Fast wave ultrasonic propagation in trabecular bone: numerical study of the influence of porosity and structural anisotropy

Effect of collagen fibril orientation on the anisotropic properties of peri-implant bone

In orthopedic and dental surgery, the implantation of biomaterials within the bone to restore the integrity of the treated organ has become a standard procedure. Their long-term stability relies on the osseointegration phenomena, where bone grows onto and around metallic implants, creating a bone-implant interface. Bone is a highly hierarchical material that evolves spatially and temporally during this healing phase. A deeper understanding of its biomechanical characteristics is needed, as they are determinants for surgical success. In this context, we propose a multiscale homogenization model to evaluate the effective elastic properties of bone as a function of the distance from the implant, based on the tissue's structure and composition at lower scales. The model considers three scales: hydroxyapatite foam (nanoscale), ultrastructure (microscale), and tissue (mesoscale). The elastic properties and the volume fraction of the elementary constituents of bone matrix (mineral, collagen, and water), the orientation of the collagen fibril relative to the implant surface, and the mesoscale porosity constitute the input data of the model. The effect of a spatiotemporal variation in the collagen fibrils' orientation on the bone anisotropic properties in the proximity of the implant was investigated. The findings revealed a strong variation of the components of the effective elasticity tensor of the bone as a function of the distance from the implant. The effective elasticity appears to be primarily sensitive to the porosity (mesoscale) rather than to the collagen fibrils' orientation (sub-micro scale). However, the orientation of the fibrils has a significant influence on the isotropy of the bone. When analyzing the symmetry properties of the effective elasticity tensor, the ratio between the isotropic and hexagonal components is determined by a combination of the porosity and the fibrils' orientation. A decrease in porosity leads to a decrease in bone isotropy and, in turn, an increase in the impact of the fibrils' orientation. These results demonstrate that the collagen fibril orientation should be taken into account to properly describe the effective elastic anisotropy of bone at the organ scale.

What kind of waves are measured in trabecular bone?

The paper discusses the fundamental mechanisms underlying the interaction between ultrasound and trabecular bone, which is considered a two-phase material. When fluid-saturated cancellous bone is interrogated by ultrasound, in some cases, one or two wave modes are observed. Many authors claim that these waves correspond to the fast and slow waves predicted by Biot's theory of elastic wave propagation in fluid-saturated porous media. Within our analysis of the physical conditions, predictions of the existing two-phase models of the propagation of ultrasonic waves in the material as well as numerical simulations for fluid-saturated trabecular bone were performed. On the basis of the theoretical results (from numerical studies) and arguments presented in this paper, we aimed to answer the question of whether two waves observed in ultrasonic wave transmission studies can be interpreted as the fast and slow waves predicted by Biot's theory.

Analytical modeling of the interaction of an ultrasonic wave with a rough bone-implant interface

Quantitative ultrasound can be used to characterize the evolution of the bone-implant interface (BII), which is a complex system due to the implant surface roughness and to partial contact between bone and the implant. The determination of the constitutive law of the BII would be of interest in the context of implant acoustical modeling in order to take into account the imperfect characteristics of the BII. The aim of the present study is to propose an analytical effective model describing the interaction between an ultrasonic wave and a rough BII. To do so, a spring model was considered to determine the equivalent stiffness K of the BII. The stiffness contributions related (i) to the partial contact between the bone and the implant and (ii) to the presence of soft tissues at the BII during the process of osseointegration were assessed independently. K was found to be comprised between 10¹³ and 10¹⁷ N/m³ depending on the roughness and osseointegration of the BII. Analytical values of the reflection and transmission coefficients at the BII were derived from values of K. A good agreement with numerical results obtained through finite element simulation was obtained. This model may be used for future finite element bone-implant models to replace the BII conditions.

Evaluation of Structural Anisotropy in a Porous Titanium Medium Mimicking Trabecular Bone Structure Using Mode-Converted Ultrasonic Scattering

The mode-converted (longitudinal to transverse, L-T) ultrasonic scattering method was utilized to characterize the structural anisotropy of a phantom mimicking the structural properties of trabecular bone. The sample was fabricated using metal additive manufacturing from high-resolution computed tomography (CT) images of a sample of trabecular horse bone with strong anisotropy. Two focused transducers were used to perform the L-T ultrasonic measurements. A normal incidence transducer was used to transmit longitudinal ultrasonic waves into the sample, while the scattered transverse signals were received by an oblique incidence transducer. At multiple locations on the sample, four L-T measurements were performed by collecting ultrasonic scattering from four directions. The amplitude of the root mean square (rms) of the collected ultrasonic scattering signals was calculated for each L-T measurement. The ratios of rms amplitudes for L-T measurements in different directions were calculated to characterize the anisotropy of sample. The results show that the amplitude of L-T converted scattering is highly dependent on the direction of microstructural anisotropy. A strong anisotropy of the microstructure was observed, which coincides with simulation results previously published on the same structure as well as with the anisotropy estimated from the CT images. These results suggest the potential of mode-converted ultrasonic scattering methods to assess the anisotropy of materials with porous, complex structures, including trabecular bone.

Reflection of an ultrasonic wave on the bone-implant interface: Effect of the roughness parameters

Quantitative ultrasound can be used to characterize the evolution of the bone-implant interface (BII), which is a complex system due to the implant surface roughness and to partial contact between bone and the implant. The aim of this study is to derive the main determinants of the ultrasonic response of the BII during osseointegration phenomena. The influence of (i) the surface roughness parameters and (ii) the thickness W of a soft tissue layer on the reflection coefficient r of the BII was investigated using a two-dimensional finite element model. When W increases from 0 to 150 μm, r increases from values in the range [0.45; 0.55] to values in the range [0.75; 0.88] according to the roughness parameters. An optimization method was developed to determine the sinusoidal roughness profile leading to the most similar ultrasonic response for all values of W compared to the original profile. The results show that the difference between the ultrasonic responses of the optimal sinusoidal profile and of the original profile was lower to typical experimental errors. This approach provides a better understanding of the ultrasonic response of the BII, which may be used in future numerical simulation realized at the scale of an implant.

