h qℎ𝑞 and 𝑂 hℎ , which work as interpolators of the 𝐷 D and 𝜓(3770) states, respectively. To extract the matrix element 𝑥 from the lattice, we need to study the asymptotic behaviour of the correlators whose diagrams are shown in fig. 3 [6],

C 𝜓 (3770) → 𝜓 (3770) (𝑡) ≡ 𝑂 hℎ (𝑡)𝑂 † hℎ (0) -→ 𝑍 2 𝜓 (3770) 𝑒 -𝑚 𝜓 (3770) 𝑡 , C DD →DD (𝑡) ≡ 𝑂 h qℎ𝑞 (𝑡)𝑂 † h qℎ𝑞 (0) -→ 𝑍 2 DD 𝑒 -𝐸 DD 𝑡 , C 𝜓 (3770) →DD (𝑡) ≡ 𝑂 h qℎ𝑞 (𝑡)𝑂 † hℎ (0) -→ ∑ 𝑡 𝑖 𝑍 𝜓 (3770) 𝑒 -𝑚 𝜓 (3770) 𝑡 𝑖 𝑥 𝑒 -𝐸 DD (𝑡-𝑡 𝑖 ) 𝑍 DD , (3.1)
where the full correlator for DD → DD process is

C DD →DD = C 2 D→D + C DD →DD, box . (3.2)
Here, hℎ(1 - , 𝑛 = 3) is the second radial excitation of the vector quarkonium while 𝐻 𝑞 is the ground-state pseudoscalar heavy-light meson. At 𝑚 ℎ = 𝑚 𝑐 , hℎ (1 - , 𝑛 = 3) ≡ 𝜓(3770) and 𝐻 𝑞 ≡ 𝐷.

Figure 3:

Diagrams needed for our study [START_REF] Mcneile | Hadronic decay of a vector meson from the lattice[END_REF].

The asymptotic behaviour of C 𝜓 (3770) →DD (𝑡) can be derived using the transfer matrix formalism. Calling 𝜓 (𝑛) the 1 -quarkonia states and 𝐷 D (𝑛) the 𝐷 D mesons 𝑃-wave states, both series of states are eigenmodes of the transfer matrix. Then, if we assume the threshold condition 𝑚 𝜓 (𝑛=3) = 𝐸 𝐷 D (𝑚=1) and Ω 𝑂 hℎ 𝐷 D (𝑛) = Ω 𝑂 h qℎ𝑞 𝜓 (𝑚) = 0, we have

𝑂 h qℎ𝑞 (𝑡)𝑂 † hℎ (0) = ∑ 0≤𝑡 1 <𝑡 ∑ 𝑚,𝑛 Ω 𝑂 h qℎ𝑞 𝐷 D (𝑛) 𝐷 D (𝑛) 𝜓(𝑚) 𝜓(𝑚)|Ω 𝑒 -𝑚 𝜓 (𝑚) 𝑡 1 𝑒 -𝐸 𝐷 D (𝑛) (𝑡-𝑡 1 ) + . . . = ∑ 0≤𝑡 1 <𝑡 Ω 𝑂 h qℎ𝑞 𝐷 D (1) 𝐷 D (1) 𝜓(3) 𝜓(3)|Ω 𝑒 -𝐸 𝐷 D (1) 𝑡 + . . . ∼ Ω 𝑂 h qℎ𝑞 𝐷 D (1) 𝜓(3)|Ω 𝑥 𝑡 𝑒 -𝐸 𝐷 D (1) 𝑡 . (3.3)
To extract 𝜓 (3770) and 𝑚 𝜓 (3770) , we solve a Generalised Eigenvalue Problem (GEVP) where we consider several Gaussian smearing levels of the quark field, covariant derivatives and Dirac structures. Setting 𝐸 DD = 𝑚 𝜓 (3770) allows to find the appropriate P𝜃BC ℎ(𝑥 + 𝐿 ì 𝑒 𝑖 ) = 𝑒 𝑖 𝜃 𝑖 𝐿 ℎ(𝑥) for the heavy field obeying the threshold condition.

We may use the asymptotic behaviour in eq. (3.1) , lattice geometry and spacing, pion mass, 𝑚 𝜋 𝐿, number of configurations in our study, and 𝜅 values for the bare light, strange and charm quarks (see main text for more details). Moreover, we know that at leading order the finite volume effect decays exponentially fast with 𝑚 𝜋 𝐿, and 𝑚 𝜋 𝐿 ≥ 4 reduces significantly its impact. Finally, we plan to extend our analysis to F7 to study the pion mass dependence of our results, but no statistics is available yet.

lattice [START_REF] Ukqcd Collaboration | Mixing of scalar glueballs and flavor singlet scalar mesons[END_REF][START_REF] Mcneile | Hadronic decay of a vector meson from the lattice[END_REF][START_REF] Ukqcd Collaboration | Hybrid meson decay from the lattice[END_REF]. The first method is to consider the decay 𝜓 (3770) → DD directly,

C 𝜓 (3770) →DD (𝑡) √ C DD →DD (𝑡) C 𝜓 (3770) → 𝜓 (3770) (𝑡) -→ 𝑡/𝑎 1 𝑡𝑥 + const, (3.4)
where the dominant excited-state contamination appears in the constant term. The factor

Ω 𝑂 h qℎ𝑞 𝐷 D (1) 𝜓(3)|Ω 𝑒 -𝐸 𝐷 D (1) 𝑡 (3.5)
of C 𝜓 (3770) →DD (𝑡) simplifies versus the denominator on the left-hand side of eq. (3.4). The other method is to only use the information from DD → DD,

C DD →DD, box C DD →DD, direct -→ 𝑡/𝑎 1 1 2 𝑥 2 𝑡 2 + O (𝑡). (3.6)
Here, it is interesting to study the dependence (if any) of the residual linear term on the twist angle 𝜃 used for the boundary conditions.

Preliminary Findings

In this section, we show the status of our analysis on ensemble D5, an 𝑁 𝑓 = 2 Coordinated Lattice Simulations CLS simulation, whose details appear in table 1. 𝑁 𝑓 = 2 CLS ensembles employ the Wilson plaquette for the gauge action, as well as O (𝑎)-improved Wilson quarks for the fermionic action. The gauge configurations were computed using the DD-HMC algorithm, the light-quark hopping parameter yields pion masses between 190 MeV < 𝑚 𝜋 < 630 MeV, the strange-quark mass is fixed to its physical value by setting 𝑚 2 K / 𝑓 2 K and 𝑚 2 𝜋 / 𝑓 2 K to their physical values, and the charm-quark mass is chosen such that 𝑚 D s = 𝑚 D s ,phys . For more details on the ensemble simulations, see [START_REF] Fritzsch | The strange quark mass and Lambda parameter of two flavor QCD[END_REF][START_REF] Heitger | Charm quark mass and D-meson decay constants from two-flavour lattice QCD[END_REF].

