Matthieu Kauffmann

Nathalie Tancret

Francis Abraham

Pascal Roussel
email: pascal.roussel@ensc-lille.fr

Kauffmann Matthieu

Tancret Nathalie

Abraham Francis

Roussel Pascal

New cobaltite materials containing CdI 2 -type layers: synthesis and structures of Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7

Keywords: Cobaltites, Barium cobalt oxyhalide, Hexagonal perovskite, Maximum Entropy Method (MEM), Crystal structure

come

Introduction

Numerous perovskite-related materials exhibit unusual and interesting properties such as high Tc superconductivity, piezo-electricity or colossal magneto-resistance among others [START_REF] Galasso | Perovskites and High T Superconductors[END_REF][START_REF] Rao | [END_REF]. These phenomena are still insufficiently understood and the comprehension of the mechanism leading to these spectacular properties is of particular interest for the scientists working in the area of new materials. Therefore perovskite-type compounds are intensively studied. In this way, for instance, the well-known manganites have been largely investigated because of their interesting properties: a wide range of valence states and the possibility to adopt several polyhedral configurations as, for example, octahedral or tetrahedral environments. The research on manganites materials has led, for example, to the discovery of colossal magneto-resistance [3,4]. Compared to these compounds, the cobaltites exhibit one additional degree of freedom provided by the cobalt spin-state. Indeed, Co 3+ atoms (3d 6 ) in O 6 octahedral environment can adopt either the low spin LS (t 2g 6 e g 0 , S=0), the intermediate spin IS (t 2g 5 e g 1 , S=1) or the high spin HS (t 2g 4 e g 2 , S=2) configurations [5,6]. The possibility to stabilize several spin configurations is of great interest, as illustrated for instance in the LnBaCo 2 O 5.5 series where the metal-insulator transition is due to a spin-blockade mechanism attributed to the presence of high-spin and low-spin of Co 3+ ions [7]. Moreover, the discovery of large thermopower properties in Na x CoO 2 [8] has reinforced the interest of the research in cobaltites materials. In this series of layered cobalt oxides, candidates for thermoelectric materials, CdI 2 -type layers were proposed to be at the origin of these interesting physical properties.

Recently, and Ba 2 Co 4 BrO 7 respectively.

Crystal structures were determined for both compounds in the R 3 m space group by direct methods using SIR97 program [START_REF] Altomare | SIR97 A Package for Crystal Structure Solution by Direct Methods and Refinement[END_REF], which readily established the heavy atom positions (Ba, Co and Cl or Br). Oxygen atoms positions were found by difference Fourier map calculations. Anisotropic displacement parameters were attributed to all atoms in the last cycles of refinement. Full-matrix least-squares structures refinements against F were carried out using the JANA2000 program [START_REF] Petricek | The Crystallographic Computing System JANA2000[END_REF].

As already reported in one of our previous paper [START_REF] Kauffmann | [END_REF], the introduction of bromine atoms in this type of layered compounds can lead to structural disorder. We thus undertook a Maximum Entropy Method (MEM) analysis of the diffraction data collected on Ba 2 Co 4 ClO 7

and Ba 2 Co 4 BrO 7 . MEM is a model-free method which is used to calculate accurate electron densities in solids using experimental phased structure factors as input. This method has been proved to be particularly suited to determine structural aspects of disorder and anharmonic vibrations [16]. To calculate the precise electron density distribution, the MEM analysis was carried out using the computer program BAYMEM [START_REF] Palatinus | BAYMEM -A computer program for application of the Maximum Entropy Method in reconstruction of electron densities in arbitrary dimension[END_REF]. The total number of electrons in the unit cell has been fixed to F(000) values (1758 e-and 1866 e-for Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 respectively) and the unit cell was divided in a grid of 72*72*486 pixels to ensure a good resolution (better than 0.1 Å for the two studied compounds). All calculations were performed with an initial flat electron density with all the independent reflections. The reliability factor of the MEM,

∑ ∑ - = obs MEM obs MEM F F F R
is given in Table 1 (F obs is obtained by the structural refinement and F MEM is the structure factor calculated from the electron density obtained by the MEM).

Results and discussion

The atomic coordinates and anisotropic displacement parameters deduced from the single crystal refinements are listed in Table 2 and Table 3 reported in oxychlorides and oxybromides of ruthenium with hexagonal perovskite-type structures [START_REF] Kauffmann | [END_REF]20].

Another way to describe the oxyhalides under study is to refer to the well known series of spinel oxides [START_REF] Fischer | [END_REF]. Indeed, spinel can be described as the alternate stacking of different types of polyhedral layers: a cation deficient octahedral layer (whose cationic lattice is called Kagomé, see hereafter) and a mixed layer built up from corner-sharing octahedra and tetrahedra. The (Ba 2 Co 8 O 14 ) 2-blocks can thus be described using the formalism developed for the spinel oxides:

-The central block is formed by the (AB) stacking of [O 4 ] layers, thus creating a sheet of edge-sharing cobalt-octahedra (Figure 3a). This arrangement can be considered as a Kagomé lattice (Figure 3b), denoted in the literature (Oc 3 ), in which all octahedral sites are fully occupied. Co(1) atoms are in the real octahedral sites of the Kagomé lattice whereas Co [START_REF] Rao | [END_REF] occupy the Kagomé windows. This distribution leads to a complete octahedral layer of the CdI 2 -type but with two different crystallographic sites for the cobalt atoms. By analogy to the (Oc 3 ) layer, we will note this fully occupied Kagomé lattice (Oc 4 ). Numerous compounds are constituted of (Oc 3 ) octahedral layers, like for example, the spinel oxide MgAl 2 O 4 [22], the olivine (Mg,Fe) 2 SiO 4 [23] or the double hexagonal LiFeSnO 4 [24]. In the present compounds, the calculated valence bond sums for Co [START_REF] Galasso | Perovskites and High T Superconductors[END_REF] [9].

-The second type of sheet is built up from corner-sharing octahedra and tetrahedra, resulting from the (AB) stacking of [O 4 ] and [BaO 3 ] layers (Figure 4a). Each tetrahedron is connected to three octahedra through the oxygen of the [BaO 3 ] layer. This mixed-polyhedra block can be considered as a (Te 2 Oc) layer (Figure 4b) in which one half of tetrahedral sites are empty.

This kind of (TeOc) layer has been already evidenced in the LiLnMo 3 O 8 compound for example [25]. In the literature, two kinds of junction have been observed to connect an (Oc 3 ) block with a (Te 2 Oc) layer:

-the junction J K/T is obtained by the occlusion of the Kagomé windows by a tetrahedron of the (Te 2 Oc) layer (Figure 5a). In this configuration, the tetrahedron shares three corners with the octahedra of the Kagomé windows. This kind of J K/T junction is observed, for example, in the spinel oxide structure [22].

-the junction J K/O exists when a Kagomé windows is blocked by an octahedron of the (Te 2 Oc) layer by sharing three corners (Figure 5b). This type of J K/O junction has been evidenced for double hexagonal LiFeSnO 4 [24].

In the same manner, J K/T and J K/O junctions have been reported for connecting (Oc 3 ) and (TeOc) layers (Figure 5c and5d). For example, in LiLnMo 3 O 8 [25], the (Oc 3 ) and (TeOc) layers are connected alternatively by J K/T and J K/O junctions.

In the compounds reported here, the junction between (Oc 4 ) and (TeOc) layers is ensured by sharing the corners of polyhedra, the oxygen atoms of

Conclusion

To summarize the main findings of the work presented here, we have determined the structures of two new isostructural cobaltites. They are built by the stacking of fluorite-type layers and interesting blocks containing (CoO 2 ) layers. The role of the halide seems to be of particular interest for the modification of the structures: replacing the chlorine by a bromine atoms leads to the creation of disordered layers. This last point confirms the previously studies made on barium ruthenates with either chlorine-or bromine-based layers [START_REF] Kauffmann | [END_REF]20].

Syntheses of pure powders suitable for a full physical characterization are in progress.

Supporting Information Available:

Further details concerning the nuclear structures determinations and the Maximum Entropy Method analysis may be obtained upon request from the authors.

[O 4 ]

 4 layers making the connexion between the two blocks. Central O(1) oxygen of tetrahedral Co(4) units is the common oxygen of three edge-sharing Co(1) octahedra while O(2) oxygen of octahedral Co(3) units ensures the connexion between three Co(2) octahedra. To our knowledge, it is the first time that this kind of junction is observed for the connection of (Oc 3 )-like (= (Oc 4 )) and (TeOc) layers. In that way, the connections of polyhedral layers in Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 , as compared to the different types of junctions previously reported in the literature, can be considered as a J K/O junction translated by (a+b)/2 (Figure 6a), hereafter noted J K/O * . In summary, the (Ba 2 Co 8 O 14 ) 2-block viewed along [010] (Figure 6b) can be schematized as a (TeOc)-(Oc 4 )-(TeOc) stacking, with a J K/O * junction between each layer. The (Ba 2 Cl 2 ) 2+ double layers are completely related to the reference structure fluorite (CaF 2 ) [26]. This later consists of infinite planes of Ca and F, thus making a compact framework of FCa 4 tetrahedra. On Figure 7 is presented the (Ba 2 Cl 2 ) 2+ block of Ba 2 Co 4 ClO 7 viewed along [010] (a) and the fluorite structure viewed along [011] (b). One should note the perfect similitudes between the (Ba 2 Cl 2 ) 2+ double layers and an isolated (Ca 2 F 2 2+ ) block of fluorite. In this way, we can consider the (Ba 2 Cl 2 ) 2+ blocks in Ba 2 Co 4 ClO 7 as fluorite-type double layers. The (Ba 2 Br 2 ) 2+ double layers of Ba 2 Co 4 BrO 7 are strongly related to the (Ba 2 Cl 2 ) 2+blocks but with some interesting differences. While Ba(1) atoms occupy exactly the same position (6c), the positions of halides are modified, considering Cl or Br atoms. The most significant two-and three-dimensional electron density (ED) images in the region of the chlorine and bromine atoms are shown in Figure8. These images clearly shows that while the chlorine atom Cl(1) is localised on its 6c site (a), bromine is split over two distinct positions:Br(1) displays a triangular shape around a 6c site (b), occupying a 18h site at 25%, and Br(2) is split on a 6c position occupied at 25 % (c). Note that the occupations have been constrainedtogether to fulfill the chemical composition Ba 2 Co 4 BrO 7 . In summary, atoms in the area of the (Ba 2 Br 2 ) 2+ double layers are affected by a structural disorder, while the other atoms are well ordered. This aspect has already been pointed out for other compounds containing bromine atoms[START_REF] Kauffmann | [END_REF].

