Velocity dispersion in trabecular bone: influence of multiple scattering and of absorption

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.

In Vivo Comparison of Backscatter Techniques for Ultrasonic Bone Assessment at the Femoral Neck

Ultrasonic techniques are being developed to detect changes in cancellous bone caused by osteoporosis. The goal of this study was to test the relative in vivo performance of eight backscatter parameters developed over the last several years for ultrasonic bone assessment: apparent integrated backscatter (AIB), frequency slope of apparent backscatter (FSAB), frequency intercept of apparent backscatter (FIAB), normalized mean of the backscatter difference (nMBD), normalized slope of the backscatter difference (nSBD), normalized intercept of the backscatter difference (nIBD), normalized backscatter amplitude ratio (nBAR) and backscatter amplitude decay constant (BADC). Backscatter measurements were performed on the left and right femoral necks of 80 adult volunteers (age = 25 ± 11 y) using an imaging system equipped with a convex array transducer. For comparison, additional ultrasonic measurements were performed at the left and right heel using a commercially available heel-bone ultrasonometer that measured the stiffness index. Six of the eight backscatter parameters (all but nSBD and nIBD) exhibited similar and highly significant (p < 0.000001) left-right correlations (0.51 ≤ R ≤ 0.68), indicating sensitivity to naturally occurring variations in bone tissue. Left-right correlations for the stiffness index measured at the heel (R = 0.75) were not significantly better than those produced by AIB, FSAB and FIAB. The short-term precisions of AIB, nMBD, nBAR and BADC (7.8%-11.7%) were comparable to that of the stiffness index measured with the heel-bone ultrasonometer (7.5%).

Mechanisms of Interaction of Ultrasound With Cancellous Bone: A Review

Ultrasound is now a clinically accepted modality in the management of osteoporosis. The most common commercial clinical devices assess fracture risk from measurements of attenuation and sound speed in cancellous bone. This review discusses fundamental mechanisms underlying the interaction between ultrasound and cancellous bone. Because of its two-phase structure (mineralized trabecular network embedded in soft tissue-marrow), its anisotropy, and its inhomogeneity, cancellous bone is more difficult to characterize than most soft tissues. Experimental data for the dependencies of attenuation, sound speed, dispersion, and scattering on ultrasound frequency, bone mineral density, composition, microstructure, and mechanical properties are presented. The relative roles of absorption, scattering, and phase cancellation in determining attenuation measurements in vitro and in vivo are delineated. Common speed of sound metrics, which entail measurements of transit times of pulse leading edges (to avoid multipath interference), are greatly influenced by attenuation, dispersion, and system properties, including center frequency and bandwidth. However, a theoretical model has been shown to be effective for correction for these confounding factors in vitro and in vivo. Theoretical and phantom models are presented to elucidate why cancellous bone exhibits negative dispersion, unlike soft tissue, which exhibits positive dispersion. Signal processing methods are presented for separating "fast" and "slow" waves (predicted by poroelasticity theory and supported in cancellous bone) even when the two waves overlap in time and frequency domains. Models to explain dependencies of scattering on frequency and mean trabecular thickness are presented and compared with measurements. Anisotropy, the effect of the fluid filler medium (marrow in vivo or water in vitro), phantoms, computational modeling of ultrasound propagation, acoustic microscopy, and nonlinear properties in cancellous bone are also discussed.

Influence of the microstructure of two-dimensional random heterogeneous media on propagation of acoustic coherent waves

Multiple scattering of waves arises in all fields of physics in either periodic or random media. For random media the organization of the microstructure (uniform or nonuniform statistical distribution of scatterers) has effects on the propagation of coherent waves. Using a recent exact resolution method and different homogenization theories, the effects of the microstructure on the effective wave number are investigated over a large frequency range (ka between 0.1 and 13.4) and high concentrations. For uniform random media, increasing the configurational constraint makes the media more transparent for low frequencies and less for high frequencies. As a side but important result, we show that two of the homogenization models considered here appear to be very efficient at high frequency up to a concentration of 60% in the case of uniform media. For nonuniform media, for which clustered and periodic aggregates appear, the main effect is to reduce the magnitude of resonances and to make network effects appear. In this case, homogenization theories are not relevant to make a detailed analysis.

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.

