Cécile Heidsieck
email: cecile.heidsieck@ensam.eu

Laurent Gajny

Christophe Travert

Jean-Yves Yves Lazennec

Wafa Skalli

Cécile

Effect of postural alignment alteration with age on vertebral strength

Keywords: vertebral strength, finite element method, barycentremetry, global sagittal alignment, osteoporosis Declarations

Introduction

Vertebral fragility fractures associated with osteoporosis are a major health problem, because they increase morbidity and mortality [START_REF] Johnell | An estimate of the worldwide prevalence and disability associated with osteoporotic fractures[END_REF][START_REF] Bliuc | Mortality risk associated with low-trauma osteoporotic fracture and subsequent fracture in men and women[END_REF]. A precise fracture risk estimation is a key factor to limit the negative consequences of those fractures and offer the appropriate treatment to the population at risk. Areal bone mineral density (aBMD) measurement with dual-energy X-ray absorptiometry (DXA) is the reference technique to evaluate fracture risk, but its sensitivity is low [START_REF] Löffler | Improved prediction of incident vertebral fractures using opportunistic QCT compared to DXA[END_REF][START_REF] Cosman | Spine fracture prevalence in a nationally representative sample of US women and men aged ≥40 years: results from the National Health and Nutrition Examination Survey (NHANES) 2013-2014[END_REF]. The conditions leading to vertebral fractures are complex and still to be explored. Depending on the studies, the causes of 27% to 38% fragility fractures are not identified [START_REF] Oudshoorn | Emergency department visits due to vertebral fractures in the Netherlands, 1986-2008: Steep increase in the oldest old, strong association with falls[END_REF][START_REF] Wang | Epidemiology of spinal fractures among the elderly in Chongqing, China[END_REF]. Other parameters than the aBMD can be taken into account to improve the fracture risk estimation. Recent studies showed that sagittal alignment could have an impact on vertebral fracture occurrence. Hu et al [START_REF] Hu | Global sagittal alignment in elderly patients with osteoporosis and its relationship with severity of vertebral fracture and quality of life[END_REF] showed a negative correlation between an alteration of the sagittal alignment and the quality of life. Also, an abnormal forward spinal inclination was found in subjects consulting for osteoporosis, even for those without vertebral fractures [START_REF] Fechtenbaum | Sagittal balance of the spine in patients with osteoporotic vertebral fractures[END_REF]. This inclination increases with age [START_REF] Bassani | Physiological variations in the sagittal spine alignment in an asymptomatic elderly population[END_REF][START_REF] Amabile | Invariance of head-pelvis alignment and compensatory mechanisms for asymptomatic adults older than 49 years[END_REF], translating a tendency to have a more forward position in the elderly population.

Finite element models (FEM) based on quantitative computer tomography (qCT) were developed to allow vertebral strength estimation with personalized mechanical properties and geometry. In brief, these models use the personalized geometry of a vertebra reconstructed from the CT-scan. The vertebra is then split into small elements, typically around 1 millimeter wide. Then each element is given specific mechanical laws depending on the volumetric BMD (vBMD) measured in this specific location in the vertebra. Then a compression force is simulated on the vertebral body to estimate its mechanical behavior and more specifically its strength. verterbral qCT-based FEM were validated with experimental data [START_REF] Crawford | Finite element models predict in vitro vertebral body compressive strength better than quantitative computed tomography[END_REF][START_REF] Li D | Novel, fast and efficient image-based 3D modeling method and its application in fracture risk evaluation[END_REF][START_REF] Choisne | Vertebral strength prediction from Bi-Planar dual energy x-ray absorptiometry under anterior compressive force using a finite element model: An in vitro study[END_REF]. They are now used in clinical studies to evaluate strength changes following osteoporosis treatments [START_REF] Imai | Vertebral fracture risk and alendronate effects on osteoporosis assessed by a computed tomography-based nonlinear finite element method[END_REF]. Vertebral fracture incidence prediction accuracy through qCT based FEM was recently assessed. It was compared to aBMD and volumetric BMD measurements. The strength obtained through qCT based FEM showed to be more correlated to vertebral fracture incidence than aBMD values but equivalent to vBMD measurements [START_REF] Allaire | Prediction of incident vertebral fracture using CT-based finite element analysis[END_REF]. Parameters taken into account by qCT based FEM might not be enough to fully predict fragility fractures. Indeed, the occurrence of a fracture depends as much on the strength of the bone as on the load that is applied to it. Subject-specific loading conditions could provide complementary information.

Musculoskeletal models have been used to estimate spinal loads in subjects with spinal deformities. Briggs et al [START_REF] Briggs | Thoracic Kyphosis Affects Spinal Loads and Trunk Muscle Force[END_REF] indicated that subjects with hyperkyphosis could experience compressive loads 14% greater than subjects with lower kyphosis in a relaxed position. Compensatory mechanisms to prevent the gravity line from shifting forward can limit the phenomenon [START_REF] Bruno | The effect of thoracic kyphosis and sagittal plane alignment on vertebral compressive loading[END_REF]. However, studies have shown that loading conditions on the spine were critical [START_REF] Buckley | Relative strength of thoracic vertebrae in axial compression versus flexion[END_REF][START_REF] Travert | Sensitivity of patient-specific vertebral finite element model from low dose imaging to material properties and loading conditions[END_REF]. A sensitivity study on loading conditions on finite element modelled vertebrae revealed that displacing the load application anteriorly by 1cm resulted in a decrease up to 60% of the vertebral strength [START_REF] Travert | Sensitivity of patient-specific vertebral finite element model from low dose imaging to material properties and loading conditions[END_REF].

In the upright position, loads on the vertebrae result from the balance between gravity loads due to the upper body weight and the forces of muscles surrounding the spine. The gravity loads supported by the intervertebral disks and the vertebral bodies depend on the mass and center of mass of the upper body segments, which results from the upper body shape and alignment [20]. Barycentremetry is a method to estimate the mass and center of mass of each body segment [20]. It uses a combination of the three-dimensional body geometry reconstruction and body segment density estimation. Recent models have allowed the computation of the whole body mass as well as the center of mass (CoM) position based on barycentremetry methods from biplanar X-Rays. First, the simultaneous acquisition of frontal and sagittal head to feet X-rays allows 3D reconstruction of the spine [START_REF] Mitton | 3D reconstruction of the pelvis from bi-planar radiography[END_REF][START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF] and body envelope [START_REF] Nérot | A 3D reconstruction method of the body envelope from biplanar X-rays: Evaluation of its accuracy and reliability[END_REF] in the standing position. Then, density models for each body segment are used to estimate the mass and CoM of the body parts above a given vertebra [START_REF] Thenard | Analysis of Center of Mass and Gravity-Induced Vertebral Axial Torque on the Scoliotic Spine by Barycentremetry[END_REF].

No study has yet combined FEM with barycentremetry to evaluate posture and morphology effect on potential vertebral fracture occurrence. The aim of this study is to investigate the impact of the aging process on sagittal alignment and on the forces applied on a vertebra, and therefore on its risk of fracture. We assume that morphological and postural modifications with age can be quantified and should be considered in the vertebral fracture risk prediction.

Materials and methods

Population inclusion criteria:

Retrospective clinical data from previous studies were used. Subjects included were asymptomatic volunteers who had low dose bi-planar X-ray radiographs after approval by the Ethics Committee [Comité de Protection des Personnes CPP No 2010/113 and CPP No 06036]. The bi-planar X-ray images were obtained with the EOS system (EOS imaging, Paris, France) [START_REF] Dubousset | Eos: a new imaging system with low dose radiation in standing position for spine and bone & joint disorders[END_REF], a low dose system which can obtain simultaneously two head to feet radiographs in the frontal and the sagittal plan with two X-ray sources. Subjects had to stand in the standardized freestanding position [START_REF] Faro | Evaluation of a functional position for lateral radiograph acquisition in adolescent idiopathic scoliosis[END_REF] with hand resting either on the mandibles or on the clavicles and had to look horizontally. Clinical exams were performed to ensure that the subjects presented no symptomatic lower limb nor spine pathology that could affect the spinal alignment and patient with a frontal cobb angle more than 20° were excluded. In total, 117 subjects (59 males, 58 females, all Europeans) were included. The mean age was 43 (1SD = 19, min = 20, max=83). We divided the volunteers in three age groups: 62 young (32 males, 30 females, mean age = 27 , 1SD =5, min = 20, max = 39), 26 intermediate (1 males, 12 females, mean age = 49, 1SD 6, min = 40, max = 59), and 29 elderly (13 males, 16 females, mean age = 70, 1SD = 5, min = 62, max = 83) subjects. The body mass index (BMI) was computed for the subjects of which the height and weight were available (90/117).

3D reconstruction:

The radiographs were processed to perform 3D reconstructions of the spine [START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF][START_REF] Gajny | Quasi-automatic 3D reconstruction of the full spine from low-dose biplanar X-rays based on statistical inferences and image analysis[END_REF], pelvis [START_REF] Mitton | 3D reconstruction of the pelvis from bi-planar radiography[END_REF], odontoid process (OD) location and external body envelope [START_REF] Nérot | A 3D reconstruction method of the body envelope from biplanar X-rays: Evaluation of its accuracy and reliability[END_REF], using validated methods.