In fig. 4, we plot the DD state energy as a function of the P𝜃BC used for the charm quark. We observe that the data is well described by the continuum dispersion relation for two free D mesons, depicted by the grey line. Note that at 𝜃 = 0, corresponding to a DD system with each meson at rest, the system on ensemble D5 is heavier than its physical value. This is to be expected, due to the unphysical pion mass 𝑚 𝜋 = 440 MeV. So far, our main obstacle has been identifying

the 𝜓 (3770) state from the charmonium spectrum. We use a basis of operators belonging to the 𝑇 -- 1 representation of the H(3) symmetry group, which overlaps with states of quantum numbers 𝐽 𝑃𝐶 = 1 --, 3 --, 4 --, . . . [START_REF] Mohler | 𝐷𝜋 scattering and 𝐷 meson resonances from lattice QCD[END_REF], and we employ different levels of Gaussian smearing. At this stage, we have already computed eq. (3.6) and fitted the function 𝑎 0 𝑡 + 1 2

𝑎 2 1 𝑡 2 to the result. The matrix element we seek corresponds to 𝑎 1 , which is plotted together with 𝑎 0 in fig. 5 as a function of the twist angle. As expected, we observe the linear term parameter 𝑎 0 to be small. Moreover, we observe a certain behaviour of 𝑎 0 and 𝑎 1 at 𝜃 = 1, which we cannot explain at this point. Once we extract 𝑚 𝜓 (3770) and the twist angle at the resonance, we could determine if this behaviour is correlated to the resonance, or not. For the moment we only include statistical errors estimated using the package pyerrors [START_REF] Joswig | pyerrors: a python framework for error analysis of Monte Carlo data[END_REF], which is based on the Γ-method [START_REF] Wolff | Monte Carlo errors with less errors[END_REF].

Outlook

Currently, our efforts revolve around an accurate determination of the 𝜓 (3770) state mass. We are probing different interpolators and smearing levels for the GEVP. Afterwards, we will be able to determine the resonant twist angle, fit to eq. (3.4), and extract the corresponding decay width.

Then, we will extend our calculations to more ensembles, starting with F7 (see table 1), to study the quark mass and lattice spacing dependence of our results. In particular, the quark mass dependence will serve to probe the validity of the 3 𝑃 0 quark model described in [START_REF] Le Yaouanc | Naive quark pair creation model and baryon decays[END_REF].

Figure 1 :

 1 Figure 1: 𝜓 (3770) → DD decay Feynman diagram.

Figure 2 :

 2 Figure 2: Experimental spectrum of vector charmonium and 𝐷 D [2].

Figure 4 :

 4 Figure 4: Left: Dispersion relation of the DD system as a function of the twist angle 𝜃 [1/𝐿], assuming 𝐸 DD = 2𝐸 𝐷 and 𝑚 𝐷 = 𝐸 𝐷 (𝜃 = 0). Right: DD effective mass for the various twist angles plotted on the left-hand side as a function of the time-slice.

Figure 5 : 2 𝑎 2 1

 522 Figure 5: Fit parameters 𝑎 0 and 𝑎 1 for the model 𝑎 0 𝑡 + 1 2 𝑎 2 1 𝑡 2 used to fit the ratio C DD →DD, box /C DD →DD, direct versus the twist angle 𝜃 [1/𝐿].

Table 1 :

 1 in two different ways to extract 𝑥 from the Coordinated Lattice Simulations (CLS) 𝑁 𝑓 = 2 ensemble parameters. From left to right: Ensemble label, inverse bare coupling 𝛽 = 6/𝑔 2 0

	id	𝛽 ( 𝐿 𝑎 ) 3 × 𝑇 𝑎	𝑎 [fm] 𝑚 𝜋 [MeV] 𝑚 𝜋 𝐿 # cfgs	𝜅 ℓ	𝜅 𝑠	𝜅 𝑐
	D5 5.3 24 3 × 48 0.0653	439	4.7	150	0.13625 0.135777 0.12724
	F7	48 3 × 96		268	4.3	t.b.d. 0.13638 0.135730 0.12713

𝜓 (𝑛 = 3) ≡ 𝜓(3770), 𝐷𝐷 (𝑚 = 1) ≡ 𝐷 D. Note that we have imposed isotropic twisted boundary conditions.

Acknowledgements

Calculations for this project have been performed on the PALMA II HPC cluster at Westfälische Wilhelms-Universität Münster (WWU). The authors gratefully acknowledge the support of the WWU-IT staff. This project is supported by Agence Nationale de la Recherche under contract ANR-17-CE31-0019 (B.B. and T.S.) and by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group GRK 2149 Strong and Weak Interactions -from Hadrons to Dark Matter (J.N. and J.H.). The authors are grateful to their colleagues of the CLS effort for providing the gauge field ensembles used in this work.


Fabrice Guillet
email: fabrice.guillet@univ-nantes.fr

Angélique Villière
email: angelique.villiere@oniris-nantes.fr

S Bogosavac

G Cosne

L Grassien

J Hajjar

How to

come

• The SketchOscent mapping enables to highlight a possible aromatic palette of a product by providing a set of locations in the map, embedding its potential odorants. • Following cartographic principles, the map of all locations remains stable in order to keep the user's mental representation and facilitate the comparison between products. • This approach, can be useful to pick up relevant terms to describe perceptions or for food-pairing type applications, but may be amended to larger sectors of activities.


Ludovic Mathieu

Mourad Aïche

Paola Marini

Olivier Bouland

Carole Chatel

Serge Czajkowski

Deby Kattikat-Melcom

Teresa Kurtukian

Stephan Oberstedt

Igor Tsekhanovich

Measurement of 242 Pu(n,f) in the [1;2MeV] energy range

The design of new generation fast nuclear reactors requires highly accurate cross-section measurements in the MeV energy region. The 242 Pu fission cross section is of particular interest for Pu incineration and nuclear waste production. There are discrepancies around 1 MeV incident neutron energy between libraries and among experimental data. Some data suggest the presence of a strong structure between 1 and 1.2 MeV whereas it is barely visible on some other data and its shape is very different among evaluations. The large majority of the 242 Pu(n,f) measurements have been carried out with respect to the 235 U(n,f) secondary-standard cross section. This introduces a strong correlation between independent measurements and this cross section exhibits structures, in particular a steep increase of +10% at 1 MeV. Therefore, we aim to re-measure the 242 Pu(n,f) cross section relative to the primary-standard 1 H(n,n)p cross section, by using a proton recoil detector. This standard has a very high accuracy (0.4%), is not used for of other 242 Pu measurements, and is structureless. An experiment has been carried out in October 2022 at the MONNET facility in JRC Geel, with incident neutron energies from 0.9 MeV to 2.0 MeV. The experimental setup will be presented, and the analysis procedure will be detailed.

Introduction

The maximum of the neutron flux in a nuclear reactor is around 1-2 MeV for neutrons are produced at these energies. In thermal reactors, and due to the 1/v cross section behaviour, the thermal part of the neutron spectrum is predominant in the fission rate. However, in fast reactors, this thermal part is negligible, and the majority of the fissions are due to the 1-2 MeV neutrons. The knowledge of fission cross sections in this energy region is then of key importance for such reactors.

One of the goals of nuclear fast reactors could be to incinerate nuclear wastes and nuclear materials. The majority of waste radiotoxicity is due to Pu isotopes. With a half-life of 375.000 years, the 242 Pu is the main contributor to the long-term radiotoxicity. A precise knowledge of its behaviour in reactor is then mandatory.

