Sylvain Bertho

Ismaël Dondasse

Pascal Retailleau

Cyril Nicolas

Isabelle Gillaizeau
email: isabelle.gillaizeau@univ-orleans.fr.

beta-C(sp2)-H Alkylation of enamides using xanthate chemistry

come

Introduction

Due to the importance of electron-rich C(sp3)-H bonds found in many biologically active natural products and pharmaceuticals, 1 the development of methods that allow the coupling of sp3 centers represents a great challenge in organic synthesis. Recent advances have permitted significant progress in sp3-sp2 cross coupling particularly in those involving single-electron pathways. 2,3 Figure 1. Representative bioactive compounds containing the -amino-,unsaturated acyl scaffold.

The -C(sp2)-H alkylation of electron-rich olefins such as enamides has attracted particular attention, as the -amino-,unsaturated acyl moiety is a unique structural framework of biological interest and a polyvalent building block that can be readily derivatized under a variety of conditions (Figure 1). 4 As a result, a series of interesting methods on the direct intermolecular -C(sp2)-H alkylation of enamides has been reported (Scheme 1). 3a-b,5 Most of them were triggered by the addition of an alkyl radical which is mainly generated by direct radical initiation 6 or a single electron-transfer (SET) process. Renaud and Schubert studied the stereocontrolled addition of sulfonylmethyl radicals to chiral enamines followed by H-atom transfers (path. (a)). 7 Curran et al. described the addition of methyl malononitrile radicals to N-vinylpyrrolidinones as part of a broader study of phenylselenenyl group transfer additions to electron-rich alkene acceptors. 8 Friestad reported an unusual non-reductive coupling of N-vinyloxazolidinones with haloesters and iodoacetonitrile in presence of tri-nbutylstannane and AIBN as radical precursor. 6 Visible-light iridium photoredox-catalysis was highlighted by Yu and coworkers using electron-deficient bromides as alkylating agents (path. (b)). 9 Loh's group elegantly described a palladium-catalyzed strategy for the branch-selective alkylation of enamides (path. (c)). 10 Li and co-workers developed an efficient Mn III -mediated cross dehydrogenation coupling reaction between enamides and 1,3dicarbonyl compounds. 11 Another approach involved -diazo compounds as reported by Maas and Muller 12 or Yan 13 through a Cu II -catalyzed process and more recently upon Rh II -catalysis by Musaev and France. 5b Although some of these methods show high efficiency and good functional group tolerance, there is still great room for improvement with the perspective of developing an efficient, generally applicable method.

Complementarily, xanthates [START_REF] Zard | For reviews of the xanthate transfer chemistry[END_REF] have emerged as attractive feedstocks in radical coupling reactions and in promoting the direct alkylation of olefins, as elegantly developed by Zard, 15 Miranda 16 or Landais. 17 It has been demonstrated that xanthates display an inherent advantage in the oxidative radical substitution of het(aryl) compounds. Zard reported the C-2 alkylation of t-butyl 3-indolecarboxylate 15a using such a strategy. The oxidative direct intermolecular alkylation of several heteroaromatic systems (i.e. indoles, pyrroles, furans, and thiophenes) was observed by Miranda. 16b Furthermore, the same group demonstrated that xanthate-based radical chemistry could be used for the intermolecular alkylation of 1,3dimethyluracil. 16 Of particular interest, the addition of a radical species derived from xanthate was highlighted by Zard on azetines, 15b also once on the N-vinylpyrolidone 15c , and by Landais 17 in the carbo-alkenylation of enamides. Apart from these examples, and to the best of our knowledge, no study has showcased to date the applicability of xanthate in the direct intermolecular C(sp2)-H alkylation of various enamides. Following our interest in radical processes to promote the selective functionalization of nitrogen-containing heterocycles, we report here the application of xanthate chemistry to the -C(sp2)-H alkylation of a range of non-aromatic enamides to synthesize new -amino-,-unsaturated systems which are envisioned to be useful for further elaboration.

Results and discussion

At the outset of our study, we chose the commercially available N-vinylpyrrolidinone 1a and xanthate of ethyl acetate 2a as the coupling partner for our model reaction (Table 1). By applying Zard's guidelines, 15 we began the optimization conditions by performing the reaction with enamide 1a (1 equiv.) and 2a (2 equiv.) in 1,2-dichloroethane (1,2-DCE) using readily biodegradable lauroyl peroxide (DLP, 1.2 equiv.) as the radical precursor. The reaction mixture was refluxed until the starting material 1a had been consumed (TLC monitoring). The related -amino-,-unsaturated ester 3a was obtained in good yield (e.g., 62%, entry 1) after 4h30. The E geometry of 3a was assigned on the basis of NMR data. [START_REF]A trans coupling constant J 3 = 14.6 Hz was observed on 1 H NMR. 19[END_REF] Other radical initiators such as di-tert-butyl peroxide (DTBP), azobisisobutyronitrile (AIBN), iodobenzene diacetate or tert-butyl hydroperoxide were next studied. In most instances, the reaction did not proceed or a dramatic decrease in yield was observed (entries 2-5). The yield of 3a was also affected by the stoichiometry of the xanthate derivative (entries 6-7) and the amount of DLP (entries 8-9), demonstrating that 2 equiv. of 2 and 1.2 equiv. of DLP was mandatory. The decomposition half-life for a given radical precursor at an operating temperature is often of great significance. Lastly, portionwise (a portion every 90 min) addition of the DLP initiator was compared with its addition in a single portion (entry 10). As expected, the direct addition of DLP proved to be the optimum choice. Polar effects in radical mediated processes are much less important than in ionic processes, allowing a much wider choice of solvents. The next step was therefore to screen common solvents for radical reactions. Of note, less polar aprotic solvents such as 1,4dioxane (entry 11) or the bio-renewable 2-Me-THF (entry 12) were capable of performing the transformation, albeit with lower efficiencies, whereas with ethyl acetate (entry 13) the addition proceeded well and product 3a was isolated with the same yield as previously observed with 1,2-DCE. We avoided using 1,2-DCE for environmental reasons. Acetonitrile proved less effective (44%, entry 14). In addition, it is noteworthy that degradation was observed when conventional heating was replaced by microwave heating. On the basis of literature precedent, the following mechanism can be proposed (Figure 2). The thermal decomposition of lauroyl peroxide (DLP) initiates first the radical addition of enamide 1 by the alkyl radical A to provide the amido radical B. As observed with (het)aromatic systems and the uracil system, 16 lauroyl peroxide induces the oxidation of the latter into the corresponding cation C and its resonance structure D. A proton elimination generated the enamide 3. The transfer-xanthate Please do not adjust margins Please do not adjust margins product resulting from the trapping of the amido radical B by the xanthate was not observed. However, in some specific cases with more electron-rich enamides, the iminium species D can be trapped by a nucleophile (i.e. the released carboxylate C11H23CO2 -) leading to the difunctionalized adduct 4, isolated as a mixture of cis-trans diastereoisomers. This mechanism demonstrated the need for a stoechiometric amount of lauryl peroxide for the reaction to succeed. With the optimized conditions in hand, we next examined the -C(sp2)-H alkylation of various enamides 1a-q with 2a (Scheme 2). To our delight, a series of cyclic and acyclic enamides 1 were successfully converted to the corresponding -alkylated enamides 3. Monosubstituted 5-or 7-membered ring tertiary enamides 1a-b worked well, giving the new alkylated product 3a-b as a unique E stereoisomer. [START_REF]A trans coupling constant J 3 = 14.6 Hz was observed on 1 H NMR. 19[END_REF] The secondary enamide 1c was also suitable for this reaction and provided 3c in good yield. Cyclic (i.e. 6-or 7membered rings) di-substituted enamides 1d-i bearing various electron-withdrawing N-protecting groups were also good substrates in this reaction and led to the desired alkylated products 3d-i with moderate to good yields. It is worth noting that the corresponding ene-sulfonamide proved not to be activated enough to react. The alkylation of 3j bearing an ester group at the C-2 position, which could serve as a valuable synthetic intermediate was tolerated. Furthermore, the reaction turned out to be compatible with other functional groups such as a hydroxyl group (3k), which was amenable to further useful transformations. Notably, the vinylogous alkylester enamide 3l and pyridones 3m-n were isolated in moderate yields accompanied, in the case of 3m, with unreacted starting material. It is noteworthy that functionalized pyridones are useful partners in Diels-Alder cycloadditions. 19 Interestingly, no chemoselectivity was observed starting from the N-allyl enamide 1o; a xanthate transfer was indeed additionally observed onto the allylic olefin allowing the implementation of a potential second radical sequence (3o). The observed low yields are due mainly to a slight degradation of the reaction mixture. As previously observed by us, 20 when more electron-rich enamides were used, the di-functionalized morpholin-2-ones 4p-q were originally isolated as a cis:trans isomer mixture (1:1 ratio). In this case, we assume that no elimination step occurred as the electron donor property of the oxygen atom lowers the acidity of the nearby proton in 4p-q. The latter two examples illustrate in an interesting way the oxyalkylation reaction of enamides. 21 Scheme 2. -C(sp2)-H Alkylation of diverse enamides 1a-q using xanthate 2a. Reaction conditions: Argon-free oxygen atmosphere, enamide 1 (1 equiv., 0.33 M), xanthate 2a (2 equiv.), DLP (1.2 equiv.), AcOEt, heating under reflux for 4h30. brsm: based on recovered starting material.

To ascertain the scope of this reaction, various xanthates 2bf (Scheme 3) were prepared by substitution of the corresponding halo derivative with commercially available potassium ethyl xanthogenate. 22 Remarkably, a variety of electron-deficient primary and secondary alkyl groups successfully underwent this transformation to give a range of new substituted acyclic enamides 3ab-af bearing an -ester or -nitrile function in moderate to good yields (39-72%).