The Ability of Ultrasonic Backscatter Parametric Imaging to Characterize Bovine Trabecular Bone

The ultrasonic backscatter technique holds the promise of characterizing bone density and microstructure. This paper conducts ultrasonic backscatter parametric imaging based on measurements of apparent integrated backscatter (AIB), spectral centroid shift (SCS), frequency slope of apparent backscatter (FSAB), and frequency intercept of apparent backscatter (FIAB) for representing trabecular bone mass and microstructure. We scanned 33 bovine trabecular bone samples using a 7.5 MHz focused transducer in a 20 mm × 20 mm region of interest (ROI) with a step interval of 0.05 mm. Images based on the ultrasonic backscatter parameters (i.e., AIB, SCS, FSAB, and FIAB) were constructed to compare with photographic images of the specimens as well as two-dimensional (2D) μ-CT images from approximately the same depth and location of the specimen. Similar structures and trabecular alignments can be observed among these images. Statistical analyses demonstrated that the means and standard deviations of the ultrasonic backscatter parameters exhibited significant correlations with bone density (|R| = 0.45-0.78, p < 0.01) and bone microstructure (|R| = 0.44-0.87, p < 0.001). Some bovine trabecular bone microstructure parameters were independently associated with the ultrasonic backscatter parameters (ΔR² = 4.18%-44.45%, p < 0.05) after adjustment for bone apparent density (BAD). The results show that ultrasonic backscatter parametric imaging can provide a direct view of the trabecular microstructure and can reflect information about the density and microstructure of trabecular bone.

Modeling ultrasound attenuation in porous structures with mono-disperse random pore distributions using the independent scattering approximation: a 2D simulation study

The validity of the independent scattering approximation (ISA) to predict the frequency dependent attenuation in 2D models of simplified structures of cortical bone is studied. Attenuation of plane waves at central frequencies ranging from 1 to 8 MHz propagating in structures with mono-disperse random pore distributions with pore diameter and pore density in the range of those of cortical bone are evaluated by finite difference time domain numerical simulations. An approach to assess the multiple scattering of waves in random media is discussed to determine the pore diameter ranges at which the ISA is applicable. A modified version of the ISA is proposed to more accurately predict the attenuation in porosity ranges where it would traditionally fail. The results show that the modified ISA can model the frequency-dependent attenuation of ultrasonic wave with pore diameter and density ranges comparable to those of cortical bone.

Predicting Structural Properties of Cortical Bone by Combining Ultrasonic Attenuation and an Artificial Neural Network (ANN): 2-D FDTD Study

The goal of this paper is to predict the micro-architectural parameters of cortical bone such as pore diameter ( ϕ ) and porosity ( v ) from ultrasound attenuation measurements using an artificial neural network (ANN). Slices from a 3-D CT scan of human femur are obtained. The micro-architectural parameters of porosity such as average pore size and porosity are calculated using image processing. When ultrasound waves propagate in porous structures, attenuation is observed due to scattering. Two-dimensional finite-difference time-domain simulations are carried out to obtain frequency dependent attenuation in those 2D structures. An artificial neural network (ANN) is then trained with the input feature vector as the frequency dependent attenuation and output as pore diameter ( ϕ ) and porosity ( v ) . The ANN is composed of one input layer, 3 hidden layers and one output layer, all of which are fully connected. 340 attenuation data sets were acquired and trained over 2000 epochs with a batch size of 32. Data was split into train, validation and test. It was observed that the ANN predicted the micro-architectural parameters of the cortical bone with high accuracies and low losses with a minimum R2 (goodness of fit) value of 0.95. ANN approaches could potentially help inform the solution of inverse-problems to retrieve bone porosity from ultrasound measurements. Ultimately, those inverse-problems could be used for the non-invasive diagnosis and monitoring of osteoporosis.

Development of a simple numerical model for trabecular bone structures

PURPOSE

Advances in additive manufacturing processes are enabling the fabrication of surrogate bone structures for applications including use in high-resolution anthropomorphic phantoms. In this research, a simple numerical model is proposed that enables the generation of microarchitecture with similar statistical distribution to trabecular bone.

METHODS

A human humerus, radius, ulna, and several vertebrae were scanned on the Imaging and Medical beamline at the Australian Synchrotron and the proposed numerical model was developed through the definition of two complex functions that encode the trabecular thickness and position-dependant spacing to generate volumetric surrogate trabecular structures. The structures reproduced those observed at 19 separate axial locations through the experimental bone volumes. The applicability of the model when incorporating a two-material approximation to absorption- and phase-contrast CT was also investigated through simulation.