Radiographic parameters:

Several parameters were measured on the 3D reconstruction to describe the sagittal spine alignment. 1/ Spinal curvatures: C3-C7 cervical curvature, T4-T12 thoracic kyphosis and L1-S1 lumbar lordosis. 2/ Pelvic parameters: pelvic incidence (PI), sacral slope (SS) and pelvic tilt (PT) [START_REF] Legaye | Pelvic incidence: a fundamental pelvic parameter for three-dimensional regulation of spinal sagittal curves[END_REF]. 3/ Three global sagittal alignment parameters (figure 1): SVA, OD-HA sagittal angle and spinal inclination [OD-S1]. The.SVA (sagittal vertical axis) was defined as the sagittal distance between the center of C7 vertebral body and the posterior corner of the sacral superior endplate (figure 1a) [START_REF] Jackson | Radiographic Analysis of Sagittal Plane Alignment and Balance in Standing Volunteers and Patients with Low Back Pain Matched for Age, Sex, and Size: A Prospective Controlled Clinical Study[END_REF]. The.OD-HA (middle of the centers of each acetabulum) angle was defined as the sagittal angle between the vertical and the line across OD-HA (figure 1b) [START_REF] Amabile | A new quasi-invariant parameter characterizing the postural alignment of young asymptomatic adults[END_REF]. The spinal inclination [OD-S1], was defined as the angle between the vertical and the line of best fit across the following points: OD, all vertebral centroids between C3 and L5 and the center of the sacral superior endplate (figure 1c) [START_REF] Lafage | Pelvic Tilt and Truncal Inclination: Two Key Radiographic Parameters in the Setting of Adults With Spinal Deformity[END_REF] Barycentremetry: Estimation of the center of mass (CoM) above the vertebral level L1 was performed for each subject using barycentremetry from the EOS images, as described by Amabile et al [START_REF] Amabile | Determination of a new uniform thorax density representative of the living population from 3D external body shape modeling[END_REF]. The method is briefly reminded hereafter. The external body envelope was reconstructed using EOS images. Although the images were acquired with hands resting on the clavicle or on the mandibles, upper limbs were virtually placed in a neutral position along the body in the model to locate the CoM (figure 2a). Then, the body envelope was divided in body segments of which the volume and mass was computed using an improved Dempster Uniform density method [START_REF] Amabile | Determination of a new uniform thorax density representative of the living population from 3D external body shape modeling[END_REF]. The CoM of each body segment was then computed and used to compute the CoM of the whole body above the superior endplate of L1.

The thoraco-lumbar junction, including the L1 level, is where most of the of vertebral fractures in osteoporotic subjects occurs [START_REF] Grados | Prevalence of vertebral fractures in French women older than 75 years from the EPIDOS study[END_REF][START_REF] Melton | Prevalence and incidence of vertebral deformities[END_REF]. Therefore, we focused the study on the L1 vertebral level. This CoM is not necessarily vertically aligned with the L1 vertebra. The distance between the vertical line going through the CoM and the L1 vertebra creates a torque that adds up to the load on the vertebra, which is proportional to this distance times the magnitude of the force. This distance is called lever arm (or moment arm).It was defined as the horizontal distance between the upper body CoM and the center of the superior endplate of L1. It was measured in the lateral and anteroposterior directions. As the 3D orientation of the vertebra affects the ability of the vertebra to resist to applied strength, it also was reported in the axial, frontal and sagittal plane.

Finite element analysis:

To study the impact of the lever arm of the CoM, and the 3D orientation of the vertebra, a previously validated vertebral FEM was used [START_REF] Choisne | Vertebral strength prediction from Bi-Planar dual energy x-ray absorptiometry under anterior compressive force using a finite element model: An in vitro study[END_REF]. In the current study, since no data was acquired to study bone mineral density on the subjects, we used three different models extracted from this previous study. To choose these models, three classes of vertebrae (weak, normal and strong) were made depending on their strength in anterior compression, and one vertebral model was chosen close to the center of each class. The weak vertebra had a vBMD of 86 mg/cm3 and a strength of 1935 N, the normal vertebra had a vBMD of 86 mg/cm3 and a strength of 3229 N, and the strong vertebra had a vBMD of 190 mg/cm3 and a strength of 5091 N. The corresponding vBMD and elastic modulus of the three strength models are the following, weak: 71 mg/cm3, 192 MPa; normal: 86 mg/cm3, 240 MPa; strong: 190 mg/cm3, 576 MPa. Those three vertebral models had their own geometry and material properties These three models were used to perform simulation for all subjects of the study. We personalized only the boundary conditions to match the 3D orientation of the subject's L1 vertebra, and the lever arm of the upper body CoM.

The finite element model is detailed in [START_REF] Choisne | Vertebral strength prediction from Bi-Planar dual energy x-ray absorptiometry under anterior compressive force using a finite element model: An in vitro study[END_REF] and briefly described hereafter. Geometry of the vertebrae used for the model was obtained by a semi-automatic segmentation method [START_REF] Pennec | CT-based semi-automatic quantification of vertebral fracture restoration[END_REF] and a hexahedral mesh of the vertebrae was generated. Mechanical properties were linear elastic and were set for each element depending on the bone density measured in the corresponding voxels of the qCT-scan.

Thin layers of polymethyl methacrylate (PMMA) bone cement were virtually added on the upper and lower vertebral endplates for parallelism to ensure uniform loading conditions. The orientation of the vertebral model was adjusted to match the orientation of the subject's L1 vertebra. Lower nodes of the PMMA layer were constrained in all degree of freedom. A compressive load was applied to a node located at the position of the upper body center of mass of each subject and was joined by rigid elements to the upper PMMA layer (figure 2b).

The simulations were run on ANSYS software (ANSYS Inc, Canonsburg, PA, USA). The failure criteria chosen to describe vertebral strength was set when a contiguous region of 1 mm 3 of elements reached 1.5% deformation [START_REF] Pennec | CT-based semi-automatic quantification of vertebral fracture restoration[END_REF].

Load to strength ratio:

To integrate the body mass variations, a load to strength ratio was also computed. The load resulted in the upper body mass above L1 as estimated by the barycentremetric model and the strength was the one computed using the personalized lever arm. Theoretically, when the ratio is higher than 1, the load exceeds bone strength and a fracture will occur.

Data analysis:

Normality of the data was assessed with the Shapiro-Wilk normality test. Differences between groups were evaluated with Student tests (in case of normality) and Wilcoxon rank-sum tests (otherwise).

Results

Clinical parameters and differences between groups: All the studied parameters followed a normal distribution, except for age. The results are presented in the table 1 with mean (M) and standard deviation (SD).

The BMI tended to increase with age and was significantly higher in the older group compared to the younger group.

Regarding pelvic parameters, no significant changes in pelvic incidence with age were observed. Pelvic tilt increased significantly from 9.4 to 18° and sacral slope decreased significantly from 41.1 to 36.1° between the young and the elderly groups.

Concerning spinal parameters, a significant loss of lumbar lordosis as well as a significant increase in the thoracic kyphosis and in the cervical curvature lordosis (indicated by the transition from a positive to a negative value) were observed between the young and the older groups.

The sagittal alignment parameters (OD-HA sagittal angle, SVA, spinal inclination [OD-S1]) all showed a narrow range of variation between all age groups, but with a tendency to increase as age is higher. Significant differences between the young group and the elderly group were observed for the three parameters. However, significant differences between the young group and the intermediate group were only observed for the SVA and the spinal inclination [OD -S1].

L1 orientation and upper body CoM's lever arm regarding L1:

The axial and frontal orientation of L1 was not significantly different between the age groups, and close to zero degree. On the other hand, the sagittal orientation of L1 varied between the age groups (table 1 and figure 3). It was the highest in the intermediate group and decreased significantly for the elderly group, from 19° to 16°.

The mean antero-posterior lever arm of the upper body CoM regarding L1 varied from 1 mm posteriorly for the young group, to 10 mm and 24 mm anteriorly for the intermediate and elderly groups respectively, with significant differences between each group (figure 4a), showing an anterior shift of the upper body CoM with age. The standard deviation for the antero-posterior lever arm was high among all age groups and was the highest for the intermediate group (20 mm).

Using the mean and standard deviation values of the antero-posterior lever arm for the young group, a reference corridor was defined as values inside the following data range : [mean +/-2*standard deviation]. It was observed that 23% of the intermediate and 48% of the elderly subjects had an abnormally high antero-posterior lever arm (above mean + 2*standard deviation). The lateral lever arm had a mean value of 2 mm to the left among all the subjects with a mean standard deviation of 7 mm. It was not significantly different between the age groups.

Strength evolution with subject-specific boundary conditions:

For the high, normal and low strength models, the computed strength with subject-specific boundary conditions was significantly lower in the elderly group than for the other age groups (figure 4b). For the normal strength model, the strength resulting from the variation of the lever arm among all the subjects ranged from 860 to 4066 N. It represented a variation of 373% between the highest and the lowest highest value. A decrease by 28% and 27% in vertebral strength was observed for the normal and the weak models between the young group and the elderly, from 2527 N to 1820 N and from 1622 N to 1183 N respectively. The loss was higher for the strong model (35%, from 3710 N to 2430 N). For all strength levels, the standard deviation was the highest in the intermediate group.

The load to strength ratio was the lowest for the strong model vertebra with loading conditions of the young group and the highest for the weak model vertebra considering the elderly group loading conditions (figure 4c). For all vertebral strength levels, it was significantly higher in the elderly group than in the young group. For the normal vertebra, while a decrease of 28% of vertebral strength between those groups was observed, the load to strength ratio increased by 67%. In addition, whereas the strength of the normal vertebra was 8% lower in the intermediate group compared to the young group, the load to strength ratio raised up to 30% in those same groups.

Discussion

Vertebral fragility fractures are an increasing concern. Understanding how they occur is a critical issue for public health. The current standard for osteoporosis diagnosis is aBDM assessment by DXA. However, its vertebral fracture prediction accuracy is limited and DXA cannot be used to analyze sagittal alignment. In order to enhance fracture prediction accuracy, other parameters have to be included in the diagnosis of osteoporosis. This study aimed to investigate the direct link between sagittal alignment and vertebral strength estimation based on radiological data using barycentremetry and finite element analysis.

