Effect of trabecular bone material properties on ultrasonic backscattering signals

In this study, ultrasonic backscattering signals in cancellous bones were obtained by finite difference time domain （FDTD） simulations, and the effect of trabecular material properties on these signals was analyzed. The backscatter coefficient （BSC） and integrated backscatter coefficient （IBC） were numerically investigated for varying trabecular bone material properties, including density, Lame coefficients, viscosities, and resistance coefficients. The results show that the BSC is a complex function of trabecular bone density, and the IBC increases as density increases. The BSC and IBC increase with the first and second Lame coefficients. While not very sensitive to the second viscosity of the trabeculae, the BSC and IBC decrease as the first viscosity and resistance coefficients increase. The results demonstrate that, in addition to bone mineral density （BMD） and microarchitecture, trabecular material properties significantly influence ultrasonic backseattering signals in cancellous bones. This research furthers the understanding of ultrasonic backscattering in cancellous bones and the characterization of cancellous bone status.     

骨小梁材料特性对超声背散射信号的影响

基于时域有限差分法(FDTD)建立了松质骨的超声背散射仿真系统,研究了骨小梁材料特性对超声背散射信号的影响。首次得到松质骨中的超声背散射系数(BSC)和积分背散射系数(IBC)随骨小梁材料参数(密度、拉梅常数、黏度系数及声阻抗系数)的变化关系。研究结果表明,IBC随骨小梁密度的增加而增加;BSC和IBC随拉梅常数的增加而增加、随第一黏度系数的增加而近似线性地减小,第二黏度的变化对背散射信号的影响很小;背散射参数随阻抗系数的增加而减小。说明松质骨中的超声背散射特性不仅受骨矿密度(BMD)和骨微结构的影响,还与骨小梁的材料参数密切相关。研究结果有利于理解松质骨中超声的背散射特性,对松质骨骨质状况的评价有一定帮助。      

Ultrasonic pulse waves in cancellous bone analyzed by finite-difference time-domain methods

The trabecular frame of cancellous bone has a high degree of porosity, anisotropy and inhomogeneity. The propagation of ultrasonic waves in cancellous bone is significantly affected by the trabecular structure. In this paper, two two-dimensional finite-difference time-domain (FDTD) methods, which were the popular viscoelastic FDTD method for a viscoelastic medium and Biot's FDTD method for a fluid-saturated porous medium, have been applied to numerically analyze the ultrasonic pulse waves propagating through bovine cancellous bone in the directions parallel and perpendicular to the trabecular alignment. The Biot's fast and slow longitudinal waves, which were identified in previous experiments for the propagation parallel to the trabecular orientation, could be analyzed using Biot's FDTD method rather than the viscoelastic FDTD method. For the single wave propagation in the perpendicular direction, on the other hand, the viscoelastic FDTD result was found to be in more good agreement with the experimental result.     

Influence of cancellous bone microstructure on two ultrasonic wave propagations in bovine femur: an in vitro study

The influence of cancellous bone microstructure on the ultrasonic wave propagation of fast and slow waves was experimentally investigated. Four spherical cancellous bone specimens extracted from two bovine femora were prepared for the estimation of acoustical and structural anisotropies of cancellous bone. In vitro measurements were performed using a PVDF transducer (excited by a single sinusoidal wave at 1 MHz) by rotating the spherical specimens. In addition, the mean intercept length (MIL) and bone volume fraction (BV/TV) were estimated by X-ray micro-computed tomography. Separation of the fast and slow waves was clearly observed in two specimens. The fast wave speed was strongly dependent on the wave propagation direction, with the maximum speed along the main trabecular direction. The fast wave speed increased with the MIL. The slow wave speed, however, was almost constant. The fast wave speeds were statistically higher, and their amplitudes were statistically lower in the case of wave separation than in that of wave overlap.     

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.     

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.     

