Q Grimal

P Laugier

Quantitative ultrasound assessment of cortical bone properties beyond bone mineral density

Keywords: ultrasound, cortical bone, elasticity, porosity, thickness

The development of quantitative ultrasound (QUS) technologies to measure bone is motivated by the need to overcome the limitations of X-ray based methods, measuring bone mineral density (BMD) which is the gold standard to date for the diagnosis of osteoporosis. Because it uses mechanical waves, the ultrasound modality is a particularly relevant means to probe bone mechanical resistance. The vast majority of QUS technologies commercialized to date merely aim to provide surrogate markers for BMD.

During the past decade, innovative QUS approaches have emerged to assess bone beyond BMD. This may be achieved by (1) specifically assessing the cortical bone compartment, independently of trabecular bone, and (2) providing intrinsic bone properties such as cortical bone thickness and material properties. One specific motivation is to estimate intracortical porosity, a quantity reflected in material properties. This article aims at an overview of recent QUS developments to measure cortical bone properties. We also draw a picture of the current knowledge on bone material properties of interest for bone QUS. We discuss the potential of ultrasound to provide novel biomarkers of bone health through the assessment of material properties.

imaging; velocity

Graphical abstract

Highlights

• Bone fragility assessment would benefit from an accurate evaluation of cortical bone Preprint submitted to Elsevier October 23, 2019 • Innovative QUS technologies aim to measure intrinsic properties of cortical bone

• Available technologies measure bone thickness and bulk wave velocities

• Intracortical porosity, a fingerprint of remodeling, can be deduced from material properties

• Ex vivo documentation of material properties of pathological bone tissues is lacking

Reference for citation

Grimal Q, Laugier P. Quantitative ultrasound assessment of cortical bone properties beyond bone mineral density. IRBM. 2018 Volume: 40 Issue: 1

Pages: 16-24

Introduction

Bone fragility associated to primary or secondary osteoporosis and the consequent risk of fracture is an important medical threat. Among the population aged over 50 years old, one in three women and one in five men will suffer a fracture associated to osteoporosis. Nine million fragility fractures occurred annually worldwide at the beginning of the 21th century [1]. Fracture risk prediction is assessed based on clinical factors and, in the standard approach, dual energy X-ray absorptiometry (DXA) in order to assess bone mineral density (BMD). However, it is well accepted that BMD assessed with DXA has strong limitations, in particular it has a lack of sensitivity [2, 3],

and DXA is a ionizing method.

Our bones are comprised of two types of porous tissues: cortical bone is the dense tissue that forms the outer shell of the bones; trabecular, or spongious, bone is the more porous tissue partly filling the bones. While for several decades, bone alteration in osteoporosis has essentially been described to be trabecular bone loss [START_REF] Parfitt | Misconceptions (2): Turnover is always higher in cancellous than in cortical bone[END_REF], a focus has been placed in recent years on cortical bone which has been recognized to also play a key role in bone resistance in particular at fracture sites such as the proximal femur [START_REF] Paul M Mayhew | Relation between age, femoral neck cortical stability, and hip fracture risk[END_REF][START_REF] Holzer | Hip fractures and the contribution of cortical versus trabecular bone to femoral neck strength[END_REF] and the distal radius [START_REF] Augat | Prediction of fracture load at different skeletal sites by geometric properties of the cortical shell[END_REF]. Aging is associated with an increased cortical porosity and thinning of the cortical shell. In old age, 70% of all appendicular bone loss may arise from the cortical compartment [START_REF] Zebaze | Intracortical remodelling and porosity in the distal radius and post-mortem femurs of women: a crosssectional study[END_REF]. It follows that including an accurate evaluation of cortical bone in skeletal status assessment could improve diagnosis and treatment monitoring.

Cortical bone mechanical properties depend on the properties of the pore network (volume fraction of pores or, shortly, the porosity, and microarchitecture) and the properties of the extracellular mineralized matrix (shortly, matrix) surrounding pores. Pathologies, aging, and treatments may alter matrix material properties through modifications of collagen and mineral [START_REF] Ammann | Strontium ranelate treatment improves trabecular and cortical intrinsic bone tissue quality, a determinant of bone strength[END_REF][START_REF] Bala | Bone micromechanical properties are compromised during long-term alendronate therapy independently of mineralization[END_REF][START_REF] Imbert | Mechanical and mineral properties of osteogenesis imperfecta human bones at the tissue level[END_REF][START_REF] Kühnisch | Multiscale, converging defects of macro-porosity, microstructure and matrix mineralization impact long bone fragility in NF1[END_REF] and the pore network [START_REF] V Bousson | Distribution of intracortical porosity in human midfemoral cortex by age and gender[END_REF][START_REF] Zebaze | Intracortical remodelling and porosity in the distal radius and post-mortem femurs of women: a crosssectional study[END_REF][START_REF] Imbert | Mechanical and mineral properties of osteogenesis imperfecta human bones at the tissue level[END_REF]. In the last years, cortical bone porosity has been increasingly recognized as a fracture risk factor [START_REF] Ahmed | Measurement of cortical porosity of the proximal femur improves identification of women with nonvertebral fragility fractures[END_REF][START_REF] Bala | Role of cortical bone in bone fragility[END_REF][START_REF] Zebaze | Denosumab Reduces Cortical Porosity of the Proximal Femoral Shaft in Postmenopausal Women With Osteoporosis[END_REF].

Cortical bone thickness (CTh) is also a critical quantity for the stability of a bone during daily activities and in falls [START_REF] Paul M Mayhew | Relation between age, femoral neck cortical stability, and hip fracture risk[END_REF][START_REF] Nishiyama | Postmenopausal women with osteopenia have higher cortical porosity and thinner cortices at the distal radius and tibia than women with normal aBMD: An in vivo HR-pQCT study[END_REF]. A reduction of CTh is usually associated to an increase of porosity [START_REF] Zebaze | Denosumab Reduces Cortical Porosity of the Proximal Femoral Shaft in Postmenopausal Women With Osteoporosis[END_REF].

Techniques to assess specifically the cortical compartment of the skeleton have been developed recently. These include X-ray computed tomography (CT) [START_REF] Geusens | High-resolution in vivo imaging of bone and joints: A window to microarchitecture[END_REF], indentation to specifically probe bone matrix [START_REF] Diez-Perez | Microindentation for in vivo measurement of bone tissue mechanical properties in humans[END_REF], and quantitative ultrasound (QUS) methods. X-ray CT is extensively used for bone quantitative imaging, but essentially in clinical research. The most advanced X-ray CT technique is high-resolution peripheral computed tomography (HR-pQCT), a method available since 2004. It is a 3-D imaging technique, that allows a quantitative analysis of the cortical and trabecular bone compartments with a physical resolution of the order of 100 µm. It yields estimates of bone density, microarchitecture, and geometry. It can be used to measure CTh and, to some extent, porosity, at the distal radius and tibia. This modality, however, like conventional CT, will unlikely be used as a widespread diagnostic tool for osteoporosis due to cost issues and ionizing radiations. The focus of the present review is on bone QUS methods to measure cortical bone.