Pelvic, spinal and sagittal alignment parameters of asymptomatic adults ranging from 20 to 83 years old were studied. A combination of an increase in the thoracic kyphosis as well as the cervical lordosis and a loss of lumbar lordosis was observed with aging, which is consistent with other studies [START_REF] Bassani | Physiological variations in the sagittal spine alignment in an asymptomatic elderly population[END_REF][START_REF] Amabile | Invariance of head-pelvis alignment and compensatory mechanisms for asymptomatic adults older than 49 years[END_REF]. In addition, pelvic tilt was significantly higher in the elderly group than in the other two age groups. Pelvic retroversion and increase in cervical lordosis are compensatory mechanisms adopted to prevent the upper body from falling forward as the individuals grow older [START_REF] Amabile | Invariance of head-pelvis alignment and compensatory mechanisms for asymptomatic adults older than 49 years[END_REF] and to maintain a horizontal gaze direction. Despite those mechanisms, a more forward posture was observed for each step of aging among the studied population. The OD-HA sagittal angle, the SVA and the spinal inclination [OD-S1] increased with age [START_REF] Hu | Global sagittal alignment in elderly patients with osteoporosis and its relationship with severity of vertebral fracture and quality of life[END_REF][START_REF] Fechtenbaum | Sagittal balance of the spine in patients with osteoporotic vertebral fractures[END_REF][START_REF] Bassani | Physiological variations in the sagittal spine alignment in an asymptomatic elderly population[END_REF][START_REF] Amabile | Invariance of head-pelvis alignment and compensatory mechanisms for asymptomatic adults older than 49 years[END_REF]. Head should remain above the pelvis in order to maintain equilibrium and lowest effort. Because of that, the OD-HA angle was proven to stay within a narrow range among a healthy population [START_REF] Amabile | Invariance of head-pelvis alignment and compensatory mechanisms for asymptomatic adults older than 49 years[END_REF]. The main difference between the OD-HA sagittal angle and the spinal inclination [OD-S1] is the inclusion of the femoral heads, which results in the inclusion of the direct compensatory mechanism of the pelvis.

Using barycentremetry, a validated method used to accurately estimate the position of body CoM [START_REF] Nérot | A 3D reconstruction method of the body envelope from biplanar X-rays: Evaluation of its accuracy and reliability[END_REF], the location of the upper body CoM above L1 was computed. In this asymptomatic population, the lateral lever arm was always very close to zero. Between the young group and the elderly group, a shift of the antero-posterior lever arm of the upper body CoM from a negative to a positive value was observed, revealing a change from a backward to a forward position as age increases. It confirms the postural changes observed through the clinical parameters. Although compensatory mechanisms allowed keeping the odontoid over the axis of the femoral head relatively well, they did not keep the CoM of the upper body strictly over the L1 vertebra. Among the intermediate group, the dispersion of the upper body CoM location was higher than in the other two age groups. The transition phase of aging could be inside this age group, which might explain those results.

We used a finite element model to study how variations in the lever arm of the upper body CoM affected vertebral strength, using three vertebral models with different intrinsic strength. Those vertebral models were chosen to represent the variety of vertebral strength for the whole age spectrum. To confirm this and compare our results to the literature, uniaxial loading was also simulated for these three vertebral models (data not shown). When compared to Bouxsein et al [START_REF] Bouxsein | Age-and Sex-Specific Differences in the Factor of Risk for Vertebral Fracture: A Population-Based Study Using QCT[END_REF] who estimated vertebral strength for young and older subjects, the strength models chosen in the current study are consistent. The normal strength and strong models (4334 -7013 N) of our study are close to the young population's strength (5366 -6572 N) [START_REF] Bouxsein | Age-and Sex-Specific Differences in the Factor of Risk for Vertebral Fracture: A Population-Based Study Using QCT[END_REF]. In addition, the weak and normal strength models (3256 -4334 N) are similar to the older population's vertebral strength (3464 -4966 N) [START_REF] Bouxsein | Age-and Sex-Specific Differences in the Factor of Risk for Vertebral Fracture: A Population-Based Study Using QCT[END_REF]. In our study, for the same strength model, a variation of an average of 30% in the strength was observed resulting from the lever arm modifications related to the aging process only. We thus showed that apart from the intrinsic strength, an apparent strength including posture is altered by modifications with age and might be relevant to consider.

The load to strength ratio was computed to take into account morphological differences. In a standing position, when simulating a young individual (combining the normal and high strength level vertebrae and the loading conditions corresponding to the young group), the calculated ratio ranges from 0.07 to 0.1, which is consistent with the ratios observed [START_REF] Bouxsein | Age-and Sex-Specific Differences in the Factor of Risk for Vertebral Fracture: A Population-Based Study Using QCT[END_REF] for a population with comparable age. When an alteration of the posture was included (elderly group), translated by a shift of the antero-posterior lever arm, the ratio could be up to 1.7 times higher. Our study offers a method to integrate those elements and refine vertebral strength estimation for critical populations.

This study presents some limitations. First, the geometry and mechanical properties were fixed. Promising results have been presented to estimate the material properties of the vertebra with biplanar radiography [START_REF] Choisne | Vertebral strength prediction from Bi-Planar dual energy x-ray absorptiometry under anterior compressive force using a finite element model: An in vitro study[END_REF], and could be used in future studies. However, keeping unique models with identified strength levels was useful to highlight the effects of postural changes with age on vertebral strength independently of already observed changes in geometry or mechanical properties with age [START_REF] Kolta | Variations in vertebral body dimensions in women measured by 3D-XA: a longitudinal in vivo study[END_REF][START_REF] Riggs | Population-based study of age and sex differences in bone volumetric density, size, geometry, and structure at different skeletal sites[END_REF]. In addition, muscle forces were not taken into account in the load estimation, only gravity loads were computed. In the standing position, for asymptomatic individuals, the subject is supposedly in an effortless position, which could indicate that the muscles forces are low. Also, the results obtained with this simplified model are consistent with the literature [START_REF] Bouxsein | Age-and Sex-Specific Differences in the Factor of Risk for Vertebral Fracture: A Population-Based Study Using QCT[END_REF]. Finally, although the results were obtained on a fair sample size, it is a cross sectional-study, and a larger longitudinal set of data is needed to understand when the changes occur. An ulterior prospective study could be particularly interesting to assess the link between vertebral fracture occurrence and the current results.

In conclusion, we proposed a method to introduce in vivo loading into vertebral fracture risk assessment. The results confirmed that posture is different among older population and we were able to quantify and illustrate how those differences would affect vertebral strength. Further studies are needed to validate our observations, but those preliminary results indicate the importance of sagittal alignment when screening patients for osteoporosis. This study is a further step to reach a fully personalized finite element analysis for the evaluation of vertebral strength and fracture risk, including both personalized strength and personalized load.

Figures

Fig 1 Fig 2

 12 Fig 1 Sagittal alignment parameters, (a) SVA (mm) (post. Posterior, sup. Superior), (b) OD-HA sagittal angle (°), (c) Spinal inclination [OD-S1] (°) Fig 2 (a) Example of the reconstructed body envelope used to compute the upper body CoM above L1, segments in yellow are included in the computation (arms were place along the body for the CoM computation), the CoM and its vertical line are located on the envelope, as well as the subject's L1 vertebra, (b) example of the personalized loading conditions of the normal strength level vertebral model and the CoM location

Fig 3

 3 Fig 3 Average sagittal L1 orientation and antero-posterior lever arm of the upper body CoM above L1 against the normal strength vertebral model Fig 4 (a) Antero-posterior distance (mean and 1*SD) between the upper body CoM above L1 and the center of L1, (b) computed strength (mean and 1*SD) and (c) load to strength ratio (mean and 1*SD) resulting from the personalization of boundary conditions specific to the upper body CoM of the three FEM of L1 vertebra with identified strength level: weak (1900 N), normal (3200 N), and strong (5000 N) for 117 subjects divided in three age groups. Results of statistical tests to assess differences between the groups (*** = p<0.001, *=p<0.05)

Fig1Fig4

  Fig1Click here to access/download;Figure;Fig1.eps

Table 1 Results

 1

  Table1Click here to access/download;Table;Table1.eps

			(b)
		OD
	C7					Line of
	center					best fit	*	0.06
		Spinal parameters
	C3C7 curvature (°)	4.7 (10.3)	-2.9 (8.2) Vertical line -4.5 (12.4)	***	**	0.57
	T4-T12 thoracic kyphosis (°)	34.9 (11.4)	39.4 (9.5)		40.4 (10.8)	0.06	*	0.71
	L1-S1 lumbar lordosis (°)	57.9 (11.8)	54.6 (9)		50.3 (12.4)	0.16	**	0.14
	L1 sagittal orientation (°)	-18.3 (5.4)	-19.3 (5.2)		-15.7 (6.1)	0.45	0.06	*
		Pelvic parameters
	PI (°)	50.1 (10.9)	53.1 (8.9)		53.8 (12.4)	0.19	0.18	0.8
	SS (°) Post. corner of the sacral sup.	41.1 (8.7)	37.3 (6.9)		36.1 (9.3)	0.07	*	0.57
	PT (°) endplate	9.1 (6.3)	15.8 (4.6)		17.8 (7.9)	***	***	0.26
		Global sagittalalignment
	OD-HA sagittal angle (°)	-2.0 (1.9)	-1.7 (2.4)		0.4 (2.8)	0.49	***	**
	SVA (c m)	-1.5(2.2)	0.2(2.6)	HA	2.2(2.6)	**	***	**
	Spinal inclination [OD-S1] (°)	-2.9 (2.6)	-0.9 (3.5)		1.9 (3.5)	*	***	**

Acknowledgments

The authors thank the ParisTech BiomecAM chair program on subject-specific musculoskeletal modelling and in particular Société Générale and COVEA.

Funding This work was supported by the ParisTech BiomecAM chair program. Heidsieck, Laurent Gajny, and Christophe Travert, Jean-Yves Ethics approval The protocol was approved by the Comité de Protection des Personnes Ile-de-France VI Paris (Ethics Committee).


