The effects of frequency-dependent attenuation and dispersion on sound speed measurements: applications in human trabecular bone

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.

Despeckling of medical ultrasound images

Speckle noise is an inherent property of medical ultrasound imaging, and it generally tends to reduce the image resolution and contrast, thereby reducing the diagnostic value of this imaging modality. As a result, speckle noise reduction is an important prerequisite, whenever ultrasound imaging is used for tissue characterization. Among the many methods that have been proposed to perform this task, there exists a class of approaches that use a multiplicative model of speckled image formation and take advantage of the logarithmical transformation in order to convert multiplicative speckle noise into additive noise. The common assumption made in a dominant number of such studies is that the samples of the additive noise are mutually uncorrelated and obey a Gaussian distribution. The present study shows conceptually and experimentally that this assumption is oversimplified and unnatural. Moreover, it may lead to inadequate performance of the speckle reduction methods. The study introduces a simple preprocessing procedure, which modifies the acquired radio-frequency images (without affecting the anatomical information they contain), so that the noise in the log-transformation domain becomes very close in its behavior to a white Gaussian noise. As a result, the preprocessing allows filtering methods based on assuming the noise to be white and Gaussian, to perform in nearly optimal conditions. The study evaluates performances of three different, nonlinear filters--wavelet denoising, total variation filtering, and anisotropic diffusion--and demonstrates that, in all these cases, the proposed preprocessing significantly improves the quality of resultant images. Our numerical tests include a series of computer-simulated and in vivo experiments.

In vitro ultrasonic characterization of human cancellous femoral bone using transmission and backscatter measurements: relationships to bone mineral density

Thirty-eight slices of pure trabecular bone 1-cm thickness were extracted from human proximal femurs. A pair of 1-MHz central frequency transducers was used to measure quantitative ultrasound (QUS) parameters in transmission [normalized broadband ultrasound attenuation (nBUA), speed of sound (SOS)] and in backscatter [broadband ultrasound backscatter (BUB)]. Bone mineral density (BMD) was measured using clinical x-ray quantitative computed tomography. Site-matched identical region of interest (ROIs) of 7 x 7 mm2 were positioned on QUS and QCT images. This procedure resulted in 605 ROIs for all the specimens data pooled together. The short-term precision of the technique expressed in terms of CV was found to be 2.3% for nBUA, 0.3% for SOS and 4.5% for BUB. Significant linear correlation between QUS and BMD were found for all the 605 ROIs pooled, with r2 values of 0.73, 0.77, and 0.58 for nBUA, SOS, and BUB, respectively (all p < 0.05). For the BUB, the best regression was obtained with a polynomial fit of second order (r2 = 0.63). An analysis of measurements errors was developed. It showed that the residual variability of SOS is almost completely predicted by measurements errors, which is not the case for BUA and BUB, suggesting a role for micro-architecture in the determination of BUA and BUB.

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.

Measurements of the anisotropy of ultrasonic velocity in freshly excised and formalin-fixed myocardial tissue

The objective of this study was to quantify the anisotropy of ultrasonic velocity in freshly excised myocardial tissue and to examine the effects of formalin-fixation. Through-transmission radio-frequency-based measurements were performed on ovine and bovine myocardial specimens from 24 different hearts. A total of 81 specimens were obtained from specific locations within each heart to investigate the possibility of regional differences in anisotropy of velocity in the left ventricular wall and septum. No regional differences were observed for either lamb or cow myocardial specimens. In addition, no specific species-dependent differences were observed between ovine and bovine myocardium. Average values of velocity at room temperature for perpendicular and parallel insonification were 1556.9 +/- 0.6 and 1565.2 +/- 0.7 m/s (mean +/- standard error), respectively, for bovine myocardium (N=45) and 1556.3 +/- 0.6 and 1564.7 +/- 0.7 m/s for ovine myocardium (N=36). Immediately after measurements of freshly excised myocardium, ovine specimens were fixed in formalin for at least one month and then measurements were repeated. Formalin-fixation appears to increase the overall velocity at all angles of insonification and to increase the magnitude of anisotropy of velocity.

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.

