Sitthichok Kasemthaveechok

Laura Abella

Marion Jean

Marie Cordier

Thierry Roisnel

Nicolas Vanthuyne

Thierry Guizouarn

Olivier Cador

Jochen Autschbach

Jeanne Crassous

Ludovic Favereau

Axially and Helically Chiral Cationic Radical Bicarbazoles: SOMO-HOMO Level Inversion and Chirality Impact on the Sta-bility of Mono-and Diradical Cations

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INTRODUCTION

Stable organic molecules with open-shell electronic structures have attracted significant interest in chemistry and biochemistry due to their specific reactivity and their use as agents for in-vivo imaging. [START_REF] Ardenkjaer-Larsen | Increase in signal-to-noise ratio of > 10,000 times in liquid-state NMR[END_REF][START_REF] Williams | Biomedical EPR -Part A: Free Radicals[END_REF][START_REF] Mathies | Efficient Dynamic Nuclear Polarization at 800 MHz/527 GHz with Trityl-Nitroxide Biradicals[END_REF] This class of compounds has also received increasing attention in materials science, because of the possibility to combine electron and spin conduction in molecular electronics such as organic light-emitting diodes (OLEDs), organic fieldeffect transistors (OFETs), and organic magnets. 4,[START_REF] Zeng | Pro-aromatic and anti-aromatic π-conjugated molecules: an irresistible wish to be diradicals[END_REF][START_REF] Kumar | Recent Advances in Organic Radicals and Their Magnetism[END_REF]7,[START_REF] Gopalakrishna | From open-shell singlet diradicaloids to polyradicaloids[END_REF][START_REF] Kato | Platforms for Stable Carbon-Centered Radicals[END_REF][START_REF] Ni | 3D global aromaticity in a fully conjugated diradicaloid cage at different oxidation states[END_REF][START_REF] Liu | Global Aromaticity in Macrocyclic Polyradicaloids: Huckel's Rule or Baird's Rule?[END_REF][START_REF] Peng | Organic Light-Emitting Diodes Using a Neutral π Radical as Emitter: The Emission from a Doublet[END_REF][START_REF] Gu | A Three-Dimensionally π-Conjugated Diradical Molecular Cage[END_REF][START_REF] Ai | A Stable Room-Temperature Luminescent Biphenylmethyl Radical[END_REF][START_REF] Ai | Efficient radical-based lightemitting diodes with doublet emission[END_REF][START_REF] Sugawara | Spintronics in organic πelectronic systems[END_REF][START_REF] Sugawara | Interplay between magnetism and conductivity derived from spin-polarized donor radicals[END_REF] In this context, organic diradicals are of particular interest since they can adopt two different spin states, namely singlet and triplet, related to the interaction between the unpaired electrons within the molecular backbone. 4,[START_REF] Gallagher | High-Spin Organic Molecules[END_REF] Organic conjugated radicals also allow to gain fundamental insights on the nature of chemical bonds and their delocalization, in addition to their other features such as narrow HOMO-LUMO energy gaps, low-lying doubly-excited states, and redox amphoterism. [START_REF] Zeng | Pro-aromatic and anti-aromatic π-conjugated molecules: an irresistible wish to be diradicals[END_REF][START_REF] Ni | 3D global aromaticity in a fully conjugated diradicaloid cage at different oxidation states[END_REF][START_REF] Liu | Global Aromaticity in Macrocyclic Polyradicaloids: Huckel's Rule or Baird's Rule?[END_REF][START_REF] Teki | Excited-State Dynamics of Non-Luminescent and Luminescent pi-Radicals[END_REF] It has recently emerged that combining these electronic properties with a chiral organic -conjugated system gives rise to molecular architectures with unique magnetic and chiropto-electronic features, such as specific absorption of near infrared (NIR) circularly polarized (CP) light, [START_REF] Ueda | Chiral Stable Phenalenyl Radical: Synthesis, Electronic-Spin Structure, and Optical Properties of [4]Helicene-Structured Diazaphenalenyl[END_REF][START_REF] Wang | Radical Cation and Neutral Radical of Azathia[7]helicene with SOMO-HOMO Energy Level Inversion[END_REF][START_REF] Gliemann | Configurationally Stable Chiral Dithia-Bridged Hetero[4]helicene Radical Cation: Electronic Structure and Absolute Configuration[END_REF][START_REF] Ivanov | Isolation of a chiral anthracene cation radical: X-ray crystallography and computational interrogation of its racemization[END_REF][START_REF] Hsieh | picene: A New Stable[END_REF][START_REF] Inoue | Hash-Mark-Shaped Azaacene Tetramers with Axial Chirality[END_REF][START_REF] Narita | An Azulene-Based Chiral Helicene and Its Air-Stable Cation Radical[END_REF][START_REF] Shu | High-Spin Diradical Dication of Chiral pi-Conjugated Double Helical Molecule[END_REF][START_REF] Prince | Cethrene: A Helically Chiral Biradicaloid Isomer of Heptazethrene[END_REF][START_REF] Ravat | Dimethylcethrene: A Chiroptical Diradicaloid Photoswitch[END_REF][START_REF] Shen | Iron Alkynyl Helicenes: Redox-Triggered Chiroptical Tuning in the IR and Near-IR Spectral Regions and Suitable for Telecommunications Applications[END_REF] radical CP luminescence (CPL), [START_REF] Mayorga-Burrezo | An Enantiopure Propeller-Like Trityl-Brominated Radical: Bringing Together a High Racemization Barrier and an Efficient Circularly Polarized Luminescent Magnetic Emitter[END_REF][START_REF] Mayorga Burrezo | Organic Free Radicals as Circularly Polarized Luminescence Emitters[END_REF] and possibly enhanced spin-filtering properties. [START_REF] Shil | A high-spin organic diradical as a spin filter[END_REF] However, designing organic chiral open-shell chromophores remains a formidable challenge, owing to the high chemical reactivity of radicals and poor configurational stability. [START_REF] Tan | Frontier molecular orbital effects control the hole-catalyzed racemization of atropisomeric biaryls[END_REF] As a result, known examples remain rare and are mainly limited to monoradicals, with very few examples of stable diradicals. [START_REF] Hsieh | picene: A New Stable[END_REF][START_REF] Inoue | Hash-Mark-Shaped Azaacene Tetramers with Axial Chirality[END_REF][START_REF] Shu | High-Spin Diradical Dication of Chiral pi-Conjugated Double Helical Molecule[END_REF][START_REF] Prince | Cethrene: A Helically Chiral Biradicaloid Isomer of Heptazethrene[END_REF] So far, the chemical and configurational stabilities issues have been overcome using mainly two strategies: Shielding the unpaired electrons with bulky substituents, or enhancing their delocalization over the molecular backbone. [START_REF] Gopalakrishna | From open-shell singlet diradicaloids to polyradicaloids[END_REF][START_REF] Kato | Platforms for Stable Carbon-Centered Radicals[END_REF][START_REF] Shu | High-Spin Diradical Dication of Chiral pi-Conjugated Double Helical Molecule[END_REF][START_REF] Wang | Air-stable diradical dications with ferromagnetic interaction exceeding the thermal energy at room temperature: from a monomer to a dimer[END_REF][START_REF] Gallagher | Thermally and Magnetically Robust Triplet Ground State Diradical[END_REF][START_REF] Kanetomo | Ground Triplet Spirobiradical: 2,2',7,7'-Tetra( tert-butyl)-9,9'(10 H,10' H)spirobiacridine-10,10'-dioxyl[END_REF][START_REF] Tang | Tuning the stability of organic radicals: from covalent approaches to non-covalent approaches[END_REF] While such approaches can be efficient, they often require additional synthetic steps and may preclude significant intermolecular interactions for charge and spin transport applications. [START_REF] Sugawara | Interplay between magnetism and conductivity derived from spin-polarized donor radicals[END_REF] Recently, theoretical and experimental studies have reported that the stability of radicals can be considerably enhanced if they are in an uncommon electronic configuration, such that the energy of the singly occupied molecular orbital (SOMO) is below the highest doubly occupied molecular orbital (HOMO) level (Figure 1). Systems with such a SOMO-HOMO inversion (SHI), which has sometimes (albeit imprecisely [START_REF]There is no 'hole below the Fermi level[END_REF] ) been construed as a 'non-Aufbau' electronic configuration, [START_REF] Wang | Radical Cation and Neutral Radical of Azathia[7]helicene with SOMO-HOMO Energy Level Inversion[END_REF][START_REF] Shu | High-Spin Diradical Dication of Chiral pi-Conjugated Double Helical Molecule[END_REF][START_REF] Gryn'ova | Switching radical stability by pH-induced orbital conversion[END_REF][START_REF] Franchi | SOMO-HOMO Conversion in Distonic Radical Anions: An Experimental Test in Solution by EPR Radical Equilibration Technique[END_REF][START_REF] Kumar | SOMO-HOMO Level Inversion in Biologically Important Radicals[END_REF] have been notably successful as achiral luminescent radical emitters in OLED devices. [START_REF] Guo | High stability and luminescence efficiency in donor-acceptor neutral radicals not following the Aufbau principle[END_REF] While this approach has appeared rather interesting to increase the stability of already persistent radicals such as nitroxide and tris(2,4,6-trichlorophenyl)methyl (TTM, Figure 1), the impact of SHI on known unstable radical remains unexplored. Regarding chiral radicals, a SHI has been mentioned only for one helical derivative under its racemic form. [START_REF] Wang | Radical Cation and Neutral Radical of Azathia[7]helicene with SOMO-HOMO Energy Level Inversion[END_REF] In addition, compounds with such a peculiar electronic configuration are often viewed as promising intermediates to form triplet ground state diradicals with nondegenerate SO-MOs upon oxidation, resulting from the removal of one electron of the HOMO level. [START_REF] Wang | Radical Cation and Neutral Radical of Azathia[7]helicene with SOMO-HOMO Energy Level Inversion[END_REF][START_REF] Kumar | SOMO-HOMO Level Inversion in Biologically Important Radicals[END_REF][START_REF] Gryn'ova | Theory and practice of uncommon molecular electronic configurations[END_REF] Accordingly, a deep understanding of the electronic and steric factors that govern monoradical stability displaying a SHI is urgently needed, as it may help to develop open-shell compounds with higher stability and bring new synthetic strategies to design innovative chiral organic high-spin materials.