RESULTS

The synthetic structures, when compared with the real trabecular microarchitecture, yielded an average mean thickness error of 2 μm, and a mean difference in standard deviation of 33 μm for the humerus, 24 μm for the ulna and radius, and 15 μm for the vertebrae. Simulated absorption- and propagation-based phase contrast CT projection data were generated and reconstructed using the derived mathematical simplifications from the two-material approximation, and the phase-contrast effects were successfully demonstrated.

CONCLUSIONS

The presented model reproduced trabecular distributions that could be used to generate phantoms for quality assurance and validation processes. The implication of utilizing a two-material approximation results in simplification of the additive manufacturing process and the generation of synthetic data that could be used for training of machine learning applications.

Velocity dispersion and backscatter in marrow-filled and water-filled trabecular bone samples in vitro

The phase velocity and the backscatter coefficient were measured in 28 bovine femoral trabecular bone samples filled with marrow and water in vitro from 0.2 to 0.6 MHz. The phase velocities decreased approximately linearly with increasing frequency and the average dispersion rate of -34 ms^-1MHz^-1 in the marrow-filled samples was higher than that of -42 ms^-1MHz^-1 in the water-filled samples. The backscatter coefficients exhibited nonlinear, monotonically increasing dependences on the frequency and the average value of the exponent n = 2.92 (frequency dependence) in the marrow-filled samples was higher than the value of n = 2.79 in the water-filled samples.

Relationships among ultrasonic and mechanical properties of cancellous bone in human calcaneus in vitro

Clinical bone sonometers applied at the calcaneus measure broadband ultrasound attenuation and speed of sound. However, the relation of ultrasound measurements to bone strength is not well-characterized. Addressing this issue, we assessed the extent to which ultrasonic measurements convey in vitro mechanical properties in 25 human calcaneal cancellous bone specimens (approximately 2×4×2cm). Normalized broadband ultrasound attenuation, speed of sound, and broadband ultrasound backscatter were measured with 500kHz transducers. To assess mechanical properties, non-linear finite element analysis, based on micro-computed tomography images (34-micron cubic voxel), was used to estimate apparent elastic modulus, overall specimen stiffness, and apparent yield stress, with models typically having approximately 25-30 million elements. We found that ultrasound parameters were correlated with mechanical properties with R=0.70-0.82 (p<0.001). Multiple regression analysis indicated that ultrasound measurements provide additional information regarding mechanical properties beyond that provided by bone quantity alone (p≤0.05). Adding ultrasound variables to linear regression models based on bone quantity improved adjusted squared correlation coefficients from 0.65 to 0.77 (stiffness), 0.76 to 0.81 (apparent modulus), and 0.67 to 0.73 (yield stress). These results indicate that ultrasound can provide complementary (to bone quantity) information regarding mechanical behavior of 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.

Microstructural characterization of trabecular bone using ultrasonic backscattering and diffusion parameters

Finite differences time domain methods were utilized to simulate ultrasound propagation and scattering in anisotropic trabecular bone structures obtained from high resolution Computed Tomography (CT). The backscattered signals were collected and the incoherent contribution was extracted. The diffusion constant was calculated for propagations along and across the main direction of anisotropy, and was used to characterize the anisotropy of the trabecular microstructures. In anisotropic structures, the diffusion constant was significantly different in both directions, and the anisotropy of the diffusion constant was strongly correlated to the structural anisotropy measured on the CT images. These results indicate that metrics based on diffusion can be used to quantify the anisotropy of complex structures such as trabecular bone.

Multiple scattering by infinitely long cylindrical glass inclusions in a saturated Biot porous medium of glass beads

The resonance spectrum of sets of two to five infinitely long parallel cylindrical glass inclusions in a fluid saturated porous matrix of unconsolidated glass beads is investigated. The ratio of bead diameters to inclusion diameters is 1/5. The far field form functions and the related phase derivatives are calculated by using an exact multiple scattering formalism and by assuming that the porous medium obeys Biot's model. In order to validate this hypothesis, comparisons between theory and experiments are done in the special case of a fast incident wave on a set of two and three inclusions.

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.

Attenuated Fractional Wave Equations With Anisotropy

This paper develops new fractional calculus models for wave propagation. These models permit a different attenuation index in each coordinate to fully capture the anisotropic nature of wave propagation in complex media. Analytical expressions that describe power law attenuation and anomalous dispersion in each direction are derived for these fractional calculus models.