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Caroline Bogliotti, Frédéric Isel

(347 words)

Sign languages are gestural languages but linguistic informations can also be conveyed by vocal componants as mouthings. Mouthing usually tend to reproduce the more relevant phonetic part of the equivalent vocal word matching with the manual sign. One crucial issue in sign language processing is to understand what function may have mouthing, i.e. to which extent they contribute to the construction of the signs at a phonological or semantic lexical level. The present pilot study aimed to investigate the role of mouthing on the processing of French Sign Language (LSF). In particular, our goal was to examine whether mouthings combined with manual signs does influence in a way the comprehension of these signs. For this purpose, we designed a repetition priming experiment with 5 Deaf individuals who had learned LSF at different ages. We created five experimental conditions in which the combination of both the manual sign and the mouthing was systematically manipulated (Condition 1: both manual signs and mouthing matched; Condition 2: the signs were combined with the mouthing of a pseudo-word; Condition 3: the signs were associated with the mouthing of a sign belonging to the same lexical category; Condition 4: signs without mouthing; Condition 5: mouthing alone, i.e. without manual sign). Participants performed a lexical decision, i.e., they had to decide as fast as possible whether a picture matched with the sign presented. Descriptive data showed that while signs combined with the appropriate mouthing were processed the most rapidly, the slowest decision times were observed for signs presented with the mouthing of a pseudo-word (interference effect). Interestingly, the processing of signs without mouthing as well as mouthing without signs were lengthened, but to a lesser extent than conditions in which a mismatch were present between the signs and the mouthings. Taken together, the data of the present pilot study lend support to the hypothesis that mouthing might constitute one unity with the manual sign, and might be integrated into it. Further investigation should be conducted for consolidating the present preliminary data using a larger sample of Deaf individuals with different learning trajectory of LSF.


Delphine Battistelli

Caroline Bogliotti

Michael Filhol

Linguistic temporality in deaf children: assessment of uses of temporal adverbials in French Sign Language

come

Introduction

The vast majority of deaf children have language deficits in the understanding and the production of temporal forms, worsened by the absence of an efficient language input [START_REF] Newport | Maturational constraints on language learning[END_REF][START_REF] Morford | A reexamination of "Early Exposure" and its implications for language acquisition by eye[END_REF][START_REF] Mayberry | Linguistic ability and early language exposure: Development[END_REF][START_REF] Berk | Sensitive period effects on the Acquisition of Language : a study of language development[END_REF]. It seems crucial to use specific linguistic tools in order to help them in the well-usage of temporal forms. Our work takes place in the development of a prototype dedicated for helping the acquisition (by training exercises) of notions and structures that require the expression of temporality in all languages used by deaf children For this purpose, we first built an assessment tool composed of nine temporality linguistic tests dedicated to children aged from 5 to 12 years-old. They allow to evaluate different capacity levels in calendary dating (with an absolute, deictic or anaphoric point of view) and ordering events (with a chronological or not point of view) by use of various temporal adverbials in French and LSF.

The present study

Methodology

We aimed to compare temporality development in hearing vs deaf children and obtain first data on temporal adverbials in French and LSF.

Participants.

-

Discussion

• AC : chez les entendants, voir si les compétences s'améliorent avec l'AC .

• Deaf vs. control : voir les resultats de l'appariemment . Des éptreuves mieux réussies chez les entendants ? Chez les sourds ?

• Parler d'un point de vue descriptif du temps mis par les enfants sourds et la manière de tester le francais écrit : il faut dire qu'ils ont mis énormément de temps alorsqu'on a essayé 'alléger les épreves (demander à Sadnra)

Perspectives

• Test more hearing children -> il faudra dire qu »on teste les entendants parce qu »on part du principe que ce n'est pas la cognition qui est pobmatique • Test more deaf children: improve test which is too long (Improve our methodologies on written French assessment) • Elaborate the temporal LSF assessment tool from temporal adverbial LSF studies (Filhol, XXX) and others sources ///Ajouter capture d'écran temporalité LS (demander à Sandra qu'elle coupe des stimuli temporalité)

• Research and developmentDevelopment of an application for a training tool for ipad..

Preleminary results

Les épreuves 2, 3, 8 et 9 sont celles qui marquent le plus de différences entre les enfants entendants et les enfants sourds

Ajouter graphique 1. developpement normal (quellesc épteues ?)

Appariemment deaf control


domain: phys spi

text:
ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

  Bouchra Asbani, Camille Douard, D. Deresmes, Didier Stiévenard, Pascal Roussel, Frederic Favier, Christophe Lethien, Thierry Brousse To cite this version: Botayna Bounor, Bouchra Asbani, Camille Douard, D. Deresmes, Didier Stiévenard, et al.. Nanostructured MnO2 Films for 3D Micro-Supercapacitors: From New Insights of the Growth Mechanism to the Fine Tuning of Areal Capacitance Values. Journal of The Electrochemical Society, 2023, 170 (3), pp.030530. 10.1149/1945-7111/acbee8. hal-04062788

Supplementary Information

Nanostructured MnO 2 films for 3D micro-supercapacitors: from new insights of the growth mechanism to the fine tuning of areal capacitance values Botayna Bounor 1,2,3,4 , Bouchra Asbani 1,3 , Camille Douard 2,3 , Dominique Deresmes 1 , Didier Stiévenard 1 , Pascal Roussel 4 , Frederic Favier


Aurore Bonnet-Lebrun
email: aurore.bonnet-lebrun@ensam.eu

Agnès Linglart

Marine De Tienda

Younes Ouchrif

Jugurtha Berkenou

Ayman Assi

Philippe Wicart

Wafa Skalli

Quantitative analysis of lower limb and pelvic deformities in children with X-linked hypophosphatemic rickets

Keywords: Enfant, Membres inférieurs, XLH, Déformations osseuses tridimensionnelles, Stéréoradiographie Child, Lower Extremity, XLH, Bone 3D Deformity, Biplanar Radiography

Introduction

X-linked hypophosphatemia (XLH) is one of the most common forms of hereditary rickets, with an incidence of about 1 per 20,000 births [1-3]. It primarily manifests itself by abnormal mineralization of bone and dental tissue [1-3]. Children affected by XLH have painful deformities of the legs, which are progressive and sometimes very severe [2-3]. These deformities appear when they start walking and can be partially corrected during growth with appropriate non-surgical treatment [2-3]. Once the growth phase has ended, only orthopedic surgery can improve the alignment of the lower limbs [2-3]; thus is it vital to have objective and quantitative information about these deformities and their progression during growth.

While several studies have been done on bone quality [4-7], very few have focused on the bone geometry of the legs and pelvis in children with XLH. Bone deformity is currently evaluated by clinical measurement of intercondylar distances (ICD) and intermalleolar distances (IMD) and conventional 2D radiography [START_REF] Haffner | Clinical practice recommendations for the diagnosis and management of X-linked hypophosphataemia[END_REF][START_REF] Gizard | Outcomes of orthopedic surgery in a cohort of 49 patients with X-linked hypophosphatemic rickets (XLHR)[END_REF][START_REF] Adams | Radiology of Rickets ans Osteomalacia[END_REF][START_REF] Lempicki | Magnetic Resonance Imaging Features as Surrogate Markers of X-Linked Hypophosphatemic Rickets Activity[END_REF]. This provides general information about the deformities, specifically in the frontal plane (varus/valgus) and sagittal plane (flessum/recurvatum). While the ICD and IMD measurements are useful for prospective monitoring by a clinician, they are not very reproducible [3] [START_REF] Gupta | Angular deformities of lower limb in children: Correction for whom, when and how?[END_REF]. Two scores have also been used to evaluate rickets on 2D radiographs. The Thacher score, developed for deficiencyrelated rickets, analyzes the alterations in the growth plate at the wrist and knees on a scale of 1 to 10 [9]. The "Radiographic Global Impression of Change", which is rated from -3 to +3, evaluates the deformity of the legs and progression of the rickets lesions between two radiographs taken at least 3 months apart [START_REF] Whyte | Validation of a Novel Scoring System for Changes in Skeletal Manifestations of Hypophosphatasia in Newborns, Infants, and Children: The Radiographic Global Impression of Change Scale[END_REF]. But these two scores are not quantitative and are based on a subjective evaluation. Moreover, using 2D images to study complex three-dimensional (3D) deformities is subject to projection bias [10-11].

The EOS® system (EOS Imaging, France) uses simultaneous biplanar acquisition to produce a personalized 3D model of the skeleton of a standing individual. This technique, which has been validated in adults and children [10-13], has been used to establish normality corridors for geometric parameters of the legs and pelvis in children of different age groups [START_REF] Labelle | Pediatric sagittal alignment[END_REF][START_REF] Massaad | Three-dimensional evaluation of skeletal deformities of the pelvis and lower limbs in ambulant children with cerebral palsy[END_REF][START_REF] Gaumétou | EOS analysis of lower extremity segmental torsion in children and young adults[END_REF][START_REF] Rampal | Lower-limb lengths and angles in children older than six years: Reliability and reference values by EOS® stereoradiography[END_REF]. We hypothesized that quantifying the bone parameters will help to document and monitor these deformities in children with XLH.

The aim of this cross-sectional study was to characterize the deformities of the legs and pelvis in a population of children with XLH with the aim of identifying the radiological parameters that could be useful in the diagnosis or monitoring of patients.

Material and Methods

Patients

Thirty-five growing children with XLH, between 5 and 14.5 years of age were included in this crosssectional study (Table 1); none had undergone surgery on their lower limbs. All were receiving nonsurgical treatments. This study was approved by the local research ethics board (CPP06001) and consent was gathered from each patient and their parents. A control group, consisting of 40 asymptomatic agematched children (20 girls, 20 boys), was used to define the reference values for the geometric parameters of the legs.

Radiographs & reconstructions

Radiographs of the entire body were made with the EOS ® imaging system. Patients were placed in the standardized position defined by Chaibi et al. [10]: standing, right hallux at the level of the left plantar arch, hands on the mandible. A single trained operator did all the 3D reconstructions of the legs and pelvis using the improved Chaibi et al. method [10][16]: points of interest were selected by the operator, used to construct the first model which was back-projected on the AP/lateral radiographs, then adjusted by the operator until the contours of the back-projected model matched the image contours (figure 1).

From the 3D reconstruction, several parameters were calculated automatically (table 2, figure 2). For the pelvis, these parameters are the sacral slope, pelvic incidence and pelvic tilt [START_REF] Labelle | Pediatric sagittal alignment[END_REF] and for the acetabulum, these parameters are anteversion, coverage and inclination [START_REF] Massaad | Three-dimensional evaluation of skeletal deformities of the pelvis and lower limbs in ambulant children with cerebral palsy[END_REF].