Figure 1. a) Examples of reported stable monoradicals with a SOMO-HOMO inversion (SHI) where the molecular fragment providing the HOMO level above the SOMO one is highlighted in red, and schematic illustration of triplet ground-state diradical obtained from the oxidation of a radical with a SHI; 42 b) Novel axially and helically mono-and diradicals cationic carbazole derivatives. The unoccupied counterpart of the SOMO in the SHI systems is calculated to be above the near-degenerate energies of the spin orbitals constituting the HOMO level, i.e., there is no electron hole among the occupied levels. Herein, we report a simple molecular design of persistent chiral organic cationic radicals based on C2-symmetric bicarbazole derivatives. Investigations of their photophysical, chiroptical and electrochemical properties revealed a high chemical stability for unprotected carbazole axial radical 1 •+ , in comparison to unstable helical 2 •+ (Figure 1b). Interestingly, both compounds exhibit SHI, indicating that such an electronic configuration is not a sufficient prerequisite to provide radical stability. Our results rather suggest that in addition to the SHI, the electronic coupling between the radical center and the electron rich unit on which the HOMO is centered plays a crucial role in the radical reactivity. In 1 •+ and 2 •+ , this aspect depends on the dihedral angle between the carbazole moieties, which is governed by the axial and helical chirality of the bicarbazole systems. [START_REF]Helical chirality is often used when the considered molecule adopts a helical structure around the axis of chirality, rather than showing an 90° angle arrangement between the substituents of the axis (see Mayor et al[END_REF] Such findings bring new insights regarding the stability of organic radicals and allowed us to further design and isolate chemically and configurationally stable chiral diradical 3 2•2+ , which displayed promising near-infrared chiroptical properties and nearly degenerate singlet-triplet ground states.

RESULTS AND DISCUSSION

Synthesis and Structural characterization. We first prepared bicarbazole derivatives 1 and 2. The synthesis was inspired from that of BICOL, a ligand previously used in asymmetric catalysis. [START_REF] Botman | Synthesis and Resolution of BICOL, a Carbazole Analogue of BINOL[END_REF][START_REF] Hoen | New approaches in asymmetric rhodium-catalyzed hydrogenations with monodentate phosphoramidites University of Groningen[END_REF] Briefly, carbazole CBzOMe was first hydrolyzed and then engaged in an oxidative homocoupling reaction using vanadium complex as catalyst to afford 4,4'-bicarbazole-3,3'-diol, BICOL, derivative (Scheme 1). The latter was then alkylated both at the nitrogen and oxygen positions to give rac-1, which was resolved to its corresponding axially chiral enantiomers (+)-/(-)-1 using chiral HPLC (see ESI for details). Helical (+)-/(-)-2 were successively obtained from (+)-and (-)-1 through hydrolysis and subsequent bridging of the hydroxy groups using diiodomethane in 90% yield. Further functionalization of monocarbazole CBzOMe afforded N-methylated 1', which was then converted to sterically hindered 3' with t-Butyl group at the 6-position (Scheme 1). Steric protection was also applied to (+)-/(-)-1 and gave (+)-/(-)-3, in 90% yield. Under similar conditions, helical bicarbazole 2 did not afford the expected bis t-Butyl derivative 4 but rather a complex mixture of oligomers with multiple t-Butyl groups. As a result, compound 4 was obtained from 3 using similar synthetic strategy than for 2, in an overall 25% yield. Scheme 1. Schematic synthetic route for 1, 1', 2, 3, 3' and 4. Reaction conditions: i) Me2SO4, 22M NaOH, Acetone, reflux, 77%; ii) BBr3, CH2Cl2, rt, >95%; iii) VO(acac)2, MeCN, O2, rt, 50%; iv) t-BuCl, ZnCl2, MeNO2, rt, >90%; v) CH2I2, K2CO3, acetone, Ar, reflux, >90%. ORTEP drawing of 1, 2, 3 and 4 with the corresponding dihedral angle between the carbazole units. Hydrogen atoms are omitted for clarity. X-Ray structures analyses provide further structural insights for rac-1 and rac-2, which crystallized in the P-1 and C 2/c space groups, respectively. Dihedral angles of 87.1° and 52.4° between the carbazole units were measured for 1 and 2, respectively, which for the latter appeared to be reminiscent of the one found for carbo [6]helicene. [START_REF] Gingras | One hundred years of helicene chemistry. Part 2: stereoselective syntheses and chiral separations of carbohelicenes[END_REF][START_REF] Chen; Shen | Helicenes Chemistry: From Synthesis to Applications[END_REF][START_REF] Dhbaibi | Enantioenriched Helicenes and Helicenoids Containing Main-Group Elements[END_REF] This structural difference was confirmed by density functional theory (DFT) calculations (Figure S49), and also probed by the 1 H NMR spectra. Protons H 5 (Scheme 1, doublet signal in Figure S33, S37) are more shielded for 1 than for 2 (6.53 and 6.95 ppm, respectively), owing to an unequal current ring effect coming from either the opposite aromatic phenyl ring in 1 or the N-heterocycle one in 2. Rac-3 and rac-4 crystallized in the P21/n space group and dis-played dihedral angles between 98 and 102° for rac-3 (two molecules within the same unit cell) and of 55.5° for rac-4. However, two conformers were found for 3 in the calculations, with dihedral angles of 65° and 106.6°, the latter being higher in energy by 1.8 kcal•mol -1 (Boltzmann population of 4% at 298 K). Therefore, due to crystal packing, 3 does not crystallize in its lowest-energy conformer. The calculated properties of 3 further support the higher abundance of the conformer with dihedral angle of 65° in solution by comparison with the experimental chiroptical properties (vide infra). All these observations indicate that the electronic coupling between the two carbazole units of these chiral compounds should differ significantly, depending on the dihedral angle, which may impact their corresponding photophysical, chiroptical and electronic properties.

Computational Details. Kohn-Sham Density functional theory (DFT) as implemented in the Gaussian (G16) package was used for all computations, [START_REF] Frisch | Gaussian 16, Revision B.01[END_REF] employing the PBE0 functional [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] and the def2-SV(P) basis. [START_REF] Weigend | Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy[END_REF][START_REF] Weigend | Accurate Coulomb-fitting basis sets for H to Rn[END_REF] For neutral and oxidized species, solvent effects were considered by means of the polarizable continuum model (PCM) for dichloromethane and acetonitrile, [START_REF] Scalmani | Continuous surface charge polarizable continuum models of solvation. I. General formalism[END_REF] respectively, to match the experimental conditions. 'D3' dispersion corrections were included in the calculations. [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF] Excited state energies, transition moments, excited state structures and their vibrational normal modes were obtained from time-dependent DFT (TD-DFT) response theory. For the absorption and electronic circular dichroism (ECD) spectra we calculated the 200 lowest-energy vertical spin-allowed electronic excitations. The transitions were subsequently Gaussianbroadened with  = 0.20 eV to simulate the spectral envelopes. For overviews of the theoretical approach to model natural optical activity by quantum chemical calculations, in particular via TD-DFT, see, for example, available reviews. [START_REF] Autschbach | Timedependent density functional response theory for electronic chiroptical properties of chiral molecules[END_REF][START_REF] Srebro-Hooper | Calculating Natural Optical Activity of Molecules from First Principles[END_REF] Different functionals and solvent effects were tested for these compounds; see the Supporting Information for details.

Photophysical properties of 1, 2 and 1'. Carbazole 1'

and bicarbazoles 1 and 2 display similar UV-vis spectra, with two maxima of absorption at 270 and 300 nm ( ~ 3.7  10 4 and 3.0  10 4 M -1 cm -1 ), respectively, and a vibronic band between 325 and 390 nm ( ~ 1.0  10 4 M -1 cm -1 , Figure 2). Whereas 1 and 2 show the expected higher absorption intensity, about twice as for 1', the UV-vis spectrum of 2 exhibits red-shifted absorption bands at 330 and 370 nm, indicating a stronger electronic interaction between the bridged carbazole systems, as expected from the smaller dihedral angle. For both bicarbazole compounds, the lowest-energy excitation with sizeable oscillator strength (f) corresponds to the HOMO-LUMO transition at 327 and 355 nm for 1 and 2 respectively. In each case, the HOMO and LUMO are delocalized over the entire molecule, and the LUMO exhibits nodes at the nitrogens (Figure 2). Both compounds have quasi-degenerate HOMO and HOMO-1 levels, separated by 0.112 eV and 0.149 eV for 1 and 2, respectively. Visual inspection of the MO isosurface plots reveals that the HOMO-1 and HOMO are essentially in-phase and out-ofphase (+/-) linear combinations of individual carbazole fragment frontier orbitals (FFOs). The different energetic splitting of the HOMO-1 and HOMO indicates that the FFO interactions are more pronounced in compound 2, which goes along with the significant red-shift and slightly larger intensities for the calculated transitions at 285-290 nm and 327-355 nm in comparison to 1 (Figure 2 and Tables S7-S8). The chiroptical properties of 1 and 2 have also been studied, both experimentally and theoretically, and are detailed in the supporting information (Figure S5 andS54). Bicarbazoles 1 and 2 display moderately intense fluorescence around 400 nm, with quantum yields of ~20% (Figure 2). The enhanced electronic communication between the carbazole units in 2 remains also present in the excited state as indicated by the fact that its emission peak maximum is red-shifted relative to the emission peak of 1, similar to how the absorption of 2 is red-shifted. In fact, 1 displays an emission profile close to the one of 1', evidencing a weak electronic coupling between the carbazole -systems also in the excited-state.