In the past three decades, researchers have developed QUS methods to measure trabecular and cortical bone [START_REF] Laugier | Instrumentation for in vivo ultrasonic characterization of bone strength[END_REF][START_REF] Raum | Ultrasound to assess bone quality[END_REF] [START_REF] Mccloskey | Predictive ability of heel quantitative ultrasound for incident fractures: an individual-level meta-analysis[END_REF] measuring in transmission the heel bone (mainly composed of trabecular bone), as well as recently developed pulse-echo techniques targeting different skeletal sites such as the spine and hip [START_REF] Casciaro | New perspectives in echographic diagnosis of osteoporosis on hip and spine[END_REF][START_REF] Karjalainen | Pulse-echo ultrasound method for detection of post-menopausal women with osteoporotic BMD[END_REF]. The ultrasound surrogate for BMD predicts fracture risk, although, compared to DXA, it has not shown its superiority [START_REF] Glüer | A new quality of bone ultrasound research[END_REF][START_REF] Laugier | Instrumentation for in vivo ultrasonic characterization of bone strength[END_REF] to date.

Another strategy in bone QUS research has been to measure intrinsic bone properties that convey information beyond BMD, such as CTh and material properties, such as bulk wave velocities. In that vein, approaches have been proposed during the past decade such as pulse-echo [START_REF] Karjalainen | Ultrasonic assessment of cortical bone thickness in vitro and in vivo[END_REF] and axial transmission [START_REF] Petro Moilanen | Ultrasonic guided waves in bone[END_REF] techniques which we cover in this review. The development of such QUS approaches requires a priori knowledge of physiological values of bone material properties and of their expected ranges variation in different pathologies.

This paper reviews the recent progress in QUS approaches aiming at the evaluation of bone fragility through the measurement of the thickness and material properties of cortical bone. We also draw a picture of the current knowledge on bone material properties of interest for in vivo ultrasound eval-uation. After this introduction as background, we review in section 2 cortical bone microanatomy and material properties. Section 3 presents QUS approaches to measure cortical bone. In Section 3.2, we account for the recent research aiming at a concurrent assessment of cortical bone anatomy and material properties. In Section 4, we discuss the potential of ultrasound to provide novel biomarkers of bone health through the assessment of material properties.

Cortical bone tissue

Anatomy

At its highest level of hierarchical organization, i.e., the millimeter (mm)scale, or mesoscale [START_REF] Grimal | A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties[END_REF], cortical bone can be considered as a two-phase composite material: a mineralized collagen matrix pervaded by a porous network [START_REF] Cooper | Cortical Bone Porosity: What Is It, Why Is It Important, and How Can We Detect It?[END_REF] (Fig. 2(d)). Bone mechanical properties are determined by the properties of the two phases: (1) pore structure and relative volume, and

(2) matrix composition and microstructure.

At the nanometer scale, collagen molecules form fibrils which are progressively mineralized forming the elementary blocks of the matrix. Arrays of mineralized collagen fibrils assemble into lamellae to form cylindrical structures called osteons, which are the most prominent motifs at the highest microstructural level in cortical bone. In long bones, osteons align roughly parallel to the axis of the diaphysis (central tube-like part of long bones)

and to the bone outer surface [START_REF] Petrtyl | Spatial organization of the haversian bone in man[END_REF]. The vascular pore network is comprised of the roughly cylindrical Haversian canals (median size ∼ 50 µm) occupying the center of osteons, connected transversely by the Volkman's canals.

Unbalanced bone remodeling occurring, e.g., in aging tend to change the porosity and the morphology of the pores. While pores occupy more or less 5% of the bone material volume before 40 years old, after 40, differences are reported between women and men, the inter-individual variations increase, and, in women, the average porosity may reach about 15% [START_REF] V Bousson | Distribution of intracortical porosity in human midfemoral cortex by age and gender[END_REF][START_REF] Thomas | Increase in pore area, and not pore density, is the main determinant in the development of porosity in human cortical bone[END_REF]. Also, porosity is known to be spatially heterogeneous within a bone, e.g., around the circumference and in the radial direction [START_REF] V Bousson | Distribution of intracortical porosity in human midfemoral cortex by age and gender[END_REF][START_REF] Malo | Longitudinal elastic properties and porosity of cortical bone tissue vary with age in human proximal femur[END_REF][START_REF] Perilli | Regional Heterogeneity in the Configuration of the Intracortical Canals of the Femoral Shaft[END_REF].

Notations

Anatomical directions in a bone diaphysis are conveniently associated to a Cartesian reference frame where directions 1, 2, and 3 stand for radial, circumferential, and axial (along the diaphysis) directions (Fig. 2).

Ultrasound propagation in bone comprises the propagation of dilatational and shear waves and is affected by material anisotropy leading to a where ρ is the mass density, are denoted respectively radial BWV and axial BWV.

Elastic anisotropy

Cortical bone is most often described as an orthotropic material, that is, the material has a plane of symmetry associated to each anatomical direction. Such a material is characterized by nine distinct elastic moduli.

Where orthotropy is weak as in the central portion of the diaphysis, a transversely isotropic material model, characterized by five moduli only [START_REF] David J Rudy | Anatomic variation in the elastic inhomogeneity and anisotropy of human femoral cortical bone tissue is consistent across multiple donors[END_REF][START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF],

is usually assumed. In long bones, the typical anisotropy ratio between longitudinal coefficients in the axial and radial (or circumferential) directions is between 1.3 and 2.5 [START_REF] David J Rudy | Anatomic variation in the elastic inhomogeneity and anisotropy of human femoral cortical bone tissue is consistent across multiple donors[END_REF][START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF]. Anisotropy of cortical bone is due to both the preferential orientation of the vascular pores and the elastic anisotropy depending on the pore volume fraction) when the mineralized matrix is assumed to be isotropic [START_REF] Grimal | Derivation of the mesoscopic elasticity tensor of cortical bone from quantitative impedance images at the micron scale[END_REF][START_REF] Andrew P Baumann | The relative influence of apatite crystal orientations and intracortical porosity on the elastic anisotropy of human cortical bone[END_REF][START_REF] William J Parnell | Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone[END_REF].

Bone material properties

A large number of ex vivo studies have reported elasticity values in cortical bone. The average Young's modulus along the diaphysis is typically around 14-20 GPa [START_REF] Duchemin | Prediction of mechanical properties of cortical bone by quantitative computed tomography[END_REF][START_REF] Mirzaali | Mechanical properties of cortical bone and their relationships with age, gender, composition and microindentation properties in the elderly[END_REF][START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF]. The average Young's modulus perpendicular to the diaphysis is around 11 GPa [START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF] but is much less documented.

The average shear modulus corresponding to a torsion experiment around an axis parallel to the diaphysis is about 4-6 GPa [START_REF] Mirzaali | Mechanical properties of cortical bone and their relationships with age, gender, composition and microindentation properties in the elderly[END_REF][START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF].

The mass density of cortical bone ranges typically between 1.6 and 2 g.cm -3 . The variations of mass density are due to a combination of variations of the bone volume fraction in a volume of interest (i.e., the bone is more or less porous) and the variations of the mass density of the extracellular mineralized matrix. The latter are usually small because the volume fraction of mineral in mature bone remains relatively constant [START_REF] Roschger | Bone mineralization density distribution in health and disease[END_REF] .