The 242 Pu fission cross section has been measured several times in the past decades. There was a renewed interest since the 2000's when it was included in the HPRL [START_REF]NEA Nuclear Data High Priority Request List[END_REF][START_REF]OECD/NEA Working Party on Evaluation and Co-operation (WPEC) Subgroup 26 Final Report[END_REF], with a target uncertainty of 5% in the 1-2 MeV energy range. However, there are some discrepancies between experimental dataset and among evaluations. In particular, a structure at 1.2 MeV is visible on ToF data of Weigman [START_REF] Weigman | [END_REF], Tovesson [4] and Koegler [5] but do not appear as clearly in quasi mono-energetic data of Salvador-Castiniera [6]. This feature cannot be explained by the nuclear structure in 243 Pu* compound nucleus, but could be linked to the 242 Pu structure itself [START_REF] Chatel | Section efficace de fission du 242 Pu : progrès théoriques et expérimentaux[END_REF]. Major evaluations are based on ToF data and then all show the same structure at 1.2 MeV. They are however in disagreement on the average value, with JEFF-3.3 being the highest (+3%) and JENDL-5 the lowest (-4%).

Nearly all experimental measurements have been performed with respect to the 235 U(n,f) cross section, apart from Salvador-Castiniera et al. where 235 U(n,f), 238 U(n,f) and 237 Np(n,f) cross sections were used. Every measurement performed using 235 U(n,f) standard are then strongly correlated. These standard cross sections are well confirmed by integral measurements, but discrepancies can be seen here and there. For instance, the 235 U(n,f) cross section show a "threshold-like" structures around 1 MeV, with a +9% increase within 150 keV, and discrepancies between experimental data set can reach up to 5%.

Another method is to use the 1 H(n,n)p cross section as reference, known as proton recoil method. This cross section is a primary standard, with an uncertainty of less than 0.4% on a large energy range [START_REF] Carlson | [END_REF], and without any structure. Measurements using this standard can add precious accurate and uncorrelated values to help to improve the cross section knowledge. In 2012 a 242 Pu(n,f) measurement was carried out by our team using this standard [9]. Technical issues prevented us from obtaining enough energy points to bring new information about the structure at 1.2 MeV. Nevertheless, these results are compatible with the lower JENDL-5 evaluation, and much less with JEFF-3.3 or ENDF/B-VIII.

A new experiment has been carried out in 2022 to re-measure the 242 Pu(n,f) in the 1-2 MeV energy range. This paper present the experimental setup and the main stages of the ongoing analysis.

Experimental setup

Facility and sample

The experiment took place in October 2022 at the new MONNET facility in JRC Geel [10]. This facility had two major advantages that justified this choice: -it allows the use of 3 H beam target, which enables to produce neutrons with energies covering the energy range of interest ; -the JRC has a radioactive sample laboratory, and the 242 Pu target made there for previous experiments was still available. This 242 Pu sample has an activity of 98.4 kBq for a diameter of 29.95 mm and is electroplated on a 0.25mm-thick Al-backing.

The neutron production reaction was 3 H(p,n) 3 He, producing neutrons with an energy from 0.9 to 2.0 MeV. Two different TiT targets were used with thicknesses of 930 and 260 µg/cm². The beam intensity varied from 15 to 40 µA. A very light water-cooling device was used on the TiT target, ensuring a low quantity of scattered neutrons.

The first element of the experimental setup was placed 90 mm away from the neutron production point. A closer distance of 60 mm was used with the thin production target.

Fission setup

Fission events were measured via the detection of a single Fission Fragment (FF) by two photovoltaic cells of 40x20 mm². The Pu target is located in a compact fission device, with two photovoltaic cells placed in front of it at a distance of 5.4 mm. The cells frame is used as a collimator with a square aperture of 34x34 mm², and cells are 0.5 mm apart.

The interaction energy on the 242 Pu target depends on two factors: the TiT target thickness and the angle of aperture of the Pu target. The further setup distance of 90 mm was used to reduce the energy spread when using the thick TiT target.

Proton recoil setup

The proton recoil technique consists in irradiating an H-rich material with neutron, in order to convert a small part of the neutron flux into easy-to-detect protons. For this purpose, a polypropylene (PP) foil was placed downstream the fission setup. The recoil protons were emitted by neutron scattering in every direction downstream. A silicon detector was placed 95 mm further, with a 15 mm diameter collimator, in order to collect protons emitted with a very small angle. The proton energy is given by:

= cos² × (1)
with θ the proton scattering angle with respect to the incident neutron direction. The protons also lose energy going through the PP foil itself, depending of their starting point. To reduce this phenomenon, the PP foil thickness had to be kept small. A 4 µm-tick PP foil was used, ensuring a proton energy loss lower than 15% in the worst case: for the lowest neutron energy (0.9 MeV) and a recoil proton emitted at the beginning of the PP foil.

The detected protons energy is then close to that of incident neutrons, and can be more easily discriminated from the background. In particular, recoil protons from scattered neutrons will have a lower energy due to a lower incident neutron energy and a different incoming direction (see equation 1).

To subtract recoil protons coming from other sources, a tantalum screen could be remotely placed downstream the PP foil to stop all recoil protons coming from the foil.

Data Analysis

Cross section calculation

The 242 Pu fission cross section is measured relative to the 1 H(n,n) cross section, with several correction factors:

, = × × × × , (2)
with:

-NFF and Np the numbers of fission fragments (FF) and recoil protons detected respectively; -εp and εFF the detection efficiencies of recoil protons and FF respectively; -ΩPP and Ω242Pu the solid angles of the PP foil and 242 Pu target respectively; -NPP and N242Pu the number of hydrogen atoms in the PP foil and 242 Pu atoms in the Pu target respectively; -σH(n,n) the neutron elastic scattering cross section on 1 H.

All these correction factors take into account the differences in counting rates, efficiencies, irradiation and atom quantities between the 242 Pu target and the reference PP foil.

Counting rates

Fission spectra in photovoltaic cells show less background and a clear alpha/FF discrimination, as can be seen in Figure 1. However, 242 Pu undergoes spontaneous fission that has to be subtracted (see following sub-section). In addition, there is also a small contribution of a constant background, which has been measured without Pu target at the end of the experiment.

Due to the amount of structure materials in the neutron production target, the reaction chamber and the structure materials, some scattered neutrons may irradiate the fission target and induce parasitic fissions. The scattered neutron energy is usually much lower than the one of direct neutrons, due to the production angle and the scattering process. An MCNP simulation has to be carried out to subtract this contribution. In the 2012 experiment, this represented 5 to 15% of the fission rate, depending on the neutron energy and the cooling system used. A low value of about 5% is expected for this experiment, as the neutron production device and cooling system was very light. Proton recoil spectra are polluted with a strong low energy background coming from electrons generated by the neutron production process [11]. When the proton energy is high enough, the proton peak can easily be discriminated from the background. However, this discrimination is trickier at low energies such as 1 MeV.

In addition, some hydrogen contaminations on structure materials can emit recoil protons up to the maximum proton energy. To reduce such contributions, efforts have been made to minimize and clean hydrogen contamination of the experimental device. A measurement with a tantalum screen downstream the PP foil enables to measure most of this background. Typical spectra are shown in Figure 2. Measurement runs and tantalum screen runs have to be normalized one to each other by a combination of fission number and neutron flux (via MONNET long-counters).

