Resonant ultrasound spectroscopy measurements of the elastic constants of human dentin

The technique of resonant ultrasound spectroscopy (RUS) was used to measure the second-order elastic constants of hydrated human dentin. Specimens were placed between two transducers, and the resonant frequencies of vibration were measured between 0.5 and 1.4 MHz. The elastic constants determined from the measured resonant frequencies in hydrated dentin exhibited slight hexagonal anisotropy, with the stiffest direction being perpendicular to the axis of the tubules (E11 = 25.1GPA) This hexagonal anisotropy was small (E33/E11 = 0.92), and almost disappeared when the specimens were dried. In addition, there was a pronounced anisotropy in the Poisson's ratio of wet dentin: v21 = 0.45; v31 = 0.29. With drying in air, this anisotropy vanished: v21 = v31 = 0.29. The isotropic Young's modulus of dried dentin was 28.1 GPa. RUS shows promise for determining the elastic constants in mineralized tissues.     

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.     

Estimation of the effective transversely isotropic elastic constants of a material from known values of the material's orthotropic elastic constants

A method is illustrated for determining the effective transversely isotropic (or isotropic) elastic constants from measured orthotropic elastic constants. This method consists of constructing upper and lower bounds on the effective transversely isotropic (or isotropic) elastic constants using the known orthotropic values. This method is illustrated using three sets of elastic constants for bone. Fortunately, the upper and lower bounds are very close. Thus very good approximations for the effective transversely isotropic (or isotropic) elastic constants for cortical and cancellous bone are obtained from previously published data on the orthotropic elastic constants for those tissue types. This work is undertaken to build a greater database for the transversely isotropic elastic constants of bone with the intention of employing them in a transversely isotropic model of bone poroelasticity. An interesting aspect of the present result is that the Voigt and Reuss bounds are very tight for these anisotropic materials. This is not always the case for these bounds. Received: 14 November 2001 / Accepted: 25 February 2002     

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.     

A global relationship between trabecular bone morphology and homogenized elastic properties.

An alternative concept of the relationship between morphological and elastic properties of trabecular bone is presented and applied to human tissue from several anatomical locations using a digital approach. The three-dimensional morphology of trabecular bone was assessed with a microcomputed tomography system and the method of directed secants as well as the star volume procedure were used to compute mean intercept length (MIL) and average bone length (ABL) of 4 mm cubic specimens. Assuming isotropic elastic properties for the trabecular tissue, the general elastic tensors of the bone specimens were determined using the homogenization method and the closest orthotropic tensors were calculated with an optimization algorithm. The assumption of orthotropy for trabecular bone was found to improve with specimen size and hold within 6.1 percent for a 4 mm cube size. A strong global relationship (r2 = 0.95) was obtained between fabric and the orthotropic elastic tensor with a minimal set of five constants. Mean intercept length and average bone length provided an equivalent power of prediction. These results support the hypothesis that the elastic properties of human trabecular bone from an arbitrary anatomical location can be estimated from an approximation of the anisotropic morphology and a prior knowledge of tissue properties.     

Finite element analysis of a three-dimensional open-celled model for trabecular bone

Based on a regular array of cubic unit cells, each containing a body-centered spherical void, we created an idealized three-dimensional model for both subchondral trabecular bone and a class of porous foams. By considering only face-to-face stacking of unit cells, the inherent symmetry was such that, except at the surface, the displacements and stresses within any one unit cell were representative of the entire porous structure. Using prescribed displacements the model was loaded in both uniaxial compressive strain and uniaxial shear strain. Based on the response to these loads, we found the tensor of elastic constants for an equivalent homogeneous elastic solid with cubic symmetry. We then compared the predicted modulus with our experimental values for bovine trabecular bone and literature values for an open-celled latex rubber foam.     