With the objective of increasing the molecular diversity, and based on the proposed mechanism depicted in Figure 2, we next envisaged that a nucleophile introduced into the reaction mixture could be trapped by the iminium species of type D (Figure 2), leading to enamides 5 in good yields. Intra-and This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins intermolecular processes can be envisioned (Scheme 4, (a) and (b)). The tandem difunctionalization of enamides involving the simultaneous formation of C(sp3)-C(sp3) and C-O or C-CN bonds has proven to be a powerful tool for the synthesis of various motifs. 23,24a Firstly, starting from 5-or 7-membered exocyclic enamides 1a-b and through a sequential one-pot reaction, 10 equivalents of ethanol or phenol were used to intermolecularly trap the iminium ion intermediate leading to 5a-h in good yields. In addition to this oxyalkylation reaction, cyanoalkylation was observed using TMSCN as the nucleophile. 24 Scheme 3. Alkylation reactions of enamide 1a with diverse xanthates 2b-f. Reaction conditions: Argon-free oxygen atmosphere, enamide 1a (1 equiv., 0.33 M), xanthate 2b-f (2 equiv.), DLP (1.2 equiv.), AcOEt, heating under reflux for 4h30.

A large array of transformations can thus be envisaged to introduce further diversity and complexity into the structures. Starting from the enamide 1s bearing a chain with a pendant dimethoxyphenyl group as the nucleophile, spontaneous cyclization occurred, leading directly to the new tricyclic enamide 5i with an overall yield of 30% for the three-step sequence. As the limiting step is the intramolecular addition of the nucleophilic aryl moiety onto the iminium ion intermediate, conducting the reaction the absence of ethanol resulted in the degradation of reaction mixture. Additionally, a twostep sequence can be achieved by taking advantage of the iminium ion intermediate reactivity in acidic media (Scheme 4, (c)) from 4r allowing the formation of the original tricyclic and diversely substituted morpholine derivative 5j. The compound 5j was isolated as a unique trans diastereoisomer; The relative stereochemistry was unambiguously confirmed by X-ray crystallography. [START_REF]CCDC NUKBAZ (5j) contains the supplementary crystallographic data for this paper. This data is provided free of charge by the Cambridge Crystallographic Data Centre[END_REF] Furthermore, the -ester function in the enamide 3h was reduced with good yields in the corresponding primary alcohol 6 in presence of LiAlH4. This approach thus illustrated the potential of this strategy to provide access to a wide range of nitrogen containing heterocycles decorated with various functional groups.

Conclusions

In summary, we have developed a mild, simple and efficient access to the -amino-,-unsaturated acyl scaffold by applying xanthate chemistry to enamide. This original reaction is totally regioselective and exhibits broad substrate scope, good functional group tolerance and thus demonstrates its potent application in the synthesis of versatile N-containing building blocks. The large availability of xanthates is advantageous to the scope of the reaction which combines a radical process and a Please do not adjust margins Please do not adjust margins polar reaction. Further applications based on this chemistry are in progress in our laboratory.

Experimental General information

Unless otherwise noted, all reagents and solvents were purchased from commercial sources and used as received. All manipulations were conducted under argon. The reactions were monitored by thin-layer chromatography (TLC) using silica gel (60 F254) plates. Compounds were visualized using a UV lamp (254 nm) and/or by potassium permanganate stain. Flash column chromatography was carried out on silica gel 60 (230-400 mesh, 0.040-0.063 mm). Melting points (mp [°C]) were taken on samples in open capillary tubes and are uncorrected. The infrared spectra of compounds were recorded on a Thermo Scientific Nicolet iS10. 1 H, 13 C and 19 F NMR spectra were recorded on a spectrometer at 250 MHz ( 13 C, 62.9 MHz) or 400 MHz ( 13 C, 100 MHz; 19 F: 376 MHz CPD). High-resolution accurate mass measurements (HRAM) were recorded with a Maxis Bruker 4G instrument and were performed in positive mode with an ESI source on a Q-TOF mass spectrometer with an accuracy tolerance of 2 ppm by the "Fédération de Recherche" ICOA/CBM (FR2708) platform.

General Procedure for the Synthesis of Enamides 1d-i, 1k, 1l and 1o-1q (G.P. A). An oven-dried single-necked roundbottomed flask under argon atmosphere was charged with dry toluene, the related imide 1 (1.0 equiv., 0.49 M) and a magnetic stir bar. The reaction vessel was cooled to -78 °C (dry ice/acetone bath) and a 1 m solution of LiEt3BH in THF (1.1 equiv.) was then added dropwise. The mixture was stirred further at -78 °C for 1 h. Next, N,N-Diisopropylethylamine (DIPEA, 5.7 equiv.) and a catalytic amount of 4-Dimethylaminopyridine (DMAP, 0.03 equiv.) were added, followed by the dropwise addition of Trifluoroacetic anhydride (TFAA, 1.2 equiv.) and the reaction mixture was allowed to warm up to room temperature (ca. 20 °C). The mixture was stirred for 3 h at the same temperature and it was quenched by the addition of water. The aqueous phase was then extracted twice with EtOAc, combined organic phases were washed (sat. aq. NaCl), dried over MgSO4 and filtered through a cotton plug. The solvents were evaporated under reduced pressure and the resulting crude enamide derivative was purified by column chromatography (SiO2).

General Procedure for the Direct Oxidative Radical β-C(sp2)-H Monoalkylation of Enamides (G.P. B

). An oven-dried 2-5 mL microwave vial under argon atmosphere was charged with the enamide substrate (1.0 equiv., 0.33 M), ethyl acetate, corresponding xanthate (2.0 equiv., 0.66 M) and a magnetic stir bar. The solution was degassed 3 times (vacuum/argon cycles), the reaction vessel was capped and it was placed in a preheated oil bath for 5 min at 90 °C. Next, the vial was uncapped and dilauroyl peroxide (DLP, 1.2 equiv.) was added. The vessel was sealed and the reaction mixture was heated at 78 °C for 4.5 h under argon atmosphere. After cooling to rt (ca. 20 °C), the solvent was concentrated under reduced pressure. The residue was taken up in dichloromethane and dry silica was added (approximately 10 times the mass of the sample). The solvents were evaporated in vacuo until the silica is dry and free-flowing and the coated support was packed on top of a silica gel column. The crude product was purified (SiO2) to give the desired monoalkylated enamide derivative in moderate to good yield.

General Procedure for the Radical Difunctionalisation of Enamides With Xanthates in Presence of a Nucleophile (G.P. C). An oven-dried 2-5 mL microwave vial under argon atmosphere was charged with the enamide substrate (1.0 equiv., 0.33 M), related xanthate derivative (2.0 equiv., 0.66 M), ethyl acetate and a magnetic stir bar. The solution was degassed 3 times (vacuum/argon cycles), the reaction vessel was capped and it was placed in a pre-heated oil bath for 5 min at 90 °C. Next, the vial was uncapped and DLP (1.2 equiv.) followed by corresponding nucleophile (10 equiv.) were added. The vessel was sealed and the reaction mixture was heated at 78 °C for 4.5 h under argon atmosphere. After cooling to rt (ca. 20 °C), the solvent was concentrated under reduced pressure. The residue was taken up in dichloromethane and dry silica was added (approximately 10 times the mass of the sample). The solvents were evaporated in vacuo until the silica is dry and free-flowing and the coated support was packed on top of a silica gel column. The crude product was purified (SiO2) to give the desired dialkylated enamide compound in moderate to good yield.

General procedure D (G.P. D). Alternatively, after solvent concentration in G.P. B and G.P. C, addition of cold acetonitrile could be performed at 0 °C, leaving decomposition products from DLP The precipitate was filtered through a sintered glass Büchner funnel and the mother liquor was recovered. Then, the acetonitrile was removed by rotary evaporation and the product residue was purified further by flash SiO2-column chromatography.

Figure 2 .

 2 Figure 2. Proposed mechanism.

Scheme 4 .

 4 Scheme 4. (a) and (b) Difunctionalization of enamide 1a or 1b with xanthate 2a or 2b in presence of nucleophiles. Reaction conditions: Argon-free oxygen atmosphere, enamide1a or 1b (1 equiv., 0.33 M), xanthate 2a or 2b (2 equiv.), DLP (1.2 equiv.), related nucleophile (10 equiv.), AcOEt, heating under reflux for 4h30 to 18h. (c) Cyclization of 4r to 5j in presence of trifluoroacetic acid. Ortep view of the crystal structure 5j (CCDC NUKBAZ). Displacement ellipsoids are drawn at the 30% probability level and hydrogen atoms with an arbitrary radius size. (d) Reduction of the -amino-,-unsaturated ester 3h to the primary alcohol 6.

Table 1 .

 1 Optimization of the -C(sp2)-H alkylation of enamide 1a using xanthate 2a.

	Entry	Radical	Equiv.	Solvent	Yield [%]
		initiator			of 3a b
	1	DLP	1.2	1,2-DCE	62
	2	DTBP	1.2	1,2-DCE	0
	3	AIBN	1.2	1,2-DCE	0
	4	PhI(OAc)2	1.2	1,2-DCE	10
	5	t-BuOOH [5M]	1.2	1,2-DCE	0
	6 c	DLP	1.2	AcOEt	58
	7 d	DLP	1.2	AcOEt	62
	8	DLP	0.5	1,2-DCE	30
	9	DLP	1	1,2-DCE	50
	10 e	DLP	1.2	1,2-DCE	35
	11	DLP	1.2	1,4-Dioxane	32
	12	DLP	1.2	2-Me-THF	31
	13	DLP	1.2	AcOEt	62
	14	DLP	1.2	MeCN	44

a Typical reaction conditions: Argon-free oxygen atmosphere, compound 1a (56 mg, 1 equiv.), xanthate 2a (2 equiv.), radical initiator (1.2 equiv.), solvent (1.5 mL), heating under reflux for 4h30. b Isolated yield after purification by column chromatography (SiO2, petroleum ether/EtOAc). c 1.5 equiv. of xanthate 2a was used. d 2.5 equiv. of xanthate 2a was used. e A portion (0.4 equiv.) of the radical initiator was added every 90 min.

Acknowledgements

The authors thank the Labex SynOrg ANR-11-LABX-0029 and the Region Centre-Val de Loire for financial support and fellowships (S.B., I.D.).

Notes and references

Conflicts of interest

There are no conflicts of interest to declare.


