Influence of viscoelastic and viscous absorption on ultrasonic wave propagation in cortical bone: Application to axial transmission

Unsupervised Learning-Based Measurement of Ultrasonic Axial Transmission Velocity in Neonatal Bone

OBJECTIVES

To develop a robust algorithm for estimating ultrasonic axial transmission velocity from neonatal tibial bone, and to investigate the relationships between ultrasound velocity and neonatal anthropometric measurements as well as clinical biochemical markers of skeletal health.

METHODS

This study presents an unsupervised learning approach for the automatic detection of first arrival time and estimation of ultrasonic velocity from axial transmission waveforms, which potentially indicates bone quality. The proposed method combines the ReliefF algorithm and fuzzy C-means clustering. It was first validated using an in vitro dataset measured from a Sawbones phantom. It was subsequently applied on in vivo signals collected from 40 infants, comprising 21 males and 19 females. The extracted neonatal ultrasonic velocity was subjected to statistical analysis to explore correlations with the infants' anthropometric features and biochemical indicators.

RESULTS

The results of in vivo data analysis revealed significant correlations between the extracted ultrasonic velocity and the neonatal anthropometric measurements and biochemical markers. The velocity of first arrival signals showed good associations with body weight (ρ = 0.583, P value <.001), body length (ρ = 0.583, P value <.001), and gestational age (ρ = 0.557, P value <.001).

CONCLUSION

These findings suggest that fuzzy C-means clustering is highly effective in extracting ultrasonic propagating velocity in bone and reliably applicable in in vivo measurement. This work is a preliminary study that holds promise in advancing the development of a standardized ultrasonic tool for assessing neonatal bone health. Such advancements are crucial in the accurate diagnosis of bone growth disorders.

Ultrasonic Guided Waves in Bone: A Decade of Advancement in Review

The use of guided wave ultrasonography as a means to assess cortical bone quality has been a significant practice in bone quantitative ultrasound for more than 20 years. In this article, the key developments within the technology of ultrasonic guided waves (UGW) in long bones during the past decade are documented. The covered topics include data acquisition configurations available for measuring bone guided waveforms, signal processing techniques applied to bone UGW, numerical modeling of ultrasonic wave propagation in cortical long bones, formulation of inverse approaches to extract bone properties from observed ultrasonic signals, and clinical studies to establish the technology's application and efficacy. The review concludes by highlighting specific challenging problems and future research directions. In general, the primary purpose of this work is to provide a comprehensive overview of bone guided-wave ultrasound, especially for newcomers to this scientific field.

Ex Vivo Assessment of Cortical Bone Properties Using Low-Frequency Ultrasonic Guided Waves

The early diagnosis of osteoporosis through bone quality assessment is a major public health challenge. Research in axial transmission using ultrasonic guided waves has shown the method to be sensitive to the geometrical and mechanical properties of the cortical layer in long bones. However, because of the asymmetric nature of cortical bone, the introduction of a more elaborate numerical model than the analytical plate and cylinder models, as well as its inversion, continues to be challenging. The aim of this article is, therefore, to implement a bone-like geometry using semianalytical finite-element (SAFE) modeling to perform the inverse characterization of ex vivo radii at low frequencies (< 60 kHz). Five cadaveric radiuses were taken from donors aged between 53 and 88 and tested using a typical axial transmission configuration at the middle of the diaphysis. The dispersion curves of the propagating modes were measured experimentally and then systematically compared with the solutions obtained with the SAFE method. For each sample, four parameters were estimated using a parameter identification procedure: 1) the bulk density; 2) the thickness; 3) the outer diameter; and 4) a shape factor (SF). The results showed a moderate agreement between the predicted bulk density and the average voxel value calculated from X-ray computed tomography images. Furthermore, a good agreement was observed between the geometrical parameters (thickness, outer diameter, and SF) and the reference images.

