Finite element simulation of ultrasonic wave ...

Biomech Model Mechanobiol DOI 10.1007/s10237-015-0651-7

ORIGINAL PAPER

Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment Romain Vayron · Vu-Hieu Nguyen · Romain Bosc · Salah Naili · Guillaume Haïat

Received: 10 September 2014 / Accepted: 10 January 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract Dental implant stability, which is an important parameter for the surgical outcome, can now be assessed using quantitative ultrasound. However, the acoustical propagation in dental implants remains poorly understood. The objective of this numerical study was to understand the propagation phenomena of ultrasonic waves in cylindrically shaped prototype dental implants and to investigate the sensitivity of the ultrasonic response to the surrounding bone quantity and quality. The 10-MHz ultrasonic response of the implant was calculated using an axisymetric 3D finite element model, which was validated by comparison with results obtained experimentally and using a 2D finite difference numerical model. The results show that the implant ultrasonic response changes significantly when a liquid layer is located at the implant interface compared to the case of an interface fully bounded with bone tissue. A dedicated model based on experimental measurements was developed in order to account for the evolution of the bone biomechanical properties at the implant interface. The effect of a gradient of material properties on the implant ultrasonic response is determined. Based on the reproducibility of the measurement, the results indicate that the device should be sensitive to the effects of a healing duration of less than one week. In all cases, the amplitude of the implant response is shown to decrease when the dental implant primary and secondary stability increase, which is consistent with the experimental results. This study paves the way for the development of a quantitative ultrasound method to evaluate dental implant stability. R. Vayron · V.-H. Nguyen · R. Bosc · S. Naili · G. Haïat (B) CNRS, Laboratoire Modélisation et Simulation Multi-Échelle, UMR CNRS 8208, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France e-mail: [email protected]

Keywords Finite element · Quantitative ultrasound (QUS) · Dental implant · Osseointegration · Biomechanical stability · Guided waves

1 Introduction Titanium implants are now widely used in oral and maxillofacial surgery for the replacement of missing teeth in fully or partially edentulous patients. Modern treatments aim at a rapid, strong and long-lasting attachment between the implant and bone tissue. However, implant failures, which may have dramatic consequences, still occur and are difficult to anticipate. The evolution of the implant biomechanical stability (Mathieu et al. 2014) is the main determinant of the surgical success. Assessing the stability of an implant is a complex problem due to the multiscale and time-evolving (Frost 2003) nature of bone tissue as well as to the presence of an interface. The implant stability immediately after surgery is referred to as primary stability and depends on the surgical procedure and on the implant geometry (Rabel et al. 2007). The secondary stability depends on remodeling processes occurring several weeks/months after surgery. The biomechanical properties of bone tissue in intimate contact with the implant [around 100–200 µm (Huja et al. 1999; Luo et al. 1999)] are the critical parameters for the assessment of both types of implant stability (Haïat et al. 2014). An accurate estimation of the biomechanical properties of bone tissue around the interface could lead to information of interest since it could help the surgeon to adapt the surgical strategy in a patient-specific manner and determine the time point for implant load bearing. In particular, a compromise must be found between a sufficiently rapid loading of the implant with the prosthesis in order to stimulate osseoin-