Application of an effective medium theory for modeling ultrasound wave propagation in healing long bones

Quantitative ultrasound has recently drawn significant interest in the monitoring of the bone healing process. Several research groups have studied ultrasound propagation in healing bones numerically, assuming callus to be a homogeneous and isotropic medium, thus neglecting the multiple scattering phenomena that occur due to the porous nature of callus. In this study, we model ultrasound wave propagation in healing long bones using an iterative effective medium approximation (IEMA), which has been shown to be significantly accurate for highly concentrated elastic mixtures. First, the effectiveness of IEMA in bone characterization is examined: (a) by comparing the theoretical phase velocities with experimental measurements in cancellous bone mimicking phantoms, and (b) by simulating wave propagation in complex healing bone geometries by using IEMA. The original material properties of cortical bone and callus were derived using serial scanning acoustic microscopy (SAM) images from previous animal studies. Guided wave analysis is performed for different healing stages and the results clearly indicate that IEMA predictions could provide supplementary information for bone assessment during the healing process. This methodology could potentially be applied in numerical studies dealing with wave propagation in composite media such as healing or osteoporotic bones in order to reduce the simulation time and simplify the study of complicated geometries with a significant porous nature.

Multiple scattering in porous media: comparison with water saturated double porosity media

Multiple scattering in a poroelastic medium obeying Biot's theory is studied; the scatterers are parallel identical cylindrical holes pierced at random in the medium. The paper focuses first on the influence, on the effective wavenumbers, of the mode conversions that occur at each scattering event. The effect of the holes on the dispersion curves is then examined for two different values of the ratio of their radius to the pores mean radius. Depending on the latter, the dispersion curves of the pierced material are compared, for the fast and shear waves, with those of either a more porous medium or a double porosity medium.

Measurement of speed of sound in poly(lactic acid)-clay composite

We measured longitudinal speed of sound for matrix[poly(lactic acid)]-additive(clay particles) composite rectangular-solid specimen prepared by injection molding. It was found that the speed of sound measured in the direction along the longer side of the specimen was the highest at the middle of the specimen. This trend corresponded with that for crystallinity determined through differential scanning calorimetry (DSC). A cross section view of the specimen parallel to its longer side showed that there was a transverse flow trace of resin in the vicinity of the injection gate while the flow trace along the direction of the longer side spread wider as getting far from the gate toward the middle of the specimen. The high crystallinity appeared in the middle of the specimen was inferred to come from the promotion of crystallization by molecular orientation induced with the above flow trace parallel to the direction along the longer side of the specimen.

Numerical simulation of effective phase velocity and attenuation of shear elastic wave propagation in unidirectional composite materials

In this paper, a simple simulation approach is presented for calculating the effective phase velocity and attenuation coefficient of elastic shear waves propagating in composite materials with randomly distributed unidirectional inclusions. As an application of the developed numerical approach, the phase velocities and attenuation coefficients of the coherent waves in four different types of composite material are simulated for various incident frequencies up to π. Numerical results are compared with theoretical predictions obtained from three representative theoretical models. While all theoretical results agree very well with numerical values at low incident frequencies, the discrepancies increase with the increased incident frequency and volume fraction of inclusions. It has been found that within the frequency and volume fraction ranges considered in this work, the generalized self-consistent model by Kanaun and Levin [18] seems to provide the most accurate estimations.

Phase velocity and attenuation predictions of waves in cancellous bone using an iterative effective medium approximation

The quantitative determination of wave dispersion and attenuation in bone is an open research area as the factors responsible for ultrasound absorption and scattering in composite biological tissues have not been completely explained. In this study, we use the iterative effective medium approximation (IEMA) proposed in [1] so as to calculate phase velocity and attenuation in media with properties similar to those of cancellous bones. Calculations are performed for a frequency range of 0.4-0.8 MHz and for different inclusions' volume concentrations and sizes. Our numerical results are compared with previous experimental findings so as to assess the effectiveness of IEMA. It was made clear that attenuation and phase velocity estimations could provide supplementary information for cancellous bone characterization.

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.

Two-wave propagation imaging to evaluate the structure of cancellous bone

The two-wave phenomenon reflects not only bone mass but also the complex bone structure of cancellous bone. We propose a new simple imaging technique based on the two-wave phenomenon for investigating the anisotropic structure of cancellous bone. A cylindrical specimen of cancellous bone was obtained from a bovine femur. The structure (alignment of trabeculae) of the specimen was obtained from 3-D X-ray micro computed tomography imaging. Using a conventional ultrasonic pulse technique, we rotated the receiver around the specimen to investigate the ultrasonic fields after propagation within the specimen. The ultrasonic propagation image clearly showed the effect of the bone structure. We found that the fast wave showed apparent refraction, whereas the slow wave did not. Fast-wave propagation imaging is thus a simple and convenient technique for easy interpretation of the anisotropic structure. This imaging technique has the potential to become a powerful tool to investigate the structure of trabeculae during in vivo measurements.

