Numerical analysis of variability in ultrasound propagation properties induced by trabecular microstructure in cancellous bone

The manner by which the trabecular microstructure affects the propagation of ultrasound waves through cancellous bone is numerically investigated by finite difference time-domain (FDTD) simulation. Sixteen 3-D numerical models of 6.45times6.45times6.45 mm with a voxel size of 64.5 mum are reconstructed using a 3-D microcomputed tomographic (muCT) image taken from a bovine cancellous bone specimen of approximately 20times20times9 mm. All cancellous bone models have an oriented trabecular structure, and their trabecular elements are gradually eroded to increase the porosity using an image processing technique. Three erosion procedures are presented to realize various changes in the trabecular microstructure with increasing porosity. FDTD simulations of the ultrasound pulse waves propagating through the cancellous bone models at each eroded step are performed in 2 cases of the propagations parallel and perpendicular to the major trabecular orientation. The propagation properties of the wave amplitudes and propagation speeds are derived as a function of the porosity, and their variability due to the trabecular microstructure is revealed. To elucidate an effect of the microstructure, the mean intercept length (MIL), which is a microstructural parameter, is introduced, and the correlations of the propagation properties with the MILs of the trabecular elements and pore spaces are investigated.     

Two-wave propagation imaging to evaluate the structure of cancellous bone

The two-wave phenomenon reflects not only bone mass but also the complex bone structure of cancellous bone. We propose a new simple imaging technique based on the two-wave phenomenon for investigating the anisotropic structure of cancellous bone. A cylindrical specimen of cancellous bone was obtained from a bovine femur. The structure (alignment of trabeculae) of the specimen was obtained from 3-D X-ray micro computed tomography imaging. Using a conventional ultrasonic pulse technique, we rotated the receiver around the specimen to investigate the ultrasonic fields after propagation within the specimen. The ultrasonic propagation image clearly showed the effect of the bone structure. We found that the fast wave showed apparent refraction, whereas the slow wave did not. Fast-wave propagation imaging is thus a simple and convenient technique for easy interpretation of the anisotropic structure. This imaging technique has the potential to become a powerful tool to investigate the structure of trabeculae during in vivo measurements.     

Development of a numerical cancellous bone model for finite-difference time-domain simulations of ultrasound propagation

The trabecular frame in cancellous bone has numerous porous spaces of various sizes and shapes. Their continual arrangement changes with position in the bone. Assuming that the complicated pore space is the aggregation of spherical pores, in this study, the trabecular structure was analyzed using a three-dimensional (3-D) X-ray microcomputed tomography (muCT) image. Analysis involved a 3-D cancellous bone model developed for numerical simulations of ultrasound propagation. In this model, the trabecular structure was simplified by regularly arranging spherical pores in a solid bone. Using a viscoelastic, finite-difference, time-domain (FDTD) method with the simplified cancellous bone model, ultrasound pulse waveforms propagating through cancellous bone were simulated in two cases of the propagations parallel and perpendicular to the main trabecular orientation. The porosity dependences of the propagation properties, attenuation, and propagation speed were derived from the simulated waveforms. Comparisons with simulated results using the realistic cancellous bone model reconstructed from a 3-D muCT image, assisted to further validate this simplified model.     

Application of the Biot model to ultrasound in bone: inverse problem

This paper concerns the ultrasonic characterization of human cancellous bone samples by solving the inverse problem using experimentally measured signals. The inverse problem is solved numerically by the least squares method. Five parameters are inverted: 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 ultrasonic propagation in cancellous bone is modelled using the Biot theory modified by the Johnson-Koplik-Dashen 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 waveforms. 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.     

Role of structural anisotropy of biological tissues in poroelastic wave propagation

Ultrasound waves have a broad range of clinical applications as a non-destructive testing approach in imaging and in the diagnoses of medical conditions. Generally, biological tissues are modeled as an homogenized equivalent medium with an apparent density through which a single wave propagates. Only the first wave arriving at the ultrasound probe is used for the measurement of the speed of sound. However, the existence of a second wave in tissues such as cancellous bone has been reported and its existence is an unequivocal signature of Biot type poroelastic media. To account for the fact that ultrasound is sensitive to microarchitecture as well as density, a fabric-dependent anisotropic poroelastic ultrasound (PEU) propagation theory was recently developed. Key to this development was the inclusion of the fabric tensor – a quantitative stereological measure of the degree of structural anisotropy of bone – into the linear poroelasticity theory. In the present study, this framework is extended to the propagation of waves in several soft and hard tissues. It was found that collagen fibers in soft tissues and the mineralized matrix in hard tissues are responsible for the anisotropy of the solid tissue constituent through the fabric tensor in the model.     