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tegration phenomena and a sufficiently low loading time in order not to destroy newly formed bone tissue around the implant interface. To solve this problem, dental surgeons usually use empirical methods based on palpation and patient sensation (Serra et al. 2008). Histomorphometry is the gold standard to assess osseointegration (Franchi et al. 2007) of an implant, but it cannot be used in vivo. Therefore, noninvasive methods have been developed to tackle this problem. The resolution of X-ray (Blanes et al. 2007; Wang et al. 2010b; Zhao et al. 2009) and magnetic resonance imaging (MRI)-based techniques around the implant interface is limited due to distortion effects related to the presence of titanium (Gill and Shellock 2012; Shalabi et al. 2007). X-ray and MRI-based methods cannot be used per operatively to retrieve information on the bone–implant mechanical contact conditions, which plays an essential part in the implant stability (Arndt et al. 2012; Mathieu et al. 2014; Souffrant et al. 2012; Wang et al. 2010a). As a consequence, biomechanical methods have been developed. Absence of ionizing radiation, inexpensiveness, portability and noninvasiveness are potential advantages of biomechanical techniques over X-ray and MRI-based modalities. The Periotest (Bensheim, Germany) (Schulte et al. 1983; Van Scotter and Wilson 1991) is based on the estimation of the deceleration time of a metallic rod impacting the implant. Osstell (Gothenburg, Sweden) (Valderrama et al. 2007) consists in modal analysis of the first bending resonance frequency (between 5 and 15 kHz) (Meredith et al. 1996) of the bone–implant system. However, Periotest (Meredith et al. 1998) and Osstell (Pattijn et al. 2007) methods are dependent on the orientation or fixation of the device. Moreover, both techniques give access to the characteristics of the entire bone–implant system and not only to the bone– implant interface (Aparicio et al. 2006), which is the critical parameter for the assessment of the implant stability (Bardyn et al. 2009; Nkenke et al. 2003; Turkyilmaz et al. 2006, 2007, 2008), even if advanced modeling and modal analysis techniques allow to derive specific characteristics of the bone–implant interface (Capek et al. 2009; Pattijn et al. 2006; Winter et al. 2010). An alternative method to assess implant stability consists in the use of quantitative ultrasound (QUS) (de Almeida et al. 2007), the implant acting as a waveguide. A 10-MHz ultrasonic device has been developed by our group and validated first ex vivo using cylindrical implants (Mathieu et al. 2011b), in vitro using dental implant inserted in a biomaterial (Vayron et al. 2013) and then in bone tissue (Vayron et al. 2014a), and eventually in vivo (Vayron et al. 2014b). The development of acoustical modeling and the associated numerical simulation is mandatory in order to understand the interaction between an ultrasonic wave and the

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bone–implant system because it may help improving the performances of the device under development. Moreover, using modeling approaches is the only solution in order to separate and discriminate the effect of the different bone parameters (such as bone structure, geometry and material properties) on the ultrasonic response of the implant, which is nearly impossible to achieve in vivo because all parameters vary in parallel. Interestingly, an analytical model considering the propagation of direct and lateral waves was developed in order to understand the ultrasonic propagation in a cylindrical implant mimicking a real dental implant (Mathieu et al. 2011a). Then, the ultrasonic propagation in the same simple implant model was modeled using 2D finite difference time domain (FDTD) simulation, and the numerical and experimental results were compared (Mathieu et al. 2011a). However, some discrepancies were obtained between the experimental and numerical results, which may be due to (1) the 2D nature of the simulation, (2) the assumption made on the modelization of the environment on the upper surface of the implant and (3) the limitation related with the use of FDTD when taking into account strong gradient of material properties which may occur at the bone–implant interface (Chang et al. 2003). The aim of this paper was to provide a more accurate modeling of the ultrasonic wave propagation in the prototype titanium cylindrical implants used in Mathieu et al. (2011a). To this end, a 3D finite element model is employed and the geometrical configuration is assumed to be axisymetric. The sensitivity of the ultrasonic response of the implant to variations of the quantity and quality of bone tissue in contact with the implant is assessed under realistic conditions. Moreover, the effect of the presence of a liquid layer of varying thickness between bone tissue and the implant is assessed and the effect of osseointegration phenomena is estimated.

2 Material and methods 2.1 Modeling wave propagation in the bone–implant system 2.1.1 Geometry and material properties The geometrical configuration used in the simulation is shown schematically in Fig. 1. The bone sample is modeled as a bilayer structure which consists of a 1-mm-thick layer of cortical bone (Ωc ) and a trabecular bone halfscape (Ωt ). The titanium cylindrical implant (Ωi ) (length L = 7 mm and diameter D = 4 mm) is buried at a depth of 6 mm in the bone sample. This configuration models the experimental configuration described in Mathieu et al. (2011a). Underneath the implant, a conical space (ΩH ) filled with water mimics the volume created by the apex of the drill during the preparation of the sample. The region (ΩL ) is defined by a

Ultrasonic evaluation of implants Table 1 Material properties used in the numerical simulations λ (GPa)

µ (GPa)

ρ (kg m−3 )

Water

2.25

10−4

1,000

Titanium

63.5a

42.3a

4,420a

Mature cortical bone tissue

17.6

5.99

1,850b

Trabecular bone tissue

1.28

0.85

1,170a,c

a See

Pattijn et al. (2006, 2007) Haïat et al. (2007) and Njeh et al. (1999) c See Sasso et al. (2008a) b See

