Jean-Yves Lazennec

Marc-Antoine Rousseau

Adrien Brusson

Dominique Folinais

Maria Amel

Ian Clarke

Aidin Eslam Pour

Total Hip Prostheses in Standing, Sitting and Squatting Positions: An Overview of Our 8 Years Practice Using the EOS Imaging Technology

Keywords: cup anteversion, femoral offset, EOS imaging, hip spine relation, limb length, pelvic obliquity, pelvic tilt, sagittal balance, sitting position, standing position, pelvic rotation, total hip arthroplasty (THA)

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INTRODUCTION

More total hip arthroplasty is performed in younger and more active patients compared to earlier years. The focus of adult reconstruction research in the future will be on extending the durability and survivorship of the implants and optimizing the radiological follow-up of these patients as loosening, osteolysis, wear and dislocation persist as potential complications. Another objective is the detection of patients at risk for complications and the optimization of planning to reduce the incidence of implant malposition and leg-length discrepancy. It is also important to better understand and manage other intricate orthopedic disorders *Address correspondence to this author at the Department of Orthopaedic Surgery, University of Michigan, 2912 Taubman Center, Box 5328, 1500 East Medical Center Drive, Ann Arbor, MI 48109-5328, USA; Tel: 734-936-5780; E-mail: aeslampo@med.umich.edu in patients who undergo THA (scoliosis, sagittal and frontal imbalance), because these disorders may affect the surgical outcome and complication rate. Most radiographic analyses of THA patients are based on anteroposterior (AP) radiographs in the standing position and computed tomography (CT) scans in the lying position. Despite its better accuracy, CT imaging cannot be routinely performed because it is expensive and exposes the patient to more radiation. Moreover, CT must be performed in the supine position. Recent literature expresses potential interest in standing and sitting radiographs and the importance of lateral evaluation of THA in these functional situations [START_REF] Lazennec | Hip spine relationships: application to total hip arthroplasty[END_REF][START_REF] Lazennec | Acetabular anteversion with CT in supine, simulated standing, and sitting positions in a THA patient population[END_REF]. Variations in pelvic parameters that are commonly measured in spine surgery (pelvic tilt PT, sacral slope SS and pelvic incidence I) [START_REF] Vialle | Radiographic analysis of the sagittal alignment and balance of the spine in asymptomatic subjects[END_REF] and changes in the orientation of the anterior pelvic plane APP (the Lewinnek plane) [START_REF] Lewinnek | Dislocations after total hip-replacement arthroplasties[END_REF] are relevant parameters for planification and navigation as they significantly modify the prosthetic hip biomechanics [START_REF] Sariali | Modification of pelvic orientation after total hip replacement in primary osteoarthritis[END_REF]. In addition, these data appear to be important during the course of THA follow-up [START_REF] Lazennec | Hip-spine relationship: a radio-anatomical study for optimization in acetabular cup positioning[END_REF][START_REF] Philippot | Pelvic balance in sagittal and Lewinnek reference planes in the standing, supine and sitting positions[END_REF][START_REF] Rousseau | Optimization of total hip arthroplasty implantation: is the anterior pelvic plane concept valid?[END_REF]. The EOS ® system (EOS Imaging, Paris, France) is an innovative slot-scanning radiograph system allowing the acquisition of radiograph images while the patient is standing, sitting or even squatting with less irradiation than standard imaging (Fig. 1). This technology is based on studies of gaseous detectors by Georges Charpak, who won the Nobel Prize in physics in 1992 [START_REF] Dubousset | A new 2D and 3D imaging approach to musculoskeletal physiology and pathology with low-dose radiation and the standing position: the EOS system[END_REF]. EOS was originally used in the peri-and postoperative management of idiopathic scoliosis considering the need for repeated radiographs and the importance of limiting the radiation dose [START_REF] Illes | Breakthrough in threedimensional scoliosis diagnosis: significance of horizontal plane view and vertebra vectors[END_REF]. Spinal deformities can be studied in standing position with AP and lateral radiographic images of equivalent quality in all sections of the spine and furthermore, three-dimensional (3D) models using the Stereos ® software [START_REF] Wade | A systematic review of the clinical effectiveness of EOS 2D/3D X-ray imaging system[END_REF].

Regarding the field of THA cases, the EOS imaging system provides coupled AP and lateral high-definition images in functional position, allowing the measurement of the pelvic parameters simultaneously with the acetabular and femoral parameters [START_REF] Lazennec | THA patients in standing and Sitting Positions : A prospective Evaluation Using the Low Dose "full body" EOS imaging system[END_REF]. It has been demonstrated recently that EOS is accurate and reliable for the measurement of pelvic, acetabular and femoral parameters [START_REF] Lazennec | Pelvis and total hip arthroplasty acetabular component orientations in sitting and standing positions: measurements reproductibility with EOS imaging system versus conventional radiographies[END_REF]. In addition, for the first time, full-body images can be obtained for more accurate analysis of current problems, including pelvic obliquity, spine deformity and imbalance, limb-length discrepancy, impingements and dislocations [START_REF] Journe | New method for measuring acetabular component positioning with EOS imaging: feasibility study on dry bone[END_REF][START_REF] Buck | Femoral and Tibial Torsion Measurements With 3D Models Based on Low-Dose Biplanar radiographs in comparison with standard CT measurements[END_REF][START_REF] Folinais | Measuring femoral and rotational alignment: EOS system versus computed tomography[END_REF].

Here we describe the contributions of the EOS imaging technology to our standard practice in the context of our first 300 THA cases. In addition to data accuracy on the precise orientation of the implants in functional standing, sitting or squatting positions, this review paper discusses our experience with EOS imaging to study the hip-spine relationship and the practical impact of the detection of outliers and risky patients for THA complications. In particular, the analysis of complex postural situations will be addressed (association with scoliosis, leg-length discrepancy and frontal and/or sagittal imbalance). We will also discuss application of the EOS system for postoperative follow-up, especially for the assessment of postural changes induced by THA implantation.

MATERIAL AND METHODS

In this cohort study, 300 consecutive THA cases in 224 patients (174 females, mean age 68 years (range 34-90), mean BMI 29, 76 bilateral THA) were included from our routine practice for clinics consultation. 196 of the patients underwent primary. 117 of the cups were cemented polyethylene cups and the remaining 183cups were cementless. 183 of the stems were cemented.

For the evaluation of the standing position using the EOS ® machine, each subject adapted a comfortable position, the reproducibility of which has been reported [START_REF] Pour | Reproducibility of the standing position for repeated assessment of the position of the lower limbs: a new approach using the EOS imaging system[END_REF]. For the sitting position each subject adapted a comfortable position on a stool with adjustable height and backrest, hands on the knees and both feet flat on the floor. The results were obtained using the Stereos ® software from EOS Imaging.

RESULTS

Hip-Spine Relations and Lumbo-Pelvic Parameters

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Although the concept of spino-pelvic balance is well accepted today [START_REF] Lazennec | Hip-spine relations and sagittal balance clinical consequences[END_REF][START_REF] Polkowski | Does standing affect acetabular component inclination and version after THA?[END_REF][START_REF] Lazennec | Lumbar-pelvic-femoral balance on sitting and standing lateral radiographs[END_REF], and although the hip is a highly mobile joint, surgical concepts are still based on the static AP view of the pelvis in standing or supine position to assess cup orientation with the measurement of the coronal inclination and the planar (or radiological) anteversion. The sagittal orientation of the cup using lateral radiographs is poorly documented [START_REF] Kennedy | Effect of acetabular component orientation on recurrent dislocation, pelvic osteolysis, polyethylene wear, and component migration[END_REF], while the lateral view of the pelvis describes relevant parameters, namely the anterior pelvic plane (APP), pelvic incidence (I), sacral slope (SS) and pelvic tilt(PT), the variations of which significantly influence "normal" function and failures in THA [START_REF] Legaye | Influence of the sagittal balance of the spine on the anterior pelvic plane and on the acetabular orientation[END_REF][START_REF] Pinoit | Low accuracy of the anterior pelvic plane to guide the position of the cup with imageless computer assistance: variation of position in 106 patients[END_REF]. SS, PT and I are currently described as linked using the formula I = SS + PT (Fig. 2). This geometric relationship has been confirmed using simple regression models for standing position (PI=0,97 + 0,972 SS + 1,001 PT) and described recently for the sitting position (PI =0,985+ 1,021 SS + 0,972 PT) [START_REF] Lazennec | THA patients in standing and Sitting Positions : A prospective Evaluation Using the Low Dose "full body" EOS imaging system[END_REF].

•

Our results are reported in Table 1. In this series, the pelvic incidence I, which does not change with the posture, is 56° in average (SD: 12°; range 30° to 89°). The data are in accordance with the previous literature for standing position, the exploration of the sitting position remaining unusual despite its interest for some unstable THA [START_REF] Vialle | Radiographic analysis of the sagittal alignment and balance of the spine in asymptomatic subjects[END_REF][START_REF] Lazennec | Hip-spine relationship: a radio-anatomical study for optimization in acetabular cup positioning[END_REF][START_REF] Philippot | Pelvic balance in sagittal and Lewinnek reference planes in the standing, supine and sitting positions[END_REF][START_REF] Lazennec | Pelvis and total hip arthroplasty acetabular component orientations in sitting and standing positions: measurements reproductibility with EOS imaging system versus conventional radiographies[END_REF][START_REF] Lazennec | Lumbar-pelvic-femoral balance on sitting and standing lateral radiographs[END_REF][START_REF] Pinoit | Low accuracy of the anterior pelvic plane to guide the position of the cup with imageless computer assistance: variation of position in 106 patients[END_REF]].

•

The standing (ST) position involves a forward tilt of the pelvic block considered as pelvic flexion. This anterior rotation of the pelvis is sometimes described in aconfusing way as pelvic "anteversion" (Fig. 2).

On lateral view, the sacral slope (SS-Sta) angle is about 40° (SD: 16°; range 33° to 63°) and the pelvic tilt (PT-Sta) is about 18° (SD: 11°; range 11° to 32°). For a given incidence angle, the pelvic tilt can be calculated as PT = I -SS [START_REF] Murray | The definition and measurement of acetabular orientation[END_REF]. One can observe a significant variation in standing SS when comparing this series to asymptomatic patients. The reduction of SS-Sta indicates a relativebackward tilt of the pelvis due to the ages of THA patients, as an associated aging spine is frequently observed [START_REF] Itoi | Roentgenographic analysis of posture in spinal osteoporotics[END_REF]. This is in accordance with imaging and navigation measurements that have demonstrated changes in the sagittal balance of the spine over time with progressive pelvic posterior rotation in standing position and decreasing SS.

