Measurements of phase velocity and group velocity in human calcaneus

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.

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

The goal of this work was to show that the apparent negative dispersion of ultrasonic waves propagating in bone can arise from interference between fast and slow longitudinal modes, each exhibiting positive dispersion. Simulations were carried out using two approaches: one based on the Biot-Johnson model and one independent of that model. Results of the simulations are mutually consistent and appear to account for measurements from many laboratories that report that the phase velocity of ultrasonic waves propagating in cancellous bone decreases with increasing frequency (negative dispersion) in about 90% of specimens but increases with frequency in about 10%.

Measurement of speed of sound dispersion in soft tissues using a double frequency continuous wave method

We introduce a method for measuring the speed of sound dispersion. It combines a short pulse transmission followed by a long burst comprised of two frequencies, one being double that of the other. The method allows the determination of the speed of sound dispersion using a single transmission. To validate the method, the dispersion was first measured in plastic samples and then in in vitro soft tissues samples. The results obtained for Perspex samples are in excellent agreement with values reported in the literature. The dispersion index in soft tissues ranged for a bovine heart from 0.63 +/- 0.24 (m/s.MHz) at 1.5 MHz to 0.27 +/- 0.05 (m/s.MHz) at 4.5 MHz and for a turkey breast from 1.3 +/- 0.28 (m/s.MHz) at 1.75 MHz to 0.73 +/- 0.1 (m/s.MHz) at 3.8 MHz. The significant difference in the speed of sound dispersion index between the studied materials indicates that dispersion may be used as a new index for soft tissue characterization by ultrasound.

Determination of ultrasound phase velocity in trabecular bone using time dependent phase tracking technique

Ultrasound velocity is one of the key acoustic parameters for noninvasive diagnosis of osteoporosis. Ultrasound phase velocity can be uniquely measured from the phase of the ultrasound signal at a specified frequency. Many previous studies used fast Fourier transform (FFT) to determine the phase velocity, which may cause errors due to the limitations of FFT. The new phase tracking technique applied an adaptive tracking algorithm to detect the time dependent phase and amplitude of the ultrasound signal at a specified frequency. This overcame the disadvantages of FFT to ensure the accuracy of the ultrasound phase velocity. As a result, the new method exhibited high accuracy in the measurement of ultrasound phase velocity of two phantom blocks with the error less than 0.4%. 41 cubic trabecular samples from sheep femoral condyles were used in the study. The phase velocity of the samples using the new method had significantly high correlation to the bulk stiffness of the samples (r = 0.84) compared to the phase velocity measured using fast Fourier transform FFT (r = 0.14). In conclusion, the new method provided an accurate measurement of the ultrasound phase velocity in bone.

The frequency dependence of ultrasonic velocity and the anisotropy of dispersion in both freshly excised and formalin-fixed myocardium

The objectives of this study were to measure the frequency dependence of the ultrasonic velocity in myocardium and to quantify the frequency dependence of phase velocity as a function of the insonification angle relative to the predominant direction of the myofibers. Broadband phase spectroscopy data were acquired, spanning a frequency range of 3 to 8 MHz. Measurements were made on 36 tissue specimens cored from 12 freshly excised lamb hearts and were repeated after fixation with formalin. Measured phase velocities were found to be well characterized by a logarithmic fit. For freshly excised myocardium, the dispersion over the 3 to 8 MHz bandwidth was dependent on the direction of insonification, ranging from 1.2 m/s change for perpendicular insonification (across the myofibers) to 3.7 m/s for parallel insonification (along the myofibers). The effects of formalin-fixation resulted in a significant increase in dispersion for perpendicular insonification, but did not appreciably alter the dispersion for parallel insonification.

Effects of frequency-dependent attenuation and velocity dispersion on in vitro ultrasound velocity measurements in intact human femur specimens

Numerous studies have shown that ultrasonic velocity measured in bone provides a good assessment of osteoporotic fracture risk. However, a lack of standardization of signal processing techniques used to compute the speed of sound (SOS) complicates the comparison between data obtained with different commercial devices. In this study, 38 intact femurs were tested using a through-transmission technique and SOS determined using different techniques. The resulting difference in measured SOS was determined as functions of the attenuation and the velocity dispersion. A numerical simulation was used to explain how attenuation and dispersion impact two different SOS measurements (group velocity, velocity based on the first zero crossing of the signal). A new method aimed at compensating for attenuation was devised and led to a significant reduction in the difference between SOS obtained with both signal processing techniques. A comparison between SOS and X-ray density measurements indicated that the best correlation was reached for SOS based on the first zero crossing apparently because it used a marker located in the early part of the signal and was less sensitive to multipath interference. The conclusion is that first zero crossing velocity may be preferred to group velocity for ultrasonic assessment at this potential fracture site.