∂σrr 1 ∂σr z σrr − σθθ − − =0 ∂r r ∂z r σr z ∂σzz − = 0, ρ u¨ z − ∂z r

ρ u¨ r −

(1) (2)

where ρ is the mass density; u r and u z are the radial and axial components of the displacement vector; σrr , σθθ , σzz , σr z are the components of the stress tensor denoted by σ . The constitutive relation using the Hooke’s laws is given by: σ = λtr()I + 2μ Fig. 1 Section view of the 3D axisymmetric geometrical configuration used in the numerical simulations. ΩH and ΩL correspond to the regions where the material properties are assumed to vary spatially and temporally during bone healing. Ωi , Ωc and Ωt denote the implant, cortical bone and trabecular bone, respectively. Ωca and Ωta correspond to absorbing layer associated with trabecular bone and cortical bone, respectively

400-µm-thick domain located between the implant external surface and bone tissue. Absorbing regions [Ωca (for cortical bone) and Ωta (for trabecular bone)] are located at the ends of the model in order to avoid reflections from boundary of the model. All media considered in this model are assumed to have homogeneous isotropic mechanical properties. Mechanical properties of the Ti-6Al-4V-ELI implant, cortical and trabecular bone materials are given in Table 1 (Haïat et al. 2007; Njeh et al. 1999; Pattijn et al. 2006; Sasso et al. 2008a). The material properties of bone tissue located in (ΩL ) and (ΩH ) are assumed to vary in order to model the effect of osseointegration phenomena (see Sect. 2.2.3). 2.1.2 Axisymmetric elastic wave equations Due to the symmetry with respect to the implant’s axis of the geometry and the boundary conditions, an axisymmetric assumption is made to model the wave field propagating in the system. As shown in Fig. 1, we denote by z and r the axial and radial coordinates, respectively. By neglecting the gravity force, the dynamic equations in each subdomain read:

(3)

where λ, μ denotes the Lamé constants and  is the strain tensor of which the nonzero components are given by:   ∂u z ∂u r ur 1 ∂u r , θθ = , r z = + , rr = ∂r r 2 ∂z ∂r ∂u z zz = (4) ∂z The experimental device uses a contact planar 10-MHz transducer positioned at the emerging surface of the implant (Vayron et al. 2013, In press-a, b). As a consequence, a plane wave is emitted from upper surface of the implant in the simulation. The emitted signal from the planar ultrasonic contact transducer is modeled as a time pulse uniform pressure applied on top surface of the implant. Here, a time-dependent func2 tion defined by p(t) = Ae−4( f c t−1) sin(2π f c t) is used to define this pulse, where A is a constant representing the signal amplitude and f c = 10 MHz is the center frequency which has the same value as the one used in the experiments (Vayron et al. 2013, 2014b). 2.1.3 Finite element simulation Discretization of the equations presented in the last section leads to a transient linear dynamic problem which may be expressed in a general form as follows: M U¨ + C U˙ + K U = F(t),

(5)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, U is the displacement vector, and F is the

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external force vector. The commercial software COMSOL Multiphysics (Stockholm, Sweden) was used because it is a general-purpose finite element code capable of solving these systems of equations which had already been used previously by our group for bone quantitative ultrasound (Haïat et al. 2011, 2009; Naili et al. 2010). The implicit generalized-α method was employed for solving the system (5). Temporal and spatial resolution of the finite element model is critical for the convergence of the numerical results. For the simulations presented in what follows, the element’s sizes in each domain Ωd (d = {t, c, i, H, L}) were chosen equal to λmin /10 = cmin /(10 f max ), where λmin corresponds to the shortest wavelength in the domain which may be calculated from the slowest wave velocity (for the isotropic elastic media considered here, cmin is the S wave velocity) and f max the maximum value of the frequency range. As the wavelengths are significantly different between domains due to the strong contrast of material properties, the region around the interfaces between two materials should be meshed with smoothly varied sizes of elements. The time step was chosen by using the stability condition:   d t ≤ α min h e /cdp where α = 0.5 is a constant and cdp , h de is the size of the elements in d , is the P wave velocity in the domain Ωd (d = {t, c, i, H, L}) (Hughes 2000). For simulations presented here, the time step is set at 1.5×10−9 s In order to prevent the nonphysical reflected wave generated from the lateral and bottom boundaries of the bone domains, an absorbing layer has been added to the model, as shown in Fig. 1. In the absorbing layer, a quadratically depth-varying damping term is added to smoothly attenuate the outgoing waves. The mesh of the considered model contains around 106 degrees of freedom, and all simulations were performed during 15 µs. 2.1.4 Signal processing The ultrasonic response of the implant being measured using an echographic mode, the output rf signal was determined by computing the spatial average of the pressure at the upper surface of the implant following:  D/2 U (r˙ )r dr (6) s(t) = 2π 0