•

In the sitting position (Fig. 2), the pelvis tilts backward during the transition to the sitting (Sit) posture; it is considered as pelvic extension. This pelvis posterior rotationis sometimes described in a confusing way as pelvic "retroversion". Sacral slope (SS-Sit) diminishes to a mean of 20° (±12°), and sometimes as low as a negative value (-10°).

Depending on the individual morphology and associated hip or spinal pathology, a variable degree of posterior pelvic tilt (PT-Sit) is observed (36°± 15°), with the sacrum more or less vertical. One can observe a greater dispersion of these parameters in the sitting position. Significant outliers can be observed in these measurements. Potential complications can be suspected in THA as unexpected functional orientation of the cups can be observed in some cases (Fig. 3).

•

In the squatting position, pelvic posterior tilt (pelvic posteriorrotationor pelvic extension)is increased. Due to the fact that squatting position is not common among our population of Caucasian patients, we rarely explored this functional position. The decrease of SS can reach 10° more with a consequent equivalent increase of pelvic tilt increase due to the short acquisition time with EOS, this protocol can be realized even in aged patient with poor body stability. Due to the limitation of hip flexion in some individuals, the positioning of the hips can be asymmetrical, sometimes with atypical postures (Fig. 4).

•

The variation in SS angle determines the range of PT, and this variation influences the orientation of the APP or the Lewinnek plane classically used as a reference for adjusting the position of the acetabular cup and for the postoperative evaluation of its orientation. (Fig. 5). The angular variations of the Lewinnek plane inclination between standing (APPI-ST) and sitting (APPI-SIT) are about 22°.

•

In our experience the sitting position is more difficult to evaluate in obese patient. The APP measurement is mainly affected regarding the detection of the antero superior iliac process. We faced this problem in 10% of the cases in standing position (31 patients) and 19% in the sitting position (57 cases) with more than 5° (6 to 11°) in accordance with two successive measurements by two different operators.

Acetabular Implant Position

• Three key parameters are considered for acetabular evaluation: frontal and sagittal inclination and acetabular anteversion. These parameters can be considered in two ways: as "morphological parameters" with reference to the bone frame of the pelvis as described by Murray in the lying position [START_REF] Murray | The definition and measurement of acetabular orientation[END_REF], and in relation to the "functional orientation" of the pelvis inducing significant variations in acetabular orientation and the values of these parameters in THA implantation for standing, sitting, squatting or pelvic rotation. In our current practice, we mainly use the "functional" point of view, as the orientation of the body (and therefore the acetabular cup) significantly changes with functional position when comparing supine, standing and sitting postures as highlighted by the EOS imaging system. Whatever the position, the reference plane remains the horizontal transverse plane for all the measurements. The EOS imaging system provides 2D and 3D information regarding the acetabular orientation, but to date, we mainly use the 2D measures as they represented a good transition from our previous use of classical plane radiographs (Fig. 6).

Case A Case B

A.P.P.$ A.P.P.$ A.P.P.$ A.P.P.$

Acetabular Frontal and Sagittal Inclinations

• Our results for frontal and sagittal inclinations are summarized in Table 2. One can observe some limitations for the measures on the lateral view. Globally, the measures could be done only in 260 cases for standing position (85%), and in 225cases in sitting position (75%). The main problem encountered has been the limitation induced by the 76 bilateral THA. Only for 44 of these bilateral cases, a correct evaluation of the sagittal inclination angle could be obtained in standing position.Due to the superposition of the 2 cups, the measures could not be done in 32 THAs. The 44 remaining cases were more favorable due to a significant rotation of the pelvis inducing a gap between the 2 cups. In addition to these problematic cases, we faced difficulties in 9 obese patients (BMI >31).

In sitting position, the difficulties were more significant in 95 THAs (superposition of the 2 cups for 44 cases, difficulties for assessing the cup profile in 41 cases).

•

The relationship between the acetabular sagittal inclination angle (ISA) and SS is clear on lateral EOS images of the lumbo-sacral junction in standing and sitting postures. In the standing position, SS is high and acetabular inclination low. In the sitting position, SS decreases and acetabular inclination increases. On both lateral and AP views, the acetabular THA component is more vertical in the sitting position than in the standing position [START_REF] Lazennec | Hip spine relationships: application to total hip arthroplasty[END_REF]. These variations in acetabular tilt contribute to the change in anterior opening of the acetabulum and thus in the orientation of the functional mobility cone of the hip joint.

•

The coherence between sagittal acetabular tilt and SS is expressed by the sacro-acetabular angle (SAA), defined as the tangent of the sacral endplate and the axis of the acetabular ellipse on lateral view (the latter defining the angle of sagittal acetabular inclination to the horizontal). The SAA is fixed and empirically applied by the surgeon to the acetabular component in

Standing Sitting

THA [START_REF] Lazennec | Hip-spine relationship: a radio-anatomical study for optimization in acetabular cup positioning[END_REF][START_REF] Lazennec | Lumbar-pelvic-femoral balance on sitting and standing lateral radiographs[END_REF] (Fig. 7). Geometrically, SAA = SS +ASI.

Our results demonstrate a wide variation of this anatomical parameter (mean 69°, range 31° to 129°). In this series, the consequences of poor adjustments in outlier cases induced atypical positioning of the cup and impingement situations in standing (12 cases) (Fig. 8) or sitting positions (14 cases) (Fig. 9).

•

The lack of variation in PT between the standing and sitting positions (less then 5°) has been observed in

Acetabular Anteversion

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The concept of acetabular anteversion (AV) is more confusing as two classical analyses of postoperative anteversion of the cup can be proposed: "radiological anteversion" using plane X-rays (and then AP EOS images) and "sectional anteversion" using CT or Stereos software measures from EOS data. This confusion explains the heterogeneity of the literature regarding the data for the optimization of cup anteversion [START_REF] Wassilew | 3D CT analysis of combined cup and stem anteversion in case of cup navigation hip arthroplasty[END_REF].

• Using the Pradhan technique [START_REF] Pradhan | Planar anteversion of the acetabular cup as determined from plain anteroposterior radiographs[END_REF] for the measurement of Radiographic Anteversion on EOS AP views, we previously reported [START_REF] Lazennec | THA patients in standing and Sitting Positions : A prospective Evaluation Using the Low Dose "full body" EOS imaging system[END_REF] significant changes in the anteversion values between standing and sitting positions.Standing radiological anteversion was 17.2° (SD: 10.2 ; range -15° to 46.3°). In the sitting position, radiological anteversion was 35.4° (SD: 13.1; range 0.3° to -62.4°).

In addition, this study noted the questionable value of the Lewinnek safe zone, as the APP is not always vertical and cannot be systematically considered as the patient coronal plane for the determination of the "Lewinnek safe zone".

•

CT is now the current gold standard for assessing the position of the implants [START_REF] Stem | Computed tomography analysis of acetabular anteversion and abduction[END_REF].These measures are obtained in supine position, and only provide an instantaneous assessment of the "functional sectional anteversion" in supine position. Time-consuming reconstructions allow the evaluation of the "sectional anteversion" with reference to the APP. Because posterior dislocations of the hip mostly occur in the sitting position, and some anterior dislocations in the standing position [START_REF] Lembeck | Pelvic tilt makes acetabular cup navigation inaccurate[END_REF], knowing the acetabular version in these positions could be important, and measuring cup version of the supine patient on a CT scan could result in misleading information as apparent acetabular anteversion increases 1° for every 2° increase of posterior pelvic tilt (pelvic posterior rotation or pelvic extension) [START_REF] Kalteis | In vitro investigation of the influence of pelvic tilt on acetabular cup alignment[END_REF][START_REF] Anda | Computed tomography measurements of the acetabulum in adult dysplastic hips: which level is appropriate?[END_REF].

•

The notion of a relationship between sectional acetabular AV and the position of the pelvis has been previously reported in native hips [START_REF] Zilber | Variations of caudal, central, and cranial acetabular anteversion according to the tilt of the pelvis[END_REF] and in THA [START_REF] Nishihara | Measurements of pelvic flexion angle using three-dimensional computed tomography[END_REF][START_REF] Siebenrock | Effect of pelvic tilt on acetabular retroversion: a study of pelves from cadavers[END_REF]. A forward tilt of the pelvis (pelvis anterior rotation or pelvic flexion) induces retroversion of the acetabulum, whereas a backward tilt of the pelvis (pelvis posterior rotation or pelvic extension) results in an anteversion of the acetabulum (Fig. 12). Lumbosacral junction mobility is the main parameter influencing the variations of AV between the standing and sitting positions. This parameter seems strongly relevant as degenerative phenomena in the aging spine induce progressive changes in sagittal balance with loss of lumbar lordosis and pelvic extension [START_REF] Lazennec | Hip-spine relations and sagittal balance clinical consequences[END_REF][START_REF] Polkowski | Does standing affect acetabular component inclination and version after THA?[END_REF]. This evolution could partially explain the cumulative increase of dislocation risk with time [START_REF] Siebenrock | Effect of pelvic tilt on acetabular retroversion: a study of pelves from cadavers[END_REF].

• The EOS imaging system provides the measurement of the acetabular anteversion in a "sectional way" [START_REF] Dubousset | A new 2D and 3D imaging approach to musculoskeletal physiology and pathology with low-dose radiation and the standing position: the EOS system[END_REF][START_REF] Lazennec | Pelvis and total hip arthroplasty acetabular component orientations in sitting and standing positions: measurements reproductibility with EOS imaging system versus conventional radiographies[END_REF] according to the APP (and then to the pelvic frame, in a plane perpendicular to the APP) or in the horizontal plane of the patient allowing "functional measures" in standing and sitting positions. This direct approach of the functional sectional anteversion in the horizontal plane is a significant improvement for a more accurate analysis of THA complications (mainly dislocation or subluxation) [START_REF] Lazennec | THA patients in standing and Sitting Positions : A prospective Evaluation Using the Low Dose "full body" EOS imaging system[END_REF]. Our results are summarized in Table 3 from 205 cases.

•

We could observe the good correlation between the anteversion values measured in the horizontal plane in the standing position and those evaluated in the plane perpendicular to the APP. It is logical as in the majority of cases the APP is more or less vertical. On the contrary the values are completely different in the sitting position with a significant increase of the values due to pelvic extension(pelvic posterior rotation).

Femoral Parameters

Two parameters are routinely used for the evaluation of femoral implants: femoral offset and femoral anteversion. Two other parameters are interesting for the evaluation of the femoral orientation. The femoral version (or femoral tilt) in the AP view (adduction or abduction of the femur) and the lateral view is poorly documented, as the long-leg

A B

radiographs are rarely performed in current practice. In addition, the concept of cumulative anteversion (the sum of sectional acetabular anteversion and femoral anteversion) is considered a relevant but also poorly investigated parameter.