Figure 1 : 7 Figure 2 :

 172 Figure 1: View along [010] of (a) Ba 2 Co 4 ClO 7 and (b) Ba 2 Co 4 BrO 7

Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 345678 Figure 3: Schematic representations of (a) central block of edge-sharing octahedra (Oc 4 ) of

  and CdI 2 -type oxide layers. These CoO 2 layers (of CdI 2 -type) are also present in the misfit layered cobaltites series [Tl 1-x Sr 2+x-y Co y O 3 ][CoO 2 ] z and seem to be responsible of the very interesting high thermopower or low resistivity properties of these materials[10].In the present paper, we report on the synthesis and structural characterization of two new isostructural oxyhalide cobaltites, Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 , which adopt hexagonal 18R structures (18R meaning that 18 layers are necessary to describe the unit cell in space group R 3 m). These structures can be described by the stacking along [001] of fluorite-type double layers (Ba 2 Cl 2 ) 2+ or (Ba 2 Br 2 ) 2+ and blocks (Ba 2 Co 8 O 14 ) 2-built up from CdI 2 -type oxide layers, which make them potential candidates for thermoelectric materials.

	quantitative analysis realized on several points of the crystals confirmed the formula of each
	compounds.
	2.2 Crystal structure determination
	For the structure determination, single crystals of Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 were
	selected, mounted on a glass fibre and aligned on a Bruker X8 APEX2 diffractometer.
	Intensities were collected at room temperature using MoKα radiation (λ=0.71073 Å) selected
	by a graphite monochromator. The ω-scan angle and the Dx parameter (distance between the
	single crystal and the CCD detector) were fixed, respectively, to 0.3°/frame and 54 mm for
	Ba 2 Co 4 ClO 7 and to 0.5°/frame and 40 mm for Ba 2 Co 4 BrO 7 , depending on the crystal quality
	2. Experimental and spot shapes. For both compounds, acquisition time was 20 seconds per frame. The
	2.1 Synthesis
	Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 single crystals were grown using a flux technique. A
	1:1:10 molar mixture of BaCO 3 , Co 3 O 4 and BaCl 2 ,2H 2 O or BaBr 2 ,2H 2 O was well ground in
	an agate mortar and heated in air at 1100°C for 48 hours in an alumina crucible. The mixture
	was then slowly cooled at 30°C/h to room temperature. After dissolving the excess of BaCl 2
	or BaBr 2 with hot water, black hexagonal plate-like crystals of approximate size 0.1 -2 mm
	were isolated from the reaction product.
	Well-developed single crystals were analyzed by energy-dispersive X-ray
	spectroscopy on a JEOL JSM-5300 scanning microscope equipped with an IMIX system of
	Princeton Gamma Technology. Measurements revealed the presence of barium, cobalt,
	oxygen and chlorine or bromine for Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 respectively. Semi-

Sun et al. have reported the existence of a new barium cobaltite series Ba n+1 Co n O 3n+3 (Co 8 O 8 ), in which Ba 2 Co 9 O 14 and Ba 3 Co 10 O 17 (n=1 and n=2 terms, respectively) where isolated

[9]

. These structures have been described as intergrowth of perovskite layers diffracted intensities were collected up to 2θ=62.46° with 2763 reflections and a redundancy of 4.86 in the space group R 3 m for Ba 2 Co 4 ClO 7 and 2θ=56.88° with 3512 reflections and a redundancy of 7.47 in the space group R 3 m for Ba 2 Co 4 BrO 7 . After data collection, the intensities were integrated and corrected for Lorentz, polarisation and background effects using the Saint 7.12 software

[START_REF]Bruker Analytical X-ray system[END_REF]

. The SADABS 2006/1 program

[START_REF] Scheldrick | SADABS, Bruker-Siemens Area Detector Absorption and Other Correction[END_REF]

was used to correct absorption effects using a semi-empirical method based on redundancy. Details of the data collection and refinements are given in Table

1

. The hexagonal unit-cell parameters were refined to a=5.716(2) Å, c=45.01(3) Å and a=5.7434(5) Å, c=46.151(9) Å for Ba 2 Co 4 ClO 7

Table 5 .

 5 for Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 , Views of the structures along (010) are represented Figure 1a and 1b. The two compounds are isostructural and can be described as perovskite-related materials with a 18layers stacking along the c-axis of [O 4 ], [BaO 3 ] and [BaX] (X=Cl or Br) layers. In the stacking sequence, two [O 4 ] layers are sandwiched by two [BaO 3 ] layers, leading to the creation of octahedral and tetrahedral sites where the cobalt atoms are localized. These blocks are separated by double fluorite-type (Ba 2 X 2 ) 2+ layers. Thus, the structure can be considered as the intergrowth of two different types of blocks represented by a (cccch'h') 3 stacking sequence with the notation of Katz and Ward [18] (where h' denotes a lacunar [BaX] layer): one hexagonal close-packed block (Ba 2 Co 8 O 14 ) 2-and one fluorite-type double layers (Ba 2 Cl 2 ) 2+ or (Ba 2 Br 2 ) 2+ . (Ba 2 Co 8 O 14 ) 2-blocks have recently been evidenced in Ba 2 Co 9 O 14 and Ba 3 Co 10 O 17 [9, 19]. These compounds have been described as intergrowth structures containing CdI 2 -type layers (formed from two close-packed oxygen layers) and perovskite-type layers (built from [BaO 3 ] layers) with either a single-octahedral layer (Ba 2 Co 9 O 14 ) or a double-octahedral layer (Ba 3 Co 10 O 17 ). The CdI 2 -type and the perovskite-type layers are connected together through a so-called "interface layer" containing octahedral and tetrahedral sites.Figure 2 presents the similarities between the structures of Ba 2 Co 4 XO 7 (X=Cl or Br) and those obtained for Ba 2 Co 9 O 14 and Ba 3 Co 10 O 17 viewed along [010]. It is thus possible to make an analogy with the compounds described by Sun et al.: in the oxyhalides, the (Ba 2 Co 8 O 14 ) 2-blocks are composed by a CdI 2 -type block and the "interface layer" of the Ba 2 Co 9 O 14 and Ba 3 Co 10 O 17 compounds. The main difference between these two classes of materials lies in the connection between the blocks: while in Ba 2 Co 9 O 14 and Ba 3 Co 10 O 17 , the (Ba 2 Co 8 O 14 ) 2-blocks are linked together by one or two octahedra respectively (in white on Figure 2), in Ba 2 Co 4 XO 7 they are clearly disconnected the ones from the others by double (Ba 2 Cl 2 ) 2+ or (Ba 2 Br 2 ) 2+ layers. One could thus imagine the formation of Ba 2 Co 9 O 14 by removing the (Ba 2 Cl 2 ) 2+ or (Ba 2 Br 2 ) 2+ layers of Ba 2 Co 4 XO 7 (X=Cl or Br) and connecting the "interface layers" via an octahedral CoO 6 slice. Such types of relations between oxides and oxyhalides have been recently

respectively. Selected bond lengths and bond valence sums (BVS) are reported in Table

4 and

  The bond valence sums calculation for octahedral Co(3) suggest a Co 3+ valence state. The Co-O bond distance of 1.946(8) Å for both Ba 2 Co 4 ClO 7 and Ba 2 Co 4 BrO 7 materials is similar to the Co(1) 3+ -O bond length of the Kagomé layer, confirming the suggested Co(3) 3+ valence state. Finally, the BVS calculation for Co(4) indicates a slightly over-bonded character. However, as commonly observed is this kind of materials, one should note the possibility of oxygen vacancies, thus leading to a possible Co(4) 2+ tetrahedral cobalt.

Acknowledgements:

M. K. is indebted to "CNRS" and "Région Nord Pas-de-Calais" for financial support.

The "Fonds Européen de Développement Régional (FEDER)", "CNRS", "Région Nord Pasde-Calais" and « Ministère de l'Education Nationale, de l'Enseignement Supérieur et de la Recherche" are acknowledged for fundings of X-ray diffractometers. The authors want to thank Dr Olivier Mentré for fruitful discussion.


Claire Faget

Patrice Taourel

Jonathan Charbit

Alban Ruyer

Chakib Alili

Nicolas Molinari

Ingrid Millet
email: i-millet@chu-montpellier.fr

Value of CT to predict surgically important bowel and/or mesenteric injury in blunt trauma: performance of a preliminary scoring system

Keywords: MDCT, Bowel injury, Mesenteric injury, Blunt trauma, Scoring system

Introduction

Blunt bowel and mesenteric injuries (BBMI) occur in about 5 % of blunt abdominal trauma patients [START_REF] Watts | Incidence of hollow viscus injury in blunt trauma: an analysis from 275,557 trauma admissions from the East multi-institutional trial[END_REF]. Accurate and timely diagnosis is required because delays in diagnosis as short as 8-12 h after injury may increase the morbidity and mortality [START_REF] Killeen | Helical computed tomography of bowel and mesenteric injuries[END_REF][START_REF] Scaglione | Blunt trauma to the gastrointestinal tract and mesentery: is there a role for helical CT in the decision-making process?[END_REF]. Identifying BBMI remains a challenge for trauma care providers as physical examination may be unrealistic in patients with multiple injuries, and it often takes hours before clinically apparent peritonitis signs and symptoms appear.

Multidetector computed tomography (MDCT) is the current accepted standard imaging modality for abdominal trauma [START_REF] Frellesen | Topogram-based automated selection of the tube potential and current in thoraco-abdominal trauma CT -a comparison to fixed kV with mAs modulation alone[END_REF]. It is highly reliable for intra-abdominal solid organ injury diagnosis and is now considered accurate in the diagnosis of bowel and mesenteric injuries complicating blunt abdominal trauma [START_REF] Brofman | Evaluation of bowel and mesenteric blunt trauma with multidetector CT[END_REF]. Many CT findings of BBMI have been reported, some of which are specific, but most are only suggestive, and the exact diagnostic performance of CT remains debatable, with experienced readers yielding more accurate diagnoses [START_REF] Killeen | Helical computed tomography of bowel and mesenteric injuries[END_REF][START_REF] Janzen | Diagnostic accuracy of helical CT for detection of blunt bowel and mesenteric injuries[END_REF][START_REF] Malhotra | Blunt bowel and mesenteric injuries: the role of screening computed tomography[END_REF][START_REF] Mirvis | Rupture of the bowel after blunt abdominal trauma: diagnosis with CT[END_REF][START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF][START_REF] Ekeh | Diagnosis of blunt intestinal and mesenteric injury in the era of multidetector CT technology-are results better?[END_REF][START_REF] Elton | Accuracy of computed tomography in the detection of blunt bowel and mesenteric injuries[END_REF][START_REF] Matsushima | Blunt hollow viscus and mesenteric injury: still underrecognized[END_REF]. Moreover, some BBMI, e.g. superficial bowel tear or mesenteric haematoma without bleeding, could sometimes be successfully managed with observational therapy. There is still controversy as to how reliably CT alone could help identify those BBMI requiring surgery. Lastly, some authors have underlined the high negative predictive value of CT for BBMI [START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF][START_REF] Petrosoniak | Detection of significant bowel and mesenteric injuries in blunt abdominal trauma with 64-slice computed tomography[END_REF][START_REF] Tan | Computed tomography has an important role in hollow viscus and mesenteric injuries after blunt abdominal trauma[END_REF], but the real predictive value of CT is unknown as, given the low incidence of BBMI, the studies were designed as case-control studies [START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF], or as surgical cohorts of patients treated by laparotomy for trauma [START_REF] Ekeh | Diagnosis of blunt intestinal and mesenteric injury in the era of multidetector CT technology-are results better?[END_REF][START_REF] Petrosoniak | Detection of significant bowel and mesenteric injuries in blunt abdominal trauma with 64-slice computed tomography[END_REF][START_REF] Tan | Computed tomography has an important role in hollow viscus and mesenteric injuries after blunt abdominal trauma[END_REF][START_REF] Wu | Contrast-enhanced multiphasic computed tomography for identifying life-threatening mesenteric hemorrhage and transmural bowel injuries[END_REF] but not in consecutive patients with abdominal trauma.

Our study was thus designed to assess the predictive value of CT signs for diagnosis of BBMI requiring surgery (Bsurgical BBMI^) in consecutive patients with abdominal blunt trauma with the aim of developing and evaluating the performance of a CT diagnostic score for therapy planning.

Materials and methods

Study population

Institutional research ethics review board approval was obtained for this retrospective observational study and informed consent was waived.

The trauma database of the intensive care unit of our level 1 regional trauma centre was retrospectively reviewed to select all patients with an abbreviated injury score (AIS) for the abdomen-pelvis area of ≥1 between April 2004 and December 2011. Among 805 patients who fulfilled this criterion, 249 were excluded for reasons detailed in Fig. 1. Finally, 556 consecutive patients were included, 409 male (73.6 %) and 147 female (26.4 %), with a median age of 29 years (interquartile range, IQR, 21-46 years). The median injury severity score (ISS) was 14 (IQR 9-22).

