Yu Qiu

Ming-Jia Li

Ya-Ling He

Ze-Dong Cheng

OPTICAL PERFORMANCE ANALYSIS OF A POROUS OPEN VOLUMETRIC AIR RECEIVER IN A SOLAR POWER TOWER

Keywords: Solar power tower, Open volumetric air receiver, Optical performance, Monte Carlo ray tracing

An optical model for a solar power tower with an open volumetric air receiver (OVAR) was developed in this paper, and the optical performance and characteristics of the OVAR were studied based on the model. Firstly, the detailed distributions of the non-uniform solar flux(q sr ) on the aperture and the solar source(S sr ) in the OVAR were studied. The solar flux was found to be relatively uniform across the aperture for a single OVAR. However, the incident angle of the rays varies between 0 o and 42 o , and this indicates that the parallel assumption of the rays, which is usually made in the performance analysis of OVAR, is not appropriate at this condition. Furthermore, the S sr in the absorber decreases from the inlet to the outlet, and the maximum source(S sr,max ) of 2.414×10 8 W•m -3 appears at the inlet. Moreover, the S sr,max was found to appear at the region near the wall rather than the center of the receiver as usual for the combined effect of the non-parallel incident rays and the diffuse reflection on the wall. In addition, study on effects of the porous parameters indicates that the solid emissivity influences both the S sr distribution and the receiver efficiency significantly. However, the pore diameter and the porosity influence S sr distribution importantly, but have negligible effect on the efficiency. Finally, the optical efficiency of 86.70%, reflection loss of 13.20% and transmission loss of 0.10% were found to be achieved by the OVAR under the design condition.

INTRODUCTION

Solar power tower (SPT) using the open volumetric air receiver (OVAR) as the solar-thermal conversion module is considered as a promising technology for solar energy utilization. Because the porous absorber of the OVAR was reported to be burned down for the local high solar flux or unstable flow, studies on the solar radiation transfer in the SPT and the solar energy density distribution in the OVAR can offer help to the safe operation and accurate performance prediction of the plant.

Many studies have focused on simulating the solar radiation transfer in the heliostat field, and some codes have been developed, such as UHC, DELSOL, HFLCAL, MIRVAL, HFLD and SOLTRACE [START_REF] Garcia | Codes for solar flux calculation dedicated to central receiver system applications: A comparative review[END_REF]. For the radiation transfer in the OVAR, some studies have also been conducted, where the solar radiation absorption on the porous absorber inlet is usually assumed to a surface phenomenon [START_REF] Wang | Thermal performance analyses of porous media solar receiver with different irradiative transfer models[END_REF]. Several studies also assumed that the incident solar radiation is parallel beam and calculated the radiation transfer in the OVAR by modified P1 model [START_REF] Roldán | Computational fluid dynamics evaluation of the operating conditions for a volumetric receiver installed in a solar tower[END_REF][START_REF] Chen | Thermal performance analysis on a volumetric solar receiver with double-layer ceramic foam[END_REF], where the effects of the solar radiation's direction distribution were ignored. Only a few studies have directly studied an OVAR in a dish collector without the above assumptions [START_REF] Chen | Integrated analysis on the volumetric absorption characteristics and optical performance for a porous media receiver[END_REF], but *Corresponding Author: yalinghe@xjtu.edu.cn TFEC-IWHT2017-17576 this work cannot be applied in SPT directly. It is the current status that no studies about the solar radiation transfer simulation from the heliostat field to the OVAR have been reported.

To provide better studies on the optical performance of the SPT with an OVAR, this work focuses on developing a comprehensive optical model using Monte Carlo Ray Tracing (MCRT) [START_REF] Qiu | Thermal performance analysis of a parabolic trough solar collector using supercritical CO2 as heat transfer fluid under non-uniform solar flux[END_REF] method. Based on the model, the solar heat source distribution in the porous absorber, and the solar radiation transfer and absorption characteristics in the OVAR were simulated and discussed.

PHYSICAL MODEL

The DHAN heliostat field located at 40.4°N, 115.9°E in Beijing is taken as the concentrating module, where 100 heliostats are installed as shown in Fig. 1. A square OVAR is taken as the energy conversion module and is shown in Fig. 2, where the optical processes are also illustrated. The detailed parameters are given in Table 1. The optical errors of the heliostat in PS10 [START_REF] Rinaldi | Comparison of linear and point focus collectors in solar power plants[END_REF] are applied in present model due to lack of DAHAN's data. Fig. 3 shows the details of the solar radiation transfer processes from the field to the OVAR.

X r Y r O Z r Z g (zenith) H Z h α h A h A s α s θ h + + - - I R Sun Sun rays G δ I R Z i Heliostat Ⅰ Heliostat Ⅱ H O λ h α r θ i Z i X i Y i X i Y i Ray P h X g (south)
Fig. 3 Sketch of the SPT with an OVAR showing the solar radiation transfer and coordinate systems.

TFEC-IWHT2017-17576

OPTICAL MODEL

For developing an optical model to simulate the solar radiation transfer in the system, several righthanded coordinate systems are established in Fig. 3, which are the receiver system (X r Y r Z r ), the incidentnormal system (X i Y i Z i ), the heliostat system (X h Y h Z h ), and the ground system (X g Y g Z g ). For X r Y r Z r , the center of the receiver aperture O is the origin. X r points to the east, and Y r points upwards. Z r is normal to X r Y r plane. For X i Y i Z i , the point hit by the ray is the origin, and Z i points towards the sun. X i is horizontal and perpendicular to Z i , and Y i is normal to X i Z i plane and points upwards. For X h Y h Z h , the center of the heliostat H is the origin. X h is horizontal, and Y h is perpendicular to the tangent plane at H and points upwards. Z h is normal to X h Y h plane. For X g Y g Z g , the base of the tower G is the origin, and X g , Y g , and Z g point to the south, the east, and the zenith, respectively. The transformation matrixes among these systems can be found in our previous work [START_REF] Qiu | A comprehensive model for analysis of real-time optical performance of a solar power tower with a multi-tube cavity receiver[END_REF].

Simulation of the solar radiation transfer in the heliostat field is as follows. The heliostat will track the sun when the sun shines on the field, where all the heliostats aim at the OVAR's aperture center O. The photons are randomly initialized on the heliostats, where the sun shape effect which induces a non-parallel angle of 4.65 mrad for the solar rays is considered [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF], and the incident vector is expressed as

I i in X i Y i Z i .
The Direct Normal Irradiance(DNI) of the solar radiation is calculated by a clear sky model [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF]. The reflected vector R h on the initialized location P h in X h Y h Z h is calculated with Fresnel's law by transforming I i to X h Y h Z h , where the total equivalent slope error is calculated by s s s + + [START_REF] Qiu | A comprehensive model for analysis of real-time optical performance of a solar power tower with a multi-tube cavity receiver[END_REF]. The atmospheric attenuation is computed as a function of the distance between O and H for each heliostat [START_REF] Schmitz | Assessment of the potential improvement due to multiple apertures in central receiver systems with secondary concentrators[END_REF]. The shading and blocking are also calculated, and the blocking is illustrated here to decide whether a ray is blocked or not. First, P h and R h on heliostat I are transformed to X h Y h Z h (II) as shown in Fig. 3. Then, the intersection of the ray and heliostat II's surface is calculated. Finally, if the intersection is within heliostat II, the ray is blocked. Further information about the modeling of the solar radiation transfer in the heliostat field can be found in Ref. [START_REF] Qiu | A comprehensive model for analysis of real-time optical performance of a solar power tower with a multi-tube cavity receiver[END_REF]. Simulation of the solar radiation transfer in the OVAR is conducted in X r Y r Z r as follows. When a ray hits the receiver aperture (inlet), firstly, P h and R h will be transferred to X r Y r Z r and expressed as P h,r and I r . Then the intersection P a,r in X r Y r Z r will be calculated. The porous absorber is treated as an isotropic semitransparent media, and the solid phase is assumed to be gray and diffuse for solar radiation. The absorption coefficient (β a ), the scattering coefficient (β s ), and the extinction coefficient (β e ) of the absorber TFEC-IWHT2017-17576 are defined in Eq.( 1) [START_REF] Vafai | Handbook of porous media[END_REF]. Because the solar rays just transfer in the air phase of the absorber, so the refractive index (n) is assumed to be equal to that of air.

where ε p is the emissivity of the solid phase; Φ is the porosity; D p is the pore diameter. When a solar photon(ray) hits the absorber aperture, the photon will interact with the porous solid phase directly. The optical interactions including scattering and absorption between the photon and the absorber are simulated in the following way. When a photon reaches an interaction site, a fraction of the photon's energy (Δe p ) will be absorbed at this site by Eq.( 2) and counted in the local grid. When a photon is scattered, the travel distance (d) and unit directional vector (R r ) of the scattered photon will be calculated by Eq.( 3) and Eq.( 4) [START_REF] Wang | MCML-Monte Carlo modeling of light transport in multi-layered tissues[END_REF], respectively. When a photon is reflected by the wall, the R r will be calculated by Lambert law, where the wall is assumed to be gray and diffuse for solar radiation.

( )

1 e ln / d ξ β = - (3)
s s i i i i i i i i i i i i s s i i i s
θ ϕ α γ γ β γ α β γ γ α γ β θ ϕ γ γ γ θ θ ϕ θ ϕ γ γ θ   -       <             =     >     ⋅   R            (4)
[ ]

T r i i i cos cos cos α β γ = I (5) 3 2 2 s 2 s 2 3 2 1 , 0 2 , cos = 1 1 1 , 0 2 1 2 g g g g g g g ξ ϕ πξ θ ξ - =      =   -    + - ≠    -+         (6)
where I r is the incident unit vector at the interaction site; ξ is a uniformly distributed random number in the interval (0,1); θ S and φ S are the deflection angle and azimuthal angle of the photon, respectively; g is the anisotropy coefficient, and g=0 when the media is isotropic; SIGN(x) returns 1 when x>0, and returns -1 when x<0.

