Using ultrasonic attenuation in cortical bone to infer distributions on pore size.

In this work we infer the underlying distribution on pore radius in human cortical bone samples using ultrasonic attenuation data. We first discuss how to formulate polydisperse attenuation models using a probabilistic approach and the Waterman Truell model for scattering attenuation. We then compare the Independent Scattering Approximation and the higher-order Waterman Truell models' forward predictions for total attenuation in polydisperse samples. Following this, we formulate an inverse problem under the Prohorov Metric Framework coupled with variational regularization to stabilize this inverse problem. We then use experimental attenuation data taken from human cadaver samples and solve inverse problems resulting in nonparametric estimates of the probability density function on pore radius. We compare these estimates to the "true" microstructure of the bone samples determined via microCT imaging. We find that our methodology allows us to reliably estimate the underlying microstructure of the bone from attenuation data.

Nonlinear elastodynamics in micro-inhomogeneous solids observed by head-wave based dynamic acoustoelastic testing.

Dynamic acoustoelastic testing provides a more complete insight into the acoustic nonlinearity exhibited by micro-inhomogeneous media like granular and cracked materials. This method consists of measuring time of flight and energy modulations of pulsed ultrasonic waves induced by a low-frequency standing wave. Here pulsed ultrasonic head waves were employed to assess elastic and dissipative nonlinearities in a region near the surface of a solid. Synchronization of the ultrasound pulse sequence with the low-frequency excitation provided instantaneous variations in the elastic modulus and the attenuation as functions of the instantaneous low-frequency strain. Weak quadratic elastic nonlinearity and no dissipative nonlinearity were detected in duralumin. In limestone, distinction between tensile and compressive behaviors revealed an asymmetry in the acoustic nonlinearity and hysteresis in both the elastic modulus and the attenuation variations. Measured nonlinear acoustical parameters are in good agreement with values obtained by different techniques. Reversible acoustically induced conditioning modified the acoustic nonlinearity both quantitatively and qualitatively. It reduced tension-compression asymmetry, suggesting a nonequilibrium modification of the sources of acoustic nonlinearity. Additionally to the metrology of the acoustic nonlinearity, head wave based dynamic acoustoelastic testing may be a useful tool to monitor changes in the microstructure or the accumulation of damage in solids.

In vivo performance evaluation of bi-directional ultrasonic axial transmission for cortical bone assessment.

Our objective was to assess a new quantitative ultrasound device suitable for the measurement of speed of sound in radius. The so-called "bidirectional" technique allows an accurate estimation of velocity based on a compensation for soft tissue effects implemented directly inside the probe. Velocity measurements at 1 MHz of the first arriving signal were performed at the one third distal radius in 358 enrolled women. The average velocity by age decade increases to a peak velocity of 4043 m/s in the class 30-39 y (n = 19) and decreases thereafter. Fracture discrimination was investigated on the subset of the population for which dual-energy x-ray absorptiometry measurement was available, in addition to first arriving signal velocity measurements. The study group consisted of 122 postmenopausal women without history of fracture (group NF) and 44 postmenopausal patients (group F) with osteoporotic fractures (hip, spine, Colles fracture). When adjusted for age and bone mass index, the odds ratio (OR) for fracture prediction by ultrasound velocity, was 1.81 (1.21; 2.70) and OR associated to neck femur BMD was 2.07 (1.31-3.29). For the full model including age and body mass index as cofactors, the area under the receiver operating characteristic curve was 0.77, either for ultrasound velocity or neck femur bone mineral density. Despite the small population and the variety of fractures in the fracture group, our data indicate that the velocity of the first arriving signal measured by bidirectional technique discriminates patients with osteoporotic fracture from controls.

Nonlinear ultrasound can detect accumulated damage in human bone.

Bone micro-damage is commonly accepted as a relevant parameter for fracture risk assessment, but there is no available technique for its non-invasive characterization. The objective of this work is to study the potential of nonlinear ultrasound for damage detection in human bone. Ultrasound is particularly desirable due to its non-invasive and non-ionizing characteristics. We show results illustrating the correlation of progressive fatigue of human bone samples to their nonlinear dynamical response. In our experiments, damage was induced in 30 samples of diaphyseal human femur using fatigue cycling. At intervals in the cycling, the nonlinear response of the samples was assessed applying Nonlinear Resonant Ultrasound Spectroscopy (NRUS). The nonlinear parameter alpha, which in other materials correlates with the quantity of damage, dramatically increased with the number of mechanical testing cycles. We find a large spread in alpha in the pristine samples and infer that the spread is due to damage differences in the sample population. As damage accumulates during cycling, we find that alpha is much more sensitive to damage than other quantities measured, including the slope and hysteresis of the load/displacement curve, and the dynamic wavespeed. To our knowledge, this study represents the first application of the concept of nonlinear dynamic elasticity to human bone. The results are promising, suggesting the value of further work on this topic. Ultimately, the approach may have merit for in vivo bone damage characterization.