It follows that in practice mass density and porosity are highly correlated.

The above interval ([1.6-2] g.cm -3 ) approximately corresponds to a range of porosity between 30% (extremely high porosity for cortical bone) and a few percents. In this range of density values, the elastic constants vary of ±30-50% around their nominal values [START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF] (Fig. 3).

In a large number of ex vivo studies, BWVs in cortical bone specimens have been measured together with mass density in order to derive elasticity [START_REF] R B Ashman | A continuous wave technique for the measurement of the elastic properties of cortical bone[END_REF]. However, the BWVs values were not reported as such. Overall, compared to elasticity, BWV is much less documented. Eneh et al. [START_REF] Eneh | Effect of porosity, tissue density, and mechanical properties on radial sound speed in human cortical bone[END_REF] measured radial BWV ex vivo on parallelepiped samples of femoral bone from 18 donors and found 3202 m/s (±77)(Fig. 4). Lefevre et al. [START_REF] Lefèvre | Analyzing the anisotropic Hooke[U+05F3]s law for children[U+05F3]s cortical bone[END_REF] measured radial and axial BWVs ex vivo in samples of fibula from 16 donors and found, respectively, 3137 m/s (±486) and 3994 m/s (±178). Grondin et al.

[47], combining measurements in several quadrants of the femur of 4 donors found 3976 m/s (±72) for axial BWV. Mathieu et al. [START_REF] Mathieu | Radial anatomic variation of ultrasonic velocity in human cortical bone[END_REF] investigated the radial variation of axial BWV in 11 femurs and found 3586 m/s (±255). In As explained in sec 3, the design of future bone QUS methods to measure material properties rely on reference data at the typical measurement sites (radius, tibia). Some ex vivo studies have provided values for cortical bone material properties such as elasticity, density and BWVs. However, the large majority of this data was obtained from the diaphysis of the femur, which is not a site measured with QUS, and, to a lesser extent, from the tibia. Little data exists on radius bone due to the difficulty to measure small samples.

The available data was obtained on cadaveric bones, usually from elderly donors without documentation on the existence of bone pathologies.

During childhood, there is a well documented effect of age on bone mineral density [START_REF] Adam | Bone mineral accrual from 8 to 30 years of age: an estimation of peak bone mass[END_REF] . The changes over age of material properties are not covered in the present review, although a bone QUS method can be dedicated to measure children's bones.

To conclude, elastic properties and BWVs at radius and tibia need to be better documented. In particular, it is not clear whether or not bone material has distinct characteristics at these two sites (tibia, as opposed to radius, is a weight-bearing bone) and if these sites are comparable to the femur for which much more data is available. A better documentation of the tibia and radius bones may be obtained with resonant ultrasound spectroscopy (RUS) [START_REF] Bernard | Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy[END_REF][START_REF] Peralta | A critical assessment of the in-vitro measurement of cortical bone stiffness with ultrasound[END_REF] a technique that allows retrieving the full stiffness tensor from small-sized rectangular parallelepiped specimens.

Cortical bone quantitative ultrasound

We review below cortical bone QUS approaches aiming at the measurement of cortical bone thickness (CTh) and material properties. These QUS approaches are essentially developed for the radius and tibia [START_REF] Laugier | Bone Quantitative ultrasound[END_REF] because these sites are easily accessible to ultrasound. The radius, but not the tibia, is an osteoporotic fracture site. Osteoporosis is a systemic disease, accordingly, measuring bone at any site is expected to have a clinical value, which has been proven with several approaches implemented at the heel, phalanx, tibia and radius.

In general, the measurement of CTh with ultrasound in vivo relies on an a priori knowledge, or a joined measurement, of one or several material properties. When measuring bone in the radial direction with a pulse-echo method (Fig. 2(c)), the ultrasound raw data, in terms of time-of-flight, couple information on material properties (BWVs) and anatomy (CTh). With the axial transmission method, measuring the propagation of guided waves in the cortical envelope of the diaphysis (Fig. 2(b)), information on bulk wave velocities and CTh are also coupled (except in the limit case of a sufficiently large CTh and high frequency, in which case a lateral wave can be measured [START_REF] Camus | Analysis of the axial transmission technique for the assessment of skeletal status[END_REF], giving access directly to the axial BWV without requiring the knowledge of thickness).

At QUS measurement sites, CTh shows a large range of values across individuals: typically 1 to 4 mm in radius and 1.5 to 5 mm in tibia [START_REF] Karjalainen | Ultrasonic assessment of cortical bone thickness in vitro and in vivo[END_REF][START_REF] Vallet | In vivo characterization of cortical bone using guided waves measured by axial transmission[END_REF].

In contrast, the range of variations of material properties is smaller, e.g., typically, the mass density varies between 1.8 and 2 g.cm -3 , and the BVWs and elastic coefficients may respectively vary of ± 10% and ± 50% around an average value. Because it is intrinsically difficult to retrieve concurrently several bone characteristics (namely CTh and material properties), to date, clinical implementations of QUS approaches only provide CTh assuming fixed values of material properties. Several approaches are under development to overcome this limitation.

Current approaches

Karjalainen et al. [START_REF] Karjalainen | Ultrasonic assessment of cortical bone thickness in vitro and in vivo[END_REF], following Greenfield et al. [START_REF] Greenfield | Measurement of the Velocity of Ultrasound in Human Cortical[END_REF] and Wear [START_REF] Wear | Autocorrelation and cepstral methods for measurement of tibial cortical thickness[END_REF] implemented a pulse-echo method (Fig. 2(c)) to measure CTh at the tibia and radius using a fixed value of the radial BWV (3565 m/s). This value was chosen such that CTh determined from the QUS measurement matches the reference CTh obtained from HR-pQCT in healthy volunteers.

Otani et al. developed a method to measure the distal radius in throughtransmission, i.e., the wrist is placed between a pair of confocally aligned transducers, [START_REF] Sai | Novel ultrasonic bone densitometry based on two longitudinal waves: Significant 23 correlation with pQCT measurement values and age-related changes in trabecular bone density, cortical thickness, and elastic modulus of trabecular bone in a normal Japanese po[END_REF] (Fig. 2(a)). Ultrasound passes through both the lateral and medial cortical layers and through the trabecular bone in the metaphysis.

Assuming a layered model of bone and some fixed material properties (radial BWV in cortical bone is set to 3300 m/s), the method yields, among other parameters, the sum of the cortical thicknesses at the inlet and outlet sides of the the US beam.

So-called axial transmission measurements involve guided waves propagating in the cortical layer in the direction of the bone axis (Fig. 2(b)).

Guided waves propagation is highly sensitive to variations of CTh [START_REF] Petro Moilanen | Ultrasonic guided waves in bone[END_REF][START_REF] Tatarinov | Comparative Examination of Human Proximal Tibiae In Vitro by Ultrasonic Guided Waves and pQCT[END_REF].