Efficiencies

The fission efficiency is quite difficult to infer: there is a non-sensitive collection grid in the front face of the cell, and polycrystalline cells may have some unconnected grains which won't participate to the energy collection. The fission efficiency is then deduced from the spontaneous fission (SF) emissions. Thus, SF runs have been performed almost every night or weekend. The SF efficiency has the advantage to also include the real target features (thickness and homogeneity). The results are presented in Figure 3. Some runs are still to be analysed to fill the gaps, and to conclude about the general trend of this efficiency. To obtain the fission efficiency from the SF efficiency, one has to take into account several physical effects: irradiation profile in flux and energy, kinematic effect and fission anisotropy. These corrections are calculated with a simulated efficiency, and then applied on the experimental SF efficiency.

The proton detection efficiency is determined via simulations, considering source characteristics, PP irradiation profile in flux and energy, and kinematic effects.

Irradiation

The third term in equation 2 deals with the amount of neutron passing through the fission target and the PP foil. For an isotropic neutron emission, this term equals the ratio of respective solid angles. The accuracy of this ratio strongly depends on the uncertainties on the distances, lengths and radii knowledge of the experimental setup. In particular, the larger distance used for the thick neutron production target helps to obtain a lower uncertainty on this term. As the neutron production is anisotropic, a simulation is carried out to calculate this ratio.

Material quantities

The fourth term in equation 2 corrects for the different number of nuclei between the fission target and the PP foil. As a radioactive isotope, the number of 242 Pu atoms is measured via alpha spectrometry with a high accuracy. Measuring the H content in the PP foil is trickier, relying either on a micro-weight measurement or on a thickness measurement. In any case, preventing H-contamination of the PP foil is mandatory as the subtraction of induced recoil protons is difficult and introduce an additional uncertainty [9,11].

Preliminary results

Few steps of the analysis have been done so far, and several corrections still need to be applied. Nevertheless, the general shape of the 242 Pu cross section begins to appear, as can be seen in Figure 4. As can be seen, a thorough check of each point is necessary. Indeed, an incorrect background subtraction in the proton recoil spectrum may lead to extremely high or low values. In addition, some corrections may change the global shape of the cross section. In particular, the scattered neutrons correction is different from one energy to the other, as well as the proton or fission efficiencies. Other corrections, like material quantities and clean background fission subtraction, influence mainly the absolute value of the cross section. At the end, the cross section will be obtained for about fifteen different energy points.

Conclusion

The 242 Pu fission cross section is an important nuclear data for the use of future nuclear fuel. Some discrepancies still exist among data and/or evaluations. This cross section has been remeasured at the MONNET facility at JRC Geel. The aim of this measurement is to provide a cross section value strongly independent from the existing data: different reference cross section, different measurement method and different type of detectors.

New data will bring valuable information about the structure around 1.2 MeV. The analysis is still ongoing and it is too early to draw any conclusion from the results.

Fig. 1 .

 1 Fig. 1. Fission spectrum in one photovoltaic cell.

Fig. 2 .

 2 Fig. 2. Proton recoil spectra with (measurement) and without (background) PP foil, for an incident neutron energy of 2 MeV.

Fig. 3 .

 3 Fig. 3. The spontaneous fission efficiency for different SF runs is shown. Still several runs has to be analysed. The mean value with its uncertainty is indicated by the orange and red lines respectively.

Fig. 4 .

 4 Fig. 4. Fission cross section of 242 Pu with preliminary result from this experiment is shown. Error bars of these points do not include systematic uncertainties.

Acknowledgement

We wish to acknowledge the MONNET team for providing support and a high quality beam during the experiment. This work is supported by the European Atomic Energy Community (EURATOM) under the European Union's Horizon 2020 SANDA, grant agreement No 847552.


Sergey Kirgizov

Khaydar Nurligareev

Asymptotics of strongly overlapping permutations

Keywords: asymptotics, permutations, consecutive patterns, overlapping

published or not. The documents may come

Introduction

The study of pattern-avoiding permutations is a dynamically developing area of mathematics that has been exciting scientists since the mid-1980s. Since the first systematic enumeration by Simion and Schmidt [START_REF] Simion | Restricted permutations[END_REF], interest in permutation patterns has grown steadily, which can be explained by the presence of extensive connections with other mathematical fields. One of the important variations of this study concerns the notion of consecutive patterns, first systematically treated by Elizalde and Noy [START_REF] Elizalde | Consecutive patterns in permutations[END_REF]. To obtain generating functions and distributions of pattern occurrences in various combinatorial objects, one can employ powerful Goulden-Jackson cluster method [START_REF] Goulden | An inversion theorem for cluster decompositions of sequences with distinguished subsequences[END_REF][START_REF] Goulden | Combinatorial enumeration[END_REF][START_REF] Noonan | The goulden-jackson cluster method: extensions, applications and implementations[END_REF]]. As a particular example, we would like to cite the proofs of Elizalde-Noy and Nakamure conjectures [START_REF] Elizalde | The most and the least avoided consecutive patterns[END_REF].

Understanding the overlaps of combinatorial sub-structures is of core importance in the "cluster" method. This need is reflected in the concept of non-overlapping permutations, also known as minimally overlapping [START_REF] Bóna | Non-overlapping permutation patterns[END_REF][START_REF] Duane | Minimal overlapping patterns in colored permutations[END_REF][START_REF] Pan | Asymptotics for minimal overlapping patterns for generalized euler permutations, standard tableaux of rectangular shape, and column strict arrays[END_REF]: a permutation of length n is non-overlapping if it does not contain isomorphic prefix and suffix of size k ∈ {2, . . . , n -1}. Here, we call two sequences of distinct integers

π = π 1 π 2 • • • π k and σ = σ 1 σ 2 • • • σ k isomorphic if π i > π j ⇔ σ i > σ j
for all possible i and j. For instance, permutation 132 is non-overlapping, but 1324 is overlapping, since its prefix 13 is isomorphic to its suffix 24. The counting sequence of non-overlapping permutations starts with 1, 2, 4, 12, 48, 280, 1 864, 14 840, 132 276, 1 323 504, . . . , see A263867 entry in Sloane's Encyclopedia [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF].

Motivated by the ongoing study [START_REF] Biane | Endhered patterns in matchings and rna[END_REF] related to distributions of special kind of patterns in RNA secondary structures with allowed pseudoknots, modelled as fixed-point free involutions, in this paper we introduce a notion of strongly overlapping permutations, which is similar yet different from Bóna's overlapping permutations. Definition 1.1. A permutation σ ∈ S n is strongly overlapping, if there is an integer 1 ≤ k < n called overlapping range such that the following three conditions hold:

1. the interval {1, . . . , k} is invariant under σ, 2. the interval {n -k + 1, . . . , n} is invariant under σ, 3. the first and last k consecutive positions of σ form isomorphic patterns.

Otherwise, we call σ strongly non-overlapping. We designate by s n and n n , respectively, the numbers of strongly overlapping and strongly non-overlapping permutations of size n.

