Anomalous negative dispersion in bone can result from the interference of fast and slow waves

Fetal Tethered Spinal Cord: Diagnostic Features and Its Association with Congenital Anomalies

OBJECTIVE

We assessed the frequency and type of associated congenital anomalies encountered with fetal tethered spinal cord (TSC) determined prenatally.

METHOD

A retrospective review was conducted based on the associated fetal abnormalities following diagnosis of low-lying fetal conus medullaris during the prenatal ultrasound.

RESULTS

Of the 26 fetuses with low-lying conus medullaris, four were solitary TSC and 22 had TSC combined with associated congenital malformations, including four cases with spina bifida occulta, four cases with spina bifida aperta, one case with severe hydrocephalus, and 13 cases with multisystem congenital malformations. Among all the 13 cases with combined multisystem congenital malformations, four cases had vertebral defects, anal anomalies, cardiac defects, trachea-esophageal fistula, renal anomalies, and limb anomalies (VACTERL) syndrome, two cases had combined kidney development abnormalities, one case had cloacal exstrophy (OEIS syndrome), and six cases had chromosomal abnormalities (one case of chromosome 7q deletion, two cases of trisomy 13 syndrome, one case of trisomy 18 syndrome, one case of trisomy 9 syndrome, and one case of chromosome 4p deletion).

CONCLUSIONS

Low-lying conus medullaris found during prenatal ultrasound examination were often associated with neural tube malformations or multi-systemic complex developmental malformations. The frequency of chromosomal abnormalities was 23.1%.

Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

In this paper, the influence of the transverse wave on sound propagation in a porous medium with a flexible structure is considered. The study is carried out in the time domain using the modified Biot theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The viscous exchanges between the fluid and the structure are described by fractional calculus. When a sound pulse arrives at normal incidence on a porous material with a flexible structure, the transverse waves interfere with the longitudinal waves during propagation because of the viscous interactions that appear between the fluid and the structure. By performing a calculation in the Laplace domain, the reflection and transmission operators are derived. Their time domain expressions depend on the Green functions of the longitudinal and transverse waves. In order to study the effects of the transverse wave on the transmitted longitudinal waves, numerical simulations of the transmitted waves in the time domain by varying the characteristic parameters of the medium are realized whether the transverse wave is considered or not.

Establishing density-dependent longitudinal sound speed in the vertebral lamina

Focused ultrasound treatments of the spinal cord may be facilitated using a phased array transducer and beamforming to correct spine-induced focal aberrations. Simulations can non-invasively calculate aberration corrections using x-ray computed tomography (CT) data that are correlated to density (ρ) and longitudinal sound speed (c_L). We aimed to optimize vertebral lamina-specific c_L(ρ) functions at a physiological temperature (37 °C) to maximize time domain simulation accuracy. Odd-numbered ex vivo human thoracic vertebrae were imaged with a clinical CT-scanner (0.511 × 0.511 × 0.5 mm), then sonicated with a transducer (514 kHz) focused on the canal via the vertebral lamina. Vertebra-induced signal time shifts were extracted from pressure waveforms recorded within the canals. Measurements were repeated 5× per vertebra, with 2.5 mm vertical vertebra shifts between measurements. Linear functions relating c_L with CT-derived density were optimized. The optimized function was c_L(ρ)=0.35(ρ-ρ_w)+ c_L,w m/s, where w denotes water, giving the tested laminae a mean bulk density of 1600 ± 30 kg/m³ and a mean bulk c_L of 1670 ± 60 m/s. The optimized lamina c_L(ρ) function was accurate to λ/16 when implemented in a multi-layered ray acoustics model. This modelling accuracy will improve trans-spine ultrasound beamforming.

Bayesian inference of human bone sample properties using ultrasonic reflected signals

The non-intrusiveness and low cost of ultrasonic interrogation is motivating the development of new means of detection of osteoporosis and other bone deficiencies. Bone is a porous media saturated with a viscous fluid and could thus be well characterized by the Biot model. The main purpose of this work is to present an in vitro methodology for the identification of the properties and structural parameters of the bone, adopting a statistical Bayesian inference technique using ultrasonic reflected signals at normal incidence. It is, in this respect, a companion paper to a previous work [J. Acoust. Soc. Am. 146, 3 (2019), pp. 1629-1640], where ultrasonic transmitted signals were considered. This approach allows the retrieval of some important parameters that characterize the bone structure and associated uncertainties. The method was applied to seven samples of bone extracted from femoral heads, immersed in water, and exposed to ultrasonic signals with a center frequency of ≈500 kHz. For all seven samples, signals at different sites were acquired to check the method robustness. The porosity, pore mean size and standard deviation, and the porous frame bulk modulus were all successfully identified using only ultrasonic reflected signals.