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.

Fast decomposition of two ultrasound longitudinal waves in cancellous bone using a phase rotation parameter for bone quality assessment: Simulation study

Ultrasound signals that pass through cancellous bone may be considered to consist of two longitudinal waves, which are called fast and slow waves. Accurate decomposition of these fast and slow waves is considered to be highly beneficial in determination of the characteristics of cancellous bone. In the present study, a fast decomposition method using a wave transfer function with a phase rotation parameter was applied to received signals that have passed through bovine bone specimens with various bone volume to total volume (BV/TV) ratios in a simulation study, where the elastic finite-difference time-domain method is used and the ultrasound wave propagated parallel to the bone axes. The proposed method succeeded to decompose both fast and slow waves accurately; the normalized residual intensity was less than -19.5 dB when the specimen thickness ranged from 4 to 7 mm and the BV/TV value ranged from 0.144 to 0.226. There was a strong relationship between the phase rotation value and the BV/TV value. The ratio of the peak envelope amplitude of the decomposed fast wave to that of the slow wave increased monotonically with increasing BV/TV ratio, indicating the high performance of the proposed method in estimation of the BV/TV value in cancellous bone.

Three-dimensional Simulation of Quantitative Ultrasound in Cancellous Bone Using the Echographic Response of a Metallic Pin

Degenerative discopathy is a common pathology that may require spine surgery. A metallic cylindrical pin is inserted into the vertebral body to maintain soft tissues and may be used as a reflector of ultrasonic wave to estimate bone density. The first aim of this paper is to validate a three-dimensional (3-D) model to simulate the ultrasonic propagation in a trabecular bone sample in which a metallic pin has been inserted. We also aim at determining the effect of changes of bone volume fraction (BV/TV) and of positioning errors on the quantitative ultrasound (QUS) parameters in this specific configuration. The approach consists in coupling finite-difference time-domain simulation with X-ray microcomputed tomography. The correlation coefficient between experimental and simulated speed of sound (SOS)-respectively, broadband ultrasonic attenuation (BUA)-was equal to 0.90 (respectively, 0.55). The results show a significant correlation of SOS with BV/TV ( R = 0.82), while BUA values exhibit a nonlinear behavior versus BV/TV. The orientation of the pin should be controlled with an accuracy of around 1° to obtain accurate results. The results indicate that using the ultrasonic wave reflected by a pin has a potential to estimate the bone density. SOS is more reliable than BUA due to its lower sensitivity to the tilt angle.

Unusually High Calcaneal Speed of Sound Measurements in Children with Small Foot Size

The purpose of this clinical note is to describe the performance of the Lunar Achilles Insight device in assessing bone quality at the calcaneus in 142 children between the ages of 5 and 11 y accessing healthcare in Johannesburg, South Africa. We observed an asymmetric bimodal distribution in speed of sound (SOS). The minor mode consisted of unusually high SOS values (≥1625 m/s), which were primarily observed among children with foot size <19 cm and height <119 cm. Cortical regions of the bone may have been inadvertently included in the region of interest for smaller feet, causing unusually high SOS values. The unusually high SOS values indicate that the validity of SOS in this device, as it is currently used for measuring bone quality in young children, is questionable. Future studies using this device in young children should develop new methodology to account for smaller foot size.

Experimental observation of piezoelectric effect in cancellous bone generated by ultrasound irradiation

In this study, piezoelectric cells (PE-cells) of cancellous bone were experimentally produced to receive an ultrasound wave. In the PE-cell, a bovine cancellous bone specimen, in which the pore spaces were saturated with air, was electrically shielded to prevent electromagnetic noise. As a result, the piezoelectric signal generated in the cancellous bone specimen by irradiating an ultrasound burst wave at 1.0 MHz could be clearly observed in water. The experimental results showed that the ultrasound sensitivity per unit area of cancellous bone was estimated to be below 1/100 and 1/100 000 of cortical bone and poly(vinylidene fluoride), respectively.

Micro-scale finite element modeling of ultrasound propagation in aluminum trabecular bone-mimicking phantoms: A comparison between numerical simulation and experimental results

The present study investigated the accuracy of micro-scale finite element modeling for simulating broadband ultrasound propagation in water-saturated trabecular bone-mimicking phantoms. To this end, five commercially manufactured aluminum foam samples as trabecular bone-mimicking phantoms were utilized for ultrasonic immersion through-transmission experiments. Based on micro-computed tomography images of the same physical samples, three-dimensional high-resolution computational samples were generated to be implemented in the micro-scale finite element models. The finite element models employed the standard Galerkin finite element method (FEM) in time domain to simulate the ultrasonic experiments. The numerical simulations did not include energy dissipative mechanisms of ultrasonic attenuation; however, they expectedly simulated reflection, refraction, scattering, and wave mode conversion. The accuracy of the finite element simulations were evaluated by comparing the simulated ultrasonic attenuation and velocity with the experimental data. The maximum and the average relative errors between the experimental and simulated attenuation coefficients in the frequency range of 0.6-1.4 MHz were 17% and 6% respectively. Moreover, the simulations closely predicted the time-of-flight based velocities and the phase velocities of ultrasound with maximum relative errors of 20 m/s and 11 m/s respectively. The results of this study strongly suggest that micro-scale finite element modeling can effectively simulate broadband ultrasound propagation in water-saturated trabecular bone-mimicking structures.

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.

A generalized recursive convolution method for time-domain propagation in porous media

An efficient numerical method, referred to as the auxiliary differential equation (ADE) method, is proposed to compute convolutions between relaxation functions and acoustic variables arising in sound propagation equations in porous media. For this purpose, the relaxation functions are approximated in the frequency domain by rational functions. The time variation of the convolution is thus governed by first-order differential equations which can be straightforwardly solved. The accuracy of the method is first investigated and compared to that of recursive convolution methods. It is shown that, while recursive convolution methods are first or second-order accurate in time, the ADE method does not introduce any additional error. The ADE method is then applied for outdoor sound propagation using the equations proposed by Wilson et al. in the ground [(2007). Appl. Acoust. 68, 173-200]. A first one-dimensional case is performed showing that only five poles are necessary to accurately approximate the relaxation functions for typical applications. Finally, the ADE method is used to compute sound propagation in a three-dimensional geometry over an absorbing ground. Results obtained with Wilson's equations are compared to those obtained with Zwikker and Kosten's equations and with an impedance surface for different flow resistivities.