CT technique

The CT studies were performed in our institution within 2 h of admission to the trauma centre (n=521, 93.7 %), or in another institution before the patient was transferred to our trauma centre unit (n=35, 6.3 %). All but 35 CTs were performed using a LightSpeed VCT 16-or 64-detector row scanner (GE Healthcare, Milwaukee, Wis), at 120 kVp, and the amperage setting ranged from 130 to 700 mA, according to the body habitus. The CT images were reconstructed at 3-mm section thickness in the axial, coronal and sagittal plane with native images available for interpretation. Intravenous contrast material (iohexol [Omnipaque 300], GE Healthcare; or iobitridol [Xénétix 350], Guerbet) was administrated at a rate of 3-4 mL/s via a power injector. The CT protocol at our institution included multi-phase acquisition with an arterial phase initiated using an automatic bolus-tracking program and a venous phase at a 70-80 s delay. A delayed sequence (3-5 min) was performed at the emergency radiologist's discretion. For the 35 CTs that had been performed outside the institution, the arterial phase was only available in 8 patients (22.9 %), but venous and delayed phases were available in all cases. CT was performed without oral contrast in all cases.

CT analysis

All 556 CT examinations were retrospectively and independently reviewed by two radiologists (C.F., A.R.) who had respectively 6 and 4 years of experience in abdominal trauma imaging. Reviewers were blinded to the original radiology report, surgical findings and final outcome, but they were aware that the research study involved diagnosis of BBMI at CT.

The CT images were screened for a number of signs, as reported in Table 1. Haemoperitoneum was defined by highattenuation peritoneal fluid (35-60 HU) and was evaluated according to its amount, as proposed by Federle and Jeffrey [START_REF] Federle | Hemoperitoneum studied by computed tomography[END_REF]. Bowel enhancement was assessed subjectively by visual comparison and was classified as abnormal if it was decreased compared to the enhanced adjacent bowel loops. Bowel walls were considered thick if they were greater than 3 mm for the small bowel and 5 mm for the colon [START_REF] Brofman | Evaluation of bowel and mesenteric blunt trauma with multidetector CT[END_REF][START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF][START_REF] Wisner | Blunt intestinal injury. Keys to diagnosis and management[END_REF]. Bowel wall thickening was considered as focal if it was less than 10 cm in length, and non-focal if it was longer than 10 cm. Diffuse bowel wall thickening with findings of hypovolaemic shock was not considered. Active arterial mesenteric vessel extravasation was considered if the extravasation appeared in the arterial phase and expanded at the later phases. When CT had not been performed with an arterial phase, it was considered when extravasation increased considerably (at least doubled) between the venous and delayed phases with coexistence of high and low density areas in the haemorrhage region, suggesting a high rate of active bleeding, as reported by Murakami et al. [START_REF] Murakami | Active extravasation of the abdomen and pelvis in trauma using 64MDCT[END_REF]. Sentinel clot sign was defined by focal high-density collection having an average CT density greater than 50 HU [START_REF] Shanmuganathan | Value of contrastenhanced CT in detecting active hemorrhage in patients with blunt abdominal or pelvic trauma[END_REF]. Pneumoperitoneum was defined as Bmesenteric^if the extra luminal air was seen only trapped in the mesentery, and as Bfree^if it was extended to the anterior part of the abdominal cavity, under the anterior parietal peritoneal layer. Mesenteric stranding was defined by a streaky soft-tissue infiltration of normal mesenteric fat. Beaded appearance of mesenteric vessels was defined as an irregular contour, and abrupt termination indicated by a lack of continuity or tapering of the artery or vein [START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF].

Injuries to solid visceral organs, bladder, large abdominal vessels, spine and pelvic bones were documented. Anterior abdominal wall injury was noted if there was stranding of subcutaneous adipose tissue or if there was a rupture in the abdominal muscular wall or an anterior muscular haematoma.

Reference standard

Two of the authors (I.M. and J.C.) reviewed the trauma database information, official surgical notes and discharge summaries of all patients. Patient demographics, injury severity score and duration of hospitalization were obtained from the trauma registry.

The final diagnosis was established based on the surgical findings and pathology reports, which served as the reference standard for surgical BBMI. As proposed by Atri et al. [START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF], a surgical bowel injury was defined as a full-thickness perforation or seromuscular tear or devascularized bowel. A surgical mesenteric injury was defined as involving active mesenteric bleeding or mesenteric injury resulting in an ischemic bowel loop. Surgical BBMI required therapeutic laparotomy. Serosal tears of the bowel, bowel wall haematomas without tear, and mesenteric haematoma in the absence of active bleeding did not require surgical intervention and were thus considered as negative for surgical BBMI.

All patients referred to our trauma centre were admitted for at least 24 h of observation (more than 3 days for 95 % of the total population). All patients were followed up for 2 months after discharge. Patients treated with non-operative management and discharged alive were thus considered as true negative for BBMI requiring surgery, as none of them returned for an occult BBMI.

Statistical analysis

Interobserver agreement for CT findings was determined with the κ statistic and classified as follows: κ=0-0.2, slight agreement; κ=0.21-0.4, fair agreement; κ=0.41-0.6, moderate agreement; κ=0.61-0.8, substantial agreement; and κ=0.81-1, almost perfect agreement.

Disagreements between the two reviewers were resolved by consensus. Consensual data were then used for final statistics.

We compared epidemiological data and CT signs between patients with surgical BBMI and those without surgical BBMI. Pearson Chi square or Fisher tests were used for All CT findings with a univariate p value≤0.1 were entered into a multivariate logistic regression model to gauge their independent association with surgical BBMI. A stepwise procedure was used to select the final model. To establish a score, a rounded up numerical value was assigned to each of the significant variables included in the final prediction model, in relation to their β parameter (logistic regression estimates). A receiver operating characteristic (ROC) curve was plotted to estimate the best cut-off of this model and its diagnostic performance. We used the bootstrap method to internally validate the score by sampling with replacement for 1000 iterations.

Statistical significance for all tests was set at p<0.05. Computer software (SAS, version 9.3, SAS Institute Cary, NC and R, version 3.0.2, R Foundation for Statistical Computing) was used to perform the statistical analyses.

Results

Study population

There was no significant difference for age or sex ratio between patients with or without surgical BBMI [26.5 years (IQR 19-47.5) and 29 years (IQR 21-46), p=0.36; H/F 3.3 and 2.7, p=0.56, respectively]. The hospitalisation duration was significantly longer in the surgical BBMI group (median 21.5 days, IQR 11-32) vs. no surgical BBMI (median 14 days, IQR 8-25) with p=0.005. The ISS was significantly higher in patients with surgical BBMI (18.5 vs. 14, p=0.0036). There were seven deaths, two of which involved surgical BBMI (one devascularized bowel and one seromuscular tear).

Reference standard

A total of 103 patients (18.5 %) underwent laparotomy, including 56 patients (54.4 %) with surgical BBMI. Eightyseven patients (82.1 %) were operated on within 24 h from the time of CT, including 50 patients (57.5 %, 50/87) with BBMI requiring surgery, whereas 16 patients were operated on 24 h after CT, including 6 patients (37.5 %, 6/16) with BBMI requiring surgery.

Surgical findings for the 56 patients with BBMI requiring surgery were distributed as follows: 3 had mesenteric lesions which corresponded to active arterial bleeding, 24 bowel injuries including 17 bowel perforation, 6 seromuscular tears and 1 ischemic bowel wall; 29 had associated bowel and mesenteric injuries including 14 mesenteric bleeding, 20 bowel ischemia, 9 seromuscular tears and 10 bowel perforation. Among these 56 patients, 19 needed multiple surgical repairs and 37 single repairs, and treatments were as follows: 33 bowel resections (21 for ischemic bowel and 12 for multiples perforations), 30 bowel sutures and 17 surgical haemostasis for active bleeding. There was no BBMI requiring surgery in the 47 others patients who underwent laparotomy, and surgical treatments were as follows: 27 splenectomies, 5 nephrectomies, 5 hepatic repairs (suture), 8 hepatic packing and 2 diaphragmatic sutures.

The remaining patients, after excluding the five deaths (n= 448, 80.6 %), were discharged alive without abdominal surgical procedure during the hospitalization or surgery during follow-up.

CT univariate analysis

Tables 2 and3 show results of univariate analysis, and diagnostic performance of each CT finding. All CT findings evaluated were significantly associated with an increased probability of surgical BBMI, except for length of bowel wall thickening (p=0.22).

Bowel wall discontinuity was noted in 20 patients; there was only one false positive discontinuity in the colic bowel in a patient who had undergone laparotomy with no abdominal lesion and who died just after surgery because of nonabdominal haemorrhagic shock.

A free pneumoperitoneum was lacking in 12/27 (44.4 %) patients with a perforated bowel, whereas a mesenteric pneumoperitoneum was lacking in 19/27 (70.4 %). In 8/27 (29.6 %) patients with small bowel perforation, there was neither free nor mesenteric pneumoperitoneum.

There were six cases of false positives for arterial mesenteric extravasation. Among them, four underwent laparotomy without active mesenteric bleeding mentioned in the surgical report or without any haemostasis procedure, but two of them died after the surgical procedure due to haemorrhagic shock. The last two had active bleeding noted in the meso-sigmoid and the other in the small omentum, and both were conservatively treated favourably with spontaneous regression of bleeding.

Among the associated intra-abdominal lesions, only trauma of the spleen and of the anterior abdominal wall were significantly associated with surgical BBMI (p = 0.0077 and <0.0001, respectively).

Multivariate analysis

Nine CT signs were independently associated with surgical BBMI, as listed in Table 4.

The ROC curve derived from our scoring system (Fig. 2) showed an AUC of 0.98 (0.96-1) with a cut-off≥5 for the best discriminative diagnostic performance. Using a cut-off≥5, our CT score had 96.4 % sensitivity, 91.5 % specificity, 56.2 % positive predictive value (PPV), 99.6 % negative predictive value (NPV), 11.4 positive likelihood ratio (PLR) and 0.04 negative likelihood ratio (NLR) in our cohort.

The internal bootstrap validation gave the following 95 % confidence intervals (CI) for the performance of our scoring system: sensitivity,91.7-100 %; specificity, 85.7-97.6 %; PPV, 41.4-82.3 %; NPV, 98.9-100 %; and AUC, 97.5-99.3 %.

Reproducibility

Interobserver agreement is reported in Table 3. Interobserver agreement was substantial to almost perfect for 6/7 direct CT signs of BBMI which belonged to our CT score, and moderate for the bowel wall thickening sign (κ=0.60). It was almost perfect for both detection of anterior abdominal wall injury (κ=0.97, CI 0.94-1) and splenic injury (κ=1).

Discussion

We developed a reliable scoring system which can be easily implemented and could allow surgeons to optimize decisions to operate on multi-trauma patients. To our knowledge, this is the largest reported cohort of BBMI requiring surgery with CT in blunt trauma (n=56), which compares favourably with the findings of the studies by Atri et al. [START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF], Steenburg et al. [20] and Wu et al. [START_REF] Wu | Contrast-enhanced multiphasic computed tomography for identifying life-threatening mesenteric hemorrhage and transmural bowel injuries[END_REF], including 38, 15 and 13 patients with surgical BBMI, respectively. Moreover, this is the largest consecutive cohort of abdominal trauma cases with CT that has been entirely screened to assess the predictive value of CT findings for surgical BBMI.

Risk scoring systems are widely used in trauma centres and ideally have the triple advantage of predicting patient outcome, facilitating decision making and enabling comparisons when benchmarking the performance of clinical units. They . This is of paramount importance because the most common missed injuries in the trauma imaging era are in the bowel region [START_REF] Matsushima | Blunt hollow viscus and mesenteric injury: still underrecognized[END_REF], and misdiagnosis of surgical BBMI could delay appropriate management and often results in significant morbidity and mortality. While maintaining this high NPV level, we obtained a PPV ranging from 41.4 to 82.3 % as estimated by the bootstrap calculation, thus limiting the number of non-therapeutic laparotomies.

