Estimation of critical and viscous frequencies for Biot theory in cancellous bone

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.

The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone

Cancellous bone is a highly porous material, and two types of waves, fast and slow, are observed when ultrasound is used for detecting bone diseases. There are several possible stimuli for bone remodelling processes, including bone fluid flow, streaming potential, and piezoelectricity. Poroelasticity has been widely used for elucidating the bone fluid flow phenomenon, but the combination of poroelasticity with charge density has not been introduced. Theoretically, general poroelasticity with a varying charge density is employed for determining the relationship between wave velocity and attenuation with charge density. Fast wave velocity and attenuation are affected by porosity as well as charge density; however, for a slow wave, both slow wave velocity and attenuation are not as sensitive to the effect of charge density as they are for a fast wave. Thus, employing human femoral data, we conclude that charged ions gather on trabecular struts, and the fast wave, which moves along the trabecular struts, is significantly affected by charge density.

Model-data comparison of high frequency compressional wave attenuation in water-saturated granular medium with bimodal grain size distribution

Several acoustic models, such as the poro-elastic model, visco-elastic model, and multiple scattering model, have been used for describing the dispersion relation in a porous granular medium. However, these models are based on continuum or scattering theory, and therefore cannot explain the broadband measurements in cases where scattering and non-scattering losses co-exist. Additionally, since the models assume that the porous granular medium consists of grains of identical size (unimodal size distribution), the models does not account for the behavior of wave dispersion in a medium that has a distribution of differing grain sizes. As an alternative approach, this study proposes a new broadband attenuation model that describes the high frequency dispersion relation for the p-wave in the case of elastic grain scatterers existing in the background fluid medium. The broadband model combines the Biot-Stoll plus grain contact squirt and shear flow (BICSQS) model and the quasicrystalline approximation (QCA) multiple scattering model. Additionally, distribution of grain size effect is examined rudimentarily through consideration of bimodal grain size distribution. Through the quantitative analysis of the broadband model and measured data, it is shown that the model can explain the attenuation dependencies of frequency and grain size distribution for a water-saturated granular medium in the frequency range from 350kHz to 1.1MHz. This study can be applied to the high frequency acoustic SONAR modeling and design in the water-saturated environment.

Simultaneous measurements of transport and poroelastic properties of rocks

A novel laboratory apparatus has been developed for simultaneous measurements of transport and poroelastic rock properties. These transport and poroelastic properties at reservoir pressure and temperature conditions are required inputs for various geoscience applications, such as reservoir simulation, basin modeling, or modeling of pore pressure generation. Traditionally, the transport and poroelastic properties are measured separately using, for example, the oscillating pore pressure method to measure hydraulic transport properties, static strain measurements for elastic properties, and pore volumometry for storage capacity. In addition to time, the separate set of measurements require either aliquot cores or subjecting the same core to multiple pressure tests. We modified the oscillating pore pressure method to build an experimental setup, capable of measuring permeability, storage capacity, and pseudo-bulk modulus of rocks simultaneously. We present here the test method, calibration measurements (capillary tube), and sample measurements (sandstone) of permeability and storage capacity at reservoir conditions. We establish that hydraulically measured storage capacities were overestimated by an order of magnitude when compared to elastically derived ones. Our concurrent measurement of elastic properties during the hydraulic experiment provides an independent constraint on storage capacity.

A multiscale poromicromechanical approach to wave propagation and attenuation in bone

Ultrasonics is an important diagnostic tool for bone diseases, as it allows for non-invasive assessment of bone tissue quality through mass density-elasticity relationships. The latter are, however, quite complex for fluid-filled porous media, which motivates us to develop a rigorous multiscale poromicrodynamics approach valid across the great variety of different bone tissues. Multiscale momentum and mass balance, as well as kinematics of a hierarchical double porous medium, together with Darcy's law for fluid flow and micro-poro-elasticity for the solid phase of bone, give access to the so-called dispersion relation, linking the complex wave numbers to corresponding wave frequencies. Experimentally validated results show that 2.25 MHz acoustical signals transmit healthy cortical bone (exhibiting a low vascular porosity) only in the form of fast waves, agreeing very well with experimental data, while both fast and slow waves transmit highly osteoporotic as well as trabecular bone (exhibiting a large vascular porosity). While velocities and wavelengths of both fast and slow waves, as well as attenuation lengths of slow waves, are always monotonously increasing with the permeability of the bone sample, the attenuation length of fast waves shows a minimum when considered as function of the permeability.