Several performance parameters including the solar radiation heat source in grid i of the absorber S sr (i), solar flux absorbed in grid j on the wall or shined on the aperture q sr (j), the Local Concentration Ratio (LCR), the optical efficiency (η opt ), the reflection loss rate (η R ), the transmission loss rate (η T ) are defined in Eq. [START_REF] Rinaldi | Comparison of linear and point focus collectors in solar power plants[END_REF].

The grid number used in the simulation for the absorber is 78400 (Width×height×thickness=56×56×25), and the details of the grid are shown in Fig. 4. The checked photon number is around 2×10 11 for the whole field and 1×10 9 for shining on the OVAR aperture.

( ) ( ) ( )

sr sr sr p w R T opt R T A A A , , / / , , , / i i j j S i Q V q j Q A LCR q j DNI Q Q Q Q Q Q Q η η η = = = + = = = (7)
TFEC-IWHT2017-17576

where Q i is the power absorbed in grid i; Q j is the power absorbed on the wall or shined on the aperture in grid j; V i is the volume of the grid i; A j is the area of the grid j; Q p and Q w are the power absorbed by the porous absorber and the wall, respectively; Q R and Q T are the power reflected back from the aperture and transmits through the outlet, respectively; Q A is the power shined on the aperture.

X r Z r Y r X r Z r Y r
(a) the porous absorber (b) the walls Fig. 4 The grids for the absorber and the walls used in the computation.

MODEL VALIDATION

To validate the model, firstly, the present LCR distribution computed by MCRT on the aperture plane of the OVAR is compared with that computed by SOLTRACE [START_REF] Garcia | Codes for solar flux calculation dedicated to central receiver system applications: A comparative review[END_REF] as shown in Fig. 5, where solar azimuth A s =120°, solar altitude α s =25°, and the atmospheric attenuation is ignored. From Fig. 5(a), it is seen that the LCR contours agree quite well with each other. Figure 5(b) illustrates the details of LCR at X r =0 and Y r =0, and it is found that both relative errors of the curves are within 1.3%. Then, the light transfer in a semitransparent slab was simulated, where n =1.0, β a = 1000 m -1 , β s = 9000 m -1 , g= 0.75, and the slab thickness is 0.2 mm [START_REF] Wang | MCML-Monte Carlo modeling of light transport in multi-layered tissues[END_REF]. The angularly diffuse reflectance (R d ) and transmittance (T d ) [START_REF] Wang | MCML-Monte Carlo modeling of light transport in multi-layered tissues[END_REF] of the slab are shown in Fig. 6. It is observed that the present results are in good agreement with the published data.

All the above results indicate that the MCRT model is appropriate for modeling the solar radiation transfer in both the heliostat field and the porous absorber.

NUMERICAL RESULTS AND DISCUSSION

Solar Radiation Distribution in OVAR

Figure 7 illustrates the solar radiation distribution at the aperture plane. From Fig. 7(a), it is seen that the solar flux(q sr ) on the aperture plane is non-uniform. The q sr decreases from the center to the margin, and the maximum flux of 2.490×10 6 W•m -2 appears at the center O which is the aim point of the heliostats. Figure 7(b) shows the flux at the OVAR aperture which locates at the square region in Fig. 7(a). It is seen that the highest q sr still appears at the center, however the largest difference of the fluxes at different locations in the aperture region is less than 1.8%. These results indicate that although the flux at the large aperture plane is significantly non-uniform, the flux at the aperture for a single OVAR is quite uniform. Figure 7(c) shows the incident angle distribution of solar rays on the aperture. It is seen that the incident angle of the non-parallel rays varies between 0 o and 42 o . It indicates that the parallel assumption of the rays, which is usually made in the performance analysis of OVAR, is not appropriate and should be avoided. Figure 8 illustrates the solar radiation distributions in the absorber and on the walls of the OVAR. It is seen that both the solar flux on the wall and the solar heat source (S sr ) in the porous decrease from the inlet to the outlet, because the solar radiation is absorbed gradually by the porous media along the incident direction. It is also found that the maximum q sr of 1.680×10 5 W•m -2 and maximum S sr of 2.414×10 8 W•m -3 appear at the inlet. The power absorbed by walls and the power absorbed in absorber are 762 W and 42,023 W, respectively.

Effects of Non-Parallel Rays and Absorption on Wall

Figure 9 shows the effects of the non-parallel rays and the absorption on the wall on the solar heat source (S sr ) distribution in the absorber. From Fig. 9(a), it is seen that the typical S sr distribution at the cross section can be divided into 3 regions: (1) beside the wall region, (2) the hot spot region, and (3) the center TFEC-IWHT2017-17576 region. It is found that the S sr at the beside wall region is lower than other regions, which can be explained for the reason that the incident radiation for a point at this region are mainly from the half space, e.g., point A in Fig. 9 (a)), while it would be the full space, e.g., point B in Fig. 9 (a) at other regions. As a result, small heat source appears beside the wall.

It is also seen that the maximum source (S sr,max ) appears at the hot spot region which is near the wall rather than the center of the receiver as usual, and this phenomenon which is counterintuitive can be explained as follows. Firstly, a point at this region can accepts radiation from the full space, so the S sr will not be as small as that in the beside the wall region. Secondly, the combined effects of the non-parallel incident rays and the diffuse reflection on the wall create the hot spot, because most of the rays which hit the wall will be reflected and reabsorbed by the absorber at this region. If the rays are parallel, this phenomenon will be greatly weaken as shown in Fig. 9 (b). If there is no reflection on the wall, this phenomenon will even be eliminated as shown in Fig. 9 (c). For the reason that the highest solar source locates at the region near the wall where the air velocity is lower than that in the center, so it can be inferred that the mismatching the solar source and the heat transfer ability of the air would occurs at this region, which may leads to the local overheating and results in the failure of the OVAR. The revelation of this phenomenon could offer help to the heat transfer analysis and safe operation of the OVAR in the future. TFEC-IWHT2017-17576

0 Y r / m 0.

Effects of Porous Absorber Parameters

The radiation properties (β a , β s , β e ) of the absorber depend on three parameters including the solid emissivity (ε p ), pore diameter (D p ), and the porosity (Φ), and these properties determine the transfer step of the photon and the absorbed power at each site. So, for predicting the solar heat source and avoiding the local overheating which may makes the absorber burn down, the effects of the three parameters were studied. Figure 10 illustrates the effects of the three parameters on the solar heat source (S sr ) distribution at the centerline of the absorber. Table 2 shows the effects of these parameters on the receiver efficiencies.

From Fig. 10(a) and Table 2, it can be observed that the S sr near the inlet and the optical efficiency (η opt ) increase with increasing ε p due to the increasing β a , and the S sr far from the inlet and the reflection loss (η R ) decrease with increasing ε p due to the decreasing β s . It can also been seen that S sr decreases sharply along the centerline due to the large β e of the porous media, and transmission loss (η T ) is very small, i.e. η T <0.4%.

From Fig. 10(b), it can be seen that the maximum source (S sr,max ) decreases with increasing D p . And, S sr,max decreases from 6.83×10 8 W•m -3 for D p =0.5 mm to 1.55×10 8 W•m -3 for D p =3.0 mm. The is because that both β a and β s decrease with increasing D p , and the penetration distance of the solar photon becomes larger.

From Fig. 10 (c), it can be found that the S sr,max decreases with increasing Φ for the decreases in β a and β s . And, S sr,max decreases from 8.40×10 8 W•m -3 for Φ=0.4 to 2.25×10 8 W•m -3 for Φ=0.90.

From Table 2, it can be concluded that the increases of D p and Φ influence little on the optical efficiency, respectively, owing to the simultaneous reduction in β a and β s . Furthermore, the η opt of 86.70% is achieved at designed parameters, and the corresponding η R and η T are 13.20% and 0.10%, respectively. TFEC-IWHT2017-17576

CONCLUSION

A comprehensive optical model for the SPT with an OVAR was developed and validated in this work, and the optical characteristics of the OVAR were studied based on the model. The following conclusions can be derived.

(1) The detailed distributions of the non-uniform solar flux (q sr ) on the aperture and the heat source (S sr ) in the OVAR were revealed. The flux with the maximum value of 2.490×10 6 W•m -2 at the aperture is found to be relatively uniform across the aperture, however the incident angle of the rays varies between 0 o and 42 o , which indicates that the parallel assumption of the rays is not appropriate for the OVAR in SPT. The S sr in the porous absorber decreases from the inlet to the outlet, and the maximum source (S sr,max ) of 2.414×10 8 W•m -3 appears at the inlet at the design parameters.

(2) The maximum source (S sr,max ) was found to appear at the region near the wall rather than the center of the receiver as usual for the combined effects of the non-parallel incident rays from the field and the diffuse reflection on the wall. The revelation of this phenomenon could offer help to the heat transfer analysis and safe operation of the OVAR in the future.

(3) Study on the effects of the porous parameters indicates that the solid emissivity influences both the S sr distribution and the receiver efficiency significantly. However, the pore diameter and the porosity influence the source distribution importantly, but have negligible effect on the efficiency. Moreover, the optical efficiency (η opt ) of 86.70%, reflection loss rate (η R ) of 13.20% and transmission loss rate (η T ) of 0.10% are achieved at designed parameters.

NOMENCLATURE

Fig. 1 Fig. 2

 12 Fig.1Heliostat field of DAHAN plant[START_REF] Yu | Simulation and analysis of the central cavity receiver's performance of solar thermal power tower plant[END_REF].Fig.2Sketch of the OVAR showing the radiation transfer.

1 1 Fig. 6

 116 Fig. 6 Comparisons of the angularly diffuse reflectance (R d ) and transmittance (T d ).

7 .

 7 (a) Aperture plane. Incident power=7.737×10 8 W rays in a angle range / % Incident angle of the rays on the aperture θ in / o Percentage of the rays / % (b) OVAR Aperture. q sr,max =2.490×10 6 W•m -2 , Incident power=48,470 W (c) Incident angle distribution of solar rays on the aperture. Fig. Solar radiation distribution at the aperture of the OVAR at 12:00, spring equinox (DNI=961 W•m -2 ).

