Comparison of acoustic characteristics predicted by Biot's theory and the modified Biot-Attenborough model in cancellous bone

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.

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.

Ultrasound temporal-spatial phase-interference in complex composite media; a comparison of experimental measurement and simulation prediction

The propagation of ultrasound through solid:liquid complex composite media such as cancellous bone suffers from a lack of a comprehensive understanding of the dependence upon density and structure. Assuming that a propagating ultrasound wave may be considered as an array of parallel sonic rays, we may determine the transit time of each by the relative proportion of the two constituents. A transit time spectrum (TTS) describes the proportion of sonic rays having a particular transit time between the minimum (tmin) and maximum (tmax) values; representing, for example, entire bone tissue and marrow respectively in the case of cancellous bone. Langton has proposed that the primary ultrasound attenuation mechanism in such media is phase-interference. The phase-interference of two or more ultrasound pulses detected at a phase-sensitive transducer has both temporal and spatial components. The temporal component is primarily dependent upon the transit time difference (dt) between the pulses and the propagating pulse-length (PL). The spatial component is primarily dependent upon the lateral separation (ds) of the detectedpulses of differing transit time and the lateral dimension of the ultrasound receive transducer aperture (dL). The aim of the paper was to explore these temporal and spatial dependencies through a comparison of experimental measurement and computer simulation in solid:liquid models of varying temporal and spatial complexity. Transmission measurements at nominal ultrasound frequencies of 1MHz and 5MHz were performed, thereby investigating the dependency upon period. The results demonstrated an overall agreement between experimental measurement and computer simulation of 87±16% and 85±12% for temporal and spatial components respectively. It is envisaged that a comprehensive understanding of ultrasound propagation through complex structures such as cancellous bone could provide an improved non-invasive tool for osteoporosis assessment.

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.

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.

Scattering of acoustic waves by macroscopically inhomogeneous poroelastic tubes

Wave propagation in macroscopically inhomogeneous porous materials has received much attention in recent years. For planar configurations, the wave equation, derived from the alternative formulation of Biot's theory of 1962, was reduced and solved recently: first in the case of rigid frame inhomogeneous porous materials and then in the case of inhomogeneous poroelastic materials in the framework of Biot's theory. This paper focuses on the solution of the full wave equation in cylindrical coordinates for poroelastic tubes in which the acoustic and elastic properties of the poroelastic tube vary in the radial direction. The reflection coefficient is obtained numerically using the state vector (or the so-called Stroh) formalism and Peano series. This coefficient can then be used to straightforwardly calculate the scattered field. To validate the method of resolution, results obtained by the present method are compared to those calculated by the classical transfer matrix method in the case of a two-layer poroelastic tube. As an example, a long bone excited in the sagittal plane is considered. Finally, a discussion is given of ultrasonic time domain scattered field for various inhomogeneity profiles, which could lead to the prospect of long bone characterization.

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.

Novel ultrasonic bone densitometry based on two longitudinal waves: significant correlation with pQCT measurement values and age-related changes in trabecular bone density, cortical thickness, and elastic modulus of trabecular bone in a normal Japanese population

A reference database for trabecular bone density, cortical thickness, and elastic modulus of trabecular bone for a novel ultrasonic bone densitometry system (LD-100) based on two longitudinal waves (fast and slow) was determined over a wide age range in a normal Japanese population.

INTRODUCTION

A novel ultrasonic bone densitometry system (LD-100 system) was applied to create a reference database for trabecular bone density (TBD), cortical thickness (CoTh), and elastic modulus of trabecular bone (EMTb) for this device over a wide age range in a normal Japanese population.

METHODS

In a comparative study between LD-100 and peripheral quantitative computed tomography (pQCT) systems, 52 individuals were examined by both systems at the same radius simultaneously. To create a reference database, a total of 2,380 healthy subjects (1,179 men, 1,201 women), ages 18-99 years, were examined using the LD-100 system.

RESULTS

Highly significant correlations between the LD-100 and pQCT systems were found in TBD (r = 0.877, p < 0.001) and CoTh (r = 0.723, p < 0.001). For the reference database, peak values of TBD, CoTh, and EMTb were observed at 30-34 years (255.09 mg/cm(3)), 20-24 years (5.23 mm), and 20-24 years (4.09 GPa) in men, and at 25-29 years (209.24 mg/cm(3)), 25-29 years (3.98 mm), and 20-24 years (3.33 GPa) in women, respectively. The TBD fell significantly (p < 0.05) beginning at 55-59 years in both sexes, with a relatively rapid decrease in women. The CoTh showed a significant decrease beginning at 40-44 years in men and 50-54 years in women. The EMTb showed a significant decrease beginning at 40-44 years in men and 55-59 years in women.

CONCLUSIONS

The LD-100 system is a useful bone densitometry device and the database of age-related changes in TBD, CoTh, and EMTb established in this study will provide fundamental data for future studies related to bone status.

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.

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.

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).

Measurement of human trabecular bone by novel ultrasonic bone densitometry based on fast and slow waves

SUMMARY

Two longitudinal transmitted waves, fast and slow waves, were observed by employing a new quantitative ultrasound (QUS) method. The trabecular bone measurements generated by this method reflect three-dimensional structural information, and the new QUS parameters were able to identify vertebral fractures.

INTRODUCTION

The aims were to identify new quantitative ultrasound (QUS) parameters that based on new QUS method reflecting not only bone volume but also the microstructures of trabecular bone ex vivo and to observe how much they predict fracture risk in vivo.