Prediction of density and mechanical properties of human trabecular bone in vitro by using ultrasound transmission and backscattering measurements at 0.2-6.7 MHz frequency range

The ultrasound (US) backscattering method has been introduced as an alternative for the through-transmission measurement of sound attenuation and speed in diagnosis of osteoporosis. Both attenuation and backscattering depend strongly on the US frequency. In this study, 20 human trabecular bone samples were measured in transmission and pulse-echo geometry in vitro. The aim of the study was to find the most sensitive frequency range for the quantitative ultrasound (QUS) analyses. Normalized broadband US attenuation (nBUA), speed of sound (SOS), broadband US backscatter (BUB) and integrated reflection coefficient (IRC) were determined for each sample. The samples were spatially scanned with five pairs of US transducers covering a frequency range of 0.2-6.7 MHz. Furthermore, mechanical properties and density of the same samples were determined. At all frequencies, SOS, BUB and IRC showed statistically significant linear correlations with the mechanical properties or density of human trabecular bone (0.51 < r < 0.82, 0.54 < r < 0.81 and 0.70 < r < 0.85, respectively). In contrast to SOS, IRC and BUB, nBUA showed statistically significant correlations with mechanical parameters or density at the centre frequency of 1 MHz only. Our results suggest that frequencies up to 5 MHz can be useful in QUS analyses for the prediction of bone mechanical properties and density. Since the use of higher frequencies provides better axial and spatial resolution, improved structural analyses may be possible. While extensive attenuation of high frequencies in trabecular bone limits the clinically feasible frequency range, selection of optimal frequency range for in vivo QUS application should be carefully considered.

Phase unwrapping for 2-D blind deconvolution of ultrasound images

In most approaches to the problem of two-dimensional homomorphic deconvolution of ultrasound images, the estimation of a corresponding point-spread function (PSF) is necessarily the first stage in the process of image restoration. This estimation is usually performed in the Fourier domain by either successive or simultaneous estimation of the amplitude and phase of the Fourier transform (FT) of the PSE This paper addresses the problem of recovering the FT-phase of the PSF, which is an important reconstruction problem by itself. The purpose of this paper is twofold. First, it provides a theoretical framework, establishing that the FT-phase of the PSF can be effectively estimated by a proper smoothing of the FT-phase of the appropriate radio-frequency (RF) image. Second, it presents a novel approach to the estimation of the FT-phase of the PSF, by solving a continuous Poisson equation over a predefined smooth subspace, in contrast to the discrete Poisson equation solver used for the classical least mean squares phase unwrapping algorithms, followed by a smoothing procedure. The proposed approach is possible due to the distinct properties of the FT-phases, among which the most important property is the availability of precise values of their partial derivatives. This property overcomes the main disadvantage of the discrete schemes, which routinely use wrapped (principal) values of the phase in order to approximate its partial derivatives. Since such an approximation is feasible subject to the restriction that the partial phase differences do not exceed pi in absolute value, the discrete schemes perform satisfactory only for few practical situations. The proposed approach is shown to be independent of this restriction and, thus, it performs for a wider class of the phases with significantly lower errors. The main advantages of the novel method over the algorithms based on discrete schemes are demonstrated in a series of computer simulations and for in vivo measurements.

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.

Robust estimation of ultrasound pulses using outlier-resistant de-noising

A different approach to the problem of estimation of the ultrasound pulse spectrum, which usually arises as a part of ultrasound image restoration algorithms, is presented. It is shown that this estimation problem can be reformulated in terms of a de-noising problem. In this formulation, the log-spectrum of a radio-frequency line (RF-line) is viewed as a noisy measurement of the signal that needs to be estimated, i.e., the ultrasound pulse log-spectrum. The log-spectrum of the tissue reflectivity function (i.e., tissue response) is considered as the noise to be rejected. The contribution of the paper is twofold. First, it provides statistical description of the reflectivity function log-spectrum for the case, when the samples of the reflectivity function are independent identically distributed (i.i.d.) Gaussian random variables. Moreover, it is shown that the problem of the pulse spectrum recovery is essentially a de-noising problem. Consequently, it is suggested to solve the problem within the framework of the de-noising by wavelet shrinkage. Second, a computationally efficient algorithm is proposed for the pulse-spectrum estimation, which can be viewed as a modified version of the classical Donoho's three-step de-noising procedure. This modification is necessary, because of specific properties of the noise to be rejected. It is shown, that whenever the samples of the reflectivity function can be assumed to be i.i.d. Gaussian random variables, the samples of its log-spectrum obey the Fisher-Tippet distribution. For this type of noise, straightforward implementation of the standard de-noising can cause serious estimation errors. In order to overcome this difficulty, an outlier-resistant de-noising is performed. The unique properties of this modified de-noising algorithm allow estimating the pulse spectrum adaptively to its properties, as they are continuously influenced by the frequency-dependent attenuation process. The performance of the proposed algorithm is examined in a series of computer-simulations. It is shown that this algorithm, developed on the assumption of the "Gaussian" reflectivity function, remains applicable for broader classes of distributions. The results obtained in a series of in vivo experiments reveal superior performance of the novel approach over some of alternative estimation techniques, e.g., cepstrum-based estimation.