The inspection of anisotropic single-crystal components using a 2-D ultrasonic array

Single-crystal metal alloys are used extensively in the manufacture of jet engine components for their excellent mechanical properties at elevated temperatures. The inspection of these components using 2-D ultrasonic arrays potentially allows the detection of subsurface defects in threedimensions from one inspection location. Such methods are not currently suitable for the inspection of single-crystal components because the high elastic anisotropy of single-crystal materials causes directional variation in ultrasonic waves. In this paper, a model of wave propagation in anisotropic material is used to correct an ultrasonic imaging algorithm and is applied to a single-crystal test specimen. For this correctedalgorithm, the orientation of the crystal in a specimen must be known before the inspection. Using the same ultrasonic array to measure the orientation and perform the defect inspection offers the most practical solution. Therefore, potential crystallographic orientation methods using 2-D ultrasonic arrays are also developed and evaluated.     

Application of the biot model to ultrasound in bone: direct problem

Ultrasonic wave propagation in human cancellous bone is considered using Biot's theory modified by the Johnson-Koplik-Dashen model for viscous exchange between fluid and structure. The transmission coefficient is derived for a slab of porous material. Experimental results for fast and slow waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.     

Influence of density, elasticity, and structure on ultrasound transmission through trabecular bone cylinders

The aim of this in vitro study is to evaluate the potentiality of quantitative ultrasound (QUS) to separate information on density, elasticity, and structure on specimens of trabecular bone. Fifteen cylinders of spongy bone extracted from equine vertebrae were progressively demineralized and subjected to QUS, micro computed tomography (muCT), Dual energy X-ray absorptiometry (DXA) at various mineralization levels. Eventually all cylinders underwent a compression test to calculate the Young's modulus. Correlation analysis shows that speed of sound (SOS) is strictly associated to bone mineral density (BMD), Young's modulus, and all muCT parameters except for degree of anisotropy (DA). Fast wave amplitude (FWA) is directly correlated with bone surface and total volume ratio (BS/TV) and trabecular separation (Tb Sp), and inversely correlated with trabecular number (Tb N). Because muCT parameters were strictly correlated to BMD and Young's modulus data, partial correlation analysis was performed between SOS, FWA, and structural and elastic data in order to eliminate the effect of density. SOS was significantly correlated to bone volume and total volume ratio (BV/TV), BS/TV, and Young's modulus, and FWA was significantly correlated to Tb Sp only. These results show that SOS is strongly influenced by volumetric mineral bone density and elastic modulus of the specimen, and FWA is mainly affected by trabecular separation independently on density. Therefore, SOS and FWA are able to provide different and complementary information, at least on trabecular bone samples.     

The correlation between the SOS in trabecular bone and stiffness and density studied by finite-element analysis

For the clinical assessment of osteoporosis (i.e., a degenerative bone disease associated with increased fracture risk), ultrasound has been proposed as an alternative or supplement to the dual-energy X-ray absorptiometry (DEXA) technique. However, the interaction of ultrasound waves with (trabecular) bone remains relatively poorly understood. The present study aimed to improve this understanding by simulating ultrasound wave propagation in 15 trabecular bone samples from the human lumbar spine, using microcomputed tomography-based finite-element modeling. The model included only the solid bone, without the bone marrow. Two structural parameters were calculated: the bone volume fraction (BV/TV) and the structural (apparent) elastic modulus (E_s), and the ultrasound propagation parameter speed of sound (SOS). Relations between BV/TV and E_s were similar to published experimental relations. At 1 MHz, correlations between SOS and the structural parameters BV/TV and E_s were rather weak, but the results can be explained from the specific features of the trabecular structure and the intrinsic material elastic modulus E_i. In particular, the systematic differences between the three main directions provide information on the trabecular structure. In addition, at 1 MHz the correlation found between the simulated SOS values and those calculated from the simple bar equation was poor when the three directions are considered separately. Hence, under these conditions, the homogenization approach - including the bar equation - is not valid. However, at lower frequencies (50-300 kHz) this correlation significantly improved. It is concluded that detailed analysis of ultrasound wave propagation through the solid structure in various directions and with various frequencies, can yield much information on the structural and mechanical properties of trabecular bone.     

Nondestructive Characterization of Planar Defects Using Laser-Generated Ultrasonic Shear Waves

This paper introduces a laser scanning technique to characterize internal planar defects in a specimen with parallel boundaries. Based on the principles of laser-based ultrasonic shear waves and shadowing, a procedure to determine flaw location, size, and orientation is described. The key feature of this scheme is the use of an optimum wave propagation angle where the maximum shear wave propagates. The feasibility of the approach is evidenced by testing specimens with various controlled and natural internal flaws. The experimental results are promising, in that the flaw characteristics can be determined with good accuracy. It is found that the scheme is especially useful for characterizing transverse-type flaws. The limitations of the technique are also addressed.     