In fact, two CT signs (mesenteric pneumoperitoneum and bowel wall discontinuity) were sufficient to confirm the presence of surgical BBMI, with a score equal to 5 (Figs. 3 and4). Mesenteric pneumoperitoneum was highly associated with surgical BBMI (OR 140.5, 95 % CI 9.3->999.9). Although free pneumoperitoneum also has very good reliability for the diagnosis of traumatic perforation, it does not appear significant in our multivariate analysis, as there was a statistical relationship between free and mesenteric pneumoperitoneum.

There are classical causes of free pneumoperitoneum which are not due to bowel wall trauma, such as rupture of bladder with Foley catheter in place or diffusion of a pneumothorax in patients with diaphragmatic rupture [START_REF] Brofman | Evaluation of bowel and mesenteric blunt trauma with multidetector CT[END_REF]. Furthermore, in traumatic bowel perforation, mesenteric pneumoperitoneum may be useful for localizing the damaged intestinal tract occurring more commonly in the small bowel [START_REF] Kim | Traumatic bowel perforation: analysis of CT findings according to the perforation site and the elapsed time since accident[END_REF]. In contrast, mesenteric air is more commonly missing than free pneumoperitoneum in bowel perforation (these two findings were lacking in 29.6 % of bowel perforations in our study), as already reported in the literature [START_REF] Mirvis | Rupture of the bowel after blunt abdominal trauma: diagnosis with CT[END_REF][START_REF] Kim | Traumatic bowel perforation: analysis of CT findings according to the perforation site and the elapsed time since accident[END_REF]. Visualization of a bowel wall discontinuity had the best positive predictive values of surgical BBMI (PPV=95 %) Fig. 2 ROC curve built from our scoring system (a), and ROC curve derived from the bootstrap analysis (b) Fig. 3 Axial abdominal contrast material-enhanced CT image acquired at arterial (a), venous (b, d) and delayed (c) phases, in a 35year-old man after motor vehicle crash. Note the active mesenteric vessel extravasation appearing in the arterial phase and expanded at the later phases (arrow), and the bowel wall discontinuity in an ileal loop (arrowhead) with free pneumoperitoneum (asterisk). Findings at laparotomy included an ileal perforation requiring suture, and an inferior mesenteric artery rupture requiring haemostasis and left hemicolectomy for ischemia among all CT signs studied, as already reported [20]. This is a direct CT sign of a perforated loop.

Arterial mesenteric extravasation had an odds ratio of 16.8 (95 % CI 3.7-75.7). The logistic regression analysis assigned a value of 3, which is inferior to the cut-off of 5 determined by our ROC curve analysis. This could be surprising since it is well established that active mesenteric vessel extravasation is a finding of BBMI requiring surgical exploration both for stopping the bleeding and for investigating the bowel because of the risk of ischemia. However mesenteric vessel extravasation is always associated with at least mesenteric stranding if not an abundant haemoperitoneum. Hence, the combination of both signs results in a score of at least 5, which is predictive of surgical BBMI.

Mesenteric haematoma and mesenteric stranding had very limited reliability for diagnosing surgical BBMI. These two findings had a positive predictive value for the diagnosis of surgical BBMI inferior to 50 %. This confirms that although a mesenteric finding is suggestive and mesenteric haematoma is specific of mesenteric injury, these signs do not always indicate a need for surgery [START_REF] Brofman | Evaluation of bowel and mesenteric blunt trauma with multidetector CT[END_REF].

Although we did not consider diffuse thickening of the bowel with other CT findings of shock as a finding of bowel trauma, bowel wall thickness had a positive predictive value for surgical BBMI inferior to 50 % and was assigned a score of 2, insufficient in itself to predict a surgical BBMI. Different criteria have been used in the literature to diagnose small bowel or colon wall thickening [START_REF] Macari | CT of bowel wall thickening: significance and pitfalls of interpretation[END_REF]. By using the same criteria to define bowel wall thickening as used in our study, Atri et al. [START_REF] Atri | Surgically important bowel and/or mesenteric injury in blunt trauma: accuracy of multidetector CT for evaluation[END_REF] have already shown that bowel wall thickening may be present despite the lack of important bowel injury requiring surgery. Numerous causes may explain bowel thickening in patients without surgical BBMI: non-significant bowel injuries such as a haematoma or a tear limited to the serosa [START_REF] Brofman | Evaluation of bowel and mesenteric blunt trauma with multidetector CT[END_REF], collapsed bowel lumen, or normal variants [START_REF] Macari | CT of bowel wall thickening: significance and pitfalls of interpretation[END_REF][START_REF] Wittenberg | Algorithmic approach to CT diagnosis of the abnormal bowel wall[END_REF].

Areas of decreased or absent bowel wall contrast enhancement, which is classically indicative of ischemic bowel wall in abdominal emergencies [START_REF] Geffroy | Increased unenhanced bowel-wall attenuation at multidetector CT is highly specific of ischemia complicating small-bowel obstruction[END_REF], were significantly associated with surgical BBMI (OR 4.4, 95 % CI 1.2-17). However, there were 11 false positives (33.3 %) in our study. This could be explained by the difficulty of analysing bowel wall enhancement in multiple trauma patients with heart failure, abundant haemoperitoneum and sometimes hypoperfusion complex [START_REF] Ames | CT hypotension complex (shock bowel) is not always due to traumatic hypovolemic shock[END_REF], because in such cases mucosal bowel enhancement is often intense and diffuse, which makes comparisons tricky with the seemingly less enhanced normal adjacent enhanced bowel.

We note an association between surgical BBMI and abdominal wall injuries. This highlights one of the well-known mechanisms of BBMI due to direct impact [START_REF] Hughes | The pathophysiology and management of bowel and mesenteric injuries due to blunt trauma[END_REF] and anterior compression of the abdomen classically caused by a fastened seatbelt, whereby the bowel loop may be crushed between the seatbelt and the lumbar spine, leading to bowel perforation. We think that abdominal anterior injury visualized in CT examination is helpful for predicting surgical BBMI, when combined with other CT score findings. By contrast, we report an inverse association between spleen lesion and surgical BBMI. Although the weight of this inverse association is weak (assigned score of -1), we think that in patients with an abundant haemoperitoneum (score of 3), the presence of a spleen lesion decreases the risk of surgical BBMI, probably because haemoperitoneum is explained by splenic injury and not by a mesenteric injury.

Our study had some limitations. First, it was a retrospective study with its inherent bias and was conducted in a single centre, but it included consecutive patients with blunt abdominal trauma investigated by CT. Secondly, there was a higher prevalence of surgical BBMI in our study (10.1 %) than reported in previous studies, i.e. twofold lower [START_REF] Malhotra | Blunt bowel and mesenteric injuries: the role of screening computed tomography[END_REF][START_REF] Wisner | Blunt intestinal injury. Keys to diagnosis and management[END_REF]. This was due to patient selection bias, as patients had more serious injuries in this study (average ISS 14) and were selected on the basis of an abdominal trauma that did not reflect the entire trauma population. The prevalence in our study is actually close to that documented in other cohorts of blunt abdominal trauma patients, with a reported prevalence of surgical BBMI of 12.3 % in the study by Wu et al. [START_REF] Wu | Contrast-enhanced multiphasic computed tomography for identifying life-threatening mesenteric hemorrhage and transmural bowel injuries[END_REF], and 15.3 % in the study by Tan et al. [START_REF] Tan | Computed tomography has an important role in hollow viscus and mesenteric injuries after blunt abdominal trauma[END_REF]. Interestingly, in our study, surgical BBMI constituted the majority of blunt abdominal trauma patients explored by laparotomy, and currently represents the main reason for laparotomy [START_REF] Sorensen | Is delayed laparotomy for blunt abdominal trauma a valid quality improvement measure in the era of nonoperative management of abdominal injuries?[END_REF]. Thirdly, although the internal validation of our score indicated high diagnostic performance, an external validation set is essential to confirm the diagnostic value before generalization. In conclusion, we developed an easy to implement and reliable scoring system based on CT findings, which is highly predictive of surgical BBMI. When the CT score is at least 5, then a surgical procedure should be carried out given the high probability of BBMI requiring surgical repair. Conversely, when the CT score is less than 5, then non-operative management should be prescribed along with clinical follow-up and possibly repetition of abdominal CT examination within 8-12 h.

Fig. 1

 1 Fig.1 Study flow chart

Fig. 4

 4 Fig.4 Axial abdominal contrast-enhanced CT image acquired at venous phase in a 59-year-old woman after motor vehicle crash. Note the mesenteric pneumoperitoneum (arrowhead) close to a focal small bowel wall thickening (arrow) and with subcutaneous fat stranding along the course of the fastened seat belt (asterisk). Findings at laparotomy included a 2mm jejunal perforation and a transverse colic perforation, all requiring suture

Table 1

 1 CT signs evaluated

	Mesenteric sign	Bowel sign
	Haemoperitoneum	Bowel wall discontinuity
	Mesenteric stranding	Free intra-or retroperitoneal air
	Sentinel clot sign	Bowel wall thickening (site and
	(=mesenteric haematoma)	length)
	Active arterial mesenteric	Decreased bowel wall
	vessel extravasation	enhancement
	Irregular beading of the mesenteric	Mesenteric pneumoperitoneum
	vessel
	Abrupt termination of the
	mesenteric vessel

comparison of categorical variables, and Student t test or Wilcoxon rank-sum test for comparison of continuous variables as appropriate. The diagnostic performance of each CT finding was also calculated.

Table 2

 2 Univariate analysis of each CT finding in the general population and according to the presence of surgical BBMI

	Overall	Surgical BBMI	No surgical BBMI	p value
	N=556	N=56	N=500

a 2 missing data because of no abdominal contrast due to heart failure b Fisher tests, others are chi-square tests must be developed using simple, reliable and reproducible parameters. The CT scoring system we developed fulfils the requirements and goals of trauma scoring systems. It

Table 3

 3 Diagnostic performance and interobserver reproducibility of each CT finding

		Sensitivity	Specificity	PPV	NPV	kappa 95 % CI kappa
	Haemoperitoneum					0.79 0.72-0.87
	1 (small amount) 25 (14-36)	56.2 (52-61) 6 (3-9)	87 (83-91)
	2 (abundant)	69.6 (58-82) 74.6 (70-78) 23.5 (17-31) 95.6 (94-98)
	Free pneumoperitoneum	30.4 (18-42) 99.2 (98-100) 81 (64-98)	92.7 (91-95) 0.89 0.77-1
	Mesenteric pneumoperitoneum	17.9 (8-28) 99.6 (99-100) 83 (68-100) 91.5 (89-94) 0.72 0.53-0.92
	Bowel wall thickness	67.9 (56-80) 91.2 (89-94) 46.3 (36-57) 96.2 (95-98) 0.60 0.48-0.70
	Bowel wall thickening length					0.61 0.50-0.72
	Focal	35.7 (23-48) 94.2 (92-96) 40.8 (27-55) 92.9 (91-95)
	Non-focal	32.1 (20-44) 97 (95-98)	54.5 (36-72) 92.7 (90-95)
	Arterial mesenteric vessel extravasation	26.8 (15-38) 98.8 (98-100) 71.4 (53-90) 92.3 (90-95) 0.90 0.78-1
	Mesenteric haematoma	42.9 (30-56) 94 (91-96)	42.9 (30-56) 93.6 (91-96) 0.75 0.62-0.88
	Mesenteric stranding	58.9 (46-72) 89.6 (87-92) 38.8 (28-49) 95.1 (93-97) 0.64 0.53-0.75
	Abrupt termination of mesenteric vessel	25 (14-36)	99.2 (98-100) 77.8 (52-94) 92.1 (90-94) 0.57 0.28-0.86
	Beaded mesenteric vessel	12 (4-21)	98.8 (98-100) 53.8 (27-81) 90.9 (89-93) 0.59 0.37-0.81
	Decreased bowel wall enhancement	39.3 (26-52) 97.8 (96-99) 66.7 (51-83) 93.5 (91-96) 0.68 0.50-0.85
	Bowel wall discontinuity	33.9 (21-46) 99.8 (99-100) 95 (85-100) 93.1 (91-95) 0.75 0.58-0.91
	Data are percentages and numbers in brackets are 95 % confidence intervals