Kramers-Kronig analysis of attenuation and dispersion in trabecular bone

A restricted-bandwidth form of the Kramers-Kronig dispersion relations is applied to in vitro measurements of ultrasonic attenuation and dispersion properties of trabecular bone specimens from bovine tibia. The Kramers-Kronig analysis utilizes only experimentally measured properties and avoids extrapolation of ultrasonic properties beyond the known bandwidth. Compensation for the portions of the Kramers-Kronig integrals over the unknown bandwidth is partially achieved by the method of subtractions, where a subtraction frequency acts as an adjustable parameter. Good agreement is found between experimentally measured and Kramers-Kronig reconstructed dispersions. The restricted-bandwidth approach improves upon other forms of the Kramers-Kronig relations and may provide further insight into how ultrasound interacts with trabecular bone.

Optimal prediction of bone mineral density with ultrasonic measurements in excised human femur

Bone mineral density (BMD) measured with dual energy X-ray absorptiometry (DXA) techniques is the current gold standard for osteoporotic fracture risk prediction. Quantitative ultrasound (QUS) techniques in transmission measurements are, however, increasingly recognized as an alternative approach. It is feasible to select different QUS methods, one type being optimized to assess microarchitectural properties of bone structure and another to assess BMD. Broadband ultrasonic attenuation (BUA) and ultrasonic velocity (UV) measured on the proximal human femur have been shown to be both significantly correlated with BMD. However, a great diversity of algorithms has been reported to measure the time-of-flight used to derive UV values. The purpose of this study was to determine which procedure results in the optimal BMD prediction at the proximal femur from ultrasound measurements. Thirty-eight excised human femurs were measured in transmission with a pair of focused 0.5-MHz central frequency transducers. Two-dimensional scans were performed and radiofrequency (RF) signals were recorded digitally at each scan position. BUA was estimated and eight different signal processing techniques were performed to estimate UV. For each signal-processing technique UV was compared to BMD. We show that the best prediction of BMD was obtained with signal-processing techniques taking into account only the first part of the transmitted signal (r2BMD-SOS = 0.86). Moreover, we show that a linear multiple regression using both BUA and speed of sound (SOS) and applied to site-matched regions of interest improved the accuracy of BMD predictions (r2BMD-SOS/BUA = 0.95). Our results demonstrate that selecting specific signal-processing methods for QUS variables allows optimal assessment of BMD. Correlation is sufficiently high that this specific QUS method can be considered as a good surrogate of BMD.

The dependencies of phase velocity and dispersion on trabecular thickness and spacing in trabecular bone-mimicking phantoms

Frequency-dependent phase velocity was measured in trabecular-bone-mimicking phantoms consisting of two-dimensional arrays of parallel nylon wires (simulating trabeculae) with thicknesses ranging from 152 to 305 microm and spacings ranging from 700 to 1000 microm. Phase velocity varied approximately linearly with frequency over the range from 400 to 750 kHz. Dispersion was characterized by the slope of a linear least-squares regression fit to phase velocity versus frequency data. The increase in phase velocity (compared with that in water) at 500 kHz was approximately proportional to the (1) square of trabecular thickness, (2) inverse square of trabecular spacing, and (3) volume fraction occupied by nylon wires. The first derivative of phase velocity with respect to frequency was negative and exhibited nonlinear, monotonically decreasing dependencies on trabecular thickness and volume fraction. The dependencies of phase velocity and its first derivative on volume fraction in the phantoms were consistent with those reported in trabecular bone.

In vitro speed of sound measurement at intact human femur specimens

Quantitative ultrasound has been recognized as a useful tool for fracture risk prediction. Current measurement techniques are limited to peripheral skeletal sites. Our objective was to demonstrate the in vitro feasibility of ultrasonic velocity measurements on human proximal femur and to investigate the relationship between velocity and bone mineral density (BMD). Sound velocity images were computed from 2-D scans performed on 38 excised human femurs in transmission at 0.5 MHz. Different regions-of-interest were investigated. Dual x-ray absorptiometry scans have been achieved for BMD measurements in site-matched regions. Our study demonstrates the feasibility of ultrasonic velocity measurements at the hip with reasonable precision (coefficient of variation of 0.3%). The best prediction of BMD was reached in the intertrochanter region (r(2) = 0.91, p < 10(-4)), with a residual error of 0.06 g/cm(2) (10%). Because BMD measured at the femur is the best predictor of hip fracture risk, the highly significant correlation and small residual error found in this study suggest that speed of sound measurement at the femur might be a good candidate for hip fracture risk prediction.