For the legs, the parameters of interest are the femoral head diameter, femoral neck length, neck-shaft angle, femoral mechanical angle, tibial mechanical angle, tibiofemoral angle, angle between the mechanical axis and the femoral shaft (HKS), tibial and femoral anteversion, ratio of femoral length to tibial length [14-15].

Statistical analysis

The leg and pelvis parameters from the children with XLH were compared to those of the control children with Student's t-test or the Wilcoxon-Mann-Whitney test, depending on whether the distribution was normal. The significance threshold was set at p = 0.05. The normality of the distributions was evaluated with the Lilliefors test.

To refine the analysis, the distribution of XLH patients in the normality corridor was calculated for each parameter. A normal value is one that falls within one standard deviation of the mean value of the control group. The value was subnormally high (low) when it was between +1 and +2 SD (-2 or -1 SD) and abnormally high (low) above this (below this).

Linear correlations between various parameters were determined by Pearson's method.

Lastly, the legs of the XLH patients were placed into three subgroupsgenu varum, genu valgum and straightbased on the values of the tibiofemoral angle. The mean value of the control group was used as a reference [15] [START_REF] Salenius | The development of the tibiofemoral angle in children[END_REF]. The patients were classified as genu varus when the tibiofemoral angle was greater than 1 SD of the mean of the control group, while it was genu valgum when lower. The parameters of the subgroup of patients with XLH and the control patients were compared with ANOVA or Kruskal-Wallis tests depending on their distribution.

Results

Figure 3 shows the wide range of deformities observed in children with XLH. Of note, certain patients have large femoral shaft curvatures, both in varus/valgus and in flessum.

Comparison with the normal range [12-15] is provided in Table 3 For the pelvis, 34% of acetabular anteversion values were abnormally low while 44% of sacral slope values were abnormally or subnormally high in patients with XLH. Statistical tests confirmed a significant difference between XLH and controls for these two parameters.

The main abnormalities in the lower limb were neck-shaft angle, ratio of femoral length to tibial length and tibial torsion with more than 35% of patients having abnormally low values, while 48% of patients had abnormally high HKS values. The diameter of the femoral head and neck length were not significantly different.

For parameters that were markedly different between the XLH patients and control, Figure 4 shows the distribution by age of the points corresponding to the legs of patients with XLH (n = 70) and the control group. There was a clear distinction between the two populations for the femur/tibia length ratio (F/Tmean,XLH = 1.05, SDXLH = 0.04; F/Tmean,C = 1.15, SDC = 0.03). The genu varum (n=23), genu valgum (n=25) and aligned (n=22) subgroups had significant differences in the tibiofemoral angle, femoral mechanical angle and HKS.

The correlation coefficient relating the femoral mechanical angle to the HKS and the femoral mechanical angle to the tibiofemoral angle was higher in the XLH patients than in the controls (r² = 0.73 versus r² = 0.30 and r² = 0.69 versus r² = 0.21, respectively) (Figure 5).

Discussion

The aim of this study was to define the bone deformities in the legs and pelvis of children with XLH rickets to identify relevant radiological parameters for their diagnosis and follow-up. We used 3D reconstructions to get around the projection bias related to 2D radiographs, particularly for complex 3D deformities like those induced by XLH [START_REF] Chaibi | Fast 3D reconstruction of the lower limb using a parametric model and statistical inferences and clinical measurements calculation from biplanar X-rays[END_REF][START_REF] Gheno | Three-dimensional measurements of the lower extremity in children and adolescents using a low-dose biplanar X-ray device[END_REF]. Weightbearing radiographs allowed us to study positional parameters such as pelvic tilt along with morphological parameters [11], [START_REF] Massaad | Three-dimensional evaluation of skeletal deformities of the pelvis and lower limbs in ambulant children with cerebral palsy[END_REF], [START_REF] Quijano | Three-dimensional reconstruction of the lower limb from biplanar calibrated radiographs[END_REF].

This cohort of 35 children with XLH rickets that is undergoing non-surgical treatment is representative of the current global population of children with XLH who are treated as soon as they are diagnosed [START_REF] Haffner | Clinical practice recommendations for the diagnosis and management of X-linked hypophosphataemia[END_REF]. The characteristics of this set of children, which is of comparable size to the largest study groups for this disease [5], are consistent with what is expected from this pathology (Table 1) [START_REF] Haffner | Clinical practice recommendations for the diagnosis and management of X-linked hypophosphataemia[END_REF], particularly in the 1/3 boy and 2/3 girl distribution.

Our study found a large diversity of 3D deformities in children with XLH, and hence, the benefit of characterizing each patient individually.

Several lower limb parameters, already studied in children in other pathological situations [12-15], are modified because of XLH. In particular, parameters related to the femoral diaphysis such as the HKS, neck-shaft angle and femur/tibia length ratio are significantly different between the XLH patients and control children. This finding is consistent with existing qualitative studies [3], [START_REF] Thacher | The Effect of Nutritional Rickets on Bone Mineral Density[END_REF]. Conversely, the local parameters at the hip (femoral head diameter and neck length) do not appear to be affected. While no significant difference was found, likely because of the large variability even within the control group, tibial torsion appears reduced on average in XLH patients.

Looking at the genu varum, genu valgum and aligned subgroups and the correlation analysis revealed the predominant role of the femur in the tibiofemoral deformity in the frontal plane.

At the pelvis, the abnormalities were not as pronounced but still impacted the sacral slope and acetabular anteversion.

Based on this study, the most relevant markers to analyze the leg deformity in children with XLH at a given time are the femoral anglesfemoral mechanical angle, neck-shaft angle and HKSalong with general parameters such as the femur/tibia length ratio and the tibiofemoral angle. Since the femoral mechanical angle appears to be correlated with the tibiofemoral angle and the HKS, calculating only the femoral mechanical angle, neck-shaft angle and femur/tibia length ratio appears to be sufficient to capture the main deformities in the legs. These highly reproducible markers [10] [START_REF] Rampal | Lower-limb lengths and angles in children older than six years: Reliability and reference values by EOS® stereoradiography[END_REF], could be incorporated in routine clinical and radiological evaluation of patients with XLH. The other radiological parameters presented in this article can refine the characterization of the deformity as needed.

These findings provide quantitative information about the bone deformities, but not allow us to evaluate the impact of these deformities on locomotion. A gait analysis of this population is currently underway.

Conclusion

Quantifying bone parameters in 3D revealed the nature of the deformities in children with XLH for the first time without projection bias. For all these patients, the deformities are mainly located in the femoral shaft, but not limited to it. Some of these radiological parameters may be useful for the diagnosis and monitoring of XLH patients. Moreover, the large range of deformities that exist in these XLH patients justifies characterizing each patient individually. By following these patients over several years, we will be able to quantify the effect of treatments on the changes in bone deformities during growth.

Clinical laboratory

Value (SD) Reference ALP (1) , IU/L 344 (122) 150-450 Phosphate (2) mmol/L 0.8 (0.2) 1.2-1.9 25 OH Vitamin D (1) , ng/mL 38 ( 16) 30-80 PTH (1) , ng/L 53 (25) 14-75 TRP (1) , % 87 [START_REF] Gheno | Three-dimensional measurements of the lower extremity in children and adolescents using a low-dose biplanar X-ray device[END_REF] 80-100 6MWTajusted (1) [18], Z-score -2.34 (1.08) -

(1) at enrollment, (2) at diagnosis SD: standard deviation, BMI: body mass index, ALP: alkaline phosphatase, PTH: parathormone, TRP: tubular reabsorption of phosphate, 6MWTadjusted: 6-minute walking test adjusted to age. For each pair of parameters, the regression line, its confidence intervals and the square of the correlation coefficient are shown. The latter indicates how well the sample points match with the regression line.

Figure 1 :

 1 Figure 1: Reconstruction of the lower limbs and pelvis using the method described by Chaibi et al. [10]

Figure 2 :

 2 Figure 2: Lower limb and pelvis parameters

Figure 3 :

 3 Figure 3: Example of the wide range of profiles found in this study

Figure 4 :Figure 5 :

 45 Figure 4: Values of the lower limb parameters in the XLH population and the control group relative to age

Table 1 :

 1 Patient characteristics

	GENERAL PARAMETERS
		Value (SD)
	Number	35
	Sex	23F/12M
	Age (years)	9.9 (2.3)
	Height (Z-score)	-0.48 (1.10)
	Growth rate, cm/year	6.7 (1.8)
	PHEX mutation	28 Yes /6 No /1 Unknown
	BMI (Z-score)	0.54 (0.89)

Table 2 :

 2 Definition of parameters related to the pelvis [14] and the lower limbs[START_REF] Labelle | Pediatric sagittal alignment[END_REF][START_REF] Massaad | Three-dimensional evaluation of skeletal deformities of the pelvis and lower limbs in ambulant children with cerebral palsy[END_REF]

	Pelvis
	Type	Name	Component 1	Component 2	Comment
	Angle	Sacral slope	Tangent to sacral growth	Horizontal	S_pelvis*
			plate
		Pelvic	Vertical	Sacro-acetabular line**	S_pelvis*
		incidence
		Pelvic tilt	Normal to sacral growth	Sacro-acetabular line**	S_pelvis*
			plate
		Acetabular	Frontal axis of pelvis	Tangent line to acetabulum	Parallel plane to
		anteversion	projected on the sacral	borders	sacral growth plate
			growth plate
		Acetabular	Normal to sacral growth	Normal to tangent plane to
		inclination	plate	acetabular borders
	Ratio	Acetabular	Surface of acetabulum	Sphere modeling the
		coverage		acetabulum
	Lower limbs
	Type	Name	Component 1	Component 2	Comment
	Angle	Neck-shaft	Axis of superior diaphysis	Neck axis of femur	F_knee***
		angle
		Femoral	Femur's mechanical axis	Tangent to distal ends of	F_knee***
		mechanical		condyles	Medial angle
		angle
		Tibial	Tibia's mechanical axis	Tangent to tibial plateau	F_knee***
		mechanical			Medial angle
		angle
		Tibiofemoral	Femur's mechanical axis	Tibia's mechanical axis	F_knee***
		angle			Lateral angle
		HKS	Axis of distal diaphysis	Femur's mechanical axis
		Femoral torsion Neck axis	Tangent posterior to	Transverse plane of
				condyles	femur
		Tibial torsion	Tangent posterior to tibial	Bimalleolar axis	Transverse plane of
			plateaus		tibia
	Ratio	Femur/tibia	Length of femur between	Length of tibia between the
			center of femoral head and	intercondylar eminence and
			trochlear groove	pilon
	Length	Diameter of femoral head	Sphere modeling the femoral head
		Length of neck	Defined between the center of the femoral head and intersection of the diaphyseal
			and neck axes
	* S_pelvis: sagittal plane of pelvis
	** Joins the center of the sacral growth plate to a point in the middle of the two acetabulums
	*** F_knee: frontal plane of knee

