Propagation of two longitudinal waves in a cancellous bone with the closed pore boundary

Ultrasound propagation in cancellous bone (porous media) under the condition of closed pore boundaries was investigated. A cancellous bone and two plate-like cortical bones obtained from a racehorse were prepared. A water-immersion ultrasound technique in the MHz range and a three-dimensional elastic finite-difference time-domain (FDTD) method were used to investigate the waves. The experiments and simulations showed a clear separation of the incident longitudinal wave into fast and slow waves. The findings advance the evaluation of bones based on the two-wave phenomenon for in vivo assessment.     

A numerical study on the propagation of Rayleigh and guided waves in cortical bone according to Mindlin's Form II gradient elastic theory

Cortical bone is a multiscale heterogeneous natural material characterized by microstructural effects. Thus guided waves propagating in cortical bone undergo dispersion due to both material microstructure and bone geometry. However, above 0.8 MHz, ultrasound propagates rather as a dispersive surface Rayleigh wave than a dispersive guided wave because at those frequencies, the corresponding wavelengths are smaller than the thickness of cortical bone. Classical elasticity, although it has been largely used for wave propagation modeling in bones, is not able to support dispersion in bulk and Rayleigh waves. This is possible with the use of Mindlin's Form-II gradient elastic theory, which introduces in its equation of motion intrinsic parameters that correlate microstructure with the macrostructure. In this work, the boundary element method in conjunction with the reassigned smoothed pseudo Wigner-Ville transform are employed for the numerical determination of time-frequency diagrams corresponding to the dispersion curves of Rayleigh and guided waves propagating in a cortical bone. A composite material model for the determination of the internal length scale parameters imposed by Mindlin's elastic theory is exploited. The obtained results demonstrate the dispersive nature of Rayleigh wave propagating along the complex structure of bone as well as how microstructure affects guided waves.     

Ultrasonic wave propagation in cancellous and cortical bone: prediction of some experimental results by Biot's theory.

Pulse transmission ultrasound was used to determine the longitudinal wave speed along the direction of trabecular alignment in 32 water-saturated anisotropic tibial bovine cancellous bone samples and in one cortical bone sample also from the bovine tibia. These results are compared to published ultrasound wave-speed data obtained from bovine femoral specimens. Nonlinear regression was used to fit Biot's theory to the data. The correlation coefficient for regression analysis between the experimental ultrasound velocities and the velocities predicted by Biot's theory was r = 0.78.     

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.     

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.     

椎弓根钉植入针道上骨组织光谱特性研究

运用光谱技术研究了椎骨组织不同位置的特征识别因子.光谱采集系统由双光纤手钻一体式探头(光纤芯径200μm,中心距离0.5mm)、卤素光源(波长360～2 000nm)、光纤光谱仪(检测波长为200～1 100nm)和计算机组成,可以同时获得生物组织的漫反射光谱和约化散射系数.以猪椎骨为实验对象,测量椎弓根螺钉植入针道上不同骨组织的漫反射光谱和约化散射系数,并对光谱进行特定波长的峰值、面积、斜率分析,获得特性识别因子.研究发现,椎弓根钉植入针道上不同骨组织的光谱表现出不同的变化特性.其中峰值的变化比约化散射系数的变化高1.88倍,面积的变化比约化散射系数的变化高2.05倍.在495～505nm处,骨密质和骨疏质的光谱斜率都为正值;在520～535nm处,骨密质光谱的斜率为正值,而骨疏质光谱的斜率为负值.结果表明,通过光谱特性分析获得的峰值、面积和斜率因子能够有效地区分针道上骨密质与骨疏质的差异.     

Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1 MHz: a finite-difference time domain study

Finite-difference time domain (FDTD) numerical simulations coupled to real experimental data were used to investigate the propagation of 1 MHz pure bulk wave propagation through models of cortical bone microstructures. Bone microstructures were reconstructed from three-dimensional high resolution synchrotron radiation microcomputed tomography (SR-muCT) data sets. Because the bone matrix elastic properties were incompletely documented, several assumptions were made. Four built-in bone matrix models characterized by four different anisotropy ratios but the same Poisson's ratios were tested. Combining them with the reconstructed microstructures in the FDTD computations, effective stiffness coefficients were derived from simulated bulk-wave velocity measurements. For all the models, all the effective compression and shear bulk wave velocities were found to decrease when porosity increases. However, the trend was weaker in the axial direction compared to the transverse directions, contributing to the increase of the effective anisotropy. On the other hand, it was shown that the initial Poisson's ratio value may substantially affect the variations of the effective stiffness coefficients. The present study can be used to elaborate sophisticated macroscopic computational bone models incorporating realistic CT-based macroscopic bone structures and effective elastic properties derived from muCT-based FDTD simulations including the cortical porosity effect.     

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.     

Two-wave behavior under various conditions of transition area from cancellous bone to cortical bone

The two-wave phenomenon, the wave separation of a single ultrasonic pulse in cancellous bone, is expected to be a useful tool for the diagnosis of osteoporosis. However, because actual bone has a complicated structure, precise studies on the effect of transition conditions between cortical and cancellous parts are required. This study investigated how the transition condition influenced the two-wave generation using three-dimensional X-ray CT images of an equine radius and a three-dimensional simulation technique. As a result, any changes in the boundary between cortical part and trabecular part, which gives the actual complex structure of bone, did not eliminate the generation of either the primary wave or the secondary wave at least in the condition of clear trabecular alignment. The results led us to the possibility of using the two-wave phenomenon in a diagnostic system for osteoporosis in cases of a complex boundary.     