Ultrasonic wave properties of human bone marrow in the femur and tibia

Ultrasonic wave properties of human bone marrow obtained in the femur and tibia were measured using an ultrasound pulse technique. The measured frequency range was 4-10 MHz, and the temperature range was 30 °C-40 °C. The sound velocity was 1410 m/s, and the attenuation coefficient was 4.4 dB/cm at 36 °C (10 MHz). These values decreased with temperature. Site dependence and individual differences in elderly human bone marrow were negligible. The slopes of the attenuation coefficient were estimated by a power law. The values of the exponent n were 2.0 (30 °C-38 °C) and 2.3 (40 °C).

Numerical Analysis of Ultrasound Backscattered Waves in Cancellous Bone Using a Finite-Difference Time-Domain Method: Isolation of the Backscattered Waves From Various Ranges of Bone Depths

Using a finite-difference time-domain method, ultrasound backscattered waves inside cancellous bone were numerically analyzed to investigate the backscatter mechanism. Two bone models with different thicknesses were modeled with artificial absorbing layers positioned at the back surfaces of the model, and an ultrasound pulse wave was transmitted toward the front surface. By calculating the difference between the simulated waveforms obtained using the two bone models, the backscattered waves from a limited range of depths in cancellous bone could be isolated. The results showed that the fast and slow longitudinal waves, which have previously been observed only in the ultrasound waveform transmitted through the bone, could be distinguished in the backscattered waveform from a deeper bone depth when transmitting the ultrasound wave parallel to the main orientation of the trabecular network. The amplitudes of the fast and slow backscattered waves were more closely correlated with the bone porosity [R2 = 0.84 and 0.66 (p < 0.001), respectively] than the amplitude of the whole (nonisolated) backscattered waves [R2 = 0.48 (p < 0.001)]. In conclusion, the nonisolated backscattered waves could be regarded as the superposition of the fast and slow waves reflected from various bone depths, returning at different times.

Nonlinear attenuation and dispersion in human calcaneus in vitro: statistical validation and relationships to microarchitecture

Through-transmission measurements were performed on 30 human calcaneus samples in vitro. Nonlinear attenuation and dispersion measurements were investigated by estimating 95% confidence intervals of coefficients of polynomial expansions of log magnitude and phase of transmission coefficients. Bone mineral density (BMD) was measured with dual x-ray absorptiometry. Microarchitecture was measured with microcomputed tomography. Statistically significant nonlinear attenuation and nonzero dispersion were confirmed for a clinical bandwidth of 300-750 kHz in 40%-43% of bone samples. The mean linear coefficient for attenuation was 10.3 dB/cm MHz [95% confidence interval (CI): 9.0-11.6 dB/cm MHz]. The mean quadratic coefficient for attenuation was 1.6 dB/cm MHz(2) (95% CI: 0.4-2.8 dB/cm MHz(2)). Nonlinear attenuation provided little information regarding BMD or microarchitecture. The quadratic coefficient for phase (which is related to dispersion) showed moderate correlations with BMD (r = -0.65; 95% CI: -0.82 to -0.36), bone surface-to-volume ratio (r = 0.47; 95% CI: 0.12-0.72) and trabecular thickness (r = -0.40; 95% CI: -0.67 to -0.03). Dispersion was proportional to bone volume fraction raised to an exponent of 2.1 ± 0.2, which is similar to the value for parallel nylon-wire phantoms (2.4 ± 0.2) and supports a multiple-scattering model for dispersion.

Fast and slow wave detection in bovine cancellous bone in vitro using bandlimited deconvolution and Prony's method

Fast and slow waves were detected in a bovine cancellous bone sample for thicknesses ranging from 7 to 12 mm using bandlimited deconvolution and the modified least-squares Prony's method with curve fitting (MLSP + CF). Bandlimited deconvolution consistently isolated two waves with linear-with-frequency attenuation coefficients as evidenced by high correlation coefficients between attenuation coefficient and frequency: 0.997 ± 0.002 (fast wave) and 0.986 ± 0.013 (slow wave) (mean ± standard deviation). Average root-mean-squared (RMS) differences between the two algorithms for phase velocities were 5 m/s (fast wave, 350 kHz) and 13 m/s (slow wave, 750 kHz). Average RMS differences for signal loss were 1.6 dB (fast wave, 350 kHz) and 0.4 dB (slow wave, 750 kHz). Phase velocities for thickness = 10 mm were 1726 m/s (fast wave, 350 kHz) and 1455 m/s (slow wave, 750 kHz). Results show support for the model of two waves with linear-with frequency attenuation, successful isolation of fast and slow waves, good agreement between bandlimited deconvolution and MLSP + CF as well as with a Bayesian algorithm, and potential variations of fast and/or slow wave properties with bone sample thickness.

The finite element method for micro-scale modeling of ultrasound propagation in cancellous bone

Quantitative ultrasound for bone assessment is based on the correlations between ultrasonic parameters and the properties (mechanical and physical) of cancellous bone. To elucidate the correlations, understanding the physics of ultrasound in cancellous bone is demanded. Micro-scale modeling of ultrasound propagation in cancellous bone using the finite-difference time-domain (FDTD) method has been so far utilized as one of the approaches in this regard. However, the FDTD method accompanies two disadvantages: staircase sampling of cancellous bone by finite difference grids leads to generation of wave artifacts at the solid-fluid interface inside the bone; additionally, this method cannot explicitly satisfy the needed perfect-slip conditions at the interface. To overcome these disadvantages, the finite element method (FEM) is proposed in this study. Three-dimensional finite element models of six water-saturated cancellous bone samples with different bone volume were created. The values of speed of sound (SOS) and broadband ultrasound attenuation (BUA) were calculated through the finite element simulations of ultrasound propagation in each sample. Comparing the results with other experimental and simulation studies demonstrated the capabilities of the FEM for micro-scale modeling of ultrasound in water-saturated cancellous bone.