Table 4

 4 Significant CT signs in multivariate analysis to predict the risk of surgical BBMI and their assigned numerical CT score according to the logistic regression estimates (β parameter)

		CT signs	p value	Odds ratio	95 % CI odds ratio	β estimate	Score
	1	Haemoperitoneum	0.0017
		Small amount		3.3	0.4-23.8	1.1944	1
		Abundant		16.3	2.4-111.4	2.7904	3
	2	Mesenteric pneumoperitoneum	0.0003	140.5	9.3->999.9	4.9456	5
	3	Bowel wall thickness	0.0001	9.8	3.1-31.4	2.2878	2
	4	Arterial mesenteric vessel extravasation	0.0002	16.8	3.7-75.7	2.8225	3
	5	Mesenteric stranding	0.0019	5.2	1.8-14.9	1.6541	2
	6	Reduced bowel wall enhancement	0.0306	4.4	1.2-17.0	1.4856	1
	7	Bowel wall discontinuity	0.0003	128.9	9.5->999.9	4.8593	5
	8	Splenic injury	0.0467	0.3	0.1-0.9	-1.1123	-1
	9	Anterior abdominal wall injury	0.0002	8.3	2.7-25.2	2.1188

Acknowledgements The scientific guarantor of this publication is Prof. Taourel. The authors of this manuscript declare no relationships with any companies whose products or services may be related to the subject matter of the article. The authors state that this work has not received any funding. One of the authors has significant statistical expertise. Institutional review board approval was obtained. Written informed consent was waived by the institutional review board. Methodology: retrospective, diagnostic or prognostic study, performed at one institution.


Isabelle Gruais

Dan Poliševski

Alina S ¸tefan

Two-temperature homogenized eigenfunctions of conduction through domains with jump interfaces

Keywords: interfacial jump, conduction, eigenvalue, two-scale convergence, homogenization MSC 2010: 35B27, 35P15, 49R05, 74A50, 80M40

In this paper we study the asymptotic behavior of the eigenvalue problem solutions of the conduction process in an ε-periodic domain formed by two components separated by a first-order jump interface. We prove that when ε → 0 the limits of the eigenvalues and eigenfunctions of this problem verify a certain (effective) two-temperature eigenvalue problem. Moreover, we show that the effective eigenvalue problem has only eigenvalues which come from the homogenization process.

Introduction

During the last decades there was a steady interest for the homogenization of problems with interfacial thermal barriers (see [START_REF] Auriault | Macroscopic modelling of heat transfer in composites with interfacial thermal barrier[END_REF], [START_REF] Monsurrò | Homogenization of a two-component composite with interfacial thermal barrier[END_REF], [START_REF] Donato | Homogenization of two heat conductors with interfacial contact resistance[END_REF], [START_REF] Bunoiu | Homogenization of a thermal problem with flux jump[END_REF]) or equivalent (see [START_REF] Lipton | Heat conduction in fine scale mixtures with interfacial contact resistance[END_REF], [START_REF] Hummel | Homogenization for heat transfer in polycrystals with interfacial resistances[END_REF], [START_REF] Ene | Model of diffusion in partially fissured media[END_REF], [START_REF] Poliševski | The Regularized Diffusion in Partially Fractured Porous Media[END_REF]). Meanwhile, it was also studied the asymptotic behavior of the eigenvalue problems in ε-periodic domains (see [START_REF] Kesavan | Homogenization of elliptic eigenvalue problems[END_REF]) , even for ε-periodically perforated domains (see [START_REF] Vanninathan | Homogenization of eigenvalue problems in perforated domains[END_REF]), where the key ingredient was the prolongation operator which was introduced by [START_REF] Cioranescu | Homogenization in open sets with holes[END_REF].

Here we continue our works [START_REF] Gruais | Heat transfer models for two-component media with interfacial jump[END_REF], [START_REF] Gruais | Model of two-temperature convective transfer in porous media[END_REF] and [START_REF] Poliševski | Heat transfer with firstorder interfacial jump in a biconnected structure[END_REF] by studying the asymptotic behavior of the eigenvalue problem solutions of the conduction process in an ε-periodic domain formed by two components separated by a first-order jump interface. We prove that when ε → 0 the limits of the eigenvalues and eigenfunctions of this problem verify a certain (effective) two-temperature eigenvalue problem. Moreover, we show that the effective eigenvalue problem has only eigenvalues which come from the homogenization process. Our key ingredient is a pair of prolongation operators, corresponding to each component of the domain. Otherwise, we mainly follow the methods of the two-scale theory (see [START_REF] Allaire | Homogenization and two-scale convergence[END_REF]). We have to remark that the procedure presented here can straightforwardly be generalized to any n-component ε-periodic domain with interfacial jumps of the first order.

The paper is organized as follows: in Section 2 we study the eigenvalue problem in the ε-periodic domain; in Section 3 we present the key prolongation operators, the a priori estimates and the specific compactness results; Section 4 is devoted to the derivation of the effective eigenvalue problem and its connection with the homogenization process.

The eigenvalue problem

Let Ω be an open connected bounded set in R N (N ≥ 3), locally located on one side of the boundary ∂Ω, a Lipschitz manifold composed of a finite number of connected components. For any ε ∈ (0, 1), Ω has two ε-periodically ditributed components. For convenience, the periodicity is described by using the cube Y = (0, 1) N , as follows:

Let

Y a ⊂⊂ Y be a Lipschitz open set such that Y b = Y \ Y a is connected. For any ε ∈ (0, 1) we denote Z ε = {k ∈ Z N : εk + εY ⊆ Ω}. ( 1
)
The two ε-periodic components of Ω are defined by:

Ω εa = int   k∈Zε (εk + εY a )   (2)
Ω εb = Ω \ Ω εa . (3)
Denoting Γ := ∂Y a = ∂Y a ∩ ∂Y b , the interface between Ω εa and Ω εb have the property:

Γ ε := k∈Zε (εk + εΓ) = ∂Ω εa = ∂Ω εa ∩ ∂Ω εb . (4)
Let us remark that Ω εb is connected and all the boundaries are at least locally Lipschitz. Also, the inward normal on ∂Y a , denoted by ν, has the property

ν ε (x) = ν ε -1 x , ∀x ∈ Γ ε , (5)
where ε -1 x is formed by the fractional parts of the components of ε -1 x.

We have to introduce the Hilbert space

H ε = u ∈ L 2 (Ω) : u Ωεa ∈ H 1 (Ω εa ), u Ω εb ∈ H 1 (Ω εb ), u = 0 on ∂Ω (6)
endowed with the scalar product

(u, v) Hε = Ωεa ∇u∇v + Ω εb ∇u∇v + ε Γε [u][v], (7)
where [u] = γ εa u -γ εb u and γ εa u, γ εb u are the traces of u on Γ ε defined in H 1 (Ω εa ) and H 1 (Ω εb ), respectively. Our domain has the following well-known property [START_REF] Ene | Model of diffusion in partially fissured media[END_REF]:

Lemma 2.1. For any v ∈ H ε there exists C > 0, independent of ε, such that |v| L 2 (Ωε b ) ≤ C |∇v| L 2 (Ωε b ) , (8)
ε 1/2 |γ εα v| L 2 (Γε) ≤ C |v| L 2 (Ωε α ) + ε |∇v| L 2 (Ωε α ) , α ∈ {a, b}, (9)
|v| L 2 (Ωε a ) ≤ C ε 1/2 |γ εa v| L 2 (Γε) + ε |∇v| L 2 (Ωε a ) . (10)
Remark 2.1. Taking in account the L 2 -norm of the jump on Γ ε the results of the previous Lemma have important consequences:

ε 1/2 [v] L 2 (Γε) ≤ C v L 2 (Ω) + ε ∇v L 2 (Ωεa) + ε ∇v L 2 (Ω εb ) , (11)
|v| L 2 (Ωεa) ≤ C |v| Hε , ∀v ∈ H ε . (12)
Next, we introduce the data of our problem: the transmission factor h ε (x) = h(x/ε) and the symmetric conductivities

a ε ij (x) = a ij (x/ε) and b ε ij (x) = b ij (x/ε)
, where h, a ij and b ij belong to L ∞ per (Y ) and have the property that there exists δ > 0 such that

h ≥ δ, a.e. on Y, (13)
a ij ξ j ξ i ≥ δξ i ξ i and b ij ξ j ξ i ≥ δξ i ξ i , ∀ξ ∈ R N , a.e. on Y. (14)
We consider the following eigenvalue problem:

Find λ ε ∈ R * such that ∃u ε ∈ H ε \ {0} verifying the equations -div (a ε ∇u ε ) = λ ε u ε , in Ω εa , (15)
-div

(b ε ∇u ε ) = λ ε u ε , in Ω εb , (16)
and the transmission conditions

a ε ij ∂u ε ∂x j ν ε i = b ε ij ∂u ε ∂x j ν ε i = εh ε (γ εa u ε -γ εb u ε ) on Γ ε . (17)
The variational formulation of the problem ( 15)-( 17) is the following:

Find λ ε ∈ R * such that ∃u ε ∈ H ε \ {0} verifying G ε (u ε , v) := Ωεa a ε ij ∂u ε ∂x j ∂v ∂x i + Ω εb b ε ij ∂u ε ∂x j ∂v ∂x i + ε Γε h ε [u ε ][v] = λ ε (u ε , v), ∀v ∈ H ε (18)
where (•, •) denotes the inner product in L 2 (Ω).

Using the procedure of [START_REF] Babuška | Eigenvalue Problems, Handbook of numerical analysis[END_REF], we introduce the operator

T ε ∈ L L 2 (Ω), H ε by denotig with T ε (u), for u ∈ L 2 (Ω), the unique solution of the problem Ωεa a ε ij ∂T ε (u) ∂x j ∂v ∂x i + Ω εb b ε ij ∂T ε (u) ∂x j ∂v ∂x i +ε Γε h ε [T ε (u)][v] = Ω uv, ∀v ∈ H ε . Defining T ε : L 2 (Ω) → L 2 (Ω) by T ε = J ε • T ε , where J ε is the inclusion of H ε into L 2 (Ω), we see that the eigenvalue problem (18) is equivalent to the eigenvalue problem T ε u ε = µ ε u ε , via µ ε = 1 λ ε . Lemma 2.2. The inclusion J ε : H ε → L 2 (Ω) is a compact operator.
Proof. As for any v ∈ H ε we have

J ε v L ε (Ω) = v Ωεa + v Ω εb ≤ C v Hε ,
is sufficient to prove that the bounded sequences from H ε contain a convergent subsequence in

L 2 (Ω). Let {v n } n a bounded sequence in H ε . We note v a n = v n Ωεa and v b n = v n Ω εb
. Since {v a n } n is a bounded sequence in H 1 (Ω εa ), from the Rellich's theorem there exist v a ∈ H 1 (Ω εa ) and a subsequence, still denoted by {n}, such that

v a n → v a strongly in L 2 (Ω εa ). Further, {v b n } n being bounded in H 1 (Ω εb ), again the Rellich's theorem implies the existence of some v b ∈ H 1 (Ω εb ) such that on a sub-subsequence it holds v b n → v b strongly in L 2 (Ω εb ). It follows that v 0 = v a in Ω εa , v b in Ω εb ⇒ v 0 ∈ H ε ⊂ L 2 (Ω).
The proof is completed as

v n -v 0 2 L 2 (Ω) = v a n -v a 2 L 2 (Ωεa) + v b n -v b 2 L 2 (Ω εb ) → 0.
We see now that T ε is a self-adjoint, compact operator in L 2 (Ω) and recalling for instance [START_REF] Brezis | Analyse fonctionnelle. Théorie et applications[END_REF] it follows that there exist {λ ε k } k , eigenvalues of the problem (18), with the property

0 < λ ε 1 ≤ λ ε 2 ≤ ... → ∞
and {u ε k } k , the corresponding eigenfunctions, which are complete and orthonormal in L 2 (Ω).

