Fatigue evaluation of long cortical bone using ultrasonic guided waves

Bone fatigue fracture is a progressive disease due to stress concentration. This study aims to evaluate the long bone fatigue damage using the ultrasonic guided waves. Two-dimensional finite-difference time-domain method was employed to simulate the ultrasonic guided wave propagation in the long bone under different elastic modulus. The experiment was conducted on a 3.8 mm-thick bovine bone plate. The phase velocities of two fundamental guided modes, A1 and S1, were measured by using the axial transmission technique. Simulation shows that the phase velocities of guided modes A1 and S1 decrease with the increasing of the fatigue damage. After 20,000 cycles of fatigue loading on the bone plate, the average phase velocities of A1 and S1 modes were 6.6% and 5.3% respectively, lower than those of the intact bone. The study suggests that ultrasonic guided waves can be potentially used to evaluate the fatigue damage in long bones.     

Compliance calibration for fracture testing of equine cortical bone

An experimental compliance calibration method for measuring crack length in fracture toughness tests of cortical bone was developed. Calibration tests were conducted on twenty compact type fracture specimens machined from the mid-diaphysis of five pairs of equine third metacarpal bones. Specimens were oriented for crack propagation in a direction transverse to the longitudinal axis of the bone. Specimen compliance was determined from the load vs. crack opening displacement record over a range of crack lengths from 0.48 to 0.75 times the specimen width. The results demonstrate that the compliance calibration method developed for isotropic materials can be used to determine crack length in bone, which is transversely isotropic. However, specimens from lateral and dorsal regions exhibited significantly different compliance calibrations even after differences in elastic modulus were taken into account in the normalized compliance.     

Anisotropic propagation of Ca2+ waves in isolated cardiomyocytes.

Digital imaging microscopy of fluor-3 fluorescence was used to study the propagation of intracellular Ca2+ waves in isolated adult rat cardiomyocytes from 17 to 37 degrees C. Ca2+ waves spread in both transverse and longitudinal direction of a myocyte. Transverse propagation was pronounced in waves starting from a focus at the edge of a myocyte and in waves following an irregular, curved path (spiral waves). For the former type of waves, propagation velocities were determined. Both transverse and longitudinal wave components propagated at constant velocity ranging from 30 to 125 micron/s. Myocytes were anisotropic with respect to wave propagation: waves propagated faster in the longitudinal than in the transverse direction. The ratio between longitudinal and transverse velocity increased from 1.30 at 17 degrees C to 1.55 at 37 degrees C. Apparent activation energies for transverse and longitudinal wave propagation were estimated to be -20 kJ/mol, suggesting that these processes are limited by diffusion of Ca2+. Direction-dependent propagation velocities are interpreted to result from the highly ordered structure of the myocytes, especially from the anisotropic arrangement of diffusion obstacles such as myofilaments and mitochondria.     

Accuracy of elastic property measurement in mandibular cortical bone is improved by using cylindrical specimens

Ultrasonic determination of elastic properties in human craniofacial cortical bone is problematic because of a lack of information about the principal material axes, and because the cortex is often thinner than in long bones. This study investigated solutions that permit reasonable determination of elastic properties in the human mandible. We tested whether ultrasonic velocities could be reliably measured in cylindrical samples of aluminum and mandibular bone, and the effects of reduced specimen thickness. Results indicted that (1) varying shape had minimal effects on ultrasonic velocities or derived elastic properties, and (2) ultrasonic velocities have relatively increased measurement error as propagation distances decreased. The increased error in velocity measurements of mandibular cortical specimens of less than 1.2 mm in thickness should be considered when assessing the reliability of single measurements.     