Measurements of acoustic dispersion on calcaneus using spilt spectrum processing technique

The speed of sound (SOS) has become a useful tool in osteoporosis assessment, since it represents a combination of density and compressibility of bone tissue and should provide better information on bone quality and an estimate the fracture risk. In general, the speed of sound on dispersive material, such as bone tissue, depends strongly on frequency. Therefore, a measurement of velocity dispersion magnitude (VDM) might provide more important bone structure information than measurements of bone mineral density (BMD), SOS or broadband ultrasound attenuation (BUA). To obtain the velocity dispersion magnitude requires a sequence of pulses that have a frequency that is different from that used in conventional approaches. The measurement is complicated by the fact that pulse waveform will distort as the pulses propagate through the frequency-dependent medium. Alternatively, the phase velocity and velocity dispersion measurements also can be obtained on frequency-domain processing. However, the accuracy of those techniques is affected by the 2mpi ambiguity in the phase unwrapping process in frequency domain. And the spectrum approach is highly dependent on the gating window selection in time domain signals. The time-domain split spectrum processing (SSP) technique is proposed here to measure the phase velocity and the VDM. The SSP technique is also used to measure the SOS and VDM of two commercial calcaneus phantoms. Simulation results are in good agreement with the preset parameters of a model-based signal obtained using the SSP technique. In addition, in vitro SSP measurements agree with the manufacturer's specifications for two commercial calcaneus phantoms. The negative dispersion is also found in in vivo measurements on human heel. Finally, an approach based on the time domain SSP technique has potential clinical applications for osteoporosis diagnosis.

Comparison of measurements of phase velocity in human calcaneus to Biot theory

Biot's theory for elastic propagation in porous media has previously been shown to be useful for modeling the dependence of phase velocity on porosity in bovine cancellous bone in vitro. In the present study, Biot's theory is applied to measurements of porosity-dependent phase velocity in 53 human calcanea in vitro. Porosity was measured using microcomputed tomography for some samples (n = 23) and estimated based on bone mineral densitometry for the remaining samples (n = 30). The phase velocity at 500 kHz was measured in a water tank using a through-transmission technique. Biot's theory performed well for the prediction of the dependence of sound speed on porosity. The trend was quasilinear, but both the theory and experiment show similar slight curvature. The root mean square error (RMSE) of predicted versus measured sound speed was 15.8 m/s.

Transient propagation in media with classical or power-law loss

This paper addresses the problem of small-signal transient wave propagation in media whose absorption coefficient obeys power-law frequency dependence, i.e., alpha infinity omega n. Our approach makes use of previously derived relations between the absorption and dispersion based on the Kramers-Kronig relations. This, combined with a recently obtained solution to a causal convolution wave equation enable expressions to be obtained for one-dimensional transient propagation when n is in the range 0 < n < 3. For n = 2, corresponding to no dispersion, straightforward analytical expressions are obtained for a delta-function and a sinusoidal step function sources and these are shown to correspond to relations previously derived. For other values of n, the effects of dispersion are accounted for by using Fourier transforms. Examples are used to illustrate the results for normal and anomalous dispersive media and to examine the question as to the conditions under which the effects of dispersion should be accounted for, especially for wideband ultrasound pulses of the type used in B-mode tissue imaging. It is shown that the product of the attenuation and total propagation path can be used as a criterion for judging whether dispersion needs to be accounted for.

Quantitative assessment of osteoporosis from the tibia shaft by ultrasound techniques

Bone mineral density (BMD) is used as a clinical estimate of the risk of fracture. Ultrasound provides an alternative or complement to X-ray based methods of bone densitometry for determining BMD. Among ultrasonic characteristics, the speed of sound (SOS) is a useful tool for assessment of osteoporosis because, as recently reported, it represents a combination of density and compressibility of bone tissue. Thus, it might provide better information on bone quality to estimate the fracture risk. In this paper, a dual-transducer ultrasound technique was employed to measure the mean ultrasound propagation speed of the cortical layer as well as the cancellous layer at the tibia shaft. Encouraging results from 18 outpatients showed a high correlation (r = 0.93) between measurements of BMD and those from dual energy X-ray absorptiometry (DEXA).