Determination of orthotropic bone elastic constants using FEA and modal analysis

Finite element models have been widely employed in an effort to quantify the stress and strain distribution around implanted prostheses and to explore the influence of these distributions on their long-term stability. In order to provide meaningful predictions, such models must contain an appropriate reflection of mechanical properties. Detailed geometrical and density information is now readily available from CT scanning. However, despite the use of phantoms, a method of determining mechanical properties (or elastic constants) from bone density has yet to be made available in a usable form.In this study, a cadaveric bone was CT scanned and its natural frequencies were measured using modal analysis. Using the geometry obtained from the CT scan data, a finite element mesh was created with the distribution of density established by matching the mass of the FE bone model with the mass of the cadaveric bone. The maximum values of the orthotropic elastic constants were then established by matching the predictions from FE modal analyses to the experimental natural frequencies, giving a maximum error of 7.8% over 4 modes of vibration. Finally, the elastic constants of the bone derived from the analyses were compared with those measured using ultrasound techniques. This produced a difference of <1% for both the maximum density and axial Young's Modulus. This study has thereby produced an orthotropic finite element model of a human femur. More importantly, however, is the implication that it is possible to create a valid FE model by simply comparing the FE results with the measured resonant frequency of the CT scanned bone.     

Low frequency cMUT technology: Application to measurement of brain movement and assessment of bone quality

Following recent advances in medical ultrasound imaging methods almost all human tissues can currently be examined. There are, however, two exceptions: the human skeleton and the brain, because bone tissue is a strongly attenuating and defocusing medium, rendering classical pulse-echo imaging methods inappropriate. Specific imaging approaches within low frequency bands, i.e. 200 kHz–2 MHz, have therefore recently been developed and the results are very promising: (1) the technique for the bone is axial transmission measurement, which consists of using elastic guided modes to characterize all elastic constants of the medium; (2) for brain exploration, it has been demonstrated that brain movement can be measured (i.e. brain pulsatility) with elastography techniques. However, there are certain limitations in the fabrication of low frequency probes with classical technology, which involve finding an alternative to the traditional PZT. Capacitive Micromachined Ultrasonic Transducers (cMUTs) can overcome these limitations and greatly improve these new imaging modalities. The study presented here represents technological development with several goals: (1) the design and fabrication of two different low frequency linear arrays for bone and brain exploration, respectively the testing of axial transmission measurements with a cMUT probe and; (2) comparison with a PZT probe; (3) the development of an imaging method based on the elastography of brain pulsatility, its implementation in a commercial ultrasound scanner and clinical trials for the validation. The results obtained with cMUT and PZT probes are compared.     

Inactivation of glucose-6-phosphate dehydrogenase in solution by low- and high-frequency ultrasound

Kinetics of inactivation of glucose-6-phosphate dehydrogenase (G6PDH, EC 1.1.1.49) in 0.1 M phosphate buffer (pH 7.4) within temperature range from 36 to 50 degrees C was studied comparatively under conditions of exposure of enzyme solution to low-frequency (LF, 27 kHz, 60 W/cm2) or high-frequency (HF, 880 kHz, 1.0 W/cm2) ultrasound (USD). Inactivation of G6PDH was characterized by effective first-order rate constants: (k(in)) total (summarized) inactivation; (k(in)*) thermal inactivation; and (k(in)(usd)) ultrasonic inactivation. Dilution of enzyme solution from 20 to 3 nM was accompanied by a significant increase in the values of the three rate constants. The following inequality was valid in all cases: k(in) > k(in)*. The rate constants increased upon increasing the temperature. The Arrhenius plots of the temperature dependencies of k(in) and k(in) (usd) have a salient point at 44 degrees C. The activation energy (Eact) of the total inactivation of G6PDH was higher than Eact for the process of ultrasonic inactivation of this enzyme. The two values were found to depend on USD frequency: Eact in case of inactivation with low-frequency ultrasound (LF-USD) was higher than in case of inactivation with high-frequency ultrasound (HF-USD). The rate of the ultrasonic induced inactivation of this enzyme substantially decreased in the presence of low concentrations of traps of radicals HO. (dimethylformamide, ethanol, and mannitol). This fact supports the conclusion that free radicals are involved in the mechanism of the G6PDH inactivation in solutions exposed to LF-USD and HF-USD. Ethanol was an effective protector of G6PDH inactivation in enzyme solutions exposed to USD.     

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.     

Characterization of vascular strain during in-vitro angioplasty with high-resolution ultrasound speckle tracking

Background

Ultrasound elasticity imaging provides biomechanical and elastic properties of vascular tissue, with the potential to distinguish between tissue motion and tissue strain. To validate the ability of ultrasound elasticity imaging to predict structurally defined physical changes in tissue, strain measurement patterns during angioplasty in four bovine carotid artery pathology samples were compared to the measured physical characteristics of the tissue specimens.

Methods

Using computational image-processing techniques, the circumferences of each bovine artery specimen were obtained from ultrasound and pathologic data.