8 .

 8 (a) Wall, absorbed power = 762 W (b) Absorber, absorbed power = 42023 W Fig. Solar radiation distribution in the absorber and on the wall of the OVAR at 12:00, spring equinox (DNI=961 W•m -2 ).

Fig. 9

 9 Fig.[START_REF] Qiu | A comprehensive model for analysis of real-time optical performance of a solar power tower with a multi-tube cavity receiver[END_REF] Effects of non-parallel rays and absorption on the wall at 12:00, spring equinox (DNI=961 W•m -2 ).

Fig. 10 .

 10 Fig. 10. Effects of porous absorber parameters.

Table 1

 1 Parameters and assumptions of the model[START_REF] Rinaldi | Comparison of linear and point focus collectors in solar power plants[END_REF][START_REF] Yu | Simulation and analysis of the central cavity receiver's performance of solar thermal power tower plant[END_REF].

	Parameters	Dim.	Parameters	Dim.
	Heliostat number	100	Absorber thickness	0.05 m
	Heliostat shape	Spherical	Heliostat reflectivity	0.90
	Heliostat width	10 m	Heliostat cleanliness	0.97
	Heliostat height	10 m	Altitude tracking error σ te,1= σ te	0.46 mrad
	Heliostat center height	6.6 m	Azimuth tracking error σ te,2= σ te	0.46 mrad
	Tower height	118 m	Heliostat slope error σ se	1.3 mrad
	Tower radius	10 m	SiC absorber emissivity ε p	0.92
	Receiver Height H O	78 m	Wall emissivity ε w	0.30
	Receiver altitude	25°	Porosity Φ	0.90
	Absorber height	0.14 m	Pore diameter D p	2.0 mm
	Absorber width	0.14 m

Table 2 .

 2 Effects of solid emissivity (ε p ), pore diameter (D p ) , and porosity (Φ) on the receiver efficiencies.

		D p =2.0 mm, Φ=0.90		ε p =0.92, Φ=0.90			D p =2.0 mm, ε p =0.92
	ε p	η opt /%	η R /% η T /%	D p	η opt /%	η R /%	η T /%	Φ	η opt /%	η R /% η T /%
	0.50	75.25	24.44 0.31	0.5	86.57	13.43	0.00	0.40	86.54	13.46 0.00
	0.60	78.72	21.06 0.22	1.0	86.64	13.36	0.00	0.50	86.55	13.45 0.00
	0.70	81.63	18.21 0.16	1.5	86.70	13.29	0.01	0.60	86.57	13.43 0.00
	0.80	84.12	15.75 0.13	2.0	86.70	13.20	0.10	0.70	86.60	13.40 0.00
	0.92	86.70	13.20 0.10	2.5	86.48	13.13	0.39	0.80	86.64	13.36 0.00
	0.95	87.27	12.64 0.09	3.0	85.95	13.07	0.98	0.90	86.70	13.20 0.10

ACKNOWLEDGEMENTS

The study is supported by the Key Project of National Natural Science Foundation of China(No. 51436007), the Major Program of the National Natural Science Foundation of China(No. 51590902), and the 111 Project (B16038).


Florence Zara
email: florence.zara@univ-lyon1.fr

Olivier Dupuis
email: olivier.dupuis@chu-lyon.fr

Chap 15: Uterus -Biomechanical modeling of uterus. Application to a childbirth simulation

Keywords: Uterus, vaginal delivery, 3D reconstruction, measurements of mechanical properties, Finite Element model, biomechanical simulation

This chapter focuses on the uterus, a major organ of the female reproductive tract. The uterus is a complex organ due to the evolution of its mechanical properties during the life. These modifications are due to the age, the hormonal changes, the number of pregnancies, and the fact that during any gestation, the size of the uterus is multiplied by more than four. The route of delivery (i.e. vaginal delivery versus cesarean section) may also impact the uterus mechanical properties. All these modifications make that the same woman will have different wombs during her life. Thus, the mechanical properties of the uterus are complex which involves some difficulties to propose an accurate simulation of its mechanical behavior.

In this context, we present in this chapter some researches made to simulate the mechanical behavior of the uterus. One of the interest of simulating its mechanical behavior is to improve the knowledge of its attachment system (i.e. ligaments binding the uterus to the pelvis wall). In particular, in this case, the goal is to evaluate damages that could occur during vaginal delivery in order to prevent or to improve treatment of uterine prolapse. The simulation of the mechanical behavior of the uterus could also lead to preventive strategies in order to prevent preterm birth caused by a short cervix. In this case, we have to understand which kind of anatomical changes lead to a cervical shortening.

We present two approaches to realize such a numerical simulation of the uterus: the first approach focuses only on the uterus (without considering others organs); the second approach simulates in real-time a vaginal delivery. This latter technique considers all the organs interacting during a childbirth but with a lot of

Introduction

In this chapter, we focus on the uterus which is a major organ of the female reproductive tract. In the literature, work has been presented to improve the knowledge of its anatomy and mechanical properties. In particular, the aim is to study the ligaments which bind the uterus into the pelvis. The idea is to evaluate their damages during vaginal delivery in order to prevent uterine prolapse or to propose adequate medical treatments if necessary. An other interest is to prevent preterm birth by a better understanding of the causes and the consequences of cervical shortening. One of the main difficulty remains the fact that the mechanical properties of the uterus vary according to age, hormonal changes, number of pregnancies and the fact that during any gestation the size of the uterus is multiplied by more than four. The route of delivery (i.e. vaginal delivery versus cesarean section) may also impact the uterus mechanical properties.

In the following, we will present two approaches to model and simulate the mechanical behavior of the uterus.

• To obtain an accurate simulation of the uterus behavior, some work [1,2,3] focused only on the uterus without considering the organs around it.

• We also present our work about a biomechanical simulation of a real-time childbirth delivery. In this work, the aim is to couple a numerical simulation to a physical interface in order to propose a complete simulator for medical training. Thus, our numerical simulation concerns the principal organs involved in a childbirth including naturally the uterus, organs of the reproductive tract and the fetus. The numerical simulation includes some simplifications in the modeling of each organ to obtain a realist global behavior that could be used to teach vaginal instrumental delivery.

These two approaches of work seem completely opposite, but they are complementary to obtain a realist simulation and for evaluating the pertinence of each part of the model (for example to evaluate the role of each ligament connected to the uterus). In both case, the usual pipeline is used to realize such a biomechanical simulation. First of all, we have to extract the organ from medical images to create its geometrical representation (for example a mesh). We also have to measure its mechanical properties (like stiffness parameters) and to determine its constitutive law using experimentations. Then, a more or less complex physical model (usually based on the Finite Element method) is defined to reproduce its mechanical behavior according to experimental results obtained and determined boundary conditions.

For the following of this chapter, section 2 presents the anatomical and functional description of the uterus. Section 3 presents previous work proposed: (i) to reconstruct a geometrical model of the uterus from medical images, (ii) to measure its mechanical properties, and (iii) to simulate its mechanical behavior and connections to ligaments. We also present in this section some work performed to improve laparoscopic uterine surgery by adding on the laparoscopic images the hidden structures. Section 4 focuses on our work to simulate the descent of the fetus during a childbirth delivery in real-time. Then, to finish this chapter, section 5 gives a conclusion about the work proposed in the study of the uterus. Anatomical description of the uterus. The uterus is a fibro-muscular organ of the female reproductive system. It can be seen as a thin closed membrane in which the fetus develops during pregnancy. It is pear-shaped and it is about 7.6 cm long, 4.5 cm broad (side to side) and 3.0 cm thick. During the pregnancy, the size of the uterus will increase from 8 cm long to 35 cm. The uterus can be anatomically divided into two parts: the corpus or body and the cervix. The two opposite parts of the uterus are connected to the others organs: the cervix opens into the vagina, while the other part of the uterus is connected to the Fallopian tubes.

Anatomical and functional description of the uterus

A complex structure of ligaments to fix the uterus to the pelvis. The uterus is a mobile organ which moves posteriorly under the pressure of a full bladder, or anteriorly under the pressure of a full rectum. The mobility is conferred to it by musculo-fibrous apparatus that consists of suspensory and sustentacular part. The ligaments that attach the uterus to the pelvic wall is a complex structure. The fundus is attached on each side (right and left) to a tripod structure: namely from anterior to posterior the round ligament, the Fallopian tube and the utero ovarian ligament. From a mechanical point of view, the Fallopian tube is a mobile structure and does not seem to interact significantly. The two others ligaments, round ligament and utero ovarian ligament, play a significant mechanical role.

When one follows the side of the uterus fundus, we find the peritoneal flap that does not seem to play any significant role. Lower, we find on each side the Mackenrodt ligament that goes around the uterine artery and the ureter. This ligament is a complex 3D structure. By going deeper in the pelvis one reaches the cervix and finds posteriorly the utero sacral ligament and anteriorly the vaginal cuff. Finally, the vaginal cuff is itself attached to the pelvic wall by the levator ani muscles and perineal floor muscles.

Modification of the uterus during pregnancy. Before pregnancy and during the first trimester of the pregnancy, the uterus is a two segments organ: the lower part being the cervix and the higher part being the uterine body. Both part are separated by a virtual border named the uterine "isthmus". Then, at 32 weeks of pregnancy the lower segment appears. This segment is between the cervix and the uterine body. At term, it will be a 10 cm high new part of the uterus. Thus, at the end of the pregnancy, the uterus will no longer be a two parts organ but a three parts organ (with the uterine body, the lower segment and the cervix).

The previous work published in the literature about the uterus

Several studies have been proposed to study the uterus during these last years [5,6,7,8]. The final aims of these studies are not necessary the same, but it can be interesting to combine them to achieve a more accurate simulation of the mechanical behavior of the uterus. To discuss this previous work, we follow the classical pipeline which permits to create a biomechanical simulation. We also point out in this section, specific work performed to add hidden structures on laparoscopic images for assisting and improving laparoscopic uterine surgery.