The waveguide thickness is retrieved by resorting to an optimization algorithm to solve an inverse problem. Precisely, the cortical bone layer is modeled as a plate of given (fixed) material properties and unknown thickness (CTh), to be determined by fitting the simulated ultrasonic behavior of the plate to experimental data. Moilanen et al. [START_REF] Petro Moilanen | Assessment of the cortical bone thickness using ultrasonic guided waves: Modelling and in vitro study[END_REF] demonstrated on ex vivo radii that CTh can be retrieved from the signal of a 200 kHz guided wave; in this work, bone material was assumed isotropic with fixed properties (BWV=4000 m/s). In a subsequent study, Vallet et al. [START_REF] Vallet | In vivo characterization of cortical bone using guided waves measured by axial transmission[END_REF] exploited several guided modes [START_REF] J G Minonzio | Guided wave phase velocity measurement using multi-emitter and multi-receiver arrays in the axial transmission configuration[END_REF] to retrieve CTh using US signals centered at 1MHz and transverse isotropic fixed properties (radial BWV=3024 m/s; axial BWV=3753 m/s).

During the last few years, the first clinical studies with the above approaches have been conducted. CTh was found to be different in fractured versus non fractured patients, all with impaired kidney function [START_REF] Mishima | Decreased cortical thickness, as estimated by a newly developed ultrasound device, as a risk for vertebral fracture in type 2 diabetes mellitus patients with eGFR of less than 60 mL/min/1.73 m2[END_REF]. Sai et al. [START_REF] Sai | Novel ultrasonic bone densitometry based on two longitudinal waves: Significant 23 correlation with pQCT measurement values and age-related changes in trabecular bone density, cortical thickness, and elastic modulus of trabecular bone in a normal Japanese po[END_REF] observed the expected decrease of CTh with age in an healthy population and the higher thickness of males compared to females. The pulse echo method of Karjalainen et al. [START_REF] Karjalainen | Ultrasonic assessment of cortical bone thickness in vitro and in vivo[END_REF], combining measurement of CTh and patient's characteristics, was shown to predict femoral neck BMD with good accuracy [START_REF] Karjalainen | Multi-site bone ultrasound measurements in elderly women with and without previous hip fractures[END_REF] and to discriminate patients with hip osteoporosis from controls [START_REF] Schousboe | Prediction of hip osteoporosis by DXA using a novel pulse-echo ultrasound device[END_REF][START_REF] Karjalainen | Pulse-echo ultrasound method for detection of post-menopausal women with osteoporotic BMD[END_REF].

Future of cortical bone QUS

In all the above-mentioned approaches, cortical bone material properties are assumed to be identical for all subjects. This a limitation as tissue properties may vary between individuals and between sites (section 2). Not only this likely impairs the accuracy of the determination of CTh but also material properties themselves may give a valuable additional information on bone quality. In particular, material properties are strongly related to porosity, which is a recognized fracture risk factor [START_REF] Patsch | Increased cortical porosity in type 2 diabetic postmenopausal women with fragility fractures[END_REF][START_REF] Bala | Role of cortical bone in bone fragility[END_REF] (see sec. 4).

Innovative methods

One perspective for cortical bone QUS, based on existing approaches, is to couple the determination of CTh and that of material properties. This can be achieved with several approaches. In pulse-echo mode, methods using a transfer function approach [START_REF] Longo | Simultaneous Determination of Acoustic Velocity and Density of a Cortical Bone Slab: Ultrasonic Model-Based Approach[END_REF][START_REF] Loosvelt | A Wavelet-Based Processing method for simultaneously determining ultrasonic velocity and material thickness[END_REF][START_REF] Tasinkevych | Simultaneous estimation of cortical bone thickness and acoustic wave velocity using a multivariable optimization approach: Bone phantom and in-vitro study[END_REF] an entire transverse cross-section of a long bone with a tomography setup, using scattering theory and Born approximation [START_REF] Lasaygues | Assessing the cortical thickness of long bone shafts in children, using two-dimensional ultrasonic diffraction tomography[END_REF] or using a full waveform approach [START_REF] Bernard | Ultrasonic computed tomography based on full-waveform inversion for bone quantitative imaging[END_REF].

Models of material properties for cortical QUS

Depending on the QUS measurement approach and the type of waves involved (shear and dilatational waves in different anatomical directions), more or less material parameters are involved in the processing of ultrasound signals. While only the radial dilatational BWV is involved in pulse-echo methods, no less than four material parameters may be involved in axial transmission configuration (e.g., one BWV and three elastic anisotropy ratios [START_REF] Foiret | Combined estimation of thickness and velocities using ultrasound guided waves: a pioneering study on in vitro cortical bone samples[END_REF] ). The latter parameters are the quantities directly accessible from a measurement. In the methods implemented for clinical applications, nor mass density nor elastic coefficients can be inferred without resorting to a model of bone material properties (e.g., [START_REF] William J Parnell | Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone[END_REF] ) relating these quantities and those directly accessible from a measurement.

A priori knowledge of bone material properties is mandatory to solve the coupled problem of the determination of CTh and material properties.

Such information is all the more important that the number of parameters to retrieve is large. It has been pointed out that the different elastic coefficients are correlated (Fig. 3) and strongly depend on density [START_REF] J Y Rho | Relations of mechanical properties to density and CT numbers in human bone[END_REF][START_REF] Duchemin | Prediction of mechanical properties of cortical bone by quantitative computed tomography[END_REF][START_REF] Orías | Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur[END_REF][START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF][START_REF] Peralta | A critical assessment of the in-vitro measurement of cortical bone stiffness with ultrasound[END_REF],

and porosity [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF] (see also sec. 4). As a consequence, a simplified model of cortical bone elastic properties with a limited number of parameters [START_REF] Grimal | A two-parameter model of the effective elastic tensor for cortical bone[END_REF][START_REF] Hellmich | can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independant phase properties and their interactions?[END_REF] could be used in order to reduce the number of unknowns when solving the QUS inverse problem. Such an approach was implemented by Bochud et al. [START_REF] Bochud | Genetic algorithms-based inversion of multimode guided waves for cortical bone characterization[END_REF] where cortical bone was modeled as a pore network of variable porosity embedded in a matrix with fixed elastic properties [START_REF] William J Parnell | Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone[END_REF][START_REF] Granke | To what extent can cortical bone millimeter scale elasticity be predicted by a two phase composite model with variable porosity?[END_REF] (an implementation of the model is available online [START_REF] Grimal | BonHom: program for homogenization[END_REF]; the model predictions are plotted against ex vivo elasticity measurements in Fig. 6).

Measuring material properties as potential biomarkers of bone health

Managing bone health often starts by assessing the risk of fracture of an individual. This depends on many factors related to the risk of an individual to fall, and to the ability of a bone to resist a low trauma. The latter depends both on bone size and geometry, and material properties. In this section, we briefly review the relationships between, on the one hand, material properties that may be derived in vivo from QUS measurements and, on the other hand, strength, fracture risk factors, and quality of the extracellular matrix.

Bone resistance

As mentioned in the introduction, one potential advantage of ultrasound over X-ray approaches is to assess material properties beyond the mere quantity of bone reflected in BMD measured with an X-ray based technique.

Bone resistance to fracture is typically characterized by strength (i.e., ultimate stress before rupture) and toughness (i.e., resistance to crack propagation) [START_REF] Zimmermann | The fracture mechanics of human bone: influence of disease and treatment[END_REF]. It is yet unclear to which extent ultrasound, probing bone at very small strains in a linear regime, may yield information on bone resistance.