Two-wave behavior under various conditions of transition area from cancellous bone to cortical bone

The two-wave phenomenon, the wave separation of a single ultrasonic pulse in cancellous bone, is expected to be a useful tool for the diagnosis of osteoporosis. However, because actual bone has a complicated structure, precise studies on the effect of transition conditions between cortical and cancellous parts are required. This study investigated how the transition condition influenced the two-wave generation using three-dimensional X-ray CT images of an equine radius and a three-dimensional simulation technique. As a result, any changes in the boundary between cortical part and trabecular part, which gives the actual complex structure of bone, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. The results led us to the possibility of using the two-wave phenomenon in a diagnostic system for osteoporosis in cases of a complex boundary.

Measurements of ultrasound velocity and attenuation in numerical anisotropic porous media compared to Biot's and multiple scattering models

This article quantitatively investigates ultrasound propagation in numerical anisotropic porous media with finite-difference simulations in 3D. The propagation media consist of clusters of ellipsoidal scatterers randomly distributed in water, mimicking the anisotropic structure of cancellous bone. Velocities and attenuation coefficients of the ensemble-averaged transmitted wave (also known as the coherent wave) are measured in various configurations. As in real cancellous bone, one or two longitudinal modes emerge, depending on the micro-structure. The results are confronted with two standard theoretical approaches: Biot's theory, usually invoked in porous media, and the Independent Scattering Approximation (ISA), a classical first-order approach of multiple scattering theory. On the one hand, when only one longitudinal wave is observed, it is found that at porosities higher than 90% the ISA successfully predicts the attenuation coefficient (unlike Biot's theory), as well as the existence of negative dispersion. On the other hand, the ISA is not well suited to study two-wave propagation, unlike Biot's model, at least as far as wave speeds are concerned. No free fitting parameters were used for the application of Biot's theory. Finally we investigate the phase-shift between waves in the fluid and the solid structure, and compare them to Biot's predictions of in-phase and out-of-phase motions.

Time-domain separation of interfering waves in cancellous bone using bandlimited deconvolution: simulation and phantom study

In through-transmission interrogation of cancellous bone, two longitudinal pulses ("fast" and "slow" waves) may be generated. Fast and slow wave properties convey information about material and micro-architectural characteristics of bone. However, these properties can be difficult to assess when fast and slow wave pulses overlap in time and frequency domains. In this paper, two methods are applied to decompose signals into fast and slow waves: bandlimited deconvolution and modified least-squares Prony's method with curve-fitting (MLSP + CF). The methods were tested in plastic and Zerdine(®) samples that provided fast and slow wave velocities commensurate with velocities for cancellous bone. Phase velocity estimates were accurate to within 6 m/s (0.4%) (slow wave with both methods and fast wave with MLSP + CF) and 26 m/s (1.2%) (fast wave with bandlimited deconvolution). Midband signal loss estimates were accurate to within 0.2 dB (1.7%) (fast wave with both methods), and 1.0 dB (3.7%) (slow wave with both methods). Similar accuracies were found for simulations based on fast and slow wave parameter values published for cancellous bone. These methods provide sufficient accuracy and precision for many applications in cancellous bone such that experimental error is likely to be a greater limiting factor than estimation error.

Multichannel instantaneous frequency analysis of ultrasound propagating in cancellous bone

An ultrasonic pulse propagating in cancellous bone can be separated into two waves depending on the condition of the specimen. These two waves, which are called the fast wave and the slow wave, provide important information for the diagnosis of osteoporosis. The present study proposes to utilize a signal processing method that extracts the instantaneous frequency (IF) of waveforms from multiple spectral channels. The instantaneous frequency was expected to be able to show detailed time-frequency properties of ultrasonic waves being transmitted through cancellous bone. The employed method, termed the multichannel instantaneous frequency (MCIF) method, showed robustness against background noise as compared to the IF that was directly derived from the original waveform. The extracted IF revealed that the frequency of the fast wave was affected by both the propagation distance within the specimen and the bone density, independently. On the other hand, the alternation of the center frequency of the originally transmitted wave did not produce proportional changes in the extracted IF values of the fast waves, suggesting that the fast wave IF mainly reflected the thickness of the specimens. These findings may provide the possibility of obtaining a more precise diagnosis of osteoporosis.

An experimental study on the ultrasonic wave propagation in cancellous bone: waveform changes during propagation

Wave propagation in a trabecular bone was experimentally investigated using an acoustic tube. For the purposes of this study, a cubic sample was gradually filed so the waveform change due to the sample thickness could be observed. The initial sample showed clear two-wave separation. As the sample became thinner, the fast and slow waves gradually overlapped. The apparent frequencies and amplitudes of the fast waves obtained from the time domain data decreased significantly for the smaller thicknesses. This indicates an increase in the apparent attenuation at the initial stage of the propagation. Next the authors investigated the distribution of the ultrasonic field after the transmission through the cancellous bone sample. In addition to a large aperture receiver, a needle-type ultrasonic transducer was used to observe the ultrasonic field. Within an area of the same size of the large transducer, the waveforms retrieved with the needle sensor exhibited high spatial variations; however, the averaged waveform in the plane was similar to the waveform obtained with the large aperture receiver. This indicates that the phase cancellation effect on the surface of the large aperture receiver can be one of the reasons for the strong apparent attenuation observed at the initial stages of the propagation.