A high-resolution technique for ultrasound harmonic imaging using sparse representations in Gabor frames

Over the last few decades there were dramatic improvements in ultrasound imaging quality with the utilization of harmonic frequencies induced by both tissue and echo-contrast agents. The advantages of harmonic imaging cause rapid penetration of this modality to diverse clinical uses, among which myocardial perfusion determination seems to be the most important application. In order to effectively employ the information, comprised in the higher harmonics of the received signals, this information should be properly extracted. A commonly used method of harmonics separation is linear filtering. One of its main shortcomings is the inverse relationship between the detectability of the contrast agent and the axial resolution. In this paper, a novel, nonlinear technique is proposed for separating the harmonic components, contained in the received radio-frequency images. It is demonstrated that the harmonic separation can be efficiently performed by means of convex optimization. It performs the separation without affecting the image resolution. The procedure is based on the concepts of sparse signal representation in overcomplete signal bases. A special type of the sparse signal representation, that is especially suitable for the problem at hand, is explicitly described. The ability of the novel technique to acquire "un-masked," second (or higher) harmonic images is demonstrated in series of computer and phantom experiments.

A Gaussian framework for modeling effects of frequency-dependent attenuation, frequency-dependent scattering, and gating

For a wide range of applications in medical ultrasound, power spectra of received signals are approximately Gaussian. It has been established previously that an ultrasound beam with a Gaussian spectrum propagating through a medium with linear attenuation remains Gaussian. In this paper, Gaussian transformations are derived to model the effects of scattering (according to a power law, as is commonly applicable in soft tissues, especially over limited frequency ranges) and gating (with a Hamming window, a commonly used gate function). These approximations are shown to be quite accurate even for relatively broad band systems with fractional bandwidths approaching 100%. The theory is validated by experiments in phantoms consisting of glass particles suspended in agar.

Blind deconvolution of ultrasound sequences using nonparametric local polynomial estimates of the pulse

The problem of reconstructing the reflectivity of a biological tissue is examined by means of blind deconvolution of the echo ultrasound signals. It is shown that the quality of the reconstruction procedure can be significantly improved when initially the ultrasonic pulse is accurately estimated. A new approach to the estimation of the ultrasound pulse echo sequences is proposed, using local polynomial approximation, which is closely related to the wavelet transform theory. This approach can be viewed as a modification of homomorphic deconvolution, by using bases different from the Fourier basis of the space of square-integrable functions L2. The bases used here are the orthogonal compactly supported wavelet bases. It is shown that the locality of the estimate can be extremely useful in number of cases of practical interest, resulting in estimates with smaller root-mean squared (rms) errors, as compared with estimates employing the Fourier basis. This approach is applied to ultrasound signals, for estimation of the ultrasound pulse log-spectrum from the log-spectrum of radio-frequency (RF) sequences. It is shown, conceptually and experimentally, that the proposed approach can provide robust and rapidly computed estimates of the ultrasound pulses from the RF-sequences, as obtained in the process of tissue scanning. The pulse phase was recovered using the minimum-phase assumption, which was found to hold for the transducers in use. The obtained pulse estimates are used for the deconvolution of the RF-sequences, which result in stable estimates of the tissue reflectivity function, fairly independent of the properties of the imaging system. Simulated data, data obtained from several phantoms and from in vitro experiments have been processed and the results seem to be quite promising.

Simultaneous measurement of sound velocity and wall thickness of a tube

A method for simultaneously measuring the sound propagation velocity and the thickness of each wall on the opposite sides of a tube is presented. The method uses a pair of ultrasound transducers to produce two reflected pulses from the outer and inner surfaces of the tube wall on the each side, and two transmitted pulses, one with and one without the tube sample between the two transducers. Using the time-domain analysis, sound velocity and wall thickness of the tube are determined from the time delays between the three pairs of ultrasound pulses, whereas using the frequency-domain analysis, phase velocity, group velocity, and wall thickness of the tube are determined from the phase differences between the three pairs of ultrasound pulses. Results of measurements on five tube samples are reported.