Femoral Offset

The importance of the offset restoration has been demonstrated. Due to the 2D nature of plain radiograph, the projection effect of the femoral anteversion and the external rotational contracture of the degenerative hip result in limited accuracy of plain pelvic radiographs. Plain AP pelvic radiographs could underestimate the offset value by 8% to 13% [START_REF] Merle | Femoral offset is underestimated on anteroposterior radiographs of the pelvis but accurately assessed on anteroposterior radiographs of the hip[END_REF][START_REF] Lecerf | Femoral offset: anatomical concept, definition, assessment, implications for preoperative templating and hip arthroplasty[END_REF]. CT with can also measure offset and hip geometry, but it exposes the patient to significant radiation dose. Due to the simultaneous capture of two orthogonal AP and lateral images (like standard radiographs), low-dose EOS imaging provides the opportunity to compare 2D and 3D data in standing position, without additional imaging study. In comparison to standard radiographs and CT scans, it has recently been demonstrated that EOS is accurate and reliable without significant inconvenience caused by the metallic artifacts of implants [START_REF] Pour | Restoration of femoral offset after total hip replacement: assessment by EOS imaging in 100 patients[END_REF]. According to our data on 100 patients, 2D measurements underestimate the offset by an average of 3.5 mm in implanted hips in comparison with 3D measurements (4.08 (± 0.76)/ 4.43 ( ± 0.70). The main factor explaining this discrepancy is hip rotational positioning, as the mean values for femoral anteversion were within the range previously published. These results are in accordance with previously published data about femoral offset and highlight the relevance of 3D EOS reconstructions (Fig. 13).

Femoral Sagittal Tilt (Femoral Sagittal Inclination Angle or Femoral Sagittal Version)

• On the lateral view, the femoral sagittal tilt (FST) or femoral sagittal inclination angle is defined as the angle subtended by the vertical and the femoral axis between the center of the femoral head and the summit of Blumensaat's line [START_REF] Lazennec | Hip-spine relations and sagittal balance clinical consequences[END_REF] (Fig. 14). It is greater in case of flexion contracture of the hip in standing position, and can be negative in hyperextension of the hip. Femoral sagittal tilt correlates statistically with pelvic tilt: the greater the pelvic tilt, the greater the FST angle [START_REF] Lazennec | Lumbar-pelvic-femoral balance on sitting and standing lateral radiographs[END_REF]. Generally the flexion contracture is mainly assessed clinically or qualitatively as the quality and feasibility of long-leg X-rays on lateral view is rarely sufficient to obtain precise angles and values.

•

We analyzed the femoral sagittal tilt angle using the EOS lateral images of the whole series. The mean value for FST was 0.5° (SD 11.6° range -29.3° to 36.2°). The wide range of values was explained by some complex postural situations due to the interrelation of spine, hips and knees (limb length discrepancy, frontal imbalance, hip or knee flexion contracture). These values illustrate the compensation mechanisms to maintain an adaptive sagittal balance. Their interpretation was useful to understand anterior subluxation cases in 9 cases.

Femoral Anteversion and Combined Femoro-Acetabular Anteversion

•

We measured the femoral torsions using the Stereos ® software in 227 patients. Mean value was 12.7° (SD 15.2°; range: -42° to 33°). In 3 cases the high value of retroversion (-42°, -38°,-34°)was explained by a rotation of the femoral stem due to loosening. In 73 patients the measure was not possible due to the superposition of the femoral condyles in the lateral view. Then, the position of the patient during the acquisition of the standing position can be a limitation.

Fig. (12).

A forward tilt of the pelvis (pelvis anterior rotation or pelvic flexion) as in standing position induces retroversion of the acetabulum, whereas a backward tilt of the pelvis (pelvis posterior rotation or pelvic extension) as in sitting position results in an anteversion of the acetabulum. The forward tilt of the pelvis is expressed by the low value of sacral slope (SS) in standing position, whereas the backward tilt is associated with a higher sacral slope angle.

standing( si)ng(

S"S" S"S"

• Many studies have proposed optimal alignment of the acetabular cup. By contrast, only a few studies have dealt with optimization of the prosthetic alignment on the femoral side [START_REF] Bargar | Femoral anteversion in THA and its lack of correlation with native acetabular anteversion[END_REF]. The concept of cumulative anteversion (CA) between the acetabular cup and the femur is considered a key factor to assess the appropriateness of overall prosthetic alignment to optimize the motion while avoiding prosthetic impingement and instability. This concept, however, is based on CT measurements in a supine position, which do not account for the functional dimension of the problem according to the standing or sitting position. For Jolles et al. [START_REF] Jolles | Factors predisposing to dislocation after primary total hip arthroplasty[END_REF] the dislocation rate increased 6.9 times in THA patients when the CA value was outside the range of 40° and 60°. Based on the results of computerized 3D model analysis, Widmer and Zurfluh [START_REF] Widmer | Compliant positioning of total hip components for optimal range of motion[END_REF] proposed the formula, cup AV + 0.7 stem AV, to calculate the CA value and considered the ideal value to be 37.3°. Yoshimine [START_REF] Yoshimine | The safe-zones for combined cup and neck anteversions that fulfill the essential range of motion and their optimum combination in total hip replacements[END_REF] proposed that the sum of the cup AV, plus 0.8 times the stem AT, plus the cup inclination should be 90.8° (AV + 0.8AT + RI = 90.8°). Hisatome and Doi [START_REF] Hisatome | Theoretically optimum position of the prosthesis in total hip arthroplasty to fulfill the severe range of motion criteria due to neck impingement[END_REF] proposed that the sum of the cup AV plus 0.7 times the stem AV should be 42° (AV + 0.7AT = 42°) with a cup inclination of 45° for a head diameter of more than 32 mm.

•

According to the good correlation between supine and standing measures of acetabular sectional anteversion, we calculated the cumulative anteversions using the standing measures by EOS. Mean cumulative anteversion was 42° (SD 20.5° range -23°to 77°). All of the 38 unstable THA in this series were outliers.

The study of the cup orientation could clearly explain instability in 18 cases (7 anterior instability in standing position, 11 posterior instability in sitting position). In 11 cases, the analysis of the femoral torsion alone could explain the instability (4 anterior instability in standing position,7 posterior instability in sitting position). In 9 cases the combined analysis of acetabular and femoral anteversion was useful to explain the instability.

EOS imaging of THA in specific situations

THA and Leg-Length Discrepancy (LLD)

• Leg-length discrepancy after THA is frequently associated with general dissatisfaction and potential complications, but the magnitude of a clinically significant LLD after THA is still a matter of controversy. Numerous radiological methods for measuring LLD have been described but the results are variable with a diverging outcome. Although fullleg standing AP radiographs may provide a clear LLD is expressed as the difference of the distance between a femoral and a pelvic landmark on both sides. Goodman et al. [START_REF] Goodman | The acetabular teardrop and its relevance to acetabular migration[END_REF] recommended the use of the teardrop points as a landmark for measurements because the vertical positions of the teardrop points is not affected significantly by rotation of the pelvis. This pelvic evaluation provides information regarding the "projected anatomical length" of the superior part of the femurs but does not describe the functional position of the lower limbs. Functional limb length is the result of a complex interaction of the anatomic lengths of bones, implants, and also soft tissue contracture. An abduction, adduction, or flexion contracture should be assessed and quantified because of the potential influence on perceived length. A flexion contracture can lead to an overestimation of shortening. Common causes of a perceived long limb include scoliosis, fixed pelvic tilt, and contralateral limb deformity. Full-leg standing AP radiographs provide better information but limitations are due to the lack of lateral views analyzing the global sagittal posture. The degree of validity of CT is well documented, but CT can give only the anatomic length of the lower limbs without aiding assessment of the functional measurements, as the examination is performed in supine position.

•

Using EOS full-body images AP and lateral, we were able to analyze 98 cases of "oblique pelvis", flexion contracture or hyperextension of lower limbs: comparison of standing and sitting AP views allows differentiation of oblique pelvis linked to lower limb discrepancy, as in the sitting position the frontal pelvic tilt disappears (41 cases) (Fig. 15). On the contrary fixed oblique pelvis are not influenced by the sitting position. We could test the impact of heel compensation to correct LLD in 12 cases (Fig. 16) as well as the consequences of THA revision (Fig. 17).

Evolution of Pelvic Orientation After THA Implantation

•

This issue is particularly controversial. Indeed the validity of preoperative planning and intraoperative navigation can be impacted by significant changes in

A B C D

the sagittal pelvic orientation postoperatively. The literature is not abundant on this topic because most postoperative controls rely on AP radiographs. CT scan reconstructions have focused on the pelvic bone reference frame regardless of variations in functional orientation in the sagittal plane. Some articles refer to changes in the anterior pelvic plane, but we know the problems associated with measurement uncertainties of the standard lateral view (obesity, pelvic rotation) [START_REF] Blondel | Pelvic tilt measurement before and after total hip arthroplasty[END_REF][START_REF] Murphy | Pelvic tilt is minimally changed by total hip arthroplasty[END_REF]. For others, the appreciation of the pelvic orientation is based on variations of the sacral slope, a more reliable benchmark that can be correlated with other pelvic parameters, including the morphological parameter of pelvic incidence.

• A classical adaptation mechanism has been describedfor flexion contracture of the hip related to osteoarthritis. The loss of extension capacity in the affected hip induces anterior pelvic tilt when the subject tries to stand. In this situation, the spine, when possible, adapts by increasing the lumbar lordosis. In these cases, THA implantation can improve the extension ability and a decrease of pelvic flexion(pelvic anterior rotation) can be noted. In other cases, a posterior pelvic extension(pelvic posterior rotation)can be observed. It is often associated with postural imbalance to compensate for a forward tilt of the trunk, especially with spinal aging. This postural adaptation verticalizes the acetabulum frontally and laterally, putting the hips in "upright hyperextension" with a potential risk for posterior impingement. This at-risk configuration may be aggravated by poorly adjusted lumbar fusion. A postoperative decrease of pelvic flexion (correction of pelvic anterior rotation) has been reported but without return to normal [START_REF] Sariali | Modification of pelvic orientation after total hip replacement in primary osteoarthritis[END_REF]. This phenomenon regarding the SS has already been described in the case of reduction of knee flexion contracture [START_REF] Lee | Effect of correction of the contractured flexed osteoarthritic knee on the sagittal alignment by total replacement[END_REF]. Other authors describe a progressive increase of the pelvic extension after THA over 2 to 4 years and mainly in older women [START_REF] Taki | Change in pelvic tilt angle 2 to 4 years after total hip arthroplasty[END_REF]; this accords with the progressive spinal aging phenomena already described [START_REF] Itoi | Roentgenographic analysis of posture in spinal osteoporotics[END_REF].