Radical 1 •+ and 2 •+ . Cyclic voltammetry (CV) revealed two irreversible oxidation processes for 1' around 0.90 V and 1.70 V, and two distinct oxidation events for both 1 and 2 in the range of 0.8 to 1.4 V (vs. SCE, Figure 3). Compound 1 exhibits a first fully reversible one-oxidation process at 0.92 V and a second less reversible one at 1.11 V with the appearance of a small reduction peak at 0.70 V in the reverse scan. In stark contrast, 2 showed two highly irreversible oxidation processes at +1.14 and +1.40 V, as classically observed for 3-and/or 6-unprotected carbazole derivatives like 1', which form unstable cations under oxidative conditions. [START_REF] Chiu | Electrochemical and Spectral Characterizations of 9-Phenylcarbazoles[END_REF][START_REF] Karon | Carbazole electrochemistry: a short review[END_REF][START_REF] Mallick | Oxidative Coupling of Carbazoles: A Substituent-Governed Regioselectivity Profile[END_REF] The unusual reversibility of the first oxidation event of 1, as well as the marked difference of ~200 mV with the one of 2, prompted us to deeper investigate monoradical cation 1 •+ . For comparison purpose, 3and 6-protected monocarbazole 3' was selected as a persistent carbazole radical reference since its CV gives one fully reversible oxidation process at 0.87 V vs. SCE (Figure 3). + BF4 -; bottom :ECD spectra of (+)-1 (black) and (+)-1 •+ BF4 -(red) with their corresponding enantiomers in dash line in dichloromethane at 298 K with an extension of the NIR-ECD spectra of (+)-and (-)-1 •+ BF4 -. Addition of AgBF4 as a chemical oxidant to a solution of 3' afforded persistent radical 3' •+ BF4 -, as expected for 3,6-protected carbazole, with a characteristic vibronic visible signature at 696 nm and 771 nm (Figure 3). [START_REF] Chiu | Electrochemical and Spectral Characterizations of 9-Phenylcarbazoles[END_REF][START_REF] Karon | Carbazole electrochemistry: a short review[END_REF]62 Using the same conditions, 1 was quantitatively converted to 1 •+ BF4 -, which showed similar optical signature than 3' •+ BF4 -, with an additional weak broad IR band ( ~ 500 M -1 cm -1 at 1000 nm), and a half-life evaluated to 3 hours in air saturated CH2Cl2. Under inert conditions, this half-life extends to 14 hours, indicating a persistent character for 1 •+ BF4 -. Indeed, mass spectroscopy and UV-vis analysis of the resulting mixture after decomposition of the radical compound reveal the presence of bicarbazole oligomers of different molecular size, which further confirm the kinetic stability of 1 •+ BF4 -. The fact that the half-life of the radical decreases in presence of oxygen indicates that other decomposition pathways may also take place. The paramagnetic nature of these two monoradical cations was also assessed by EPR measurements, which display a similar intense doublet signal of an organic nitrogen based radical for both compounds (g = 2.00, Figure 3 and Figure S16). The identical optical and magnetic signatures of 3' •+ BF4 -and 1 •+ BF4, and the lack of intense intervalence absorption transition for the latter, suggest that the unpaired electron is localized mainly on one carbazole unit in 1 •+ BF4 -, and only slightly delocalized over the second one, contrary to class II and III of organic mixed valence compounds. [START_REF] Hankache | Organic Mixed Valence[END_REF] This situation is rather surprising since it leads to an unprecedented persistent carbazole centered radical without any steric protection at the reactive 6-position. According to the photophysical experiments and theoretical calculations (Figure 2), the two carbazole fragment orbitals in 1 showed negligible interaction due to their relative nearly perpendicular orientation, affording almost degenerate HOMO and HOMO-1 levels in the neutral state. As a result, the first two oxidation events occurring for 1 are the sequential oxidations of each carbazole unit, which partially lift the pseudo-degeneracy of the HOMO in the monoradical cationic state 1 •+ . These two anodic signals are followed by a third irreversible one at +1.63 V, attributed to the second oxidation of one CBz •+ unit, in accordance with the sequential oxidations observed for monocarbazole 3' at +0.85 (3' •+ ) and +1.53 V (3' 2•2+ , Figure 3). Following these considerations, 1 •+ BF4 -adopts an uncommon electronic configuration since its SOMO becomes lower in energy than the remaining HOMO of the second non-oxidized carbazole fragment, partially isolated from the radical center due to the high dihedral angle between the two -systems. Further insights regarding the SHI were obtained by theoretical analysis of the electronic configuration of 1 •+ BF4 -. As depicted in Figure 4, the calculated spin densities are represented by spin-polarized Kohn-Sham (KS) systems where the α-spin HOMO has a matching occupied β-spin HOMO, with both having very similar orbital energies, while the SOMO lies energetically below the HOMO-1. Unlike the HOMO and HOMO-1 on the neutral systems, the frontier orbitals of 1 •+ are localized on either one of the fragments. The calculated spin density of 1 •+ clearly reflects the excess α-spin represented by the SOMO and the corresponding β-spin hole represented by the LUMO. As mentioned already, this SHI has been claimed as a stabilizing strategy for radicals, [START_REF] Wang | Radical Cation and Neutral Radical of Azathia[7]helicene with SOMO-HOMO Energy Level Inversion[END_REF][START_REF] Gryn'ova | Switching radical stability by pH-induced orbital conversion[END_REF][START_REF] Franchi | SOMO-HOMO Conversion in Distonic Radical Anions: An Experimental Test in Solution by EPR Radical Equilibration Technique[END_REF][START_REF] Kumar | SOMO-HOMO Level Inversion in Biologically Important Radicals[END_REF][START_REF] Guo | High stability and luminescence efficiency in donor-acceptor neutral radicals not following the Aufbau principle[END_REF][START_REF] Kusamoto | Realization of SOMO-HOMO Level Conversion for a TEMPO-Dithiolate Ligand by Coordination to Platinum(II)[END_REF] and our results suggest that such effects may be also responsible for the observed stability of 1 •+ BF4 -. It is important to note, however, that 2 •+ BF4 -appeared experimentally unstable, despite an electronic configuration implying a SHI, and instantaneously affords a complex mixture of oligomers. This high stability contrast represents therefore a unique opportunity to gain more insights about the key parameters that lead to stable radical with SHI configuration. We note that different functionals have been tested to ensure that the SHI for 1 •+ and 2 •+ does not only show up with one specific functional (see SI). Since the structure of 1 and 2 differ by the dihedral angle between the two carbazole systems, the difference of reactivity between 1 •+ BF4 -and 2 •+ BF4 -may be related either to steric arguments or to the electronic coupling between the radical center and the second carbazole fragment. As mentioned above, both radicals 1 •+ BF4 -and 2 •+ BF4 -show a similar decomposition pathway implying the formation of oligomers occurring presumably at the remaining free 6 and 6' positions, as it occurs for classical non protected carbazole radicals. Since the second carbazole unit is not attached next to this reactive position of the carbazole radical unit, and less well oriented for 1 •+ BF4 -than for 2 •+ BF4 - (related to their dihedral angles, Figure S49), the substituent steric hindrance seems too far to play a decisive role in the difference of stability between 1 •+ BF4 -and 2 •+ BF4 -. On the contrary, the electronic properties between 1 and 2 clearly differ, as evidence by the photophysical and chiroptical characterizations, which show that the electronic interaction between the two carbazole units is negligible for 1 in comparison to 2. This is fur-ther supported by the separation of the two first oxidation potentials, ΔEOx, equal to 190 mV and 260 mV for 1 and 2, respectively (Figures 3 and4). [START_REF] Hankache | Organic Mixed Valence[END_REF][START_REF] Heckmann | Organic Mixed-Valence Compounds: A Playground for Electrons and Holes[END_REF] In addition, the second carbazole unit does not help to stabilize the radical by resonance effect due to its meta-position regarding the nitrogen atom of the radical unit. On the contrary, it induces a negative inductive effect which destabilizes more radical 2 •+ BF4 -than 1 •+ BF4 -, due to the greater electronic communication between the carbazole systems in 2 •+ BF4 -. This is experimentally confirmed by the facility of 1 to form the corresponding monoradical 1 •+ (EOx = 0.92 V vs. SCE) as opposed to 2 •+ (EOx = 1.14 V vs. SCE, Figure 4), and theoretically reproduced by the corresponding calculated vertical ionization potentials, which indicate a higher value for 2 (5.42 eV), in comparison to 1 (5.16 eV, Figure 4). The difference amounts to 25 kJ/mol. Another notable difference between 1 •+ and 2 •+ , is the orbital energy gap  between the HOMO and the SOMO of 0.8 eV and 0.4 eV, respectively. Moreover, the energy difference between the HOMO of 2 and 2 •+ is 0.54 eV, much larger than the difference of 0.18 eV for 1 and 1 •+ , confirming the stronger interaction between the radical center and the second carbazole in 2 •+ (Figures 4, S57 andS58). The obtained results bring further evidence of the crucial role of the electronic coupling between the radical center and the neutral electron-rich fragment on stabilizing a molecular open-shell electronic state with SOMO-HOMO energy inversion. [START_REF] Guo | High stability and luminescence efficiency in donor-acceptor neutral radicals not following the Aufbau principle[END_REF] In addition to its chemical stability, 1 •+ BF4 -remains also configurationally stable, presumably because of the high enantiomerization barrier of its corresponding neutral form (G  = 173.5 kJ mol -1 , 180°C in 1,2-dichlorobenzene), which allowed to record the ECD of the radical enantiomers (+)-/ (-)-1 •+ BF4 -. As depicted in Figure 3, the ECD signals of the neutral form experienced a decrease in intensity and a global red shift of around 10 nm in the radical state, together with the appearance of new ECD active bands across the visible and the IR regions. For instance, the intense signal at 302 nm (Δ = + 40 M -1 cm -1 ) and 360 nm (Δ = -14 M -1 cm -1 ) for neutral (+)-1 decreased and broadened to 312 and 370 nm (Δ = + 24 and -4.5 M -1 cm -1 , respectively) upon formation of (+)-1 •+ BF4 -, all these changes being well reproduced by calculated ECD spectra (Figure S60). Importantly, new mirror-image Cotton effects at 570, 715 and 860 nm (Δ = + 0.3, -0.6 and + 1.6 M -1 cm -1 , respectively) can be unambiguously associated to radical absorption transitions, highlighting the intrinsically chiral nature of this new persistent radical.

Mono and diradical of 3 and 4 and related characterization.