Estimation of fast and slow wave properties in cancellous bone using Prony's method and curve fitting

The presence of two longitudinal waves in poroelastic media is predicted by Biot's theory and has been confirmed experimentally in through-transmission measurements in cancellous bone. Estimation of attenuation coefficients and velocities of the two waves is challenging when the two waves overlap in time. The modified least squares Prony's (MLSP) method in conjuction with curve-fitting (MLSP + CF) is tested using simulations based on published values for fast and slow wave attenuation coefficients and velocities in cancellous bone from several studies in bovine femur, human femur, and human calcaneus. The search algorithm is accelerated by exploiting correlations among search parameters. The performance of the algorithm is evaluated as a function of signal-to-noise ratio (SNR). For a typical experimental SNR (40 dB), the root-mean-square errors (RMSEs) for one example (human femur) with fast and slow waves separated by approximately half of a pulse duration were 1 m/s (slow wave velocity), 4 m/s (fast wave velocity), 0.4 dB/cm MHz (slow wave attenuation slope), and 1.7 dB/cm MHz (fast wave attenuation slope). The MLSP + CF method is fast (requiring less than 2 s at SNR = 40 dB on a consumer-grade notebook computer) and is flexible with respect to the functional form of the parametric model for the transmission coefficient. The MLSP + CF method provides sufficient accuracy and precision for many applications such that experimental error is a greater limiting factor than estimation error.

NUMERICAL SIMULATIONS OF CHANGE IN TRABECULAR STRUCTURE DUE TO BONE REMODELING UNDER ULTRASOUND PROPAGATION

Bone remodeling is defined as the coupling of bone formation and resorption on the bone surface. Numerical simulations of the remodeling in cancellous bone were performed to reproduce the change in the trabecular structure. Assuming that the formation/resorption in cancellous bone could be generated on the trabecular surface, where the local stress under the mechanical load was larger/smaller than the averaged stress on the surrounding surface, voxel trabecular elements in a numerical model of bovine cancellous bone were added/removed. An ultrasound continuous wave in the frequency range 0.1–1.0 MHz was applied as the mechanical load, and then, the local stress was analyzed using a finite-difference time-domain (FDTD) method. Using the remodeling simulations, both changes in the trabecular structure could be reproduced with decreasing and increasing porosity. In changes, the trabecular elements and the pore spaces became strongly oriented in the direction of ultrasound propagation. In addition, the remodeling simulations indicated that both bone formation and resorption lessened as the frequency increased.

Simulations of ultrasound propagation in random arrangements of elliptic scatterers: occurrence of two longitudinal waves

Ultrasound propagation in clusters of elliptic (two-dimensional) or ellipsoidal (three-dimensional) scatterers randomly distributed in a fluid is investigated numerically. The essential motivation for the present work is to gain a better understanding of ultrasound propagation in trabecular bone. Bone microstructure exhibits structural anisotropy and multiple wave scattering. Some phenomena remain partially unexplained, such as the propagation of two longitudinal waves. The objective of this study was to shed more light on the occurrence of these two waves, using finite-difference simulations on a model medium simpler than bone. Slabs of anisotropic, scattering media were randomly generated. The coherent wave was obtained through spatial and ensemble-averaging of the transmitted wavefields. When varying relevant medium parameters, four of them appeared to play a significant role for the observation of two waves: (i) the solid fraction, (ii) the direction of propagation relatively to the scatterers orientation, (iii) the ability of scatterers to support shear waves, and (iv) a continuity of the solid matrix along the propagation. These observations are consistent with the hypothesis that fast waves are guided by the locally plate/bar-like solid matrix. If confirmed, this interpretation could significantly help developing approaches for a better understanding of trabecular bone micro-architecture using ultrasound.

Cancellous bone fast and slow waves obtained with Bayesian probability theory correlate with porosity from computed tomography

A Bayesian probability theory approach for separating overlapping ultrasonic fast and slow waves in cancellous bone has been previously introduced. The goals of this study were to investigate whether the fast and slow waves obtained from Bayesian separation of an apparently single mode signal individually correlate with porosity and to isolate the fast and slow waves from medial-lateral insonification of the calcaneus. The Bayesian technique was applied to trabecular bone data from eight human calcanei insonified in the medial-lateral direction. The phase velocity, slope of attenuation (nBUA), and amplitude were determined for both the fast and slow waves. The porosity was assessed by micro-computed tomography (microCT) and ranged from 78.7% to 94.1%. The method successfully separated the fast and slow waves from medial-lateral insonification of the calcaneus. The phase velocity for both the fast and slow wave modes showed an inverse correlation with porosity (R(2) = 0.73 and R(2) = 0.86, respectively). The slope of attenuation for both wave modes also had a negative correlation with porosity (fast wave: R(2) = 0.73, slow wave: R(2) = 0.53). The fast wave amplitude decreased with increasing porosity (R(2) = 0.66). Conversely, the slow wave amplitude modestly increased with increasing porosity (R(2) = 0.39).

Comparison of phase velocity in trabecular bone mimicking-phantoms by time domain numerical (EFIT) and analytical multiple scattering approaches

The corrected Waterman-Truell model and the Elastodynamic Finite Integration Technique were used to analyze the ultrasonic wave dispersion in trabecular bones mimicking phantoms. A simple two-phase model of the trabecular bone is assumed; the trabeculae structure and the bone marrow. The phase velocity for frequencies within the range from 400kHz to 800kHz were computed for different scatterer arrays varying their dimensions and number. The theoretical and numerical results were compared to experimental published data, obtained from a mimicking phantom composed by a periodic array of nylon shreds (trabeculae array) immersed in a water tank. Our results showed an excellent consistency when compared to experimental data. The negative dispersions of -8.48m/s/MHz and -9.16m/s/MHz were computed by the multiple scattering method and the numerical approach, respectively, where the latter is closer to the experimental dispersion of -12.09m/s/MHz. Similar result has been reported in the literature, where the dispersion predicted by the Generalized Self-Consistent Method [J. Acoust. Soc. Am. 124 (2008) 4047] is -9.96m/s/MHz.