Nondestructive Characterization of Planar Defects Using Laser-Generated Ultrasonic Shear Waves

Abstract

This paper introduces a laser scanning technique to characterize internal planar defects in a specimen with parallel boundaries. Based on the principles of laser-based ultrasonic shear waves and shadowing, a procedure to determine flaw location, size, and orientation is described. The key feature of this scheme is the use of an optimum wave propagation angle where the maximum shear wave propagates. The feasibility of the approach is evidenced by testing specimens with various controlled and natural internal flaws. The experimental results are promising, in that the flaw characteristics can be determined with good accuracy. It is found that the scheme is especially useful for characterizing transverse-type flaws. The limitations of the technique are also addressed.     

The dependence of ultrasonic backscatter on trabecular thickness in human calcaneus: theoretical and experimental results

Trabecular thickness within cancellous bone is an important determinant of osteoporotic fracture risk. Noninvasive assessment of trabecular thickness potentially could yield useful diagnostic information. Faran's theory of elastic scattering from a cylindrical object immersed in a fluid has been used to predict the dependence of ultrasonic backscatter on trabecular thickness. The theory predicts that, in the range of morphological and material properties expected for trabecular bone, the backscatter coefficient at 500 kHz should be approximately proportional to trabecular thickness to the power of 2.9. Experimental measurements of backscatter coefficient were performed on 43 human calcaneus samples in vitro. Mean trabecular thicknesses on the 43 samples were assessed using micro computed tomography (CT). A power law fit to the data showed that the backscatter coefficient empirically varied as trabecular thickness to the 2.8 power. The 95% confidence interval for this exponent was 1.7 to 3.9. The square of the correlation coefficient for the linear regression to the log transformed data was 0.40. This suggests that 40% of variations in backscatter may be attributed to variations in trabecular thickness. These results reinforce previous studies that offered validation for the Faran cylinder model for prediction of scattering properties of cancellous bone, and provide added evidence for the potential diagnostic utility of the backscatter measurement.     

Stress-wave displacement polarizations and aftenuation in unidirectional composites: Theory and experiment

The wave propagation mechanism of changes in displacement polarizations was studied in unidirectional graphite/epoxy composite materials. Change in Displacements can be large enough to cause a transition in the mode or displacement polarizations from longitudinal to transverse. These unusual mode transitions are a result of the peculiar elastic anisotropy observed in only a few crystals and unidirectional graphite/epoxy composities at high-fiber volume fractions Theoretical calculation of these mode transitions were compared with experimental measurements Mode transitions occur when the wave vector orientation is varied from 51.9° to 74.4° in unidirectional samples of T300/5208 graphite/epoxy composite with a 0.6°-fiber volume fraction. Energy flux deviation and particle displacement directions and amplitudes also were compared with theory. To show this mode transition, an attenuation study was performed. The attenuation coefficient, measured in units of reciprocal time, does not appear to depend on the wave vector orientation and the wave type (quasi-transverse and quasi-longitudinal waves) at 5-MHZ frequency. But the attenuation coefficient, expressed in units of reciprocal length, does depend on the wave type and the wave vector orientation because the wave velocity is included in the calculation of this coefficient. Previous studies have focused on how anisotropy and attenuation influence the stress wave speed (eigenvalues), but in this study we focused more on how the same parameters influence the displacement polarizations (eigenvectors) of the same propagating waves. Because eigenvalues and their corresponding eigenvectors are both solutions of the same eigenvalue problem, more attention should be given to measurement of the eigenvectors.     

Bidirectional axial transmission can improve accuracy and precision of ultrasonic velocity measurement in cortical bone: a validation on test materials

The axial transmission technique uses a linear arrangement of ultrasonic emitters and receivers placed on a same side of a cortical bone site in contact with the skin, involving ultrasonic propagation along the axis of bone. The velocity of the waves radiated from bone has been shown to reflect bone status. The thickness and composition of soft tissue may vary along the length of the bone, between different skeletal sites, or between subjects. Hence, accurate estimates of velocity require first to eliminate the effect of the overlying soft tissue that is traversed by the ultrasound wave. To correct for such bias without measuring soft tissue properties, we designed new ultrasonic probes in the 1-2 MHz frequency range. It is based on propagation along the bone surface in two opposite directions from two sources placed on both sides of a unique group of receivers. The aim is to obtain an unbiased estimate of the velocity without any intermediate calculation of soft tissue properties, such as thickness variation or velocity. Validation tests were performed on academic material such as Perspex or aluminium. We found that head wave velocity values could be biased by more than 10% for inclination of a few degrees between the test specimen surface and the probe. On test materials, the compensation procedure implemented in our probe led to a relative precision error on velocity measurement lower than 0.2 to 0.3%. These results suggest that the correction procedure allows measuring in vivo velocities independently of soft tissue properties.     