Effects of extracorporeal shock wave therapy and radial pressure wave therapy on elasticity and microstructure of equine cortical bone

OBJECTIVE

To measure changes in the modulus of elasticity (E) and describe histologic findings after extracorporeal shock wave therapy and radial pressure wave therapy on equine cortical bone specimens.

SAMPLE POPULATION

16 bone specimens from the proximodorsal cortex of an equine third metacarpal or metatarsal bone.

PROCEDURE

Baseline E was determined by the density (p) and unidirectional ultrasound transmission velocity (C) of each specimen according to the equation E = pC2. Eight specimens were treated with 500 pulses of 0.15 mJ/mm2 of extracorporeal shock wave therapy, and 8 specimens were treated with 500 pulses of 0.16 mJ/mm2of radial pressure wave therapy. After treatment, C was determined again. Four treatment sessions resulted in 2,000 pulses and 5 C measurements. The p of each sample was measured again. Mean post-treatment E was calculated for each group. Nondecalcified sections of all specimens were stained with toluidine blue or basic fuchsin for histologic evaluation.

RESULTS

Overall treatment group effect was not significant for C or E. Final E was not different from baseline values for extracorporeal shock wave therapy and radial pressure wave therapy. No histologic changes could be attributed to either treatment modality.

CONCLUSIONS AND CLINICAL RELEVANCE

Extracorporeal shock wave therapy and radial pressure wave therapy did not affect the material properties of equine bone at the energy and pulse values used in this study.

In vitro acoustic waves propagation in human and bovine cancellous bone

The acoustic behavior of cancellous bone with regard to its complex poroelastic nature has been investigated. The existence of two longitudinal modes of propagation is demonstrated in both bovine and human cancellous bone. Failure to take into account the presence of these two waves may result in inaccurate material characterization.

INTRODUCTION

Acoustic wave propagation is now a commonly used nondestructive method for cancellous bone characterization. However, wave propagation in this material may be affected by fluid-solid interactions inherent to its poroelastic nature, resulting in two different longitudinal waves. This phenomenon has been demonstrated in previous studies and is in agreement with Biot's theory. The purpose of this paper is to extend these findings to human trabecular bone and to thoroughly investigate these two waves.

MATERIALS AND METHODS

Sixty human and 14 bovine cancellous bone cubic specimens were tested in vitro in three different directions using an immersion acoustic transmission method. Original procedures were developed to quantify both velocity and attenuation characteristics of each wave. In term of attenuation, a modified broadband ultrasound attenuation (BUA), describing the rate of change of the frequency-dependent attenuation, was defined for each wave (FDUA).

RESULTS

Both waves were identified in most of the specimens. The fast wave velocities demonstrated a negative linear correlation with porosity (1500-2300 m/s, R2 = 0.44, p < 10(-3)), whereas the slow wave velocities exhibited two different behaviors: (1) a first set of data clearly dependent on porosity showing a positive linear correlation (1150-1450 m/s, R2 = 0.26, p < 10(-3)) and (2) a second group independent on porosity. The fast wave FDUA (20-140 dB/cmMHz) showed a parabolic behavior and reached a maximum for 75% porosity (second degree relationship R2 = 0.41,p < 10(-3)), whereas a positive linear behavior was observed for the slow wave FDUA (15-40 dB/cmMHz; R2 = 0.15, p < 10(-2)).

CONCLUSIONS

Existence of two wave propagation modes were demonstrated in human cancellous bone. Our data suggest that, in some cases, the amplitude of the slow wave is much larger than the amplitude of the fast wave. For this reason, care should be taken when using measurement systems that incorporate simple threshold detection because the fast wave could remain undetected. Moreover, failure to consider the presence of these two waves could result in an inaccurate quantification of cancellous bone physical properties.

Acoustic wave propagation in bovine cancellous bone: application of the Modified Biot-Attenborough model

Acoustic wave propagation in bovine cancellous bone is experimentally and theoretically investigated in the frequency range of 0.5-1 MHz. The phase velocity, attenuation coefficient, and broadband ultrasonic attenuation (BUA) of bovine cancellous bone are measured as functions of frequency and porosity. For theoretical estimation, the Modified Biot-Attenborough (MBA) model is employed with three new phenomenological parameters: the boundary condition, phase velocity, and impedance parameters. The MBA model is based on the idealization of cancellous bone as a nonrigid porous medium with circular cylindrical pores oriented normal to the surface. It is experimentally observed that the phase velocity is approximately nondispersive and the attenuation coefficient linearly increases with frequency. The MBA model predicts a slightly negative dispersion of phase velocity linearly with frequency and the nonlinear relationships of attenuation and BUA with porosity. The experimental results are in good agreement with the theoretical results estimated with the MBA model. It is expected that the MBA model can be usefully employed in the field of clinical bone assessment for the diagnosis of osteoporosis.