Correlations have been reported between bone resistance and elastic properties as for engineering and natural materials in general [START_REF] Ashby | Engineering materials 1[END_REF]. Indeed, post-yield and elastic properties are all determined by the biochemical composition and the microstructure of bone. It is thus expected that elasticity and BWVs reflect material resistance to some extent. Pooling results of compression testing of children and adult bone, Öhman et al. [START_REF] Öhman | Compressive behaviour of child and adult cortical bone[END_REF] found a high correlation between yield stress and Young's modulus (R 2 =0.88). Weaker correlations have been found in studies considering only specimens from adult donors (R 2 =0.25 in three-point bending tests [START_REF] Grimal | Assessment of cortical bone elasticity and strength: Mechanical testing and ultrasound provide complementary data[END_REF] ; R 2 =0.53 and 0.56 in tension and compression tests, respectively [START_REF] Duchemin | Prediction of mechanical properties of cortical bone by quantitative computed tomography[END_REF]). Further studies should elucidate more precisely how much of bone material resistance (strength or toughness) can be learned from the measurement of elastic properties and BWVs.

Elasticity and bulk wave velocities reflect porosity and matrix properties

Porosity, a fracture risk factor [START_REF] Bala | Role of cortical bone in bone fragility[END_REF], is an important determinant of stiffness variations [START_REF] Currey | The effect of porosity and mineral content on the Young's modulus of elasticity of compact bone[END_REF][START_REF] Schaffler | Stiffness of compact bone: effects of porosity and density[END_REF]. Several authors have reported correlations between porosity and elastic moduli, although the range of correlation coefficients is quite large. For example, Mirzaali et al. [START_REF] Mirzaali | Mechanical properties of cortical bone and their relationships with age, gender, composition and microindentation properties in the elderly[END_REF] found R 2 =0.08 (Young's modulus) and R 2 =0.47 (shear modulus), Granke et al. [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF] and Cai et al.

[88] found R 2 in the range 0.70-0.84 for all shear and longitudinal stiffness coefficients. Relatively large relative variations of stiffness in the porosity range [2.9-26.9%] were reported [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF]: 58%, 34%, 48%, and 59% for C 11 , C 33 , C 44 , and C 66 , respectively (Fig. 6). Overall, as porosity increases, C 11 (longitudinal elasticity transverse to Haversian canals) decreases more compared to C 33 (longitudinal elasticity in the direction of Haversian canals). This is consistent with results of theoretical studies [START_REF] Baron | Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1 MHz: a finite-difference time domain study[END_REF][START_REF] Hellmich | can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independant phase properties and their interactions?[END_REF][START_REF] William J Parnell | Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone[END_REF].

Data suggest that porosity is also a strong determinant of BWVs. In axial direction, porosity was found to explain about 30% of BWV variation [START_REF] Grondin | Relative contributions of porosity and mineralized matrix properties to the bulk axial ultrasonic wave velocity in human cortical bone[END_REF][START_REF] Mathieu | Radial anatomic variation of ultrasonic velocity in human cortical bone[END_REF]. In radial direction it was found to explains about 50% of BWV variations [START_REF] Eneh | Effect of porosity, tissue density, and mechanical properties on radial sound speed in human cortical bone[END_REF]. There is few data suggesting that variations of material properties of the matrix may be reflected in QUS signals in human bones [START_REF] Raum | Bone microstructure and elastic tissue properties are reflected in QUS axial transmission measurements[END_REF]. There is, however, ex vivo data on prepared specimens pointing at an effect of matrix properties variations on mesoscale elasticity. For instance, Rho et al. [START_REF] J Y Rho | Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nano-indentation[END_REF] found that variations of the mesoscale axial Young's modulus (R 2 =0.49) correlated to the variations of the matrix elasticity (probed with nanoindentation). Granke et al. [START_REF] Granke | Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women[END_REF] found that mesoscale elasticity was correlated with matrix acoustical impedance (a proxy for stiffness) probed with acoustic microscopy (R 2 < 0.25). In another study, variations of matrix impedance has been found to explain as much as 52% of axial BWV [START_REF] Grondin | Relative contributions of porosity and mineralized matrix properties to the bulk axial ultrasonic wave velocity in human cortical bone[END_REF]. Eneh et al. [START_REF] Chibuzor | Inter-individual changes in cortical bone three-dimensional microstructure and elastic coefficient have opposite effects on radial sound speed[END_REF] showed, with simulations performed on a limited number of samples, that the correlation of BWV in the radial direction with porosity may be lost due to inter-individual variations of matrix properties.

The main determinant of matrix stiffness variations is commonly thought to be the mineral content. This is well evidenced considering a large variety of bone samples taken from different species [START_REF] Currey | The effect of porosity and mineral content on the Young's modulus of elasticity of compact bone[END_REF], and theoretical calculations predict that a change of 10% of mineral volume fraction leads to a change of typically more than 20% of matrix elastic coefficients [START_REF] Hellmich | Mineral-collagen interactions in elasticity of bone ultrastructure -a continuum micromechanics approach[END_REF][START_REF] J M Deuerling | Specimenspecific multi-scale model for the anisotropic elastic constants of human cortical bone[END_REF][START_REF] Grimal | A two-parameter model of the effective elastic tensor for cortical bone[END_REF].

In a recent study on femoral bone specimens from 19 elderly donors, Cai [START_REF] Cai | Multiscale investigation of the elastic properties of human cortical bone measured by resonant ulrasound spectroscopy[END_REF] found that more than 50% of mesoscale elasticity variations were associated to variations of mineral content. Collagen fibers mechanical quality and organizational patterns have also been proposed as possible determinants of mesoscopic properties. As far as we are aware, there is no data for human bone showing an effect of a pathological alteration of collagen on mesoscale elastic properties. However, artificial degradation of the collagen with chemical treatments is known to alter elastic properties [START_REF] Mehta | Bone elasticity and ultrasound velocity are affected by subtle changes in the organic matrix[END_REF].

From a mechanical standpoint, mesoscale cortical bone material properties such as density, elasticity, or BWVs are fully determined by the properties of the pore network and of the matrix. Whether or not a change of these properties is reflected at the mesoscale in a given dataset critically depends on the range of variations of the properties at the different scales. The data reported above was obtained from the tissues of donors with no documented medical history, hence the conclusions drawn from these studies only pertain to this type of population and cannot be extrapolated to groups of subjects carrying specific bone diseases. To conclude, for these bones from non-targeted populations, the available data suggest that a large part of the elasticity and BWVs variations is explained by the variation of porosity. This is consistent with the prediction of theoretical models which assume that the matrix properties have limited inter-individual variations and that porosity varies in a relatively large interval [START_REF] Grimal | A two-parameter model of the effective elastic tensor for cortical bone[END_REF][START_REF] Granke | To what extent can cortical bone millimeter scale elasticity be predicted by a two phase composite model with variable porosity?[END_REF] (sec. 3.2.2). Variations of the properties of the matrix (mineral content, impedance, elasticity) also impact mesoscale elastic properties and BWVs, however, only limited data is available. Some pathologies involving a low mineral content or a weak alignment of collagen fibers are expected to strongly affect mesoscale properties through modifications of both the porosity and the matrix. This calls for more studies designed to investigate the variations of BWVs and elasticity in different targeted populations.