相关函数算法在超声导波计量管道内液体流速中的应用

通过对超声导波技术在充液管道内流量计量的研究,提出了一种新的供热管路中热量表流量计量在线检测方法。试验中,因超声导波在充液管道内传播速度较快,为了准确地测量管道内液体流速对导波传播速度的影响,需要非常准确地提取导波的传播时间。分析了相关函数法在计算导波传播时间的应用原理,并通过理论分析和实验验证了相关函数法的准确性,为采用超声导波测量管道内液体流量提供理论基础。     

Guided Wave Damage Characterization via Minimum Variance Imaging with a Distributed Array of Ultrasonic Sensors

Guided wave imaging with a distributed array of inexpensive transducers offers a fast and cost-efficient means for damage detection and localization in plate-like structures such as aircraft and spacecraft skins. As such, this technology is a natural choice for inclusion in condition-based maintenance and integrated structural health management programs. One of the implementation challenges results from the complex interaction of propagating ultrasonic waves with both the interrogation structure and potential defects or damage. For example, a guided ultrasonic wave interacts with a surface or sub-surface defect differently depending on the angle of incidence, defect size and orientation, excitation frequency, and guided wave mode. However, this complex interaction also provides a mechanism for guided wave imaging algorithms to perform damage characterization in addition to damage detection and localization. Damage characterization provides a mechanism to help discriminate actual damage (e.g. fatigue cracks) from benign changes, and can be used with crack propagation models to estimate remaining life. This work proposes the use of minimum variance imaging to perform damage detection, localization, and characterization. Scattering assumptions used to perform damage characterization are obtained through both analytical and finite element models. Experimental data from an in situ distributed array are used to demonstrate feasibility of this approach using a through-hole and two through-thickness notches of different orientations to simulate damage in an aluminum plate.     

Nondestructive Evaluation of Anisotropy in Composite Materials Via Acoustic Birefringence

A linearly polarized ultrasonic shear wave propagating in fiber-reinforced composites produces an elliptical vibration pattern due to birefringence. The acoustic birefringence of shear waves is analyzed and applied for evaluation of in-plane stiffness anisotropy of uni-axial nonfabric, biaxial, and quadra-axial fabric composite materials. The parameters of the elliptical motion are determined by measurements of the amplitude and phase of the transmitted/reflected shear wave as functions of the receiver polarization angle. The strength of birefringence is derived from the measured elliptical pattern and used to quantify the in-plane stiffness asymmetry in glass and carbon fiber-reinforced nonfabric and fabric composites. The technique is shown to be sensitive enough to detect damage induced variation in stiffness anisotropy. The correlation between the asymmetry of shear stiffness and orientation of impact cracking in composites is also discussed.     

Depth measurement for cracks in corners using ultrasonic Rayleigh waves

The scattering of ultrasonic Rayleigh waves incident normally on corners containing cracks is considered by using elastodynamic ray theory. Detailed calculations are presented for vertical and horizontal cracks in right-angle corners in aluminium. It is shown that crack depth can be measured simply from the spacing of interference fringes in the high-frequency spectra of either the back- or forward-scattered Rayleigh waves, given only a knowledge of the Rayleigh wave speed. Use of the back-scattered wave is preferable because its fringes show stronger modulation, and because an experiment requires a single transducer and access to only one face of the specimen. The technique is applicable without modification to the more general case of a crack at any angle in a corner of any angle.     

Single-Sided Ultrasonic Technique to Characterize Thick FRP Composites

This paper reports on the development of a single-sided, laser ultrasonic technique that is capable of measuring all of the engineering constants of a pultruded FRP composite structural member. The complex nature of these members (which are used in civil infrastructure applications) precludes the determination of all of the engineering constants using mechanical tests, while immersion ultrasonic techniques have certain drawbacks. The proposed technique, which uses a combination of contact piezoelectric and optical methodologies, does not call for cutting the specimen or placing it in an immersion tank. Optical generation and detection of ultrasound enble the direct measurement of the longitudinal and surface acoustic wave speeds in the pultrusion direction, without requiring access to the ends of the specimen, or relying on reflected quasilongitudinal waves. The measured engineering constants are compared to immersion technique results, and the advantages of the proposed technique are clearly demonstrated.