A deconvolution method for deriving the transit time spectrum for ultrasound propagation through cancellous bone replica models

The acceptance of broadband ultrasound attenuation for the assessment of osteoporosis suffers from a limited understanding of ultrasound wave propagation through cancellous bone. It has recently been proposed that the ultrasound wave propagation can be described by a concept of parallel sonic rays. This concept approximates the detected transmission signal to be the superposition of all sonic rays that travel directly from transmitting to receiving transducer. The transit time of each ray is defined by the proportion of bone and marrow propagated. An ultrasound transit time spectrum describes the proportion of sonic rays having a particular transit time, effectively describing lateral inhomogeneity of transit times over the surface of the receiving ultrasound transducer. The aim of this study was to provide a proof of concept that a transit time spectrum may be derived from digital deconvolution of input and output ultrasound signals. We have applied the active-set method deconvolution algorithm to determine the ultrasound transit time spectra in the three orthogonal directions of four cancellous bone replica samples and have compared experimental data with the prediction from the computer simulation. The agreement between experimental and predicted ultrasound transit time spectrum analyses derived from Bland–Altman analysis ranged from 92% to 99%, thereby supporting the concept of parallel sonic rays for ultrasound propagation in cancellous bone. In addition to further validation of the parallel sonic ray concept, this technique offers the opportunity to consider quantitative characterisation of the material and structural properties of cancellous bone, not previously available utilising ultrasound.

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.

Simulations of ultrasound propagation in random arrangements of elliptic scatterers: occurrence of two longitudinal waves

Ultrasound propagation in clusters of elliptic (two-dimensional) or ellipsoidal (three-dimensional) scatterers randomly distributed in a fluid is investigated numerically. The essential motivation for the present work is to gain a better understanding of ultrasound propagation in trabecular bone. Bone microstructure exhibits structural anisotropy and multiple wave scattering. Some phenomena remain partially unexplained, such as the propagation of two longitudinal waves. The objective of this study was to shed more light on the occurrence of these two waves, using finite-difference simulations on a model medium simpler than bone. Slabs of anisotropic, scattering media were randomly generated. The coherent wave was obtained through spatial and ensemble-averaging of the transmitted wavefields. When varying relevant medium parameters, four of them appeared to play a significant role for the observation of two waves: (i) the solid fraction, (ii) the direction of propagation relatively to the scatterers orientation, (iii) the ability of scatterers to support shear waves, and (iv) a continuity of the solid matrix along the propagation. These observations are consistent with the hypothesis that fast waves are guided by the locally plate/bar-like solid matrix. If confirmed, this interpretation could significantly help developing approaches for a better understanding of trabecular bone micro-architecture using ultrasound.

Comparison of phase velocity in trabecular bone mimicking-phantoms by time domain numerical (EFIT) and analytical multiple scattering approaches

The corrected Waterman-Truell model and the Elastodynamic Finite Integration Technique were used to analyze the ultrasonic wave dispersion in trabecular bones mimicking phantoms. A simple two-phase model of the trabecular bone is assumed; the trabeculae structure and the bone marrow. The phase velocity for frequencies within the range from 400kHz to 800kHz were computed for different scatterer arrays varying their dimensions and number. The theoretical and numerical results were compared to experimental published data, obtained from a mimicking phantom composed by a periodic array of nylon shreds (trabeculae array) immersed in a water tank. Our results showed an excellent consistency when compared to experimental data. The negative dispersions of -8.48m/s/MHz and -9.16m/s/MHz were computed by the multiple scattering method and the numerical approach, respectively, where the latter is closer to the experimental dispersion of -12.09m/s/MHz. Similar result has been reported in the literature, where the dispersion predicted by the Generalized Self-Consistent Method [J. Acoust. Soc. Am. 124 (2008) 4047] is -9.96m/s/MHz.

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

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

IN VITROACOUSTIC WAVE PROPAGATION IN HUMAN AND BOVINE CANCELLOUS BONE AS PREDICTED BY BIOT'S THEORY

Recent in vitro studies have provided evidence of the propagation of two different longitudinal wave modes at ultrasonic frequencies in cancellous bone. The genesis of these two plane waves in fluid-saturated porous media is predicted by the poroelastic approach to wave propagation originally developed by Biot. However, wave velocity is usually analyzed as a function of bone mass density only; therefore, the influence of the cancellous bone microstructure over the wave velocity is not taken into account. In the present study, a descriptor of the microstructure is considered in Biot's theory. This model is used to evaluate the large experimental variability of both fast and slow wave velocities measured on randomly oriented human and bovine cancellous bone samples. The role of the anisotropic solid structure and fluid in the behavior of fast and slow wave velocities is examined. Experimental and theoretically predicted velocities are found in close agreement when analyzed as a function of both porosity and structural index. This model has the potential to be used to determine an acoustically derived structural index in cancellous bone.

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.

High frequency poroelastic waves in hydrogels

In this work a continuum model for high frequency poroelastic longitudinal waves in hydrogels is presented. A viscoelastic force describing the interaction between the polymer network and the bounded water present in such materials is introduced. The model is tested by means of ultrasound wave speed and attenuation measurements in polyvinylalcohol hydrogel samples. The theory and experiments show that ultrasound attenuation decreases linearly with the increase in the water volume fraction beta of the hydrogel. The introduction of the viscoelastic force between the bounded water and the polymer network leads to a bi-phasic theory, showing an ultrasonic fast wave attenuation that can vary as a function of the frequency with a non-integer exponent in agreement with the experimental data in literature. When beta tends to 1 (100% of interstitial water) due to the presence of bounded water in the hydrogel, the ultrasound phase velocity acquires higher value than that of pure water. The ultrasound speed gap at beta=1 is confirmed by the experimental results, showing that it increases in less cross-linked gel samples which own a higher concentration of bounded water.