From a geometric point of view, if a permutation σ ∈ S n is strongly overlapping with an overlapping range k, then its plot possesses two congruent blocks of size k × k: one of them is located at the lower left corner, while the other is in the upper right one. For example, the permutation σ = 214365 is strongly overlapping with overlapping ranges k = 2 and k = 4 (Figure 1). Figure 2 illustrates all strongly overlapping permutation of size n ≤ 4. Observing these and other examples, the reader may give a guess that a strongly overlapping permutation possesses an overlapping range of no more than half its size. In Section 2, we show that this is, indeed, the case (Lemma 2.1). This observation gives rise to a structural decomposition of permutations into direct sums of strongly non-overlapping ones (Theorem 2.4). In its turn, the decomposition leads to relations on generating functions (Theorem 2.5), that allows us to establish counting sequences (s n ) and (n n ). The first few values of the sequence (s n ) are A single glance at these numbers is enough to conjecture that with high probability a typical large permutation is strongly non-overlapping. We prove this conjecture at Section 3. Moreover, we establish a complete asymptotic expansion of the probability that a uniform random permutation is strongly overlapping (Theorem 3.1). It turns out that the coefficients involved into the expansion admit combinatorial interpretation: they are n n again. Furthermore, the expansion under consideration resembles another one related to indecomposable permutations. This observation raises several questions that are discussed in Section 4.

s n = 0,
2 Structure and enumeration Lemma 2.1. Any strongly overlapping permutation of size n admits an overlapping range of size at most n/2.

Proof. Suppose that an overlapping range k of a strongly overlapping permutation σ ∈ S n is greater than n/2. In this case, the lower left and upper right blocks of size k × k are congruent and have non-empty intersection. The intersection is a (2k-n)×(2k-n) block, which is congruent to the lower left and upper right blocks of the same size. Therefore, σ admits overlapping range (2k

-n) < k. If (2k -n) < n/2
, we are done. Otherwise, repeat the procedure until the overlapping block sizes are less than n/2 (Figure 3). Recall that the direct sum of permutations π ∈ S m and τ ∈ S n is the permutation π ⊕ τ of length (m + n) defined by

(π ⊕ τ )(i) = π(i) for 1 ≤ i ≤ m, τ (i -m) + m for m + 1 ≤ i ≤ n.
Lemma 2.1 allows us to decompose strongly overlapping permutations into a direct sum of smaller permutations (schematically, the summands are represented by consecutive non-intersecting blocks, see the right part of Figure 3). More precisely, we get the following structural result. Lemma 2.2. Any strongly overlapping permutation σ can be uniquely decomposed into a direct sum

σ = π ⊕ τ ⊕ π,
where π is strongly non-overlapping permutation and τ is arbitrary (possibly, empty) permutation.

Proof. Let k be the minimal overlapping range of σ. According Lemma 2.1, k ≤ n/2. Hence, σ possesses three consecutive invariant intervals: {1, . . . , k}, {k+1, . . . , n-k} and {n-k+1, . . . , n} (it may happen that the second of them is empty). This gives us the above decomposition σ = π ⊕τ ⊕π. Note that the permutation π ∈ S k is strongly non-overlapping due to minimality of the overlapping range k. Indeed, if π is strongly overlapping with an overlapping range l, then l < k is the overlapping range of σ too, which leads to a contradiction (see Figure 4). Corollary 2.3. The counting sequence (s n ) of strongly overlapping permutations satisfies

s n = [n/2] k=1 n k • (n -2k)!.
Theorem 2.4. Any permutation σ can be uniquely decomposed into a direct sum of strongly nonoverlapping permutations,

σ = π 1 ⊕ . . . ⊕ π m ⊕ τ ⊕ π m ⊕ . . . ⊕ π 1 , (1)
where the permutation τ is, possibly, empty.

Proof. If σ is strongly non-overlapping, then decomposition ( 1) is done with m = 0 and τ = σ.

Otherwise, let us apply Lemma 2.2 iteratively. In other words, express σ as π 1 ⊕ τ 1 ⊕ π 1 . If τ 1 is strongly overlapping, then express it as π 2 ⊕ τ 2 ⊕ π 2 , etc. Thus, after a finite number of iterations, we obtain decomposition (1). Now, let us show the uniqueness of the above decomposition. Suppose the contrary, that is, suppose that we have two different decompositions of form (1):

π 1 ⊕ . . . ⊕ π m ⊕ τ ⊕ π m ⊕ . . . ⊕ π 1 = π 1 ⊕ . . . ⊕ π m ⊕ τ ⊕ π m ⊕ . . . ⊕ π 1 .
Without loss of generality, we can assume that π 1 = π 1 , which means that they are of different sizes, say, l and l . However, if l < l , then π 1 is strongly overlapping with the overlapping range l (see Figure 5). The case l < l leads to a similar contradiction. Thus, π 1 = π 1 , and decomposition (1) is unique. Let us consider the generating functions S(z) and N (z) of strongly overlapping and strongly non-overlapping permutations, respectively:

π 1 π 1 . . .
S(z) = ∞ n=1 s n z n and N (z) = ∞ n=1 n n z n .
We formally assume that an empty permutation is neither strongly overlapping nor strongly nonoverlapping. Hence, the generating function P (z) of all permutations can be represented as the following sum:

P (z) = 1 + S(z) + N (z) = ∞ n=0 n! z n . (2)
This observation, together with Lemma 2.2, helps us establish relations that determine the behavior of S(z) and N (z).

Theorem 2.5. The generating functions S(z) and N (z) satisfy the following relations:

N (z) = P (z) 1 -N (z 2 ) -1 (3)
and

S(z) = 1 + N (z) 1 -N (z 2 ) • N (z 2 ). (4)
Proof. In terms of generating functions, Lemma 2.2 can be interpreted as the following relation:

S(z) = P (z) • N (z 2 ).
At the same time, it follows from (2) that

S(z) = P (z) -N (z) -1.
Equating the right-hand sides of these two identities, we obtain relation [START_REF] Atkinson | Restricted permutations and the wreath product[END_REF]. Meanwhile, to get (4), it is sufficient to take the first of the identities and replace P (z) with the expression obtained via relation (3).

Remark 2.6. Relation (3) rewritten as

P (z) = 1 + N (z) 1 -N (z 2 ) (5)
is another face of Theorem 2.4. Indeed, the factor 1 + N (z) corresponds to the central block of decomposition (1), while 1 -N (z 2 ) -1 represents the sequence of strongly non-overlapping permutations.

Note that the argument z 2 reflects the fact that each element of the sequence is taken twice. Relation (4) carries a similar meaning. The only difference is the presence of the additional factor N (z 2 ) that guarantees that the decomposition consists of at least two blocks.

3 Asymptotics Theorem 3.1. For any positive integer r, the probability that a uniform random permutation σ ∈ S n is strongly overlapping, as n → ∞, satisfies

P(σ is strongly overlapping) = r-1 k=1 n k (n) 2k + O 1 n 2r , (6)
where (n) k = n(n -1) . . . (n -k + 1) are the falling factorials.

Proof. It follows from Corollary 2.3, that the counting sequence of strongly overlapping permutations satisfies

s n ≤ r-1 k=1 n k • (n -2k)! + [n/2] k=r k! • (n -2k)!.
Let us estimate the second summand. For n large enough and r < k ≤ n/2, we have

k! • (n -2k)! ≤ (r + 1)! • (n -2r -2)!.
As a consequence,

[n/2] k=r k! • (n -2k)! ≤ r! • (n -2r)! + (n -r) • (r + 1)! • (n -2r -2)! = O (n -2r)! ,
which leads us to

s n = r-1 k=1 n k • (n -2k)! + O (n -2r)! .
Now, since P(σ is strongly overlapping) = s n /n!, it is sufficient to divide the obtained inequality by n!.