Elasticity–density and viscoelasticity–density relationships at the tibia mid-diaphysis assessed from resonant ultrasound spectroscopy measurements

Cortical bone tissue is an anisotropic material characterized by typically five independent elastic coefficients (for transverse isotropy) governing shear and longitudinal deformations in the different anatomical directions. It is well established that the Young’s modulus in the direction of the bone axis of long bones has a strong relationship with mass density. It is not clear, however, whether relationships of similar strength exist for the other elastic coefficients, for they have seldom been investigated, and the results available in the literature are contradictory. The objectives of the present work were to document the anisotropic elastic properties of cortical bone at the tibia mid-diaphysis and to elucidate their relationships with mass density. Resonant ultrasound spectroscopy (RUS) was used to measure the transverse isotropic stiffness tensor of 55 specimens from 19 donors. Except for Poisson’s ratios and the non-diagonal stiffness coefficient, strong linear correlations between the different elastic coefficients \((0.7 < {r^{2}} < 0.99)\) and between these coefficients and density \((0.79 < {r^{2}} < 0.89)\) were found. Comparison with previously published data from femur specimens suggested that the strong correlations evidenced in this study may not only be valid for the mid-tibia. RUS also measures the viscous part of the stiffness tensor. An anisotropy ratio close to two was found for damping coefficients. Damping increased as the mass density decreased. The data suggest that a relatively accurate estimation of all the mid-tibia elastic coefficients can be derived from mass density. This is of particular interest (1) to design organ-scale bone models in which elastic coefficients are mapped according to Hounsfield values from computed tomography scans as a surrogate for mass density and (2) to model ultrasound propagation at the mid-tibia, which is an important site for the in vivo assessment of bone status with axial transmission techniques.     

The bone tissue of the canine mandible is elastically isotropic

This paper reports experimental measurements which show that canine mandibular bone tissue is elastically isotropic. Earlier work has established that human, canine and bovine cortical bone tissue of the femur, tibia and skull are elastically anisotropic and therefore the reported isotropy of mandibular tissue was unexpected. The isotropic elastic moduli of the canine mandible are represented by a Young's modulus of 7.5 GPa and a Poisson's ratio of 0.4. Earlier work gave the three orthotropic Young's moduli of the cortical one of the canine femur as 12.8 GPa, 15.6 GPa and 20.1 GPa. The experimental technique employed is elastic wave propagation at ultrasonic frequencies.     

Elastic properties of external cortical bone in the craniofacial skeleton of the rhesus monkey

Knowledge of elastic properties and of their variation in the cortical bone of the craniofacial skeleton is indispensable for creating accurate finite-element models to explore the biomechanics and adaptation of the skull in primates. In this study, we measured elastic properties of the external cortex of the rhesus monkey craniofacial skeleton, using an ultrasonic technique. Twenty-eight cylindrical cortical specimens were removed from each of six craniofacial skeletons of adult Macaca mulatta. Thickness, density, and a set of longitudinal and transverse ultrasonic velocities were measured on each specimen to allow calculation of the elastic properties in three dimensions, according to equations derived from Newton's second law and Hooke's law. The axes of maximum stiffness were determined by fitting longitudinal velocities measured along the perimeter of each cortical specimen to a sinusoidal function. Results showed significant differences in elastic properties between different functional areas of the rhesus cranium, and that many sites have a consistent orientation of maximum stiffness among specimens. Overall, the cortical bones of the rhesus monkey skull can be modeled as orthotropic in many regions, and as transversely isotropic in some regions, e.g., the supraorbital region. There are differences from human crania, suggesting that structural differences in skeletal form relate to differences in cortical material properties across species. These differences also suggest that we require more comparative data on elastic properties in primate craniofacial skeletons to explore effectively the functional significance of these differences, especially when these differences are elucidated through modeling approaches, such as finite-element modeling.     

Wave propagation in bone media

The composite nature of bone dictates the use of a model for bone which is transversely isotropic. We solve the associated sets of partial differential equations governing the dynamic elastic behavoor of a two-layered cylindrical-shaped bone. The solution is analyzed for long, short, and intermediate length waves. The special case of compact bone is treated for long and short wave lengths and a numerical example is worked out to determine the wave speeds (for short wave lengths) given a set of elastic constants, determined by ultrasonic methods, and the bone density, wave frequency, and radius.     