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.

Bayesian inference of a human bone and biomaterials using ultrasonic transmitted signals

Ultrasonic techniques could be good candidates to aid the assessment of osteoporosis detection, due to their non-intrusiveness and low cost. While earlier studies made use of the measured ultrasonic phase velocity and attenuation inside the bone, very few have considered an inverse identification of both the intrinsic pore microstructure and the mechanical properties of the bone, based on Biot's model. The main purpose of this work is to present an in vitro methodology for bone identification, adopting a statistical Bayesian inference technique using ultrasonic transmitted signals, which allows the retrieval of the identified parameters and their uncertainty. In addition to the bone density, Young's modulus and Poisson's ratio, the bone pore microstructure parameters (porosity, tortuosity, and viscous length) are identified. These additional microstructural terms could improve the knowledge on the correlations between bone microstructure and bone diseases, since they provide more information on the trabecular structure. In general, the exact properties of the saturating fluid are unknown (bone marrow and blood in the case of bone study) so in this work, the fluid properties (water) are identified during the inference as a proof of concept.

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.

Experimental observation of ultrasound fast and slow waves through three-dimensional printed trabecular bone phantoms

In this paper, ultrasound measurements of 1:1 scale three-dimensional (3D) printed trabecular bone phantoms are reported. The micro-structure of a trabecular horse bone sample was obtained via synchrotron x-ray microtomography, converted to a 3D binary data set, and successfully 3D-printed at scale 1:1. Ultrasound through-transmission experiments were also performed through a highly anisotropic version of this structure, obtained by elongating the digitized structure prior to 3D printing. As in real anisotropic trabecular bone, both the fast and slow waves were observed. This illustrates the potential of stereolithography and the relevance of such bone phantoms for the study of ultrasound propagation in bone.

Conventional, Bayesian, and Modified Prony's methods for characterizing fast and slow waves in equine cancellous bone

Conventional, Bayesian, and the modified least-squares Prony's plus curve-fitting (MLSP + CF) methods were applied to data acquired using 1 MHz center frequency, broadband transducers on a single equine cancellous bone specimen that was systematically shortened from 11.8 mm down to 0.5 mm for a total of 24 sample thicknesses. Due to overlapping fast and slow waves, conventional analysis methods were restricted to data from sample thicknesses ranging from 11.8 mm to 6.0 mm. In contrast, Bayesian and MLSP + CF methods successfully separated fast and slow waves and provided reliable estimates of the ultrasonic properties of fast and slow waves for sample thicknesses ranging from 11.8 mm down to 3.5 mm. Comparisons of the three methods were carried out for phase velocity at the center frequency and the slope of the attenuation coefficient for the fast and slow waves. Good agreement among the three methods was also observed for average signal loss at the center frequency. The Bayesian and MLSP + CF approaches were able to separate the fast and slow waves and provide good estimates of the fast and slow wave properties even when the two wave modes overlapped in both time and frequency domains making conventional analysis methods unreliable.

Fast characterization of two ultrasound longitudinal waves in cancellous bone using an adaptive beamforming technique

The received signal in through-transmission ultrasound measurements of cancellous bone consists of two longitudinal waves, called the fast and slow waves. Analysis of these fast and slow waves may reveal characteristics of the cancellous bone that would be good indicators of osteoporosis. Because the two waves often overlap, decomposition of the received signal is an important problem in the characterization of bone quality. This study proposes a fast and accurate decomposition method based on the frequency domain interferometry imaging method with a modified wave transfer function that uses a phase rotation parameter. The proposed method accurately characterized the fast and slow waves in the experimental study, and the residual intensity, which was normalized with respect to the received signal intensity, was less than -20 dB over the bone specimen thickness range from 6 to 15 mm. In the simulation study, the residual intensity was less than -20 dB over the specimen thickness range from 3 to 8 mm. Decomposition of a single received signal takes only 5 s using a laptop personal computer with a single central processing unit. The proposed method has great potential to provide accurate and rapid measurements of indicators of osteoporosis in cancellous bone.

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.

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.

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.