Artificial neural network to estimate micro-architectural properties of cortical bone using ultrasonic attenuation: A 2-D numerical study

The goal of this study is to estimate micro-architectural parameters of cortical porosity such as pore diameter (φ), pore density (ρ) and porosity (ν) of cortical bone from ultrasound frequency dependent attenuation using an artificial neural network (ANN). First, heterogeneous structures with controlled pore diameters and pore densities (mono-disperse) were generated, to mimic simplified structure of cortical bone. Then, more realistic structures were obtained from high resolution CT scans of human cortical bone. 2-D finite-difference time-domain simulations were conducted to calculate the frequency-dependent attenuation in the 1-8 MHz range. An ANN was then trained with the ultrasonic attenuation at different frequencies as the input feature vectors while the output was set as the micro-architectural parameters (pore diameter, pore density and porosity). The ANN is composed of three fully connected dense layers with 24, 12 and 6 neurons, connected to the output layer. The dataset was trained over 6000 epochs with a batch size of 16. The trained ANN exhibits the ability to predict the micro-architectural parameters with high accuracy and low losses. ANN approaches could potentially be used as a tool to help inform physics-based modelling of ultrasound propagation in complex media such as cortical bone. This will lead to the solution of inverse-problems to retrieve bone micro-architectural parameters from ultrasound measurements for the non-invasive diagnosis and monitoring osteoporosis.

Effect of intracortical bone properties on the phase velocity and cut-off frequency of low-frequency guided wave modes (20-85 kHz)

The assessment of intracortical bone properties is of interest since early-stage osteoporosis is associated with resorption in the endosteal region. However, understanding the interaction between ultrasonic guided waves and the cortical bone structure remains challenging. The purpose of this work is to investigate the effect of intracortical bone properties on the ultrasonic response obtained at low-frequency (<100 kHz) using an axial transmission configuration. The semi-analytical finite element method was used to simulate the propagation of guided waves in a waveguide with realistic geometry and material properties. An array of 20 receivers was used to calculate the phase velocity and cut-off frequency of the excited modes using the two-dimensional Fourier transform. The results show that the position of the emitter around the circumference of the bone is an important parameter to control since it can lead to variations of up to 10 dB in the amplitude of the transmitted modes. The cut-off frequency of the high order modes was, however, only slightly affected by the circumferential position of the emitter, and was sensitive mainly to the axial shear modulus. The phase velocity and cut-off frequency in the 20-85 kHz range are promising parameters for the assessment of intracortical properties.

Reflection of an ultrasonic wave on the bone-implant interface: A numerical study of the effect of the multiscale roughness

Quantitative ultrasound is used to characterize and stimulate osseointegration processes at the bone-implant interface (BII). However, the interaction between an ultrasonic wave and the implant remains poorly understood. This study aims at investigating the sensitivity of the ultrasonic response to the microscopic and macroscopic properties of the BII and to osseointegration processes. The reflection coefficient R of the BII was modeled for different frequencies using a two-dimensional finite element model. The implant surface roughness was modeled by a sinusoidal function with varying amplitude h and spatial frequency L. A soft tissue layer of thickness W was considered between bone tissue and the implant in order to model non-mineralized fibrous tissue. For microscopic roughness, R is shown to increase from around 0.55 until 0.9 when kW increases from 0 to 1 and to be constant for kW > 1, where k is the wavenumber in the implant. These results allow us to show that R depends on the properties of bone tissue located at a distance comprised between 1 and 25 μm from the implant surface. For macroscopic roughness, R is highly dependent on h and this dependence may be explained by phase cancellation and multiple scattering effects for high roughness parameters.

Semi-analytical finite-element modeling approach for guided wave assessment of mechanical degradation in bones

Numerical models based on the Semi Analytical Finite-Element method are used to study the characteristics of guided wave modes supported by bone-like multi-layered tubular structures. The method is first validated using previous literature and experimental studies on phantoms mimicking healthy and osteoporotic conditions of cortical bone, and later used to study a trilayer marrow–bone–tissue system at varying mechanical degradation levels. The results show that bone condition strongly affects the modal properties of axially propagating guided waves and indicates that L(0,3) and F(1,6) are suitable modes for assessing the mechanical condition of the bone. The work here reports suitable modal selection and their dispersion properties which would the aid in development of a transduction mechanism for mechanical assessment of bones.

Simulation of acoustic guided wave propagation in cortical bone using a semi-analytical finite element method

Axial transmission techniques have been extensively studied for cortical bone quality assessment. However, the modeling of ultrasonic guided waves propagation in such a complex medium remains challenging. The aim of this paper is to develop a semi-analytical finite element method to simulate the propagation of guided waves in an irregular, multi-layer, and heterogeneous bone cross-section modeled with anisotropic and viscoelastic material properties. The accuracy of the simulations was verified against conventional time-domain three-dimensional finite element. The method was applied in the context of axial transmission in bone to investigate the feasibility of first arrival signal (FAS) to monitor degradation of intracortical properties at low frequency. Different physiopathological conditions for the intracortical region, varying from healthy to osteoporotic, were monitored through FAS velocity using a 10-cycle tone burst excitation centered at 32.5 kHz. The results show that the variation in FAS velocity is mainly associated with four of the eight modes supported by the waveguide, varying with velocity values between 550 and 700 m/s along the different scenarios. Furthermore, the FAS velocity is shown to be associated with the group velocity of the mode with the highest relative amplitude contribution at each studied scenario. However, because of the evolution of the mode with the highest contribution, the FAS velocity is shown to be limited to discriminate intracortical bone properties at low frequency.