The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity

The objective of this study was to examine the dependence of the elastic properties of cortical bone as a transversely isotropic material on its porosity. The longitudinal Young's modulus, transverse Young's modulus, longitudinal shear modulus, transverse shear modulus, and longitudinal Poisson's ratio of cortical bone were determined from eighteen groups of longitudinal and transverse specimens using tensile and torsional tests on a servo-hydraulic material testing system. These cylindrical waisted specimens of cortical bone were harvested from the middle diaphysis of three pairs of human femora. The porosity of these specimens was assessed by means of histology. Our study demonstrated that the longitudinal Young's and shear moduli of human femoral cortical bone were significantly (p<0.01) negatively correlated with the porosity of cortical bone. Conversely, the elastic properties in the transverse direction did not have statistically significant correlations with the porosity of cortical bone. As a result, the transverse elastic properties of cortical bone were less sensitive to changes in porosity than those in the longitudinal direction. Additionally, the anisotropic ratios of cortical bone elasticity were found to be significantly (p<0.01) negatively correlated with its porosity, indicating that cortical bone tended to become more isotropic when its porosity increased. These results may help a number of researchers develop more accurate micromechanics models of cortical bone.     

Sensitivity analysis of ultrasonic guided waves propagating in trilayered bone models: a numerical study

The fundamental ultrasonic guided modes are consistently observed in long bones ex vivo and in vivo. However, the responses of ultrasonic guided waves to the changes of cortical thickness, cortical elastic parameters, and thickness of the overlying soft tissues are not comprehensively understood. This paper systematically presents a sensitivity analysis of leaky Lamb modes to the geometry and material characteristics of layered bone model by means of semi-analytical finite element modeling. The stratified bone model is consisted of a transversely isotropic cortex with an overlying soft tissue and underlying marrow. The study is important as it offers guidance to the parameter inversion process about the optimal selection of guided modes and regions of sensitivity for better inversion results.     

Elastic properties of human supraorbital and mandibular bone

Elastic constants, including the elastic modulus, the shear modulus, and Poisson's ratio, were measured on human craniofacial bone specimens obtained from the supraorbital region and the buccal surfaces of the mandibles of unembalmed cadavers. Constants were determined using an ultrasonic wave technique in three directions relative to the surface of each sample: 1) normal, 2) tangential, and 3) longitudinal. Statistical analysis of these elastic constants indicated that significant differences in the relative proportions of elastic properties existed between the regions. Bone from the mandible along its longitudinal axis was stiffer than bone from the supraorbital region. Directional differences in both locations demonstrated that cranial bone was not elastically isotropic. It is suggested that differences in elastic properties correspond to regional differences in function. © 1993 Wiley-Liss, Inc.     

On the theory of Cherenkov emission in a plasma

The Cherenkov emission of transverse-longitudinal waves in an anisotropic plasma is considered by applying a Hamiltonian method and by drawing an analogy between the equations for the Cherenkov emission of purely transverse and purely longitudinal waves in isotropic media and the equations for the emission of transverse-longitudinal electromagnetic waves in a highly anisotropic medium (a magnetized plasma). A formula for the emitted power is derived, as well as an expression for the directional pattern of the emitted waves in an anisotropic plasma.     

Elastic properties of cancellous bone: measurement by an ultrasonic technique

The high degree of porosity of cancellous bone makes elastic property measurement difficult by traditional mechanical testing methods. An ultrasonic technique is described with which mechanical properties of anisotropic, rigid, porous materials, such as cancellous bone, can be measured. The technique utilizes unique piezoelectric transducers operated in a continuous wave mode at a frequency of approximately 50 kHz. Both longitudinal and shear waves can be propagated and received with the transducers allowing both Young's moduli and shear moduli to be determined with the technique. A comparison between moduli measured with the ultrasonic technique and moduli measured with traditional mechanical testing shows the new method to be quite accurate in elastic property determination, (r2 = 0.935, Emech = 1.00E1dt + 23.3 MPa) (r2 = 0.656, Gmech = 1.08 Gult--3.3MPa).     