Estimation of multipath transmission parameters for quantitative ultrasound measurements of bone

When applying quantitative ultrasound (QUS) measurements to bone for predicting osteoporotic fracture risk, the multipath transmission of sound waves frequently occurs. In the last 10 years, the interest in separating multipath QUS signals for their analysis awoke, and led to the introduction of several approaches. Here, we compare the performances of the two fastest algorithms proposed for QUS measurements of bone: the modified least-squares Prony method (MLSP), and the space alternating generalized expectation maximization algorithm (SAGE) applied in the frequency domain. In both approaches, the parameters of the transfer functions of the sound propagation paths are estimated. To provide an objective measure, we also analytically derive the Cramér-Rao lower bound of variances for any estimator and arbitrary transmit signals. In comparison with results of Monte Carlo simulations, this measure is used to evaluate both approaches regarding their accuracy and precision. Additionally, with simulations using typical QUS measurement settings, we illustrate the limitations of separating two superimposed waves for varying parameters with focus on their temporal separation. It is shown that for good SNRs around 100 dB, MLSP yields better results when two waves are very close. Additionally, the parameters of the smaller wave are more reliably estimated. If the SNR decreases, the parameter estimation with MLSP becomes biased and inefficient. Then, the robustness to noise of the SAGE clearly prevails. Because a clear influence of the interrelation between the wavelength of the ultrasound signals and their temporal separation is observable on the results, these findings can be transferred to QUS measurements at other sites. The choice of the suitable algorithm thus depends on the measurement conditions.

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.

Transient ultrasound propagation in porous media using Biot theory and fractional calculus: application to human cancellous bone

A temporal model based on the Biot theory is developed to describe the transient ultrasonic propagation in porous media with elastic structure, in which the viscous exchange between fluid and structure are described by fractional derivatives. The fast and slow waves obey a fractional wave equation in the time domain. The solution of Biot's equations in time depends on the Green functions of each of the waves (fast and slow), and their fractional derivatives. The reflection and transmission operators for a slab of porous materials are derived in the time domain, using calculations in the Laplace domain. Their analytical expressions, depend on Green's function of fast and slow waves. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

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).

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.

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.

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.

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.

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.

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.

The dependencies of phase velocity and dispersion on volume fraction in cancellous-bone-mimicking phantoms

Frequency-dependent phase velocity was measured in eight cancellous-bone-mimicking phantoms consisting of suspensions of randomly oriented nylon filaments (simulating trabeculae) in a soft-tissue-mimicking medium (simulating marrow). Trabecular thicknesses ranged from 152 to 356 mum. Volume fractions of nylon filament material ranged from 0% to 10%. Phase velocity varied approximately linearly with frequency over the range from 300 to 700 kHz. The increase in phase velocity (compared with phase velocity in a phantom containing no filaments) at 500 kHz was approximately proportional to volume fraction occupied by nylon filaments. The derivative of phase velocity with respect to frequency was negative and exhibited nonlinear, monotonically decreasing dependence on volume fraction. The dependencies of phase velocity and its derivative on volume fraction in these phantoms were similar to those reported in previous studies on (1) human cancellous bone and (2) phantoms consisting of parallel nylon wires immersed in water.

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

Speed of sound measurements are widely used clinically to assess bone strength. Trabecular bone is an attenuating composite material in which negative values of velocity dispersion have been measured, this behavior remaining poorly explained physically. The aim of this work is to describe the ultrasonic propagation in trabecular bone modeled by infinite cylinders immersed in a saturating matrix, and to derive the physical determinants of velocity dispersion. A homogenization model accounting for the coupling of multiple scattering and absorption phenomena allows the computation of phase velocity and of dispersion while varying bone properties. The present model is adapted from the generalized self-consistent method developed in the work of Yang and Mal [(1994). "Multiple-scattering of elastic waves in a fiber-reinforced composite," J. Mech. Phys. Solids 42, 1945-1968]. It predicts negative values of velocity dispersion, in agreement with experimental results obtained in phantoms mimicking trabecular bone. In trabecular bone, mostly negative and also positive values of velocity dispersion are predicted by the model, which span within the range of values measured experimentally. Scattering effects are responsible for the negative values of dispersion, whereas the frequency dependence of the absorption coefficient in bone marrow and/or in the trabeculae results in an increase in dispersion, which may then become positive.

Interference between wave modes may contribute to the apparent negative dispersion observed in cancellous bone

Previous work has shown that ultrasonic waves propagating through cancellous bone often exhibit a linear-with-frequency attenuation coefficient, but a decrease in phase velocity with frequency (negative dispersion) that is inconsistent with the causality-imposed Kramers-Kronig relations. In the current study, interfering wave modes similar to those observed in bone are shown to potentially contribute to the observed negative dispersion. Biot theory, the modified Biot-Attenborogh model, and experimental results are used to aid in simulating multiple-mode wave propagation through cancellous bone. Simulations entail constructing individual wave modes exhibiting a positive dispersion using plausible velocities and amplitudes, and then summing the individual modes to create mixed-mode output wave forms. Results of the simulations indicate that mixed-mode wave forms can exhibit negative dispersion when analyzed conventionally under the assumption that only one wave is present, even when the individual interfering waves exhibit positive dispersions in accordance with the Kramers-Kronig relations. Furthermore, negative dispersion is observed when little or no visual evidence of interference exists in the time-domain data. Understanding the mechanisms responsible for the observed negative dispersion could aid in determining the true material properties of cancellous bone, as opposed to the apparent properties measured using conventional data analysis techniques.