René Guinebretière
email: rene.guinebretiere@unilim.fr

Stephan Arnaud

Nils Blanc

Nathalie Boudet

Elsa Thune

David Babonneau

Olivier Castelnau

Full reciprocal space mapping up to 2000 K under controlled atmosphere: the multi-purpose QMAX furnace

Keywords: Reciprocal space mapping, high temperature, in situ, Structural and microstructural evolutions under extreme conditions, controlled atmosphere

This article presents the capability of the QMAX furnace devoted to reciprocal space mapping through X-ray scattering at high temperature up to 2000 K.

Introduction

One of the current issues in material science is to be able to determine in situ the structural and microstructural evolutions of materials as a function of external constraints (Tao & Salmeron, 2011;Villanova et al., 2017). X-ray scattering (diffuse scattering, small angle X-ray scattering, diffraction) or absorption (tomography, absorption spectroscopy) experiments are non-destructive and very well suited for the characterization of actual materials during their elaboration or in operating conditions.

Nevertheless, dynamical experimental studies able to evidence in convenient time-scale the in situ evolution of materials under very high temperature or pressure, mechanical constrains or environmental atmosphere variations through gas flows, require the use of high flux X-ray sources. It is well-known that the 3 rd generation of synchrotron radiation sources are very useful facilities in this field [START_REF] Montano | situ synchrotron radiation research in materials science[END_REF]. Nowadays, the association of such X-ray sources and 2D-position sensitive detectors allows the collection of scattering or diffraction patterns in a second or even shorter time scale (Basolo et al., 2007;Bergamaschi et al., 2010;Ors et al., 2019) and full 3D exploration of the reciprocal space can be done around any reciprocal lattice node (RLN) within one hour (Cornelius et al., 2012) or less (Leake et al. 2019). For many years, the structural or microstructural evolutions (i.e. phase transitions, thermal expansion, grain growth, defect mobility etc.) as a function of temperature or pressure are studied through in situ X-ray scattering based methods. Nevertheless, in many cases, the samples have to be prepared in such a way that they fit the experimental constrains related to the use of high-temperature furnaces or diamond anvil cells.

Consequently, the characterized samples are often very different, especially from the microstructural point of view, from the actual materials under consideration. It is thus an on-going work to develop Xray scattering set-ups well suited to capture the structural evolutions of single crystal surfaces, thin films, nanostructures as well as 3D bulk polycrystalline materials under reactive or non-reactive atmosphere at temperature higher than 1250 K with a convenient time scale resolution.

Chemists who synthesize new compounds or develop crystal chemistry studies commonly realize high temperature X-ray diffraction (XRD) experiments. In most cases, such studies are done on powdered samples often put into capillary sample holders. The use of such capillary sample holders under the Debye-Scherrer diffraction geometry [START_REF] Guinebretière | X-ray diffraction on polycrystalline materials[END_REF] associated with one-dimensional position sensitive gaseous (Benard et al., 1996;Muller et al., 1997) or solid state (Pickup et al., 2000;Sarin et al., 2009) curved detectors allows high angular resolution diffraction patterns to be collected in a short time at temperatures as high as 1250 K. Nevertheless, such an approach is far from the characterization of single crystal surfaces, thin films or bulk materials. On the other hand, cylindrical furnaces where a platinum foil strips act simultaneously as heating element and sample holder have been developed for many years (Koppelhuber-Bitschnau et al. 1996). However, a quite large expansion of this metallic strips and subsequent sample displacement are induced by the temperature increase. Although such setup has been largely used, accurate sample positioning as a function of

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3 temperature is still a difficult task (Brown et al., 1993;Misture et al., 2002;Beck & Mittemeijer, 2002) and it is well known that sample displacement with respect to the goniometer axes has a great influence on the accuracy of the d-spacing measurements (Masson et al., 1996). This drawback can be solved using cylindrical furnaces in which the samples are put on top of an alumina disk located on the spinning stage that corresponds to the azimuthal rotation around the sample surface (Gualtieri et al., 1999;Estermann et al., 1999;[START_REF] Guinebretière | [END_REF]. Heating of the samples is usually realized using conventional high temperature heaters located around the sample holder. Association of this set-up with a translational stage along the azimuthal ϕ-rotation axis, allows one to continuously compensate the sample holder thermal expansion (Gualtieri et al., 1999;[START_REF] Guinebretière | [END_REF].

One of the most significant achievement using this configuration is certainly the furnace built by M.

Yashima et al. on the 4B2 beamline at the Photon Factory in Japan (Yashima et al., 2006). They demonstrated that the combination of such a cylindrical furnace and a multiple detector allows highangular resolution XRD patterns to be recorded at temperatures as high as 1900 K on a number of ceramic materials and the corresponding structures to be described (Yashima, 2002;Yashima & Tanaka 2004).

The devices that we briefly described above are build-up in such a way that whatever the detectors are, the X-rays scattered by the sample can be recorded only if they are scattered in the so-called "diffraction plane", i.e. the plane containing both the direction of the primary X-ray beam and the normal to the sample surface. Taking into account the limited width of the window of the cylindrical furnaces, very few X-ray beams scattered out of this plane can reach the detector. This is the main drawback of such set-ups, which induces two different consequences: firstly it strongly limits the interest of the use of any 2D-detector that are nowadays more and more used especially for 2D or 3D exploration of the reciprocal space; Secondly it forbids in-plane measurements such as Grazing Incidence Small angle X-ray Scattering (GISAXS) or Grazing Incidence Diffraction (GID).

During the last years, we developed a new furnace allowing reciprocal space maps (RSM) to be recorded up to 2000 K either close to the centre of the reciprocal space, i.e. in the small angle scattering regime, or around any diffraction nodes. We report here on the use of this set-up, called the "QMAX furnace" that we have implemented on the French Collaborating Research Group (F-CRG) D2AM beamline at the ESRF (Chahine et al., 2019). The QMAX furnace works in reflection mode and it allows GISAXS and GID to be collected as well as conventional out-of-plane diffraction patterns under symmetric or asymmetric configurations. Beside the very high temperatures that are under consideration, one of the main characteristics of the QMAX furnace is that the experiments are realized under controlled gas flux. In situ reactive experiments can thus easily be performed.

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4

Description of the QMAX furnace

A global view of the QMAX furnace is reported in fig. 1a and a cross-sectional drawing is reported fig. 1b. The implementation on the D2AM beamline goniometer is illustrated in fig. 1c. In order to be able to characterize bulk polycrystalline samples as well as thin films or single crystal surfaces, the furnace has been designed for experiments in the reflection geometry under asymmetric incidence angles on flat samples [START_REF] Guinebretière | X-ray diffraction on polycrystalline materials[END_REF]. The confinement of the atmosphere around the sample is realized using a hemispherical beryllium dome and the furnace is designed in such a way that the detector can collect all the X-rays scattered in the half-space above the sample. One can notice that this geometry is similar to that chosen by the Anton-Paar company for its "DHS" heating stage series (Kotnik et al., 2006). Nevertheless, the maximal temperature accessible using these furnaces is, under optimal conditions, 1350 K that is far below to

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5 the temperature of the sample is.

The QMAX furnace internal temperature is measured using a Pt -10%Rh-Pt thermocouple located at roughly one millimeter underneath the sample. The temperature evolution is driven by a Nanodac controller from Eurotherm by Schneider Electric company. It is linked to the computer driving all the experimental set-up and thus the temperature is recorded in a global data file. The PID furnace parameters can be fully controlled by the computer and adjusted as a function of the temperature. The heating rate can be as high as 50 °C/min. The furnace itself is mounted above a goniometric motorized head made of two crossed cradles and two translations put on top of the kappa stage of the D2AM goniometer (Chahine et al., 2019). All along this article, the description of the experiments will be done using the classical convention with respect to the goniometer rotation axis and reciprocal space vector components [START_REF] Guinebretière | X-ray diffraction on polycrystalline materials[END_REF]. The ω-axis gives the incidence angle of the primary beam onto the sample surface, the ϕ-axis is orthogonal to the ω-axis and normal to a reference plane chosen by the experimentalists (the sample surface, any diffracting plane of a single crystal, etc.), and the χ-axis is orthogonal to the two others. After convenient positioning process of the sample with respect to these goniometer axes using the motorized goniometric head, the , and components of the scattering vectors are respectively collinear to the χ-axis, ω-axis and ϕ-axis unit vectors.

Using such a set-up, flat samples located into the furnace can be carefully aligned before the X-ray scattering measurements at any temperature with respect to their surface or any of their diffracting planes with an angular accuracy of a few thousandths of degree (Boulle et al., 2001). Translational displacement of the samples due to the thermal expansion can be fully compensated using a ztranslation situated below the goniometric head. After a first determination of the thermal expansion of the sample, an automatic compensation of this expansion can be implemented in the code driving the goniometer. Thus, whatever the temperature is, the surface of the sample contains the cross point of the ω, ϕ and χ rotation axes of the diffractometer. This allows solving the well-known sample displacement problem underlined by numerous authors (see above). As already said, the QMAX furnace is designed for the realization of experiments at atmospheric pressure under inert or reactive gas flow. Chemical reactions can be followed under a given gas or a mixture of two different gases of any kind introduced into the furnace through computer-driven flowmeters allowing a gas flux up to one litre per minute with a few millilitres per minute accuracy. Finally, the main technical characteristics of the QMAX furnace are given Table 1.

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6 One of the main characteristics of a furnace is of course the maximal temperature that can be reached during the experiment. It is important to note that the temperature of interest is that one of the sample and not the value of the temperature set-point. We made a large number of experiments and we observed that the maximal accessible temperature is depending on the nature of the samples and more specifically on their thermal properties that themselves are related to the composition of the samples but also to their state (i.e. if it is a single crystal, a powder or a bulk sample). We report on fig. 2

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7 axis of alumina. The main diffraction peaks observed in the patterns are that of both monoclinic and tetragonal zirconia. Zooming with respect to the q-axis between 17.5 and 18.5 nm -1 allows evidencing the displacement of the 012 diffraction peak of alumina with respect to the increase of the temperature. Of course increasing of the temperature induces a decrease of the q-value of the peak position. At higher temperature, this position is roughly equal to 17.755 nm -1 . According to Touloukian et al. [START_REF] Touloukian | Thermophysical properties of matter[END_REF], this is corresponding to 2090 K and this is the higher sample temperature that we reached. We nevertheless fix the maximal working temperature of the QMAX furnace to 2000 K (see table 1).