Finite element modeling of impulsive excitation and shear wave propagation in an incompressible,transversely isotropic medium

Elastic properties of materials can be measured by observing shear wave propagation following localized, impulsive excitations and relating the propagation velocity to a model of the material. However, characterization of anisotropic materials is difficult because of the number of elasticity constants in the material model and the complex dependence of propagation velocity relative to the excitation axis, material symmetries, and propagation directions. In this study, we develop a model of wave propagation following impulsive excitation in an incompressible, transversely isotropic (TI) material such as muscle. Wave motion is described in terms of three propagation modes identified by their polarization relative to the material symmetry axis and propagation direction. Phase velocities for these propagation modes are expressed in terms of five elasticity constants needed to describe a general TI material, and also in terms of three constants after the application of two constraints that hold in the limit of an incompressible material. Group propagation velocities are derived from the phase velocities to describe the propagation of wave packets away from the excitation region following localized excitation. The theoretical model is compared to the results of finite element (FE) simulations performed using a nearly incompressible material model with the five elasticity constants chosen to preserve the essential properties of the material in the incompressible limit. Propagation velocities calculated from the FE displacement data show complex structure that agrees quantitatively with the theoretical model and demonstrates the possibility of measuring all three elasticity constants needed to characterize an incompressible, TI material.     

Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.     

Ultrasound propagation in anisotropic soft tissues: the application of linear elastic theory

Linear elastic theory has served well in modeling the mechanical properties of numerous materials. In modeling ultrasonic wave propagation in biological soft tissues, an isotropic model has usually been employed. Many tissues, however, possess a lower order of symmetry, and the speed of sound in muscle is known to vary with the direction of propagation. In this study, by applying linear regression to acoustic microscopic data from seven frog sartorius specimens, four observable elastic constants associated with a transversely isotropic model were obtained. The average values of these constants were c11 = 2.64, c13 = 3.39 and c33 = 4.40 Nm-2 for resting muscles and c11 = 2.65, c13 = 3.43 and c33 = 4.57 Nm-2 for muscles undergoing tetanic contraction, where '1' and '3' represent the transverse and longitudinal axes, respectively. In all cases, c44 was 0, indicating a minimal contribution from longitudinal shear. For all seven specimens, the model of transverse isotropy provided a better fit of the data than that of isotropy.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

The estimated elastic constants for a single bone osteonal lamella

Micromechanical estimates of the elastic constants for a single bone osteonal lamella and its substructures are reported. These estimates of elastic constants are accomplished at three distinct and organized hierarchical levels, that of a mineralized collagen fibril, a collagen fiber, and a single lamella. The smallest collagen structure is the collagen fibril whose diameter is the order of 20 nm. The next structural level is the collagen fiber with a diameter of the order of 80 nm. A lamella is a laminate structure, composed of multiple collagen fibers with embedded minerals and consists of several laminates. The thickness of one laminate in the lamella is approximately 130 nm. All collagen fibers in a laminate in the lamella are oriented in one direction. However, the laminates rotate relative to the adjacent laminates. In this work, all collagen fibers in a lamella are assumed to be aligned in the longitudinal direction. This kind of bone with all collagen fibers aligned in one direction is called a parallel fibered bone. The effective elastic constants for a parallel fibered bone are estimated by assuming periodic substructures. These results provide a database for estimating the anisotropic poroelastic constants of an osteon and also provide a database for building mathematical or computational models in bone micromechanics, such as bone damage mechanics and bone poroelasticity.     