Ultrasonic velocity dispersion in bovine cortical bone: an experimental study

Cortical bone quality is determinant in bone fragility and its ultrasonic evaluation has become possible in clinical practice. However, the interaction between a broadband ultrasonic pulse and this complex multiscale medium remains poorly understood. The frequency dependence of phase velocity, which may impact clinical measurements, has been sparsely investigated. Our objective is to evaluate the determinants of the frequency dependence of phase velocity in bovine femoral cortical bone samples using an in vitro ultrasonic transmission device. The apparent phase velocity varies quasilinearly on a 1 MHz restricted bandwidth around 4 MHz. After compensating for diffraction effects, significant differences in velocity dispersion are obtained according to the anatomical location. The microstructure of each sample is determined using an optical microscope, which allows assessing the dependence of dispersion on the type of bone microstructure. Mostly positive but also negative values of dispersion are measured. Negative dispersion is mainly obtained in samples constituted of mixed microstructure, which may be explained by phase cancellation effects due to the presence of different microstructures within the same sample. Dispersion is shown to be related to broadband ultrasonic attenuation values, especially in the radial direction. Results are compared with the local Kramers-Kronig relationships.

Negative dispersion in bone: the role of interference in measurements of the apparent phase velocity of two temporally overlapping signals

In this study the attenuation coefficient and dispersion (frequency dependence of phase velocity) are measured using a phase sensitive (piezoelectric) receiver in a phantom in which two temporally overlapping signals are detected, analogous to the fast and slow waves typically found in measurements of cancellous bone. The phantom consisted of a flat and parallel Plexiglas plate into which a step discontinuity was milled. The phase velocity and attenuation coefficient of the plate were measured using both broadband and narrowband data and were calculated using standard magnitude and phase spectroscopy techniques. The observed frequency dependence of the phase velocity and attenuation coefficient exhibit significant changes in their frequency dependences as the interrogating ultrasonic field is translated across the step discontinuity of the plate. Negative dispersion is observed at specific spatial locations of the plate at which the attenuation coefficient rises linearly with frequency, a behavior analogous to that of bone measurements reported in the literature. For all sites investigated, broadband and narrowband data (3-7 MHz) demonstrate excellent consistency. Evidence suggests that the interference between the two signals simultaneously reaching the phase sensitive piezoelectric receiver is responsible for this negative dispersion.

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.

A device for in vivo measurements of quantitative ultrasound variables at the human proximal femur

Quantitative ultrasound (QUS) at the calcaneus has similar power as a bone mineral density (BMD)- measurement using DXA for the prediction of osteoporotic fracture risk. Ultrasound equipment is less expensive than DXA and free of ionizing radiation. As a mechanical wave, QUS has the potential of measuring different bone properties than dual X-ray absorptiometry (DXA,) which depends on X-ray attenuation and might be developed into a tool of comprehensive assessment of bone strength. However, site-specific DXA at the proximal femur shows best performance in the prediction of hip fractures. To combine the potential of QUS with measurements directly at the femur, we developed a device for in vivo QUS measurements at this site. Methods comprise ultrasound transmission through the bone, reflection from the bone surface, and backscatter from the inner trabecular structure. The complete area of the proximal femur can be scanned except at the femoral head, which interferes with the ilium. To avoid edge artifacts, a subregion of the proximal femur in the trochanteric region was selected as measurement region. First, in vivo measurements demonstrate a good signal to noise ratio and proper depiction of the proximal femur on an attenuation image. Our results demonstrate the feasibility of in vivo measurements. Further improvements can be expected by refinement of the scanning technique and data evaluation method to enhance the potential of the new method for the estimation of bone strength.