In the following sections we shall study the behaviour of (λ ε k , u ε k ) when ε → 0.

A priori estimates

We begin this section by proving the boundedness of {λ ε k } ε , the eigenvalues of (18). For every k ∈ N let us denote

H ε,k = {S subspace of H ε , dim S = k}.
Applying the Minimum-maximum principle (see [START_REF] Babuška | Eigenvalue Problems, Handbook of numerical analysis[END_REF]), λ ε k can be estimated via the Rayleigh quotient:

λ ε k = min S∈H ε,k max v∈S G ε (v, v) v 2 L 2 (Ω) ≤ ≤ C min S∈H ε,k max v∈S ∇v 2 L 2 (Ωεa) + ∇v 2 L 2 (Ω εb ) + ε [v] 2 L 2 (Γε) v 2 L 2 (Ω) . (19)
In order to further estimate λ ε k with respect to ε we introduce two prolongation operators.

First, for any v ∈ H 1 (Y a ) let w ∈ H 1 (Y b ) be the only solution of the Dirichlet problem:

-∇w = 0 in Y b , (20)
w = 0 on ∂Y, w = v on Γ. ( 21
)
We have to introduce here P a (v) ∈ H 1 (Y ) by

P a (v) = v in Y a , w in Y b .
It has the property

P a (v) H 1 (Y ) ≤ C v H 1 (Ya) . (22)
Denoting u ε k (y) := u(εk + εy), for any k ∈ Z ε , y ∈ Y a and u ∈ H 1 (Ω εa ), we define our first prolongation operator, P ε a : H 1 (Ω εa ) → H 1 0 (Ω), by

P ε a (u)(x) =    u(x) for x ∈ Ω εa P a (u ε k ) { x ε } for x ∈ (εk + εY b ), k ∈ Z ε 0 for x ∈ Ω \ k∈Zε (εk + εY )
Rescaling (22) we easily obtain

P ε a (u) H 1 (Ω) ≤ C u L 2 (Ωεa) + ε ∇u L 2 (Ωεa) , ∀v ∈ H 1 (Ω εa ). (23) Second, for any v ∈ H 1 (Y b ) let w ∈ H 1 (Y b )

be the only solution of the Dirichlet problem:

-

∇w = 0 in Y a , (24)
w = v on Γ.

(25)

For this component we introduce P b (v) ∈ H 1 (Y ) by

P b (v) = w in Y a , v in Y b ,
which has a property similar to (22):

P b (v) H 1 (Y ) ≤ C v H 1 (Y b ) . (26)
Denoting u ε k (y) := u(εk + εy), for any k ∈ Z ε , y ∈ Y b , u ∈ H 1 (Ω εb ), u = 0 on ∂Ω, we define our second prolongation operator, P ε b : {u ∈ H 1 (Ω εb ), u=0 on ∂Ω} → H 1 0 (Ω), by

P ε b (u)(x) = u(x) for x ∈ Ω εb P b (u ε k ) { x ε } for x ∈ (εk + εY a ), k ∈ Z ε .
Rescaling (26) we get

P ε b (u) H 1 (Ω) ≤ C u L 2 (Ω εb ) + ε ∇u L 2 (Ω εb ) , ∀u ∈ H 1 (Ω εb ), u=0 on ∂Ω. (27)
We can now estimate the terms of (19). The first term gives:

∇v 2 L 2 (Ωεa) v 2 L 2 (Ω) ≤ ∇P ε a (v) 2 L 2 (Ωεa) P ε a (v) 2 L 2 (Ω) • P ε a (v) 2 L 2 (Ω) v 2 L 2 (Ω) < C P ε a (v) 2 L 2 (Ω) v 2 L 2 (Ω)
, the constant being the first eigenvalue of the problem (20)-(21). Using (23) we obtain

∇v 2 L 2 (Ωεa) v 2 L 2 (Ω) < C 1 + ε 2 C 2 Ωεa a ε ∇v∇v v 2 L 2 (Ω) . (28)
The second term gives:

∇v 2 L 2 (Ω εb ) v 2 L 2 (Ω) ≤ ∇P ε b (v) 2 L 2 (Ω εb ) P ε b (v) 2 L 2 (Ω) • P ε b (v) 2 L 2 (Ω) v 2 L 2 (Ω) < C P ε b (v) 2 L 2 (Ω) v 2 L 2 (Ω)
, the constant being the first eigenvalue of the problem (24)-(25). Using (27) we obtain

∇v 2 L 2 (Ω εb ) v 2 L 2 (Ω) < C 1 + ε 2 C 2 Ω εb b ε ∇v∇v v 2 L 2 (Ω) . ( 29
)
The third term can be estimated from [START_REF] Hummel | Homogenization for heat transfer in polycrystals with interfacial resistances[END_REF], that is

[v] 2 L 2 (Γε) v 2 L 2 (Ω) < C 1 + ε 2 C 2 Ωεa a ε ∇v∇v v 2 L 2 (Ω) + ε 2 C 2 Ω εb b ε ∇v∇v v 2 L 2 (Ω) . ( 30
)
Adding the estimates ( 28)-(30) we get

λ ε k ≤ C 1 + ε 2 C 2 λ ε k ,
which obviously implies that λ ε k is bounded for sufficiently small ε. Setting v = u ε k in (18) and using the coerciveness of G ε (•, •) and the orthonormality of {u ε k } k we find

G ε (u ε k , u ε k ) = Ωεa a ε ij ∂u ε k ∂x j ∂u ε k ∂x i dx + Ω εb b ε ij ∂u ε k ∂x j ∂u ε k ∂x i dx+ +ε Γε h ε [u ε k ] 2 ds = λ ε k , ∀u ε ∈ H ε . (31)
As {λ ε k } ε is bounded, we find that

{u ε k } ε is bounded in H ε . ( 32
)
Using the previously obtained result together with inequalities ( 8)- [START_REF] Gruais | Model of two-temperature convective transfer in porous media[END_REF], it follows that there exists C > 0 such that

|∇u ε k | L 2 (Ωεa) ≤ C, |∇u ε k | L 2 (Ω εb ) ≤ C, |[u ε k ]| L 2 (Γε) ≤ C. ( 33
)
Applying to the properties of the two-scale convergence theory [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], a specific compactness result follows.

Theorem 3.1. There exist

λ k ∈ R + , u a k ∈ H 1 (Ω), u b k ∈ H 1 0 (Ω) and η α k ∈ L 2 Ω; H 1 per (Y α ) , α ∈ {a
, b}, such that the following convergences hold on some subsequence

χ ε α u ε k 2s χ α u α k , (34)
χ ε α ∇u ε k 2s χ α (∇ x u α k + ∇ y η α k (•, y)) , (35)
λ ε k → λ k , (36)
where

χ ε α : L 2 (Ω εα ) → L 2 (Ω) and χ α : L 2 (Ω × Y α ) → L 2 (Ω × Y ), α ∈ {a,
b}, denote the straight prolongations with zero; sometimes they can be identified with the corresponding characteristic functions.

The homogenization process

Passing (18) to the limit on the subsequence on which the convergences of Theorem 3.1 hold, we obtain like in [START_REF] Ene | Model of diffusion in partially fissured media[END_REF]:

Lemma 4.1. For any ϕ a ∈ C ∞ ( Ω), ϕ b ∈ D(Ω) and ψ α ∈ D(Ω; C ∞ per (Y α )), α ∈ {a, b}, it holds Ω×Ya a ij ∂u a k ∂x j + ∂η a k ∂y j ∂ϕ a ∂x i + ψ a ∂y i + Ω×Y b b ij ∂u b k ∂x j + ∂η b k ∂y j ∂ϕ b ∂x i + ψ b ∂y i + + h Ω (u a k -u b k )(ϕ a -ϕ b ) = λ k Ω×Y χ a u a k ϕ a + χ b u b k ϕ b ( 37
)
where h is defined by

h = Γ h(y)ds. (38)
Proof. For ϕ α and ψ α , α ∈ b} like in the hypotheses, we set v in (18) as follows:

v(x) = ϕ a (x) + εψ a x, x ε , ϕ b (x) + εψ b x, x ε . (39)
We obtain

λ ε (χ ε a u ε , ϕ a ) + λ ε (χ ε a u ε , ϕ b ) + O(ε) = α∈{a,b} Ωεα α ε ij ∂u ε ∂x j ∂ϕ α ∂x i + ∂ψ α ∂y i + +ε Γε h ε (γ εa u ε -γ εb u ε ) [ϕ a -ϕ b + ε(ψ a -ψ b )] . ( 40
)
The proof is completed by following the same steps as in the proof presented in [START_REF] Ene | Model of diffusion in partially fissured media[END_REF].

In order to present the next results we have also to introduce the Hilbert space

H := H 1 (Ω) × H 1 0 (Ω) × L 2 (Ω, H 1 per (Y a ) × L 2 (Ω, H 1 per (Y b )) , (41)
endowed with the scalar product

(((u a , u b ), (η a , η b )) , ((ϕ a , ϕ b ), (ψ a , ψ b ))) H = α∈{a,b} Ω ∇u α ∇ϕ α + + Ω (u a -u b ) (ϕ a -ϕ b ) + α∈{a,b} Ω×Yα ∇ y η α ∇ y ψ α . (42)
Using density arguments, Lemma 4.1 yields:

Theorem 4.1. λ k ∈ R * + and (u a k , u b k ), (η a , η b ) ∈ H \ {0} verify: Ω×Ya a ij ∂u a k ∂x j + ∂η a k ∂y j ∂ϕ a ∂x i + ψ a ∂y i + Ω×Y b b ij ∂u b k ∂x j + ∂η b k ∂y j ∂ϕ b ∂x i + ψ b ∂y i + + h Ω (u a k -u b k )(ϕ a -ϕ b ) = λ k Ω×Y χ a u a k ϕ a + χ b u b k ϕ b , ∀ ((ϕ a , ϕ b ), (ψ a , ψ b )) ∈ H. ( 43
)
We can present now the main result of this paper.

Theorem 4.2. If (λ ε k , u ε k
) is a solution of (18) then the limits of the convergences (34)-(36), that is

λ k ∈ * + and (u a k , u b k ) ∈ H 1 (Ω) × H 1 0 (Ω) \ {0}
, put together a solution of the following effective eigenvalue problem:

Find (λ k , (u a k , u b k )) ∈ R * + × H 1 (Ω) × H 1 0 (Ω) \ {0} such that G hom ((u a k , u b k ), (ϕ a , ϕ b )) = λ k Ω m u a k ϕ a + (1 -m)u b k ϕ b , ∀ (ϕ a , ϕ b ) ∈ H 1 (Ω) × H 1 0 (Ω), (44)
where m= Y a ,

G hom ((u a k , u b k ), (ϕ a , ϕ b )) = Ω âij ∂u a k ∂x j ∂ϕ a ∂x i + Ω bij ∂u b k ∂x j ∂ϕ b ∂x i + h Ω (u a k -u b k )(ϕ a -ϕ b ), (45)
the effective coefficients αij , α ∈ {a, b}, are defined by

αij = Yα α ij + α ik ∂e a j ∂y k dy, ∀i, j ∈ {1, 2, ..., N } , (46)
and

e α k ∈ H 1 per (Y α ), k ∈ {1, 2, ..., N } , is the unique solution of the local- periodic problem - ∂ ∂y i α ij ∂ (e α k + y k ) ∂y j = 0 in Y α , (47)
α ij ∂ (e α k + y k ) ∂y j ν i = 0 on Γ. ( 48
)
Moreover, there are no eigenvalues of the problem (44) except those obtained as limits of the homogenization process.