Numerical and experimental study on the wave attenuation in bone--FDTD simulation of ultrasound propagation in cancellous bone

In cancellous bone, longitudinal waves often separate into fast and slow waves depending on the alignment of bone trabeculae in the propagation path. This interesting phenomenon becomes an effective tool for the diagnosis of osteoporosis because wave propagation behavior depends on the bone structure. Since the fast wave mainly propagates in trabeculae, this wave is considered to reflect the structure of trabeculae. For a new diagnosis method using the information of this fast wave, therefore, it is necessary to understand the generation mechanism and propagation behavior precisely. In this study, the generation process of fast wave was examined by numerical simulations using elastic finite-difference time-domain (FDTD) method and experimental measurements. As simulation models, three-dimensional X-ray computer tomography (CT) data of actual bone samples were used. Simulation and experimental results showed that the attenuation of fast wave was always higher in the early state of propagation, and they gradually decreased as the wave propagated in bone. This phenomenon is supposed to come from the complicated propagating paths of fast waves in cancellous bone.     

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

The use of Biot theory for modelling ultrasonic wave propagation in porous media involves the definition of a "critical frequency" above which both fast and slow compressional waves will, in principle, propagate. Critical frequencies have been evaluated for healthy and osteoporotic cancellous bone filled with water or marrow, using data from the literature. The range of pore sizes in bone gives rise to a critical frequency band rather than a single critical frequency, the mean of which is lower for osteoporotic bone than normal bone. However, the critical frequency is a theoretical concept and previous researchers considered a more realistic "viscous frequency" above which both fast and slow waves may be experimentally observed. Viscous frequencies in bone are found to be several orders of magnitude greater than calculated critical frequencies. Whereas two waves may well be observed at all ultrasonic frequencies for water-filled cancellous bone at 20 degrees C, it is probable megahertz frequencies would be needed for observation of two waves in vivo.     

In vitro estimation of fast and slow wave parameters of thin trabecular bone using space-alternating generalized expectation-maximization algorithm

In testing cancellous bone using ultrasound, two types of longitudinal Biot’s waves are observed in the received signal. These are known as fast and slow waves and their appearance depend on the alignment of bone trabeculae in the propagation path and the thickness of the specimen under test (SUT). They can be used as an effective tool for the diagnosis of osteoporosis because wave propagation behavior depends on the bone structure. However, the identification of these waves in the received signal can be difficult to achieve.In this study, ultrasonic wave propagation in a 4 mm thick bovine cancellous bone in the direction parallel to the trabecular alignment is considered. The observed Biot’s fast and slow longitudinal waves are superimposed; which makes it difficult to extract any information from the received signal. These two waves can be separated using the space alternating generalized expectation maximization (SAGE) algorithm. The latter has been used mainly in speech processing.In this new approach, parameters such as, arrival time, center frequency, bandwidth, amplitude, phase and velocity of each wave are estimated. The B-Scan images and its associated A-scans obtained through simulations using Biot’s finite-difference time-domain (FDTD) method are validated experimentally using a thin bone sample obtained from the femoral-head of a 30 months old bovine.     

Numerical investigation of ultrasound reflection and backscatter measurements in cancellous bone on various receiving areas

In this study, new ultrasound reflection and backscatter measurements in cancellous bone using a membrane-type hydrophone are proposed. A membrane hydrophone made of a piezoelectric polymer film mounted on an annular frame allows an incident ultrasound wave to pass through its aperture because it has no backing material. Therefore, in measurements using the membrane hydrophone, the receiving area could be located independently from the transmitting area. In addition, the size and shape of the receiving area, which corresponded to those of the electrode deposited on the piezoelectric film, could be arranged in various ways. To investigate the validity of the proposed measurements, before bench-top experiments, the reflected and backscattered waves from cancellous bone were numerically simulated using a finite-difference time-domain method. The reflection and backscatter parameters were measured on various receiving areas, and their correlation coefficients with the structural parameters in the cancellous bone were derived. The simulated results suggested that appropriate receiving areas for the reflection and backscatter measurements could exist and that the proposed measurements could be more effective for evaluating bone properties than conventional measurements.     