Biomarkers of material heterogeneity

Aging may be associated to an increase of the heterogeneity of the distribution and size of the pores in the cortical bone layer, resulting in a gradient of porosity: high porosity close to the marrow and relatively low porosity close to the external surface of the bone [START_REF] Zebaze | Intracortical remodelling and porosity in the distal radius and post-mortem femurs of women: a crosssectional study[END_REF] . In terms of mechanics of materials, this raises the question of the existence of a representative volume element of cortical bone material [START_REF] Grimal | A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties[END_REF]. If the local variations of porosity are too strong, the cortical bone material can not be evaluated per se and the cortex needs to be considered as a structure. In case of a mild heterogeneity of porosity, it may be relevant to model the cortex material as a heterogeneous field of material properties. This issue has in part been theoretically addressed in an axial transmission QUS configuration [START_REF] Haiat | Influence of a gradient of material properties on ultrasonic wave propagation in cortical bone: Application to axial transmission[END_REF][START_REF] Baron | Using the Gradient of Human Cortical Bone Properties to Determine Age-Related Bone Changes Via Ultrasonic Guided Waves[END_REF] but has not been implemented in clinical practice as far as we know. Since the heterogeneity of material properties within the cortex is expectedly associated to a reduced mechanical competence, it could be interesting to develop ultrasound biomarkers reflecting heterogeneity.

Conclusion

QUS technologies to measure cortical bone thickness, a proven biomarker of bone health, are available and used in vivo. Improvements of these technologies and disruptive technologies are expected to be available in a near future, which will achieve a coupled assessment of cortical thickness and material properties. One motivation is to estimate intracortical porosity, a quantity hardly directly measurable in vivo. Assessing porosity with ultrasound would be a significant progress because porosity is a recognized fracture risk factor and because it is a fingerprint of the remodeling activity. One route to infer porosity is to use empirical relationships, or material models, relating quantities measured with QUS and porosity. Other routes are currently being explored such as imaging blood perfusion using ultrasound contrast agent [START_REF] Du | Imaging of cortical pores using ultrasound contrast agents: Phantom and ex vivo studies[END_REF] and measuring ultrasonic attenuation assuming it has a strong relationship with pore properties [START_REF] Omid Yousefian | The effect of pore size and density on ultrasonic attenuation in porous structures with mono-disperse random pore distribution: A two-dimensional in-silico study[END_REF].

Probing the quality of the mineralized collagen matrix in vivo with ultrasound is a far-reaching goal. It may be a reasonable objective in targeted pathologies providing that alterations of porosity and matrix properties are

Figure[

  Figure 1: * [Graphical abstract] Bone anatomy and QUS measurement configurations. (a) Measurement in bone radial direction with a through transmission approach (courtesy of OYO Electric CO., LTD., Japan); (b) Measurement in bone axial direction with bidirectional axial transmission to measure guided waves (courtesy of AZALEE, France) ; (c) Measurement in bone radial direction with a pulse-echo approach (courtesy of Bindex, Finland) ; (d) image of cortical bone microanatomy obtained with synchrotron radiation microtomography showing the vascular porosity mainly oriented along the bone axis ; (e) X-ray image of a cross-section of the distal radius with a depicted region of interest for a typical QUS measurement with axial transmission.

Figure 2 :

 2 Figure 2: Bone anatomy and QUS measurement configurations. (a) Measurement in bone radial direction with a through transmission approach (courtesy of OYO Electric CO., LTD., Japan); (b) Measurement in bone axial direction with bidirectional axial transmission to measure guided waves (courtesy of AZALEE, France) ; (c) Measurement in bone radial direction with a pulse-echo approach (courtesy of Bindex, Finland) ; (d) image of cortical bone microanatomy obtained with synchrotron radiation microtomography showing the vascular porosity mainly oriented along the bone axis ; (e) X-ray image of a cross-section of the distal radius with a depicted region of interest for a typical QUS measurement with axial transmission.

  of the mineralized matrix due to the orientation of the mineralized collagen fibers. Theoretical studies have shown that bone mesoscale anisotropy mostly stems from the anisotropy of the matrix, the preferential orientation of the pores leading to moderate values of anisotropy (C 33 /C 11 ∼ 1.1-1.3

Figure 3 :

 3 Figure3: This figure, reproduced from[START_REF] Bernard | Elasticity-density and viscoelasticitydensity relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements[END_REF], illustrates the ranges of variations of elastic coefficients (ex vivo measurements). Stiffness and engineering coefficients of 55 cortical bone specimens from the tibia, are plotted as a function of C66 to illustrate the interdependency of the different elastic coefficients. The range of mass density values is [1.6-2.0] g.cm -3 for the different specimens.

Figure 4 :

 4 Figure 4: Reproduced from [45]. Histogram (numbers represent the center of bins that are 60 m/s in width) of the average values of radial bulk wave velocity (denoted SOS for 'speed of sound' in the figure) in 44 cortical bone samples from the femur diaphysis. The black and white bars represent radial BWV measured at 2.25 and 5.00 MHz, respectively.

  could in principle be designed to retrieve BWVs, density, attenuation, and CTh exploiting the times-of-flight and amplitudes in the reflected signals. The multimode axial transmission technique allows retrieving CTh and material properties or porosity, which has been demonstrated ex vivo [68, 69, 70]. Obtaining ultrasound images of the internal structure of cortical bone is another exciting perspective. Conventional ultrasound scanners are used in clinical practice to image the outer surface of bones allowing for the diagnosis of bone fractures [71]. However, these ultrasound systems fail to reveal the internal structure of bones because (1) the algorithm used to construct the image assumes that ultrasound follow a straight path and that BWVs do not vary along the path; (2) attenuation in cortical bone is relatively large; and (3) important energy loss occur at the soft tissue-bone interface caused by the large acoustic impedance mismatch. Using wave scattering theory to model the wave path, Zheng et al. [72] obtained ex vivo an image of the cortical layer of a bovine femur. Taking advantage of the tremendous performance improvement of hardware electronics in ultrasound scanners in the last years and developing a dedicated image reconstruction technique, Renaud et al. [73] have recently obtained in vivo quantitative images of the cortical layer of human radius and tibia (Fig. 5). The velocity of bulk dilatational waves in the different anatomical directions is recovered by combining a measurement of the lateral wave and optimizing image quality.

Figure 5 :

 5 Figure 5: Reproduced from[START_REF] Renaud | In vivo ultrasound imaging of the bone cortex[END_REF]. Ultrasound image of the cortical layer of a radius in vivo. Red and blue lines correspond to the periosteum and endosteum. Straight lines (doted black line) approximate the interfaces. The cortical thickness, defined as the mean distance between these lines is found to be 3.5 mm for this acquisition and is in agreement with the thickness measured with HR-pQCT.