Let s

(m) n be the number of permutations of size n whose decomposition (1) consists of (2m + 1) summands (in particular, s

(0) n = n n ). Define also n (m) n
to be the number of permutations that can be represented as a direct sum of m strongly non-overlapping permutations (in particular, n

(0) 0 = 1, n (0)
n = 0 for n > 0, and n

(1) n = n n ).
Theorem 3.2. For any positive integer r, the probability that decomposition (1) of a uniform random permutation σ ∈ S n consists of (2m + 1) summands, as n → ∞, satisfies

P(σ consists of 2m + 1 blocks) = r-1 k=1 n (m) k -n (m+1) k (n) 2k + O 1 n 2r , (7)
where (n) k = n(n -1) . . . (n -k + 1) are the falling factorials.

Proof. Due to Lemma 2.2 and Theorem 2.4, the number of permutations whose decomposition (1) consists of at least (2m + 1) summands is equal to

[n/2] k=1 n (m) k • (n -2k)!.
Therefore, the sequence s

(m) n satisfies s (m) n = [n/2] k=1 n (m) k -n (m+1) k • (n -2k)!.
Similarly to the previous theorem, for any tail of this sum, we have an estimation

[n/2] k=r n (m) k -n (m+1) k • (n -2k)! = O (n -2r)! .
Thus, dividing the above relation by n!, we obtain asymptotic expression [START_REF] Comtet | Sur les coefficients de l'inverse de la série formelle n!t n[END_REF].

Conclusion

As we have seen in Section 2, each permutation can be represented as a direct sum of strongly non-overlapping ones. Therefore, we can consider non-overlapping permutations as a sort of basic irreducible blocks that can serve for constructing the whole class of permutations. There are other examples of this kind in the literature related to permutations. One of them concerns so called indecomposable permutations, that is, permutations with no proper invariant interval of the form {1, . . . , k} (see, for example, the paper of Cori [START_REF] Cori | Indecomposable permutations, hypermaps and labeled Dyck paths[END_REF]). It is known that each permutation can be represented as a sequence of indecomposable ones. In particular, the generating function I(z) of indecomposable permutations satisfies

I(z) = 1 - 1 P (z)
.

In the class of permutations, the strongly non-overlapping ones form the majority. In other words, almost all permutations are strongly non-overlapping. The same is true for indecomposable permutations. Moreover, this analogy can be extended to complete asymptotic expansions. It was Comtet [START_REF] Comtet | Sur les coefficients de l'inverse de la série formelle n!t n[END_REF] who first studied the probability that a large permutation is indecomposable, and established its expansion over the basis 1/(n) k . Later, it turned out that the involved coefficients have combinatorial meaning. More precisely, the following result was recently shown [START_REF] Nurligareev | Irréductibilité des objets combinatoires : probabilité asymptotique et interprétation[END_REF]: for any positive integer r, the probability that a uniform random permutation σ ∈ S n is indecomposable satisfies

P σ is indecomposable = 1 - r-1 k=1 2i k -i (2) k (n) k + O 1 n r , (8)
where (i k ) is the counting sequence of indecomposable permutations, and (i

k ) counts permutations with exactly two indecomposable parts. From Theorem 3.1, we can see that the asymptotic expansion of the probability that a uniform random permutation is strongly non-overlapping has the same spirit:

P(σ is strongly non-overlapping) = 1 - r-1 k=1 n k (n) 2k + O 1 n 2r . (9)
Another example of irreducibilities involves simple permutations that do not map non-trivial intervals onto intervals (see [START_REF] Albert | The enumeration of simple permutations[END_REF]). The generating function M (z) of simple permutations of size at least 4 can be found iteratively from the equation

P (z) -P 2 (z) 1 + P (z) = z + M P (z) .
The asymptotic expansion of the probability that a uniform random permutation σ ∈ S n is simple was established by Borinsky [START_REF] Borinsky | Generating asymptotics for factorially divergent sequences[END_REF]:

P(σ is simple) = 1 e 2 1 - 4 n + 2 (n) 2 - 40 3(n) 3 - 182 3(n) 4 -. . . . (10)
Again, we observe the role of the basis 1/(n) k . However, compared to the cases related to strongly non-overlapping and indecomposable permutations, this expansion has two important differences. First, the leading term is not 1 anymore: the proportion of simple permutations tends to e -2 . Second, the involved coefficients are not integers anymore. These observations raise a number of questions. First of all, it would be of interest to unite the three above discussed permutation classes under the same theory that could explain both the similarities and differences. A priori, this question is quite complicated. It would be natural to try employing the wreath product introduced by Atkinson and Stitt [START_REF] Atkinson | Restricted permutations and the wreath product[END_REF] for studying restricted permutations. Indeed, constructing the class of permutations from the simple ones admits description in terms of the wreath product [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF]; that is true for indecomposable permutations as well. However, it looks like the strongly non-overlapping permutations do not allow such a description.

Another question is the following: is it possible to naturally modify the basis in asymptotics [START_REF] Elizalde | The most and the least avoided consecutive patterns[END_REF], so that the coefficients become integers? At first glance, it seems that it could be sufficient to divide the basis vectors 1/(n) k by k!. This change suggests switching from the ordinary generating functions to the exponential ones. That trick works well for establishing complete asymptotic expansions of indecomposable permutations and indecomposable perfect matchings, see [START_REF] Nurligareev | Irréductibilité des objets combinatoires : probabilité asymptotique et interprétation[END_REF]Chapter 9]. The potential of its applicability for simple permutations needs to be verified.

Finally, if we get a positive answer to the previous question, what is the combinatorial interpretation of these coefficients? Unfortunately, there is no clue in the OEIS [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF]. The only thing one can be sure of is that the coefficients cannot be interpreted as a counting sequence of combinatorial class: some of them are negative. Thus, we would rather expect linear combinations of different counting sequences, as is the case for indecomposable permutations [START_REF] Cori | Indecomposable permutations, hypermaps and labeled Dyck paths[END_REF].

Figure 1 :

 1 Figure 1: Overlapping blocks in the permutation 214365.

Figure 2 :

 2 Figure 2: Strongly overlapping permutations of size at most 4.

Figure 3 :

 3 Figure 3: Schema for reducing an overlapping range. If overlapping blocks have a non-empty intersection (gray areas on the figure), then this intersection can be slided to get smaller overlapping blocks (blue and violet areas).

Figure 4 :

 4 Figure 4: Structure of strongly overlapping permutations. If the overlapping block π is a strongly overlapping permutation itself, then its size is not the minimal overlapping range of the permutation σ = π ⊕ τ ⊕ π.

π 1 π 1 Figure 5 :

 15 Figure 5: Uniqueness of decomposition (1).

Acknowledgements

Authors were partially supported by the grant ANR-22-CE48-0002 funded by l'Agence Nationale de la Recherche and the project ANER ARTICO funded by Bourgogne-Franche-Comté region (France).


M.D Nicolas Massart

M.D. a , Pierre Fillatre

PharmD Florian Lemaitre

PhD. b Arnaud Mari

M.D. a , Camille Tron
email: camille.tron@chu-rennes.fr

Pharmacokinetics of baricitinib in critically ill COVID-19 patients

Keywords: Critical care, baricitinib, pharmacokinetics, COVID-19

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

METHODS

This retrospective observational study was conducted in critically ill patients with COVID-19 treated with baricitinib 4 mg/day. Plasma concentrations were measured at predose (C0), 1 h (C1) and 3 h (C3) after the drug intake. PK and area under the curve (AUC) were estimated using non-compartmental pharmacokinetic analysis.