In vitro estimation of fast and slow wave parameters of thin trabecular bone using space-alternating generalized expectation-maximization algorithm

In testing cancellous bone using ultrasound, two types of longitudinal Biot's waves are observed in the received signal. These are known as fast and slow waves and their appearance depend on the alignment of bone trabeculae in the propagation path and the thickness of the specimen under test (SUT). They can be used as an effective tool for the diagnosis of osteoporosis because wave propagation behavior depends on the bone structure. However, the identification of these waves in the received signal can be difficult to achieve. In this study, ultrasonic wave propagation in a 4mm thick bovine cancellous bone in the direction parallel to the trabecular alignment is considered. The observed Biot's fast and slow longitudinal waves are superimposed; which makes it difficult to extract any information from the received signal. These two waves can be separated using the space alternating generalized expectation maximization (SAGE) algorithm. The latter has been used mainly in speech processing. In this new approach, parameters such as, arrival time, center frequency, bandwidth, amplitude, phase and velocity of each wave are estimated. The B-Scan images and its associated A-scans obtained through simulations using Biot's finite-difference time-domain (FDTD) method are validated experimentally using a thin bone sample obtained from the femoral-head of a 30 months old bovine.

Investigation of the influence of reflection on the attenuation of cancellous bone

The model proposed in this paper is based on the fact that the reflection might have a significant contribution to the attenuation of the acoustic waves propagating through the cancellous bone. The numerical implementation of the mentioned effect is realized by the development of a new representative volume element that includes an infinitesimally thin 'transient' layer on the contact surface of the bone and the marrow. This layer serves to model the amplitude transformation of the incident wave by the transition through media with different acoustic impedances and to take into account the energy loss due to the reflection. The proposed representative volume element together with the multiscale finite element is used to simulate the wave propagation and to evaluate the attenuation coefficient for samples with different effective densities in the dependence of the applied excitation frequency. The obtained numerical values show a very good agreement with the experimental results. Moreover, the model enables the determination of the upper and the lower bound for the attenuation coefficient.

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.

Role of structural anisotropy of biological tissues in poroelastic wave propagation

Ultrasound waves have a broad range of clinical applications as a non-destructive testing approach in imaging and in the diagnoses of medical conditions. Generally, biological tissues are modeled as an homogenized equivalent medium with an apparent density through which a single wave propagates. Only the first wave arriving at the ultrasound probe is used for the measurement of the speed of sound. However, the existence of a second wave in tissues such as cancellous bone has been reported and its existence is an unequivocal signature of Biot type poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as density, a fabric-dependent anisotropic poroelastic ultrasound (PEU) propagation theory was recently developed. Key to this development was the inclusion of the fabric tensor - a quantitative stereological measure of the degree of structural anisotropy of bone - into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of waves in several soft and hard tissues. It was found that collagen fibers in soft tissues and the mineralized matrix in hard tissues are responsible for the anisotropy of the solid tissue constituent through the fabric tensor in the model.

Properties of ultrasonic waves in bovine bone marrow

We investigated the properties of ultrasonic waves in bovine bone marrow. Six bone marrow samples were obtained from different parts (proximal, middle and distal) of the shafts of two bovine femora without destruction. The measured frequency range was 3 to 10 MHz, and the temperature range was 15 to 40°C. Both wave velocity and attenuation coefficient in bone marrow always decreased as temperature increased. The velocity ranged from 1400 to 1610 m/s and attenuation coefficient ranged from 4 to 16 dB/cm. Wave velocities in bone marrow were similar to those in water, whereas the temperature dependences were different, and the wave attenuation coefficients were much higher than those in water. The dependence of velocity on temperature changed slightly around 23-24°C, where a transition from soft gel to oily liquid occurred. The transition temperature was confirmed by differential scanning calorimetry (DSC). Below this transition temperature, positive velocity dispersion was observed.

Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone

Previous studies have shown that interference between fast waves and slow waves can lead to observed negative dispersion in cancellous bone. In this study, the effects of overlapping fast and slow waves on measurements of the apparent attenuation as a function of propagation distance are investigated along with methods of analysis used to determine the attenuation properties. Two methods are applied to simulated data that were generated based on experimentally acquired signals taken from a bovine specimen. The first method uses a time-domain approach that was dictated by constraints imposed by the partial overlap of fast and slow waves. The second method uses a frequency-domain log-spectral subtraction technique on the separated fast and slow waves. Applying the time-domain analysis to the broadband data yields apparent attenuation behavior that is larger in the early stages of propagation and decreases as the wave travels deeper. In contrast, performing frequency-domain analysis on the separated fast waves and slow waves results in attenuation coefficients that are independent of propagation distance. Results suggest that features arising from the analysis of overlapping two-mode data may represent an alternate explanation for the previously reported apparent dependence on propagation distance of the attenuation coefficient of cancellous bone.

Propagation of two longitudinal waves in a cancellous bone with the closed pore boundary

Ultrasound propagation in cancellous bone (porous media) under the condition of closed pore boundaries was investigated. A cancellous bone and two plate-like cortical bones obtained from a racehorse were prepared. A water-immersion ultrasound technique in the MHz range and a three-dimensional elastic finite-difference time-domain (FDTD) method were used to investigate the waves. The experiments and simulations showed a clear separation of the incident longitudinal wave into fast and slow waves. The findings advance the evaluation of bones based on the two-wave phenomenon for in vivo assessment.

Numerical investigation of ultrasound refraction caused by oblique orientation of trabecular network in cancellous bone

Ultrasound propagation through cancellous bone can be greatly affected by the trabecular structure. In the present study, the ultrasound propagation for the oblique orientation of the trabecular network was numerically investigated using 3-D finite-difference time-domain (FDTD) simulations. The models of cancellous bone were reconstructed from X-ray microcomputed tomographic (μCT) images of a bovine bone. Cancellous bone models with various orientations of the trabecular network were realized by cutting the μCT images rotated from 0 to 90°. Ultrasound waveforms propagating through these cancellous bone models were simulated while changing the receiving position. The refraction of the ultrasound wave for the oblique angle of the main orientation was investigated on the basis of the variation in the arrival time and peak amplitude. As the propagation direction approached the direction parallel to the main orientation, the arrival time of the first peak became less and the peak amplitude became smaller. This means that the wave of the first peak, which corresponded to a fast wave, propagated in the direction perpendicular to the main orientation. In addition, a strong correlation between the first-peak amplitude and the arrival time was observed in the porosity range of 0.68 to 0.85, in which the slope of the amplitude with respect to time increased linearly with porosity.