Multiple scattering in a trabecular bone: influence of the marrow viscosity on the effective properties

The Foldy and the Waterman and Truell approximations are used to determine the effective properties of the coherent wave that emerges after multiple scattering of a plane longitudinal fast wave by the largest pores in a trabecular bone. The unit scattering cell considered is either a single pore or two close cylindrical pores (cluster), at a fixed overall bone porosity. In the cluster case, the effective attenuation is about twice that obtained with one single pore per scatterer. It is shown that taking into account the marrow viscosity leads only to minor differences on the effective dispersion and attenuation.

Characterization of trabecular bone using the backscattered spectral centroid shift

Ultrasonic attenuation in bone in vivo is generally measured using a through-transmission method at the calcaneus. Although attenuation in calcaneus has been demonstrated to be a useful predictor for osteoporotic fracture risk, measurements at other clinically important sites, such as hip and spine, could potentially contain additional useful diagnostic information. Through-transmission measurements may not be feasible at these sites due to complex bone shapes and the increased amount of intervening soft tissue. Centroid shift from the backscattered signal is an index of attenuation slope and has been used previously to characterize soft tissues. In this paper, centroid shift from signals backscattered from 30 trabecular bone samples in vitro were measured. Attenuation slope also was measured using a through-transmission method. The correlation coefficient between centroid shift and attenuation slope was -0.71. The 95% confidence interval was (-0.86, -0.47). These results suggest that the backscattered spectral centroid shift may contain useful diagnostic information potentially applicable to hip and spine.

Scattering by a fluid cylinder in a porous medium: application to trabecular bone

In a trabecular bone, considered as a nondissipative porous medium, the scattering of an incident wave by cylindrical pores larger than the wavelength is studied. The goal is to know if scattering alone may cause such a high attenuation as that observed in calcaneus. The porous medium is modelized via Biot's theory and the scattering by a single pore is characterized from the definition of a scattering matrix. An approximation of weakly disordered medium is then discussed to estimate the effective attenuation and dispersion as a function of frequency. These effective properties are shown to be different of those measured on calcaneus, due to the neglect of wave conversions during the scattering process.

Acoustical characterization of bone using a cylindrical model and time of flight method: edge reconstruction and ultrasound velocity determination in cortical bone and in medullar marrow

Our objective is to evaluate the external and internal dimensions of bone diaphysis and the speed of sound in cortical bone and in medullar marrow. The diaphysis is modelled by a cylindrical hollow tube. The theory of rays is used and an approximation allows us to break free from the data gained by ultrasonic field amplitude. Then, acoustical and dimensional parameters are only related to the time of flight of reflected and transmitted acoustic echoes in the tube. From the arrival time of particular echoes, the inverse problem resolution then allows us to experimentally determine the sought parameters. This method is validated in vitro on a bovine femur and gives satisfactory results.

A stratified model to predict dispersion in trabecular bone

Frequency-dependent phase velocity (dispersion) has previously been measured in trabecular bone by several groups. In contrast to most biologic tissues, phase velocity in trabecular bone tends to decrease with frequency. A stratified model, consisting of alternating layers of bone and marrow (in vivo) or water (in vitro), has been employed in an attempt to explain this phenomenon. Frequency-dependent phase velocity was measured from 300 to 700 kHz in 1) phantoms consisting of regularly spaced thin parallel layers of polystyrene sheets in water and 2) 30 calcaneus samples in vitro. For the polystyrene phantoms, the agreement between theory and experiment was good. For the calcaneus samples, the model has some limited usefulness (uncertainty of about 5%) in predicting average phase velocity. More importantly, the model seems to perform consistently well for predicting the frequency dependence of phase velocity in calcaneus.