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).

Application of Biot's theory to ultrasonic characterization of human cancellous bones: determination of structural, material, and mechanical properties

This paper is devoted to the experimental determination of distinctive macroscopic structural (porosity, tortuosity, and permeability) and mechanical (Biot-Willis elastic constants) properties of human trabecular bones. Then, the obtained data may serve as input parameters for modeling wave propagation in cancellous bones using Biot's theory. The goal of the study was to obtain experimentally those characteristics for statistically representative group of human bones (35 specimens) obtained from a single skeletal site (proximal femur). The structural parameters were determined using techniques devoted to the characterization of porous materials: electrical spectroscopy, water permeametry, and microcomputer tomography. The macroscopic mechanical properties, Biot-Willis elastic constants, were derived based on the theoretical consideration of Biot's theory, micromechanical statistical models, and experimental results of ultrasonic studies for unsaturated cancellous bones. Our results concerning structural parameters are consistent with the data presented by the other authors, while macroscopic mechanical properties measured within our studies are situated between the other published data. The discrepancies are mainly attributed to different mechanical properties of the skeleton frame, due to strong structural anisotropy varying from site to site. The results enlighten the difficulty to use Biot's theory for modeling wave propagation in cancellous bone, implying necessity of individual evaluation of input parameters.

Model-based estimation of quantitative ultrasound variables at the proximal femur

To improve the prediction of the osteoporotic fracture risk at the proximal femur we are developing a scanner for quantitative ultrasound (QUS) measurements at this site. Due to multipath transmission in this complex shaped bone, conventional signal processing techniques developed for QUS measurements at peripheral sites frequently fail. Therefore, we propose a model-based estimation of the QUS variables and analyze the performance of the new algorithm. Applying the proposed method to QUS scans of excised proximal femurs increased the fraction of evaluable signals from approx. 60% (using conventional algorithms) to 97%. The correlation of the standard QUS variables broadband ultrasound attenuation (BUA) and speed of sound (SOS) with the established variable bone mineral density (BMD) reported in previous studies is maintained (BUA/BMD: r(2) = 0.69; SOS/BMD: r(2) = 0.71; SOS+BUA/BMD: r(2) = 0.88). Additionally, different wave types could be clearly detected and characterized in the trochanteric region. The ability to separate superimposed signals with this approach opens up further diagnostic potential for evaluating waves of different sound paths and wave types through bone tissue.

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

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

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

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

Investigation of an anisotropic tortuosity in a biot model of ultrasonic propagation in cancellous bone

The modeling of ultrasonic propagation in cancellous bone is relevant to the study of clinical bone assessment. Historical experiments revealed the importance of both the viscous effects of bone marrow and the anisotropy of the porous microstructure. Of those propagation models previously applied to cancellous bone, Biot's theory incorporates viscosity, but has only been applied in isotropic form, while Schoenberg's anisotropic model does not include viscosity. In this paper we present an approach that incorporates the merits of both models, by utilizing the tortuosity, a key parameter describing pore architecture. An angle-dependent tortuosity for a layered structure is used in Biot's theory to generate the "Stratified Biot Model" for cancellous bone, which is compared with published bone data. While the Stratified Biot model was inferior to Schoenberg's model for slow wave velocity prediction, the proposed model improved agreement fast wave velocity at high propagation angles, particularly when sorted for porosity. An attempt was made to improve the fast wave agreement at low angles by introducing an angle-dependent Young's Modulus, which, while improving the agreement of predicted fast wave velocity at low angles, degraded agreement at high angles. In this paper the utility of the tortuosity in characterizing the architecture of cancellous bone is highlighted.

What is ultrasound?

This paper is based on material presented at the start of a Health Protection Agency meeting on ultrasound and infrasound. In answering the question 'what is ultrasound?', it shows that the simple description of a wave which transports mechanical energy through the local vibration of particles at frequencies of 20 kHz or more, with no net transport of the particles themselves, can in every respect be misleading or even incorrect. To explain the complexities responsible for this, the description of ultrasound is first built up from the fundamental properties of these local particle vibrations. This progresses through an exposition of the characteristics of linear waves, in order to explain the propensity for, and properties of, the nonlinear propagation which occurs in many practical ultrasonic fields. Given the Health Protection environment which framed the original presentation, explanation and examples are given of how these complexities affect issues of practical importance. These issues include the measurement and description of fields and exposures, and the ability of ultrasound to affect tissue (through microstreaming, streaming, cavitation, heating, etc.). It is noted that there are two very distinct regimes, in terms of wave characteristics and potential for bioeffect. The first concerns the use of ultrasound in liquids/solids, for measurement or material processing. For biomedical applications (where these two processes are termed diagnosis and therapy, respectively), the issue of hazard has been studied in depth, although this has not been done to such a degree for industrial uses of ultrasound in liquids/solids (sonar, non-destructive testing, ultrasonic processing etc.). However, in the second regime, that of the use of ultrasound in air, although the waves in question tend to be of much lower intensities than those used in liquids/solids, there is a greater mismatch between the extent to which hazard has been studied, and the growth in commercial applications for airborne ultrasound.

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.