The SHI found for 1 •+ BF4 -gives an interesting opportunity to generate the corresponding organic diradical. As illustrated in Figure 1, oxidation of monoradical displaying a SHI may result in a diradical with triplet ground state multiplicity, in accordance with Hund's first rule, or an open-shell singlet with antiferromagnetic coupling of the unpaired spins (Figure S20). In addition, as 1 •+ BF4 -can be considered as an innovative molecular design of chiral organic spin-polarized donor, [START_REF] Sugawara | Spintronics in organic πelectronic systems[END_REF][START_REF] Izuoka | Spin Alignment in Singly Oxidized Spin-Polarized Diradical Donor: Thianthrene Bis(nitronyl nitroxide)[END_REF][START_REF] Sakurai | Design, Preparation, and Electronic Structure of High-Spin Cation Diradicals Derived from Amine-Based Spin-Polarized Donors[END_REF][START_REF] Nakazaki | Design and preparation of pyrrole-based spin-polarized donors[END_REF][START_REF] Komatsu | Synthesis and properties of TSF-based spin-polarized donor[END_REF], 70 it appears interesting to investigate the magnetic interaction between the two radical units upon oxidation of the second carbazole unit. In order to evaluate the possibility to obtain stable chiral diradicals, oxidation of 1 •+ was investigated by using tris(4-bromophenyl)aminium hexachloridoantimonate ("magic blue" oxidant, (4-BrPh)3NSbCl6). [START_REF] Connelly | Chemical Redox Agents for Organometallic Chemistry[END_REF] However, UV-vis titration indicated instantaneous reactivity of 1 2•2+ 2SbF6 -, presumably through polymerization reaction as indicated by the formation of a broad IR band between 800 and 1800 nm (Figure S18) and anticipated by the non-fully reversible second oxidation process of 1 for in CV (Figure 3). To circumvent this aspect and increase radical persistence, we turned our attention to 3 and 4, which are the 6 and 6' sterically protected analogues of 1 and 2. These new chiral systems show similar chiroptical and photophysical properties as 1 and 2, but slightly red-shifted, likely owing to the presence of the donor t-butyl groups (Figure S5, S6). Regarding the possible presence of two conformers of 3 with different dihedral angles, the calculated ECD spectra of the conformer with the small dihedral angle (67°) agree well with the experimental ones (Figure S61), confirming its higher abundance in solution over the conformer with the large dihedral angle (106°). As for 1 and 2, CVs of 3 and 4 afford two oxidation events for both compounds (Figure 5), suggesting a SHI for corresponding monoradicals 3 •+ and 4 •+ , as in the case of 1 •+ and 2 •+ . Interestingly, the reversibility of the oxidations processes was strongly enhanced in comparison to the ones for 1 •+ and 2 •+ , highlighting the benefits of the steric protection on the carbazoles 6 and 6' positions. Indeed, 3 shows full reversibility for both the first and second oxidation signals (EOx = 0.87 and 1.10 V vs. SCE) with no degradation after cycling scans. In contrast, while the first oxidation process of 4 exhibits a fully reversible response at 1.06 V vs. SCE, its second oxidation at 1.37 V shows only a partial reversibility and leads to the appearance of a weak new reduction peak at 0.85 V in the reverse scan, suggesting some instability for diradical 4 2•2+ , as in the case of 1. The UV-vis-NIR spectra of monoradicals 3 •+ and 4 •+ were investigated using spectroelectrochemistry and show similar photophysical signatures than 1 •+ in the UV-vis region but significant changes in the low energy part of the spectra (Figure 4). For both 3 •+ and 4 •+ , the corresponding NIR absorption bands red-shift to 812 and 850 nm respectively, and show much less vibronic structure than 1 •+ . In addition, the broad NIR band that extends up to 1800 nm for both radicals, appears more intense than for 1 •+ , with  = 760 M -1 cm -1 and 1100 M -1 cm -1 for 3 •+ and 4 •+ , respectively. These observed optical differences suggest that the radical center is not anymore localized on one carbazole fragment for both 3 •+ and 4 •+ but interacts with the second donor fragment to some extent. Moreover, the electronic coupling between the two carbazole seems higher for helical radical 4 •+ (ΔEOx = 310 mV) than for axial 3 •+ (ΔEOx = 230 mV), in line with its smaller dihedral angle (54° vs. 65° for 4 •+ and 3 •+ , respectively). Despite the presence of the t-butyl groups, addition of one equivalent of magic blue to an acetonitrile solution of 4 did not lead to 4 •+ SbCl6 -, but rather polymeric materials as indicated by the obtained broad NIR band centered at 1440 nm which clearly differs from the optical signature recorded by spectroelectrochemistry (Figure S10). This result further indicates that the steric protection at the 6 and 6' positions of the investigated bicarbazole systems is not a decisive factor for reaching radical stability. On the contrary, 3 •+ BF4 -was quantitatively obtained and displays a high stability with a measured half-life of 4.1 days in solution (Figure S28, Table S6). For comparison, 3' •+ BF4 -shows a half-life of 3.5 days under the same conditions, highlighting a synergetic effect between the steric protection and the SHI on the radical stability. Frontier molecular orbitals of 3 •+ further show that the SHI goes along with a (HOMO/SOMO) energy gap of 0.47 eV, which is in between the ones for 1 •+ and 2 •+ . This follows the trend for the calculated dihedral angles between the -conjugate systems (77° and 52° for 1 •+ and 2 •+ , respectively). These results further confirm the importance of minimizing the electronic interaction between the radical center and the additional donor unit to increase stability of a radical displaying SHI. The high stability found for 3 •+ SbCl6 -prompted us to further explore the formation of diradical 3 2•2+ 2SbCl6 -. Additions of two equivalents of magic blue to the solution of 3 results in the quantitative formation of diradical dication 3 2•2+ 2SbCl6 -, which represents one of the rare examples of persistent intrinsically organic chiral diradicals, and thus embedded unique magnetic and chiroptical properties (Figure 6). 3 2•2+ 2SbCl6 -displayed similar UV-vis-NIR signature than 3 •+ SbCl6 -, with twice higher absorption intensity, as expected from the presence of two oxidized carbazole units (Figure 6). The UV-vis-NIR spectrum of 3 2•2+ 2SbCl6 - shows a 20 nm blue shift of the signal at 800 nm in comparison to 3 •+ SbCl6 -, along with an increase and red-shift of the broad IR band ( = 2250 M -1 cm -1 at 1400 nm), suggesting some interactions between the radical centers. The calculated spectrum of 3 2•2+ reproduced well the experimental one and notably allowed us to assign the low-energy bands between 700 and 1200 nm to transitions among the set HOMO-1, HOMO and LUMO, LUMO+1 (Figure S66 and Table S14). As for 1 •+ BF4 -, enantiomers of 3 •+ SbCl6 -and 3 2•2+ 2SbCl6 -remain also configurationally stable, and corresponding mirrorimage ECD spectra show similar profile up to 1100 nm. Both (+)-3 •+ SbCl6 -and (+)-3 2•2+ 2SbCl6 -displayed Cotton effects assigned to radical transitions at 660, 820 and 900-950 nm (Δ = + 0.7, -2.2 and + 2.8 M -1 cm -1 , respectively, Figure S21), which are red-shifted in comparison to (+)-1 •+ BF4 -, reflecting the higher electronic coupling between the carbazole -systems. Importantly, diradical 3 2•2+ 2SbCl6 -showed high chiroptical properties in the NIR region, as indicated by its absorption anisotropy factors, gabs= 1.0  10 -3 at 920 nm (Figure 6), which is one of the highest measured for molecular organic diradicals. [START_REF] Inoue | Hash-Mark-Shaped Azaacene Tetramers with Axial Chirality[END_REF][START_REF] Shu | High-Spin Diradical Dication of Chiral pi-Conjugated Double Helical Molecule[END_REF] The magnetic properties of 3 2•2+ 2SbCl6 -were investigated by EPR, which resulted in an intense and broad doublet without any hyperfine coupling, nor m = 2 half-field transition, presumably due to a weak interaction between the two radicals (g = 2.08, Figure 6). [START_REF] Eaton | Use of the ESR half-field transition to determine the interspin distance and the orientation of the interspin vector in systems with two unpaired electrons[END_REF] Given the high stability of 3 2•2+ 2SbCl6 - (more than 66 hours in solid state), magnetometry measurement were also performed powder of 3 2•2+ 2SbCl6 -at a magnetic field strength of 2 kOe below 20 K and 10 kOe above to get further insight about the interaction between radicals within the chiral environment. χMT is equal 0.73 cm 3 K mol -1 at 298 K for 6), which almost perfectly matches with the contribution for two uncoupled single electrons with g = 2.00. This value started to decrease for temperatures below 170 K, indicating an antiferromagnetic coupling between spins. χM - 1 vs. T plot can be fitted with Curie-Weiss law (χM = C/(T-) with C = 0.79 cm 3 K mol -1 and  = -24 K (Figure 6). The Singlet-Triplet gap (EST) estimated from  at 67 cm -1 (0.19 kcal mol -1 ) reveals a weak antiferromagnetic intramolecular spin-exchange interaction. Interestingly, these findings confirm the expected formation of a diradical upon oxidation of the highest doubly occupied molecular orbital of a monoradical with a SHI. For 3 2•2+ 2SbCl6 -, the two unpaired electrons are localized on each carbazole fragments, with a weak overlap between the corresponding two SOMOs. [START_REF] Kumar | SOMO-HOMO Level Inversion in Biologically Important Radicals[END_REF][START_REF] Gryn'ova | Theory and practice of uncommon molecular electronic configurations[END_REF] Likewise, the calculations showed that 3 2•2+ has two unpaired electrons and a spin-triplet electronic state instead of a singlet configuration, including two 𝛼-spin SOMOs and, below in energy, the set of spin-paired orbitals (Figure S64). The isosurfaces of the two SOMOs (as well as for LUMO and LUMO+1) are seen to represent in-phase and outof-phase linear combinations of frontier fragment orbitals with an energetic splitting of 0.127 eV, whose combined densities are reflected by the spin density calculations (Figure S65 and ESI for details). However, spin state energetics are not easy to deal with in DFT calculations. We found that an open-shell singlet configuration became effectively iso-energetic with the triplet when the fractions of the exact exchange in the functional went to zero, and weak interactions between the electron spins in adjacent units of 3 2•2+ in the solid may further complicate the picture. We thus hypothesize that, given the weak electronic coupling between the carbazole fragments in 3 2•2+ 2SbCl6 -, the two SOMOs could be considered as disjoint, which ultimately favored singlet ground state, in analogy with previous 3,3′-Dimethylenebiphenyl diradicals molecular design and using Ovchinnikov parity models. [START_REF] Gallagher | High-Spin Organic Molecules[END_REF][START_REF] Ovchinnikov | Multiplicity of the ground state of large alternant organic molecules with conjugated bonds[END_REF][START_REF] Rajca | Intramolecular Antiferromagnetic vs Ferromagnetic Spin Coupling through the Biphenyl Unit[END_REF]

3 2•2+ 2SbCl6 -(Figure

CONCLUSION

In conclusion, we described here the first persistent sterically unprotected axially chiral bicarbazole radical and its diradical analogue, along with their experimental and theoretical optical, chiroptical and magnetic properties. We investigated the relationship-structure properties of organic chiral monoradicals based on bicarbazole molecular architectures with either axial and helical chirality and found that only the former is chemically and configurationally stable, with a half-life up to 3 hours under air conditions. We attributed this unprecedented result to a complex interplay of SHI and electronic coupling between the radical center and the remaining neutral carbazole fragment, which is directly linked to the carbazoles dihedral angle and related to axial and helical chirality. Such unique electronic features provided an opportunity to design and isolate a persistent, intrinsically chiral, diradical, which displayed near infrared electronic circular dichroism intensity up to 1100 nm and almost degenerate singlet-triplet ground states with weak antiferromagnetic interactions evaluated by SQUID experiments (EST ~ 0.05 kcal mol -1 ). The reported findings may bring new insights on the key parameters governing the chemical and configurational stabilities of chiral organic radicals, which may offer new opportunity to design innovative organic high-spin materials.