Relationships among calcaneal backscatter, attenuation, sound speed, hip bone mineral density, and age in normal adult women

The present study was undertaken in order to investigate the use of calcaneal ultrasonic backscatter for the application of diagnosis of osteoporosis. Broadband ultrasonic attenuation (BUA), speed of sound (SOS), the average backscatter coefficient (ABC), and the hip bone mineral density (BMD) were measured in calcanea in 47 women (average age: 58 years, standard deviation: 13 years). All three ultrasound variables had comparable correlations with hip BMD (around 0.5). As reported previously by others, BUA and SOS were rather highly correlated with each other. The logarithm of the ABC was only moderately correlated with the other two. The three ultrasound parameters exhibited similar moderate negative correlations with age. These results taken collectively suggest that the ABC may carry important diagnostic information independent of that contained in BUA and SOS and, therefore, may be useful as an adjunct measurement in the diagnosis of osteoporosis.

Experimental validation of the use of Kramers-Kronig relations to eliminate the phase sheet ambiguity in broadband phase spectroscopy

The technique of broadband phase spectroscopy proposed in 1978 by Sachse and Pao [J. Appl. Phys. 49, 4320-4327 (1978)] determines the phase velocity as a function of frequency from the Fourier transforms of a received reference and through-sample signal. Although quite successful, this approach can be influenced by an ambiguity in the phase velocity calculation which stems from the boundedness of the inverse tangent operation used to calculate phase. Several empirical approaches to resolve the phase ambiguity have been reported. An alternative approach that has not previously been considered appeals to the causal nature of the measurements. This article experimentally validates a method which uses the causally consistent Kramers-Kronig relations to eliminate the ambiguity in phase spectroscopy-derived phase velocity calculations. Broadband pulse and narrow-band tone burst measurements were performed on three gelatin-based phantoms containing different concentrations of graphite particles (0%, 10%, and 20% by volume). The phantoms were constructed to have attenuation coefficients which vary approximately linear-with-frequency, a dependence exhibited by many soft tissues. The narrow-band phase velocity measurements do not suffer from a phase ambiguity, and thus they serve as a "gold standard" against which the broadband phase velocity measurements are compared. The experimental results illustrate that using the Kramers-Kronig dispersion relations in conjunction with phase spectroscopy-derived phase velocity measurements is an effective means by which to resolve the phase sheet ambiguity in broadband phase spectroscopy.

Ultrasonic attenuation in human calcaneus from 0.2 to 1.7 MHz

Ultrasonic attenuation has been demonstrated to be a useful measurement in the diagnosis of osteoporosis. Most studies have employed ultrasound in a range of frequencies from about 200 kHz-300 kHz to 600 kHz-1 MHz, and many have assumed a linear dependence of attenuation on frequency. In order to investigate the attenuation properties of human calcaneus at higher frequencies, 16 defatted human calcanea were interrogated in vitro using two matched pairs of transducers with center frequencies of 500 kHz and 2.25 MHz. The linear dependence of attenuation on frequency seems to extend up to at least 1.7 MHz. The correlation between attenuation coefficient and frequency from 400 kHz to 1.7 MHz was r = 0.999 (95% confidence interval, CI, = 0.998-1.00). The measurements suggest that some deviations from linear frequency dependence of attenuation may occur at lower frequencies (below 400 kHz), however.

A numerical method to predict the effects of frequency-dependent attenuation and dispersion on speed of sound estimates in cancellous bone

Many studies have demonstrated that time-domain speed-of-sound (SOS) measurements in calcaneus are predictive of osteoporotic fracture risk. However, there is a lack of standardization for this measurement. Consequently, different investigators using different measurement systems and analysis algorithms obtain disparate quantitative values for calcaneal SOS, impairing and often precluding meaningful comparison and/or pooling of measurements. A numerical method has been developed to model the effects of frequency-dependent attenuation and dispersion on transit-time-based SOS estimates. The numerical technique is based on a previously developed linear system analytic model for Gaussian pulses propagating through linearly attenuating, weakly dispersive media. The numerical approach is somewhat more general in that it can be used to predict the effects of arbitrary pulse shapes and dispersion relationships. The numerical technique, however, utilizes several additional assumptions (compared with the analytic model) which would be required for the practical task of correcting existing clinical databases. These include a single dispersion relationship for all calcaneus samples, a simple linear model relating phase velocity to broadband ultrasonic attenuation, and a constant calcaneal thickness. Measurements on a polycarbonate plate and 30 human calcaneus samples were in good quantitative agreement with numerical predictions. In addition, the numerical approach predicts that in cancellous bone, frequency-dependent attenuation tends to be a greater contributor to variations in transit-time-based SOS estimates than dispersion. This approach may be used to adjust previously acquired individual measurements so that SOS data recorded with different devices using different algorithms may be compared in a meaningful fashion.