Assessment of the biomechanical stability of a dental implant with quantitative ultrasound: A three-dimensional finite element study

Dental implant stability is an important determinant of the surgical success. Quantitative ultrasound (QUS) techniques can be used to assess such properties using the implant acting as a waveguide. However, the interaction between an ultrasonic wave and the implant remains poorly understood. The aim of this study is to investigate the sensitivity of the ultrasonic response to the quality and quantity of bone tissue in contact with the implant surface. The 10 MHz ultrasonic response of an implant used in clinical practice was simulated using an axisymmetric three-dimensional finite element model, which was validated experimentally. The amplitude of the echographic response of the implant increases when the depth of a liquid layer located at the implant interface increases. The results show the sensitivity of the QUS technique to the amount of bone in contact with the implant. The quality of bone tissue around the implant is varied by modifying the bone biomechanical properties by 20%. The amplitude of the implant echographic response decreases when bone quality increases, which corresponds to bone healing. In all cases, the amplitude of the implant response decreased when the dental implant stability increased, which is consistent with the experimental results.

Ultrasound Speed of Sound Measurements in Trabecular Bone Using the Echographic Response of a Metallic Pin

Bone quality is an important parameter in spine surgery, but its clinical assessment remains difficult. The aim of the work described here was to demonstrate in vitro the feasibility of employing quantitative ultrasound to retrieve bone mechanical properties using an echographic technique taking advantage of the presence of a metallic pin inserted in bone tissue. A metallic pin was inserted in bone tissue perpendicular to the transducer axis. The echographic response of the bone sample was determined, and the echo of the pin inserted in bone tissue and water were compared to determine speed of sound, which was compared with bone volume fraction. A 2-D finite-element model was developed to assess the effect of positioning errors. There was a significant correlation between speed of sound and bone volume fraction (R(2) = 0.6). The numerical results indicate the relative robustness of the measurement method, which could be useful to estimate bone quality intra-operatively.

Modeling of transient wave propagation in a heterogeneous solid layer coupled with fluid: application to long bones

The aim of this work is to evaluate the effects of the heterogeneity and anisotropy of material properties of cortical bone on its ultrasonic response obtained by using axial transmission method. The heterogeneity and anisotropy of material properties are introduced by using a parametric probabilistic model. The geometrical configuration of the tested sample is described by a tri-layer medium composed of a heterogeneous and anisotropic solid layer sandwiched between two acoustic fluid layers of which one of these layers is excited by an acoustic linear source. The numerical results focus on studying of an interest quantity, called velocity of the first arriving signal, showing that it strongly depends on the dispersion induced by statistical fluctuations of stochastic elasticity field.

Finite element simulation of ultrasonic wave propagation in a dental implant for biomechanical stability assessment

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.

Propagation of time-reversed Lamb waves in bovine cortical bone in vitro

The present study aims to investigate the propagation of time-reversed Lamb waves in bovine cortical bone in vitro. The time-reversed Lamb waves were successfully launched at 200 kHz in 18 bovine tibiae through a time reversal process of Lamb waves. The group velocities of the time-reversed Lamb waves in the bovine tibiae were measured using the axial transmission technique. They showed a significant correlation with the cortical thickness and tended to follow the theoretical group velocity of the lowest order antisymmetrical Lamb wave fairly well, consistent with the behavior of the slow guided wave in long cortical bones.