The integral given in Eq. 6 was computed using a spatial discretization approximately equal to 15 µm, and we verified that slightly changing the discretization step did not modify the results. The same signal processing technique as the one employed in Vayron et al. (2013) was used to derive a quantitative indicator I. Note that the variations of I have already been related to dental implant stability (Mathieu et al. 2011a, b, 2012; Vayron et al. 2013, 2014a, b). The indicator I was devised

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to quantitatively estimate the average amplitude of the signal between 0 and 15 µs. Therefore, the envelop S(t) of the rf signal s(t) recorded at the emerging surface of the transducer was determined and the indicator I was given by: I =

1,500 

S(i T0 )

(7)

i=1

where T0 = 0.01 µs corresponds to the sampling period. The physical meaning of the indicator I is given by the energy of the signal contained in the corresponding time window. The value of the indicator I was determined for each numerical configuration. 2.2 Sensitivity study Acoustical modeling constitutes a powerful approach to realize sensitivity studies, which is the first step toward the resolution of the inverse problem. In what follows, the different configurations of interest considered are described. 2.2.1 Modeling bone quantity around the implant Different geometrical configurations corresponding to the experimental results described in Mathieu et al. (2011a) are studied in order to assess the influence of bone quantity in contact with the implant surface on the ultrasonic response of the implant. Studying this configuration is of interest because i) it allows an experimental validation of the finite element model developed and ii) it corresponds to a situation of interest from a physiological point of view, since the amount of bone in contact with the implant is related to the implant primary stability (Mathieu et al. 2014). The four experimental geometrical configurations considered in Mathieu et al. (2011a, b) are shown schematically in Fig. 2. The difference between the configurations lies in the length of the 400-µm-thick region surrounding the lateral boundaries of the implant, providing different amounts of bone in contact with the implant (see Fig. 2). Configuration #1 corresponds to an implant in contact with bone tissue over its entire surface. For configuration #2 (respectively, #3 and #4), a 2-mm (respectively, 4-mm and 6-mm)-long and 400-µm-thick region filled with water (contained in ΩL ) was considered on the lateral boundaries of the implant. The four configurations model a progressive variation of anchorage depth of the implant in bone tissue. In configuration #4, all lateral boundaries of the implant are distant from bone tissue, which corresponds to a fully unstable implant. The value of the indicator I was determined for each configuration #n(n = {1, 4}) represented in Fig. 2: (1) for the experimental rf signals obtained in Mathieu et al. (2011b) (denoted I1n ), (2) for the numerical results obtained using the 2D FDTD simulation obtained in Mathieu et al. (2011a) (denoted I2n ) and (3) for the numerical results obtained using

Corcal bone

4 mm

6 mm

2 mm

1 mm

Ultrasonic evaluation of implants

Trabecular bone

400 μm

Water

Configuraon #2

Configuraon #3

6 mm

4 mm

Configuraon #1

Configuraon #4

Fig. 2 Schematic illustrations of the four geometrical configurations, which only differ by the amount of bone in contact with the lateral boundaries of the implant

the finite element method described in Sect. 2.1.3 of the present paper (denoted I3n ). The ratios RI n1 = I1n /I11 , RI n2 = I2n /I21 and RI n3 = I3n /I31 are then defined in order to compare the respective sensitivity obtained experimentally and with the two numerical approaches to the amount of bone surrounding cylindrical implants. 2.2.2 Modeling a liquid layer around the implant The effect of the presence of fibrous tissue, which can be modeled as a liquid, between bone tissue and the implant is an important parameter affecting implant stability. Therefore, the influence of the thickness of a liquid layer located around the implant interface on the ultrasound response of the implant is assessed. To do so, a liquid layer in contact with the implant lateral boundary having a thickness comprised between 0 and 500 µm is considered in ΩL . The maximum value of the water thickness was chosen because it corresponds to the distance below which bone properties are known to affect the implant stability (Viceconti et al. 2001, 2006). 2.2.3 Modeling bone healing around the implant One advantage of using a finite element method compared with FDTD lies in the more accurate treatment of gradient of material properties, which are known to occur dur-