Table 3 :

 3 Comparison to normality brackets for the pelvis [12-13] and lower limbs [14-15] of patients with XLH

		Abnormally	Subnormally	Normal Subnormally	Abnormally
		low	low		high	high
	Pelvic parameters
	Sacral slope	5%	3%	49%	30%	14%
	Incidence	0%	11%	62%	16%	11%
	Version	3%	35%	41%	19%	3%
	Acetabular parameters
	Anteversion	34%	23%	41%	3%	0%
	Inclination	1%	5%	74%	16%	3%
	Coverage	1%	18%	72%	9%	0%
	Lower limb parameters
	Femoral torsion	19%	23%	39%	11%	9%
	Tibial torsion	36%	24%	33%	4%	4%
	Femoral mechanical angle	28%	9%	20%	10%	34%
	Tibial mechanical angle	23%	16%	40%	13%	9%
	Tibiofemoral angle	6%	8%	45%	18%	24%
	Neck-shaft angle	39%	38%	23%	1%	0%
	HKS	14%	10%	20%	9%	48%
	Diameter of femoral head	0%	5%	69%	19%	8%
	Length of neck	0%	5%	60%	33%	3%
	Femur/tibia length ratio	41%	36%	23%	0%	0%
	Distribution of patients with XLH into normal, subnormal and abnormal values
	Note: in a normal asymptomatic population, 68% of values are normal, 27% are subnormal and 5% are
	abnormal.

Funding

Kyowa Kirin Pharma provided financial support for this study.

Conflict of interest

Professor Linglart received grants and fee reimbursement from Kyowa Kirin.

Professor Skalli has a patent related to the biplanar imaging system but does not personally receive any financial benefit from it.

Royalties from EOS Imaging are paid to the Georges Charpak Human Biomechanics Institute to fund research and education.

Author contributions

A. Bonnet-Lebrun: study design, data acquisition and processing, writing of article A. Linglart: study design, patient enrolment, scientific analysis, contributed to writing article M. De Tienda, Y. Ouchrif: contributed to data acquisition and processing J. Berkenou: patient enrolment, contributed to analyzing the clinical data P. Wicart: study design, contributed to data acquisition, scientific analysis and writing the article A. Assi: data from control subjects, contributed to data processing and scientific analysis W. Skalli: study design, contributed to data processing, scientific analysis and writing the article


Paul Chaillou

Jialei Shi

Alexandre Kruszewski

Isabelle Fournier

Helge A Wurdemann
email: h.wurdemann@ucl.ac.uk

Christian Duriez
email: christian.duriez@inria.fr

Reduced finite element modelling and closed-loop control of pneumatic-driven soft continuum robots

The introduction of soft robots has led to the development of inherently safe and flexible interventional tools for medical applications, when compared to their traditionally rigid counterparts. In particular, robot-assisted surgery is one of the medical applications that benefits from the inherent properties of soft instruments. However, robust control and reliable manipulation of soft tools remain challenging. In this paper, we present a new method based on reduced finite element method model and closed-loop inverse kinematics control for a fiber-reinforced soft robot. The highly flexible, pneumatically driven soft robot has three fully fiber-reinforced chamber pairs. The outer diameter is 11.5 mm. An inner working channel of 4.5 mm provides a free lumen for in-vivo cancer imaging tools during minimally invasive interventions. Here, the manipulator is designed in order to retrieve a tissue biopsy which can then be investigated for cancerous tissue. Simulation and experimental results are compared to validate the model and control methods, using one-module and two-module robots. The results show a real-time control is achievable using the reduced model. Combing the closed-loop control, the median position tracking errors are generally less than 2 mm.

I. INTRODUCTION

For a long time, one way of ensuring positioning accuracy in robotics included the increase of stiffness in an articulated robot structure. But the metrics used, especially absolute positioning in space, were derived from use cases in industrial environments. In these settings, traditional robotic arms were made of rigid materials. For surgical robotics applications, the environment is fundamental different and the metrics have naturally evolved, giving a particularly interesting use case for soft robots. The need for performance, however, remains, which is sometimes challenging to obtain.

In this paper, we focus on a use case involving a soft robot for an automatic scanning tool during a laparoscopic procedure. To guarantee a precise scanning procedure, the robot requires a good relative positioning accuracy of its tip with respect to an organ surface. At the same time, this interventional application inside the human body demands an inherently safe robotic tool, which advocates for the usage of inherently soft materials. The challenge is to increase the relative positioning accuracy for a robotic manipulator made of soft material. Hence, the contribution of this paper lies in a new method based on reduced finite element method (FEM) model and its closed-loop inverse kinematics control, adapted for a fiber-reinforced soft robot for laparoscopy.

A. Background

Compared to rigid-link robots, soft continuum robots have a variety of design morphology and actuation principles [START_REF] Rus | Design, fabrication and control of soft robots[END_REF]. The generalisation of a kinematic or dynamic modelling framework for soft robots is challenging [START_REF] Armanini | Soft robots modeling: a literature unwinding[END_REF]. One prominent modelling approach is the constant curvature (CC) model. The curvature along the back bone of a soft manipulator is assumed to be constant. Hence, the kinematic shape can be described by the curve length, curvature and out-of-plane angle [START_REF] Webster | Design and kinematic modeling of constant curvature continuum robots: A review[END_REF]. The CC model can be extended to a piece-wise constant curvature (PCC) model, where a continuum robot is discretised into sections. Each section has a constant curvature [START_REF] Hannan | Kinematics and the implementation of an elephant's trunk manipulator and other continuum style robots[END_REF]. In such way, a robot can have sections of different curvatures [START_REF] Mahl | A variable curvature continuum kinematics for kinematic control of the bionic handling assistant[END_REF]. Based on the (P)CC assumptions, the dynamics model can be established, e.g., considering a robot is composed of virtual prismatic or revolute joints [START_REF] Katzschmann | Dynamic motion control of multi-segment soft robots using piecewise constant curvature matched with an augmented rigid body model[END_REF]. In addition, general variable curvature models have been explored, for instance, the Cosserat rod models [START_REF] Rucker | A geometrically exact model for externally loaded concentric-tube continuum robots[END_REF], [START_REF] Till | Real-time dynamics of soft and continuum robots based on Cosserat rod models[END_REF], the piecewise constant strain model [START_REF] Renda | Discrete cosserat approach for multisection soft manipulator dynamics[END_REF], the beam mechanics [START_REF] Chikhaoui | Comparison of modeling approaches for a tendon actuated continuum robot with three extensible segments[END_REF] or FEM models [START_REF] Duriez | Control of elastic soft robots based on real-time finite element method[END_REF]. As some approaches are computationally expensive, the shape interpolation [START_REF] Sadati | Real-time dynamics of concentric tube robots with reduced-order kinematics based on shape interpolation[END_REF] technique or reducedorder models have been explored [START_REF] Gandarias | Open-loop position control in collaborative, modular variable-stiffness-link (VSL) robots[END_REF] to improve efficiency.

Once the forward kinematics or dynamics models are determined, the model-based inverse control can be implemented. Based on the (P)CC assumption, analytical inverse approaches can be calculated, for instance, via the closed form [START_REF] Abidi | Highly dexterous 2-module soft robot for intra-organ navigation in minimally invasive surgery[END_REF] or the differential Jacobian [START_REF] Jones | Kinematics for multisection continuum robots[END_REF]. Della Santina et. al described dynamic impedance control when a soft robot is in physical interaction with the environment [START_REF] Della Santina | Modelbased dynamic feedback control of a planar soft robot: trajectory tracking and interaction with the environment[END_REF]. A real-time simulation and control method of soft robots with self-collisions is proposed in [START_REF] Goury | Real-time simulation for control of soft robots with self-collisions using model order reduction for contact forces[END_REF], based on a reduced-order FEM and implemented using SOFA [START_REF] Duriez | Framework for online simulation of soft robots with optimizationbased inverse model[END_REF]. Alternatively, the optimisation-based method can be used, e.g., constructing the initial conditions by including the actuation variables, the shooting method can solve the inverse control by minimising the errors of the boundary based on the Cosserat rod model [START_REF] Till | Efficient computation of multiple coupled cosserat rod models for real-time simulation and control of parallel continuum manipulators[END_REF]. Apart from model-based inverse control, learning approaches can be advantageous to deal with the un-modelled uncertainties and external disturbances [START_REF] Thuruthel | Learning closed loop kinematic controllers for continuum manipulators in unstructured environments[END_REF]. Those modelling and control advances have facilitated the development of soft robots. In particular, the medical sector has emphasised increasing levels of autonomy to achieve safe and efficient robot-assisted surgeries [START_REF] Yang | Medical robotics-regulatory, ethical, and legal considerations for increasing levels of autonomy[END_REF]. In this case, robust and reliable control and manipulation of medical soft robots are of paramount importance.