Experimental observation of cumulative second-harmonic generation of lamb waves propagating in long bones

The experimental observation of cumulative second-harmonic generation of fundamental Lamb waves in long bones is reported. Based on the modal expansion approach to waveguide excitation and the dispersion characteristics of Lamb waves in long bones, the mechanism underlying the generation and accumulation of second harmonics by propagation of the fundamental Lamb waves was investigated. An experimental setup was established to detect the second-harmonic signals of Lamb wave propagation in long bones in vitro. Through analysis of the group velocities of the received signals, the appropriate fundamental Lamb wave modes and the duration of the second-harmonic signals could be identified. The integrated amplitude of the time-domain second-harmonic signal was introduced and used to characterize the efficiency of second-harmonic generation by fundamental Lamb wave propagation. The results indicate that the second-harmonic signal generated by fundamental Lamb waves propagating in long bones can be observed clearly, and the effect was cumulative with propagation distance when the fundamental Lamb wave mode and the double-frequency Lamb wave mode had the same phase velocities. The present results may be important in the development of a new method to evaluate the status of long bones using the cumulative second harmonic of ultrasonic Lamb waves.

A study on Rayleigh wave dispersion in bone according to Mindlin's Form II gradient elasticity

The classical elasticity cannot effectively describe bone's mechanical behavior since only homogeneous media and local stresses are assumed. Additionally, it cannot predict the dispersive nature of the Rayleigh wave which has been reported in experimental studies and was also demonstrated in a previous computational study by adopting Mindlin's Form II gradient elasticity. In this work Mindlin's theory is employed to analytically determine the dispersion of Rayleigh waves in a strain gradient elastic half-space. An isotropic semi-infinite space is considered with properties equal to those of bone and dynamic behavior suffering from microstructural effects. Microstructural effects are considered by incorporating four intrinsic parameters in the stress analysis. The results are presented in the form of group and phase velocity dispersion curves and compared with existing computational results and semi-analytical curves calculated for a simpler case of Rayleigh waves in dipolar gradient elastic half-spaces. Comparisons are also performed with the velocity of the first-order antisymmetric mode propagating in a dipolar plate so as to observe the Rayleigh asymptotic behavior. It is shown that Mindlin's Form II gradient elasticity can effectively describe the dispersive nature of Rayleigh waves. This study could be regarded as a step toward the ultrasonic characterization of bone.

Inverse problems of ultrasound tomography in models with attenuation

We develop efficient methods for solving inverse problems of ultrasound tomography in models with attenuation. We treat the inverse problem as a coefficient inverse problem for unknown coordinate-dependent functions that characterize both the speed cross section and the coefficients of the wave equation describing attenuation in the diagnosed region. We derive exact formulas for the gradient of the residual functional in models with attenuation, and develop efficient algorithms for minimizing the gradient of the residual by solving the conjugate problem. These algorithms are easy to parallelize when implemented on supercomputers, allowing the computation time to be reduced by a factor of several hundred compared to a PC. The numerical analysis of model problems shows that it is possible to reconstruct not only the speed cross section, but also the properties of the attenuating medium. We investigate the choice of the initial approximation for iterative algorithms used to solve inverse problems. The algorithms considered are primarily meant for the development of ultrasound tomographs for differential diagnosis of breast cancer.

Numerical investigation of ultrasonic attenuation through 2D trabecular bone structures reconstructed from CT scans and random realizations

In this paper, we compare ultrasound interrogations of actual CT-scanned images of trabecular bone with artificial randomly constructed bone. Even though it is known that actual bone does not have randomly distributed trabeculae, we find that the ultrasound attenuations are close enough to cast doubt on any microstructural information, such as trabeculae width and distance between trabeculae, being gleaned from such experiments. More precisely, we perform numerical simulations of ultrasound interrogation on cancellous bone to investigate the phenomenon of ultrasound attenuation as a function of excitation frequency and bone porosity. The theoretical model is based on acoustic propagation equations for a composite fluid-solid material and is solved by a staggered-grid finite-difference scheme in the time domain. Numerical experiments are performed on two-dimensional bone samples reconstructed from CT-scanned images of real human calcaneus and from random distributions of fluid-solid particles generated via the turning bands method. A detailed comparison is performed on various parameters such as the attenuation rate and speed of sound through the bone samples as well as the normalized broadband ultrasound attenuation coefficient. Comparing results from these two types of bone samples allows us to assess the role of bone microstructure in ultrasound attenuation. It is found that the random model provides suitable bone samples for ultrasound interrogation in the transverse direction of the trabecular network.