Table 2 Young’s moduli E and mass density ρ used to model the mechanical properties of mature and newly formed bone tissue at 4, 7 and 13 weeks of healing 4 weeks

7 weeks

13 weeks

∞

Healing time

2 weeks

E (GPa)

15.02

15.35*

15.85*

17.82*

20.66*

ρ (kg m−3 )

1,408

1,495

1,600*

1,811*

1,850a

∞ indicates mature bone tissue. The data marked with a asterisk are taken from Vayron et al. (2012, 2014c) a See Haïat et al. (2007) and Njeh et al. (1999)

ing the implant healing period (Chang et al. 2003). Recent studies by our group (Mathieu et al. 2011c; Vayron et al. 2011, 2012, 2014c) show that the material properties of bone increase during healing. In what follows, a simple model of bone healing around the implant is described and is used in order to determine the effect of healing time on the ultrasonic response of the implant. Bone is considered as a medium with homogenized material properties (Sansalone et al. 2010, 2012). Table 2 shows the values used for the material properties of mature and newly formed bone tissue as a function of healing time. The values marked with a asterisk are directly derived from Mathieu et al. (2011c) and Vayron et al. (2011, 2012, 2014c). A linear regression analysis was used to infer the values of bone mass density for 2 and 4 weeks of healing time (respectively, of the Young’s modulus for 2 weeks of healing time) using the data obtained at 7 and 13 weeks (respec-

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Fig. 3 Schematic description of the osseointegration model used to study the influence of a gradient of mechanical properties at the bone– implant interface

tively, 4, 7 and 13 weeks) of healing time. Moreover, the Poisson’s ratio was considered as homogeneous and equal to 0.25

because its dependence on healing time remains unknown. Once the variations of the bone material properties as a function of healing time are determined, a geometrical model corresponding to the implant osseointegration was derived. It consists in assigning a spatial distribution of material properties which varies as a function of healing time. The spatial dependence of the bone Young’s modulus E and mass density ρ is modeled as follows. All material properties are assumed to be the same as the one described in Sect. 2.1.1 in all regions except for the domains ΩL and ΩH , which correspond to regions where osseointegration phenomena are assumed to occur. For a fully osseointegrated implant (which corresponds to an “infinite” healing duration), bone material properties in ΩL and ΩH are assumed to be equal to the one shown in the last column of Table 2. For healing duration equal to 2, 4, 7 and 13 weeks, a linear gradient of E and ρ is assumed, as evidenced experimentally in Chang et al. (2003) (see Fig. 3). To do so, we use the following assumptions:

t=0.2μs

(a)

t=0.6μs

(b)

t=1μs

(c)

t=1.5μs

(d)

t=2.3μs

(e)

t=3.5μs

(f)

Fig. 4 Cartography of isovalues of amplitude of Vr (geometrical configuration #1) at six different times after the emission

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Indicator I (arbitrary units)

3

2.5

2

1.5

1 1

10

100

1000

Water thickness (µm)

Fig. 6 Indicator I as function of the thickness of the water layer at the periphery of the implant. The dashed line corresponds to the case where the bone–implant interface is fully bounded Fig. 5 Variation of the ratio of normalized value of the indicator RI ni as a function of the configuration number n corresponding to the implant stability for the results obtained experimentally (black line), with the 2D FDTD simulation (light gray) and 3D finite element model (dark gray)