Assessment of bone structure and acoustic impedance in C3H and BL6 mice using high resolution scanning acoustic microscopy

Two hundred-MHz time-resolved scanning acoustic microscopy was applied for the investigation of acoustic and structural bone properties of mice from two inbred strains. Transverse sections of femur taken from 5 C57BL/6J@Ico and 5 C3H/HeJ@Ico mice were explored. Both strains had the same bone diameter, but the C3H/HeJ@Ico mice had greater cortical thickness, smaller cancellous diameter, and greater acoustic impedance values than C57BL/6J@Ico mice. The strong differences in the measured acoustic impedances among the two inbred strains indicate that the impedance is a good parameter to detect genetic variations of the skeletal phenotype in small animal models.     

Prediction of backscatter coefficient in trabecular bones using a numerical model of three-dimensional microstructure

A model of ultrasonic backscattering for cancellous bone saturated by water is proposed. This model assumes that scattering is caused by the solid trabeculae and describes the cancellous bone as a weak scattering medium. The backscatter coefficient is related to the spatial Fourier transform of bone microarchitecture and to the density and compressibility fluctuations between the solid trabeculae and the saturating fluid. The computations of the model make use of three-dimensional numerical images of bone microarchitecture, obtained by tomographic reconstructions with a 10 microm spatial resolution. With this model, the predictions of the frequency dependence and of the magnitude of the backscatter coefficient are reasonably accurate. The theoretical predictions are compared to experimental data obtained on 19 specimens. An accuracy error of approximately 1 dB was found (difference between the averaged experimental values and theoretical predictions). One limit of the model may come from inaccurate values of trabecular bone characteristics needed for the computations (density and longitudinal velocity), which are yet to be precisely determined for human trabecular bone. However, the model is only slightly sensitive to variations of bone material properties. It was found that an accuracy error of 2.2 dB at maximum resulted from inaccurate a priori values of bone material properties. A computation of the elastic mean free path in the medium suggests that multiple scattering plays a minor role in the working frequency bandwidth (0.4-1.2 MHz). It follows from these results that a weak scattering medium model may be appropriate to describe scattering from trabecular 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.     

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。      

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.     

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.     

Ultrasonic characterization of human cancellous bone using the Biot theory: inverse problem

This paper concerns the ultrasonic characterization of human cancellous bone samples by solving the inverse problem using experimental transmitted signals. The ultrasonic propagation in cancellous bone is modeled using the Biot theory modified by the Johnson et al. model for viscous exchange between fluid and structure. The sensitivity of the Young modulus and the Poisson ratio of the skeletal frame is studied showing their effect on the fast and slow wave forms. The inverse problem is solved numerically by the least squares method. Five parameters are inverted: the porosity, tortuosity, viscous characteristic length, Young modulus, and Poisson ratio of the skeletal frame. The minimization of the discrepancy between experiment and theory is made in the time domain. The inverse problem is shown to be well posed, and its solution to be unique. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.     

Measurements of tortuosity in stereolithographical bone replicas using audiofrequency pulses

The tortuosity of five air-filled stereolithographical cancellous bone replicas has been obtained from measurements using audiofrequency pulses in a rectangular waveguide. The data obtained from the replicas yields information about anisotropy with respect to orthogonal axes of the passages that would be marrow filled in vivo. A strong relationship has been found between the acoustically measured tortuosity and the independently measured porosity. Use of stereolithographical bone replicas has the potential to simulate perforation and thinning of cancellous bone and hence evaluate the dependence of acoustic properties on cancellous bone microstructure. As an "extreme" illustration of such use, "inverses" of the original replicas have been manufactured and acoustic measurements have been made on them. The data reveal significantly greater tortuosity of the passages that are geometrically equivalent to the original solid bone structures.     

Numerical simulation of the dependence of quantitative ultrasonic parameters on trabecular bone microarchitecture and elastic constants

Finite-difference numerical simulation of ultrasound propagation in complex media such as cancellous bone represents a fertile alternative to analytical approaches because it can manage the complex 3D bone structure by coupling the numerical computation with 3D numerical models of bone microarchitecture obtained from high-resolution imaging modalities. The objective of this work was to assess in silico the sensitivity of ultrasound parameters to controlled changes of microarchitecture and variation of elastic constants. The simulation software uses a finite-difference approach based on the Virieux numerical scheme. An incident plane wave was propagated through a volume of bone of approximately 5 x 5 x 8 mm(3). The volumes were reconstructed from high-resolution micro-computed tomography data. An iterative numerical scenario of "virtual osteoporosis" was implemented using a dedicated image processing algorithm in order to modify the initial 3D microstructures. Numerical computations of wave propagation were performed at each step of the process. The sensitivity to bone material properties was also tested by changing the elastic constants of bone tissue. Our results suggest that ultrasonic variables (slope of the frequency-dependent attenuation coefficient and speed of sound) are mostly influenced by bone volume fraction. However, material properties and structure also appear to play a role. The impact of modifications of the stiffness coefficients remained lower than the variability caused by structural variations. This study emphasizes the potential of numerical computations tools coupled to realistic 3D structures to elucidate the physical mechanisms of interaction between ultrasound and bone structure and to assess the sensitivity of ultrasound variables to different bone properties.     

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.