Radial anatomic variation of ultrasonic velocity in human cortical bone

Quantitative ultrasound techniques can be used to retrieve cortical bone quality. The aim of this study was to investigate the anatomic variations in speed of sound (SOS) in the radial direction of cortical bone tissue. SOS measurements were realized in 17 human cortical bone samples with a 3.5-MHz transverse transmission device. The radial dependence of SOS was investigated in a direction perpendicular to the periosteum. For each sample, bone porosity was measured using an X-ray micro-computed tomography device. The mean SOS was 3586 ± 255 m/s. For 16 of 17 specimens, similar radial variations in SOS were observed. In the periosteal region, SOS first decreased in the direction of the endosteum and reached a minimum value approximately in the middle of the cortical bone. SOS then increased, moving to the endosteal region. A significant negative correlation was obtained between SOS and porosity (R = -0.54, p = 0.02).

Modeling elastic waves in coupled media: Estimate of soft tissue influence and application to quantitative ultrasound

The effect of medium coupling on propagation of elastic waves is a general concern in a variety of engineering and bio-medical applications. Although some theories and analytical models are available for describing waves in multi-layered engineering structures, they do not focus on canvassing ultrasonic waves in human bones with coupled soft tissues, where the considerable differences in acoustic impedance between bone and soft tissue may pose a challenge in using these models (the soft tissues having an acoustic impedance around 80% less than that of a typical bone). Without proper treatment of this coupling effect, the precision of quantitative ultrasound (QUS) for clinical bone assessment can be compromised. The coupling effect of mimicked soft tissues on the first-arriving signal (FAS) and second-arriving signal (SAS) in a series of synthesized soft-tissue-bone phantoms was investigated experimentally and calibrated quantitatively. Understanding of the underlying mechanism of the coupling effect was supplemented by a dedicated finite element analysis. As revealed, the medium coupling impacts influence on different wave modes to different degrees: for FAS and SAS, the most significant changes take place when the soft tissues are initially introduced, and the decrease in signal peak energy continues with increase in the thickness or elastic modulus of the soft tissues, but the changes in propagation velocity fluctuate within 5% regardless of further increase in the thickness or elastic modulus of the soft tissues. As an application, the calibrated effects were employed to enhance the precision of SAS-based QUS when used for predicting the simulated healing status of a mimicked bone fracture, to find prediction of healing progress of bone fracture based on changes in velocity of the FAS or the SAS is inaccurate without taking into account the effect of soft tissue coupling, entailing appropriate compensation for the coupling effect.

Anatomical distribution of the degree of mineralization of bone tissue in human femoral neck: impact on biomechanical properties

Osteoporotic hip fractures represent a major public health problem associated with high human and economic costs. The anatomical variation of the tissue mineral density (TMD) and of the elastic constants in femoral neck cortical bone specimens is an important determinant of bone fragility. The purpose of this study was to show that a Synchrotron radiation microcomputed tomography system coupled with a multiscale biomechanical model allows the determination of the 3-D anatomical dependence of TMD and of the elastic constants (i.e. the mechanical properties of an anisotropic material) in human femoral neck. Bone specimens from the inferior femoral neck were obtained from 18 patients undergoing standard hemiarthroplasty. The specimens were imaged using 3-D synchrotron micro-computed tomography with a voxel size of 10.13 μm, leading to the determination of the anatomical distributions of porosity and TMD. The elastic properties of bone tissue were computed using a multiscale model. The model uses the experimental data obtained at the scale of several micrometers to estimate the components of the elastic tensor of bone at the scale of the organ. Statistical analysis (ANOVA) revealed a significant effect of the radial position on porosity and TMD and a significant effect of axial position on TMD only. Porosity was found to increase in the radial direction moving from the periosteum inwards (p<10(-5)). At any given distance from the periosteum, porosity does not vary noticeably along the bone axis. TMD was found to be significantly higher (p<10(-5)) in the periosteal region than in other bone locations and decreases from the periosteal to the endosteal region with an average slope of 10.05 g.cm(-3).m(-1), the decrease being faster in the porous part of the samples (average slope equal of 30.04 g.cm(-3).m(-1)) than in dense cortical bone. TMD was found to decrease from the distal to the proximal part of the femur neck (average slope of 6.5 g.cm(-3).m(-1)). Considering TMD variations in the radial direction induces weak changes of bone properties compared to constant TMD. TMD variations in the axial direction are responsible for a significant variation of elastic constants. These results demonstrate that the anatomical variations of TMD affect the bone elastic properties, which could be explained by the complex stress field in bone affecting bone remodeling. TMD spatial variations should be taken into account to properly describe the spatial heterogeneity of elastic coefficients of bone tissue at the organ scale.