Generation of the geometrical models

The first step to realize a biomechanical simulation is to construct the geometrical models for the considered organs.

Extraction from MRI data. Classically, the meshes of the organs are obtained after the segmentation of MRI (Magnetic Resonance Imaging) images. Then, as some important discontinuities can appear due to the resolution of medical images, the surface meshes need to be smoothed to obtain regular meshes which are needed for the computation of the numerical simulations. These two steps are performed using appropriate softwares.

Quality of the geometrical model. The level of details of the geometrical model varies according to the aim of the simulation. For a teaching tool for which the interactive time is an important matter, a simple geometrical model is chosen (i.e. it does not consider the suspensory ligaments of the uterus for example) with a limited number of nodes to decrease the computational time of the simulation. Fig. 2-3 present the geometrical model of the uterus proposed in [5,6] respectively. Fig. 4 presents the FE models proposed in [9] to simulate a vaginal delivery.

ation curves is shown in Fig. 3. different locations of the uterus s variability manifests itself even om the same anatomical type of n in Fig. 4 exhibiting the stage at cavum relation follows closely a s mentioned above, the excessive ication of the model in a simulator required.

sents an example uterine shape method. Deformations resulting hown to the right. Note that these t derived automatically with the uterine cavity are the basis of the er to increase the realism of the ves of various uterine samples tested in he text. Enormous variability was again ed from a uterus exhibiting a large myoma. with a weakly compressible, non-linear, mined by inverse FE fitting. The ''myoma'' ttom and coincide with on of the ''fundus'' rmission from Weiss et al. [START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF].

Conclusion and Future Work

In this paper we have presented the methods developed for the full simulation of hydrometra during hysteroscopy. Both the organ's deformation and the fluid flow in the cavity are modeled with highest possible realism while still meeting the real-time requirements of a simulator.

The fluid simulation will be further validated. We plan to perform an accurate flow simulation with exactly the same boundary conditions with an available commercial packages offline and quantitatively compare to our own results. The increased computational power of future hardware can be used to enhance the For a simulation which focuses on the role of the different structures of the pelvic floor (in particular to investigate the role of the different structures on pelvic statics dysfunctions), a more complete anatomical model including all the structures is naturally requested. Fig. 5 presents the pelvic floor proposed in [10].

of the history of childbirth studies, addressing the various techniques used to gain insights into this complex process. We then discuss recent developments in computational models of childbirth, comparing the results and the modeling frameworks. Lastly, we describe some of the main challenges in this field and propose possible solutions for further improvements.

HISTORY OF CHILDBIRTH STUDIES

The mechanism of labor was first described by obstetricians early in the eighteenth century, and much of our understanding of the process is due to these early pioneers. In particular, William Smellie (1697-1763) documented that the primary considerations for vaginal delivery were the size and shape of the pelvis and the fetal head. He measured fetal skull diameters and pelvic diameters, and described the process of internal rotation of the fetal head. These were subsequently published in a set of anatomical tables in 1793. 17 Many of the early observations made by Smellie were confirmed by Borell and Fernstrom in 1950s and 1960s in their published studies on childbirth and the mechanism of labor. 18-22 By observing the movements of the fetal head with respect to the bony pelvis through a series of X-ray images, 18-22 they suggested that the shape of the bony pelvis had the most influence on the motion of the fetal head. In addition, they emphasized the importance of the LA in the labor process. Their investigations initiated a new era in childbirth studies using imaging techniques. However, such studies are unlikely to be repeated because of ethical restrictions. 23 Ultrasound and MRI are the standard imaging modalities in modern obstetrics and gynecology. 5,24-27 They are considered superior to X-ray or computed tomography (CT), which expose subjects to harmful radiation during the imaging process. Ultrasound is more commonly employed because of its low cost, images suffer from poor signal-to-noise ratio and significant geometric distortions, making this modality less suitable for the creation of mathematical models. In contrast, MRI can be readily transformed into computer models for mechanical analysis (Figure 1). 28-31 Such techniques have significantly enhanced the representation of pelvic floor structures. However, the limiting factor with MRI is the relatively low spatial resolution, which fails to give a reasonable representation of detailed pelvic floor structures such as muscle fibers and connective tissues.

In order to address this issue, Janda et al. 32 carried out a study on the morphological parameters of the pelvic floor using an embalmed female cadaver. The study quantified in detail the muscle fiber length, fiber directions, sarcomere length, and cross-sectional area. However, before these data can be used in an individual-specific finite element (FE) model, some adjustments need to be applied to these descriptions in order to compensate for the loss of muscle tone prior to the embalming process.

MODELS OF CHILDBIRTH

With the advance in computational technology, the ability to study complex biomechanical processes, such as childbirth, can now be achieved on conventional commodity computers. Several research groups have developed childbirth models in recent years. Table 1 provides a brief summary of their work with highlights of their models.

First stage of labor

There have been very few studies involving computational models of the first stage of labor. Early FE studies investigated the material properties of the fetal head, and simulated fetal head deformation by applying pressure boundary conditions. 33-36

461

Figure 5: The pelvic floor model presented in [10] to simulate vaginal delivery.

For a numerical simulation that only focuses on the uterus, we need a complete geometrical model of the uterus with a high number of nodes. Fig. 6 presents a FE model of the uterus proposed in [5]. It has 20,000 elements.

Mechanical tissue properties

A typical set of loading-relaxation curves is shown in Fig. 3. issue samples were taken from different locations of the uterus see legend). Again, an enormous variability manifests itself even n samples which were taken from the same anatomical type of issue.

.

Finite element model

A model of the uterus is shown in Fig. 4 exhibiting the stage at ull hydrometra. The pressure-cavum relation follows closely a ample curve (Fig. 2, bottom). As mentioned above, the excessive xecution time prevents the application of the model in a simulator here real-time capabilities are required.

.

Extended statistical model

The left image in Fig. 5 presents an example uterine shape ynthesized with the discussed method. Deformations resulting rom increasing hydrometra are shown to the right. Note that these eometries are not calculated but derived automatically with the tatistical model.

.

Visual representation

The triangular surfaces of the uterine cavity are the basis of the nteractive visualization. In order to increase the realism of the ig. 3. Exemplary stress-relaxation curves of various uterine samples tested in niaxial compression as described in the text. Enormous variability was again ound. The ''myoma'' sample was recovered from a uterus exhibiting a large myoma. he constitutive parameters associated with a weakly compressible, non-linear, sotropic and viscoelastic law were determined by inverse FE fitting. The ''myoma'' nd ''corpus'' curves are located at the bottom and coincide with on of the ''fundus'' urves. The figure is reproduced with permission from Weiss et al. [START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF].

Mechanical tissue properties

A typical set of loading-relaxation curves is shown in Fig. 3. Tissue samples were taken from different locations of the uterus (see legend). Again, an enormous variability manifests itself even in samples which were taken from the same anatomical type of tissue.

Finite element model

A model of the uterus is shown in Fig. 4 exhibiting the stage at full hydrometra. The pressure-cavum relation follows closely a sample curve (Fig. 2, bottom). As mentioned above, the excessive execution time prevents the application of the model in a simulator where real-time capabilities are required.

Extended statistical model

The left image in Fig. 5 presents an example uterine shape synthesized with the discussed method. Deformations resulting from increasing hydrometra are shown to the right. Note that these geometries are not calculated but derived automatically with the statistical model.

Visual representation

The triangular surfaces of the uterine cavity are the basis of the interactive visualization. In order to increase the realism of the Fig. 3. Exemplary stress-relaxation curves of various uterine samples tested in uniaxial compression as described in the text. Enormous variability was again found. The ''myoma'' sample was recovered from a uterus exhibiting a large myoma. The constitutive parameters associated with a weakly compressible, non-linear, isotropic and viscoelastic law were determined by inverse FE fitting. The ''myoma'' and ''corpus'' curves are located at the bottom and coincide with on of the ''fundus'' curves. The figure is reproduced with permission from Weiss et al. [START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF]. Manual reconstruction. The segmentation of the MRI data is still difficult for the pelvic system and it remains hard to extract all the details such as the ligaments connected to the uterus. In this case, the partial segmentations obtained from MRI data have to be completed using a manual reconstruction [11]. This step is performed according to anatomical descriptions. It concerns the vagina, the pelvic floor, floor muscles, ligaments (with some difficulties for some ligaments where there is no consensus) and the cervix (according to the considering dilatation step in the case of the realization of a childbirth simulation).

Variable anatomical models. The use of solid models to generate geometrical meshes facilitates the addition of elements (like ligaments) into these models. This is an important issue when all the structures are not visible in the medical images, or when we want to adjust the models according to educational criteria for a training simulator, or to better understand the role of each structure of the female pelvic system (which is illustrated in Fig. 1). For example, House [7] constructed geometrical models of the uterine fundus and cervix (which are presented in Fig. 7) to understand the deformation mechanisms which lead to cervical shortening during pregnancy. The construction of the geometrical models was performed in two steps. Firstly, solid models have been generated from MRI data using a solid modeling software. Secondly, the numerical models (appropriate for the simulation) have been generated by exporting the solid models in a finite element software. A 3D ultrasound technique applied on pregnant volunteer patients was also used to obtain imaging volumes. most significant limitation of our approach wa required several days to convert ultrasound data a usable numerical model. Through a combination of in vivo imag mechanical characterization of cervical tissue numerical simulation, we examined the key fac that affect the cervical function and dysfunction d ing pregnancy. An improved understanding of th factors will aid efforts to develop rational thera that aim to improve cervical function and de preterm birth.