• We used the EOS ® 2D images for the follow-up of 48 THA patients including 15 bilateral THA and 11 THA revisions (Figs. 18,19). In standing position, we did not observe a significant change of sagittal tilt for 28 patients including 4 bilateral THA (group 1). Other patients experienced significant changes for sagittal pelvic orientation (group 2): an increase of SS (> 5°) in 7 cases and a decrease of SS ( >5°) in 13 cases. For sitting position, the group 1 did not experience changes in SS. For the other cases, changes were not related to those in standing position. Despite the postoperative alterations to pelvic orientation that could not be anticipated, none of these patients experienced an instability.

THA in Scoliosis Patients: The Influence of Pelvic Rotation

• This topic is poorly documented. However, a rotation or atypical pelvic tilt is frequently observed in scoliosis patients. Normal pelvic posture is classically defined on AP images in a subject in a strictly anatomic posture, with both iliac wings projecting symmetrically with respect to the longitudinal axis of the trunk. The standard criteria of sagittal balance are described on lateral views with both femoral heads perfectly superimposed. Pelvic rotation can be extreme in scoliosis involving the pelvic vertebra. The impact on acetabular orientation can be significant, due to altered functional anteversion in both standing and sitting postures [START_REF] Rousseau | Assessment of the axial rotation of the pelvis with the EOS ® imaging system: intra-and inter-observer reproducibility and accuracy study[END_REF]. In addition, some publications associate functional or anatomic LLD to pelvic torsion, but there is no clear consensus about the relation between the direction of ilium rotation and the observation of LLD [START_REF] Cooperstein | The relationship between pelvic torsion and anatomical leg length inequality: a review of the literature[END_REF][START_REF] Rhodes | The relationship between pelvic torsion and anatomical leg length inequality : a review of the literature[END_REF].

•

The EOS ® standing and sitting views of patients in their "normal or comfortable position" can detect this pelvic rotation with forward displacement of one side of the pelvis with corresponding backward displacement of the other side. On AP view, this is seen as an asymmetric projection of the iliac wings, with the anteriorized wing appearing thinner than the other. On lateral view, the two femoral heads and two iliac wings are not superimposed. Such "twisting" is hard to quantify on plain radiograph due to the conical spread of the radiographs, giving a misleading aspect to the femoral head more remote from the source. It is, in contrast, well analyzed on EOS ® imaging in the standing and sitting positions, with good 3D visualization of the position of the pelvis [START_REF] Rousseau | Assessment of the axial rotation of the pelvis with the EOS ® imaging system: intra-and inter-observer reproducibility and accuracy study[END_REF]. Three-D reconstruction clearly shows how the forward displacement of one half of the pelvis with respect to the other increases the "forward opening" of the hip joint (increased functional anteversion of the acetabulum). Conversely, the relative backward displacement of the other semi-pelvis induces acetabular functional retroversion on that side (Fig. 20).

•

We explored 30 patients (42 THA) with significant pelvic rotation (more than 10°). 12 patients had bilateral THA (24 hips). In these cases we could observe a significant asymmetry regarding the implants orientation in standing and sitting positions (Fig. 21).

CONCLUSION

Low-dose EOS imaging is an innovative slot-scanning radiograph system providing valuable data for the current practice of the THA surgeon as well as for spinal surgery. The simultaneous capture of two orthogonal AP and lateral images (like standard X-rays) while the patient is standing provides 2D and 3D measurements for the analysis of functional positions in THA patients. EOS has recently demonstrated accuracy and reliability in comparison to standard radiographs and CT scans, without significant inconvenience caused by the metallic artifacts of implants. We are aware of the limitations of this preliminary experience (limited number of patients, medical team with EOS experience) but some interesting information can be drawn.

For the first time, a global analysis can be performed in the true standing, sitting and even squatting This innovative approach to the relationship between hip joint and spine, and even between knee joint and spine, is the manifestation of a postural strategy conditioned by anatomic and functional characteristics that can differ greatly from person to person.

The relation between the position of the spine and the acetabulum has a direct influence on the real functional range of motion of the hips. Anterior pelvic plane and sacral slope variations are relevant parameters for planning and navigation, but the comparison of standing and sitting positions demonstrates the importance of outlier detection. The EOS ® imaging system provides valuable information regarding the pelvis functional anatomy in THA patients with clear applications for the study of unstable cases and potentially for the exploration of wear phenomena. Analysis of sagittal balance is relevant for the hip surgeon performing hip replacements in elderly subjects or in those with abnormal sagittal posture. This analysis must therefore be individual and integrated into the comprehensive evolution of the subject over time, because the phenomenon of an aging spine is frequently associated with the process of aging hips. New perspectives are open for more precise understanding of complex cases as limb length discrepancies or cases with pelvic rotation and obliquity as encountered in scoliotic patients. «"sky"view"»":" 3D"reconstruc4on" 3D""AP"pelvic"view" 3D""le9"lateral"pelvic"view" 3D"pelvic"view"from"below"

Fig. ( 1 )

 1 Fig. (1). Simultaneous AP and lateral EOS images in standing sitting and squatting positions.

Fig. ( 2 ).

 2 Fig. (2).Changes in SS and PT according to the standing or sitting position. "I" is an anatomical (morphological) angle and then no modification are observed between standing and sitting positions. From standing to sitting position, the lumbar lordosis reduces, and the lumbar spine becomes more flat according to the modification of the pelvic tilt. SS reduces in sitting position while PT increases.

Fig. ( 3 ).

 3 Fig. (3).Significant outliers can be observed. In this case, a moderate hip flexion can be observed in standing position. For sitting position, the hip flexion is very limited: pelvic extension is needed as a compensation (we can observe a simultaneous and coherent spinal kyphosis).

Fig. ( 4 ).

 4 Fig. (4).Standing sitting and squatting positions: the difference in SS corresponds to the available flexion of the lumbo-sacral junction (available extrinsic pelvic flexion), as distinct from potential hip-joint flexion (available intrinsic pelvic flexion). In the squatting position, pelvic extension is increased (case 1). Due to the limitation of hip flexion in some individuals, the positioning of the hips can be asymmetrical (case 2).

Fig. ( 6 ).

 6 Fig. (6). Comparison in standing and sitting positions for Frontal Acetabular Inclination angles (FAIA) on AP views and Sagittal Acetabular Inclination angles (SAIA) on lateral views.

Fig. ( 7 ).

 7 Fig.[START_REF] Philippot | Pelvic balance in sagittal and Lewinnek reference planes in the standing, supine and sitting positions[END_REF]. Definition of the Sacro-Acetabular Angle (SAA). In this standard case SAA is 75°.

Fig. ( 9

 9 Fig. (9). Abnormal SAA value: the cup is too horizontal (SAA is 47°).

Fig. ( 11

 11 Fig. (11).A fused lumbo-sacral junction can induce impingement as the pelvis remains with the same tilt in standing and sitting position: the main adjustment parameter is the hips range of motion. These 2 patients are standing as in a sitting position (posterior pelvic tilt or pelvic extension).

Fig. ( 13 )C

 13 Fig. (13). EOS 2D measurements (A) using exclusively AP view underestimate the offset in comparison with 3D measurements (B, C) using the combination of AP and lateral views

Fig. ( 14 ).

 14 Fig. (14). On the lateral view, the femoral sagittal tilt (FST) or femoral sagittal inclination angle is defined as the angle subtended by the vertical and the femoral axis between the center of the femoral head and Blumensaat's line. A: standard patient in standing position. B: hip flexion in standing position. C: hip hyper extension in standing position.

Fig. ( 15 ).

 15 Fig. (15). Comparison of standing and sitting AP views allows differentiation of oblique pelvis linked to lower limb discrepancy, as in the sitting position the frontal pelvic tilt disappears.

Fig. ( 17 )

 17 Fig. (17). EOS comparison before and after left THA revision: correction of the length of the left lower extremity; the left flexion has disappeared.

Fig. ( 19

 19 Fig.[START_REF] Polkowski | Does standing affect acetabular component inclination and version after THA?[END_REF]. THA revision (in this case bilateral acetabular revision) can induce complex modification of the posture in the standing position but significant increase of pelvic extension for sitting position (decrease of SS from 26° to 7°).

Fig. ( 20 ).

 20 Fig.[START_REF] Lazennec | Lumbar-pelvic-femoral balance on sitting and standing lateral radiographs[END_REF]. Pelvic rotation in standing position (asymmetric projection of the iliac wings, with the posteriorized wing appearing thicker than the other): in this case the backward displacement of the left hip with THA induces a decrease in the cup anterior opening.

Fig. ( 21 )

 21 Fig. (21).Bilateral THA case with thoraco-lumbar scoliosis (previous lumbo sacral fusion): Frontal imbalance in standing position is associated with significant pelvic rotation. On the lateral view the 2 femoral heads are not superimposed. One can observe the asymmetric projection of the iliac wings, with the anteriorized wing appearing thinner than the other: in this case the forward displacement of the left hip with THA induces an increase of the cup anterior opening (the patient experienced anterior subluxations in standing position).

Table 1 . Pelvic parameters in standing and sitting positions (300 THA cases).

 1

			Standing			Sitting
		Mean	SD	Min Max Mean	SD	Min	Max
	PT	18,3	11,2	-16	46,7	36	15,2	0	63
	SS	40,01 10,1	9	74	20,4	12	-10	55
	APPI
	(from 269	0,9	8,6	-29,7	20	23	13	-11	47
	cases )
	APPI
	(from
	243
	cases )

Table 2 . Results for acetabular inclination ( frontal and sagittal) in 300 cases. Standing Sitting Mean SD Min Max Mean SD Min Max

 2

	FAIA	47,4	10,1	19	87	58	14,4	20	86
	SAIA st
	260
	THAs	41,3	16	0	92	53	15,2	16	84
	available
	(85%)
	SAIA st
	225 THAs
	available
	(75%)

Table 3 . Results for acetabular inclination ( frontal and sagittal) in 205 cases. Mean SD Min Max Mean SD Min Max

 3

	Anatomical	34,7 15,4 -15	58
	anteversion
	Functional	36	16,7 -20	62	44	15,2	-15	85
	anteversion
	standing

Functional anteversion sitting.

ACKNOWLEDGEMENTS

Declared none.

CONFLICT OF INTEREST

The authors confirm that this article content has no conflict of interest.