a. For each healing time tH , the material properties at the interface between the implant ΩL and ΩH are equal to the one corresponding to tH shown in Table 2. b. For all healing durations, the material properties at the interface between bone tissue and ΩL and ΩH (indicated by dotted lines in Fig. 3) are equal to the one corresponding to mature bone (last column of Table 2). The two aforementioned conditions lead to a gradient of material properties in the radial direction in ΩL , while the gradient direction is more complex in ΩH . 3 Results 3.1 Qualitative results Figure 4 shows the cartography of isovalues of amplitude of Vr , the projection of the particle velocity on the r direction corresponding to the geometrical configuration #1 at six different times after the emission. After 0.6 µs (Fig. 4b), a transverse wave front is observed, resulting from mode conversion of the direct wave on the lateral boundaries of the implant. After 1 µs (Fig. 4c), the direct wave reaches the bottom of the implant, whereas the transverse wave front propagates with an angle α, relative to the length of the implant. After 1.5 µs (Fig. 4d), the direct wave is reflected at the bottom of the implant and the transverse wave front reaches the side opposite to the side of its formation. After 2.3 µs (Fig. 4e), the reflection of the transverse wave front on the lateral sides of the implant leads, through mode conversion, to the emergence of a longitudinal wave front, called lateral wave. After 3.5 µs (Fig. 4f), the lateral wave reaches the top of the implant, whereas the direct wave

approaches the bottom. The results are qualitatively similar with those obtained with the 2D FDTD model (Mathieu et al. 2011a). 3.2 Effect of bone quantity around the implant Figure 5 shows the results obtained for the ratio of the indicator RI ni (for the four configurations considered, n ∈ {1, 4}) for the experiments (i = 1), the FDTD simulation (i = 2) and the finite element model (i = 3). For both simulation methods, RI ni increase when the implant stability decreases, this result being consistent with the experimental results. Although the 3D FEM underestimates the experimental results, the agreement with the experimental results is better for 3D FEM than for 2D FDTD. 3.3 Effect of the presence of a liquid layer around the implant Figure 6 shows the variation of the indicator I as a function of the thickness of the liquid layer located at the implant interface in ΩL . The value of the indicator I is significantly higher when water is considered at the implant interface compared to the case of an interface fully bounded with bone tissue. Moreover, the value of I weakly depends on the thickness of the layer at the implant interface. 3.4 Effect of bone quality around the implant Figure 7 shows the results obtained for the indicator I for the different configurations described in Sect. 2.2.3 corresponding to different healing durations. The value of the indicator I is shown to decrease as a function of healing time. The dotted line corresponds to the case of full healing.

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Indicator I (Arbitrary units)

1.2 1.15 1.1 1.05 1 0.95 0.9 0

2

4

7

13

Healing time (weeks)

Fig. 7 Variation of the indicator I as function of healing time. The study was conducted by considering a gradient of material properties in ΩL and ΩH , following the method described in Sect. 2.2.3. The dotted line corresponds to the case of full healing

4 Discussion The present study is the first one to use 3D numerical simulation in order to investigate ultrasonic wave propagation occurring in an implant. This situation is of interest in the framework of the experimental device developed by our group, which aims at investigating the implant stability. de Almeida et al. (2007) have carried out a study employing FDTD simulation and modeling another ultrasonic device dedicated to the estimation of the implants stability. However, their study did not take into account the presence of cortical bone, which is determinant for the implant stability (Merheb et al. 2010). Moreover, the use of a 1-MHz frequency does not allow to measure time-resolved echoes. In the present study, a frequency of 10 MHz allows higher time resolution for the investigation of the sensitivity of QUS parameters to variations of parameters being of primary importance for the assessment of both primary and secondary stabilities of an implant (Seong et al. 2009) and more specifically of (1) water thickness around the implant interface, (2) the amount of bone in contact with the implant and (3) biomechanical properties of bone tissue around the implant. Another originality of this study lies in the model developed to account for osseointegration phenomena and corresponding to the evolution of the spatial distribution of bone properties around the implant as a function of healing time. The advantage of performing numerical simulations lies in the possibility to consider a controlled situation where each parameter varies independently, which is nearly impossible when performing experiments because of the complex nature of bone tissue and the difficulty of achieving a perfectly controlled geometrical configuration. Numerical simulations provide more insight on the sensitivity of the implant ultrasonic response to each parameter independently.