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.

Relationships of quantitative ultrasound parameters with cancellous bone microstructure in human calcaneus in vitro

Ultrasound parameters (attenuation, phase velocity, and backscatter), bone mineral density (BMD), and microarchitectural features were measured on 29 human cancellous calcaneus samples in vitro. Regression analysis was performed to predict ultrasound parameters from BMD and microarchitectural features. The best univariate predictors of the ultrasound parameters were the indexes of bone quantity: BMD and bone volume fraction (BV/TV). The most predictive univariate models for attenuation, phase velocity, and backscatter coefficient yielded adjusted squared correlation coefficients of 0.69-0.73. Multiple regression models yielded adjusted correlation coefficients of 0.74-0.83. Therefore attenuation, phase velocity, and backscatter are primarily determined by bone quantity, but multiple regression models based on bone quantity plus microarchitectural features achieve slightly better predictive performance than models based on bone quantity alone.

Correlations of group velocity, phase velocity, and dispersion with bone density in bovine trabecular bone

The present study investigated the correlations of the group velocity, the phase velocity, and the velocity dispersion with the apparent bone density in bovine trabecular bone in vitro. The phase velocity exhibited the negative dispersion, consistent with the behavior in human trabecular bone. The group and the phase velocities were found to increase with increasing apparent bone density, respectively, exhibiting similar high correlations of r=0.94 and 0.96. The negative dispersion rate exhibited a decreasing dependence on the apparent bone density, with a significant correlation of r=-0.86.

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.

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.

Two-dimensional simulation of linear wave propagation in a suspension of polymeric microcapsules used as ultrasound contrast agents

A generation of tissue-specific stable ultrasound contrast agent (UCA) composed of a polymeric capsule with a perfluorocarbone liquid core has become available. Despite promising uses in clinical practice, the acoustical behavior of such UCA suspensions remains unclear. A simulation code (2-D finite-difference time domain, FDTD) already validated for homogeneous particles [Galaz Haiat, Berti, Taulier, Amman and Urbach, (2010). J. Acoust. Soc. Am. 127, 148-154] is used to model the ultrasound propagation in such UCA suspensions at 50 MHz to investigate the sensitivity of the ultrasonic parameters to physical parameters of UCA. The FDTD simulation code is validated by comparison with results obtained using a shell scatterer model. The attenuation coefficient (respectively, the sound velocity) increases (respectively, decreases) from 4.1 to 58.4 dB/cm (respectively, 1495 to 1428 m/s) when the concentration varies between 1.37 and 79.4 mg/ml, while the backscattered intensity increases non-linearly, showing that a concentration of around 30 mg/ml is sufficient to obtain optimal backscattering intensity. The acoustical parameters vary significantly as a function of the membrane thickness, longitudinal and transverse velocity, indicating that mode conversions in the membrane play an important role in the ultrasonic propagation. The results may be used to help manufacturers to conceive optimal liquid-filled UCA suspensions.

Ultrasonic wave propagation in cementitious materials: a multiphase approach of a self-consistent multiple scattering model

This paper examines ultrasonic wave propagation through strongly heterogeneous materials such as cementitious materials, and deals meanly with the formulation of a multiphase approach of a self-consistent multiple scattering model, the so-called dynamic generalized self-consistent model (DGSCM) proposed by Yang [J. Appl. Mech. 70(2003) 575-582]. This extended model can describe the influence of the size and volume fraction of aggregates on cementitious materials, as well as the interaction, contribution, and influence of entrapped air voids together with the aggregates on frequency-dependent parameters such as the phase velocity and the attenuation coefficient. To show the performance of this approach, theoretical predictions were compared with experimental ultrasonic measurements over a wide frequency range from several mortar specimens with different features in their microstructure properties and concentrations of aggregates up to 60%. The multiphase approaches of both the DGSCM and the Waterman-Truell model (WT) were also compared. The obtained results of the multiphase DGSCM were found to be significantly better than those obtained from the N-phase WT model for ultrasonic measurements from cementitious materials at high aggregate concentrations. The feasibility of material characterization using the multiphase approach of DGSCM was also discussed.