. Numerical simulations of cervical function. The top simulation captures the well-known progres cervical funneling. The middle row shows uterine growth from 22 weeks (green) to 36 weeks (pink RI. The simulation is able to capture both the increase in uterine size but also the development of rine segment. In the bottom row, the ultrasound response to fundal pressure is compared to a simula Note that only when the correct properties of the cervix are used is the simulation able to capture l changes seen on ultrasound. This technique gives an estimate of the tissue properties of the cervi or figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Three-dimensional Anatomy of the Cervix During Pregnancy

Figure 7: FE models proposed in [7] for numerical simulations of cervical function.

In the context of the development of a medical training simulator, an other challenge is to propose a wide variety of scenarios. These scenarios are based on models for healthy organs or pathological organs. For this purpose, in order to realize a training hysteroscopic simulator (which permits to visualize the inner surface of the uterus), Sierra presented a framework to generate variable models of the uterus anatomy [12,13]. The method was based on a statistical description. The parameters of the geometrical model of the uterus concerned for example the dimension of the fundus, the dimension of the cervix and the length of the corpus (see Fig. 8). This work was completed by a method to generate pathologies found in the uterine cavity (such as leiomyomas and polyps) and a method to incorporate these pathologies into the model of the healthy uterus [14,15]. case the last generated shape is kept. The range of possible values can be set independently for each predictor through a configuration file. The Newton-Raphson optimization is a second order scheme and thus converges relatively fast, i.e., the number of correct digits is doubled in every iteration. In general, for the present implementation less than six iterations are required. Hence, new shapes can be evaluated in real time as the user moves the sliders. Even the underlying statistical analysis including the transformation of the shapes into a common coordinate system, which is performed offline as a preprocessing step, can be computed in a few seconds on a standard 3 GHz PC.

The variation of the mean shape induced by the 'fundus depth' predictor is illustrated in the upper row in Fig. 3. The values for the presented images range from 26 to 49 mm, covering about 1.5 times the standard deviation found in the present database. The measured values for all other predictors, i.e., the distances between the other landmarks, did not change after the deformation, as tions. Larger changes of these values would make the adjustment of the remaining predictors also necessary to avoid unnatural shapes. These findings may indicate that the width is correlated to the remaining predictors. However, it has to be noted that the segmentation of the lateral walls of the uterus is relatively difficult and error prone, as the arteria uterina follows exactly this landmark and strongly blurs the MRI scans. It is therefore not trivial to assign the different behavior of the predictors to a single source. To assess the results of the presented framework, a systematic analysis of the prediction performance has been carried out.

In the following, three properties of the statistical framework are discussed (Davies, 2002). First, the model should be able to represent unseen instances of the object class. i.e., it should generalize the main characteristics of the shapes based on the given set of objects. Second, the compactness of the model can be assessed, i.e., its ability to represent an object by as few parameters as possible. Third,

Study of the uterus in laparoscopic images

In recent years, searchers have developed software for assisting and improving laparoscopic uterine surgery [16,17,18,19]. The goal is to provide real-time Augmented Reality (AR), which allows the surgeon to see where important hidden structures are located, such as the uterine canal, major blood vessels and myomas (see Fig. 9). This works by aligning (or registering) a pre-operative radiological image such as MRI or CT with the real-time laparoscopic video. Once done, anatomical structures from the pre-operative image are overlaid with the laparoscopic video in real-time. Currently AR has been developed for the treatment of myomas (also called uterine fibroids) using pre-operative T2 weighted MRI images [17]. A preliminary study of the clinical impact was published in [START_REF] Bourdel | Augmented reality in gynecologic surgery: Evaluation of potential benefits for myomectomy[END_REF]. To achieve this AR, three technical challenges were solved. complementary cues involving natural anatomical landmarks, the uterus' boundary contour and feature correspondences. This is the first work which registers a pre-operative MRI of the uterus to laparoscopic images. In our ongoing research we will perform a deeper quantitative evaluation, test other deformable models, and test the system's performance on live videos. Our solution is not limited to visualising myomas. Any anatomical structure that is visible in the MRI can be visualised in the laparoscopic images, such as the cervix and uterine cavity. We expect that augmenting this information may also help the surgeon perform myomectomies and other uterine surgery, improve safety and reduce time in the operating room. Reconstruction of the 3D model. The first challenge is to reconstruct a 3D model of the uterus and myomas from the MRI data (commonly known as segmentation or delineation). Because the pre-operative image is taken before surgery (typically several weeks before), this process does not need to be solved in real-time, and was performed semi-automatically with the aid of a human operator using opensource segmentation software [START_REF] Wolf | The medical imaging interaction toolkit (mitk) a toolkit facilitating the creation of interactive software by extending VTK and ITK[END_REF].

Initial registration. The second challenge is to determine the change of shape of the uterus between the uterus in the pre-operative image and at the start of surgery (referred to as the initial registration). This is necessary to correctly register the pre-operative models with the laparoscopic video to account for soft-tissue deformation. For non-pregnant patients, soft-tissue deformation is caused by a number of factors including the patient's lying position and rate of abdominal insufflation. The initial registration was solved as follows. We record a short laparoscopic video of the uterus from different angles (the exploratory video), by having a surgical assistant move the uterus with a cannula. From the exploratory video, the 3D shape of the uterus was reconstructed using a process called Structure-from-Motion. The registration was achieved using three complementary visual cues. The first cue was the 3D reconstruction: specifically, that the surface of the visible region of the pre-operative uterus model should align closely to the 3D reconstruction. The second cue was manually-annotated registration landmarks. A "registration landmark" is a point on the uterus that can be found in both the pre-operative MRI and the laparoscopic images. The junctions between the Fallopian tubes and the fundus were used, as these could be clearly found in both modalities. The third cue was the boundary contours of the uterus. Specifically, the uterus' boundary was manually annotated in the exploratory video, and the registration was made to align the pre-operative model's boundaries with the manually-annotated boundaries. Registration was achieved with a so-called energy minimisation approach. This involves encoding the registration cues as energy terms, and numerically minimizing the energy to obtain the correct registration. To achieve this, a model of deformation must be used. Here an approximate model was used (a 3D affine model), which is not a complex deformation model. However, a compromise had to be made between using a more complex model and being able to determine the model's parameters. The affine model was shown to provide a good approximation and its parameters could be determined reliably. Note that research has continued to make the initial registration fully automatic. In [19] a method to automatically detect the uterus and the Fallopian tube junctions in laparoscopic images was proposed. In [18] a method to automatically segment the uterus in laparoscopic images was proposed. The segmentation gives the boundary contours of the uterus, which saves considerable time for a human annotator.

Alignment of the pre-operative images with the laparoscopic video. The third technical challenge to achieve AR is to update the initial registration for each new laparoscopic image and to overlay the myomas onto the laparoscopic image. Unlike the first and second challenges, this one needs to be real-time. A simplifying assumption was made, which was that the update could be modeled with a 3D rigid transform. This transform can account for changes of the laparoscopes position and global changes of the uterus' position, but it cannot account for significant soft tissue deformation. This was acceptable by instructing the surgeons to not significantly deform the uterus with surgical tools whilst AR was active. A robust, automatic solution was found that worked reliably for long durations (5 minutes or more) [16]. This operated at approximately 20 frames per second on a standard workstation PC and could handle occlusions from instruments and the uterus going out of the field-of-view. For each laparoscopic image, the registration was used to give the 3D position of the myomas in the laparoscopes field-of-view. The myomas surfaces were then rendered from the laparoscope's viewpoint, and the render was blended with the real laparoscopic image. The blend was done to give the impression that the uterus was semi-transparent and the myomas could be seen inside. Fig. 10 presents the registered models obtained in [16].

anything physical, but rather the region on the uterus onstruct shape. dense 3D models built in Phase 1. Column 2: the olumns 3&4: the registered models using WBMTR.

d Future Work liable and fast way to register the uterus in monocular vel two-phase approach. The approach di↵ers to SLAM ping and segmentation (done in Phase 1) from live ase 2). Phase 2 is achieved in realtime at approximately rdware, and does not depend on successful registration s thus simpler than EKF-SLAM and PTAM because it ing between tracking and re-localisation. We have shown ificantly outperforms EKF-SLAM for this problem. In nlarge our evaluation dataset and to explore the new method opens up for AR-assisted resection planning in does not correspond to anything physical, but rather the region on the uterus for which SfM could reconstruct shape.

Conclusion and Future Work

We have presented a reliable and fast way to register the uterus in monocular laparoscopy using a novel two-phase approach. The approach di↵ers to SLAM by decoupling 3D mapping and segmentation (done in Phase 1) from live registration (done in Phase 2). Phase 2 is achieved in realtime at approximately 26fps using standard hardware, and does not depend on successful registration in previous frames. It is thus simpler than EKF-SLAM and PTAM because it does not require switching between tracking and re-localisation. We have shown that our approach significantly outperforms EKF-SLAM for this problem. In the future we aim to enlarge our evaluation dataset and to explore the new opportunities that our method opens up for AR-assisted resection planning in uterine laparosurgery. The several techniques proposed in these papers may be used in the context of MRI data of pregnant patients to improve the extraction and the reconstruction of the uterus and the others tissues around it.

Assessment of the biomechanical properties of the uterus

To simulate the mechanical behavior of an organ, we also have to perform some experiments in vivo or in vitro on the tissues to determine its stress-strain relationship, that is a specific force per unit area (stress) versus a specific displacement from initial length (strain). This relation is the constitutive law of the organ. It characterizes the mechanical behavior of the deformable object. This mechanical behavior can be linear, non linear, isotropic, anisotropic, heterogeneous, viscoelastic or viscoplastic. We also need to determine the mechanical parameters associated to the constitutive law. Their exact values are extremely difficult to determine and they may vary by one or several order of magnitude, depending on the protocol used for their estimation. Moreover, the values obtained in vitro are usually inappropriate, and it is often difficult to perform the experiments in vivo.