RESULTS

Seven patients contributed to 22 baricitinib plasma concentration measurements after a median [range] of 3 days [START_REF] Marconi | Efficacy and Safety of Baricitinib for the Treatment of Hospitalised Adults with COVID-19 (COV-BARRIER): A Randomised, Double-Blind, Parallel-Group, Placebo-Controlled Phase 3 Trial[END_REF][START_REF] Trøseid | Efficacy and Safety of Baricitinib in Hospitalized Adults with Severe or Critical COVID-19 (Bari-SolidAct): A Randomised, Double-Blind, Placebo-Controlled Phase 3 Trial[END_REF] of treatment. Median baricitinib plasma concentrations were 2.2 ng/mL [1.4-8.0], 24.0 ng/mL [4.9-37.3] and

BACKGROUND

Despite tremendous improvements in severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infectious disease (COVID-19) management, few treatments improve survival in critically ill patients. Among them, the oral selective Janus Kinase 1/2 (JAK1/2) inhibitor baricitinib has shown a survival benefit in mechanically ventilated COVID-19 patients [START_REF] Ely | Efficacy and Safety of Baricitinib plus Standard of Care for the Treatment of Critically Ill Hospitalised Adults with COVID-19 on Invasive Mechanical Ventilation or Extracorporeal Membrane Oxygenation: An Exploratory, Randomised, Placebo-Controlled Trial[END_REF][START_REF] Marconi | Efficacy and Safety of Baricitinib for the Treatment of Hospitalised Adults with COVID-19 (COV-BARRIER): A Randomised, Double-Blind, Parallel-Group, Placebo-Controlled Phase 3 Trial[END_REF] but is not without adverse drug reactions [START_REF] Trøseid | Efficacy and Safety of Baricitinib in Hospitalized Adults with Severe or Critical COVID-19 (Bari-SolidAct): A Randomised, Double-Blind, Placebo-Controlled Phase 3 Trial[END_REF]. Surprisingly, data regarding baricitinib pharmacokinetics (PK) are scarce even in usual rheumatology indications and do not exist in the COVID-19 setting. However, some variables may modify baricitinib PK such as weight and renal function [START_REF] Kim | Pharmacokinetics, Pharmacodynamics, and Proposed Dosing of the Oral JAK1 and JAK2 Inhibitor Baricitinib in Pediatric and Young Adult CANDLE and SAVI Patients[END_REF][START_REF] Wang | Physiologically-Based Pharmacokinetic Modelling to Investigate Baricitinib and Tofacitinib Dosing Recommendations for COVID-19 in[END_REF], both parameters being highly dynamic in critically ill patients. The pharmacokinetic pathway of baricitinib is characterized by a rapid absorption orally, with a median time to peak concentrations of 1 hour and bioavailability around 79%. Baricitinib is widely distributed in tissues (Vd 76 L) and is mainly excreted unchanged in urine and feces. Renal elimination is the main mechanism of baricitinib clearance, with less than 10% eliminated by hepatic metabolism. Baricitinib is a substrate of drug transporters which may impact its PK profile and may be involved in drug-drug interactions [START_REF] Jorgensen | A Review of Pharmacology, Safety, and Emerging Clinical Experience in COVID-19[END_REF].

As baricitinib is a life-saving therapy of a severe acute infection, it is crucial to further study its PK in this context. The aim of our study is to describe real-life baricitinib plasma exposure in critically ill COVID-19 patients.

METHODS

We performed a retrospective observational study in patients treated in a polyvalent intensive care unit (ICU) in western France. Critically ill COVID-19 patients, as assessed by a SARS-CoV 2 positive retrotranscription polymerase chain reaction (RT-PCR), admitted between October 2021 and May 2022 were eligible. Only those who received baricitinib as a COVID-19 therapeutic and for whom blood concentration measurements were performed were included. Patients were informed of the study and were offered the option of declining to participate. However, as we performed a retrospective analysis of previously collected data, written informed consent was not required. The study protocol received approval from the local Ethics Committee (n° 22-137).

The therapeutic COVID-19 protocol consisted of 6 to 12 mg/day of dexamethasone (depending on inflammatory parameters) for 10 days following admission, and baricitinib was prescribed after a collegial discussion between critical care and infectious disease physicians. In this case, 4 mg of baricitinib was administered daily for 14 days, orally or in the gastric tube for intubated patients. Other medications included 3 rd generation cephalosporin in case of suspicion of co-infection, proton pump inhibitor when clinically indicated and respiratory organ support with facial mask, high flow nasal cannula or tracheal intubation when indicated. Baricitinib plasma concentrations were measured at trough (C0), 1 hour (C1) and when possible, 3 hours (C3) after the drug intake. After centrifugation (10 min, 2000 g, 10°C) of heparinized blood samples, the plasma samples were sent (frozen) to the Rennes University Hospital pharmacology laboratory for analysis. Concentrations were measured using a validated liquid-chromatography tandem mass spectrometry method (LC-MS/MS) adapted from [START_REF] Veeraraghavan | Simultaneous Quantification of Baricitinib and Methotrexate in Rat Plasma by LC-MS/MS: Application to a Pharmacokinetic Study[END_REF]. Briefly, 100 µL of plasma were mixed with 300 µL of precipitant solution (acetonitrile-zinc sulphate in water (0.05 M), 1:1 v/v) and 100 µL of internal standard diluted in acetonitrile (H 8 deuterated imatinib). Samples were vortexed for 1 min and centrifuged for 5 min at 3000g 4°C. A volume of 5µL of the supernatant was injected in a LC-MS/MS system composed of an Acquity UPLC-H-Class and a Xevo TQ-XS (Waters, Milford, USA). The quantification range was 0.5 to 100 ng/mL. When three concentration time-points were available between two administrations, AUC was estimated using non-compartmental pharmacokinetic analysis (PK Solver). Non-parametric descriptive statistics (median, range) were used to describe the data, and the Mann Whitney test was used to compare groups.

RESULTS

During the study period, 56 patients were admitted for COVID-19 confirmed by RT-PCR. Of these, 12 received baricitinib but for organizational reasons, only 7 had drug concentrations measurements. The baseline characteristics, co-medication, and outcomes are reported in Table 1. The median age was 65 years [range 63-72], the median SAPS II was 44 [range 38-48], the median PaO2/FiO2 ratio was 172 mmHg [range 160-190]; 3 patients were mechanically ventilated and the others had high flow nasal cannula support. The median body mass index (BMI) was 28.6 kg/m² [28.1-33.9] and the median glomerular filtration rate (GFR) was 108 mL/min [105-127] (Cockcroft formula). None of the patients died. Baricitinib blood measurements were performed at steady state after a median of 3 days [2-3] of treatment.

The median baricitinib plasma concentrations were 2.2 ng/mL [1.4-8.0], 24.0 ng/mL [4.9-37.3] and 14.1 ng/mL [8.3-15.1] for the trough (C0), C1 and C3 concentrations, respectively. High inter-patient variability was observed especially for C0 and C1 (coefficient of variation of 62.5% and 47.7% respectively) (Figure 1). Overall exposure between two administrations could be estimated by calculating the AUCs for 4 patients. The median AUC0-24h was 188.8 ng.h/mL [141. 3-236.3]. Overall, 2 patients who received baricitinib had liver function test perturbation that eventually led to baricitinib cessation. Both had baricitinib trough concentrations higher than the median and appeared to have atypical pharmacokinetic profiles with similar C1 and C3 concentrations, although no difference in plasma concentrations was observed between patients with or without liver cytolysis (p=0.32 for C0 and 0.19 for C1) (Table 1). Liver function tests subsequently corrected without other implications for patient management. The only patient with altered renal function (GFR of 74 mL/min) had the highest baricitinib trough concentration (Table 1).