Figure 6 :

 6 Figure 6: Reproduced from [36]. (a) Longitudinal and (b) shear mesoscopic elastic coefficients versus porosity in 21 cortical bone samples from the femur. The model based on continuum mechanics laws (solid and dotted lines), assuming fixed matrix properties and variable porosity, predicts the trend of variation of elasticity.

  well documented ex vivo. Such documentation of acoustical properties in bone tissue with different pathologies is a keystone of the future development of bone QUS methods. Acknowledgment [1] C Cooper and F Ferrari. IOF Compendium of Osteoporosis. Technical report, International Osteoporosis Foundation, 2017. [2] E S Siris, Y T Chen, T A Abbott, E Barrett-Connor, P D Miller, L E Wehren, and M L Berger. Bone mineral density thresholds for pharmacological intervention to prevent fractures. Archives of Internal Medicine, 164(10):1108-1112, 2004. [3] Karine Briot, Simon Paternotte, Sami Kolta, Richard Eastell, Dieter Felsenberg, David M. Reid, Claus C. Glüer, and Christian Roux.

FRAX R :

 : Prediction of major osteoporotic fractures in women from the general population: The OPUS study. PLoS ONE, 8(12), 2013.


Keywords: Single point incremental forming process, mechanical model, experimental test, numerical simulation, forming load

simulation of the SPIF process. In this context, the aim of this study is to evaluate the ability of a simple elasto-plastic model, identified from conventional tensile tests carried out on a T40 titanium alloy, to predict the forming loads used to manufacture a truncated cone. The evaluation of the model is performed by comparing numerical predictions with the experimental results (efforts and geometry).

Incremental forming of Titanium T40 sheet: experimental and numerical investigations

A. Abdelkefi a , D. Guines a , L. Léotoing a , S. Thuillier b a. Univ. Rennes, INSA Rennes, LGCGM -EA 3913, F-35000 Rennes, France, abir.abdelkefi@insa-rennes.fr, dominique.guines@insa-rennes.fr, lionel.leotoing@insarennes.fr b. Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France, sandrine.thuillier@univ-ubs.fr

Résumé :

Le procédé de formage incrémental mis en oeuvre avec un poinçon unique et de forme simple (SPIF) présente un fort potentiel pour la fabrication en petites séries de produits personnalisés et de formes complexes. Au cours du procédé SPIF, la tôle est bridée et un outil hémisphérique la déforme localement, de manière itérative, en suivant une trajectoire bien spécifique. L'utilisation de robots industriels pour mettre en oeuvre ce procédé a pour objectif d'améliorer sa flexibilité (

Abstract:

Single point incremental forming process (SPIF) has a great potential to manufacture small series of customized products. During SPIF process, the sheet metal is clamped and locally deformed, in an iterative way, by a hemispherical tool that follows a specified trajectory. The use of industrial robots to implement this process aims to improve its flexibility (increased volumes of work and complexity of the formed parts) and to reduce its cost. For these mechanical systems, for which the structural rigidity is low, the prediction of forming forces associated to an elastic modeling of the robot is necessary to compensate tool path errors due to the robot compliance and to ensure the geometrical quality of the final part. The forming forces can be predicted from a finite element simulation of the process and used then as input data to the elastic modeling of the robot. However, the accuracy of the calculated loads depends strongly on the constitutive material behavior of the blank introduced in the

Introduction

During the last decades, several authors pay attention to the incremental forming process (SPIF) due to its ability to manufacture complex shapes using simple tool. Usually, the forming operation is performed by classical three-axis CNC machines with high stiffness. A solution to improve the production flexibility and to reduce manufacturing costs is to use robots connected in serial or parallel. Nevertheless, they present some drawbacks such as high compliance and low positioning precision under high loading [START_REF] Belchior | A process/ Machine coupling approach: Application to robotized incremental sheet forming[END_REF]. To improve the accuracy of the robotized process, an approach coupling the Finite Element Analysis (FEA) of the process with the elastic modeling of robot structure is proposed [START_REF] Belchior | A process/ Machine coupling approach: Application to robotized incremental sheet forming[END_REF]. In order to evaluate tool path deviation, FEA is used to predict the forming loads which are used as input in the elastic model of the robot. To the author's knowledge, only a few studies have investigated the influent parameters on the forces predicted by FEA. Henrard et al. [START_REF] Henrard | Forming forces in single point incremental forming: prediction by finite element simulations, validation and sensitivity[END_REF] and Flores et al. [START_REF] Flores | Model identification and FE simulations: effect of different yield loci and hardening laws in sheet forming[END_REF] have compared the influence of different plastic behaviors (yield criterion, hardening law, hardening type) on the force prediction for AA3003-O. Results show that forming forces depend more on the hardening law than on the yield locus and the use of kinematic hardening leads to a noticeable drop (up to 20%) of the estimated forces. However, Ben Said et al. [START_REF] Said | Numerical prediction of the ductile damage in single point incremental forming process[END_REF] have also shown that the prediction of tool forces depends both on the hardening law and yield criterion. Moreover, the use of isotropic hardening leads to a high value of the predicted forces in comparison with the ones obtained from a mixed hardening model. The same conclusion is drawn by Benedetti et al. [START_REF] Benedetti | Single-point incremental forming of sheet metals: Experimental study and numerical simulation[END_REF], they found that the use of mixed hardening seems to be more suitable to predict the forming loads. From these previous studies, it should be highlighted that the selection of a suitable material model to predict accurate forming forces is still a challenging task. These difficulties mainly lie in high plastic strain levels (larger than 100%) and complex strain paths observed in this process [START_REF] Flores | Model identification and FE simulations: effect of different yield loci and hardening laws in sheet forming[END_REF]. For these reasons, material parameters should be carefully identified. Among the typical materials used to manufacture complex shapes for SPIF process, titanium stands out due to its high strength to weight ratio, inertness, and bio-compatibility [START_REF] Bannon | Titanium Alloys for Biomaterial Application: An Overview, Titanium Alloys in Surgical Implants[END_REF]. In terms of experimental characterization of titanium alloy to identify material parameters for the numerical simulation of SPIF process, detailed studies remain few in literature. Saidi et al. [START_REF] Saidi | Etude expérimentale et numérique des efforts lors du formage incrémental de tôles en titane[END_REF] and Sakhtemanian et al. [START_REF] Sakhtemanian | Numerical and experimental study on the layer arrangement in the incremental forming process of explosive-welded low-carbon steel/CP-titanium bimetal sheet[END_REF] have used tensile tests to identify T40 alloy parameters to make truncated cone and pyramid, respectively. Lino et al. [START_REF] Lino | Single point incremental forming of a medical implant[END_REF] have used tensile tests at several orientations and bulge test, for the same material, to manufacture a medical implant. In this paper, an experimental investigation covering both the mechanical characterization of T40 alloy and the manufacturing of a truncated cone are carried out in section 2. A numerical approach of the SPIF process using a simple elasto-plastic model is proposed in section 3. In section 4, the evaluation of the model is performed by comparing numerical predictions with the experimental results (efforts and geometry).