Several techniques to perform the measurements. In 2014, Mazza [8] presented a review on research made to characterize the biomechanical and microstructural properties of the cervix during pregnancy. This kind of research is made in order to prevent preterm delivery, with an established correlation between the stiffness of the cervix and the risk of preterm delivery. Three methods were privileged for these measurements. The first method, called elastography, consists to assess the deformation of the organ by measuring its displacement when a force is applied on it. The corresponding displacement of the organ is quantified by tracking the position of a large number of points of the organ during the motion. We obtain an elastogram corresponding to a color map indicating the difference between soft regions (i.e. with large displacements of the points) and hard regions (i.e. with very small displacements of the points) of the deformed organ. The second method consists to compress the tissue using ultrasound until no further deformation can be observed to determine its "maximum deformability". The third method consists to use an aspiration device to determine in vivo the stress-strain curve of the tissue.

Impact of the protocol used. In 2008, Myers [START_REF] Myers | Mechanical and biochemical properties of human cervical tissue[END_REF] proposed a study to establish a stringent protocol to collect data in order to measure the mechanical properties of cervical tissue under different loading modes (for example, in tension, confined compression, unconfined compression). The study was performed for non-pregnant and pregnant patients. To briefly sum up, a non linear behavior was observed for the cervical stroma. Moreover, one of the interest of this paper is to draw attention on the large discrepancies in the results presented in the literature. Thus, a lot of precautions are necessary to ensure the repeatability of the measurements performed on a tissue to obtain useful values for the numerical simulation and to characterize with accuracy the mechanical properties of the tissue.

Results obtained from an aspiration device. In 2006, Mazza [START_REF] Mazza | Mechanical properties of the human uterine cervix: an in vivo study[END_REF] presented a study performed with an aspiration device to characterize the mechanical properties of the human uterine cervix in vivo. The average values of the stiffness parameter vary from 0.095 to 0.24 bar/mm. The experiments have been performed on eight patients, aged from 47 to 69 years and having had 1 to 4 births. As during the pregnancy, the uterus undergoes significant changes [START_REF] Fung | Biomechanics. Mechanical properties of living tissues[END_REF], Bauer [START_REF] Bauer | Assessment of the in vivo biomechanical properties of the human uterine cervix in pregnancy using the aspiration test: A feasibility study[END_REF] presented another study in 2009 performed with the same device but on pregnant women (between 21 and 36 weeks' gestation). Stiffness values varied between 0.013 and 0.068 bar/mm. We can note that the tissue of non-pregnant patient is significantly stiffer than tissue of pregnant women. These results remained the same for both tension and compression experiments. In 2013, Badir [START_REF] Badir | A novel procedure for the mechanical characterization of the uterine cervix during pregnancy[END_REF] presented a novel procedure for the realization of measurement on the cervix during gestation. This procedure was also based on the pipette aspiration technique. As the measurement is fast and it does not caused any pain, it can be used as routine control during the pregnancy. Thus, it permits to obtain a large amount of measurements and it permits to follow the evolution of the biomechanical properties during the gestation.

To sum up, a lot of studies have been performed on the cervix biomechanical properties. Obviously, this part of the uterus was studied because of the estimated relation between cervical stiffness and the risk of preterm. The technics and protocols of measurement used are more and more precise and aim to ensure the repeatability of the measurements. But it seems still difficult to assess with high accuracy the biomechanical properties of the tissues. Indeed, the measurements naturally depend on many parameters such as localization of the assessment, degree of cervix lubrification, type of protocol used to move the tissue (mainly by applying a force on it), maternal age, number of pregnancies, or term of pregnancies.

The Finite Element models proposed to simulate the behavior of the uterus

Once the geometrical models of the organs created and their mechanical properties studied, we can create the corresponding physical model in order to perform the numerical simulation. This physical model is more or less complex according to the aim of the numerical simulation. To achieve an accurate simulation, a Finite Element model is generally chosen with a constitutive law in agreement with that of the organ (classically hyper-elastic and anisotropic for tissues). To reduce the computational time, a simpler law can be selected but at the expense of precision. We also have to define the boundary conditions of the organs involved in the simulation as well as their interactions.

Electrophysiological and molecular mechanism of uterus. Sharp [START_REF] Sharp | Computer models to study uterine activation at labour[END_REF] proposed a review on the computer models published to study the uterine activation during labor. The models were divided in three categories: (i) models of uterine electrophysiology and propagation, (ii) models of the molecular mechanisms that initiate the labor, (iii) and biomechanical models of labor. Then, the review focused only on electrophysiological and molecular mechanisms activating labor. This kind of research seems distant from work on biomechanics models but it can be interesting to follow the results obtained with this approach.

Modeling of the uterine pressure. In 2010, Li [START_REF] Li | Modeling childbirth: elucidating the mechanisms of labor[END_REF] proposed a review on finite element models developed to simulate mechanics of vaginal delivery. The described models concerned in particular the modeling of the pelvic floor (including the levator ani muscles) and the fetal head. The aim is to be able to simulate both the deformation of the head and the pelvic floor. The most challenging aspect of childbirth modeling is to take into account all the forces applied on the organs: the expelling forces induced by the uterus and the abdominal muscles Those forces were approximated by applying kinematic boundary constraints on the fetal head. This strategy avoids to explicitly simulate the complex mechanism of the uterus which is not the primary aim of those simulations.

Finite Element models of a pregnant woman. Some papers focused on the modeling of the pregnant woman to study injury mechanisms in car crashes [START_REF] Delotte | Pregnant woman and road safety: a numerical approach. application to a restrained third trimester pregnant woman in frontal impact[END_REF][START_REF] Prs | A pregnant woman model to study injury mechanisms in car crashes[END_REF]. The aim is to propose a specific safety system to decrease the risk of fetal loss. A simulation based on a Finite Element model was performed. The geometrical model of the uterus was extracted from MRI data of a pregnant woman at nine month pregnancy. Then, the gravid uterus has been incorporated into the HU-MOS (HUman MOdel For Safety) model which is a Finite Element model of a complete human body in driving position. The mechanical behavior of the uterus was modeled as a hyper-elastic Ogden material and the amniotic fluid was represented using a Euler model. Thus, the anisotropic behavior of the uterus was not considered in this study. For the validation of the simulations, the PMHS (Post Mortem Human Subjects) approach has been improved by inserting an artificial uterus made in silicon into a woman body. Four belt loading tests were realized and compared to the numerical response of the model in similar conditions. Preliminary results obtained were in agreement with the experimental curves. In these tests of car crash, the uterus did not sustain high strain.

Recently, a biomechanical model of the pregnant pelvic system has been proposed in [11]. The aim was to study the constraints applied on the pelvic components during childbirth to improve the knowledge about uterine prolapse. The pelvic system proposed (see Fig. 11) included the muscles, ligaments and pelvic organs. The geometrical models have been constructed from MRI data with a manual reconstruction of non visible structures. These manual reconstructions have been made according to anatomical descriptions. The ligaments were also added to the geometrical models with some difficulties for ones for which no consensus is established. Moreover, several models of the fetal head and uterus have been used to simulate the vaginal delivery. Thus, the volume of the geometrical models have been changed in order to measure their corresponding impacts on the ligaments. The properties of the tissues were based on previous work performed by the same team [START_REF] Rubob | Vers une modélisation de la cavité pelvienne, 18eme congrès francais de Mécanique[END_REF][START_REF] Rubod | Biomechanical properties of vaginal tissue. part 1: New experimental protocol[END_REF][START_REF] Rubod | Biomechanical properties of vaginal tissue: preliminary results[END_REF][START_REF] Brieu | A nonlinear-elastic constitutive model for soft connective tissue based on a histologic description: Application to female pelvic soft tissue[END_REF] and the tissues were modeled with an elastic linear behavior using the shell finite elements. As boundary conditions, the pelvic bone and uterine fundus were fixed; the muscle floor was fixed to the pelvic bone. Then, the simulation of childbirth was performed by imposing a trajectory of the fetal head during its descent. The simulations enabled to measure the deformation of the uterosacral ligaments during the vaginal delivery. Some improvements have to be made in order to consider the hyper-elastic and anisotropic behavior of the tissues. Nevertheless, we can note that this pregnant pelvic system is the most complete proposed in the literature. Thus, we can hope that, once the improvements performed, this pelvic system (which includes the ligaments connecting the uterus to the pelvis) could improve our knowledge of damages sustained during a vaginal delivery.

textbooks [5]. Figure 3 shows the main steps of this trajectory, imposed in the most frequent variety (left anterior occipitoiliac engagement, occipito-pubic expulsion). Simulations were realised with the finite elements software "SOFA" developed by INRIA and largely used for biomedical simulations [START_REF] Bourdel | Augmented reality in gynecologic surgery: Evaluation of potential benefits for myomectomy[END_REF]. Simulations were done with varying the size of the fetal head.

Results

We obtained a pelvic model of the pre which allows us to simulate delivery. pelvic system at different steps of th (engagement, deflexion, expulsion) and ed displacements. Displacement is ma floor muscle, vagina and uterosacral lig It becomes possible to evaluate the areas during childbirth, and especially th the suspension system. Placed as they ligaments mainly suffer compression a aged. But uterosacral ligaments will be when the head reaches the inferior ut medium-sized fetal head (50th perc uterosacral ligaments undergo a defo (Fig. 5).