Numerical simulation of ultrasonic wave propagation for the evaluation of dental implant biomechanical stability

Osseointegration of dental implants remains poorly understood. The objective of this numerical study is to understand the propagation phenomena of ultrasonic waves in prototypes cylindrically shaped implants and to investigate the sensitivity of their ultrasonic response to the surrounding bone biomechanical properties. The 10 MHz ultrasonic response of the implant was calculated using a finite difference numerical simulation tool and was compared to rf signals taken from a recent experimental study by Mathieu et al. [Ultrasound Med. Biol. 37, 262-270 (2011a)]. Reflection and mode conversion phenomena were analyzed to understand the origin of the different echoes and the importance of lateral wave propagation was evidenced. The sensitivity of the ultrasonic response of the implant to changes of (i) amount of bone in contact with the implant, (ii) cortical bone thickness, and (iii) surrounding bone material properties, was compared to the reproducibility of the measurements. The results show that, either a change of 1 mm of bone in contact with the implant, or 1.1 mm of cortical thickness or 12% of trabecular bone mass density should be detectable. This study paves the way for the investigation of the use of quantitative ultrasound techniques for the evaluation of bone-implant interface properties and implant stability.

Measurements of ultrasonic phase velocities and attenuation of slow waves in cellular aluminum foams as cancellous bone-mimicking phantoms

The water-saturated aluminum foams with an open network of interconnected ligaments were investigated by ultrasonic transmission technique for the suitability as cancellous bone-mimicking phantoms. The phase velocities and attenuation of nine samples covering three pores per inch (5, 10, and 20 PPI) and three aluminum volume fractions (5, 8, and 12% AVF) were measured over a frequency range of 0.7-1.3 MHz. The ligament thickness and pore sizes of the phantoms and low-density human cancellous bones are similar. A strong slow wave and a weak fast wave are observed for all samples while the latter is not visible without significant amplification (100x). This study reports the characteristics of slow wave, whose speeds are less than the sound speed of the saturating water and decrease mildly with AVF and PPI with an average 1469 m/s. Seven out of nine samples show positive dispersion and the rest show minor negative dispersion. Attenuation increases with AVF, PPI, and frequency except for the 20 PPI samples, which exhibit non-increasing attenuation level with fluctuations due to scattering. The phase velocities agree with Biot's porous medium theory. The RMSE is 16.0 m/s (1%) at n = 1.5. Below and above this value, the RMSE decreases mildly and rises sharply, respectively.

Inverse problems in cancellous bone: estimation of the ultrasonic properties of fast and slow waves using Bayesian probability theory

Quantitative ultrasonic characterization of cancellous bone can be complicated by artifacts introduced by analyzing acquired data consisting of two propagating waves (a fast wave and a slow wave) as if only one wave were present. Recovering the ultrasonic properties of overlapping fast and slow waves could therefore lead to enhancement of bone quality assessment. The current study uses Bayesian probability theory to estimate phase velocity and normalized broadband ultrasonic attenuation (nBUA) parameters in a model of fast and slow wave propagation. Calculations are carried out using Markov chain Monte Carlo with simulated annealing to approximate the marginal posterior probability densities for parameters in the model. The technique is applied to simulated data, to data acquired on two phantoms capable of generating two waves in acquired signals, and to data acquired on a human femur condyle specimen. The models are in good agreement with both the simulated and experimental data, and the values of the estimated ultrasonic parameters fall within expected ranges.

Influence of cancellous bone microstructure on two ultrasonic wave propagations in bovine femur: an in vitro study

The influence of cancellous bone microstructure on the ultrasonic wave propagation of fast and slow waves was experimentally investigated. Four spherical cancellous bone specimens extracted from two bovine femora were prepared for the estimation of acoustical and structural anisotropies of cancellous bone. In vitro measurements were performed using a PVDF transducer (excited by a single sinusoidal wave at 1 MHz) by rotating the spherical specimens. In addition, the mean intercept length (MIL) and bone volume fraction (BV/TV) were estimated by X-ray micro-computed tomography. Separation of the fast and slow waves was clearly observed in two specimens. The fast wave speed was strongly dependent on the wave propagation direction, with the maximum speed along the main trabecular direction. The fast wave speed increased with the MIL. The slow wave speed, however, was almost constant. The fast wave speeds were statistically higher, and their amplitudes were statistically lower in the case of wave separation than in that of wave overlap.

Cancellous bone analysis with modified least squares Prony's method and chirp filter: phantom experiments and simulation

The presence of two longitudinal waves in porous media is predicted by Biot's theory and has been confirmed experimentally in cancellous bone. When cancellous bone samples are interrogated in through-transmission, these two waves can overlap in time. Previously, the Modified Least-Squares Prony's (MLSP) method was validated for estimation of amplitudes, attenuation coefficients, and phase velocities of fast and slow waves, but tended to overestimate phase velocities by up to about 5%. In the present paper, a pre-processing chirp filter to mitigate the phase velocity bias is derived. The MLSP/chirp filter (MLSPCF) method was tested for decomposition of a 500 kHz-center-frequency signal containing two overlapping components: one passing through a low-density-polyethylene plate (fast wave) and another passing through a cancellous-bone-mimicking phantom material (slow wave). The chirp filter reduced phase velocity bias from 100 m/s (5.1%) to 69 m/s (3.5%) (fast wave) and from 29 m/s (1.9%) to 10 m/s (0.7%) (slow wave). Similar improvements were found for 1) measurements in polycarbonate (fast wave) and a cancellous-bone-mimicking phantom (slow wave), and 2) a simulation based on parameters mimicking bovine cancellous bone. The MLSPCF method did not offer consistent improvement in estimates of attenuation coefficient or amplitude.