Figure 2 .

 2 Figure 2. a) UV-vis absorption spectra of 1 (black), 2 (red) and 1' (green); b) Normalized fluorescence spectra of 1 (black), 2 (red) and 1' (green); c) Calculated (Calc.) absorption spectra for 1 (black) and 2 (red) with selected transitions and oscillator strengths indicated by 'sticks' bars and; d) Details for selected transitions and occupied (occ)-unoccupied (unocc) MO pair contributions (greater than 10%) to the transition density for (+)-1 and (+)-2. H and L indicate the HOMO and LUMO, respectively, for which isosurfaces (0.035 au) are shown in e).

Figure 3 .

 3 Figure 3. a) Cyclic voltammograms of 1' (green), 1 (black), 2 (red) and 3' (blue) versus saturated calomel electrode (SCE).; b) Synthetic route to persistent radical 1 •+ BF4 -(red) and 3' •+ BF4 -(blue); c) Top: UV-vis-NIR absorption spectra of 1 (black), 1 •+ BF4 -(red) and 3' •+ BF4 -(blue) in dichloromethane at 298 K, inset: X-band EPR (ν = 9.4858 GHz) of 1•+ BF4 -; bottom :ECD spectra of (+)-1 (black) and (+)-1 •+ BF4 -(red) with their corresponding enantiomers in dash line in dichloromethane at 298 K with an extension of the NIR-ECD spectra of (+)-and (-)-1 •+ BF4 -. Addition of AgBF4 as a chemical oxidant to a solution of 3' afforded persistent radical 3' •+ BF4 -, as expected for 3,6-protected carbazole, with a characteristic vibronic visible signature at 696 nm and 771 nm (Figure3).[START_REF] Chiu | Electrochemical and Spectral Characterizations of 9-Phenylcarbazoles[END_REF][START_REF] Karon | Carbazole electrochemistry: a short review[END_REF] 62 Using the same conditions, 1 was quantitatively converted to 1 •+ BF4 -, which showed similar optical signature than 3' •+ BF4 -, with an additional weak broad IR band ( ~ 500 M -1 cm -1 at 1000 nm), and a half-life evaluated to 3 hours in air saturated CH2Cl2. Under inert conditions, this half-life extends to 14 hours, indicating a persistent character for 1 •+ BF4 -. Indeed, mass spectroscopy and UV-vis analysis of

Figure 4 .

 4 Figure 4. Calculated spin density with an isosurface of ±0.002 au, orbital energies (in eV) and isosurfaces (±0.030) of frontier molecular orbitals computed for monoradicals 1 •+ (left) and2•+ (right). The HOMO-SOMO energy differences are averaged over the highest occupied spin orbital energies. Oxidation potentials and separation of the two first oxidation potentials in V vs. SCE, theoretical (calc.) and experimental (exp.) dihedral angles between the carbazole fragments. Another notable difference between 1 •+ and 2 •+ , is the orbital energy gap  between the HOMO and the SOMO of 0.8 eV and 0.4 eV, respectively. Moreover, the energy difference between the HOMO of 2 and 2 •+ is 0.54 eV, much larger than the difference of 0.18 eV for 1 and 1 •+ , confirming the stronger interaction between the radical center and the second carbazole in 2 •+ (Figures4, S57 and S58). The obtained results bring further evidence of the crucial role of the electronic coupling between the radical center and the neutral electron-rich fragment on stabilizing a molecular open-shell electronic state with SOMO-HOMO energy inversion.[START_REF] Guo | High stability and luminescence efficiency in donor-acceptor neutral radicals not following the Aufbau principle[END_REF] In addition to its chemical stability, 1 •+ BF4 -remains also configurationally stable, presumably because of the high enantiomerization barrier of its corresponding neutral form (G  =

Figure 5 .

 5 Figure 5. a) Cyclic voltammograms of 3 (black) and 4 (red) versus SCE; b) UV-vis-NIR absorption spectra of 1 •+ (grey),

Figure 6 .

 6 Figure 6. Synthetic pathway to form 3 •+ SbCl6 -and 3 2•2+ 2SbCl6 -; a) UV-vis-NIR absorption spectra of 3 (black), 3 •+ SbCl6 -(red) and 3 2•2+ 2SbCl6 -(blue) in acetonitrile at 298 K; b) Absorption dissymmetry factor gabs spectra of (+)-3 2•2+ 2SbCl6 - (solid line) and (-)-3 2•2+ 2SbCl6 -(dash line) in acetonitrile at 298 K; c) EPR of 3 2•2+ 2SbCl6 -(n = 9.4858 GHz) at 77 K (solid line); d) χMT vs T of solid diradical 3 2•2+ 2SbCl6 -; e) Calculated spin density for (+)-3 2•2+ with an isosurface of ±0.002 au.As for 1 •+ BF4 -, enantiomers of 3 •+ SbCl6 -and 3 2•2+ 2SbCl6 -remain also configurationally stable, and corresponding mirrorimage ECD spectra show similar profile up to 1100 nm. Both (+)-3 •+ SbCl6 -and (+)-3 2•2+ 2SbCl6 -displayed Cotton effects assigned to radical transitions at 660, 820 and 900-950 nm (Δ = + 0.7, -2.2 and + 2.8 M -1 cm -1 , respectively, FigureS21), which are red-shifted in comparison to (+)-1 •+ BF4 -, reflecting the higher electronic coupling between the carbazole -systems. Importantly, diradical 3 2•2+ 2SbCl6 -showed high chiroptical properties in the NIR region, as indicated by its absorption anisotropy factors, gabs= 1.0  10 -3 at 920 nm (Figure6), which is one of the highest measured for molecular organic diradicals.[START_REF] Inoue | Hash-Mark-Shaped Azaacene Tetramers with Axial Chirality[END_REF][START_REF] Shu | High-Spin Diradical Dication of Chiral pi-Conjugated Double Helical Molecule[END_REF] The magnetic properties of 3 2•2+ 2SbCl6 -were investigated by EPR, which resulted in an intense and broad doublet without any hyperfine coupling, nor m = 2 half-field transition, presumably due to a weak interaction between the two radicals (g = 2.08, Figure6).[START_REF] Eaton | Use of the ESR half-field transition to determine the interspin distance and the orientation of the interspin vector in systems with two unpaired electrons[END_REF] Given the high stability of 3 2•2+ 2SbCl6 - (more than 66 hours in solid state), magnetometry measurement were also performed powder of 3 2•2+ 2SbCl6 -at a magnetic field strength of 2 kOe below 20 K and 10 kOe above to get further insight about the interaction between radicals within the

ACKNOWLEDGMENT

We acknowledge the Ministère de l'Enseignement supérieur, de la Recherche et de l'Innovation, the Centre National de la Recherche Scientifique (CNRS). L. A. and J. A. thank the National Science Foundation (CHE-1855470) for financial support and the Center for Computational Research (CCR) at the University at Buffalo for computational resources. 75 The PRISM core facility (Bioge-nouest©, UMS Biosit, Université de Rennes 1 -Campus de Villejean -35043 RENNES Cedex, FRANCE) is acknowledged for the NMR characterizations and ECD measurements.

ASSOCIATED CONTENT

Supporting Information. Methods and synthetic procedures, ORTEP diagrams, table of crystal data, unit cell diagrams, HPLC separation data, additional photophysiscal and chiroptical characterizations and theoretical calculations can be found in the Electronic Supporting Information.

CORRESPONDING AUTHORS

* jochena@buffalo.edu * ludovic.favereau@univ-rennes1.fr

NOTES

The authors declare no competing financial interest.


Pierre Bauvin

Claire Delacôte

Guillaume Lassailly

Line Carolle

Ntandja Wandji

Viviane Gnemmi

Flavien Dautrecque

Alexandre Louvet

Robert Caiazzo

Violeta Raverdy

Emmanuelle Leteurtre

François Pattou

Sylvie Deuffic-Burban

Philippe Mathurin
email: philippe.mathurin@chru-lille.fr

A tool to predict progression of non-alcoholic fatty liver disease in severely obese patients

Keywords: NAFLD: Non-alcoholic Fatty Liver Disease NASH: Non-alcoholic Steatohepatitis BMI: Body Mass Index NL: Normal Liver NASH, fibrosis, Markov model, disease progression, risk factors

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

LAY SUMMARY

A predictive tool would be of great interest to manage the burden of non-alcoholic fatty liver disease (NAFLD) and in particular to identify and refer patients at the highest risk as well as estimate disease progression and adapt therapeutic strategies. We have developed this predictive tool for severely obese patients based on the history of candidates for bariatric surgery, including data on the history of overweight, obesity and diabetes.The model describes the NAFLD progression from the overweight onset and is able to predict the current disease stages, with or without information from histological or non-invasive tests, as well as progression at 5 years.