Olivier Ozenda

Pierre Saramito

Guillaume Chambon

Tensorial rheological model for concentrated non-colloidal suspensions: normal-stress differences

. A systematic sensitivity study was performed to provide insights into the physical interpretation of the different model terms. This new model is compared to a large experimental dataset involving varying volume fractions, from dilute to concentrated cases. Both apparent viscosity and normal stress differences in steady-state, are quantitatively reproduced in the whole range of volume fraction, and qualitative agreement for transient evolution of apparent viscosity during shear-reversal is obtained. Furthermore, the model is validated against particle pressure measurements that were not used for parameter identification. Even if the proposed constitutive equation for the Cauchy stress tensor is more difficult to interpret than in the minimal model, this study opens way for the use of conformation tensor rheological models in applications where the effect of normal stress differences is prominent, like elongational flows or particle migration processes.

Introduction

Concentrated suspensions of non-colloidal rigid particles present two main non-Newtonian rheological trends, namely finite normal stress differences under shear flow and transient effects (see, e.g., the recent review by [START_REF] Guazzelli | Rheology of dense granular suspensions[END_REF]. (i) Since the early work of [START_REF] Gadala-Maria | The rheology of concentrated suspensions[END_REF], normal stress differences have been investigated in different flow geometries [START_REF] Zarraga | The characterization of the total stress of concentrated suspensions of noncolloidal spheres in newtonian fluids[END_REF][START_REF] Singh | Experimental measurements of the normal stresses in sheared stokesian suspensions[END_REF][START_REF] Couturier | Suspensions in a tilted trough: second normal stress difference[END_REF][START_REF] Boyer | Dense suspensions in rotating-rod flows: normal stresses and particle migration[END_REF][START_REF] Dai | Viscometric functions for noncolloidal sphere suspensions with newtonian matrices[END_REF][START_REF] Dbouk | Normal stresses in concentrated non-brownian suspensions[END_REF]. These measurements consistently show that the second normal stress difference N 2 is negative, with an absolute value that grows with volume fraction φ. The first normal stress difference N 1 is generally found to be much smaller, in absolute value, than |N 2 |. While most studies reported negative N 1 values, few also reported positive values especially at high volume fractions. (ii) Using shear-reversal experiments, [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF] showed that non-colloidal suspensions present transient rheological responses, characterized by a rate-independent evolution of the shear viscosity over typical strains of a few units, before this quantity eventually reaches its steady-state value. These shear-reversal experiments were later revisited by [START_REF] Kolli | Transient normal stress response in a concentrated suspension of spherical particles[END_REF] and [START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF]. All these authors related the transient effects to the development of microstructure anisotropy under shear flow, and attempted to quantify this anisotropy using the pair distribution function.

Most non-Newtonian trends of concentrated suspensions can be captured by particlebased numerical simulations, which offer additional insights into the physical processes at play. Earliest approaches were based on Stokesian dynamics, in which hydrodynamic interactions are split into far-field and near-field (lubrication) contributions [START_REF] Brady | The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation[END_REF]. Numerous studies showed that essential ingredients to obtain realistic rheological behaviour in these simulations have to be added, in particular, the existence of finite normal-stress differences, is to account for particle contacts through shortrange repulsive forces [START_REF] Nott | Pressure-driven flow of suspensions: simulation and theory[END_REF][START_REF] Sierou | Rheology and microstructure in concentrated noncolloidal suspensions[END_REF]. Numerical results also demonstrated that frictional interactions should be taken into account to recover correct orders of magnitude for these normal-stress differences, i.e. |N 1 | < |N 2 | [START_REF] Sierou | Rheology and microstructure in concentrated noncolloidal suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in non-brownian suspensions[END_REF]. Similar conclusions concerning the critical role of particle friction were reached by [START_REF] Gallier | Rheology of sheared suspensions of rough frictional particles[END_REF], using a more accurate DNS method to model hydrodynamic interactions. More recently [START_REF] Peters | Rheology of non-brownian suspensions of rough frictional particles under shear reversal: A numerical study[END_REF] and [START_REF] Chacko | Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data[END_REF] were also able to successfully simulate transient effects during shear-reversal using DNS and simplified Stokesian dynamics approaches, respectively. These particlebased simulations make it possible to continuously vary physical parameters and perform numerical experiments, thereby providing datasets that are well complementary to "real" experiments. However, due to the high demand in computing power, they are still limited to small systems with a relatively low number of particles (a few thousands at most).

Unlike discrete particle-based models, continuous rheological models are suited to the simulation of large systems. However, the elaboration of constitutive laws able to reproduce both normal-stress differences and transient features has been proved a challenging task (e.g., Denn & Morris 2014). General expressions for the bulk Cauchy stress tensor of suspensions can be obtained in the frame of two-phase mixture models [START_REF] Jackson | The dynamics of fluidized particles[END_REF][START_REF] Nott | The suspension balance model revisited[END_REF][START_REF] Baumgarten | A general fluid-sediment mixture model and constitutive theory validated in many flow regimes[END_REF], and involve contributions from both the interstitial fluid and the particles. For practical applications, physical closures need to be postulated for these different contributions. [START_REF] Nott | Pressure-driven flow of suspensions: simulation and theory[END_REF] introduced the suspension balance model (later extended by [START_REF] Morris | Curvilinear flows of noncolloidal suspensions: The role of normal stresses[END_REF][START_REF] Miller | Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions[END_REF], which involves explicit empirical expressions for the particle-induced normal stress components. This model provides relatively accurate predictions for steadystate flows and, when coupled to an evolution equation for the volume fraction φ (that arises from mass and momentum conservation of the particle phase), can also account for particle migration processes. A similar type of approach based on explicit stress closures was recently followed by [START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF], who proposed a model describing the transition to shear-thickening in highly-concentrated suspensions. By construction, however, these models do not attempt to capture transient rheological effects.

Transient effects can be described by introducing a conformation tensor, denoted b e in this paper, which represents microstructure anisotropy. Note that conformation tensors are also sometimes referred to as texture tensors (see e.g. [START_REF] Lehoucq | Analysis of image vs. position, scale and direction reveals pattern texture anisotropy[END_REF]. For noncolloidal suspensions, the conformation tensor b e needs to obey an evolution equation that ensures rate-independence, i.e. this equation should have no characteristic time (e.g., [START_REF] Goddard | Memory materials without characteristic time and their relation to the rheology of certain particle suspensions[END_REF]. [START_REF] Hand | A theory of anisotropic fluids[END_REF] formulated general properties for anisotropic fluids, and expressed the Cauchy stress tensor σ as a function of both b e and the strain rate tensor γ. [START_REF] Phan-Thien | Constitutive equation for concentrated suspensions in Newtonian liquids[END_REF] proposed a differential constitutive equation for the conformation tensor that led, for the first time, to predictions qualitatively in agreement with experimental observations during shear-reversal. [START_REF] Goddard | A dissipative anisotropic fluid model for non-colloidal particle dispersions[END_REF] revisited this approach, and proposed a model involving twelve material parameters and two tensors for describing the anisotropy. By a systematic fitting procedure on a limited dataset, he obtained numerical results in quantitative agreement with experiments for transient effects and normal stress differences. [START_REF] Stickel | A constitutive model for microstructure and total stress in particulate suspensions[END_REF] proposed a simplified expression for the Cauchy stress tensor, which is linear in terms of both the strain rate and the conformation tensors (see also [START_REF] Stickel | Application of a constitutive model for particulate suspensions: Time-dependent viscometric flows[END_REF][START_REF] Yapici | Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow[END_REF]. While this latter model involves thirteen free parameters, it failed to provide quantitative comparisons with shear-reversal experiments: an unexpected sharp spike in both apparent viscosity and normal stress differences was obtained at the time of the reversal. Using a much simpler model, involving only four free parameters and a linear evolution evolution for the conformation tensor, [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF] recently obtained a good quantitative agreement with the shear-reversal measurements of [START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF] for a wide range of volume fraction φ. Furthermore, at the micro-structural scale, the model successfully reproduced both the pair distribution function and the depletion angles measured by [START_REF] Blanc | Microstructure in sheared nonbrownian concentrated suspensions[END_REF]. However, this simple model predicts unphysical normal stress differences.

The objective of this paper is to propose a continuous rheological model that provides quantitative predictions for both normal stress differences and transient effects, for dilute to concentrated suspensions. For that purpose, we revisit and extend the model proposed by [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF] based on the general expansion of [START_REF] Hand | A theory of anisotropic fluids[END_REF]. We limit consideration to rate-independent constitutive responses, i.e. to suspensions dominated by hydrodynamic (and possibly frictional) interactions. We also restrict attention to concentration ranges sufficiently below jamming, such that granular plasticity effects arising from sustained frictional contacts between particles (e.g. granular dilatancy, [START_REF] Pailha | A two-phase flow description of the initiation of underwater granular avalanches[END_REF][START_REF] Baumgarten | A general fluid-sediment mixture model and constitutive theory validated in many flow regimes[END_REF] do not come into play. In particular, our model does not include the shear-thinning and shear-thickening effects observed in more complex systems and/or close to jamming transition (see, e.g., Royer et al. 2016;[START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF]. When compared with previous tensorial approaches ( [START_REF] Phan-Thien | Constitutive equation for concentrated suspensions in Newtonian liquids[END_REF][START_REF] Goddard | A dissipative anisotropic fluid model for non-colloidal particle dispersions[END_REF][START_REF] Stickel | A constitutive model for microstructure and total stress in particulate suspensions[END_REF]), the present model involves only seven adjustable parameters, and compares well with experiments. Section 2 presents the rheological model. Section 3 is devoted to simple shear flows. The system of time-dependent equations is expanded for this case, and the stationary solution is explicitly exhibited. The dependence of the material parameters upon volume fraction φ is investigated, based on asymptotic analyses in the dilute and highly concentrated limits. In section 4, model predictions are compared with a large number of experimental results regarding both transient effects and steady-state normal stresses. As the presented model involves seven dimensionless constants, a systematic sensivity study is performed in section 5. Finally, results are summarised and discussed in section 6.

