Michal Pakula

Short ultrasonic waves in cancellous bone

A new cellular model of the propagation of short ultrasonic waves in cancellous bone is introduced. The theoretical results derived from the proposed model have been compared with the well known macrocontinual Biots theory. Independently, the experimental results obtained by the pulse transmission method for cancellous bovine bone and a model material have been reported. The data from time and frequency domain are analyzed and discussed in light of the two-phase description developed under long and short wave assumption.

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 Biots 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 Biots 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 Biots theory for modeling wave propagation in cancellous bone, implying necessity of individual evaluation of input parameters.

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.

Multiphase nature and structure of biomaterials studied by ultrasounds

The paper discusses the applicability of a two-phase model of saturated porous materials for a description of the results of broadband ultrasonic studies of wave parameters in bovine trabecular bone. The analysis is focused on the role of the internal structure of the materials in the propagation of dilatational waves within the frequency range with a significant attenuation of wave energy due to absorption and scattering. The applicability of ultrasonic studies for the determination of characteristic macro- and micro-structural parameters of biomaterials using a model-based approach is considered.

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.

Phase Velocity of Cancellous Bone: Negative Dispersion Arising from Fast and Slow Waves, Interference, Diffraction, and Phase Cancellation at Piezoelectric Receiving Elements

Frequency-dependent phase velocity measurements may prove useful in bone quality assessment. However, the physical mechanisms of ultrasonic wave propagation in cancellous bone that govern phase velocity are not yet fully understood, particularly the phenomena that lead to the observed anomalous negative dispersion. This chapter provides an overview of phase velocity studies of cancellous bone, especially negative dispersion, and proposals for resolving the apparent conflict with the causality-imposed Kramers-Kronig relations.

DECOMPOSITION OF INTERFERING ULTRASONIC WAVES IN BONE AND BONE‐MIMICKING MATERIALS

Cancellous bone is a lattice‐like arrangement of solid trabeculae surrounded by soft bone marrow. When interrogated with ultrasonic waves, the complex architecture gives rise to two longitudinal modes known as a fast and a slow wave. Depending on experimental conditions and the ultrasonic characteristics of the bone sample under investigation, the two waves may strongly overlap in the ultrasounic data. Analyzing such data conventionally, as if only one wave were present, can potentially mask or alter bone quality parameters that are commonly used in clinical sonometry.In this study, ultrasonic data were acquired on a bovine femur condyle specimen and on a plastic bone‐mimicking phantom constructed from Lucite and Lexan blocks. The acquired data were used as inputs to a program that implements a Bayesian calculation to model the ultrasound signal as two interfering plane waves and then estimates the ultrasonic parameters of the fast and slow waves. The calculations were carried out using Markov chain Monte...

Extracting fast and slow wave velocities and attenuations from experimental measurements of cancellous bone using Bayesian probability theory

The consensus among many laboratories is that the attenuation coefficient of cancellous bone exhibits an approximately linear-with-frequency dependence. In the majority of cases, the phase velocity decreases with frequency. This negative dispersion appears to be inconsistent with the causality-imposed Kramers-Kronig (KK) relations for media with a linear-with-frequency attenuation coefficient. The porous structure of cancellous bone can support two compressional waves, known as a fast wave and a slow wave, that can overlap in time. Our laboratory in St. Louis has sought to explain the observed negative dispersion as an artifact of analyzing rf data containing two interfering waves as if only one wave were present. In this study, the inverse problem of how to recover the individual fast and slow waves from interference data was addressed. Waves transmitted through bone samples were analyzed using Bayesian probability theory to recover the individual properties of the fast and slow waves. Data at nine independent sites were acquired in Paris on a bovine femur condyle sample using broadband 500 kHz center frequency transducers. Each rf line served as input to a Bayesian analysis program. In the Bayesian calculation, ultrasonic wave propagation through cancellous bone was modeled as the superposition of two plane waves characterized by a linear-with-frequency attenuation coefficient and a logarithmic-with-frequency increasing phase velocity. The calculation employed Markov chain Monte Carlo (MCMC) to obtain estimates of the joint posterior probability for all parameters in the model. In all cases where the data processed by conventional means exhibited negative dispersion, two waves with positive dispersions were recovered with Bayesian analysis. The mean ± SD fast and slow wave velocities for the nine sites analyzed were (2072 ± 43) m/s and (1518 ± 22) m/s, respectively. The mean ± SD slopes of the attenuation coefficients were (17.3 ± 9.9) dB/cm/MHz and (10.8 ± 5.1) dB/cm/MHz for the fast and slow waves, respectively. Many complicating factors, including phase cancellation at the face of a piezoelectric receiver and diffraction effects, are not explicitly accounted for in the present model. Nevertheless, the Bayesian models proved to be a reliable method for recovering fast and slow waves from data that yielded negative dispersions when processed as if a single wave were present.

Characterizing Biot constants from ultrasound through trabecular bone

Reliable quantification of mechanical constants of bone tissue is an open issue with relevance for the diagnostic of bone quality disorders, such as osteoporosis. The reconstruction of such parameters from nondestructive testing based on ultrasonic transmission of pulses and model-based solution of the identification inverse problem is proposed as a novel technique with high potential not only due to the reduced cost and its non-ionizing nature, but for the direct relationship and sensitivity of the propagation of those mechanical waves to the mechanical strength of bone, which defines the ultimate criterion for diagnosis. This work is aimed at (i) evaluating the feasibility of the model-based inverse problem to reconstruct the mechanical constants that govern Biot theory, and quantify its accuracy and error for trabecular bone specimens. A second goal is (ii) to validate to which extent the Biot theory assumed in the model of wave propagation is valid, and if any corrections can empirically be suggested to overcome inconsistencies.