B. Contribution and outline

In this paper, we first present the design of a highly flexible, pneumatically driven soft continuum robot, which is a miniaturised version of the STIFF-FLOP design [START_REF] Abidi | Highly dexterous 2-module soft robot for intra-organ navigation in minimally invasive surgery[END_REF]. The robot has three fully fiber-reinforced chamber pairs. Made of sewing thread, this fiber avoids the radial expansion. The outer diameter is 11.5 mm. An inner working channel of 4.5 mm provides a free lumen for in-vivo cancer imaging tools during minimally invasive interventions in cancer surgery. Here, a device is introduced through the inner free channel of the manipulator in order to retrieve a digital biopsy thanks to the molecular analysis of tumor and peri-tumoral tissues using a mass spectrometry based technology, the SpiderMass [START_REF] Fatou | Substrate-mediated laser ablation under ambient conditions for spatially-resolved tissue proteomics[END_REF]. We then present the formulation of a reduced FEM model with the open-loop and closed-loop control, based on the hypothesis of the inextensibility of the fiber-reinforcement layer (see Fig. 1). The challenge is to use our new movement parameterization model to do realtime control. A comparison between the computational and experimental results validates our approach.

[mm]

Inner diameter of the chambers.

[mm]

Diameter of the chamber position.

[mm]



Angle between two adjacent chambers. 60.0 [deg] Outer diameter of the chamber. The remainder of the paper is organised as follows: Section II discusses the medical background for in-vivo cancer imaging and presents the design of the medical soft instruments. Section III then details the reduced FEM model with the open-loop and closed-loop inverse kinematics control. Section IV then reports the validation results from the simulation and experiments, using one-module and twomodule robotic manipulators. Section V presents the discussions, and the conclusions are presented in Section VI.

[mm]

II. REQUIREMENTS FOR IN-VIVO CANCER INTRAOPERATIVE IMAGING IN CANCER SURGERY USING SPIDERMASS TECHNOLOGY AND ROBOT DESIGN

A. Context : In-vivo cancer imaging

Surgery is essential and remains the first frontline treatment of solid cancers. In order to determine the exact extension of the cancer, to preserve non-cancerous tissue at most, robotic imaging may give useful informations the surgeon. Our envisioned technology is based on the mini-invasive real-time mass spectrometry (SpiderMass Technology) [START_REF] Ogrinc | Robot-Assisted SpiderMass for in vivo real-time topography mass spectrometry imaging[END_REF], within Minimally Invasive Surgery (MIS) context. Compared to traditional in-vivo imaging (CTscan, IRM) and margin verification systems (Histology and immunohistochemical analysis), which are time-consuming and limited, the Spi-derMass Technology can analyse the cells in real-time and distinguish the different cell phenotypes therefore making possible to discriminate types, subtypes and grades of solid tumors [START_REF] Saudemont | Real-Time molecular diagnosis of tumors using water-assisted laser desorption/ionization mass spectrometry technology[END_REF]. It uses an optical fiber to bring the laser beam to the surface and a transfer line to bring the aerosol produced by the laser ablation up to the mass spectrometer for further analysis. Combining this real-time imaging technology with soft robots might result in a significant reduction in operation time and impact on the patient's health. Soft robots, due to their high dexterity and intrinsic safety, are very adapted to sensitive and narrow context, such as MIS. To product quick and accurate images, the point per point scanning have to be automatic, and soft properties may gives more security guarantee compared to rigid robot. This medical context leads to our design and to the constrains we want to satisfy. A hollow robot is also needed to be able to pass and protect the optic fiber and the suction tube of the SpiderMass system.

Components Diameter [mm] Descriptions

Vacuum tube 1.5 mm Retrieving the biopsy samples.

Actuation pipes * 1.0 mm Three actuation pipes.

Depth sensor 1.0 mm Checking the focal distance the laser ablation (three wires).

Optical fiber 0.7 mm Ablating the organs.

Position sensor 1.0 mm For the closed-loop control (one wire).

B. Design of one soft robotic module

Based on the introduced cancer imaging application, a pneumatically driven soft module is presented (see Fig. 1(a)), with an outer diameter of 11.5 mm (less than the diameter of the commercially available trocar port which has a diameter of 12 mm). An inner lumen of a 4.5 mm diameter is embedded. The length of one robot module is 55 mm. Connecting two robot modules in series results in a two-module robot which is highly flexible and dexterous (see Fig. 1(b)). The robot body is made of a low shore hardness silicone (Ecofex 00-50, SmoothOn). The robot has six fully fiber-reinforced chambers distributed evenly [START_REF] Fraś | New STIFF-FLOP module construction idea for improved actuation and sensing[END_REF]. Two adjacent chambers are internally connected via a 1 mm silicone pipes and actuated as one pair. The robot fabrication can be referred to [START_REF] Abidi | Highly dexterous 2-module soft robot for intra-organ navigation in minimally invasive surgery[END_REF], [START_REF] Shi | Design and characterisation of cross-sectional geometries for soft robotic manipulators with fibre-reinforced chambers[END_REF]. To keep the inner lumen free for instrumental components (see Table . I), e.g., the optic fiber and sensor wires, three 1.5 mm diameter slots are preserved along the lumen to place the actuation pipes of a 1 mm diameter (see Fig. 2). For closed loop position control, the inner lumen offers sufficient space for an additional positioning sensor and depth sensor, so that the end effector is put at the focal distance of the surface to be analysed. Details of the crosssectional geometries are shown in Fig. 2.

III. MODELLING AND CLOSED-LOOP CONTROL METHOD

The modelling and associated control methods require a trade-off between the accuracy and computational time. The quality of the control is impacted by the accuracy of the model, but in a feedback control scheme, the computation time is also of paramount importance. A high-frequency simulation will be beneficial for choosing control coefficients leading to a robust system, that is able to effectively resist perturbations and returns minimal modelling errors. In order to be able to compute real-time positioning, we derive a reduced FEM model to compute the elastic internal forces in a reduced beam space. In practice, we compute all internal forces and the impact of the actuation in the FEM space, but we solve the equilibrium of forces in this reduced space.

A. Modelling of the reduced model 1) Reduced beam kinematics: In beam theory, the motion of points follows certain assumptions. In particular, the crosssections are assumed to be rigid, so the motion of any point in the beam can be parameterised by the central axis motions. This assumption of a rigid section is motivated by the use of fiber-reinforced chambers in the design. We use this beam kinematics to reduce the size of the motion coordinates of the model. To model forces on the soft robot module, we use a 3D FEM model, meshed with nodes. Then, the position of any nodes of the FEM can be deduced from the degrees of freedom (DOFs) of the beam. This yields in [START_REF] Rus | Design, fabrication and control of soft robots[END_REF].

x = A(q) (1)
where x denotes position of all nodes (denoted by p n ) of the FEM mesh, and q denotes position of the beam's DOFs (see Fig. 3). A(q) creates a static link between the FEM nodes and beam, i.e., the position of the FEM points is static in the beam frame. For each point of the FEM, we associate a barycentric reference R bary (p n , t) by interpolation

R bary (p n , t) = slerp(q k (t), q k+1 (t), l(p n )) (2)
where q k is the position of the kth DOF of the beam, and slerp() is the interpolation function for position and orientation between two nodes of the beam. Details on the interpolation method are described in [START_REF] J.-S. Ahn | Realization of orientation interpolation of 6-axis articulated robot using quaternion[END_REF]. l(p n ) is the function that associates the p n nodes to the position of the barycentric reference between two beam nodes. In the barycentric reference associated to each point, the position of the node p n of the FEM mesh is static and described by

p n (t) R bary (pn,t) = p n (0) R bary (pn,0) (3)
This models the fiber around the pneumatic cavities. As the distance between the FEM point and the beam node is fixed, the FEM mesh does not expand laterally due to fiber constraints. Expressing p n (t) in the global frame yields

R bary (p n , t) -1 p n (t) Abs = R bary (p n , 0) -1 p n (0) Abs (4)
The subscript Abs refers to the absolute position in the global frame. As such,

p n (t) Abs = R bary (p n , t)R bary (p n , 0) -1 p n (0) Abs (5)
In practice, this link is made in the SOFA framework using BeamAdapter Plugin [START_REF] Jourdes | Visual haptic feedback for training of robotic suturing[END_REF]. For a small displacement or speed, the derivatives of node position x from (1) are described by

dx = ∂A ∂q dq and dx dt = ∂A ∂q dq dt (6)
2) Forces projection : To model the structural non-linear elasticity and the pneumatic actuation, we compute the force balance on a volume 3D FEM model. To bring the non-linear forces f (x) of the FEM mesh to the DOFs of the beam, we use the principle of virtual works which is independent from the references. The virtual displacement dq (or dx in the FEM space) makes the forces to work, this yields

dW = dx T f (x) = dq T τ (q) (7)
with τ (q) as the internal forces, expressed in this reduced space. The Jacobian matrix J of the function A(q) is introduced to achieve the coordinate transformation by J = ∂A ∂q so with [START_REF] Katzschmann | Dynamic motion control of multi-segment soft robots using piecewise constant curvature matched with an augmented rigid body model[END_REF], it is obtained : dx = Jdq (8)

By using the Jacobian matrix from ( 8), the forces in the FEM space expressed in the beam space are

J T f (x) = τ (q) (9)
In practice, a linearisation is computed at each step to compute the equilibrium.

J T f (x + dx) ≈ J T (f (x) + ∂f ∂x dx) ≈ J T f (A(q)) + J T KJdq (10)
K = ∂f ∂x is the non-linear stiffness of the structure computed by FEM. This forces equilibrium is solved at each time step in the SOFA simulation. For the dynamics, we may calculate the same kind of projection. However, the implicit integration scheme is dampening the dynamics at this frame rate.

B. Comparison with the full FEM model

To validate and to measure the performance gain, the reduced model is compared with a classical FEM model without any beam motion reduction. In this simulation, we model the impact of the reinforcement fiber using stiff springs. These springs are placed around the cavity in the FEM, to reproduce the effect of real fibers constraining lateral deformation of the module (i.e., preventing ballooning).

1) Elastic parameters: As the impact of the reinforcement fiber is directly modelled in the classical FEM, the equivalent Young's modulus of Ecofex 00-50 is 100 kPa. For the reduced model, the force projection introduces artificial stiffness in the simulation. This artificial rigidity may be due to the beam assumption, i.e., the cross-section is rigid. So, the shrinking effect of the cross-section is constrained. Hence, the Young's modulus in the reduced model is 83 kPa, derived from the gradient descent optimisation.