INTRODUCTION

Non-alcoholic fatty liver disease (NAFLD), which is associated with an increased risk of liver-related morbidity and mortality, [START_REF] Chalasani | The diagnosis and management of nonalcoholic fatty liver disease: practice guideline by the American Gastroenterological Association, American Association for the Study of Liver Diseases, and American College of Gastroenterology[END_REF][START_REF] Paradis | Hepatocellular carcinomas in patients with metabolic syndrome often develop without significant liver fibrosis: a pathological analysis[END_REF][START_REF] Povsic | A Structured Literature Review of the Epidemiology and Disease Burden of Non-Alcoholic Steatohepatitis (NASH)[END_REF] is a growing public health issue and has become one of the main etiologies in adults waiting for liver transplantation. [START_REF] Wong | Nonalcoholic steatohepatitis is the most rapidly growing indication for liver transplantation in patients with hepatocellular carcinoma in the U[END_REF][START_REF] Marcellin | Liver diseases: A major, neglected global public health problem requiring urgent actions and large-scale screening[END_REF] The disease burden increased along with the prevalence of obesity and metabolic syndrome, its two main causes. [START_REF] Lazo | Prevalence of nonalcoholic fatty liver disease in the United States: the Third National Health and Nutrition Examination Survey, 1988-1994[END_REF][START_REF] Kotronen | Liver fat in the metabolic syndrome[END_REF] The histopathological spectrum of NAFLD includes isolated steatosis, non-alcoholic steatohepatitis (NASH) and the different stages of fibrosis. [START_REF] Chalasani | The diagnosis and management of nonalcoholic fatty liver disease: practice guideline by the American Gastroenterological Association, American Association for the Study of Liver Diseases, and American College of Gastroenterology[END_REF][START_REF] Neuschwander-Tetri | Nonalcoholic steatohepatitis: summary of an AASLD Single Topic Conference[END_REF][START_REF] Starley | Nonalcoholic fatty liver disease and hepatocellular carcinoma: a weighty connection[END_REF] Natural history studies have shown that NAFLD patients with NASH disclose significantly higher risk of fibrosis progression than those with isolated steatosis. [START_REF] Ekstedt | Long-term follow-up of patients with NAFLD and elevated liver enzymes[END_REF][START_REF] Younossi | Pathologic criteria for nonalcoholic steatohepatitis: interprotocol agreement and ability to predict liver-related mortality[END_REF][START_REF] Angulo | Liver Fibrosis, but No Other Histologic Features, Is Associated With Long-term Outcomes of Patients With Nonalcoholic Fatty Liver Disease[END_REF][START_REF] Loomba | The Hierarchical Model of NAFLD: Prognostic Significance of Histologic Features in NASH[END_REF] Although fibrosis progression is slower in patients without NASH, development of cirrhosis occurs in patients with simple steatosis. [START_REF] Angulo | Liver Fibrosis, but No Other Histologic Features, Is Associated With Long-term Outcomes of Patients With Nonalcoholic Fatty Liver Disease[END_REF][START_REF] Bedossa | Systematic review of bariatric surgery liver biopsies clarifies the natural history of liver disease in patients with severe obesity[END_REF] Indeed, in NAFLD F3-F4 patients, simple steatosis is observed in around 20% and NASH in 80% of cases. [START_REF] Angulo | Liver Fibrosis, but No Other Histologic Features, Is Associated With Long-term Outcomes of Patients With Nonalcoholic Fatty Liver Disease[END_REF][START_REF] Bedossa | Systematic review of bariatric surgery liver biopsies clarifies the natural history of liver disease in patients with severe obesity[END_REF] Many cofactors are also associated with disease progression: genetic variants, metabolic syndrome, diabetes, sex and age. 15- 18 However, despite improved understanding of the natural history of NAFLD, data are still lacking on the role of the duration of exposure, an important component in chronic disease. In particular, how does the duration of exposure to overweight, obesity and/or diabetes impacts the severity of NAFLD and its evolution.

Severely obese candidates for bariatric surgery are a population of interest to analyse the impact of obesity and diabetes history on NAFLD. [START_REF] Bedossa | Systematic review of bariatric surgery liver biopsies clarifies the natural history of liver disease in patients with severe obesity[END_REF] Indeed, these patients present a wide heterogeneity in NAFLD severity, history of obesity, diabetes, and metabolic syndrome. The policy of our centre consists in performing systematic liver biopsy at surgery, and the use of questionnaires on the past history of obesity and diabetes, two crucial factors in the setting of NAFLD. Such data collection is an opportunity to provide more insights into NAFLD progression by exploring the relationship between histological features, clinical characteristics and duration of exposure to obesity and/or diabetes. An indicator of liver risk that integrates all of this relevant information is an attractive approach to identify patients at high risk of disease progression.

Bariatric surgery induces sustained weight-loss, improves metabolic comorbidities, frequently resolves NASH (80% of cases) and improves fibrosis (60% of cases). [START_REF] Lassailly | Bariatric Surgery Reduces Features of Nonalcoholic Steatohepatitis in Morbidly Obese Patients[END_REF] When considering the number of severely obese with NAFLD, there is a need to prioritize those who would benefit most from surgery. Such approach requires to optimize patients' path from general practitioners to specialized centres, and improve the decision-making process for patients already addressed in the centre. For the general practitioners, the availability of a user-friendly tool identifying those with high risk of advanced fibrosis or rapid disease progression is attractive. In specialized centres, baseline assessment of fibrosis is more and more performed by non-invasive methods such as dosage of biomarkers or measurement of liver stiffness. [START_REF] Castera | Invasive and non-invasive methods for the assessment of fibrosis and disease progression in chronic liver disease[END_REF] A tool integrating this partial information to predict the disease progression would have added value. Mathematical modelling is a method of choice for the development of such a tool. [START_REF] Beck | The Markov process in medical prognosis[END_REF][START_REF] Deuffic-Burban | Quantification of fibrosis progression in patients with chronic hepatitis C using a Markov model[END_REF] The aims of our study were to quantify the progression to steatosis, fibrosis and NASH, identify and evaluate the impact of independent disease progression cofactors and provide a tool predicting disease progression based on a modelling approach.

MATERIALS AND METHODS

We developed a Markov model to describe NAFLD progression among severely obese, using preoperative histological and clinical data from 1,801 candidates for bariatric surgery from the "Lille Bariatric Cohort". [START_REF] Marshall | MARKOV: a computer program for multi-state Markov models with covariables[END_REF] The model considered disease progression from the initial stage of the normal liver through cirrhosis. It assessed the transition probabilities from one stage to another as well as the effects of covariates on these transitions, estimated from the observed data. The resulting model predicts the current disease stage of an individual, as well as its progression in the future, taking into account the duration of being overweight, obese and individual characteristics. A web-based tool is available for both the evaluation of disease severity, and the prediction of disease progression at https://lille-model.shinyapps.io/deployedapp/.

Markov model

The progression of NAFLD to compensated cirrhosis was modelled according to the severity of fibrosis and the development of NASH through the 10 following stages of liver disease (Figure 1): normal liver (NL), simple steatosis with F0 to F3 fibrosis, NASH with F0 to F3 fibrosis, and cirrhosis. [START_REF] Angulo | Liver Fibrosis, but No Other Histologic Features, Is Associated With Long-term Outcomes of Patients With Nonalcoholic Fatty Liver Disease[END_REF][START_REF] Bedossa | Systematic review of bariatric surgery liver biopsies clarifies the natural history of liver disease in patients with severe obesity[END_REF] The progression of steatosis, fibrosis or NASH are estimated as the result of the combination of progression and regression, as in other modelling studies of NAFLD. [START_REF] Estes | Modeling the epidemic of nonalcoholic fatty liver disease demonstrates an exponential increase in burden of disease[END_REF] For more details, please refer to the Supporting Information.

We assumed that all patients had a NL when they initially became overweight, and then were at-risk of NAFLD. We assessed this risk from one available biopsy from each patient, which represented the stage of progression over the exposure time.

Study population and histological analysis

The "Lille Bariatric Cohort" consists of 2,735 severely or morbidly obese patients referred for bariatric surgery between 1994 and 2018 and prospectively included, as described in Lassailly et al (details in Supporting Information, section "Inclusion criteria"). [START_REF] Lassailly | Bariatric Surgery Reduces Features of Nonalcoholic Steatohepatitis in Morbidly Obese Patients[END_REF] Moreover, 12 (0.4%) patients were candidates for bariatric surgery and had a liver biopsy, but didn't undergo surgery because of serious conditions. They were added to the cohort to be closer to a real-life distribution of disease severity. All patients were invited to follow a nutrition program before surgery.

A standardized questionnaire was prospectively presented to each patient in the cohort at inclusion to specifically assess the history of overweight and obesity, recording the following information: period when weight gain occurred, age of onset of obesity and body mass index (BMI) at a 20 years old.

Patients were asked about their overweight history by reporting the period when weight gain occurred as: "before the age of 7" referring to kindergarten/preschool ("école maternelle") which corresponds to children up to 6 years old; "between age 7 and puberty" referring to primary school ("école primaire") which corresponds to children aged 7 to 11; "during puberty" which refers to secondary school ("collège") and corresponds to adolescents aged 12 to 15, and "post puberty", i.e. after secondary school "collège". We used these data to infer the age of overweight onset; for more details, please refer to the Supporting Information, section "Exposure time". The presence of diabetes was defined by ADA criteria. [START_REF]Expert Committee on the Diagnosis Classification of Diabetes Mellitus. Report of the expert committee on the diagnosis and classification of diabetes mellitus[END_REF] Diabetic patients were asked the age of diagnosis of diabetes. Liver biopsies were systematically planned during the surgical procedure. Diagnosis of NASH was performed by the presence of hepatocellular steatosis, ballooning/clarification and inflammation, the well-recognized histological features of NASH. [START_REF] Brunt | Nonalcoholic steatohepatitis: a proposal for grading and staging the histological lesions[END_REF] Severity of the necroinflammatory activity was assessed using the NAS score. [START_REF] Kleiner | Design and validation of a histological scoring system for nonalcoholic fatty liver disease[END_REF] Fibrosis was characterized by both the Metavir and Kleiner scores. [START_REF] Kleiner | Design and validation of a histological scoring system for nonalcoholic fatty liver disease[END_REF][START_REF] Bedossa | An algorithm for the grading of activity in chronic hepatitis C. The METAVIR Cooperative Study Group[END_REF] In the baseline analysis, we used the Metavir score for developing the model. An alternative analysis using the Kleiner score is presented in the Supporting Information section "Sensitivity analysis".

Four hundred and seventy-three patients were excluded due to past history of alcohol, age under 18, histological data lacking or because they were referred from other medical centre. Four hundred seventy-three of the remaining 2,274 patients (20.8%) were excluded because essential clinical data were missing for the glycaemia assessment or the onset of weight gain (flow-chart in Figure 2). Finally, 411 of the 1,801 remaining patients (22.8%) couldn't recall their weight at 20 (309/1,801) and/or the age at the diagnosis of diabetes (148/600 patients with diabetes). To include these patients in the model, the missing BMI at 20 years old and/or age at the onset of diabetes were imputed based on the following variables: sex, period of onset of overweight, age, BMI and the presence of diabetes at the biopsy. We used the widely used, [START_REF] Van Buuren | mice: Multivariate Imputation by Chained Equations in R[END_REF] predictive mean matching method. [START_REF] Little | Missing-Data Adjustments in Large Surveys[END_REF][START_REF] Rubin | Statistical Matching Using File Concatenation With Adjusted Weights and Multiple Imputations[END_REF] Informed written consent was obtained for all patients and the study was performed in accordance with the Declaration of Helsinki. The ethics committee approved the cohort and the study was supported by grants from the government and the French Ministry of Health (PHRC). After legal revision, new approval was obtained in 2006 (no. CP06/49, NCT01129297)

Covariates and exposure time

The effects of independent covariates were evaluated to take into account the heterogeneity of disease progression: sex, period of onset of overweight, BMI category at 20 years old and the presence of diabetes. We used data on period of onset of overweight to infer the age of overweight onset. Exposure time is the duration of exposure to overweight/obese (BMI ≥ 25), when the patient is at-risk of developing NAFLD. It was calculated as the difference between the age at biopsy and the age of overweight onset (details in Supporting Information, section "Exposure time").