Ultrasonic wave propagation in human cancellous bone: application of Biot theory

Ultrasonic wave propagation in human cancellous bone is considered. Reflection and transmission coefficients are derived for a slab of cancellous bone having an elastic frame using Biot's theory modified by the model of Johnson et al. [J. Fluid Mech. 176, 379-402 (1987)] for viscous exchange between fluid and structure. Numerical simulations of transmitted waves in the time domain are worked out by varying the modified Biot parameters. The variation is applied to the governing parameters and is about 20%. From this study, we can gain an insight into the sensitivity of each physical parameter used in this theory. Some parameters play an important role in slow-wave wave form, such as the viscous characteristic length lambda and pore fluid bulk modulus Kf. However, other parameters play an important role in the fast-wave wave form, such as solid density rhos and shear modulus N. We also note from these simulations that some parameters such as porosity phi, tortuosity alpha(infinty), thickness, solid bulk modulus Ks, and skeletal compressibility frame Kb, play an important role simultaneously in both fast and slow wave forms compared to other parameters which act on the wave form of just one of the two waves. The sensitivity of the modified Biot parameters with respect to the transmitted wave depends strongly on the coupling between the solid and fluid phases of the cancellous bone. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.     

Fast wave ultrasonic propagation in trabecular bone: numerical study of the influence of porosity and structural anisotropy

Our goal is to assess the potential of computational methods as an alternative to analytical models to predict the two longitudinal wave modes observed in cancellous bone and predicted by the Biot theory. A three-dimensional (3D) finite-difference time-domain method is coupled with 34 human femoral trabecular microstructures measured using microcomputed tomography. The main trabecular alignment (MTA) and the degree of anisotropy (DA) were assessed for all samples. DA values were comprised between 1.02 and 1.9. The influence of bone volume fraction (BV/TV) between 5% and 25% on the properties of the fast and slow waves was studied using a dedicated image processing algorithm to modify the initial 3D microstructures. A heuristic method was devised to determine when both wave modes are time separated. The simulations (performed in three perpendicular directions) predicted that both waves generally overlap in time for a direction of propagation perpendicular to the MTA. When these directions are parallel, both waves are separated in time for samples with high DA and BV/TV values. A relationship was found between the least bone volume fraction required for the observation of nonoverlapping waves and the degree of anisotropy: The higher the DA, the lower the least BV/TV.     

Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone

Previous studies have shown that interference between fast waves and slow waves can lead to observed negative dispersion in cancellous bone. In this study, the effects of overlapping fast and slow waves on measurements of the apparent attenuation as a function of propagation distance are investigated along with methods of analysis used to determine the attenuation properties. Two methods are applied to simulated data that were generated based on experimentally acquired signals taken from a bovine specimen. The first method uses a time-domain approach that was dictated by constraints imposed by the partial overlap of fast and slow waves. The second method uses a frequency-domain log-spectral subtraction technique on the separated fast and slow waves. Applying the time-domain analysis to the broadband data yields apparent attenuation behavior that is larger in the early stages of propagation and decreases as the wave travels deeper. In contrast, performing frequency-domain analysis on the separated fast waves and slow waves results in attenuation coefficients that are independent of propagation distance. Results suggest that features arising from the analysis of overlapping two-mode data may represent an alternate explanation for the previously reported apparent dependence on propagation distance of the attenuation coefficient of cancellous bone.     

Application of an effective medium theory for modeling ultrasound wave propagation in healing long bones

Quantitative ultrasound has recently drawn significant interest in the monitoring of the bone healing process. Several research groups have studied ultrasound propagation in healing bones numerically, assuming callus to be a homogeneous and isotropic medium, thus neglecting the multiple scattering phenomena that occur due to the porous nature of callus. In this study, we model ultrasound wave propagation in healing long bones using an iterative effective medium approximation (IEMA), which has been shown to be significantly accurate for highly concentrated elastic mixtures. First, the effectiveness of IEMA in bone characterization is examined: (a) by comparing the theoretical phase velocities with experimental measurements in cancellous bone mimicking phantoms, and (b) by simulating wave propagation in complex healing bone geometries by using IEMA. The original material properties of cortical bone and callus were derived using serial scanning acoustic microscopy (SAM) images from previous animal studies. Guided wave analysis is performed for different healing stages and the results clearly indicate that IEMA predictions could provide supplementary information for bone assessment during the healing process. This methodology could potentially be applied in numerical studies dealing with wave propagation in composite media such as healing or osteoporotic bones in order to reduce the simulation time and simplify the study of complicated geometries with a significant porous nature.     