Prediction of bone mechanical properties using QUS and pQCT: study of the human distal radius.

The objective was to compare the prediction of bone mechanical properties provided by axial transmission to that provided by peripheral quantitative computed tomography (pQCT) at the distal radius. The distal radius is the location for Colles' fractures, a common osteoporosis related trauma situation. Measurements of the radial speed of sound were performed using three axial transmission devices: a commercial device (Sunlight Omnisense, 1.25 MHz), a bi-directional axial transmission prototype (1 MHz), both measuring the velocity of the first arriving signal (FAS), and a low frequency (200 kHz) device, measuring the velocity of a slower wave. Co-localized pQCT measurements of bone mineral density and cortical thickness were performed. Ultrasound and pQCT parameters were compared to mechanical parameters such as failure load and Young's modulus, obtained using quasistatic compressive mechanical testing and finite elements modelling (FEM). Correlations of the ultrasound and pQCT parameters to mechanical parameters were comparable. The best predictor of failure load was the pQCT measured cortical thickness. The best predictor of Young's modulus was the bi-directional SOS. The low frequency device significantly correlated to cortical thickness and failure load. The results suggest that different axial transmission approaches give access to different bone mechanical parameters. The association of different axial transmission techniques should be able to provide a good prediction of bone mechanical parameters, and should therefore be helpful for fracture risk prediction.

Apparent Young's modulus of human radius using inverse finite-element method.

The ability to assess the elastic and failure properties of cortical bone at the radial diaphysis has a clinical importance. A new generation of quantitative ultrasound (QUS) devices and peripheral quantitative computed tomography (p-QCT) has been developed to assess non-invasively bone material and structural properties at the distal radius. This anatomical site is characterized by a thin cortical thickness that complicates traditional mechanical testing methods on specimens. Until now, mechanical properties of cortical bone at distal radius (e.g., elastic modulus, yield stress and strain) remain rarely studied probably due to experimental difficulties. The present study introduces an inverse finite-element method strategy to measure the elastic modulus and yield properties of human cortical specimens of the radial diaphysis. Twenty millimeter-thick portions of diaphysis were cut from 40 human radii (ages 45-90) for biomechanical test. Subsequently the same portion was modeled in order to obtain a specimen-specific three dimensional finite-element model (3D-FEM). Longitudinal elastic constants at the apparent level and stress characterizations were performed by coupling mechanical parameters with isotropic linear-elastic simulations. The results indicated that the mean apparent Young's modulus for radial cortical bone was 16 GPa (SD 1.8) and the yield stress was 153 MPa (SD 33). Breaking load was 12,946 N (SD 3644), cortical thickness 2.9 mm (SD 0.6), structural effective strain at the yield (epsilon(y)=0.0097) and failure (epsilon(u)=0.0154) load were also calculated. The 3D-FEM strategy described here may help to investigate bone mechanical properties when some difficulties arise from machining mechanical sample.

Bone micro-damage assessment using non-linear resonant ultrasound spectroscopy (NRUS) techniques: a feasibility study.

Non-linear resonant ultrasound spectroscopy (NRUS) is a technique exploiting the significant non-linear behavior of damaged materials, related to the presence of damage. This study shows for the first time the feasibility of this technique for damage assessment in bone. Two samples of bovine cortical bone were subjected to a progressive damage experiment. Damage accumulation was progressively induced in the samples by mechanical testing. For independent assessment of damage, X-ray CT imaging was performed at each damage step, but only helped in the detection of the prominent cracks. Synchrotron micro-CT imaging and histology using epifluorescence microscopy were performed in one of the two samples at the last damage step and allowed detection of micro-cracks for this step. As the quantity of damage accumulation increased, NRUS revealed a corresponding increase in the non-linear response. The measured change in non-linear response is much more sensitive than the change in elastic modulus. The results suggest that NRUS could be a potential tool for micro-damage assessment in bone. Further work has to be carried out for a better understanding of the physical nature of damaged bone, and for the ultimate goal of in vivo implementation of the technique where bone access will be a challenging problem.

Site-matched assessment of structural and tissue properties of cortical bone using scanning acoustic microscopy and synchrotron radiation muCT.