This first modelling shows the displac ligaments whose important role in pelvic [10]. It becomes possible to vary the de lating different situations, function of g trophicity, by varying volumes of fetal analysing potential damages in those example, shows the deformation of the when the head is inferior to the 10th pe premature birth or fetal hypotrophy at 4

Pelvic tissues have a visco-hyperelas be damaged. In 2008, Rubod et al. sho curves realised by cyclic trials with def levels. Behaviour of the pelvic soft tissu of a synthetic elastomeric material and s varies with the maximal deformation of size of the fetal head grows, more dist uterosacral ligaments and therefore pote damage that may alter pelvic statics. We the study assess quantitative damage of floor muscle (levator ani) and on the va

Discussion

Recapitulation of results

We wanted to study the constraints and d structures during childbirth to evaluate prolapse. We built an anatomical and fu pelvic system of the pregnant woman. many numerical challenges to successfu istic passage of the fetal head through vagina-pelvic floor". Uterosacral ligam an important traction during the passag the lower uterine segment, and their defo around 30 %. Previous research of ou Fig. 2 Complete geometric model of pelvic system of the pregnant woman Table 1 Mechanical properties of pelvic tissues of the nonpregnant woman already determined by our team [10] were adapted to pregnancy using literature data [18, Finite Element models of the uterus. In 1975, Mizrahi and Karni [1] have presented a mechanical model of the uterus. As the thickness of the uterus is small in comparison to its two others dimensions, they modeled the uterus as a thin closed shell. They also treated the constraints involved by the ligaments attachments and the cervix. For the definition of the shell model, they considered that the uterine muscle has no rigidity in bending and that the single stiffener is the cervix acting as a tensile anchoring ring. Moreover, the displacements of the cervix have been put to zero for a single contraction and constant during the second stage of labor when the cervix is fully dilated. They also assumed that the volume bounded by the uterus is constant during the deformation due to the incompressibility of the inter-uterine fluid. In [2], they presented a study to improve the anisotropic behavior of the uterine muscle. It is difficult to evaluate the results obtained as there is no comparison between the results obtained by simulations and real in vivo experimentations. More recently, an other Finite Element model (FEM) of the uterus was proposed in [3] to reproduce realistic mechanical and physiologic behaviors of hysteroscopy. The corresponding surgical procedure consists to distend the uterus with the accumulation of watery fluid in order to easily access and to well visualize the uterine cavity and faloppian tube. The volumetric mesh used for the hysteroscopic simulations was composed of tetrahedra with a total of 56,000 nodes. The mechanical behavior defined was based on a polynomial strain-energy function proposed by Kauer [START_REF] Kauer | inverse finite element characterization of soft tissues with aspiration experiments[END_REF]. Moreover, as boundary conditions, the cervix was fixed at the beginning of the hysteroscopy. Then, a pressure of 20 kPa was applied into the uterine cavity. But due to a lack of experimental data, it was difficult to prove the accuracy of the model for various loading situations. We can note that these simulations required about 17 hours of CPU time.

Real-time simulations of the uterus. Some work focused on the implementation of mechanical models of the uterus that run in real-time [START_REF] Székely | Modelling of soft tissue deformation for laparoscopic surgery simulation[END_REF][START_REF] Khnapfel | Endoscopic surgery training using virtual reality and deformable tissue simulation[END_REF][START_REF] Montgomery | Surgical simulator for hysteroscopy: A case study of visualization in surgical training[END_REF]12,6,5]. The idea is to propose training simulators for hysteroscopic or laparoscopic surgery. The challenge is to make adequate simplifications of mechanical models to develop pertinent simulators reproducing realistic simulations. In this context of real-time simulation, the accurate FEM proposed in [3] was combined to a Free-Forme Deformation (FFD) technique [6]. The idea is to register accurate responses of the tissue model to different fluid pressures in order to use these results during the real-time simulation. For this, some hypothesis were made to simplify the model: the interactions made on the uterus with the simulator were restricted to the fundus of the uterus; the boundary conditions of the uterus did not change during the intervention; the cavity of the uterus should not endure extreme modifications (i.e. the topology of the uterus was supposed not to change). Thus, the FEM [3] was first executed as precomputations. Then, during the use of the simulator, the results of the FEM simulation were loaded and used to compute in real-time the response of the tissue according to the interaction applied. For this purpose, a simple linear interpolation was performed to compute the new positions of the mesh. Furthermore, a fluid simulation (based on the Navier-Stokes equations) was also performed to simulate the fluid flow in the cavity in order to increase the simulator realism. In future work, the authors will plan to use the GPU to simulate the fluid with an higher accuracy. Moreover, the FEM simulation used for the computations will be improved to simulate the non-linear and anisotropic behavior of the myometrium.

Our childbirth simulation in real-time

Our work focused on a childbirth simulator for training of medical gestures during vaginal delivery. In this context, we proposed a biomechanical simulation of the descent of the fetus during the second step of the labor. The challenge was to obtain a real-time simulation. This was achieved by making a lot of simplifications in the modeling of each organ involved, since it is not possible to combine an accurate simulation of each organ. Thus, our aim was to obtain a realist trajectory of the fetus induced by the interaction of each organ despite their simplifications. All the details of this work can be found in [9].

The simulation was composed of the fetus, uterus, pelvis and abdomen of the parturient woman. During the childbirth process, the uterus is one of the most important organs of the pelvic system since it supports all the efforts applied by others organs. Thus, the uterine contractions associated to the voluntary efforts of the parturient woman induce a pressure on the fetus, pushing it into the birth canal. Throughout the descent of the fetus, the inner walls of the uterus are flattened against the fetal body, decreasing the uterine volume progressively. The height of the uterus at the end of the childbirth process is approximately one third of its height before the beginning of the labor.

Generation of the geometrical models

The geometrical models of the organs have been generated from MRI data for soft tissues and CT-scan data for the bony parts of the parturient. For this, the ITK SNAP software has been used for the segmentation of the medical images. Then, the ReMESH software has been used to simplify the dense mesh of each organ obtained after the segmentation. Finally, the volume mesh of each organ has been generated by importing the surface mesh in the Abaqus Finite Element software. The 3D mesh has been generated thanks to the use of the tetrahedralization algorithm based on the Delaunay technique. 4.2. Biomechanical modeling of the uterus and forces FE model of the uterus. For the biomechanical modeling of the uterus, the work of Mizrahi [2] showed that the behavior of uterine muscles changes during the childbirth, with an isotropic behavior in the early stages of childbirth and an anisotropic behavior at the end of the labor. To simplify our model, we considered only an anisotropic behavior for the uterine membrane and we modeled the uterus as a Neo-Hookean material. The strain energy per unit of reference volume is then defined by: W = C 10 (I 1 -3), with C 10 = G/2 where G = E/ (2 (1 + ν)) is the shear modulus, E is the Young modulus, ν the Poisson ratio, and I 1 the first deviatoric strain invariant defined as I 1 = λ 1 2 + λ 2 2 + λ 3 2 with λ i the deviatoric stretches defined by λ i = J -1 3 λ i , where J is the total volume ratio and λ i are the principal stretches. For our model, we chose a density of 950 kg/m3 and C10 = 30 kPa [START_REF] Mazza | Mechanical properties of the human uterine cervix: an in vivo study[END_REF][START_REF] Bauer | Assessment of the in vivo biomechanical properties of the human uterine cervix in pregnancy using the aspiration test: A feasibility study[END_REF].

Modeling of the uterine contractions and expulsion forces. Instead of modeling the muscle behavior of the uterus, we modeled its consequences which are the uterine contractions. These forces are applied on the uterus during labor and are involuntary. They occur 3 or 4 times every ten minutes (which corresponds to one period). The average duration of a contraction is 90 seconds and its amplitude varies between the "base tonus" (pressure prevailing in the uterus caused by strong deformation) and the intensity of the uterine contractions. The difference between these two amplitudes (called "true intensity") corresponds to the effective thrust forces of the uterine contractions applied on the uterus during the delivery [START_REF] Schaal | Mécanique et Technique Obstétricales[END_REF]. Fig. 13 illustrates these uterine contraction forces. This thrust is insufficient to make the fetal delivery by deleting the effect of the pelvic muscles which retain the fetus. Therefore, during the second stage of labor, the parturient woman must voluntarily produce a series of significant abdominal thrusts (called "expulsion forces"). These expulsion forces are caused by the contraction of the abdominal muscles (which are normally located in the lower abdomen, but lifted by the presence of the fetus) and the diaphragm. Although, these expulsion forces are about 4 times higher than the uterine contractions, they have to be synchronized with the uterine contractions. Fig. 14 illustrates this fact: to exceed the threshold necessary to overcome pelvic floor resistance the addition of these two forces are necessary. Thus, the combination of the uterine forces and the expulsion forces applied on the uterus shrink the uterine walls. This fact causes a force that expels the fetus into the vaginal canal. Consequently, in our simulation, we applied periodically the uterine contraction forces and the expulsion forces on the uterus to simulate this behavior.

For the boundary conditions of the uterus, we limited the displacements of the vaginal canal in the transverse plane (allowing the opening and closing of the vaginal canal while avoiding the descent of the organs) and we connected the displacement field in the lower part of the uterus to the pelvis (instead of modeling all the ligaments connected the uterus to the pelvis). Fig. 15 presents an illustration of the boundary conditions and the parts of the uterus where were applied the forces: in gray, the part of the uterus on which the uterine contractions were applied; in green, the part of the uterus on which the uterine contractions and the maternal expulsion forces were applied; in red the part of the uterus fixed to the pelvis.

Figure 15: The parts of the uterus where were applied the forces and the boundary conditions used for the simulation: the uterine contractions (green); the uterine contractions and expulsion maternal forces (gray); the part of the uterus fixed to the pelvis (red).

Contacts between the uterus and the others organs involved in the simulation

In our simulation, the uterus is the interface between the maternal abdomen and the fetus. Consequently, the thrust and the fetal movement were performed through the uterus. To model the contacts between the uterus and the others organs involved, we considered as a frictionless contact, the interactions between the uterus and the fetus (due to the fact that the amniotic fluid has lubricated the internal walls of the uterus). We also considered as a frictionless contact, the interactions between the uterus and the maternal abdomen (due to the viscous contact provided by the peritoneal fluid between all the organs within the great peritoneal cavity). Fig. 16 presents the boundary conditions of the organs modeled in our simulation of vaginal delivery.