Frequency decomposition and compounding of ultrasound medical images with wavelet packets

Ultrasound beams propagating in biological tissues undergo distortions due to local inhomogeneities of the acoustic parameters and the nonlinearity of the medium. The spectral analysis of the radio-frequency (RF) backscattered signals may yield important clinical information in the field of tissue characterization, as well as enhancing the detectability of tissue parenchymal diseases. In this paper, we propose a new tissue spectral imaging technique based on the wavelet packets (WP) decomposition. In a conventional ultrasound imaging system, the received echo-signals are generally decimated to generate a medical image, with a loss of information. With the proposed approach, all the RF data are processed to generate a set of frequency subband images. The ultrasound echo signals are simultaneously frequency decomposed and decimated, by using two quadrature mirror filters, followed by a dyadic subsampling. In addition, to enhance the lesion detectability and the image quality, we apply a nonlinear filter to reduce noise in each subband image. The proposed method requires simple additional signal processing and it can be implemented on any real-time imaging system. The frequency subband images, which are available simultaneously, can be either used in a multispectral display or summed up together to reduce speckle noise. To localize the different frequency response in the tissues, we propose a multifrequency display method where three different subband images, chosen among those available, are encoded as red, green, and blue intensities (RGB) to create a false-colored RGB image. According to the clinical application, different choices can evidence different spectral proprieties in the biological tissue under investigation. To enhance the lesion contrast in a grey-level image, one of the possible methods is the summation of the images obtained from narrow frequency subbands, according to the frequency compounding technique. We show that by adding the denoised subband images created with the WP decomposition, the contrast-to-noise ratio in two phantom images is largely increased.

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.

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.

Relationships of ultrasonic backscatter with ultrasonic attenuation, sound speed and bone mineral density in human calcaneus

Ultrasonic attenuation and sound speed have been investigated in trabecular bone by numerous authors. Ultrasonic backscatter has received much less attention. To investigate relationships among these three ultrasonic parameters and bone mineral density (BMD), 30 defatted human calcanei were investigated in vitro. Normalized broadband ultrasonic attenuation (nBUA), sound speed (SOS), and logarithm of ultrasonic backscatter coefficient (LBC) were measured. Bone mineral density was assessed using single-beam dual energy x-ray absorptiometry (DEXA). The correlation coefficients of least squares linear regressions of the three individual ultrasound (US) parameters with BMD were 0.84 (nBUA), 0.84 (SOS) and 0.79 (LBC). The 95% confidence intervals for the correlation coefficients were 0. 69-0.92 (nBUA), 0.68-0.92 (SOS) and 0.60-0.90 (LBC). The correlations among pairs of US variables ranged from 0.63-0.79. Variations in nBUA accounted for r(2) = 62% of the variations in LBC. Variations in SOS accounted for r(2) = 40% of the variations in LBC. These results suggest that ultrasonic backscattering properties may contain substantial information not already contained in nBUA and SOS. A multiple regression model including all three US variables was somewhat more predictive of BMD than a model including only nBUA and SOS.

Anisotropy of ultrasonic backscatter and attenuation from human calcaneus: implications for relative roles of absorption and scattering in determining attenuation

Although bone sonometry has been demonstrated to be useful in the diagnosis of osteoporosis, much remains to be learned about the processes governing the interactions between ultrasound and bone. In order to investigate these processes, ultrasonic attenuation and backscatter in two orientations were measured in 43 human calcaneal specimens in vitro at 500 kHz. In the mediolateral (ML) orientation, the ultrasound propagation direction is approximately perpendicular to the trabecular axes. In the anteroposterior (AP) orientation, a wide range of angles between the ultrasound propagation direction and trabecular axes is encountered. Average attenuation slope was 18% greater while average backscatter coefficient was 50% lower in the AP orientation compared with the ML orientation. Backscatter coefficient in both orientations approximately conformed to a cubic dependence on frequency, consistent with a previously reported model. These results support the idea that absorption is a greater component of attenuation than scattering in human calcaneal trabecular bone.

Measurements of phase velocity and group velocity in human calcaneus

Ultrasonic velocity in calcaneus correlates highly with bone mineral density, which is a good predictor of osteoporotic fracture risk. Several commercial bone sonometers perform a velocity measurement based on the transit time of a broadband pulse to assess skeletal status. This approach is somewhat problematic, however, because several authors have reported ambiguities in measurements in calcaneus. Phase velocity is an alternative that may be less dependent on device spectral characteristics. In addition, dispersion (the frequency-dependence of phase velocity) is a fundamental property worth investigating to increase understanding of interaction between ultrasound and bone. To compare two group-velocity measurement methods and one phase-velocity measurement method, a polycarbonate sample (for method validation) and 24 human calcanei were investigated in vitro. Phase velocity in calcaneus at 500 kHz was 1511 m/s +/- 30 m/s (mean +/- standard deviation). Average phase velocity decreased approximately linearly with frequency (-18 m/s MHz). The two group velocity measurements were comparable and tended to be slightly lower than phase velocity. The magnitude of dispersion showed little correlation with bone mineral density.