Development of a method for ultrasound-guided placement of pedicle screws

Abstract-Many forms of spinal fusion involve the placement of long screws through the pedicles into the vertebral body. During the procedure, there is substantial risk of damage to vital neural and vascular structures due to the limited visibility of anatomic landmarks and high anatomic variability. As an alternative to current guidance systems, we have investigated the feasibility of performing ultrasound imaging through cancellous bone for the purpose of pedicle screw guidance. Quantitative ultrasonic characterization and A-mode imaging of seven defatted vertebral cancellous bone specimens was performed along the craniocaudal axis in water with unfocused, 1-MHz and 3.5- MHz broadband transducers. The center frequency attenuation increased considerably from 10.5 +/- 4.6 dB/cm at 1 MHz to 24.1 +/- 7.2 dB/cm at 3.5 MHz, while the speed of sound exhibited moderate positive dispersion, increasing from 1489 +/- 4.7 m/s at 1 MHz to 1494 +/- 4.2 m/s at 3.5 MHz. Despite the high attenuation and large specimen thickness (1.0-1.9 cm), A-mode imaging through cancellous bone to detect an aluminum reflector was possible in 83.2% and 70.1% of the cases at 1 MHz and 3.5 MHz, respectively. Specimen boundaries were identifiable with clinically sufficient average accuracy of 1.1 mm and 0.9 mm in the 1 MHz and 3.5 MHz A-mode images, respectively.

Prediction of negative dispersion by a nonlocal poroelastic theory

The objective of this work is to show that the negative dispersion of ultrasonic waves propagating in cancellous bone can be explained by a nonlocal version of Biot's theory of poroelasticity. The nonlocal poroelastic formulation is presented in this work and the exact solutions for one- and two-dimensional systems are obtained by the method of Fourier transform. The nonlocal phase speeds for solid- and fluid-borne waves show the desired negative dispersion where the magnitude of dispersion is strongly dependent on the nonlocal parameters and porosity. Dependence of the phase speed and attenuation is studied for both porosity and frequency variation. It is shown that the nonlocal parameter can be easily estimated by comparing the theoretical dispersion rate with experimental observations. It is also shown that the modes of Lamb waves show similar negative dispersion when predicted by the nonlocal poroelastic theory.

Instrumentation for in vivo ultrasonic characterization of bone strength

Although it has been more than 20 years since the first recorded use of a quantitative ultrasound (QUS) technology to predict bone fragility, the field has not yet reached its maturity. QUS has the potential to predict fracture risk in several clinical circumstances and has the advantages of being nonionizing, inexpensive, portable, highly acceptable to patients, and repeatable. However, the wide dissemination of QUS in clinical practice is still limited and suffering from the absence of clinical consensus on how to integrate QUS technologies in bone densitometry armamentarium. Several critical issues need to be addressed to develop the role of QUS within rheumatology. These include issues of technologies adapted to measure the central skeleton, data acquisition, and signal processing procedures to reveal bone properties beyond bone mineral quantity and elucidation of the complex interaction between ultrasound and bone structure. This article reviews the state-of-the art in technological developments applied to assess bone strength in vivo. We describe generic measurement and signal processing methods implemented in clinical ultrasound devices, the devices and their practical use, and performance measures. The article also points out the present limitations, especially those related to the absence of standardization, and the lack of comprehensive theoretical models. We conclude with suggestions of future lines and trends in technology challenges and research areas such as new acquisition modes, advanced signal processing techniques, and modelization.

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%.

Group velocity, phase velocity, and dispersion in human calcaneus in vivo

Commercial bone sonometers measure broadband ultrasonic attenuation and/or speed of sound (SOS) in order to assess bone status. Phase velocity, which is usually measured in frequency domain, is a fundamental material property of bone that is related to SOS, which is usually measured in time domain. Four previous in vitro studies indicate that phase velocity in human cancellous bone decreases with frequency (i.e., negative dispersion). In order to investigate frequency-dependent phase velocity in vivo, through-transmission measurements were performed in 73 women using a GE Lunar Achilles Insight commercial bone sonometer. Average phase velocity at 500 kHz was 1489 +/- 55 m/s (mean +/- standard deviation). Average dispersion rate was -59 +/- 52 m/sMHz. Group velocity was usually lower than phase velocity, as is expected for negatively dispersive media. Using a stratified model to represent cancellous bone, the reductions in phase velocity and dispersion rate in vivo as opposed to in vitro can be explained by (1) the presence of marrow instead of water as a fluid filler, and (2) the decreased porosity of bones of living (compared with deceased) subjects.

Bayesian estimation of the underlying bone properties from mixed fast and slow mode ultrasonic signals

We recently proposed that the observed apparent negative dispersion in bone can arise from the interference between fast wave and slow wave modes, each exhibiting positive dispersion [Marutyan et al., J. Acoust. Soc. Am. 120, EL55-EL61 (2006)]. In the current study, we applied Bayesian probability theory to solve the inverse problem: extracting the underlying properties of bone. Simulated mixed mode signals were analyzed using Bayesian probability. The calculations were implemented using the Markov chain Monte Carlo with simulated annealing to draw samples from the marginal posterior probability for each parameter.