In the following sections, we will present different cases of in situ high temperature studies realized on the D2AM beamline at the ESRF using the QMAX furnace described above. Both the scientific topics that are concerned and the type of samples studied were very different. The first case addresses the self-organization processes on single crystal vicinal surfaces. The second study illustrates the capability of the set-up to follow the thermal expansion and to evidence phase transitions and crystal growth in a crystalline powder as a function of the temperature or during isothermal treatments. The third case concerns a similar approach on bulk polycrystalline samples. It evidences the relationship between phase transition and residual stress relaxation processes. In the last example, we illustrate the capability of our set-up to follow the oxidation process of metallic plates during rapid increase of the temperature (20 K/min). In all the cases, the experimental results that we present are not discussed on the point of view of their meaning with respect to material science. Such discussions are out of the scope of this article and they will be presented in forthcoming papers.

3D-Grazing Incidence Small Angle X-ray Scattering at high temperature on vicinal single crystal surfaces

Thermal treatment of vicinal surfaces of single crystals usually induces step bunching that results in the formation, through a self-organization process, of periodic hill-and-valley structures at the mesoscopic scale (Misbah et al., 2010). These ordered surfaces can be very well imaged through nearfield microscopy observations. Nevertheless, such local observations probe micrometric areas and more global characterization methods such as GISAXS are also mandatory.

Sapphire, i.e. α-alumina single crystals, is commonly used as substrate for the growth of various oxide nanostructures used in Light-Emitting Diodes (Nakamura, 1998;[START_REF] Koester | [END_REF], zinc oxide nanodots or nanowires for electroluminescent devices (Huang et al., 2001;Eaton et al., 2016) as well as lithium niobate epitaxial thin films for electro-optical modulators (Boulle et al., 2009). It has been shown that the use of sapphire vicinal substrates allows controlling the growth process and the size or the shape of the dots grown onto such surfaces (Lee, 2007;Ago et al., 2007;Bachelet et al., 2007a;Bachelet et al., 2007b;Camelio et al., 2007). The crystallographic structure of sapphire is

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described in the trigonal R3 c space group and it is usually represented in the hexagonal multiple cell.

Throughout this paper, indexations will be given with respect to this hexagonal setting.

A few years ago, we demonstrated that thermal treatment of sapphire vicinal surfaces obtained by cutting the single crystals with a miscut angle of 10° with respect to the (006) planes allows promoting the formation of well-ordered surfaces exhibiting periodic steps with a width of a few tens of nanometers, depending on the temperature, the duration and the atmosphere of the thermal treatment (Thune et al., 2017;Matringe et al., 2017). However, significant ordering effect is observed only when the thermal treatment is realized at temperatures higher than 1250 K in air or oxygen-rich atmosphere. In addition, it is worth noting that one of the most remarkable results achieved in these studies is the observation of 1D as well as 2D ordered surfaces (Matringe, 2016). The previous investigations left open the question whether the 2D structure appears by a rearrangement of the 1D one or these two structures are an alternative each for the other. Answering this question requires to be sure that the observed area of the sample is always the same for each thermal treatment condition;

it is thus very difficult to give an accurate input relative to this question through ex situ measurements, which would imply that a set of samples initially strictly identical is used. We have thus make use of the new QMAX furnace to follow the self-ordering process through in situ GISAXS measurements performed under isothermal conditions. However, observation of significant evolution for a reasonable amount of time (i.e. few hours) compatible with the availability of the synchrotron beamtime requires the achievement of measurements at least at 1450 K under pure oxygen continuous flux. While each GISAXS map corresponds roughly to a (q y ,q z ) 2D slice of the reciprocal space, 3D measurements are obtained by recording a large set of (q y ,q z ) maps at different azimuthal angles (around the ϕ-axis). It is well known that the intensity distribution collected through GISAXS 2D measurements is very sensitive to the z-position of the sample and to the angle of incidence of the primary X-ray beam onto the studied surface. It is therefore very important that these two parameters are kept constant all along the 3D measurement with an accuracy of 10 microns and 0.01 degree respectively.

We followed continuously the self-organization process of a sapphire vicinal surface during isothermal treatment at 1600 K under pure oxygen flux during 40 hours. The GISAXS signal was recorded at 8 keV with an incidence angle of 0.30° and using the 3.2 version of the XPAD 2D pixel hybrid detector (Medjoubi et al. 2010;Le Bourlot et al. 2012) located at 5050 mm from the sample.

The surface was elaborated with a miscut angle equal to 10° with respect to the (006) sapphire planes and in such a way that the step edges were parallel to the 11 0 direction. The azimuthal orientation was determined from the diffraction of the (1 1 12) planes and the zero value of the φ rotation was defined as the position where these planes were under diffraction condition, i.e. with the primary beam orthogonal to the [110] direction. Because the 11 0 direction of sapphire is orthogonal to the

[110] one, in this situation the step edges are strictly parallel to the primary X-ray beam. After

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9 convenient alignment, we were able to record 3D-RSMs made of 721 (q y ,q z ) 2D-maps collected with a 0.5° rotation step around the ϕ-axis. A full 3D-RSM was obtained with a good signal-to-noise ratio in only 15 minutes. Because significant evolution of the surface took several hours, we can consider that time-resolved measurements were thus carried out.

During the temperature increase, because of the thermal expansion not only of the sample but also of some parts of the heating sample holder, the sample surface shifts and can be more or less disoriented.

As mentioned above, the entire furnace is located on top of the goniometric head, therefore the correct height as well as the orientation of the sample can be recovered at any temperature. After reaching the target temperature, the sample surface was thus fully realigned. Furthermore, the true temperature of the sample was determined according to the thermal expansion law of the c cell parameter of sapphire [START_REF] Touloukian | Thermophysical properties of matter[END_REF] through the measurement of the position of the 0,0,6 sapphire RLN. It is noticeable that neither any shift nor twist of the sample was observed all along the roughly two-days isothermal treatment at 1600 K.

Projections along the q z axis of some of Additional structures in the scattered signal progressively appear onto the maps recorded after more than 5 hours, with specific diffuse scattering lines being observed.

According to ex situ GISAXS measurements (Matringe, 2016;Matringe et al. 2020), the presence of these diffuse scattering lines is related to the appearance of the 2D ordering.

We demonstrated with this experiment that, thanks to the QMAX furnace, 3D-GISAXS signals can be recorded in situ during a few tens of hours at very high temperatures under a controlled atmosphere. Beside its ability to allow the determination of the sample temperature, the use of this single crystalline sample holder presents other interesting features. First of all, the procedure that we described to define with a very high accuracy the orientation of a single crystalline sample ( §3) can be used for powdered samples. Such an accurate positioning of powdered samples can be of interest when rotation around the azimuthal axis is used or when the instrumental function needs to be optimized for high-resolution XRD measurements devoted to quantitative microstructural studies. The second important feature concerns the probed volume. For a number of reasons, XRD experiments realized at synchrotron radiation sources are often done using high-energy X-ray beams. The penetration depth is thus often of several hundreds of micrometers or higher. The use of a single crystalline sample holder allows the thickness of the probed volume to be strictly defined and thus the line broadening due to the sample transparency to be limited (Misture et al., 1994).

Thermal expansion, phase transition and crystal growth in crystalline powders

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11 Zirconium oxide, i.e. zirconia, is used for more than five decades in a very large number of applications spreading between high quality brick for glass furnaces, ceramic materials exhibiting high mechanical properties, thermal barrier coatings for aircrafts, solid-state ionic conductors for oxide full cells or basic compound of biomedical implants. In most cases, the production of zirconia materials requires the use of high or very high temperatures and in a large number of applications, these materials are used at high temperature. The knowledge of the thermal behavior of zirconia is thus essential. Under atmospheric pressure, pure zirconia (ZrO 2 ) exhibits two Solid-state Phase Transitions (SPTs) during cooling from the liquidus temperature. It solidifies into a cubic crystal structure (Fm3m space group) at about 3000 K, transforms to tetragonal (P4 2 /nmc space group) upon cooling to 2575 K and becomes monoclinic (P21/c space group) at 1440 K (Smirnov et al., 2003).

Because of the values of the temperatures under consideration, it is a hard task to obtain accurate experimental data on the SPTs processes. As indicated above the tetragonal zirconia crystallized under the primitive P4 2 /nmc space group. Nevertheless, this structure is commonly described in the pseudo-cubic "face-centered tetragonal" lattice in which the crystallographic directions are parallel to those of the cubic form (see for example [START_REF] Kisi | Zirconia engineering ceramics. Old challenges -new ideas[END_REF]. This setting is very efficient in order to follow the m t phase transition process. Accordingly, we adopt it all along this paper.

In 1994, R.J. Hill and M.D. Cranswick (Hill & Cranswick, 1994) High temperature XRD patterns were recorded at 17.6 keV using a 2D pixel hybrid detector (XPAD) located at 800 mm from the sample onto the goniometer 2θ arm. We followed the structural evolutions of the powder between RT and 1800 K. At each temperature, 21 images were recorded in θ-2θ mode using a one-degree 2θ step between 8° and 28°. Only 5 s were needed for recording one image and the total signal was thus obtained in less than 2 minutes. The 21 2D-images were merged and integrated using a special class (multi-geometry) of the pyFAI library (Ashiotis et al., 2015) to

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12 reconstruct 1D diffraction patterns. The part of the reciprocal space explored was between 12 and 55 nm -1 that contains all the significant diffraction peaks of the different zirconia phases.

A part of the patterns recorded between RT and 1500 K is reported in fig. 5. This picture is a very clear observation of the strong anisotropy of the thermal expansion of monoclinic zirconia. Expansion along the a and c cell vectors is very significant. Simultaneously the β angle decreases (not shown here). On the contrary, the b cell parameter is quasi constant whatever the temperature is. The discussion on this observation is out of the scope of this article, nevertheless it can be noticed that such an anisotropy will generate in zirconia-based bulk materials large and anisotropic residual stresses.