Proof. The first assertion follows from Theorem 4.1 using standard homogenization procedures.

Next, suppose that there exists an eigenvalue of the problem (44), µ ∈ R * + , µ = λ k , ∀k ≥ 1, where {λ k } k represents the eigenvalues of the problem (44) obtained as limits of the homogenization process. It follows that there exists k such that λ k < µ < λ k+1 . Let w = (w a , w b ) ∈ H 1 (Ω) × H 1 0 (Ω) \ {0} be an eigenfunction associated to µ.

Let us introduce w ε ∈ H ε , the unique the solution to the problem

G ε (w ε , v) = µ Ωεa w a v + µ Ω εb w b v, ∀v ∈ H ε . (49)
Setting v = w ε in the previous relation we find that

G ε (w ε , w ε ) ≤ C w ε L 2 (Ω) .
Applying the coercivity property of G ε , we find that the sequence {w ε } ε is bounded in H ε . Under these circumstances, there exists w * = (w * a , w * b ) ∈ H 1 (Ω) × H 1 0 (Ω) such that

P ε a w ε w * a in H 1 (( 1

 11 Ω) and P ε b w ε w * b in H 1 0 (Ω).Homogenizing the problem (49) we obtainlim ε→∞ G ε (w ε , v) = G hom (w * , v) = µ(w, v) = G hom (w, v), ∀v ∈ H 1 0 (Ω),from which followsw = w * and Ω (w ε ) 2 → Ω m(w a ) 2 + (1 -m)(w b ) 2 = 0. (50)Next, we define ŵε = w ε -k i=1 Ω w ε u ε i u ε i, where u ε i ∈ H ε is an eigenfunction associated to λ i . We note that Ω ŵε u ε j = 0, ∀j ≤ k, and henceG ε ( ŵε , ŵε ) -χ ε a )P ε b w ε u ε i , ∀i ∈ {1, ..., k}, the passage to the limit yieldsΩ w ε u ε i → Ω mw a u a i + (1 -m)w b u b i = 0,

Acknowledgements. The authors gratefully acknowledge partial support from the International Network GDRI ECO-Math.

which obviously implies

Passing (49) to the limit in we finally get

which is in contradiction with (51), via (52).

Remark 4.1. The formulation of the effective two-temperature eigenvalue problem follows from Theorem (4.2):

and the boundary condition

where n is the outward normal on ∂Ω.


David Pérez-Morales

Olivier Kermorgant
email: olivier.kermorgant@ec-nantes.fr

Salvador Domínguez-Quijada
email: salvador.dominguezquijada@ls2n.fr

Philippe Martinet
email: philippe.martinet@inria.fr

Automatic Perpendicular and Diagonal Unparking Using a Multi-Sensor-Based Control Approach

This paper explores the feasibility of a Multi-Sensor-Based Control (MSBC) approach for addressing forward nonparallel (perpendicular and diagonal) unparking problems of car-like vehicles as an alternative to classical approaches (e.g. path planning based, etc.). The results of individual cases are presented to illustrate the behavior and performance of the proposed approach as well as results from exhaustive simulations to evaluate the convergence and stability. The results presented in this work increase the versatility and validity of our MSBC approach towards a fully autonomous parking system.

I. INTRODUCTION

Even for experienced drivers, parking and unparking can be a difficult tasks, especially in big cities where the parking spots are often narrow. The search for an increase in comfort and safety when parking and unparking has led to a quite extensive literature [START_REF] Wang | Automatic parking of vehicles: A review of literatures[END_REF], having explored many different approaches to automate this bothersome task.

Despite the fact that the automobile industry has already started to roll out some commercial active parking assistants capable of actively controlling acceleration, braking and steering [START_REF] Song | Analysis and Review of State-of-the-Art Automatic Parking Assist System[END_REF], the quantity of systems able to unpark is limited.

Path planning approaches have been heavily investigated in recent years. Even if the focus of most works in the literature is on the parking task, if a path is found it can be used as well for unparking. Among the different planning techniques it is possible to distinguish between geometric approaches, with either constant turning radius [START_REF] Petrov | Path Planning and Steering control for an Automatic Perpendicular Parking Assist System[END_REF], [START_REF] Petrov | Saturated Feedback Control for an Automated Parallel Parking Assist System[END_REF] using saturated feedback controllers, or continuous-curvature planning using clothoids [START_REF] Vorobieva | Geometric Continuous-Curvature Path Planning for Automatic Parallel Parking[END_REF], [START_REF] Yi | Smooth path planning for autonomous parking system[END_REF]; heuristic approaches [START_REF] Chen | Path planning with orientationaware space exploration guided heuristic search for autonomous parking and maneuvering[END_REF] and machine learning techniques [START_REF] Notomista | Maneuver segmentation for autonomous parking based on ensemble learning[END_REF].

A well-known drawback of path planning is that it is necessary to have the knowledge on the free and occupied space of the whole environment beforehand if online replanning is not feasible, potentially leading to costly infrastructure requirements. Furthermore, the tracking performance of a given path is highly dependent on the localization performance which might get degraded on certain environments (e.g. underground parking lots without any special infrastructure) or after a few maneuvers leading to non-negligible differences between the planned path and the performed one [START_REF] Vorobieva | Geometric Continuous-Curvature Path Planning for Automatic Parallel Parking[END_REF], [START_REF] Yi | Smooth path planning for autonomous parking system[END_REF].

An interesting alternative is the use of a sensor-based control approach. It has been proven to be valid for navigation [START_REF] De Lima | Sensor-Based Control with Digital Maps Association for Global Navigation: A Real Application for Autonomous Vehicles[END_REF], dynamic obstacle avoidance [START_REF] Kang | Dynamic obstacles avoidance based on image-based dynamic window approach for humanvehicle interaction[END_REF] and for parking applications [START_REF] Morales | Autonomous parking using a sensor based approach[END_REF], [START_REF] Pérez-Morales | Laser-Based Control Law For Autonomous Parallel And Perpendicular Parking[END_REF] but has yet to be investigated for unparking tasks -where an acceptable final pose for the vehicle might not be as clearly defined when compared to parking tasks. Thus, it is worth to study the behavior and performance of unparking maneuvers using a sensor-based control approach.

Assuming that the vehicle is capable of perceiving its own parking spot, it should possible to unpark without any path planning nor prediction using a Multi-Sensor-Based Control (MSBC) approach by minimizing the error between the current value of a certain set of sensor features (i.e. a line collinear to the parking spot main axis and another collinear to the rear boundary of the parking spot) and its desired value while avoiding collision by imposing certain constraints on another set of sensor features (lines defining the boundaries of the parking spot, points at the corners of said spot, etc.).

The contribution of this paper is the exploration of a MSBC approach for forward perpendicular and diagonal unparking in one maneuver. It should be noted that, in order to decouple the performance of the controller from the perception, the sensory data is generated virtually and assumed to be available all the time.

In the next section the kinematic model of the vehicle and the multi-sensor modeling are presented. Section III describes the interaction model allowing to formalize the unparking task and the constraints for collision avoidance. Afterwards, the controller is presented in Section IV. The obtained results are presented in Section V: several cases in two different simulation environments are presented as well as exhaustive simulations results for assessing the convergence performance of the presented approach for the two different types of parking maneuvers addressed are shown. Finally, some conclusions are given in Section VI.

II. MODELING AND NOTATION

A. Car-like robot model and notation

Given that unparking maneuvers are low-speed motions, a kinematic model can be considered as accurate enough. Thus, in the present work the kinematic model for a car with rear-wheel driving is considered: where v and φ are the longitudinal and steering velocities. The point M is located at the mid-distance between the passive fixed wheels (rear) axle and the distance between the rear and the front axle is described by l wb . The generalized coordinates are q = [x, y, θ, φ] T where x and y are the Cartesian coordinates of the point M, θ is the orientation of the platform with respect to the x 0 axis and the steering angle of the steerable wheel(s) is denoted by φ (Fig. 1a).

    ẋ ẏ θ φ     =     cos θ sin θ tan φ/l wb 0     v +     0 0 0 1     φ, (1)
The turning radius ρ m around the instantaneous center of rotation (ICR) can be defined as:

ρ m = l wb tan φ (2)
The vehicle used for experimentation and simulation, represented by its bounding rectangle in Fig. 1a, is a Renault ZOE (Fig. 1b). Its relevant dimensional parameters are presented in Table I.

B. Multi-sensor modeling

The considered multi-sensor modeling is recalled in this subsection.

1) Kinematic model: Let us consider a robotic system equipped with k sensors (Fig. 2) that provide data about the environment. Each sensor S i gives a signal (sensor feature)

s i of dimension d i with k i=1 d i = d.
In a static environment, the sensor feature derivative can be expressed as follows:

ṡi = Li vi = Li i Tm vm ( 3
)
where Li is the interaction matrix [START_REF] Chaumette | Visual servo control, part I : Basic Approaches[END_REF] of s i (dim( Li ) = d i × 6) and i Tm is the 3D screw transformation matrix that allows expressing the sensor twist vi (which is expressed in its corresponding frame F i ) with respect to the robot twist vm (expressed in the control frame F m ).

F 0 F m F 1 F 2 F O S 1
ṡ = Ls vm (4)
with:

Ls = L Tm (5)
where L and Tm are obtained by concatenating either diagonally or vertically, respectively, matrices Li and i Tm

∀ i ∈ [1 . . . k].
Planar world assumption: Assuming that the vehicle to which the sensors are rigidly attached evolves in a plane and that the sensors and vehicle have vertical parallel z axes, all the twists are reduced to [v xi , v yi , θi ] T hence the reduced forms Ľ, Ľs , Ľi , vm and i Ťm of, respectively, L, Ls , Li , vm and i Tm are considered.

Ľi is of dimension

d i ×3, vm = [v xm , v ym , θm ] T and i Ťm is defined as: i Ťm =   cos( m θ i ) sin( m θ i ) x i sin( m θ i ) -y i cos( m θ i ) -sin( m θ i ) cos( m θ i ) x i cos( m θ i ) + y i sin( m θ i ) 0 0 1   (6)
where m t i = [x i , y i ] T and m θ i are, respectively, the position and orientation of S i (frame

F i ) with respect to F m expressed in F m .
Furthermore, since in the considered model the control frame F m is attached to the vehicle rear axis with origin at the point M (Fig. 1a), it is not possible to generate a velocity along y m on the vehicle frame and assuming that there is no slipping nor skidding (i.e. v ym = 0), the robot twist vm can be further reduced to:

v m = [v xm , θm ] T (7)
with v xm = v and θm = θ according to the model (1), thus it is possible to write:

ṡ = L s v m (8)
where L s is composed of the first and third columns of Ľs .

III. INTERACTION MODEL

For the interaction model, we rely on the perception of several lines L j and points from several (virtual) sensors The sensors placement can be seen in Fig. 3a. S 1 corresponds to the VLP-16 while S 2 to the 2D LiDAR placed on the rear (LMS151). S 3 to S 6 are placed on the corners of the car bounding rectangle and have the same orientation as the control frame.

As it can be seen in Fig. 3a, points p 1 to p 4 correspond to the corners of the parking spot while p 5 and p 6 are, respectively, the midpoints between (p 1 , p 4 ) and (p 2 , p 3 ). L 1 is a line that passes through p 5 and p 6 , i.e. it passes through the center of the parking spot. L 2 is a line that passes through p 1 and p 4 thus corresponding to the rear boundary of the parking spot. L 3 is a line that passes through p 3 and p 4 . All the lines are parametrized using normalized Plücker coordinates.