Experiments

Mechanical behavior of T40 alloy

The sheet material considered in this work is the T40 titanium alloy often used in aeronautical and biomedical industries due to its formability and its corrosion resistance. This sheet material is generally suitable for SPIF process and forming applications [START_REF] Saidi | Etude expérimentale et numérique des efforts lors du formage incrémental de tôles en titane[END_REF][START_REF] Sakhtemanian | Numerical and experimental study on the layer arrangement in the incremental forming process of explosive-welded low-carbon steel/CP-titanium bimetal sheet[END_REF][START_REF] Lino | Single point incremental forming of a medical implant[END_REF]. T40 sheets are received as coldrolled blank with a thickness of 0.5 mm. The chemical composition of the T40 alloy is given in Table 1. The mechanical behavior is investigated by means of uniaxial tensile tests performed at several orientations to the rolling direction, simple shear and bulge tests. However, in a first step, only isotropic hardening associated with von Mises yield criterion is considered and only the mechanical behavior in tension in the rolling direction (Fig. 1) is used. Mechanical properties are then calculated, leading as average values over three tests to R p0.2% = 215 MPa and R m = 328 MPa.

Experimental SPIF process

For the designed part, a T40 square sheet of 270 x 270 x 0.5 mm 3 is used to form a frustum cone of 45 ° wall angle (Fig. 2(a)). The depth of the truncated cone is 30 mm. The forming tool is a hemispherical punch with a 15 mm diameter. The feed rate value of the tool is 360 mm/min and the tool rotation is locked. The SPIF operation is carried out on a three axis CNC milling machine (Famup MCX600). The tool path consists of successive circular contours at constant Z. The incremental step size value is 1mm per loop. The forming forces applied by the tool are measured by a six-axes force/torque sensor ATI Omega190. The clamping system (Fig. 2 Figure 3 presents the forming force measured in the axial direction of the tool. At the end of each circular path at constant Z, the imposed tool path leads to the loss of contact between the tool and the sheet metal, which results in a return of the force to zero. At each new Z-increment, the tool-tosheet contact restoration generates a peak force, especially for the deepest passes. As shown on the zoom, figure 3, for a circular path at , the force then stabilizes quickly from the beginning of the considered path and is relatively constant over one revolution. The same tendencies have been observed in previous works [START_REF] Belchior | A process/ Machine coupling approach: Application to robotized incremental sheet forming[END_REF][START_REF] Henrard | Forming forces in single point incremental forming: prediction by finite element simulations, validation and sensitivity[END_REF][START_REF] Flores | Model identification and FE simulations: effect of different yield loci and hardening laws in sheet forming[END_REF][START_REF] Said | Numerical prediction of the ductile damage in single point incremental forming process[END_REF][START_REF] Benedetti | Single-point incremental forming of sheet metals: Experimental study and numerical simulation[END_REF]. The axial load increases gradually during successive increments of the tool Z-positions, reaches then a maximum value of 910 N and remains stable around this value until the end of forming. At the end of the forming process, the clamped part is measured by a 3D laser measurement system.

Finite element analysis (FEA)

In this section, the full process simulation is carried out using Abaqus/implicit© software. To reduce computation time, only one fourth of the blank is considered, though the problem is not symmetric. However, such a simplified model was compared to the full one, showing very comparable results. Four node shell elements with reduced integration (S4R) are employed to model the sheet. The x-axis corresponds to the rolling direction of the sheet. The X and Y axes are also the symmetry axes of the reduced model. The hemispherical punch is modeled as a rigid body. An embedding condition is applied along the outer circular edge (corresponding to the 140 mm diameter in Fig. 4). Frictionless contact is considered between the tool and the sheet. The sheet is modeled as an elastic-plastic material associated with von Mises yield criterion (section 2.1).

Figure 4. FE model with boundary conditions

Discussion

A comparative study of the experimental and numerical results of the final part geometry, before the unclamping stage, is presented in Fig. 5a. For the simulation, the part profile is taken from the median plane of the model shown in Fig. 4, which corresponds to a 45° cutting plane of the XZ plane. For the formed part, the measured profile corresponds to the bottom side of the sheet metal (on the opposite side to the tool with respect to the sheet). As shown in Fig. 5(a), FE simulation predicts well the wall region and underestimates the cone bottom. As explained in previous studies [START_REF] Henrard | Comparison of FEM simulations for the incremental forming process[END_REF][START_REF] He | Finite element modeling of incremental forming of aluminum sheets[END_REF], if the constitutive model is not a key factor to predict accurately the final experimental geometry, the use of a reduced or full numerical model seems to be more influent [START_REF] Flores | Model identification and FE simulations: effect of different yield loci and hardening laws in sheet forming[END_REF]. Differences observed in Fig. 5(a) can be explained by the modeling adopted here, i.e. one fourth of the blank is modeled and symmetry conditions are applied on X and Y axes. Figure 5b presents a comparison between experimental and simulated forces in Z-direction. The average of the vertical force over one cycle (at Z-constant) is calculated and used in the comparison between experiments and simulation. As shown in Fig. 5(b), the force reaches a steady maximum value after a Z-displacement of about 15 mm. The FE model noticeably underestimates the steady value of the forming load, with a relative gap of 11%. This mismatch can be explained by several reasons: i) shell element model seems not to be suitable to represent the transversal shear deformation which is considered not negligible by some authors [START_REF] Benedetti | Single-point incremental forming of sheet metals: Experimental study and numerical simulation[END_REF]; ii) the choice of isotropic hardening and isotropic yield criterion does not represent accurately the material behavior, iii) the use of an embedding condition may not be suitable to model the real boundary conditions applied to the blank during the SPIF process.

Embedding

Conclusion

In the present study, the SPIF process is investigated through experimental and numerical approaches. Mechanical characterization is led by means of uniaxial tensile tests to identify the material parameters of the T40 titanium alloy. A FE simulation of a SPIF operation of a truncated cone is performed using a simple elasto-plastic model. Impact of the proposed model on the final geometry and the forming load is investigated. It is concluded that the final geometry is rather well predicted but the final forming load is underestimated by 11%. In the future, in order to improve the prediction of the forming loads, additional numerical analysis of the SPIF process will be performed to check influent parameters on the predicted forces such as mechanical models, boundary conditions and element types.

Figure 1 .

 1 Figure 1. Stress strain curve of T40 in tension in the rolling direction. Necking occurs at 0.3 but the Cauchy stress is still calculated above this point.

  (b)) is composed of a blank holder screwed on a frame. The contact between tool and sheet is lubricated to minimize the value of the friction coefficient. The duration of the forming operation is about 30 min.

Figure 2 .

 2 Figure 2. Experimental setup (a) shape of 45° wall angle truncated cone (dimensions in millimeters), (b) incremental forming device

Figure 3 .

 3 Figure 3. Force measurement along the tool axis. The zoom on the right-hand side corresponds to the load evolution during a circular path at a constant Z-position.

Figure 5 .

 5 Figure 5. Comparison between experiments and FE predictions (a) geometry of the final shape (b) vertical load as a function of the tool displacement

Table 1 :

 1 Chemical composition (in wt %) of the T40 alloy.

	Fe	C	N	H	O	Ti
	0.3	0.08	0.03	0.015	0.25	bal.
				law		where	is the
	equivalent stress,		and is the equivalent plastic strain, K = 630 MPa,	205 MPa
	and n = 0.3. Swift parameters are identified from the tensile test results.

The elastic behavior of the material is defined by the Young modulus E = 110 GPa and the Poisson ratio =0.3. The plastic behavior is modeled by the Swift