Mathematical model

Problem statement

Following [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF], let us introduce the conformation tensor b e = d -2 0 ℓ ⊗ ℓ , where ℓ is the branch vector joining the centres of two neighbouring particles and d 0 is the average distance between neighbouring particle centers in an isotropic configuration at rest. In the isotropic configuration at rest, we have b e = I , where I is the identity matrix. For convenience, we introduce the tensor γ e = c 0 (b e -I), where c 0 is a dimensionless constant that will be chosen later. The tensor γ e interprets as the deformation of the micro-structure with respect to rest configuration. As in [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF], we assume a linear evolution for γ e :

D a γ e Dt + δ 1 | γ|γ e -γ = 0, (2.1a) γ = ∇u + ∇u T (2.1b)
where γ is the strain rate tensor, u is the velocity field of the suspension, and δ 1 is a positive dimensionless material parameter. The matrix norm |ξ| is defined as |ξ| 2 = (ξ : ξ)/2 for any matrix ξ, with (• : •) denoting the double contracted matrix product. We use in (2.1a) a general Gordon-Schowalter tensor derivative [START_REF] Gordon | Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions[END_REF][START_REF] Saramito | Complex fluids: modelling and algorithms[END_REF]:

D a γ e Dt = ∂γ e ∂t + (u.∇)γ e -W (u)γ e + γ e W (u) - a 2 ( γγ e + γ e γ)
where [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF] used an upper-convected derivative, corresponding to a = 1. Equation (2.1a) has no characteristic time, such that its solutions are rate-independent : In simple shear, changing the amplitude of the strain rate γ leaves the evolution of γe unchanged when expressed in terms of the dimensionless time | γ|t (that interprets as a strain).

W (u) = (∇u -∇u T )/2, and a ∈ [-1, 1]. Note that
The Cauchy stress tensor σ of the suspension is assumed to express as an analytical function of the two tensorial variables γ and γ e . Following [START_REF] Hand | A theory of anisotropic fluids[END_REF], and by virtue of Cayley-Hamilton theorem, only the zeroth, first and second powers of γ and γ e can contribute to the expression. Furthermore, to preserve rate-independence, terms in γ2 should also be discarded. After extensive tests to assess the effects of the different possible terms, we chose to consider the following expression for the Cauchy stress tensor:

σ = -p b I + η γ + η e δ 1 | γ|γ e + β γ 2 e γ + γγ 2 e 2 -| γ|γ 2 e + δ 2 γ e γ + γγ e 2 (2.1c)
where p b is the bulk pressure of the suspension, η and η e are characteristic viscosities, and β and δ 2 are additional dimensionless material parameters. Note that the terms in factor of δ 1 and δ 2 are linear in γ e , while the term in factor of β is quadratic in γ e . The present model represents an upgrade of that presented by [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF], which involved only one linear and one non-linear terms. For the sake of simplicity and of formulating a minimal model, the scalar coefficients η, η e β, η e δ 1 and η e δ 2 are assumed to be independent of the invariants of γ e , which is a restriction compared to the general theory of [START_REF] Hand | A theory of anisotropic fluids[END_REF].

Constitutive equations (2.1) are coupled with mass and momentum conservations to obtain a closed problem for three unknowns, namely the suspension pressure p b , the suspension velocity u, and the tensor describing the anisotropy of the micro-structure γ e :

D a γ e Dt + δ 1 | γ|γ e -γ = 0 (2.2a) ρ ∂u ∂t + (u.∇)u -div σ = 0 (2.2b) div u = 0 (2.2c)
where the Cauchy stress tensor σ is expressed by (2.1c) and the strain rate tensor γ is expressed by (2.1b) . Note that p b in (2.1c) should be regarded as a Lagrange multiplier associated to the incompressibility constraint (2.2c). This set of equations is closed by appropriate boundary and initial conditions for u and γ e .

From (2.2a), remark that the conformation tensor b e = c -1 0 (c 0 I + γ e ) satisfies:

D a b e Dt -c 0 δ 1 | γ|I + c 0 δ 1 | γ|b e = (1 -ac 0 ) γ
Hence, by choosing the dimensionless constant c 0 = 1/a, the right-hand-side in the previous relation is zero. In the following, we also assume a ∈ ]0, 1]. According to [START_REF] Hulsen | A sufficient condition for a positive definite configuration tensor in differential models[END_REF], and since δ 1 > 0, these choices guarantee that the conformation tensor b e is positive definite at any time, if this property is satisfied at initial time.

Simple shear and shear-reversal

The reduced problem

Let us consider a simple shear flow with uniform shear rate. The x axis is in the flow direction and the y axis is in the direction of velocity gradient, such that u(t, x, y, z) = (u x (t, y), 0, 0). Let us denote γ = ∂ y u x the uniform scalar shear rate, such that | γ| = | γ|. Evolution equation (2.2a) for γ e (t) then reduces to the following system of ordinary differential equations:

∂ t γ e,xx -(1+a) γ γ e,xy + δ 1 | γ| γ e,xx = 0 (3.1a) ∂ t γ e,yy + (1-a) γ γ e,xy + δ 1 | γ| γ e,yy = 0 (3.1b) ∂ t γ e,xy - 1+a 2 γ γ e,yy - 1-a 2 γ γ e,xx + δ 1 | γ| γ e,xy = γ (3.1c) ∂ t γ e,xz -(1+a) γ γ e,yz + δ 1 | γ| γ e,xz = 0 (3.1d ) ∂ t γ e,yz + (1-a) γ γ e,xz + δ 1 | γ| γ e,yz = 0 (3.1e) ∂ t γ e,zz + δ 1 | γ| γ e,zz = 0 (3.1f )
We assume γ e,xz = γ e,yz = γ e,zz = 0 at t = 0, such that these components remain zero at any time. The constitutive equation (2.1c) for σ becomes:

σ xx = -p b + η e δ 1 |
σ xz = σ yz = 0 (3.2e)
The system (3.1) is linear and admits explicit solutions when the shear rate γ is given. Explicit expressions of the stress components are then obtained from (3.2). In the following, we will also consider situations in which the shear stress σ xy is imposed. In this case, numerical solutions of the coupled equations (3.1)-(3.2) are computed using lsode ordinary differential equation solver [START_REF] Radhakrishnan | Description and use of lsode, the livermore solver for ordinary differential equations[END_REF] implemented within the numpy-scipy environment [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF].

Explicit solution in steady-state

The steady-state solution of (3.1a)-(3.1c) writes:

γ e,xx = 1 + a 1 -a 2 + δ 2 1 (3.3a) γ e,yy = -(1 -a) 1 -a 2 + δ 2 1 (3.3b) γ e,xy = δ 1 sgn( γ) 1 -a 2 + δ 2 1 (3.3c)
The steady-state apparent viscosity η app = σ xy / γ and normal stress differences N 1 = σ xx -σ yy and N 2 = σ yy -σ zz are then easily deduced from (3.2a)-(3.2d):

η app = η + η e 1 -a 2 + δ 2 1 δ 2 1 + aδ 2 + 1 + a 2 + δ 2 1 -2aδ 1 β 1 -a 2 + δ 2 1 (3.4a) N 1 = 2η e | γ| 1 -a 2 + δ 2 1 δ 1 - 2aβ 1 -a 2 + δ 2 1 (3.4b) N 2 = η e | γ| 1 -a 2 + δ 2 1 δ 1 (-1 + a + δ 2 ) + 2aδ 1 -(1-a) 2 + δ 2 1 β 1 -a 2 + δ 2 1 (3.4c)
We also define the particle pressure p p = p -p b , where p = -tr(σ)/3 is the total pressure of the suspension. In steady-state:

p p = - 2η e | γ| 3(1 -a 2 + δ 2 1 ) δ 1 (a + δ 2 ) + 2aδ 1 -1 + a 2 + δ 2 1 β 1 -a 2 + δ 2 1 (3.4d )

Dependence of material parameters upon volume fraction

The rheological model presented in section 2 involves six material parameters: two viscosities η and η e , and four dimensionless parameters a, δ 1 , δ 2 and β. Preliminary tests to fit experimental data for different volume fractions φ (see § 4), showed that all these parameters, except a, have to vary with φ. Following these preliminary investigations, we assumed a to be independent of φ, and the five remaining parameters to depend only upon the reduced volume fraction ψ = φ/φ m , with φ m the maximal volume fraction of the suspension. This latter assumption is required to obtain a model that can be applied to different experimental datasets characterised by different values of φ m . In practice, φ m can vary between 0.53 and 0.64, typically, depending on particle shape, particle roughness, etc (e.g., [START_REF] Guazzelli | Rheology of dense granular suspensions[END_REF].

As shown in appendix A, the parameter δ 1 is directly related to the depletion angle θ e , i.e. the angle between the x axis and the direction of the eigenvector associated to the largest eigenvalue of γ e : δ 1 = tan(2θ e ). Based on the experimental measurements of θ e provided by [START_REF] Blanc | Microstructure in sheared nonbrownian concentrated suspensions[END_REF], the following dependence law is proposed for δ 1 (see figure 1):

δ 1 (ψ) = δ1 (1 -ψ) -1 -(1 -bψ) , (3.5a)
where δ1 and b are positive constants independent of ψ. Observe that δ 1 (0) = 0 and δ 1 (1) = ∞, such that the model predicts θ e (0) = 0 and θ e (1) = π/4.

To be consistent with numerous existing results [START_REF] Maron | Application of Ree-Eyring generalized flow theory to suspensions of spherical particles[END_REF][START_REF] Morris | Curvilinear flows of noncolloidal suspensions: The role of normal stresses[END_REF][START_REF] Guazzelli | Rheology of dense granular suspensions[END_REF], the apparent viscosity η app and the second normal stress difference N 2 should behave as (1 -ψ) -2 in the ψ → 1 limit. In addition, we expect η e = 0 in the Newtonian limit ψ = 0. Accordingly, the following dependence laws are postulated for the viscosities η and η e :

η(ψ) = η 0 1 -ω + ψ 5 2 φ m -2ω + ω(1 -ψ) -2 (3.5b) η e (ψ) = η 0 ηe ψ(1 -ψ) -2 (3.5c)
where η 0 is the viscosity of the suspending fluid, and ηe > 0 is a constant independent of ψ. The constant ω ∈ ]0, 1[ is introduced in (3.5b) in order to recover [START_REF] Einstein | Eine neue bestimmung der moleküldimensionen[END_REF]'s relation when ψ → 0 (see §3.4). Note that η expresses as a convex combination of [START_REF] Maron | Application of Ree-Eyring generalized flow theory to suspensions of spherical particles[END_REF] and [START_REF] Einstein | Eine neue bestimmung der moleküldimensionen[END_REF] laws.

Similar to η e , the parameters δ 2 and β should vanish when ψ = 0, i.e. when the suspension reduces to a Newtonian fluid. For simplicity, we consider that δ 2 is proportional to δ 1 . We also assume that β behave as (1 -ψ) -2 when ψ → 1. Accordingly, the following dependence laws are postulated:

δ 2 (ψ) = δ2 (1 -ψ) -1 -(1 -bψ) (3.5d ) β(ψ) = β (1 -ψ) -1 -1 2 , i = 1, 2, 3 (3.5e)
where δ2 and β are constants independent of ψ.