Mechanical behavior of the uterus during the simulation

The simulation was performed by using the Abaqus FE software developed by Dassault Systems. We used the Euler explicit scheme, with 250 time steps. The duration of our birth simulation was 32 minutes, requiring 8 contractions to expel the fetus (with an average fetal head velocity of 0.09 mm/s). The execution time was 45 minutes on on a Intel PC Core duo, 2.4GHz, 6 GB RAM. Fig. 17 presents some pictures made at different stages of the simulation. These pictures show the evolution of the shape of the uterus during the simulation of the descent of the fetus. We can see the decreasing size of the uterus during the simulation. To evaluate this reduction, we tracked the front-sagittal trajectory of a point at the top of the uterus (noted u pt1 i ). We compared this point to a point on the lower part of the uterus (noted u pt2 i ). As shown in Fig. 18, the distance between these two points was 230 mm at the beginning of the labor phase and it was 80 mm at the fetus delivery. Consequently, the size of the uterus has been decreased of about 2/3 which is consistent with reality. We also followed two points chosen in the transversal plane of the uterus. Fig. 19 shows the movement of these points along the coronal/transversal axis. As expected, their displacements followed opposite directions, as the behavior of the uterus corresponds to the periodic behavior of the uterine contractions.

Discussion about our real-time simulation of vaginal delivery

We managed to simulate the descent of the fetus during the vaginal childbirth. The trajectory of the fetus was only induced by the interactions between the considered organs in the simulation and the expulsion forces applied on the uterus. The uterus was simulated with an anisotropic behavior. In agreement with reality, its size decreased during the vaginal delivery. In the future, we will consider a more precise simulation of the mechanical behavior of the uterus in order to increase the simulation accuracy of the cervix during the vaginal delivery. But the challenge will remain, i.e. a compromise between the precision of the simulation and its computational time, in order to couple our simulation to a haptic device.

Conclusion

We presented in this chapter, researches that focused on the study of the uterus. This work was mainly performed to improve the knowledge about this organ. To analyse this work, we followed the classical pipeline which enables to realize a biomechanical simulation of an organ.

It is still difficult to directly extract from medical MRI images all details of the uterus anatomy and geometry in particular all the ligaments that connect the uterus to the pelvis. Thus, a manual reconstruction of unseen structures is still required. We also reviewed advances that were made to measure organs mechanical properties (in particular performed on the cervix due to its direct impact into the risk of preterm childbirth). Recent procedures and protocols improve the measures repeatability and enable to better understand their limitations. Then, we provided a presentation of some finite element models used to simulate the mechanical behavior of the uterus for pregnant or non-pregnant uterus. These models are more or less accurate according to their purpose.

More specific research dealing with Augmented Reality used during laparoscopic uterine surgery are presented. This work can be used in the context of biomechanical simulation of the uterus and pelvic system.

Finally, we presented our work on real-time simulation of the fetus descent, in order to develop a training simulator of medical gestures performed during vaginal delivery. Despite the simplifications performed to obtain a real-time simulation, the deformation of the uterus is in agreement with the reality, with a decrease of its size during the childbirth.

The next challenge is to propose a patient specific simulation, which integrates the measurement of the mechanical properties of the cervix performed on the patient. The idea is to be able to quantify the risk of preterm of the patient or to prevent the uterine prolapse by quantifying the damages induced by a vaginal delivery onto the ligaments of the patients.

Fig. 1

 1 Fig. 1 illustrates the position of the uterus: it is inside the pelvis, dorsal of the urinary bladder, and ventral of the rectum.

Figure 1 :

 1 Figure 1: Structures present around the human uterus and relating to the human uterus [4].

Fig. 4 .

 4 Fig. 4. FE model of uterus with some 20,000 elements. Reduced variants (less degrees of freedom) were used in the development of the real-time models (Figs. 5 and 6).

Fig. 5 .Figure 2 :

 52 Fig. 5. Triangle meshes synthesized with the extended statistical model. The uterine cavity is deformed according to increasing distension fluid pressures (left to right).

Fig. 1 .

 1 Fig. 1. Left: Uterus with labeled vertices. Middle: Corresponding surface of tetrahedra mesh. Right: uterus with hydrometra. In all cases the frontal surface has been rendered in wireframe mode to visualize the inner cavity.

Fig. 2 .

 2 Fig. 2. Left: Visualization of the flow in the uterine cavity under limited pressure. Middle: Flow pathlines in a fully distended state. Right: Effect of additional vortices on the flow field.

Fig. 3 .

 3 Fig. 3. Left: Low hydrometra in a cavity with a myoma. Middle: Same scene under maximal distension. Right: Scene with blood streams under the influence of both inflow and vortices.

Figure 3 :

 3 Figure 3: Geometrical model of the uterus proposed in [6].

Figure 4 :

 4 Figure 4: FE model of a vaginal delivery at different steps of the simulation. Images from [9].

FIGURE 1 |

 1 FIGURE 1 | Models of the female pelvic floor structures in anterior (left) and inferior (right) views.

.

Fig. 4 .

 4 Fig. 4. FE model of uterus with some 20,000 elements. Reduced variants (less degrees of freedom) were used in the development of the real-time models (Figs. 5 and 6).

Fig. 5 .

 5 Fig. 5. Triangle meshes synthesized with the extended statistical model. The uterine cavity is deformed according to increasing distension fluid pressures (left to right).

Fig. 6 .

 6 Fig. 6. Hydrometra deformation states as shown in the simulator, resulting from increasing pressure settings of the distension fluid (left to right).

Fig. 4 .

 4 Fig. 4. FE model of uterus with some 20,000 elements. Reduced variants (less degrees of freedom) were used in the development of the real-time models (Figs. 5 and 6).

Fig. 5 .

 5 Fig. 5. Triangle meshes synthesized with the extended statistical model. The uterine cavity is deformed according to increasing distension fluid pressures (left to right).

Fig. 6 .Figure 6 :

 66 Fig. 6. Hydrometra deformation states as shown in the simulator, resulting from increasing pressure settings of the distension fluid (left to right).

  d. By combining MRI and 3D ultrasound developed anatomical models with good n to in 3D anatomy in vivo. , cervical shortening is a complex function properties, loading conditions, and anatomietry. Less clear is how a short cervix relates athogenesis of preterm birth. In patients vical insufficiency, a short cervix likely weakened tissue properties of the cervical owever, in the setting of placental abruptner et al., 2005) or intrauterine infection et al., 2010), a short cervix is caused by a was essential to detect the anatomy of interest.

Fig. 2 .

 2 Fig. 2. Graphical user interface for derivation of new uteri based on seven predictors.

Fig. 3 .

 3 Fig. 3. Upper row: variation of the uterus for different values of the 'fundus depth' predictor. Lower row: different cervix-fundus angles induced by the variation of the overall uterine length predictor.Figure 8: Generation of variable anatomical models proposed in [13].

Figure 8 :

 8 Generation of variable anatomical models proposed in[13].

Figure 9 :

 9 Figure 9: (a) Registration error vs. number of views. (b) Number of iterations vs. number of views. (c) Mean surface registration error for 2,4 and 15 views.

Figure 10 :

 10 Figure 10: Registration between the uterus in a pre-operative MRI to intra-operative laparoscopic images, and visual augmentation of two hidden myomas. The top row shows M (the mesh model of the uterus from the MRI ) overlaid in each image. The red contours denote the silhouette boundaries of M in each image. The bottom row shows the augmented myomas. Best Viewed in colour.

Figure 9 :

 9 Figure 9: Registration between the uterus in a pre-operative MRI to intra-operative laparoscopic images (top), and visual augmentation (bottom) of two hidden myomas. Images from [17].

Fig. 3 :

 3 Fig. 3: Column 1: the dense 3D models built in Phase 1. Column 2: the coagulation markers. Columns 3&4: the registered models using WBMTR.

Figure 10 :

 10 Figure 10: The registered models obtained using WBMTR (Wide-Baseline Multi-Texturemap Registration). Images from [16].

Figure 11 :

 11 Figure11: The geometrical model of pelvic system (proposed in[11]) of a pregnant woman.

Fig. 12

 12 presents the geometrical model of the uterus used in the simulation. At the left, the figure presents the initial model obtained from MRI data. It was composed of 42,811 nodes. At the middle and the right, we can see the final 3D model of the uterus obtained after smoothing the initial model. It includes the uterus body, the cervix and the vaginal canal. The cervix has been manually constructed from anatomical description. This final 3D model of the uterus was composed of 7,489 tetrahedra.

Figure 12 :

 12 Figure 12: The geometrical model of the uterus. At the left, the data (with 42,811 nodes) generated from MRI data. At the middle and the right, the 3D model (with 7,489 tetrahedra) generated after smoothing. The cervix has been manually constructed.

Figure 13 :

 13 Figure 13: Uterine contraction forces (mmHg) versus time.

Figure 14 :

 14 Figure 14: Evolution of the uterine forces: synchronized forces (left) and unsynchronized forces (right). The line corresponds to the threshold for the expulsion of the fetus. UC: uterine contraction; ABD: abdominal forces; diaph: diaphragm forces.

Figure 16 :

 16 Figure 16: The modeled organs and their boundary conditions. ABD: abdominal forces.

Figure 17 :

 17 Figure 17: Behavior of the uterus at different stages during the simulation of the fetus' descent.

Figure 18 :

 18 Figure 18: Evolution during the simulation of the trajectory of a front-sagittal point of the uterus.

Figure 19 :

 19 Figure 19: Displacement of two points chosen in the transversal plane of the uterus.

  Lapeer

	Vagina Rectum Uterus	Perineal body	Bulbospongiosus Transverse perineae
	Bladder	Bony pelvis Obturator internus	ani Levator	Coccyx	External anal sphincter Internal anal sphincter

Acknowledgements

The realization of the childbirth simulation has been performed during the Ph.D. thesis of R. Buttin under the supervision of F. Zara, B. Shariat (LIRIS, Université Lyon 1) and T. Redarce (laboratoire Ampère, INSA de Lyon). This work was partly financed by grants from GMCAO and SIMED projects of the ISLE cluster of the French Rhône-Alpes region, and supported by French state funds managed by the ANR within the SAGA project (ANR-12-MONU-0006). The medical images used have been provided by Pr C. Adamsbaum of the St Vincent de Paul Hospital (AP-HP), Paris. The segmentation of these medical data has been performed by Jérémie Anquez (TELECOM ParisTech, CNRS, UMR-5141, LTCI).