Group velocity, phase velocity, and dispersion in human calcaneus in vivo

Commercial bone sonometers measure broadband ultrasonic attenuation and/or speed of sound (SOS) in order to assess bone status. Phase velocity, which is usually measured in frequency domain, is a fundamental material property of bone that is related to SOS, which is usually measured in time domain. Four previous in vitro studies indicate that phase velocity in human cancellous bone decreases with frequency (i.e., negative dispersion). In order to investigate frequency-dependent phase velocity in vivo, through-transmission measurements were performed in 73 women using a GE Lunar Achilles Insight commercial bone sonometer. Average phase velocity at 500 kHz was 1489 +/- 55 m/s (mean +/- standard deviation). Average dispersion rate was -59 +/- 52 m/sMHz. Group velocity was usually lower than phase velocity, as is expected for negatively dispersive media. Using a stratified model to represent cancellous bone, the reductions in phase velocity and dispersion rate in vivo as opposed to in vitro can be explained by (1) the presence of marrow instead of water as a fluid filler, and (2) the decreased porosity of bones of living (compared with deceased) subjects.     

Vertebral cancellous bone turn-over: microcallus and bridges in backscatter electron microscopy

Backscatter electron microscopy (BSE) is a powerful technique for investigating cancellous bone structure. Its main function is to offer information regarding the degree of mineralization of the tissue within individual trabeculae.

To illustrate the qualitative information that can be drawn from BSE imaging technique, we present a study on human vertebral cancellous bone. This tissue is continuously remodeled through osteoclastic resorption and osteoblastic new bone apposition. It is thought that osteoclastic resorption pits are especially deleterious for vertebral bone architecture since they often perforate the thin trabeculae; the osteoblasts being unable to repair the gap. In addition, excessive stress may also disrupt the architecture in case of trabecular fracture or damage accumulation.

Waves of new bone formation were easy to identify in BSE. Often these waves were connecting both edges of a perforation and called bridges. Additionally, we present a few images of microcallus formations. A microcallus is described as a small mass of woven bone that generally repairs a trabecula. The microstructural aspects of different microcalluses are presented and discussed. Both bridges and microcallus should be considered as examples of the repair porcess since they obviously preserve the connectivity of the trabeculae. However, bridges were much more frequent than microcallus (396 vs 15). Both mechanisms probably illustrate the normal response to different local stimuli.     

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.     

Compound high-energy limb fractures with delayed union: our experience with adjuvant ultrasound stimulation (exogen)

The use of low intensity pulsed ultrasound accelerates cortical and cancellous bone fracture healing. Seventeen patients with eighteen high-energy fractures of the long bones were treated with low intensity pulsed ultrasound supplementation to surgical skeletal stabilization and tissue flaps. Sixteen fractures were healed within 13-52 weeks after starting ultrasound supplementation despite severe soft-tissue injuries and varying degrees of tissue loss. This method may be useful in the combined treatment of high-energy limb injuries.     

Use of multiple acoustic wave modes for assessment of long bones: model study

Multiple acoustic wave mode method has been proposed as a new modality in axial bone QUS. The new method is based on measurement of ultrasound velocity at different ratio of wavelength to the bone thickness, and taking into account both bulk and guided waves. It allows assessment of changes in both the material properties related to porosity and mineralization as well as the cortical thickness influenced by resorption from inner layers, which are equally important in diagnostics of osteoporosis and other bone osteopenia. Developed method was validated in model studies using a dual-frequency (100 and 500 kHz) ultrasound device. Three types of bone phantoms for long bones were developed and tested: (1) tubular specimens from polymer materials to model combined changes of material stiffness and cortical wall thickness; (2) layered specimens to model porosity in compact bone progressing from endosteum towards periosteum; (3) animal bone specimens with both cortical and trabecular components. Observed changes of the ultrasound velocity of guided waves at 100 kHz followed gradual changes in the thickness of the intact cortical layer. On the other hand, the bulk velocity at 500 kHz remained nearly constant at the different cortical layer thickness but was affected by the material stiffness. Similar trends were observed in phantoms and in fragments of animal bones.