What kind of waves are measured in trabecular bone?

The paper discusses the fundamental mechanisms underlying the interaction between ultrasound and trabecular bone, which is considered a two-phase material. When fluid-saturated cancellous bone is interrogated by ultrasound, in some cases, one or two wave modes are observed. Many authors claim that these waves correspond to the fast and slow waves predicted by Biot's theory of elastic wave propagation in fluid-saturated porous media. Within our analysis of the physical conditions, predictions of the existing two-phase models of the propagation of ultrasonic waves in the material as well as numerical simulations for fluid-saturated trabecular bone were performed. On the basis of the theoretical results (from numerical studies) and arguments presented in this paper, we aimed to answer the question of whether two waves observed in ultrasonic wave transmission studies can be interpreted as the fast and slow waves predicted by Biot's theory.

Acoustic wave propagation in equivalent fluid macroscopically inhomogeneous materials.

A one-dimensional problem of propagation of plane harmonic wave in macroscopically inhomogeneous materials is analyzed. A general description is proposed for the material of the equivalent fluid type characterized locally by two acoustical parameters: the wavenumber and the acoustical impedance. The coupled system of ordinary differential equations for amplitudes of forward and backward waves is derived. As an example the problem of wave interaction with a layer of inhomogeneous material placed between two homogeneous halfspaces is considered. The analytical solution and explicit expressions for reflection and transmission coefficients are obtained. It is shown that the presence of the inhomogeneous transition layer causes strong frequency dependence on both coefficients.

Inverse problems in cancellous bone: estimation of the ultrasonic properties of fast and slow waves using Bayesian probability theory.

Quantitative ultrasonic characterization of cancellous bone can be complicated by artifacts introduced by analyzing acquired data consisting of two propagating waves (a fast wave and a slow wave) as if only one wave were present. Recovering the ultrasonic properties of overlapping fast and slow waves could therefore lead to enhancement of bone quality assessment. The current study uses Bayesian probability theory to estimate phase velocity and normalized broadband ultrasonic attenuation (nBUA) parameters in a model of fast and slow wave propagation. Calculations are carried out using Markov chain Monte Carlo with simulated annealing to approximate the marginal posterior probability densities for parameters in the model. The technique is applied to simulated data, to data acquired on two phantoms capable of generating two waves in acquired signals, and to data acquired on a human femur condyle specimen. The models are in good agreement with both the simulated and experimental data, and the values of the estimated ultrasonic parameters fall within expected ranges.

Influence of the filling fluid on frequency-dependent velocity and attenuation in cancellous bones between 0.35 and 2.5 MHz.

The paper is focused on experiments on human cancellous bones filled with different fluids with the goal of evaluating their contribution to velocity dispersion, absorption, and scattering mechanisms. The specimens were measured first filled with marrow and subsequently, after marrow removal, with water and alcohol. No significant influence of the fluids was evidenced on the attenuation coefficient. Given the absence of impact of viscosity of the saturating fluid, the authors hypothesized that the source of attenuation is associated with viscoelastic absorption in the solid trabeculae and with scattering. Alteration of scattering obtained by changing the acoustic impedance mismatch between the fluid (alcohol vs water) and the trabeculae was reflected neither in the attenuation nor in its slope. This led the authors to suggest that longitudinal-to-shear scattering together with absorption in the solid phase are candidates as main sources for the attenuation. The differences in velocity values indicate that the elastic properties of the fluid are main determinants of the phase velocity. This finding is particularly significant in the context of /in vivo/ measurements, because it demonstrates that the subject-dependent properties of marrow may partly explain the inter-subject variability of speed of sound values.

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.