A. Line parametrization

Given two distinct 3D points i p f and i p g in homogeneous coordinates, with

i p f = [ i X f , i Y f , i Z f , i W f ] T (9a) i p g = [ i X g , i Y g , i Z g , i W g ] T , (9b)
a line passing through them can be represented using normalized Plücker coordinates as a couple of 3-vectors [START_REF] Andreff | Visual Servoing from Lines[END_REF]:

i L j = [ i u j , i h j ] T (10)
where i u j = i u j /|| i u j || (with i u j = 0) describes the orientation of the line and i h j = i r j /|| i u j || where i r j encodes the plane containing the line and the origin (interpretation plane) and the distance from the origin to the line. The two 3-vectors i u j and i r j are defined as [START_REF] Přibyl | Camera Pose Estimation from Lines using Plücker Coordinates[END_REF]:

i u T j = i W f [ i X g , i Y g , i Z g ] -i W g [ i X f , i Y f , i Z f ] (11a) i r T j = [ i X f , i Y f , i Z f ] × [ i X g , i Y g , i Z g ] (11b)
Due to the planar world assumption considered in this paper, the third element of i u j and the first and second elements of i h j are equal to zero, i.e. i u j (3) = i h j (1) = i h j (2) = 0, therefore the sensor signal s i L j and interaction matrix Ľi L j for the line i L j observed by S i are defined respectively as:

s i L j = i u j (1), i u j (2), i h j (3) T (12) Ľi L j =   0 0 i u j (2) 0 0 -i u j (1) -i u j (2) i u j (1) 0   (13)

B. Task sensor features

The set of task sensor features s t is defined as:

s t = [s t 1 , . . . , s t 6 ] T = [s 1 L 1 , s 1 L 2 ] T (14)
The corresponding interaction matrix Ľt is computed by a 2nd order approximation [START_REF] Thari | On the efficient second order minimization and image-based visual servoing[END_REF] of the form:

Ľt = ĽL + Ľ * L 2 (15)
where ĽL = [ Ľi L 1 , Ľi L 2 ] T and Ľ * L is equal to the value of ĽL at the desired pose.

Sensor-based control laws are usually defined so that the system reaches desired sensor features. In our case, the features are built on i L 1 and i L 2 . While the desired values for those two lines are clearly defined when parking, multiple choices may be available when unparking. Indeed, it depends on how we want the car to be positioned at the end of the unparking maneuver. A sensible choice would be for i L * 1 to be perpendicular to the vehicle longitudinal axis (x m -axis) and i L * 2 to be parallel to y m -axis, with each line being at a certain distance to the vehicle in order to be able to drive it outside of the parking spot.

C. Constraints sensor features

The set of constrained sensor features (Fig. 3b) used for collision avoidance s c is defined as:

s c = [s c 1 , . . . , s c 6 ] T = [s 3 , s 6 ] T (16)
with

s 3 = [ 3 h 2 (3), 3 Y 2 , 3 d lat2 ] T (17a) s 6 = [ 6 h 2 (3), 6 h 3 (3), 6 Y 3 ] T . (17b)
where the difference of radii i d lata is defined as:

i d lata = i ρ pa -ρ lat , (18)
with:

i ρ pa = ( i X a + x i ) 2 + ( i Y a + y i -ρ m ) 2 ρ lat = |ρ m | -wve 2 (19)
The interaction matrix Ľi Ya associated to i Y a is:

Ľi Ya = 0 -1 -i X a (20)
while interaction matrix associated to i d lata is defined as:

Ľi d = 0 i y i ρ 2 pa i Xa i y i ρ 2 pa ( 21
)
with i y = -| i Y a + y i -ρ m |.
The interaction matrices associated to the rest of the features used as constraints can be deduced from the third row of ( 13). The corresponding interaction matrix Ľc s is computed at each iteration.

It should be noted that some constraints must be deactivated under certain conditions as the scene topology evolves. These conditions used to obtain the results presented in this work are detailed in the following table:

TABLE II CONSTRAINTS DEACTIVATION CONDITIONS Constraint Deactivate if 3 d lat 2 φ ≥ 0 or 3 X 2 > -x i 6 h 3 (3) 6 X 3 < 0 6 Y 3 6 X 3 < 0 IV. CONTROL
The controller presented in [START_REF] Pérez-Morales | Laser-Based Control Law For Autonomous Parallel And Perpendicular Parking[END_REF] used for parking tasks is now used as well for unparking. For the sake of clarity we recall its definition and main properties. The control input of the robotized vehicle is defined as:

v r = [v, φ] T (22)
with φ, considering (1) and ( 2), being mapped to θ by

θ = v ρ m . ( 23
)
The steering angle φ is bounded by its maximum value φ max with |φ| < φ max (24)

while the longitudinal velocity of the vehicle is saturated by

|v| < v max ( 25
)
where v max is an adaptive saturation value imposing a deceleration profile based on the velocity profile shown in [START_REF] Petrov | Saturated Feedback Control for an Automated Parallel Parking Assist System[END_REF] as the vehicle approaches the final pose. Furthermore, to avoid large changes in the control signals at the current iteration n that may cause uncomfortable sensations for the passengers or surrounding witnesses and, to consider to some extent the dynamic limitations of the vehicle, the control signals are saturated as well by some increments with respect to the previous control signals (at iteration n -1) as shown below:

(v n-1 -∆ dec ) ≤ v n ≤ (v n-1 + ∆ acc ) (26a) (φ n-1 -∆ φ ) ≤ φ n ≤ (φ n-1 + ∆ φ ). (26b)
In order to automatically adapt the influence of each task feature, we use a weighting version of the feature error ( 14):

e t W = We t = W(s t -s t * ) ( 27
)
where W is a diagonal weighting matrix of dimension 6, with constant value for w 3 and w 6 but where (w i , i ∈ {1, 2, 4, 5}) are computed using a smooth weighting function (Fig. 4) based on the one presented in [START_REF] Narayanan | Vision-based adaptive assistance and haptic guidance for safe wheelchair corridor following[END_REF].

s i - s i s - s i s + s i + w i - w i + Fig. 4. Weighting function w(s)
The task interaction matrix corresponding to ( 27) is similarly weighted:

L t W = WL t s ( 28
)
The core of the control law is formulated in a quadratic programming form [START_REF] Kanoun | Kinematic control of redundant manipulators: generalizing the task priority framework to inequality tasks[END_REF] with only inequality constraints. It expresses that we want s t to approach its desired value at each iteration, but only in the vehicle velocity subspace compatible with the various constraints (obstacles, velocity and steering limits):

v m = argmin||L t W v m + λe t W || 2 s.t. Av m ≤ b (29) with: A = [L c s , -L c s ] T (30) b = [α(s c + -s c ), -α(s c --s c )] T ( 31
)
where α is a gain constant, λ is the control gain and [s c -, s c + ] is the interval in which s c should remain. Since the constraints are used for collision avoidance, only one side of the interval [s c - i , s c + i ] is considered for each feature. We now present simulation results illustrating the validity of this framework.

V. RESULTS

For the results shown in this section, the parameters in Table III are considered. The value of φ max corresponds to the maximum steering angle of the real vehicle while the rest of the parameters were determined by empirical testing. For the MATLAB implementations, the sampling time T s is equal to 0.1 s while for ROS, T s = 0.01 s.

The nonlinear solver used for MATLAB implementations is fmincon with a Sequential Quadratic Programming (SQP) algorithm while for C++ implementations the solver NLopt [START_REF] Johnson | The NLopt nonlinear-optimization package[END_REF] with a Sequential Least Squares Programming (SLSQP) algorithm is used. To illustrate the behavior of the MSBC approach, perpendicular (Fig. 5) and diagonal (Fig. 6) maneuvers are shown. The final position is marked with a colored square whose color depends on the final value of ||e t ||. It can be clearly seen that, for both cases, the car is able to unpark successfully. The control signals have a generally smooth evolution (Figs. 5b,6b) thanks to the considered bounds and saturation values. The (active) constraints imposed on s c are satisfied at each time instant (Fig. 5d) ensuring a collisionfree maneuver.

B. Exhaustive simulations

It is well known that the stability of sensor-based control laws such as (29) are very difficult to analyze. That is why various convergence analyses for the two unparking cases were conducted by means of exhaustive simulations. Due to paper length constraints, we assume that the desired final orientation of the vehicle is 0 • for the two considered cases (Figs. 7a-7b).

Since the exhaustive simulations are an aggregation of the results obtained from several simulations (like those shown in Figs. 5 and6), each figure consists of a parking spot (represented by 3 lines) adapted to each case and a scatter The green portion of each scatter plot corresponds to the region of attraction (ROA) and the red one represents the initial positions that are outside of the ROA. It should be noted that in the red regions the car ends outside of the parking spot but, due to the constrained sensor features and non-holonomic constraints, the car is not capable of reaching the desired pose.

Furthermore, it was expected for the ROA to be smaller for perpendicular maneuvers compared to the diagonals ones since, as most experienced drivers know, more space is needed to maneuver when unparking from a perpendicular spot.

C. Fast prototyping environment

A homemade fast prototyping environment using the same software architecture as the one embedded inside the car is used for simulation purposes. In addition to behaving nearly identically (from a software architecture point of view) to the real vehicle, this fast prototyping environment simulates as well the dynamics of the vehicle, leading to more realistic simulations than the MATLAB environment used for the results presented in the previous subsections.

As it can be seen in Figs. 89, the car is able to unpark successfully from the parking spot (represented by a green rectangle) in one motion while satisfying the constraints during the whole maneuver, with the evolution of the many different signals being very similar to the MATLAB cases. VI. CONCLUSIONS Following our previous work [START_REF] Pérez-Morales | Laser-Based Control Law For Autonomous Parallel And Perpendicular Parking[END_REF], we've shown how, with some minor changes (simplifications) on the interaction model, it is possible to perform unparking tasks using the same control approach as for parking tasks. These new results increase the, already important, versatility and validity of our MSBC approach towards a fully autonomous parking system. The main considered (future) improvement is on the automated definition of the desired features, that are not uniquely defined when unparking. A region-reaching control law may be suitable, as in a fully autonomous framework the car could reach the most feasible position on the trajectory to be tracked after unparking.

Fig. 1 .

 1 Fig. 1. (a) Kinematic model diagram for a car-like rear-wheel driving robot. (b) Robotized Renault ZOE used for real experimentation

Fig. 2 .

 2 Fig. 2. Multi-sensor model Denoting s = (s 1 , . . . , s k ) the d-dimensional signal of the multi-sensor system, the signal variation over time can be linked to the moving vehicle twist:

  Fig. 3. (a) General sensors' configuration and sensor features. (b) Example of the constrained sensor features

Fig. 5 .

 5 Fig. 5. Constrained forward ⊥ unparking maneuver signals: (a) performed maneuver (desired pose = (5m, 6.5m, 0 • )), (b) control signals, (c) task error signal, (d) constrained sensor signals

Fig. 6 .

 6 Fig. 6. Constrained forward diagonal unparking maneuver signals: (a) performed maneuver (desired pose = (5.5m, 4m, 0 • )), (b) control signals, (c) task error signal, (d) constrained sensor signals

Fig. 7 .

 7 Fig. 7. Exhaustive simulations. Final orientation = 0 • . Parking spot length = 4m and width = 2.7m plot of the final position of the vehicle (with a sampling step of 10cm), whose color depends on the final value of ||e t ||.The green portion of each scatter plot corresponds to the region of attraction (ROA) and the red one represents the initial positions that are outside of the ROA. It should be noted that in the red regions the car ends outside of the parking spot but, due to the constrained sensor features and non-holonomic constraints, the car is not capable of reaching the desired pose.Furthermore, it was expected for the ROA to be smaller for perpendicular maneuvers compared to the diagonals ones since, as most experienced drivers know, more space is needed to maneuver when unparking from a perpendicular spot.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Forward perpendicular unparking maneuver in simulation using a homemade fast prototyping environment

ACKNOWLEDGMENT

This work was supported by the Mexican National Council for Science and Technology (CONACYT). This paper des-cribes work carried out in the framework of the Valet project, reference ANR-15-CE22-0013-02.