METHODS

Ex vivo measurement: Three human femoral heads were used for the experiment. Attenuation of the slow wave, attenuation of the fast wave, speed of the slow wave, speed of the fast wave (SOFW), bone mass density of trabecular bone, and elastic modulus of the trabecular bone (EMTb) of each specimen were obtained using a new QUS method and compared with three-dimensional structural parameters measured by micro-computed tomography. In vivo measurement: Eighty-nine volunteers were enrolled, and the bone status in the distal radius was measured using a new QUS method. These parameters were compared with data evaluated by peripheral quantitative computed tomography and dual X-ray absorptiometry.

RESULTS

Ex vivo measurement: SOFW and EMTb showed correlations with the parameter of trabecular anisotropy. In vivo measurement: The new QUS parameters were able to identify vertebral fractures.

CONCLUSION

The newly developed QUS technique reflects the three-dimensional structure and is a promising method to evaluate fracture risk.

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

Our goal is to assess the potential of computational methods as an alternative to analytical models to predict the two longitudinal wave modes observed in cancellous bone and predicted by the Biot theory. A three-dimensional (3D) finite-difference time-domain method is coupled with 34 human femoral trabecular microstructures measured using microcomputed tomography. The main trabecular alignment (MTA) and the degree of anisotropy (DA) were assessed for all samples. DA values were comprised between 1.02 and 1.9. The influence of bone volume fraction (BV/TV) between 5% and 25% on the properties of the fast and slow waves was studied using a dedicated image processing algorithm to modify the initial 3D microstructures. A heuristic method was devised to determine when both wave modes are time separated. The simulations (performed in three perpendicular directions) predicted that both waves generally overlap in time for a direction of propagation perpendicular to the MTA. When these directions are parallel, both waves are separated in time for samples with high DA and BV/TV values. A relationship was found between the least bone volume fraction required for the observation of nonoverlapping waves and the degree of anisotropy: The higher the DA, the lower the least BV/TV.

Predictions of the modified Biot-Attenborough model for the dependence of phase velocity on porosity in cancellous bone

The modified Biot-Attenborough (MBA) model for acoustic wave propagation in porous media has been found useful to predict wave properties in cancellous bone. The present study is aimed at applying the MBA model to predict the dependence of phase velocity on porosity in cancellous bone. The MBA model predicts a phase velocity that decreases nonlinearly with porosity. The optimum values for input parameters of the MBA model, such as compressional speed c(m) of solid bone and phase velocity parameter s(2), were determined by comparing the predictions with previously published measurements in human calcaneus and bovine cancellous bone. The value of the phase velocity parameter s(2)=1.23 was obtained by curve fitting to the experimental data for 53 human calcaneus samples only, assuming a compressional speed c(m)=2500 m/s of solid bone. The root-mean-square error (RMSE) of the curve fit was 15.3m/s. The optimized value of s(2) for all 75 cancellous bone samples including 22 bovine samples was 1.42 with a value of 55 m/s for the RMSE of the curve fit. The latter fit was obtained by using of a value of c(m)=3200 m/s. Although the MBA model relies on the empirical parameters determined from experimental data, it is expected that the model can be usefully employed as a practical tool in the field of clinical ultrasonic bone assessment.

Phase velocity and normalized broadband ultrasonic attenuation in Polyacetal cuboid bone-mimicking phantoms

The variations of phase velocity and normalized broadband ultrasonic attenuation (nBUA) with porosity were investigated in Polyacetal cuboid bone-mimicking phantoms with circular cylindrical pores running normal to the surface along the three orthogonal axes. The frequency-dependent phase velocity and attenuation coefficient in the phantoms with porosities from 0% to 65.9% were measured from 0.65 to 1.10 MHz. The results showed that the phase velocity at 880 kHz decreased linearly with porosity, whereas the nBUA increased linearly with porosity. This study provides a useful insight into the relationships between ultrasonic properties and porosity in bone at porosities lower than 70%.

Empirical angle-dependent Biot and MBA models for acoustic anisotropy in cancellous bone

The Biot and the modified Biot-Attenborough (MBA) models have been found useful to understand ultrasonic wave propagation in cancellous bone. However, neither of the models, as previously applied to cancellous bone, allows for the angular dependence of acoustic properties with direction. The present study aims to account for the acoustic anisotropy in cancellous bone, by introducing empirical angle-dependent input parameters, as defined for a highly oriented structure, into the Biot and the MBA models. The anisotropy of the angle-dependent Biot model is attributed to the variation in the elastic moduli of the skeletal frame with respect to the trabecular alignment. The angle-dependent MBA model employs a simple empirical way of using the parametric fit for the fast and the slow wave speeds. The angle-dependent models were used to predict both the fast and slow wave velocities as a function of propagation angle with respect to the trabecular alignment of cancellous bone. The predictions were compared with those of the Schoenberg model for anisotropy in cancellous bone and in vitro experimental measurements from the literature. The angle-dependent models successfully predicted the angular dependence of phase velocity of the fast wave with direction. The root-mean-square errors of the measured versus predicted fast wave velocities were 79.2 m s(-1) (angle-dependent Biot model) and 36.1 m s(-1) (angle-dependent MBA model). They also predicted the fact that the slow wave is nearly independent of propagation angle for angles about 50 degrees , but consistently underestimated the slow wave velocity with the root-mean-square errors of 187.2 m s(-1) (angle-dependent Biot model) and 240.8 m s(-1) (angle-dependent MBA model). The study indicates that the angle-dependent models reasonably replicate the acoustic anisotropy in cancellous bone.