DISCUSSION

Baricitinib is approved for chronic inflammatory disease such as rheumatoid arthritis, alopecia areata and atopic dermatis, and PK data are available for rheumatology indications. Since the pandemic, baricitinib has been approved to treat severe forms of COVID-19, however, no real life PK data are available in the case of acute inflammation such as COVID-19. While the pandemic situation is currently improving, the drug still helpful in case of severe forms that still exist in vulnerable patients that are at high risk of PK variability [START_REF] Toussi | Therapeutics for COVID-19[END_REF].

To the best of our knowledge, this is the first description of baricitinib plasma exposure in critically ill COVID-19 patients. All patients received the same drug dosage (4 mg daily) that is the usual dosage used in the context of rheumatology indications. However, baricitinib PK may be different in critically ill COVID-19 patients. In our cohort, we observed high interpatient variability in C0, C1 and C3 plasma concentrations and to a lesser extent in AUC0-24h. Median C1, trough concentrations and AUC, were lower than values observed in rheumatoid arthritis (RA) patients (i.e. 6.9 ng/mL, 53.4 ng/mL, 477.6 ng.h/mL for C0, Cmax, and AUC0-24h respectively in RA [START_REF] Jorgensen | A Review of Pharmacology, Safety, and Emerging Clinical Experience in COVID-19[END_REF]. This may be explained by a higher distribution volume in critically ill patients (e.g. due to vascular filling) and to the high median BMI of our patient population. However, the small size of our cohort does not allow verifying this hypothesis and no association was found between BMI and exposure. Influence of body weight on baricitinib PK was suggested by Kim et al. in pediatrics. They reported that weight and renal function significantly influenced volume of distribution and clearance, and that higher body weight was associated with lower exposure, which is consistent with our hypothesis [START_REF] Kim | Pharmacokinetics, Pharmacodynamics, and Proposed Dosing of the Oral JAK1 and JAK2 Inhibitor Baricitinib in Pediatric and Young Adult CANDLE and SAVI Patients[END_REF]. In healthy volunteers, Cmax and AUC lower than in RA were reported suggesting an impact of chronic but not acute inflammation (COVID-19) on the exposure [START_REF] Posada | Prediction of Transporter-Mediated Drug-Drug Interactions for Baricitinib[END_REF][START_REF] Shi | The Pharmacokinetics, Pharmacodynamics, and Safety of Baricitinib, an Oral JAK 1/2 Inhibitor, in Healthy Volunteers[END_REF]. The only patient of our cohort with significant renal failure had the highest trough concentration which is consistent with a lower clearance and the elimination pathway of the drug.

Interestingly, PK of ruxolitinib, which is also a JAK1/2 inhibitor, is well described in hematological patients. Noteworthily, ruxolitinib exposure is highly increased in patients with renal or hepatic impairment or when co-administrated with CYP3A4 inhibitors [START_REF] Chen | Pharmacokinetics and Pharmacodynamics of Orally[END_REF]12]. In opposition to baricitinib, this treatment failed to demonstrate mortality reduction in mechanically ventilated COVID-19 patients [13].

Comparative PK studies of baricitinib and ruxolitinib in a critical-care setting would be of interest. Contrary to ruxolitinib, baricitinib is not extensively metabolized by the liver, so less exposure variability due to liver impairment or CYP450-mediated drug-drug interactions is expected. Nevertheless, our data suggest that PK variability and atypical PK profiles still occur with baricitinib and that one drug dosage is unlikely to fit all patients.

The main limitation of our study is the small number of patients for whom the AUC could be calculated due to the small number of sampling time-points available for each patient. In addition, the noncompartmental method used to estimate AUC could overestimate the actual value due to lack of concentration time points in the terminal elimination phase of the PK profile. Moreover, no population PK modeling was available in this population as an alternative to estimate the patients' individual PK parameters. The size of our cohort did not allow assessing accurately the exposure/effect relationship of baricitinib in critically ill COVID-19 patients but despite relatively low plasma concentrations, all patient had positive evolution of the disease. Further studies are needed to determine whether a threshold of plasma exposure is associated with clinical outcomes in COVID-19 and whether therapeutic drug monitoring of baricitinib could contribute to optimize the drug response.

CONCLUSION

  In conclusion, we observed a high inter-patient PK variability as well as relatively low trough concentrations and AUCs of baricitinib in critically ill COVID-19 patients. Further concentrationcontrolled studies of baricitinib in the critical care setting should be performed to decipher the exposureeffect relationship in the COVID-19 indication. Administered Ruxolitinib (INCB018424 Phosphate) in Renal and Hepatic Impairment Patients. Clin. Pharmacol. Drug Dev. 2014, 3, 34-42, doi:10.1002/cpdd.77. 12. Shi, J.G.; Chen, X.; Emm, T.; Scherle, P.A.; McGee, R.F.; Lo, Y.; Landman, R.R.; McKeever, E.G.; Punwani, N.G.; Williams, W.V.; et al. The Effect of CYP3A4 Inhibition or Induction on the Pharmacokinetics and Pharmacodynamics of Orally Administered Ruxolitinib (INCB018424 Phosphate) in Healthy Volunteers. J. Clin. Pharmacol. 2012, 52, 809-818, doi:10.1177/0091270011405663. 13. Rein, L.; Calero, K.; Shah, R.; Ojielo, C.; Hudock, K.M.; Lodhi, S.; Sadaka, F.; Bellam, S.; Palma, C.; Hager, D.N.; et al. Randomized Phase 3 Trial of Ruxolitinib for COVID-19-Associated Acute Respiratory Distress Syndrome. Crit. Care Med. 2022, 50, 1701-1713, doi:10.1097/CCM.0000000000005682.

Figure :

 : Figure : Plasma concentrations of baricitinib in critically ill COVID-19 patients (n=7). The four patients with three baricitinib concentrations available are displayed with empty forms. Patients with delayed Tmax are displayed in bold lines.

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. participated to acquisition of data, NM, PF, FL and CT participated to the conception and design of the study, NM and CT performed the analysis and interpretation of data and drafted the article, all authors finally approved

Pharmacokinetics of baricitinib in critically ill

 This study provides original data on baricitinib plasma concentrations observed in critically ill patients with COVID-19  High inter-patient variability of baricitinib plasma exposure was observed  The results of this observational study incite to further investigate the use of therapeutic drug monitoring to optimize the drug response BMI: body mass index; SAPS II: Simplified Acute Physiology Score (The SAPS II is a severity score used in critical care medicine, ranging from 0 to 163, used to calculate predicted admission mortality) ; SOFA: Sequential Organ Failure Assessment; PaO2/FiO2: Partial pressure of oxygen in arterial blood/Fraction of inspired oxygen; ASAT: Aspartate aminotransferase; ALAT: Alanine aminotransferase; H0: concentration measured at predose; H1: concentration measured one hour after drug uptake; H3: concentration measured three hours after drug uptake; ICU: intensive care unit