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The 3D FEM developed herein has been validated by comparison with results obtained with 2D FDTD and experimentally. The results show that the results obtained with the 3D FEM are closer to the experimental results compared to the results obtained with the 2D FDTD, which can be explained by a more realistic geometry as well as a better modeling of the interface phenomena. However, the discrepancy between the experimental results and the results obtained with 3D FEM may be due to possible experimental errors on the geometry of the configuration as well as on material properties, which remain difficult to control. More generally, the results obtained with the 3D FEM show that the value of the indicator I decreases when bone quantity around the implant surface increases, which corresponds to an increase of the implant primary stability. Similarly, the value of the indicator I decreases with healing time, which corresponds to an increase of the implant secondary stability. This decrease of the value of I as a function of the implant stability may be explained by the contrast of mechanical properties between the implant and the surrounding material, which increases when the implant stability decreases. When the gap of material properties increases, the transmission coefficient between the implant and the surrounding medium decreases and the energy leakage in the surrounding medium decreases, which explains that the acoustic energy received by the transducer is lower for stable implant than for unstable ones. The advantages of using 3D FEM simulation over the 2D FDTD are essentially due to the fact that 3D FEM allows to describe realistic geometry and boundary conditions. Firstly, the mechanisms of wave propagation occurring in a 2D rectangular model are fundamentally different compared to the 3D case, which has been described in details in Mathieu et al. (2011a) by considering the relative amplitude of the lateral wave (odd peaks) compared to the direct wave (even peaks). The results obtained in Mathieu et al. (2011a) show quantitative differences in terms of the relative amplitude of the signal amplitude of the even and odd peaks in 2D and 3D. As a consequence, it is important to model the 3D wave propagation phenomena because it plays a role on the determination of the indicator I . Secondly, the boundary and interface conditions are described more precisely in the FE modeling compared to the case of FDTD. When using 2D FDTD simulation, the upper bone surface was not free but loaded by an unrealistic fluid. This free surface has been described accurately using FEM. Moreover, another advantage of using FEM compared to FDTD method lies in a better modeling of gradient of material properties, which allows to derive the effect of osseointegration described in Sect. 2.2.3. The discrepancy obtained for configuration #2 and #3 between experimental and simulated results may be explained by the difficulty of achieving precise positioning of bone tis-

Ultrasonic evaluation of implants

sue in the experimental configuration, resulting in errors on the amount of bone in contact with the implant. The results obtained in Fig. 6 show that the implant response does not depend on the water thickness in the vicinity of the implant, but on the material properties of the medium in direct contact with the implant surface. Note that due to spatial discretization issues, computation with a water thickness lower than 1 µm could not be realized. These results may be explained by the same reasons given above, i.e., the importance of the material properties of the medium in intimate contact with the implant, which do not vary when the water thickness varies between 1 and 1,000 µm. The results indicate that a variation in thickness of the fibrous tissue layer between 1 µm and 1 mm does not influence the ultrasound amplitude. Moreover, the presence of a 1-µmthick water layer significantly influences the implant ultrasonic response compared to the case of a fully bounded interface (indicated by the dotted line in Fig. 6), which indicates that the ultrasound amplitude is sensitive to the biomechanical properties of bone tissue located at a distance of 1 µm around the interface. Taking into account, experimental errors in combination with the variation of the indicator I as a function of healing time (see Fig. 7) may lead to an estimation of the sensitivity of the device to actual variation of healing time, a parameter of interest for the clinical situation (Haïat et al. 2014). The standard deviation of I obtained experimentally in Mathieu et al. (2011a) is equal to 6 % of its average value, which corresponds approximately to a variation of ±3 weeks of healing time. However, using the ultrasonic transducer described in (Vayron et al. 2014b) leads to a significant decrease of the precision error, since the standard deviation of I drops to 0.55 % of its average value, which corresponds approximately to a variation of ±0.3 weeks of healing time. In this paper, 10 MHz was chosen based on previous work (Mathieu et al. 2011a, b, 2012; Vayron et al. 2013, 2014a, b), because this frequency is a compromise between (1) a sufficiently high frequency to obtain a good resolution at the interface (as shown in the present paper) and (2) a sufficiently low frequency to work with standard ultrasound electronics. Moreover, the attenuation coefficient of longitudinal waves in trabecular or cortical bone at 10 MHz is typically comprised between 50 and 400 dB/cm (Laugier and Haïat 2010; Sasso et al. 2007), which indicates that ultrasonic energy is localized at the bone–implant interface. Beside attenuation phenomena, guided wave effects described herein constitute an additional phenomenon concurring in the sensitivity of the measurement to the bone–implant interface properties. This study has several limitations. First, a simplistic cylindrical implant model was used in order to validate the 3D FEM simulation code by comparison with (1) experiments and (2) the 2D FDTD simulation code. However, the shapes of real implants are various and complex (e.g., threaded),