Effect of porosity distribution in the propagation direction on ultrasound waves through cancellous bone

Cancellous bone is a porous material composed of numerous trabecular elements, and its porosity changes according to its position within a bone. In this study, the effect of porosity distribution in the propagation direction on ultrasound waves through cancellous bone was numerically investigated using finite-difference time-domain (FDTD) simulations. Fifty four numerical models of cancellous bone were reconstructed from 3-D X-ray microcomputed tomographic (microCT) images at 6 positions in a bovine femoral bone. To generate trabecular structures with distinct porosity distributions, 3 erosion procedures were performed in which the trabecular elements in each cancellous bone model were eroded. In one procedure, erosion was uniformly distributed over the whole spatial region of the cancellous bone model, but in the other 2 procedures, the spatial distribution of erosion was changed in a specific direction. Fast and slow waves propagating through the 3-D microCT cancellous bone models in the porosity-distributed direction were simulated using the viscoelastic FDTD method. The wave amplitudes and propagation speeds of the fast and slow waves were measured for the cancellous bone models eroded by each procedure, and the effect of porosity distribution was investigated in terms of change in the trabecular microstructure. The results suggest that both wave amplitudes increased when porosity distribution was low and when trabecular structure was more uniform, but that the speed of the fast wave increased when porosity distribution was high and when longer trabecular elements were present.

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.

Decomposition of two-component ultrasound pulses in cancellous bone using modified least squares prony method--phantom experiment and simulation

Porous media such as cancellous bone often support the simultaneous propagation of two compressional waves. When small bone samples are interrogated in through-transmission with broadband sources, these two waves often overlap in time. The modified least-squares Prony's (MLSP) method was tested for decomposing a 500 kHz-center-frequency signal containing two overlapping components: one passing through a polycarbonate plate (to produce the "fast" wave) and another passing through a cancellous-bone-mimicking phantom (to produce the "slow" wave). The MLSP method yielded estimates of attenuation slopes accurate to within 7% (polycarbonate plate) and 2% (cancellous bone phantom). The MLSP method yielded estimates of phase velocities accurate to within 1.5% (both media). The MLSP method was also tested on simulated data generated using attenuation slopes and phase velocities corresponding to bovine cancellous bone. Throughout broad ranges of signal-to-noise ratio (SNR), the MLSP method yielded estimates of attenuation slope that were accurate to within 1.0% and estimates of phase velocity that were accurate to within 4.3% (fast wave) and 1.3% (slow wave).

Frequency dependence of average phase shift from human calcaneus in vitro

If dispersion in a medium is weak and approximately linear with frequency (over the experimental band of frequencies), then it can be shown that the constant term in a polynomial representation of phase shift as a function of frequency can produce errors in measurements of phase-velocity differences in through-transmission, substitution experiments. A method for suppressing the effects of the constant phase shift in the context of the single-wave-model was tested on measurements from 30 cancellous human calcaneus samples in vitro. Without adjustment for constant phase shifts, the estimated phase velocity at 500 kHz was 1516+/-6 m/s (mean+/-standard error), and the estimated dispersion was -24+/-4 m/s MHz (mean+/-standard error). With adjustment for constant phase shifts, the estimated mean velocity decreased by 4-9 m/s, and the estimated magnitude of mean dispersion decreased by 50%-100%. The average correlation coefficient between the measured attenuation coefficient and frequency was 0.997+/-0.0026 (mean+/-standard deviation), suggesting that the signal for each sample was dominated by one wave. A single-wave, linearly dispersive model conformed to measured complex transfer functions from the 30 cancellous-bone samples with an average root-mean-square error of 1.9%+/-1.0%.

Influence of the filling fluid on frequency-dependent velocity and attenuation in cancellous bones between 0.35 and 2.5 MHz

The paper is focused on experiments on human cancellous bones filled with different fluids with the goal of evaluating their contribution to velocity dispersion, absorption, and scattering mechanisms. The specimens were measured first filled with marrow and subsequently, after marrow removal, with water and alcohol. No significant influence of the fluids was evidenced on the attenuation coefficient. Given the absence of impact of viscosity of the saturating fluid, the authors hypothesized that the source of attenuation is associated with viscoelastic absorption in the solid trabeculae and with scattering. Alteration of scattering obtained by changing the acoustic impedance mismatch between the fluid (alcohol vs water) and the trabeculae was reflected neither in the attenuation nor in its slope. This led the authors to suggest that longitudinal-to-shear scattering together with absorption in the solid phase are candidates as main sources for the attenuation. The differences in velocity values indicate that the elastic properties of the fluid are main determinants of the phase velocity. This finding is particularly significant in the context of /in vivo/ measurements, because it demonstrates that the subject-dependent properties of marrow may partly explain the inter-subject variability of speed of sound values.

Relationship between ultrasonic parameters and apparent trabecular bone elastic modulus: a numerical approach

The physical principles underlying quantitative ultrasound (QUS) measurements in trabecular bone are not fully understood. The translation of QUS results into bone strength remains elusive. However, ultrasound being mechanical waves, it is likely to assess apparent bone elasticity. The aim of this study is to derive the sensitivity of QUS parameters to variations of apparent bone elasticity, a surrogate for strength. The geometry of 34 human trabecular bone samples cut in the great trochanter was reconstructed using 3-D synchrotron micro-computed tomography. Finite-difference time-domain simulations coupled to 3-D micro-structural models were performed in the three perpendicular directions for each sample and each direction. A voxel-based micro-finite element linear analysis was employed to compute the apparent Young's modulus (E) of each sample for each direction. For the antero-posterior direction, the predictive power of speed of sound and normalized broadband ultrasonic attenuation to assess E was equal to 0.9 and 0.87, respectively, which is better than what is obtained using bone density alone or coupled with micro-architectural parameters and of the same order of what can be achieved with the fabric tensor approach. When the direction of testing is parallel to the main trabecular orientation, the predictive power of QUS parameters decreases and the fabric tensor approach always gives the best results. This decrease can be explained by the presence of two longitudinal wave modes. Our results, which were obtained using two distinct simulation tools applied on the same set of samples, highlight the potential of QUS techniques to assess bone strength.