Analysis of superimposed ultrasonic guided waves in long bones by the joint approximate diagonalization of eigen-matrices algorithm

The parameters of ultrasonic guided waves (GWs) are very sensitive to mechanical and structural changes in long cortical bones. However, it is a challenge to obtain the group velocity and other parameters of GWs because of the presence of mixed multiple modes. This paper proposes a blind identification algorithm using the joint approximate diagonalization of eigen-matrices (JADE) and applies it to the separation of superimposed GWs in long bones. For the simulation case, the velocity of the single mode was calculated after separation. A strong agreement was obtained between the estimated velocity and the theoretical expectation. For the experiments in bovine long bones, by using the calculated velocity and a theoretical model, the cortical thickness (CTh) was obtained. For comparison with the JADE approach, an adaptive Gaussian chirplet time-frequency (ACGTF) method was also used to estimate the CTh. The results showed that the mean error of the CTh acquired by the JADE approach was 4.3%, which was smaller than that of the ACGTF method (13.6%). This suggested that the JADE algorithm may be used to separate the superimposed GWs and that the JADE algorithm could potentially be used to evaluate long bones.

A numerical study on the propagation of Rayleigh and guided waves in cortical bone according to Mindlin's Form II gradient elastic theory

Cortical bone is a multiscale heterogeneous natural material characterized by microstructural effects. Thus guided waves propagating in cortical bone undergo dispersion due to both material microstructure and bone geometry. However, above 0.8 MHz, ultrasound propagates rather as a dispersive surface Rayleigh wave than a dispersive guided wave because at those frequencies, the corresponding wavelengths are smaller than the thickness of cortical bone. Classical elasticity, although it has been largely used for wave propagation modeling in bones, is not able to support dispersion in bulk and Rayleigh waves. This is possible with the use of Mindlin's Form-II gradient elastic theory, which introduces in its equation of motion intrinsic parameters that correlate microstructure with the macrostructure. In this work, the boundary element method in conjunction with the reassigned smoothed pseudo Wigner-Ville transform are employed for the numerical determination of time-frequency diagrams corresponding to the dispersion curves of Rayleigh and guided waves propagating in a cortical bone. A composite material model for the determination of the internal length scale parameters imposed by Mindlin's elastic theory is exploited. The obtained results demonstrate the dispersive nature of Rayleigh wave propagating along the complex structure of bone as well as how microstructure affects guided waves.

Equivalent contributing depth investigated by a lateral wave with axial transmission in viscoelastic cortical bone

Cortical bone is a viscoelastic heterogeneous medium which may be assessed with axial transmission. This work aims at evaluating the average depth investigated by the lateral wave for radial variations of material properties in relatively thick cortical bone. The equivalent contributing depth (ECD) is derived from the finite element simulation results for spatial variations of a viscoelastic coefficient (η(11)) and of porosity. A value of ECD equal to around 1.6 mm is obtained for a spatial variation of η(11). The method fails to predict accurate values of the ECD for a spatial variation of porosity, because all parameters vary simultaneously.

Determination of the heterogeneous anisotropic elastic properties of human femoral bone: from nanoscopic to organ scale

Cortical bone is a multiscale composite material. Its elastic properties are anisotropic and heterogeneous across its cross-section, due to endosteal bone resorption which might affect bone strength. The aim of this paper was to describe a homogenization method leading to the estimation of the variation of the elastic coefficients across the bone cross-section and along the bone longitudinal axis. The method uses the spatial variations of bone porosity and of the degree of mineralization of the bone matrix (DMB) obtained from the analysis of 3-D synchrotron micro-computed tomography images. For all three scales considered (the foam (100 nm), the ultrastructure (5 microm) and the mesoscale (500 microm)), the elastic coefficients were determined using the Eshelby's inclusion problem. DMB values were used at the scale of the foam. Collagen was introduced at the scale of the ultrastructure and bone porosity was introduced at the mesoscale. The pores were considered as parallel cylinders oriented along the bone axis. Each elastic coefficient was computed for different regions of interest, allowing an estimation of its variations across the bone cross-section and along the bone longitudinal axis. The method was applied to a human femoral neck bone specimen, which is a site of osteoporotic fracture. The computed elastic coefficients for cortical bone were in good agreement with experimental results, but some discrepancies were obtained in the endosteal part (trabecular bone). These results highlight the importance of accounting for the heterogeneity of cortical bone properties across bone cross-section and along bone longitudinal axis.