200 MHz scanning acoustic microscopy (SAM) and synchrotron radiation muCT (SR-muCT) were used to assess microstructural parameters and tissue properties in site-matched regions of interest in cortical bone. Anterior and postero-lateral regions of ten cross sections from human cortical radius were explored. Structural parameters, including diameter and number of Haversian canals per cortical area (Ca.Dm, N.Ca/Ar) and porosity Po were assessed with both methods using a custom-developed image fusion and analysis software. Acoustic impedance Z and degree of mineralization of bone DMB were extracted separately for osteonal and interstitial tissues from the fused images. Structural parameter estimations obtained from radiographic and acoustic images were almost identical. DMB and impedance values were in the range between 0.77 and 1.28 g cm(-3) and 5.13 and 12.1 Mrayl, respectively. Interindividual and regional variations were observed, whereas the strongest difference was found between osteonal and interstitial tissues (Z: 7.2 +/- 1.1 Mrayl versus 9.3 +/- 1.0 Mrayl, DMB: 1.06 +/- 0.07 g cm(-3) versus 1.16 +/- 0.05 g cm(-3), paired t-test, p < 0.05). Weak, but significant correlations between DMB and Z were obtained for the osteonal (R(2) = 0.174, p < 10(-4)) and for the pooled (osteonal and interstitial) data. The regression of the pooled osteonal and interstitial tissue data follows a second-order polynomial (R(2) = 0.39, p < 10(-4)). Both modalities fulfil the requirement for a simultaneous evaluation of cortical bone microstructure and material properties at the tissue level. While SAM inspection is limited to the evaluation of carefully prepared sample surfaces, SR-muCT provides volumetric information on the tissue without substantial preparation requirements. However, SAM provides a quantitative estimate of elastic properties at the tissue level that cannot be captured by SR-muCT.

Comparison of three ultrasonic axial transmission methods for bone assessment.

This study compared three approaches to bone assessment using ultrasonic axial transmission. In 41 fresh human radii, velocity of the first arriving signal was measured with a commercial device (Sunlight Omnisense) operating at 1.25 MHz, a prototype based on 1-MHz bidirectional axial transmission and a low-frequency (200 kHz) prototype, also measuring the velocity of a slower wave. Cortical and trabecular bone mineral density, cortical thickness and cross-sectional area were determined by peripheral quantitative computed tomography. Significant but modest correlation between velocities reflects differences in the nature of the propagating waves and methodological differences. Of the higher frequency devices, bidirectional measurements provided stronger correlations with bone properties than did conventional measurements. High-frequency devices were less sensitive to cortical thickness than was the low-frequency device, because higher frequency waves interrogate thinner cortical layers. The results suggest that different axial transmission approaches reflect different bone properties. Therefore, a multifrequency technique might be useful in probing different bone properties.

Contrast and velocity ultrasonic tomography of long bones.

Our objective is to derive quantitative sound speed images of cortical bone using ultrasonic transmission tomography. Cortical bone is a highly refracting medium, i.e., the sound velocity changes abruptly across the interface between soft tissue and bone. It results in a loss of data compared to classical tomography in soft tissues. In order to correct for degradation by refraction effects, the classical acquisition procedure of projection data is modified: the transducers are oriented according to Snell's law of refraction with the aim of optimizing the sound propagation as parallel longitudinal rays inside the bone. This strategy allows the subsequent application of straight-ray reconstruction by the backprojection technique, which is a classical procedure in x-ray tomography. The method is validated with Plexiglas solid cylinders and tubes immersed in water. Improved sound velocity images are then derived using conventional Radon transform of the experimental time-of-flight data. The method is then extended to in vitro human femur immersed in water. The geometry of the bone cross-section is reconstructed from measurements using ultrasonic reflection tomography. The result is then introduced in the calculation of the position and orientation of the transducers, which are associated with the parallel acoustical paths in bone in the transmission measurements. The procedure leads to significant restoration enhancement over the non corrected image. The mean value of the velocity of 3,200 ms(-1) in the cortical shell is consistent with the values known from literature. These preliminary quantitative images using combined reflected and transmission ultrasound show promise for bone imaging.

The wave bivector formalism associated with circumferential leaky waves

It is shown that the circumferential internal waves propagating around an elastic cylinder can be locally seen as plane evanescent waves, at any observation point inside the surrounding fluid. This is done by direct calculation of the associated complex bivector. The exact anatomy of the wave is detailed and the phase propagation paths are found to be curved, as expected. The transition to the plane interface is achieved. Polarization ellipses associated with the acoustic displacement vector are described. The additional low evanescence assumption leads to conventional ray interpretation with identification of the ray tube divergence coefficients, and the wavefront is found to be the involute of a circle.

Analysis of the axial transmission technique for the assessment of skeletal status.

Ultrasonic wave propagation in human cortical bone has been investigated in vitro using the so-called axial transmission technique. This technique, which relies on velocity measurement of the first arriving signal, has been used in earlier investigations to study bone status during fracture healing or osteoporosis. Two quasi-point-source elements, one transmitter and one receiver (central frequency 0.5 MHz), were used to generate a wide ultrasonic beam, part of which strikes the sample surface at the longitudinal critical angle, and to receive the signals reflected from the sample surface. The analysis of the field reflected from a fluid-solid interface for an incident spherical wave predicts the existence of a lateral wave propagating along the sample surface at a velocity close to the longitudinal velocity, in addition to the ordinary reflected wave and vibration modes. The transducer-sample and the transmitter-receiver distances were chosen such that the lateral wave is the first arriving signal. Validation of the measuring technique was performed on test materials and was followed by experiments on human cortical bones. Experimental results (arrival time and velocity) strongly suggest that the first detected signal corresponds to the lateral wave predicted by theory.