At this stage, the suspension model involves seven constants independent of ψ, namely a, b, ω, ηe , δ1 , δ2 , and β that need to be determined from experimental data. Note that with the dependence laws (3.5b)-(3.5c) for η and η e , we obtain from (3.4) that the normal stress ratios

α 1 = N 1 η app | γ| and α 2 = N 2 η app | γ|
are independent of suspending fluid viscosity η 0 , in agreement with experiments [START_REF] Dbouk | Normal stresses in concentrated non-brownian suspensions[END_REF][START_REF] Guazzelli | Rheology of dense granular suspensions[END_REF].

Dilute and concentrated limits

In the dilute limit ψ → 0, a second-order Taylor expansion of (3.4a) leads to the following expression for apparent viscosity η app :

η app = η 0 1 + 5 2 φ m ψ + 3ω 2 + a(1 + b) δ2 ηe (1 -a 2 ) ψ 2 + O ψ 3 (3.6a)
At first order in ψ, the model agrees with [START_REF] Einstein | Eine neue bestimmung der moleküldimensionen[END_REF]'s relation η app ≈ η 0 (1 + 5φ/2).

The different constants associated to micro-structure evolution, a, b, ηe and δ2 , appear only at second order, consistently with the asymptotic expansion of [START_REF] Batchelor | The determination of the bulk stress in a suspension of spherical particles to order c 2[END_REF].

In the concentrated limit ψ → 1, the expansion of η app writes:

η app = η 0 η1 (1 -ψ) -2 + O (1 -ψ) -1 (3.6b) with η1 = ω + 1 + β δ2 1 ηe
Hence, η app grows as (1 -ψ) -2 , as required [START_REF] Maron | Application of Ree-Eyring generalized flow theory to suspensions of spherical particles[END_REF].

Let us now turn to the asymptotic behaviour of normal stress ratios α 1 and α 2 and particle pressure p p . In the dilute limit ψ → 0, from (3.4), we obtain:

α 1 = 2η e δ1 (1 + b) 1 -a 2 ψ 2 + O ψ 3 α 2 = - (1 -a)η e δ1 (1 + b) 1 -a 2 ψ 2 + O ψ 3 p p η app | γ| = - 2aη e δ1 (1 + b) 3(1 -a 2 ) ψ 2 + O ψ 3
Observe that α 1 , α 2 and p p all vanish for ψ = 0, as expected. Moreover, we obtain α 1 0 and α 2 0 when ψ → 0. The sign of α 2 is consistent with experimental observations [START_REF] Couturier | Suspensions in a tilted trough: second normal stress difference[END_REF][START_REF] Dbouk | Normal stresses in concentrated non-brownian suspensions[END_REF][START_REF] Dai | Viscometric functions for noncolloidal sphere suspensions with newtonian matrices[END_REF]).

In the concentrated limit ψ → 1, the expansions of normal stress ratios and particle pressure write:

α 1 = 2η e η1 δ1 (1 -ψ) + O (1 -ψ) 2 α 2 = - ηe η1 -δ1 δ2 + β δ2 1 + O (1 -ψ) p p η app | γ| = 2η e 3η 1 -δ1 δ2 + β δ2 1 + O (1 -ψ)
Hence, the ratio α 1 vanishes when ψ → 1. On the contrary, α 2 tends to a constant for ψ → 1, in agreement with experiments [START_REF] Leighton | The shear-induced migration of particles in concentrated suspensions[END_REF][START_REF] Guazzelli | Rheology of dense granular suspensions[END_REF]. Furthermore, it is sufficient to assume δ2 < 0 and β 0 to obtain α 2 0.

Quantitative comparisons with experiments

Identification of model constants

As explained above, the rheological model proposed in this paper was built to recover qualitative agreement with experimental observations in the dilute and concentrated limits. More quantitatively, the seven model constants were identified through an adjustment to different experimental datasets reporting detailed rheological measurements on non-colloidal suspensions for different volume fractions φ (in the range [0.2, 0.5]). The four following datasets were considered for the adjustment:

• Measurements of the depletion angle θ e performed by [START_REF] Blanc | Microstructure in sheared nonbrownian concentrated suspensions[END_REF]. These authors used a suspending fluid of viscosity η 0 = 0.85 Pa.s and two sets of PMMA spheres of radius r p = 80 and 90 µm. They reported a maximal volume fraction φ m = 0.57.

• Measurements of transient apparent viscosity after a shear-reversal performed by [START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF] for five different values of volume fraction φ. These authors used a suspending fluid of viscosity η 0 = 1.02 Pa.s, PMMA spheres of radius r p = 16 µm, and reported a relatively low maximal volume fraction φ m = 0.535 presumably due to the presence of a residual cellulosic surfactant.

• Measurements of normal-stress differences in steady-state performed by [START_REF] Couturier | Suspensions in a tilted trough: second normal stress difference[END_REF]. These authors used a suspending fluid with viscosity η 0 = 2.15 Pa.s, polystyrene spheres of radius r p = 35 µm, and reported a maximal volume fraction φ m = 0.62.

• Measurements of normal-stress differences in steady-state performed by [START_REF] Dai | Viscometric functions for noncolloidal sphere suspensions with newtonian matrices[END_REF]. These authors used a slightly shear-thinning suspending fluid with viscosity 1.17 η 0 1.3 Pa.s for shear rates 0.01 γ 100 s -1 , polystyrene spheres of radius r p = 20 µm, and reported a maximal volume fraction φ m = 0.62. For all these datasets, values of φ were rescaled in terms of reduced volume fraction ψ = φ/φ m by using the provided values of φ m . As described in appendix B, the set of seven model constants {a, ω, ηe , δ2 , β1 , β2 β3 } was globally identified from the datasets. The obtained values are indicated in table 1. Note that first attempts at trying to obtain a comparable quantitative agreement with both steady-state and transient measurements, failed. As a consequence, we chose to assign a stronger weight to the agreement with steady-state viscosity and normal-stress data. A lower weight was assigned to the agreement with transient experiments, although sufficient to ensure that model predictions present a correct qualitative behavior. More details on the identification procedure, including expression of the minimised cost function, are provided in appendix B.

Direct comparisons between model predictions with the constants given in table 1, and the experimental data used for the identification, are shown in figures 1, 2 and 3. Figure 1 shows that predicted steady-state depletion angle and apparent viscosity are indeed in excellent quantitative agreement with the experimental data. Figure 2 presents comparisons for the transient response, namely evolution of apparent viscosity η app (t) versus strain γ(t) = t 0 | γ(s)|ds after a shear-reversal. Let us recall that the experiments of [START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF] were performed at imposed shear stress, while γ(t) varies. Note that both model predictions and experimental data are normalised by their respective steady state-values η app (∞) in the figure. It is observed that, although the model generally tends to underestimate the amplitude of the transient response, a satisfactory qualitative behavior is nevertheless obtained for all values of φ. In particular, the successive phases present in the measurements, namely the initial brutal drop in apparent viscosity, the smooth evolution to a minimum, and the relaxation towards the steady-state, are all reproduced. The typical duration of the transient phase is also correctly captured.

Figure 3 presents the evolution of steady-state normal stress ratios α 1 and α 2 as a Values of α 1 are positive but very small for ψ 0.4, negative and decreasing with ψ for 0.4 ψ 0.7, and then increasing again towards the concentrated limit α 1 (1) = 0 (see § 3.4). Here also, these predictions appear to be in good quantitative agreement with the measurements in terms of overall magnitude, albeit the existence of non-monotonic variations cannot be confirmed from existing data due to the level of experimental noise. For comparison, experimental measurements obtained by [START_REF] Dbouk | Normal stresses in concentrated non-brownian suspensions[END_REF] (not used for the adjustment) are also shown in figure 3, using a value φ m = 0.58 for the volume fraction scaling. These measurements are fully consistent with the two other datasets and the model for α 2 , but show positive values of α 1 , thereby highlighting the need for additional characterisations of the first normal-stress difference in these systems.

Comparison with independent particle pressure measurements

Model predictions were also assessed against steady-state particle pressure measurements obtained by [START_REF] Deboeuf | Particle pressure in a sheared suspension: a bridge from osmosis to granular dilatancy[END_REF]. These authors used a suspending fluid of viscosity η 0 = 3 Pa.s and differents sets of particles with radii ranging between 40 and 140 µm.

As they did not provide the value of maximum volume fraction φ m in their systems, we used φ m = 0.58 to rescale their data. We insist that this new dataset was not used for the identification of model constants. As shown in figure 4, model predictions obtained with the values of constants given in table 1, show here also a good agreement with the experimental measurements. This remarkable result validates the introduction of reduced volume fraction ψ = φ/φ m in the dependence laws for the material parameters (see §3.3), and indicates that the model constants involved in these dependence laws can be regarded, at least to a first approximation, as independent of the experimental conditions. In other words, the influence of experimental conditions is fully encoded in the maximum volume fraction φ m used to rescale the volume fraction values.

Sensitivity study

A systematic sensitivity study of model predictions with respect to the seven model constants a, b, ω, ηe , β, , δ1 , δ2 , was performed. Figure 5 shows the evolution of steadystate normal stress ratios α 1 and α 2 and steady-state apparent viscosity η app versus ψ (upper three plot lines), as well as the transient evolution of η app after a shear-reversal (lower plot line), when model constants are varied in consistent [-50%, +100%] ranges around their reference values obtained through identification (see §4).

It is observed that a variation of any of the model constant implies a significant effect on at least one of the displayed quantities. Hence, none of the terms introduced in the Cauchy stress equation (2.1c) can be omitted without significantly degrading the match with experimental observations. We also note the existence of non-trivial couplings between these different terms, which explains the difficulties encountered in the identification procedure when trying to simultaneously reproduce steady-state and transient experimental data (see §4). Note that unphysical predictions |α 1 | |α 2 | are obtained when values of δ1 are close to zero, values of β are too high, or when a ≈ 1.

A closer look on the results provides insights into the physical interpretation of the different model constants. In steady-state, the constant ηe acts as a magnitude scaling for all stress terms that are non-colinear to the strain rate γ (see (3.4)). Accordingly, this affects both normal stress ratios α 1 and α 2 . Conversely, β only acts on terms that are quadratic in γ, and mostly affects the first normal stress ratio α 1 . Both ηe and β also play a role on the steady-state viscosity η app , but have essentially no influence on the minimum observed during the transient evolution of η app .