Procedure

The parameters of progression are estimated by the maximum likelihood estimates method, [START_REF] Jackson | Multistate Markov models for disease progression with classification error[END_REF] using the biopsy data from the "Lille Bariatric Cohort". The underlying methodology has been previously validated, [START_REF] Deuffic-Burban | Quantification of fibrosis progression in patients with chronic hepatitis C using a Markov model[END_REF][START_REF] Delacote | A Model to Identify Heavy Drinkers at High Risk for Liver Disease Progression[END_REF][START_REF] Longini | Statistical analysis of the stages of HIV infection using a Markov model[END_REF][START_REF] Marshall | Multi-state models and diabetic retinopathy[END_REF] and is described in the Supporting Information, section "Estimation of the parameters".

To assess the internal validity of the model, the total cohort was randomly divided into a training cohort (n=1,207; 67%) and a testing cohort (n=594; 33%). The training cohort was used to estimate the parameters. The testing cohort was used to evaluate the internal validity of the model, by calculating the global error score, i.e. the weighted mean of the relative differences between the number of individuals at each stage observed at the date of biopsy and the number predicted by the model throughout the period of exposure (see details in Supporting Information, section "Model evaluation"). For further validation, we used k-fold cross-validation (k=10) described in the Supporting Information.

Once the parameters are estimated, the model can predict the current state of the liver state for any hypothetical patient, by simulating the disease progression from the age s/he became overweight.

Moreover, it can also able predict the 5-year risk of disease progression, based on the current assessment of fibrosis. We chose to present the 5-year risk of the development of cirrhosis in two contexts: 1) with partial information provided by non-invasive methods, i.e. only the stage of fibrosis without information on NASH status (in this case, NASH status had to be estimated by the model);

and 2) with complete information provided by biopsy i.e. NASH status and stage of fibrosis.

The technical aspects of model building, testing and use are further detailed in the Supporting Information section "Methodological appendix". To gauge model uncertainty all of these results were provided with a plausible interval obtained from a probabilistic sensitivity analysis. [START_REF] Zhang | Probabilistic sensitivity analysis on Markov models with uncertain transition probabilities: an application in evaluating treatment decisions for type 2 diabetes[END_REF] All analyses were performed by PB with the R-software (https://www.R-project.org/) version 3.3.3 and the msm package, under the supervision of SDB and PM. [START_REF] Jackson | Multi-State Models for Panel Data: The msm Package for R[END_REF]

RESULTS

Patient characteristics

The characteristics of the population are summarized in Table 1, separately for the study cohort (n=1,801) and the two randomly selected subsets (training cohort, n=1,207 and testing cohort, n=594). In the study cohort, at biopsy 74.5% were women, severely or morbidly obese (mean BMI ± SD of 47.6 ± 8 kg/m²), with mean age of 41.2 ± 11 years old. Histological NASH was diagnosed in 183/1,801 (10.2%) cases, and cirrhosis in 27 (1.5%) patients.

Adequacy with data

The error scores for each histological stage (presented in Supporting Table 1) show that the model adequately reproduces disease progression in all the stages of NAFLD. Good accuracy is confirmed by a global error score of 4.5% in the training cohort and 8.8% in the testing cohort. If we consider error rates >15%, the model overestimates the number of patients with F2-steatosis (11 predicted versus 7 observed), with F3-steatosis (5 predicted versus 4 observed) and with F3-NASH (9 predicted versus 6 observed), while it underestimates the number of patients with F2-NASH (8 predicted, versus 11 observed). It accurately predicts the number of patients with cirrhosis (9 predicted vs 9 observed).

Transition rates

Table 2 summarizes the ten baseline transition rates and adjusted hazard ratios (HR) of the covariates estimated from the training cohort. Baseline transition rates correspond to a patient profile with the following characteristics: women who became overweight after puberty, with a BMI of 25 kg/m² at 20 years old and without diabetes. For example, with the above characteristics, the baseline transition rate and 95% confidence interval (CI) from NL to F0-steatosis is 7.72% (6.85% -8.56%), meaning that around 8 out of 100 of these patients with NL develop steatosis per year. The Markov model provides transition rates specific to each patient's profile from baseline rates and adjusted HR.

For example, patients with the above characteristics, starting overweight at 5 years old have reduced yearly transition rates toward steatosis (5.25%, 95% CI = 4.53% -6.07%) than patients with onset of overweight at 15, i.e. during puberty (6.39%, 95% CI = 5.55% -7.36%). However, as expected, the former have a greater probability of steatosis at 20 (48.32%) than the latter (26.13%), due to the fact that their duration of exposure is longer (15 years versus 5 years).

The patients with the fastest disease progression are diabetic men, who became overweight during the post-puberty period, and who are class II or III obese (BMI > 30) at 20. These patients have an annual progression rate from steatosis to NASH of 2.17% (95% CI: 1.68% -2.77%), compared to 0.61% (95% CI: 0.52% -0.73%) in standard patients (women, non-diabetic, who became overweight during the post-puberty period and who have a BMI at age 20 indicating that they are overweight).

Those results are relatively unchanged among the different sensitivity analysis (details in Supporting Information, section"Sensitivity analysis", and Supporting Tables 234567).

Evaluation of disease severity

By simulating disease progression from its starting point, this model evaluates the probability of being in a disease stage based on individual characteristics. Figure 3 shows the model-predicted risks according to three patient profiles. First, 40-year-old non-diabetic women who became overweight at 20 (i.e., during the post-puberty period, with a BMI at age 20 indicating that they are overweight and with 20 years of exposure to the disease) have a probability of 21.4% (plausible interval: 17.8% -24.9%) to still have a NL, 72.8% (68.8% -76.8%) to have steatosis, 5.7% (4.5% -7.1%) to have NASH and 0.1% (0.1% -0.2%) to have cirrhosis (Figure 3A). Furthermore, there is a 0.6% (0.4% -1.1%) probability of F3-F4 fibrosis (0.2% steatosis-F3-F4, 0.4% NASH-F3-F4).

Second, patients with the above characteristics who have been diabetic since the age of 35, have a 13.3% (11.2% -17.5%) probability of having a NL, 78.0% (74.0% -80.3%) of having steatosis, 8.4% (6.9% -9.7%) NASH and 0.3% (0.2% -0.5%) cirrhosis (Figure 3B). There is a 1.7% (1.2% -2.1%) probability of having F3-F4 fibrosis (0.6% steatosis-F3-F4, 1.1% NASH-F3-F4) in these patients.

Finally, 40-year-old men who have been overweight at 10, obese class II at 20 and diabetic at 35 have a probability of 2.7% (1.8% -4.6%) of having a NL, 77.6% (72.8% -81.0%) of having steatosis, 16.1% (13.3% -19.8%) NASH and 3.6% (2.4% -5.5%) cirrhosis (Figure 3C). These patients have a 10.0% (7.1% -13.4%) probability of having F3-F4 fibrosis (3.3% steatosis-F3-F4, 6.7% NASH-F3-F4).

Prediction of 5-year disease progression

The present model predicts the cumulative risk of cirrhosis at 5 years based on non-invasive assessment of baseline fibrosis, or biopsy. Figure 4 shows this risk according to different severities of fibrosis evaluated by non-invasive methods in 40-year-old non-diabetic women who became overweight at 20 years old. The five-year risk of cirrhosis in these patients with baseline NL or <F2 fibrosis is 0.02% (plausible interval: 0.02% -0.04%), ranging from 0.006% (0.003% -0.02%) in the absence of NASH to 0.3% (0.2% -0.6%) in those with NASH at baseline (Figure 4A). The five-year risk in the same patients with baseline ≥F2 fibrosis is 12.0% (10.2% -16.6%), ranging from 7.2% (4.8% -13.5%) in the absence of NASH to 17.8% (15.1% -21.2%) in the presence of NASH (Figure 4B).

When the disease stage is assessed by liver biopsy, providing a precise stage of fibrosis and the presence or absence of NASH, the model also predicts the probability of reaching each disease stage according to this baseline histological information. For example, previous patients with F2-steatosis have a 5-year risk of cirrhosis of 1.8% (0.7% -5.2%), compared to 6.0% (3.9% -9.1%) in F2-NASH patients and 21.5% (17.3% -26.5%) in F3-NASH patients (Figure 5). In more severe profiles (men who become overweight at 10, obese class II at 20 and diabetic at 35), F2 patients without NASH have a 5-year risk of cirrhosis of 10.3% (4.3% -25.5%), compared to 26.7% (20.0% -34.2%) in the presence of NASH, and 47.0% (40.4% -53.8%) in F3-NASH patients (not shown).

DISCUSSION

Our model provides a robust tool for predicting the course of disease in severely obese patients. The novelty of our work is that all the model parameters are estimated from preoperative liver biopsies and individual characteristics from the Lille Bariatric Cohort. Moreover, the model takes into account the global overweight history, including the age of overweight onset and BMI at 20, to assess the relationship between the progression of NAFLD and the duration of exposure to being overweight, obese and with diabetes. We modelled the patient's average disease history, by considering that, as stated elsewhere, candidates for bariatric surgery could not achieve long-term weight loss, [START_REF] Sjostrom | Effects of bariatric surgery on mortality in Swedish obese subjects[END_REF] while taking into account that they could achieve temporary regression of steatosis, fibrosis or NASH.

Another key aspect is that the resulting predictions can be obtained in the absence of information about the current stage of fibrosis or after integrating disease stage determined by either liver biopsy or non-invasive methods. Our work shows that patients with similar baseline characteristics may have marked differences in their risk of progression due to diverse past histories of obesity and diabetes.