Thermal treatments at higher temperature induce the monoclinic to tetragonal SPT and the reverse process during cooling down. XRD patterns were recorded every 10 K or even 5 K between 800 and 1800 K. Some of these patterns illustrating the SPTs are reported in fig. 6. The m t transition starts at around 1450 K and at 1630 K all the crystals are under the tetragonal symmetry. The reverse transition, t m starts close to 1450 K and ends below 1150 K. We show here that even in this powdered sample a priori free of any internal stresses, the phase transition process extends over more than 200 K. The t m phase transition in zirconia is a first order one and it is of martensitic type.

According to R.J. Hill (Hill, 1992) the mean size of the zirconia crystals in the pristine sample is equal to 62 nm. The temperature spreading of the phase transition and the thermal hysteresis are related to size and microstrain effects that we will not discuss here.

Since Scherrer's pioneering work in 1918 (Scherrer, 1918), it is well know that line profile analysis of the XRD peaks is an efficient way allowing the quantitative determination of the main microstructural parameters of powdered or bulk nanostructured polycrystalline materials.

Very often, microstructural evolutions are Figure 6. in situ observation between 1350 and 1800 K of the tetragonal monoclinic phase transition in the pure IUCr zirconia powder during respectively (a) the increase or (b) the decrease of the temperature.

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13 not reversible and it is thus important to in situ catch processes such as crystal growth, microstrains relaxation, as well as variation of the density of stacking faults or dislocations. We studied the microstructural evolution of the IUCr zirconia sample during a 9 hours isothermal treatment at 1773 K under airflow. As written above, each full pattern is recorded in less than 2 minutes. Such measurements will allow to extract with high accuracy the kinetic of crystal growth and this will be presented in a forthcoming paper. The isothermal evolution at 1773 K of the 111 tetragonal zirconia peak is reported in fig.

7a where one pattern every half an hour is shown. The diffracted intensity is normalized and reported in log-scale. We observed all along the isothermal treatment a clear condensation of the diffuse scattering. The peak narrows in width and its profile evolves with a decrease of its Lorentzian part.

As a preliminary result, a Voigt function was used to fit the diffraction peaks of the first and the last patterns recorded during the isothermal treatment. Following the Williamson and Hall approach (Williamson & Hall, 1953), we plotted, the evolution of * = cos ⁄ (where β is the full width at half maximum of the diffraction peaks) as a function of the reciprocal space vector length d* (see fig. 7b). According to the yintercepts of the straight lines, the mean size of the tetragonal crystals evolves from 190 to 290 nm during the whole isotherm at 1773 K. Moreover, when looking at the slopes of the corresponding linear fits, it seems that the amount of microstrains stays roughly constant all along the isothermal treatment.

Phase transition and diffuse scattering into bulk polycrystalline samples

The physical properties of bulk metallic alloys as well as ceramics materials are often significantly different from those of the powders of the same compounds. In fact, residual stresses appearing during the synthesis of bulk materials through fuse cast or sintering processes have very often a strong -0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 q (1/nm) coupling between strain relaxation and t m phase transition. The volume of the monoclinic cell is roughly 4% higher than that of the tetragonal cell, and therefore during heating, the m t transition is associated to stress relaxation. On the contrary, the reverse t m transition induces the appearance of huge stresses that are partially relaxed by the formation of a microcrack network around the monoclinic crystals [START_REF] Kisi | Zirconia engineering ceramics. Old challenges -new ideas[END_REF]. Description of this whole feature and of the interdependency of size and strain effects requires micromechanical modeling and is far from the topic of this paper.

Nevertheless, we have shown here that using the QMAX furnace the coupled strain and diffuse scattering evolution on polycrystalline samples can be in situ captured at high temperature.

In situ oxidation of metal under controlled atmosphere

In all the experimental cases presented above, the measurements were done under air or oxygen flux and because the studied samples were oxides, no chemical reaction between the sample and the One of the aims of the study was to follow the relative amount of monoclinic and tetragonal zirconia appearing under oxidation atmosphere. The zirconia crystals have a size between a few nanometers and a few tens of nanometers, and the relative amount of tetragonal and monoclinic phases is strongly related to the size of these nanocrystals. We were able to follow the evolution of the mean size of both tetragonal and monoclinic zirconia crystals through line profile analysis during isothermal treatment realized at different temperatures between 700 °C and 1000 °C with a time scale of ten seconds.

Concluding remarks

We built-up through a 5-years collaborating research program funded by the ANR (ANR-09-NANO-031), a unique high temperature furnace implemented at the D2AM beamline at the ESRF. X-ray scattering or diffraction measurements can be done up to 2000 K under controlled gas flux at atmospheric pressure under reflection geometry. We have shown with the above description of four strongly different cases that this QMAX furnace is well-adapted for GISAXS or GID experiments as well as asymmetric or symmetric XRD on powdered or bulk samples under inert or reactive gas mixtures. The association of the QMAX furnace and up to date 2D pixel hybrid detectors allows the full exploration of the reciprocal space everywhere above the sample plane.

Thanks to the presence of a high precision motorized goniometric head on top of the D2AM kappa diffractometer, the translational and angular positions of the samples can be precisely adjusted as a function of temperature, and at any temperature, with an accuracy of a few tens of microns and few hundredth degree. We have also shown that we were able to keep this high precision positioning during two days at a temperature as high as 1600 K.

The development of devices based on nanostructured materials requires to be able to follow the evolution of their microstructure during their elaboration process or operando. Nowadays, it is clear that in situ or operando X-ray scattering set-ups have to be built using 2D, or a least 1D, position sensitive detector. Although lot of work is currently done to obtain high angular resolution using 2D position sensitive detectors (see for example Dejoie et al., 2018), under reflection geometry the resolution remains directly linked to the beam size. We demonstrate that thanks to the high flux provided by the synchrotron radiation source and in spite of the use of a highly collimated parallel primary X-ray beam, high resolution XRD patterns can be recorded at the second time scale and full 3D RSM can be obtained in a few minutes. Such a time scale is very well suited for a large part of the

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18 X-ray scattering measurements needed in the field of material science. These results open the field of experiments devoted to the in situ determination of the microstructure evolution of nanostructured materials at high temperature. We have shown that the study of nanocrystal growth can be performed at 1775 K on the zirconia IUCr standard sample and we observed relaxation of residual stresses into bulk ceramic materials through a high temperature phase transition.

External constrains can be the temperature, it can also be the evolution of the atmosphere surrounding the samples. The QMAX furnace has been designed to work under atmospheric pressure. It is thus easy to realize any type of X-ray scattering measurements at high temperature under controlled flux of gas mixture. The chemical reactions between single crystals as well as powdered or bulk polycrystalline samples with various types of gas mixture can thus be followed during isothermal treatments or during high speed heating or cooling.

Figure 1a

  the accessible temperature with the QMAX furnace. Moreover, no accurate control of the gas flux is implemented on the DHS devices. The heating of the samples is ensured by a 80Pt-20Rh plate put onto an alumina flat holder specifically designed so that it retains its integrity despite the differential expansion associated with temperature gradients. The samples themselves are fixed onto the heating element in a way depending on their nature (see below). Very near to the sample heating stage, a flowing water cooling device made of two coaxial copper cylinders allows the steel body of the furnace to be maintained at the room temperature (RT). Additionally, an airflow shower located above the beryllium dome, i.e. above the entire furnace, allows keeping the temperature of the dome below 350 K to prevent the beryllium from oxidation whatever (a) picture of the furnace (b) cross-sectional drawing of the furnace (c) pictures showing the furnace at high temperature on the goniometer and all the associated devices.

Figure 1 .

 1 Figure 1. The QMAX furnace in its working environment at the D2AM beamline.

Figure 2 .

 2 Figure 2. Thermal expansion of α-alumina in a zirconiaalumina powder mixing. The lower Q-position of the 012 diffraction peak is corresponding to a temperature of 2100 K.

  fig. 3. Before any thermal treatment, the vicinal surface is made of steps without any order and the scattering signal reported in fig. 3a is isotropic. After 1 hour and 45 minutes, the projection of the 3D-RSM exhibits very clearly a scattering line along the q y direction which, according to the sample orientation, is due to the ordering of the steps. All along the thermal treatment, this scattering line is observed.

Figure 3 .

 3 Figure 3. Self-organization of a sapphire vicinal surface during 40 hours of isothermal treatment at 1600 K and atmospheric pressure under pure oxygen flux.

  (a) The powders sample holder (b) evolution of the temperature determined with respect to the position of the sapphire 0,0,6 RLN as a function of the set-point temperature and evolution of discrepancy between the measured and set-point temperatures.

Figure 4 .

 4 Figure 4. Sample holder for powders and measurement of the sample temperature.

  published a paper reporting on a round robin test on the structure refinement through X-ray and neutron diffraction on powdered samples and the Rietveld method. This study was realized under the auspices of the International Union of Crystallography and two different samples were used. One of them was a powder of monoclinic zirconia synthesized by wet chemical route (referred below as "IUCr sample"), and we have recently used a small amount of it to show the capability of the QMAX furnace concerning XRD on powdered samples.

Figure 5 .

 5 Figure 5. Anisotropy of the thermal expansion of monoclinic zirconia.

Figure 7 .

 7 Figure7. in situ study of the tetragonal zirconia crystals growth in the IUCr pure zirconia powder during a 9 hours isotherm at 1773 K.

  behavior of the materials under external constrains. The ability to determine the structural and microstructural evolutions in situ in bulk samples is thus often a key point for the development of new objects or devices. The QMAX furnace was built in such a way that it allows the characterization of bulk polycrystalline samples under conditions similar to those used for powdered samples. Samples exhibiting a typical surface of one square centimeter and a thickness of one millimeter can be put on top of the furnace, glued or not on a ceramic coating deposited on the heating element. As for the two first examples, the sample temperature was determined through the measurement of the thermal expansion α-alumina. Nevertheless, in that case we used a powder instead of a single crystal. Before the experiment, a thin layer of NIST SRM 676a alumina powder was spread onto the bulk samples surface and we determined the sample temperature according to the thermal evolution of both the a and c cell parameters of α-alumina.As an illustration of high-temperature in situ XRD experiments on polycrystalline bulk samples, we studied ceramic dense samples that are part of large zirconia blocks elaborated through a specific fused cast process developed by the St Gobain company in order to manufacture large bricks (sub-meter scale) used as refractory components in industrial furnaces devoted to the production of glass. The sample was made of sub-millimeter zirconia dendrites embedded into a silica based glassy phase and it contains 95 % of pure zirconia. More details about the microstructure of this material have been published elsewhere (Patapy et al., 2013). The XRD patterns were collected at 17.9 keV just below the zirconium absorption edge using the same 2D-pixel hybrid detector as for the experiments on powdered samples (see previous section). Taking into account of both the energy and the incidence angle of the X-ray beam, the penetration depth was close to one hundred of microns. Finally, according to the submillimeter size of this beam, the probed volume was in all cases smaller than one dendrite. We have shown previously (Örs(a) Part of Debye-Scherrer rings diffracted by one zirconia dendrite.(b) 3D-reciprocal space maps around the around the 1,1,1 and 1,1,1 monoclinic RLNs at high temperature.