The constant δ1 interprets as a characteristic strain controlling the length of the transients. In particular, this constant sets the position of the minimum in η app . This constant also plays a strong role on the steady-state values of apparent viscosity and normal-stress ratios. This ubiquitous role is consistent with the fact that δ 1 is the only parameter present in both the constitutive equation for σ (2.1c) and the evolution equation for γ e (2.2a).

The constants δ2 , a and b do not have much influence on the apparent viscosity η app . The constant δ2 essentially acts on the second normal stress ratio α 2 , while the constants a and b have a more pronounced effect on α 1 . However, an unphysical change of sign of α 2 is observed when a ≈ 1. These observations are consistent with equations (3.4).

Finally, the constant ω is responsible for an overall shift of the apparent viscosity η app in both transient and steady-state regimes, but only weakly affects the normal stress ratios α 1 and α 2 .

Discussion and conclusions

An improved, rate-independent rheological model for non-colloidal suspensions of rigid spheres, involving the evolution of a conformation tensor, was presented. The model upgrades that of [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF] in several aspects. While the structure of the linear evolution equation (2.1a) for the conformation tensor is unchanged, a more general Gordon-Schowalter tensor derivative is used, with a material parameter a ∈ ]0, 1[. In addition, the expression of the Cauchy stress tensor (2.1c) is modified by the inclusion of additional linear and non-linear terms. While still keeping with the goal of a minimal model involving as few adjustable parameters as possible, these amendments were required to predict both the transient evolution of apparent viscosity after a shearreversal, and realistic values for the normal stress difference N 1 and N 2 in steady-state. Although the contributions of the different terms to the overall rheological behavior of the suspension appear to be strongly coupled, a systematic sensitivity study was performed to provide insights into the physical interpretation of the different model parameters. Note however that the sensitivity study was performed for limited ranges of the parameters, and might thus not be representative of the model behavior when parameters are varied in much larger ranges.

The model was compared to different experimental datasets involving various suspending fluids and particle sizes. To that aim, the material parameters were all expressed in terms of reduced volume fraction ψ = φ/φ m , where the maximum volume fraction φ m varies among the experiments, and a of set of seven model constants. The good agreement observed with steady-state data, in terms of apparent viscosity, normal stress differences and particle pressure, suggests that the rheological model, together with the constants given in table 1, can be applied to large range of suspensions. Regarding the transient responses, only qualitative agreement was obtained with experimental data. Further improvements of these transient predictions would certainly require complementing our minimal model with additional terms, for instance polynomial dependences of the material parameters upon the invariants of the texture tensor [START_REF] Hand | A theory of anisotropic fluids[END_REF], or introduction of additional microstructural tensorial variables [START_REF] Goddard | A dissipative anisotropic fluid model for non-colloidal particle dispersions[END_REF]. Along the same line, introduction of a plasticity term in the constitutive law would open interesting perspectives to tackle highly-concentrated regimes close to, or above, the jamming transition, in which the rheological behavior becomes dominated by sustained granular contacts [START_REF] Pailha | A two-phase flow description of the initiation of underwater granular avalanches[END_REF][START_REF] Baumgarten | A general fluid-sediment mixture model and constitutive theory validated in many flow regimes[END_REF].

The capability of the presented model to predict accurate normal-stress differences in steady-state potentially opens way to computations of elongational or more complex flows. Such applications to a larger class of flows, together with a more systematic parameter identification procedure, shall also make it possible to improve the physical interpretation of the different terms introduced in the expression of the Cauchy stress tensor. Here also, detailed comparisons with experiments (e.g. [START_REF] Dai | Elongational flows of some non-colloidal suspensions[END_REF] will however be necessary to assess the validity of the model, and particularly of the dependences of material parameters upon volume fraction introduced in §3.3, beyond the simple shear configuration investigated in this study. A more in-depth analysis of the mathematical and thermodynamical properties of the model would also need to be undertaken. The new rheological model could also be coupled to an evolution equation for the volume fraction in order to address particle migration processes (e.g., [START_REF] Miller | Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions[END_REF][START_REF] Haddadi | Suspension flow past a cylinder: particle interactions with recirculating wakes[END_REF], test the effect of normal-stress differences over longer time scales, and allow for comparisons with other modelling approaches (e.g., [START_REF] Dbouk | A suspension balance direct-forcing immersed boundary model for wet granular flows over obstacles[END_REF]. Finally, let us also note that, although the focus of the present paper was mainly on trying to reproduce physical experiments, systematic comparisons with discrete simulation results (e.g. [START_REF] Chacko | Shear reversal in dense suspensions: the challenge to fabric evolution models from simulation data[END_REF][START_REF] Singh | A constitutive model for simple shear of dense frictional suspensions[END_REF]) could constitute a fruitful avenue for further validating and improving our rheological model. In particular, comparisons regarding the evolution of normal stress differences during transients, for which no experimental data are currently available, could provide useful additional insights.

Figure 1 .

 1 Figure 1. Evolution of depletion angle θe and steady-state apparent viscosity ηapp versus reduced volume fraction ψ: comparison between model predictions and experimental data of Blanc et al. (2013) (θe) and Blanc et al. (2011) (ηapp).

Figure 2 .

 2 Figure2. Evolution of apparent viscosity ηapp versus shear strain γ after shear-reversal: comparison between model prediction and experimental measurements of[START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF] for two volume fractions φ.

Figure 3 .

 3 Figure 3. Evolution of steady-state normal stress ratios α1 and α2 versus reduced volume fraction ψ: comparison between model predictions and experimental measurements of Couturier et al. (2011), Dai et al. (2013) and Dbouk et al. (2013).

Figure 4 .

 4 Figure 4. Evolution of steady-state normalized particle pressure pp/(η0| γ|) versus reduced volume fraction ψ: comparison between model predictions and experimental measurements of Deboeuf et al. (2009). The consistent data obtained for three different values of particle radius rp are shown.

Figure 5 .

 5 Figure 5. Upper three plot lines: Evolution of steady-state normal stress ratios α1, α2 and apparent viscosity ηapp versus reduced volume fraction ψ as the seven model constants ω, ηe, δ1, δ2, a, b, β vary around their reference values indicated in table 1 (see legends). Lower plot line : Idem for the transient evolution of apparent viscosity ηapp after a shear-reversal (case φ = 0.44 and φm = 0.535).

  γ|γ e,xx + β γ (γ e,xx + γ e,yy ) γ e,xy -| γ| γ 2 e,xx + γ 2 σ yy = -p b + η e δ 1 | γ|γ e,yy + β γ (γ e,xx + γ e,yy ) γ e,xy -| γ| γ 2 e,xy + γ 2

						e,xy	+ δ 2 γγ e,xy	(3.2a)
						e,yy	+ δ 2 γγ e,xy	(3.2b)
	σ zz = -p b					(3.2c)
	σ xy = η γ + η e δ 1 | γ|γ e,xy + β	γ 2	γ 2 e,xx + γ 2 e,yy + 2γ 2 e,xy -| γ| (γ e,xx + γ e,yy ) γ e,xy
	+	δ 2 2	γ	(γ e,xx + γ e,yy )	(3.2d )

Table 1 .

 1 Values of the seven dimensionless model constants identified from experimental data.

	a	b	ω	ηe	δ1	δ2	β
	0.51	1.5	0.52	0.089	0.47	-1.15	1.7

† Email address for correspondence: pierre.saramito@math.cnrs.fr

Declaration of interests

The authors report no conflict of interest.

Appendix A. Eigen directions of γ e

We prove here that, in stationary simple shear flow, the depletion angle, i.e. the angle between the x axis and the direction of the eigenvector associated to the largest eigenvalue of γ e , expresses as:

From (3.3a)-(3.3c), eigenvalues of γ e , denoted as λ min λ max , are both non-zero and write:

The corresponding eigenvectors are:

1 Accordingly, we obtain:

which is equivalent to (A 1). Observe that ∀ δ 1 > 0, θ e ∈ ]0, π/4[.

Remark that, while the eigenvalues of γ e depend on the parameter a of the tensor derivative, the eigenvectors are independent thereof. Hence, all the relations between microstructure orientation and model parameters derived in [START_REF] Ozenda | A new rate-independent tensorial model for suspensions of noncolloidal rigid particles in newtonian fluids[END_REF], notably (A 1), remain unchanged in the present model.

Appendix B. Identification of model constants

The rheological model proposed in this paper involves seven constants to be identified: a, b, ω, ηe , β, , δ1 , δ2 . Global identification of these parameters with respect to experimental data was achieved by minimising the following cost function:

The function θ obs e (ψ) is a piecewise-constant interpolation of steady-state depletion angle measurements reported by [START_REF] Blanc | Microstructure in sheared nonbrownian concentrated suspensions[END_REF] for ψ ∈ [ψ 1 , ψ 2 ], with ψ 1 = 0.088 and ψ 2 = 0.98 the range of reduced volume fraction explored in the experiments. The function η obs app (ψ) is a piecewise-constant interpolation of steady-state values of apparent viscosity measured by [START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF] for ψ ∈ [ψ 3 , ψ 4 ], with ψ 3 = 0.56 and ψ 4 = 0.93 in this case. The functions α obs 1 (ψ) and α obs 2 (ψ) are piecewise-constant interpolations of steady-state values of normal stress ratios measured by [START_REF] Couturier | Suspensions in a tilted trough: second normal stress difference[END_REF]) and [START_REF] Dai | Viscometric functions for noncolloidal sphere suspensions with newtonian matrices[END_REF] for ψ ∈ [0, ψ m ], with ψ m = 0.81 the maximal reduced volume fraction reached in the experiments. The functions η obs app,k (γ), k = 1 . . . 5, represent the transient evolutions of apparent viscosity during the five shear-reversal experiments of [START_REF] Blanc | Local transient rheological behavior of concentrated suspensions[END_REF] characterised by reduced volume fractions ψ k . Here, γ k , k = 1 . . . 5, represent the maximal strain reached in each experiment. Lastly, the function η app (ψ, γ) is associated to the transient solution of the model, while the functions θ e (ψ), η app (ψ), α 1 (ψ), and α 2 (ψ) are associated to the steady-state solution. The factors w a , w e , w s,i and w t,k are the relative weights of the different terms of the cost function J.

The minimization problem of J was solved with a Levenberg-Marquardt method [START_REF] More | User guide for minpack-1[END_REF], using the optimisation library lmfit [START_REF] Newville | Lmfit: non-linear least-square minimization and curve-fitting for python[END_REF]. As explained in §4, larger values had to be used for the weights w a , w e , w s,i , with respect to w t,k , in order to promote the agreement with steady-state measurements. In addition, after the global identification step, a secondary, one-by-one, adjustment of each model constants was performed in order to further improve the solution. Final values of identified model constants are indicated in table 1.