inducing propagation phenomena which may differ from the simple situation taken into account herein. Future study should consider real implant geometries in order to provide a better insight on the medical device under development. Moreover, the experiments already realized in vitro (Vayron et al. 2014a) [respectively, in vivo (Vayron et al. 2014b)] lead to similar results as the ones obtained herein since a decrease of the indicator was obtained when bone quantity increases (Vayron et al. 2014a) [respectively, healing time increases (Vayron et al. 2014b)]. These results constitute a further validation of the 3D FEM developed herein. Second, the model used to account for the evolution of the spatial distribution of bone material properties due to healing time only partially account for the complex osseointegration phenomena occurring during bone healing, which are not fully understood (Haïat et al. 2014). A linear gradient of material properties was considered, while the phenomena may be highly heterogeneous and dependent on the mechanical stresses occurring on the implant, which are not taken into account. Moreover, data obtained from rabbit bone tissue were used because the evolution of human bone material properties during healing remains unknown. In addition, a linear time dependence of each material property was considered, while the variation of bone mechanical quality has been shown to be nonlinear, which is not taken into account in the present study. In particular, a decrease of the Implant Stability Quotient (ISQ) measured using radiofrequency analysis (RFA) during the first few weeks after surgery has been obtained (Balshi et al. 2005; Glauser et al. 2004), but the reason for this phenomenon remains unclear (Coelho et al. 2010; Raghavendra et al. 2005). Similarly, the bone–implant contact (BIC) ratio is known to be partial over the implant surface [with values up to 60 % (Blanco et al. 2011)], while a value of 100 % was taken for the BIC ratio corresponding to a fully osseointegrated implant was considered herein. However, the simple osseointegration model described in Sect. 2.2.3 constitutes an easy way to assess the variation of the implant ultrasonic response as a function of healing time. Third, trabecular and cortical bone mechanical properties are strongly heterogeneous and anisotropic and these properties are known to strongly affect the ultrasonic propagation in bone tissue (Haïat et al. 2009, 2011; Naili et al. 2010). Moreover, scattering effects due to the porous structure of bone tissue are mostly responsible for the dispersive properties of trabecular (Haïat et al. 2008) as well as cortical bone (Sasso et al. 2007, 2008b). However, bone tissue was modeled with homogenized mechanical properties due to the difficulty of accounting for the bone microstructure with sufficient spatial resolution. Moreover, all absorption effects were also neglected in our model. Further studies are necessary to determine the effects of bone heterogeneity and attenuation on the implant ultrasonic response, which still remains unknown.

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The principle of measurement of vibration techniques is different compared to the present technique because it is based on the analysis of the first resonance frequency [between 5 and 15 kHz (Meredith et al. 1996, 1997; Pattijn et al. 2006)] in bending mode of a metallic rod inserted in the implant (Meredith et al. 1996, 1997; Pattijn et al. 2006). Here, the device is based on the propagation of ultrasonic waves within the implant at 10 MHz, so the frequency varies by almost three orders of magnitude. As shown in this paper, the advantage of ultrasound technique is that the results are sensitive to the bone–implant interface, while vibrational techniques investigate the mechanical properties of the entire bone–implant system. However, the added value of ultrasound technique compared to vibrational techniques still remains to be investigated. Note that the aim of the present paper is not to compare vibrational and ultrasound techniques but to better understand ultrasonic wave propagation in the implant in order to improve the ultrasound technique. This study highlights the potentiality of quantitative ultrasound techniques to study dental implant stability because it gives more insight on the ultrasonic propagation in the bone– implant system. Future work should focus on considering a realistic dental implant geometry and developing model accurate model of osseointegration phenomena, thus leading to a more realistic description of the medical device under a biomechanical point of view. Another perspective consists in solving the inverse problem given by the determination of the implant stability based on its ultrasonic response, the present sensitivity study constituting a first step toward its resolution. Acknowledgments This work has been supported by French National Research Agency (ANR) through the PRTS Program (Project OsseoWave ANR-13-PRTS-0015).

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