Figure 8 .

 8 Figure 8. Phase transitions, diffuse scattering and stresses relaxation at high temperature in a bulk zirconia ceramic sample.

  surrounding atmosphere was expected. As a last illustration of the capabilities of the QMAX furnace, we present an in situ reactive high-temperature time-resolved experiment. The samples were flat alloy and the general aim of the study was to understand the oxidation of this metallic alloy submitted to quick variations of the temperature up to 1450 K under oxygen rich atmosphere. According to the literature, in the range from 975 to 1450 K, the kinetics of the oxidization process of zirconium alloy strongly depends on the temperature. Very fast processes are expected at high temperature and our aim was to follow the formation of the oxide with a time resolution of about one minute during heating and one second during isothermal treatments. All the diffraction patterns were recorded using a large 2D-pixel hybrid detector (XPAD-WOS) available at the beamline and allowing a large part of the Debye-Scherrer rings to be recorded without any detector movement. This detector was fixed on the 2θ goniometer arm and the energy of the X-ray beam was fixed at 17.6 keV. The position of the detector was fixed at 850 mm from the specimen in such a way that, according to the energy of the X-ray beam, the oxide diffraction main peaks are clearly visible between the main diffraction peaks of zirconium alloy (see fig. 9a). Because of the kinetics of the oxidation process, measurement of the sample temperature was a tricky part of the experiment. It was monitored via two thermocouples, one fixed on the heating resistor, and one spotwelded directly on the specimen surface. Accurate temperature was also determined by following the evolution of the cell parameters of the NIST SRM 676a alumina powder placed top of the sample. The main stumble point with respect to the realization of reactive experiments able to capture structural evolutions of samples lies in the control of the gas atmosphere. Thanks to the use of the two gas lines linked to the furnace (see §2), we were able to monitor the composition of a (a) Part of the Debye-Scherrer rings observed at RT and close to 1050 K. Two diffraction rings corresponding to the diffraction by zirconia crystals are clearly visible at high temperature. (b) Evolution of the XRD patterns during heating at 20 K/min between RT and 1100 K.

Figure 9 .

 9 Figure 9. Oxidation of a zirconium based alloy at high temperature under controlled atmosphere of He-O 2 reactive gas mixture.

  gas mixture as a function of time and temperature through a computer-controlled process. As illustration, we report in fig.9bthe evolution of the diffraction patterns evidencing the oxidation of zirconium during heating from RT to 1050 K at a rate of 20 K/min and under a 1000 mL/min flux of 90%He-10%O 2 gas mixture. Patterns were recorded every 30 s. For sake of clarity, below 500 °C, only a few patterns are reported in fig.9b. At higher temperatures, half of the recorded patterns are shown, which thus roughly corresponds to temperature steps of 20 K.

  Figure 1c

Table 1 .

 1 Main characteristics of the QMAX furnace

	Geometry of the x-ray	Reflection mode under grazing or any incidence angle in small angle range
	scattering measurements	or around any RLN located above the sample surface
		Heating element	80 at% platinum -20 at% rhodium alloy plate
		Maximal temperature	2000 K
		Maximal heating / cooling rate 50 K/min
	Heating conditions	Working pressure	Atmospheric pressure
			Mixing of any two different gas with
		Gas flux	respect to the conventional security rules Few millilitre per minute to one litre per
			minute
	Bulk single crystalline or	Typical size of the surface	10 x 10 mm
	polycrystalline samples	Typical sample thickness	Few tenths of millimetre to few millimetres
	Powdered samples	The powder is located in counterbore (0.5 mm depth and 12.5 mm of diameter) machined on a sapphire cylindrical piece
	Global dimension of the	Stage diameter	140 mm
	furnace	Stage height	96.5 mm with the dome
	Confinement around the sample	Beryllium dome

Acknowledgements The QMAX furnace has been designed and built-up as a part of the QMAX Project No. ANR-09-NANO-031 funded by the French National Research Agency (ANR). We acknowledge the ESRF and the French Collaborating Research Group (F-CRG) for provision of synchrotron radiation facilities beamtimes. H. Song is thanked for his work on the preliminary thermomechanical calculations and the general sizing of the furnace. The build-up of the furnace has been realized under the technical supervision of D. de Barros. The experiments on the sapphire vicinal surfaces and on the IUCr zirconia powder were realized with the valuable help of C. Matringe (IRCER) and J.B. Marijon (PIMM) respectively. The authors are thankful to I. Cabodi and O. Bories (Saint-Gobain CREE) for the supply of the bulk zirconia based materials. Experiments on these bulk samples were done in the frame of the ASZTECH research program funded by the ANR (ANR-12-RMNP-0007). M. Huger and F. Gouraud from IRCER lab. and T. Örs and V. Michel from the PIMM lab. are strongly acknowledged for their implication during these experiments. The oxidation experiments were realized in the frame of a research program funded by the CEA Paris-Saclay centre in collaboration with R. Guillou, M. Lesaux, D. Menut and J.L. Bechade who are also strongly acknowledged.


Di Meng

Marilyn Keller

Edmond Boyer

Michael Black

Sergi Pujades

Learning a statistical full spine model from partial observations

Keywords: Spine statistical model, Vertebrae surface, Incomplete data

The study of the morphology of the human spine has attracted research attention for its many potential applications, such as image segmentation, bio-mechanics or pathology detection. However, as of today there is no publicly available statistical model of the 3D surface of the full spine. This is mainly due to the lack of openly available 3D data where the full spine is imaged and segmented. In this paper we propose to learn a statistical surface model of the full-spine (7 cervical, 12 thoracic and 5 lumbar vertebrae) from partial and incomplete views of the spine. In order to deal with the partial observations we use probabilistic principal component analysis (PPCA) to learn a surface shape model of the full spine. Quantitative evaluation demonstrates that the obtained model faithfully captures the shape of the population in a low dimensional space and generalizes to left out data. Furthermore, we show that the model faithfully captures the global correlations among the vertebrae shape. Given a partial observation of the spine, i.e. a few vertebrae, the model can predict the shape of unseen vertebrae with a mean error under 3 mm. The full-spine statistical model is trained on the VerSe 2019 public dataset and is publicly made available to the community for non-commercial purposes 3 .

Introduction

A reliable spine model with accurate vertebrae structures is essential in numerous medical application such as image segmentation, orthopedics, anesthesiology and pathology quantification. A full spine model can help segment 3D volumetric images such as computed tomography (CT) or magnetic resonance imaging (MRI), as well as 2D projections like X-ray images, for instance by injecting prior knowledge on the global shape of the spine as a regularizer. Statistical models have also shown their interest in the generation of synthetic data for different modalities. A full spine model can also be used as a reference structure in clinical practice supporting the localization of other organs, as well as be used in the diagnosis of spine scoliosis.

The statistical shape model is one of the most employed statistical atlas [START_REF] Heimann | Statistical shape models for 3D medical image segmentation: a review[END_REF]. Creating statistical models of shape using surface measurements has attracted a lot of attention since the early work of Blanz and Vetter [START_REF] Blanz | A morphable model for the synthesis of 3D faces[END_REF]. While early methods mostly rely on Principal Component Analysis, non-linear methods have been also proposed to overcome the fact that unseen shapes cannot be expressed by linear combinations of the training samples [START_REF] Stacklies | pcaMethodsa bioconductor package providing PCA methods for incomplete data[END_REF].

In order to create a statistical surface model of the full spine using the classical methodology, the full spine of several subjects needs to be observed. However, observations of the full spine (e.g. CT images) are rare. Practitioners usually narrow down the field of view of the spine to one specific level for a detailed diagnosis as well as to reduce the dose given to the patient. Thus most work has been done to model the shape of the vertebrae which are consistently observed in the scans. For instance, lumbar vertebrae have received a special attention for low back pain as this area supports the greatest load of the spine [START_REF] Bruners | Electromagnetic tracking for CT-guided spine interventions: phantom, ex-vivo and in-vivo results[END_REF]. Hollenbeck et al. [START_REF] Hollenbeck | Statistical shape modeling characterizes three-dimensional shape and alignment variability in the lumbar spine[END_REF] generated a statistical shape model of the lumbar region from 52 subjects using PCA with a focus on the L4-L5 and L5-S1 functional spinal units (FSU). Campbell et al. [START_REF] Campbell | Automated finite element modeling of the lumbar spine: Using a statistical shape model to generate a virtual population of models[END_REF] used automated methods for landmark identification to create a statistical shape model of the lumbar spine, applied in biomechanics and population-based finite element modeling. A nonlinear SSM based on kernel PCA was investigated for 3D active shape model segmentation [START_REF] Kirschner | 3D active shape model segmentation with nonlinear shape priors[END_REF]. They used 7 CT scans covering vertebrae level from T10 to L3.

Another difficulty in the learning of statistical models of the spine is that the relative position of vertebrae highly depends on the patient's posture during data acquisition. Overlapping and excessive separation of neighboring structures may appear in the vertebrae segmentation. In order to tackle this problem, a model of the interspace of the vertebrae was developed [START_REF] Castro-Mateos | Statistical interspace models (SIMs): application to robust 3D spine segmentation[END_REF], by considering the variations of the space between two surfaces. Similarly, our model only focuses on the relative translation of the vertebrae and disregards their relative rotation. Multi-vertebrae statistical models [START_REF] Bossa | Multi-object statistical pose+ shape models[END_REF] [START_REF] Rasoulian | Lumbar spine segmentation using a statistical multi-vertebrae anatomical shape+ pose model[END_REF] capture the variations in shape and pose simultaneously, and reduce the number of registration parameters, allowing to help in the segmentation of a section of the vertebral column. These models were tested on 32 subjects and with a focus on the lumbar vertebrae section.