As previously reported, [START_REF] Singh | Fibrosis progression in nonalcoholic fatty liver vs nonalcoholic steatohepatitis: a systematic review and meta-analysis of paired-biopsy studies[END_REF][START_REF] Matteoni | Nonalcoholic fatty liver disease: a spectrum of clinical and pathological severity[END_REF] the model shows that NASH accelerates disease progression with variations according to stage of fibrosis. The model also weights the relative impact of different independent cofactors such as diabetes, male sex, BMI at 20 years old and the age category of overweight onset. As examples, our study confirm the impact of diabetes and male sex, as previously reported, [START_REF] Pang | Diabetes, Plasma Glucose, and Incidence of Fatty Liver, Cirrhosis, and Liver Cancer: A Prospective Study of 0.5 Million People[END_REF][START_REF] Lonardo | Sex Differences in Nonalcoholic Fatty Liver Disease: State of the Art and Identification of Research Gaps[END_REF] leading to an increase of the progression rate by 202% in a diabetic woman compared to a non-diabetic woman, and 291% between a diabetic man and a non-diabetic woman, other factors being equal. An interesting insight is that age of overweight onset appears as an important predictive factor: in two patients with the same histological diagnosis at the same age, the patient who became overweight later is identified with faster disease progression, all other factors being equal.

Our model is available as a web-based tool (https://lille-model.shinyapps.io/deployedapp/). It can be used to improve the care pathway of severely obese patients, and to support the decision for bariatric surgery according to the risk of the progression of liver disease. However, it is important to note that the model does not consider other important metabolic comorbidities. It is now possible to identify which severely obese patients should be referred to specialists during the clinical evaluation based on the prediction of the stage of liver disease. Specialists working in the bariatric surgery unit can now select patients with rapid disease progression whatever their baseline fibrosis, in whom delaying bariatric surgery would be deleterious to the liver. Specialists working in the bariatric surgery unit can now select patients with rapid disease progression whatever their baseline fibrosis, in whom delaying bariatric surgery would be deleterious to the liver. For example, 50-year-old non-diabetic women, who became overweight at 20, diagnosed with a F3-NASH have a lower 5-year risk of cirrhosis (21.5%) than 50-year-old diabetic men, who become overweight at 15 years old, obese class I at 20 and diabetic at 35, and diagnosed with F2-NASH (25.8%). However, this approach opens up the unsolved issue of what is the appropriate cut-off risk of disease progression to undergo bariatric surgery. Such cut-off should be proposed by experts and would need to be estimated in a second step in prospective studies.

The present study has certain limitations. First, the model may be subject to selection bias, because the training is based on data from severely obese patients. Thus, it should only be used in these patients. Moreover, external validation was not available. Although the model was internally validated using a split into the training-testing cohorts, as well as a cross-validation procedure described in the Supporting Information, future validation is required to strengthen these conclusions. This shows the importance of recording the patients' history of obesity, which is often lacking in bariatric surgery cohorts. Second, our approach may have resulted in an undersampling of cirrhosis, as this event is often a contraindication for bariatric surgery. However, we decided to include patients with cirrhosis who were candidates but who were not selected for surgery, to limit this bias. As a result, the prevalence of compensated cirrhosis was 1.8% in NAFLD patients, which is coherent with the prevalence observed in other, non-bariatric cohorts of NAFLD patients with different demographics. [START_REF] Hagstrom | Fibrosis stage but not NASH predicts mortality and time to development of severe liver disease in biopsy-proven NAFLD[END_REF] Patients with decompensated cirrhosis were not referred for bariatric surgery in the first place, but this group is outside the scope of this study. Third, because of the absence of prospective, repeated liver biopsies, the model was based on one biopsy from living patients at the time of surgery, and assuming that patients had a normal liver when they became overweight. The only way to obtain an unbiased evaluation of the natural history of disease would be sequential biopsies from unselected patients and in the absence of treatment, which to our knowledge are not available, due to ethical reasons. In order to test the potential immortal time bias, as all patients had to survive to inform the model, a model incorporating background mortality from the literature was also tested in sensitivity analysis and have led to similar results. [START_REF] Sjostrom | Effects of bariatric surgery on mortality in Swedish obese subjects[END_REF] Fourth, our study could be limited by a memory bias. Indeed, even if the data on the history of obesity was collected during specific face-to-face interviews, we cannot rule out errors in the patients' memories of their history of overweight/obesity, especially if it occurred during childhood. As the age of onset is a key variable, we decided to exclude patients with this missing information. Moreover, a sensitivity analysis was performed to test the impact of the recall bias using a simplified age of onset of overweight and, again, similar results were obtained.

Fifth, we imputed the missing covariates "BMI at 20 years old" and "age of diagnosis of diabetes" in 411 cases. We used the predictive mean matching method which is the recommended approach and produced the least biased estimates. [START_REF] Marshall | Comparison of imputation methods for handling missing covariate data when fitting a Cox proportional hazards model: a resampling study[END_REF][START_REF] Marshall | Comparison of techniques for handling missing covariate data within prognostic modelling studies: a simulation study[END_REF] Such imputation seems to not affect the resulting model, as the results were relatively unchanged after exclusion of the patients with imputed data, as described in the Supporting Information.

In conclusion, our model can simulate and predict the progression of NAFLD from its onset to cirrhosis in severely obese patients. It shows that taking into account the duration of exposure to key risk factors is essential. This model could help in the management of NAFLD in severely obese patients, by providing a decision-making tool to help refer and select patients for bariatric surgery based on predicted liver status.

LEGEND OF FIGURES

Figure 1 -

 1 Figure 1 -Markov model of compensated non-alcoholic fatty liver disease.

Figure 2 -

 2 Figure 2 -Flow-chart.

Figure 3 -

 3 Figure 3 -Expected risk of being in one of the stages of the disease, based on individual characteristics, for A) 40-year-old women patients who became overweight at 20, and without diabetes, B) 40 year-old women patients who became overweight at 20 and with diabetes since age 35, and C) 40 year-old men patients who became overweight at 10, obese class II at 20 and with diabetes since age 35.

Figure 4 -

 4 Figure 4 -Cumulated risk of cirrhosis at 5 years, for 40 year-old women patients who became overweight at 20 and without diabetes, according to current stage assessed by a non-invasive test: (A) patients with NL or <F2 fibrosis, (B) patients with ≥F2 fibrosis. The dashed line provides a weighted risk estimated without information on NASH status, taking into account the likelihood of having NASH at current evaluation or within 5 years, evaluated by the model.

Figure 5 -

 5 Figure 5 -Cumulated risk of cirrhosis at 5 years, for 40 year-old women patients who became overweight of 20 and without diabetes, according to current stage assessed by liver biopsy: normal liver, F2-steatosis, F2-NASH, F3-steatosis, F3-NASH.

Table 1 -Clinical, biological, and histological characteristics of patients at biopsy and before

 1

			Training	Testing
		All
			cohort	cohort	P-value*
		(n=1801)
			(n=1207)	(n=594)
	Female sex, n (%)	1341 (74.5)	898 (74.4)	443 (74.6)	0.98
	At biopsy :
	Age, mean ± SD	41.2 (±11)	41.3 (±12)	41.0 (±11)	0.52
	BMI (kg/m²), mean ± SD	47.6 (±8)	47.3 (±8)	48.2 (±8)	0.03
	Diabetes, n (%)	600 (33.3)	411 (34.1)	189 (31.8)	0.37
	Steatosis, n (%)	1527 (84.8)	1021 (84.6)	506 (85.2)	0.99
	NASH, n (%)	183 (10.2)	121 (10.0)	62 (10.4)	0.85
	Fibrosis (Metavir), n (%)
	F0	1359 (75.5)	902 (74.7)	457 (76.9)	0.33
	F1	321 (17.8)	221 (18.3)	100 (16.8)	0.48
	F2	58 (3.2)	40 (3.3)	18 (3.0)	0.86
	F3	36 (2.0)	26 (2.2)	10 (1.7)	0.62
	F4	27 (1.5)	18 (1.5)	9 (1.5)	0.99
	History of overweight/obesity :
	Onset of overweight, n (%)
	Before 7 years old	395 (21.9)	265 (22.0)	130 (21.9)	0.99

Table 2 -Estimated baseline yearly transition rates for a standard patient, and adjusted hazard ratios of the covariates, impacting these transition rates. A standard patient is a woman without diabetes, who became overweight during the post-puberty period and have a BMI at age 20 indicating that they are overweight .

 2

	Parameters	95% confidence intervals

ACKNOWLEDGEMENTS

This work is supported by the French government, managed by the National Research Agency (ANR) under the Future Investment Program (PIA2) with the reference ANR-16-RHUS-0006_PreciNASH.

FUNDING

This work is supported by the French government, managed by the National Research Agency (ANR) under the Future Investment Program (PIA2) with the reference ANR-16-RHUS-0006_PreciNASH.

COMPETING INTERESTS

None: Pierre Bauvin, Claire Delacôte, Flavien Dautrecque, Line Carole Ntandja Wandji, Viviane Gnemmi, Alexandre Louvet, Emmanuelle Leteurtre. Guillaume Lassailly has received lecture fees from Gilead and Novartis.

Sylvie Deuffic-Burban has received consultancy honoraria from Intercept, and lecture fees from Abbvie and Gilead.

Philippe Mathurin was paid speaking at symposiums for Abbvie, Gilead Sciences, Ipsen, Eisai, MSD, Bayer Healthcare pharmaceutical companies. He is member of the French boards of experts in hepatology for Abbvie, MSD, Gilead Sciences, Ipsen, Eisal and Bayer Healthcare pharmaceutical companies. He is consulting for the MSD, Ipsen, Eisal, Abbvie, Sanofi, Gilead Sciences, Servier, Bayer Healthcare.

The others authors did not answer about their potential conflicts of interest at the time of submission.

AUTHOR CONTRIBUTIONS

Philippe Mathurin and Sylvie Deuffic-Burban contributed to the study idea and conceptualization.

Pierre Bauvin, Claire Delacôte, Guillaume Lassailly, Philippe Mathurin and Sylvie Deuffic-Burban contributed to the methodology, the conception and design of the analysis. All authors contributed to data curation, acquisition and interpretation. Pierre Bauvin performed the formal analysis, devised, programmed and ran the model. Pierre Bauvin, Claire Delacôte, Guillaume Lassailly, Line Carolle Ntandja Wandji, Sylvie Deuffic-Burban and Philippe Mathurin drafted the manuscript. All authors revised it critically for important intellectual content and approved the final version to be published. All authors agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.