2) Accuracy comparison: To investigate the accuracy between the two modelling techniques, three tests were conducted. 0 to 100 kPa were applied to one, two and three chamber. The positioning comparison results are displayed in Fig. 4. Globally, the positioning properties of the two models are very similar. We may see that for very large deformations, the reduced model looks to introduce artificial rigidity, that may be due to the static cross-section assumption of the robot in the reduced model. All the value are the mean value for 4 experiments, described with their standard deviation. (*FPS = Frame Per Second)

3) Computational performance comparison: To understand the computational expense, we run simulations in an open-loop control scenario, that drew a 40 mm diameter circle, as described in Experiment 2 (see Section IV-B). The simulations ran for 1000 iterations, each with a 0.1s time step. Each experiment ran 4 times. Table II summarises the mean time and FPS with the associated standard deviations. The computation of the reduced model is faster than the full FEM model. For the simulation of one module it is about 4 times faster. For the other simulations, it is about 8 times faster. Indeed, the complexity of the computation is different:

• for the full FEM model, most of the computation is dedicated to the factorization of the sparse FEM matrix system which has a theoretical complexity O(n 3 ).

• for the reduced model, most computational time is spent on the projection of matrix K in the reduced space (J T KJ shown in [START_REF] Chikhaoui | Comparison of modeling approaches for a tendon actuated continuum robot with three extensible segments[END_REF]) which has a linear complexity. The time to solve the equilibrium by factorization of the system is quick due to the fact that the size is small.

C. Inverse kinematics control

For the inverse kinematics control, we compute the inverse problem using an optimisation-based method [START_REF] Duriez | Framework for online simulation of soft robots with optimizationbased inverse model[END_REF], based on the reduced FEM model, using the available SoftRobot.Inverse Plugin in SOFA. In this way, the inverse control is open-loop because no feedback from the hardware (so y u = y d ) is obtained. To accommodate un-modelled uncertainties, a closed-loop control is implemented using a proportional-integral (PI) controller. The desired position y d is compensated by the current measured error y e , so the demand y u in the closed-loop control equals

y u = y d + k p y e + k i t 0 y e dt (11)
The open-loop and closed loop control schemes are summarised in Fig. 5. Please note that k p and k i are not scalars, instead, they have a diagonal form so the controller can use different gains to correct the position errors along three axes.

In practice, the PI parameters are determined empirically.

IV. COMPUTATIONAL SIMULATION AND EXPERIMENTAL VALIDATION RESULTS

A. Experimental setup

The chamber pressure is regulated and monitored by proportional pressure regulators (Camozzi K8P). Pressurised air is supplied to the regulators by a compressor (HYUNDAI Model HY5508). The pressure regulators are controlled by an Arudino Due. The board sends PWMs (0∼ 100%) which are proportionally converted to (0∼ 10 V) analogue control voltages. An electromagnetic tracking system (NDI Aurora) monitors the tip (and the middle position) of the (twomodule) robot using trackers. Experiment 1 -Point-to-point control (one-module robot): A point table was implemented with desired points sent out one by one (see Fig. 6). The desired height is 55 mm and 60 mm on the right-hand and the left-hand xy plane, respectively. To simulate a static condition, the desired positions were updated every 20 calculation steps. Both open-loop and closed-loop control were investigated. Each test repeated the point table 10 times.

Experiment 2 -Spatial trajectory tracking control (onemodule robot): Four different trajectories were used to validate the open-loop and closed-loop control. Two circular trajectories with a diameter of 30 mm (C30) and 40 mm (C40), and two rectangular trajectories with a side length of 20 mm (R20) and 30 mm (R30), were chosen. The desired positions in the z-axis were kept as 55 mm. Each trajectory was repetitively tracked for 10 times.

Experiment 3 -Spatial trajectory tracking control (twomodule robot): Two different trajectories were used to validate the modelling and control for a two-module robot. The trajectories included a circle shape with a diameter of 80 mm, and a rectangle shape with a side length of 60 mm. The desired positions in the z-axis were kept as 110 mm. Apart from the monitored tip position, an additional NDI tracker was embedded in the middle of the robot (see Fig. 9(a)). The robot's positions were tracked 10 times for each trajectory.

C. Simulation and experimental results

Results for Experiment 1: Fig. 6 Furthermore, as shown in Fig. 10, the median closed-loop tip errors are 1.6 mm and 1.9 mm for the circular and rectangular shapes, respectively. By comparison, the median tip errors using the open-loop control are 6.9 mm and 7.9 mm for the circular and rectangular shapes, respectively. In this work, we assumed a linear material model with a constant Young's Modulus. This assumption becomes less effective when the robot undergoes larger deformation, as the silicone material shows a nonlinear strain-stress relationship [START_REF] Marechal | Toward a common framework and database of materials for soft robotics[END_REF]. As such, exploring nonlinear hyper-elastic models might improve the modelling accuracy and further mitigate the errors from the open-loop control. For instance, it was reported that the Neo-Hookean model could be used to model the fiber-reinforced bending actuator when the elongation is less than 50% [START_REF] Polygerinos | Modeling of soft fiber-reinforced bending actuators[END_REF]. Furthermore, the cross-sectional area will shrink when the robot elongates, in particular, when the material is incompressible, as reported in [START_REF] Gazzola | Forward and inverse problems in the mechanics of soft filaments[END_REF]. Considering this cross-sectional deformation in the reduced model could also improve the modelling accuracy. 11.5 mm. The robot can be inserted through a commercially available 12 mm trocar port. The robot is equipped with an inner free lumen of a 4.5 mm diameter, which can accommodate the optic fiber and vacuum pipes for retrieving biopsies. In addition, two modules have been connected to achieve higher flexibility (see Fig. 1). We applied the reduced to model and control both the one-module and two-module soft robots using SOFA. The simulation demonstrates performances of the reduced FEM is at least 4 times faster than a classical FEM model. The simulation and experiments were both conducted to further validate the reduced FEM model and inverse control. To compensate unmodelled uncertainties in the open-loop control and improve the control accuracy, we introduced the closed-loop control using position feedback.

B. Discussions of the closed-loop control

In future work, we will first aim to improve the model accuracy which can further produce better open-loop control results, e.g., by introducing nonlinear hyper-elastic material models and considering the dynamics behaviour. In addition, we will integrate the soft instrument prototypes to the phantom and the SpiderMass device to develop and validate the proposed medical application.

Fig. 1 .

 1 Fig. 1. The physical (top) and modelled (bottom) (a) one-module and (b) two-module soft manipulators in SOFA.

Fig. 2 .

 2 Fig.2. Overview of dimensions of the cross-sectional geometry. Two adjacent chambers are actuated as one pair, which have been marked in the same colour. One segment has an outer diameter of 11.5 mm, the actuation chambers have an inner diameter of 1.5mm. The free inner lumen has an inner diameter of 4.5 mm. The length of one module is 55 mm.

Fig. 3 .

 3 Fig. 3. Scheme of the link between the FEM nodes pn and the beam DOF positions q k .

Fig. 4 .

 4 Fig. 4. Simulation comparisons between the classical and reduced FEM.

Fig. 5 .

 5 Fig. 5. The open-loop and closed-loop control scheme. y d is the desired position, yu is the input position to the inverse controller, and ym is the measured position. For open-loop control, yu equals y d without feedback correction. While yu is compensated by the feedback position using a PI controller in the closed-loop control.

  The positions are imported to SOFA as feedback information for closed-loop control. A PC runs SOFA including the simulation, inverse kinematics and closed-loop controller. SOFA sends out the real-time desired pressure to the Arduino Due board by serial communication. B. Experimental protocols Three sets of experiments were conducted to validate the reduced FEM model and the closed-loop and open-loop control methods. The desired position, measured position (from NDI trackers), simulated positions (from SOFA) and the chamber pressure were recorded. The fiber and vacuum pipes for Mass spectrometry were inserted to the central lumen during the experiments. The simulation steps in SOFA were set to 0.1 s.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Results for Experiment 1: (a) Comparison between the measured tip position, the simulated position and the desired position for the open-loop and closed-loop controller. The summarised mean absolute tip errors with respect to different desired points are shown in (b) and (c).

Figs. 7 Fig. 8 .

 78 Figs. 7(d) and 7(e) show the errors from the open-loop control become larger when tracking larger trajectories. Conversely, the closed-loop controller results in the robot consistently and stably following the desired trajectories. Furthermore, the summarised errors from the open-loop and closed-loop control are reported in Fig. 8 using box plots. In

Fig. 9 .

 9 Fig. 9. Results for Experiment 3: Trajectory tracking of the open-loop and closed-loop control for a two-module robot. (a) Measurement of the trajectory. Tracking results for (b) a rectangular shape, with a side length of 60 mm, and (c) a circular shape, with a diameter of 80 mm.

Experiments 1 -Fig. 10 . 3 PFig. 11 .

 110311 Fig. 10. Results for Experiment 3: Summarised tracking errors for (a) a circle with a diameter of 80 mm, (b) rectangle with a side length of 60 mm.

TABLE I LIST

 I OF MEDICAL TOOLS INSERTED THROUGH THE INNER WORKING CHANNEL OF THE SOFT ROBOT FOR in-vivo CANCER IMAGING.

TABLE II SUMMARY

 II OF PERFORMANCE OF THE FULL & REDUCED FEM MODEL.

		1 module	2 modules
		Time [s]	FPS*	Time [s]	FPS*
	Full FEM	179.69 (0.83)	5.56 (0.02)	607.01 (0.90)	1.64 (0.01)
	Reduced model 38.32 (2.28)	26.16 (1.48) 71.76 (0.71)	13.93 (0.14)

*Both authors contributed equally to this manuscript. This work is supported by the Engineering and Physical Sciences Research Council (grant numbers: EP/S014039/1, EP/V01062X/1), the UCL Dean's Prize, UCL Mechanical Engineering, and the China Scholarship Council.