Ultrasonic wave propagation in stereo-lithographical bone replicas

Predictions of a modified anisotropic Biot-Allard theory are compared with measurements of pulses centered on 100 kHz and 1 MHz transmitted through water-saturated stereo-lithographical bone replicas. The replicas are 13 times larger than the original bone samples. Despite the expected effects of scattering, which is neglected in the theory, at 100 kHz the predicted and measured transmitted waveforms are similar. However, the magnitude of the leading negative edge of the waveform is overpredicted, and the trailing parts of the waveforms are not predicted well. At 1 MHz, although there are differences in amplitudes, the theory predicts that the transmitted waveform is almost a scaled version of that incident in conformity with the data.

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

Cortical bone and the surrounding soft tissues are attenuating and heterogeneous media, which might affect the signals measured with axial transmission devices. This work aims at evaluating the effect of the heterogeneous acoustic absorption in bone and in soft tissues on the bone ultrasonic response. Therefore, a two-dimensional finite element time-domain method is derived to model transient wave propagation in a three-layer medium composed of an inhomogeneous transverse isotropic viscoelastic solid layer, sandwiched between two viscous fluid layers. The model couples viscous acoustic propagation in both fluid media with the anisotropic viscoelastic response of the solid. A constant spatial gradient of material properties is considered for two values of bone thicknesses (0.6 and 4 mm). In the studied configuration, absorption in the surrounding fluid tissues does not affect the results, whereas bone viscoelastic properties have a significant effect on the first arriving signal (FAS) velocity. For a thin bone, the FAS velocity is governed by the spatially averaged bone properties. For a thick bone, the FAS velocity may be predicted using a one-dimensional model.

Experimental validation of a time domain simulation of high frequency ultrasonic propagation in a suspension of rigid particles

Ultrasonic propagation in suspensions of particles is a difficult problem due to the random spatial distribution of the particles. Two-dimensional finite-difference time domain simulations of ultrasonic propagation in suspensions of polystyrene 5.3 mum diameter microdisks are performed at about 50 MHz. The numerical results are compared with the Faran model, considering an isolated microdisk, leading to a maximum difference of 15% between the scattering cross-section values obtained analytically and numerically. Experiments are performed with suspensions in through transmission and backscattering modes. The attenuation coefficient at 50 MHz (alpha), the ultrasonic velocity (V), and the relative backscattered intensity (I(B)) are measured for concentrations from 2 to 25 mg/ml, obtained by modifying the number of particles. Each experimental ultrasonic parameter is compared to numerical results obtained by averaging the results derived from 15 spatial distributions of microdisks. alpha increases with the concentration from 1 to 17 dB/cm. I(B) increases with concentration from 2 to 16 dB. The variation of V versus concentration is compared with the numerical results, as well as with an effective medium model. A good agreement is found between experimental and numerical results (the larger discrepancy is found for alpha with a difference lower than 2.1 dB/cm).

Simplified expressions of the subtracted Kramers-Kronig relations using the expanded forms applied to ultrasonic power-law systems

The Kramers-Kronig (KK) relations are a large class of integral transformations that exploit the broad principle of simple causality in order to link the physical properties of matter and materials. In applications to the complex-valued wavenumber for acoustic propagation, the method of subtractions is used to form convergent integral relations between the phase velocity and the attenuation coefficient. When the method of subtractions is applied in the usual manner, the integrands in the relations become unnecessarily complicated. In this work, an expanded form of the subtracted relations is presented, which is essentially a truncated Taylor series expansion of the Hilbert transforms. The implementation of the relations only requires the explicit evaluation of two simply expressed integrals involving the Hilbert transform kernel. These two integrals determine the values of the other terms in the subtracted relations, demonstrating the computational efficiency of the technique. The method is illustrated analytically through its application to power-law attenuation coefficients and its associated dispersion, which are observed in a wide variety of materials. This approach explicitly shows the central role of the Hilbert transform kernel in the KK relations, which can become obscured in other formulations.

Phase velocity analysis of acoustic propagation in trabecular bone

The trabecular bones are highly dispersive acoustic media composed by randomly shaped trabeculae (considered as scatterers) and surrounded by bone marrow. An acoustic wave generated by an incident ultrasonic pulse with respect to the media under study, shows that its velocity and amplitude are a function of the density size and shaped of the scatterers. Two different methods were used to theoretically study this scattering phenomena: i) a self-consistent model proposed by Sabina and Willis, and ii) the multiple-scattering theory of Waterman-Truell. These methods were able to compute the phase velocity and amplitude as a function of operating frequency, density and size of scatterers. The theoretical results were compared with experimental data already published and the phase veloctiy shows a good agreement for low concentration of scatterers.

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.