Variations in cortical material properties throughout the human dentate mandible

Material properties and their variations in individual bone organs are important for understanding bone adaptation and quality at a tissue level, and are essential for accurate mechanical models. Yet material property variations have received little systematic study. Like all other material property studies in individual bone organs, studies of the human mandible are limited by a low number of both specimens and sampled regions. The aims of this study were to determine: 1) regional variability in mandibular material properties, 2) the effect of this variability on the modeling of mandibular function, and 3) the relationship of this variability to mandibular structure and function. We removed 31 samples on both facial and lingual cortices of 10 fresh adult dentate mandibles, measured cortical thickness and density, determined the directions of maximum stiffness with a pulse transmission ultrasonic technique, and calculated elastic properties from measured ultrasonic velocities. Results showed that each of these elastic properties in the dentate human mandible demonstrates unique regional variation. The direction of maximum stiffness was near parallel to the occlusal plane within the corpus. On the facial ramus, the direction of maximum stiffness was more vertically oriented. Several sites in the mandible did not show a consistent direction of maximum stiffness among specimens, although all specimens exhibited significant orthotropy. Mandibular cortical thickness varied significantly (P < 0.001) between sites, and decreased from 3.7 mm (SD = 0.9) anteriorly to 1.4 mm posteriorly (SD = 0.1). The cortical plate was also significantly thicker (P < 0.003) on the facial side than on the lingual side. Bone was 50-100% stiffer in the longitudinal direction (E(3), 20-30 GPa) than in the circumferential or tangential directions (E(2) or E(1); P < 0.001). The results suggest that material properties and directional variations have an important impact on mandibular mechanics. The accuracy of stresses calculated from strains and average material properties varies regionally, depending on variations in the direction of maximum stiffness and anisotropy. Stresses in some parts of the mandible can be more accurately calculated than in other regions. Limited evidence suggests that the orientations and anisotropies of cortical elastic properties correspond with features of cortical bone microstructure, although the relationship with functional stresses and strains is not clear.     

Anisotropic conduction block and reentry in neonatal rat ventricular myocyte monolayers

Anisotropy can lead to unidirectional conduction block that initiates reentry. We analyzed the mechanisms in patterned anisotropic neonatal rat ventricular myocyte monolayers. Voltage and intracellular Ca (Ca(i)) were optically mapped under the following conditions: extrastimulus (S1S2) testing and/or tetrodotoxin (TTX) to suppress Na current availability; heptanol to reduce gap junction conductance; and incremental rapid pacing. In anisotropic monolayers paced at 2 Hz, conduction velocity (CV) was faster longitudinally than transversely, with an anisotropy ratio [AR = CV(L)/CV(T), where CV(L) and CV(T) are CV in the longitudinal and transverse directions, respectively], averaging 2.1 ± 0.8. Interventions decreasing Na current availability, such as S1S2 pacing and TTX, slowed CV(L) and CV(T) proportionately, without changing the AR. Conduction block preferentially occurred longitudinal to fiber direction, commonly initiating reentry. Interventions that decreased gap junction conductance, such as heptanol, decreased CV(T) more than CV(L), increasing the AR and causing preferential transverse conduction block and reentry. Rapid pacing resembled the latter, increasing the AR and promoting transverse conduction block and reentry, which was prevented by the Ca(i) chelator 1,2-bis oaminophenoxy ethane-N,N,N',N'-tetraacetic acid (BAPTA). In contrast to isotropic and uniformly anisotropic monolayers, in which reentrant rotors drifted and self-terminated, bidirectional anisotropy (i.e., an abrupt change in fiber direction exceeding 45°) caused reentry to anchor near the zone of fiber direction change in 77% of monolayers. In anisotropic monolayers, unidirectional conduction block initiating reentry can occur longitudinal or transverse to fiber direction, depending on whether the experimental intervention reduces Na current availability or decreases gap junction conductance, agreeing with theoretical predictions.