While most works have focused on the modeling of a specific part of the spine, a few works have modeled the full spine. For instance, Klinder et al. [START_REF] Klinder | Automated model-based vertebra detection, identification, and segmentation in CT images[END_REF] designed an automatic framework segmenting vertebrae from arbitrary CT images with a full spine surface model. To create the model they first scanned a commercially available plastic phantom to create the template, and then they manually registered it to ten actual scans of the full spine. Mirzaalian et al. [START_REF] Hengameh | Fast and robust 3D vertebra segmentation using statistical shape models[END_REF] learned a statistical shape model of the spine by independently learning three models, one for each level (cervical, thoracic and lumbar). Thus, their per-level models do not learn the shape correlations across the full spine.

Other probabilistic models, different from the shape surface models, such as probabilistic atlas [START_REF] Ruizespaa | Automatic segmentation of the spine by means of a probabilistic atlas with a special focus on ribs suppression[END_REF], graph models [START_REF] Schmidt | Spine detection and labeling using a partsbased graphical model[END_REF], hidden markov models [START_REF] Glocker | Automatic localization and identification of vertebrae in arbitrary fieldof-view CT scans[END_REF] and hier-archical models [START_REF] Seifert | Hierarchical parsing and semantic navigation of full body CT data[END_REF][26] [START_REF] Cai | Multimodality vertebra recognition in arbitrary views using 3D deformable hierarchical model[END_REF] have also been proposed. For example Ruiz et al. [START_REF] Ruizespaa | Automatic segmentation of the spine by means of a probabilistic atlas with a special focus on ribs suppression[END_REF] proposed a probabilistic atlas of the spine. By co-registering 21 CT scans, a probability map is created which can be used to segment and detect the vertebrae with a special focus on ribs suppression. Schmidt et al. [START_REF] Schmidt | Spine detection and labeling using a partsbased graphical model[END_REF] proposed a probabilistic graphical model for the location and identification of the vertebrae in MR images. In both cases full spines were observed at train time and the proposed methods cannot be used to infer the shape of the full spine from a partial observation.

As full spines are not publicly available, we learn our novel statistical surface model from partial spine observations using Probabilistic Principal Component Analysis (PPCA). In the medical domain, PPCA has been used for craniofacial statistical modeling [START_REF] Lüthi | Building shape models from lousy data[END_REF]. However, in their work full observations of the skull were used for training. In contrast, our proposed model never observed a complete spine. A robust shape model can capture the characteristics of individual vertebrae and the anatomy of the spine. It also provides the appearance information and relative positions of the observed objects. However, existing statistical models have focused on a specific region of the spine or were learned from full spine observations. Unfortunately, none of them are publicly available.

Our full spine statisical model is composed of 24 vertebrae including cervical, thoracic and lumbar. The model is learned from the openly available VerSe 2019 Dataset [START_REF] Sekuboyina | VerSe: A Vertebrae Labelling and Segmentation Benchmark[END_REF] containing partial observations and annotations of the spine. Our model captures the strong shape correlations across the individual vertebrae of the spine. Given an arbitrary field-of-view and incomplete scan, our model is able to predict the full spine adapting to the shape variations of the subject. The model is openly available to the community for non-commercial purposes.

Method

The inputs to our method are i) 80 manually segmented CT volumes V i , where i ∈ [START_REF] Arun | Least-squares fitting of two 3-D point sets[END_REF]80] denotes the volume index, together with the annotations of the individual vertebra from the VerSe Dataset [START_REF] Sekuboyina | VerSe: A Vertebrae Labelling and Segmentation Benchmark[END_REF] and ii) 24 artist created mesh templates T k with k ∈ [1, .., 24] being the vertebra index, one for each individual vertebra. Each template T k has N k 3D vertices and the full spine mesh including all 24 vertebrae has 127.294 vertices and 254.636 faces. All template meshes T k are symmetric with respect to the medial plane. From the segmented volumes V i we extract meshes using the Marching Cubes approach. We name them scans and note them as S i k . Let us note that: i) in the VerSe Dataset none of the volumes contains the 24 vertebra; ii) some vertebra are imaged at the boundary of the CT volume and thus only provide an incomplete observation. These scans have holes and are not watertight; we note them Si

k . An overview of the VerSe Dataset vertebrae is shown in Fig. 1.

The creation of the statistical spine shape model consists of several steps. First, in Sec. 2.1 we detail how we perform a non-rigid registration of each template T k to all watertight scans {S i k -Si k }. To bring all registrations to a coherent frame, we compute a rigid transformation [START_REF] Arun | Least-squares fitting of two 3-D point sets[END_REF] between the registration and the template. We name these coherent registrations I i k . Then, for each vertebra k, we compute Principal Component Analysis (PCA) on all the available registrations. As we have few observations, we exploit the medial plane symmetry: for each registration we create the symmetric registration. With the registrations and their symmetric we obtain 24 (symmetric) mean shapes T µ k and their 24 individual principal vectors B k . With the obtained PCA, we perform a second non-rigid registration of each template T k as explained in Sec. 2.2. This time all scans S i k , including the non-watertight, are registered. We initialize the template k vertices with the mean shape T µ k and use the computed shape space B k to constrain the registration process. We compute the rigid transformation to obtain the coherent registrations and keep the corresponding rotation R i k and translation t i k . We exploit again the medial plane symmetry and obtain a new set of registrations A i k . From these registrations, we use Probabilistic Principal Component Analysis (PPCA) (Sec. 2.3) and we learn the full-spine shape model (Sec 2.4) defined by T µ and B.

Initial registration

We first perform a non-rigid registration of each template T k to all watertight scans {S i k -Si k }. Because the scans are watertight, we want all template vertices to be close to the scan surface, and vice versa, all scan vertices to be close to the template surface. We effectively enforce this constraint by computing the vertex to surface distance from a point set to a surface and define the energy E p2m (S, T), which accounts for the distance of the vertices of S to the mesh surface T. The registration has three steps. Translation optimization, translation and rotation optimization and free form optimization. To simplify notation we drop the volume indices i and keep k to indicate the per-vertebrae process.

We start by computing a translation t, so that T k + t is close to S by minimizing

E(S, T k , t) = E p2m (S, T k + t) + λ m2s E p2m (T k + t, S) (1)
w.r.t. to t, where λ m2s = 1. Next we optimize for a 3D rotation, parametrized by a 3D Rodriguez vector r. Given a 3D vector r and a mesh T, we use R(T, r) to describe the 3D rotated mesh. The CT scans from the VerSe Dataset (see Sec. 3) have a consistent patient orientation encoded in the dicom metadata.

After reorienting all the CTs along the same orientation, we can thus initialize the rotation vector r (r = [-2, 0, 2]) with the same value for all i ∈ [START_REF] Arun | Least-squares fitting of two 3-D point sets[END_REF]80]. This value corresponds to the rotation between the template frame and the acquisition frame defined by the radiologist. Then we optimize

E(S, T k , t, r) = E p2m (S, R(T k , r) + t) + λ m2s E p2m (R(T k , r) + t, S) (2)
w.r.t. to t and r, where t is initialized with the result of (1). Next we allow all vertices of R(T k , r) + t) to freely deform to best match S i k . This free-form deformation is represented as an additive 3D vector added to each vertex that we note f . To regularize the position of these displacements we use a coupling term on edges E cpl (T, T + f ), enforcing the edges of the registration to be close to the edges of the initial template shape. To compute E cpl we use the same energy term defined in Eq. 8 from [START_REF] Bogo | Detailed full-body reconstructions of moving people from monocular RGB-D sequences[END_REF] and optimize

E(S, T k , t, r, f ) = E p2m (S, (R(T k , r) + t) + f )+ + λ m2s E p2m ((R(T k , r) + t) + f , S)+ + λ cpl E cpl ((R(T k , r) + t), (R(T k , r) + t) + f ) (3)
w.r.t. to f and by keeping t and r fixed. The weight λ cpl was empirically chosen to leverage the contribution of each term into the final energy, namely λ cpl = 0.01 for all experiments. Then we compute a rigid transformation [START_REF] Arun | Least-squares fitting of two 3-D point sets[END_REF] between the obtained registration and the template and obtain I i k . From these registrations we compute a per vertebrae mean template T µ k and perform PCA. We note by B k the 10 first PCA principal components of vertebra k.

PCA-guided registration

In the PCA-guided registration we first compute t and r for non-watertight scans Si k , by optimizing ( 1) and ( 2). We set λ m2s = 0 as not every vertex in the template must explain a scan point. Then, we use the individual shape space to parametrize a mesh as T µ k + βB k where β is a 10-dimensional vector. We solve for t, r and β to minimize

E(S, T µ k , t, r, β) = E p2m (S, R(T µ k , r) + t + βB k )+ + λ m2s E p2m (R(T µ k , r) + t + βB k , S) (4)
with λ m2s = 1 if the scan is watertight and λ m2s = 0 otherwise. Then we perform a free vertices optimization following (3) by coupling the vertices to the solution obtained from (4). We compute the rigid transformation [START_REF] Arun | Least-squares fitting of two 3-D point sets[END_REF] between the obtained registration and the mean template T µ k and obtain the final unposed registrations A i k . These registrations will be used to learn the full spine model.

PPCA on the registrations

For each volume index i, we do not have the full set of registration A i k , as some k are not observed in the volume i (see Fig. 1). Thus we use Probabilistic Principal Component Analysis (PPCA) [START_REF] Tipping | Probabilistic principal component analysis[END_REF], a variant of PCA dealing with missing data. We construct a matrix with size N × S values, where N = 24 k=1 3N k , and S = 160 is the number of volumes used (80 volumes plus their symmetric version). Each column i is the concatenation of the 24 vectors, each given by the registration A i k vertices, if it is available, or an empty vector if the data is missing. The created matrix has 57% missing values. We compute PPCA using the publicly available implementation in [START_REF]Probabilistic PCA python implementation[END_REF] and obtain the mean spine T µ and the associated shape space B.

The spine model

The architecture of the spine model is a graphical model mesh architecture parametrized by the shape parameters and the pose parameters, where each vertebrae has its own translation and rotation. The shape parameters β apply additive offsets to the initial vertebrae mesh T µ . The offsets are the PCA principal components learned with the PPCA and the poses and translations are rigidly applied to each vertebra. To constrain the locations of the vertebrae, we learn a distribution over the relative positions of neighboring vertebrae. For each pair of neighboring vertebrae k and k + 1, we use R i k , t i k and t i (k+1) to compute the relative translation of vertebra k + 1 w.r.t. vertebra k. Then for each 23 pairs k, k + 1 we fit a Gaussian model to the observed relative translations. These relative translations distribution accounts for the thickness of the inter-vertebral disk. However, as the relative rotation between vertebrae highly depends on the patient posture, we do not learn a pose prior on the relative rotations.

VerSe challenge dataset

To train our model we use the publicly available MICCAI Challenge 2019 VerSe dataset [START_REF] Sekuboyina | VerSe: A Vertebrae Labelling and Segmentation Benchmark[END_REF]. It provides 80 CT scans with manually annotated voxel-wise labels, and 40 CT scans without labels. We use the 80 annotated subjects to create the spine model. Most volumes have a voxel size 1x1x1 mm 3 and a few 1x1x2 mm 3 . We resample all volumes to have a fixed 1x1x1 mm 3 sampling. An overview of the available observations from the full spine in the VerSe datset is shown in Fig. 1. Each row in the figure represents a vertebrae, C1 to T5 from top to bottom. Each column represents a CT volume. As we can see, each CT scan only covers a few vertebrae and no full spine is observed in any CT volume. In addition, the number of volumes including a given vertebra is not balanced across different vertebrae. The dataset is dominated by lumbar observations and few cervical vertebrae are observed. The section including the junction between cervical and thoracic vertebrae is critically sparse.

In our work we aim at learning the shape correlations across the full spine and to evaluate our model we use cross validation. In order to make sure that the junction between the cervical and thoracic vertebrae is always seen at train time, we label a volume junction volume if C5 to T2 are observed in the volume, and not junction volume otherwise. We use this junction label to create a balanced cross validation 8-fold setting for all our experiments.

As our method is trained on a few samples, we carefully inspected the input data. We noticed some misannotations in the ground truth labels. For example, some misannotations appear as floating points far away from the corresponding vertebra, as well as lumbar volumes having cervical labels. We cleaned the ground truth meshes by keeping only the biggest connected component in the mesh extracted from the CT volume. This procedure is standard as the evaluation criteria of the VerSe dataset only considers the biggest component in the volume [START_REF] Sekuboyina | VerSe: A Vertebrae Labelling and Segmentation Benchmark[END_REF]. We also observed a few vertebrae containing metal implants. As their shape is severely biased, we manually removed them from the dataset.

Experiments

To demonstrate the accuracy and benefits from the spine model we perform two evaluations. In a first set of experiments we assess the metric accuracy of the registrations, the created model and the compacity of the learned shape space. In a second evaluation we assess the accuracy of our model to predict the shape of missing vertebrae.

Metric Accuracy

Before we learn the spine model, we first show that the registrations closely capture the input data shape with sub-voxel accuracy. Then we measure the compacity and the generalization power of the full spine model.

Individual vertebrae registrations accuracy. To assess the accuracy of the individual registrations obtained in Sec. 2.2 we compute the point to mesh distance between the scans S i k and their registrations A i k . For each individual vertebra we aggregate the mean distance per scan by taking the mean over all available volumes. We report the mean error and standard deviation for each vertebra in Tab. 1. We aggregate the data by weighting each vertebrae errors with the number of samples and obtained 0.24 mm errors in mean and 0.20 mm std. Let us recall that the input voxel size is 1x1x1 mm 3 , thus the registrations faithfully capture the input data shape with a sub-voxel accuracy. 1. Vertebrae registration error. me: mean error. std: standard deviation (mm). Full-Spine model accuracy. To assess the compacity of the learned spine shape space we show on the left of Fig. 2 the cumulative variance of the shape space. The fist 10 shape components capture 85% of the variance, and 90% variance is captured by the first 20. Let us recall that each registration has 127.294 vertices and 254.636 faces. To assess how well the model can capture the shape of unseen spines, we perform an 8-fold cross validation (see Sec. 3). We learn the shape space with a train set and evaluate it on the unseen test set. For an unseen, potentially partial, registration, we complete the missing vertebrae by using the learned PPCA, we then project it into the PCA space and reconstruct it with a selected number of components. We then measure the vertex to vertex error between the registration and the reconstruction. Only original data, i.e. not completed by the PPCA, is used in the metrics. On the right of Fig. 2 the generalization errors w.r.t. the number of used components are presented. Using 10 components we obtain a mean reconstruction error of 1.80 mm, and using 20 components we obtain a mean error of 1.68 mm.

Missing vertebrae shape prediction accuracy

Next, we aim at quantifying how well the model captures the global correlations across the spine. With the learned model, given a partial observation of the spine we can reconstruct the missing spine shape. We setup the following experiment to measure the accuracy of the shape of the predicted vertebrae. Fig. 3. Left: reconstructing C1 from partial spine observations. When one vertebra is hidden C1 is not observed, when 5 vertebrae are hidden, C1, C2, C3, C4 and C5 are not observed. Right: reconstructing L5 from partial spine observations. When one vertebra is hidden L5 is not observed, when 5 vertebrae are hidden, L5, L4, L3, L2 and L1 are not observed. The reconstruction errors slightly increase as less data is used for the shape prediction.

For a cross validation fold, we learn the spine model on the train test. Then we take all registrations obtained in Sec 2.2 in the test fold containing the cervical vertebrae C1 to C6. We first mask one vertebra starting from C1 and reconstruct it using the rest of the volume vertebrae. We then remove two vertebrae (C1 and C2) and reconstruct them using the rest. Progressively we remove cervical vertebrae until C5, and use the remaining ones for reconstruction. We measure the vertex to vertex distance between the reconstructed vertebrae and the original registration. In an analogous way, we perform the same experiment for lumbar vertebrae, by progressively removing L5 to L1.

We use the 8-fold cross validation scheme (see Sec. 3) and evaluate on all volumes for which the data is available. In Fig. 3 we report the per vertex mean error over all volumes for C1 and L5. In the left plot, the reconstruction errors on C1, when masking 1, 2, 3, 4 and 5 vertebrae are reported. In the right plot, the results for the L5 reconstruction experiment are presented. In Fig. 4 we present the per vertex mean errors on the mean template.

We aggregate the per vertex errors for all C1 and obtain reconstruction errors of 1.61 mm when one vertebra is masked, 1.65 mm when two vertebrae are masked, 1.68 mm when three vertebrae are masked, 1.72 mm when four vertebrae are masked and 1.75 mm when five vertebrae are masked. For the L5 vertebrae we obtain 2.57 mm when one vertebra is masked, 2.65 mm when two vertebrae are masked, 2.70 mm when three vertebrae are masked, 2.73 mm when four vertebrae are masked and 2.74 mm when five vertebrae are masked.

A trend in the increase of the error is observed as more vertebrae are masked, but the errors do not drastically increase. This trend can be seen in Fig. 3 as well as in the aggregated errors. These experiments show that the shape of all vertebrae is strongly correlated throughout the spine, and by observing a subset of the spine, the learned model can infer a plausible shape for the missing vertebrae.

Conclusion

In this paper we present a statistical surface model of the full-spine learned solely from partial and incomplete views of the spine. In order to deal with the partial observations we use probabilistic principal component analysis (PPCA) to learn a surface shape model of the full spine. The nature of the spine, specifically the correlations across the vertebrae shape, make the use of PPCA particularly well-suited. The quantitative evaluation demonstrates that the obtained model faithfully captures the shape of the population in a low dimensional space and generalizes to left out data. The model is openly available to the community for non-commercial purposes at https://gitlab.inria.fr/spine/spine_model.

While in this paper we presented an application of the model to predict the shape of missing vertebrae, future work will address the application of the model in other tasks, such as the generation of synthetic data, its registration to 2D X-ray images and its use in the segmentation and identification of vertebrae in volumetric images.

Fig. 1 .

 1 Fig. 1. Overview of the VerSe dataset. Rows represent one of the 24 individual vertebrae (C1 to L5). Column are the observed patients. No full spine is observed for any patient. The columns in red are the test set of one of the 8-fold used for cross validation.

  20 0.16 C2 0.20 0.16 C3 0.19 0.15 C4 0.19 0.15 C5 0.19 0.15 C6 0.18 0.15 C7 0.21 0.16 T1 0.21 0.17 T2 0.21 0.17 T3 0.21 0.16 T4 0.21 0.16 T5 0.21 0.17 T6 0.21 0.17 T7 0.22 0.19 T8 0.23 0.19 T9 0.23 0.20 T10 0.23 0.20 T11 0.25 0.22 T12 0.25 0.21 L1 0.25 0.21 L2 0.26 0.22 L3 0.27 0.23 L4 0.28 0.25 L5 0.29 0.26 Table

Fig. 2 .

 2 Fig. 2. Left: the cumulative variance of the full-spine model wrt the number of used shape components. Right: the generalization error of the model (cross validation) wrt to the number of used shape components.

Fig. 4 .

 4 Fig. 4. Per vertex mean reconstruction error when N ∈ [1,2,3,4,5] vertebrae are left out, visualized on the mean shape. First two rows: C1 reconstructions front and rear views. Red is 4 mm. Blue is 0mm. Third and fourth rows: L5 reconstructions front and rear views. Red is 6 mm. Blue is 0mm. Errors slightly increase as less vertebrae are used for the reconstruction.

https://gitlab.inria.fr/spine/spine_model

Acknowledgments and Disclosures

We thank Florence Forbes and Jakob Verbeek for insightful discussions. Di Meng's work was funded by the SPINE PDCA project. Sergi Pujades work was funded by the ANR SEMBA project. Michael J. Black has received research gift funds from Intel, Nvidia, Adobe, Facebook, and Amazon. While he is a part time employee of Amazon and has financial interests in Amazon and Meshcapade GmbH, his research was performed solely at, and funded solely by, MPI.