Results

Ultrasound-strain-based and pathology-based arterial circumference measurements were correlated with an R² value of 0.94 (p = 0.03). The experimental elasticity imaging results confirmed the onset of deformation of an angioplasty procedure by indicating a consistent inflection point where vessel fibers were fully unfolded and vessel wall strain initiated.

Conclusion

These results validate the ability of ultrasound elasticity imaging to measure localized mechanical changes in vascular tissue.     

Wavelet decomposition of transmitted ultrasound wave through a 1-D muscle-bone system

In the attempt for using ultrasound as a diagnostic device for osteoporosis, several authors have described the result of the in vitro experiment in which ultrasound is passed through a cancellous bone specimen placed in a water tank. However, in the in vivo setting, a patient's cancellous bone is surrounded by cortical and muscle layers. This paper considers in the one-dimensional case (1) what effect the cortical bone segments surrounding the cancellous segment would have on the received signal and (2) what the received signal would be when a source and receiver are placed on opposite sides of a structure consisting of a cancellous segment surrounded by cortical and muscle layers. Mathematically this is accomplished by representing the received signal as a sum of wavelets which go through different reflection-transmission histories at the muscle-cortical bone and cortical-cancellous bone interfaces. The muscle and cortical bone are modeled as elastic materials and the cancellous bone as a poroelastic material described by the Biot-Johnson-Koplik-Dashen model. The approach presented here permits the assessment of which possible paths of transmission and reflection through the cortical-cancellous or muscle-cortical-cancellous complex will result in significant contributions to the received waveform. This piece of information can be useful for solving the inverse problem of non-destructive assessment of material properties of bone. Our methodology can be generalized to three-dimensional parallelly layered structure by first applying Fourier transform in the directions perpendicular to the transverse direction.     

Vibrational characteristics of bone fracture and fracture repair: application to excised rat femur

BACKGROUND: The vibrational characteristics of any object are directly dependent on the physical properties of that object. Therefore, changing the physical properties of an object will cause the object to adopt changed natural frequencies. A fracture in a bone results in the loss of mechanical stability of the bone. This change in mechanical properties of a bone should result in a change of the resonant frequencies of that bone. A vibrational method for bone evaluation has been introduced. METHOD OF APPROACH: This method uses the radiation force of focused amplitude-modulated ultrasound to exert a vibrating force directly, and remotely, on a bone. The vibration frequency is varied in the range of interest to induce resonances in the bone. The resulting bone motion is recorded and the resonance frequencies are determined. Experiments are conducted on excised rat femurs and resonance frequencies of intact, fractured, and bonded (simulating healed) bones are measured. RESULTS: The experiments demonstrate that changes in the resonance frequency are indicative of bone fracture and healing, i.e., the fractured bone exhibits a lower resonance frequency than the intact bone, and the resonance frequency of the bonded bone approaches that of the intact bone. CONCLUSION: It is concluded that the proposed radiation force method may be used as a remote and noninvasive tool for monitoring bone fracture and healing process, and the use of focused ultrasound enables one to selectively evaluate individual bones.     

Non Destructive Characterization of Cortical Bone Micro-Damage by Nonlinear Resonant Ultrasound Spectroscopy

The objective of the study was to evaluate the ability of a nonlinear ultrasound technique, the so-called nonlinear resonant ultrasound spectroscopy (NRUS) technique, for detecting early microdamage accumulation in cortical bone induced by four-point bending fatigue. Small parallelepiped beam-shaped human cortical bone specimens were subjected to cyclic four-point bending fatigue in several steps. The specimens were prepared to control damage localization during four-point bending fatigue cycling and to unambiguously identify resonant modes for NRUS measurements. NRUS measurements were achieved to follow the evolution of the nonlinear hysteretic elastic behavior during fatigue-induced damage. After each fatigue step, a small number of specimens was removed from the protocol and set apart to quantitatively assess the microcrack number density and length using synchrotron radiation micro-computed tomography (SR-µCT). The results showed a significant effect of damage steps on the nonlinear hysteretic elastic behavior. No significant change in the overall length of microcracks was observed in damaged regions compared to the load-free control regions. Only an increased number of shortest microcracks, those in the lowest quartile, was noticed. This was suggestive of newly formed microcracks during the early phases of damage accumulation. The variation of nonlinear hysteretic elastic behavior was significantly correlated to the variation of the density of short microcracks. Our results suggest that the nonlinear hysteretic elastic behavior is sensitive to early bone microdamage. Therefore NRUS technique can be used to monitor fatigue microdamage progression in in vitro experiments.     

Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture

A method is presented to find orthotropic elastic symmetries and constants directly from the elastic coefficients in the overall stiffness matrix of trabecular bone test specimens. Contrary to earlier developed techniques, this method does not require pure orthotropic behavior or additional fabric measurements. The method uses high-resolution computer reconstructions of trabecular bone specimens as input for large-scale FE-analyses to determine all the 21 elastic coefficients in the overall stiffness matrix of the specimen, using a direct mechanics approach. An optimization procedure is then used to find the coordinate transformation that yields the best orthotropic representation of this matrix. The method is illustrated here relative to two trabecular bone specimens. The techniques developed here can be used to obtain a complete characterization of the mechanical properties of trabecular architecture. With the development of in vivo reconstruction techniques, even in vivo measurements will be possible.     

Theoretically proposed optimal frequency for ultrasound induced cartilage restoration

Background

Matching the frequency of the driving force to that of the system’s natural frequency of vibration results in greater amplitude response. Thus we hypothesize that applying ultrasound at the chondrocyte’s resonant frequency will result in greater deformation than applying similar ultrasound power at a frequency outside of the resonant bandwidth. Based on this resonant hypothesis, our group previously confirmed theoretically and experimentally that ultrasound stimulation of suspended chondrocytes at resonance (5 MHz) maximized gene expression of load inducible genes. However, this study was based on suspended chondrocytes. The resonant frequency of a chondrocyte does not only depend on the cell mass and intracellular stiffness, but also on the mechanical properties of the surrounding medium. An in vivo chondrocyte’s environment differs whether it be a blood clot (following microfracture), a hydrogel or the pericellular and extracellular matrices of the natural cartilage. All have distinct structures and compositions leading to different resonant frequencies. In this study, we present two theoretical models, the first model to understand the effects of the resonant frequency on the cellular deformation and the second to identify the optimal frequency range for clinical applications of ultrasound to enhance cartilage restoration.

Results

We showed that applying low-intensity ultrasound at the resonant frequency induced deformation equivalent to that experimentally calculated in previous studies at higher intensities and a 1 MHz frequency. Additionally, the resonant frequency of an in vivo chondrocyte in healthy conditions, osteoarthritic conditions, embedded in a blood clot and embedded in fibrin ranges from 3.5???4.8 MHz.

Conclusion

The main finding of this study is the theoretically proposed optimal frequency for clinical applications of therapeutic ultrasound induced cartilage restoration is 3.5???4.8 MHz (the resonant frequencies of in vivo chondrocytes). Application of ultrasound in this frequency range will maximize desired bioeffects.

     

Non-destructive assessment of human ribs mechanical properties using quantitative ultrasound

Advanced finite element models of the thorax have been developed to study, for example, the effects of car crashes. While there is a need for material properties to parameterize such models, specific properties are largely missing. Non-destructive techniques applicable in vivo would, therefore, be of interest to support further development of thorax models. The only non-destructive technique available today to derive rib bone properties would be based on quantitative computed tomography that measures bone mineral density. However, this approach is limited by the radiation dose. Bidirectional ultrasound axial transmission was developed on long bones ex vivo and used to assess in vivo health status of the radius. However, it is currently unknown if the ribs are good candidates for such a measurement. Therefore, the goal of this study is to evaluate the relationship between ex vivo ultrasonic measurements (axial transmission) and the mechanical properties of human ribs to determine if the mechanical properties of the ribs can be quantified non-destructively. The results show statistically significant relationships between the ultrasonic measurements and mechanical properties of the ribs. These results are promising with respect to a non-destructive and non-ionizing assessment of rib mechanical properties.     

Dependence of ultrasonic attenuation on bone mass and microstructure in bovine cortical bone

The development of the axial transmission technique now enables in vivo evaluation of cortical bone quality, which plays an important role in bone fragility. Cortical bone is a complex multiscale material, which may be made of different types of microstructure. The interaction between ultrasound and cortical bone remains unclear and most studies have been confined to wave speed analysis. The first aim of this study is to investigate the dependence of the frequency-dependent attenuation on the type of bone microstructure. The second goal is to determine whether broadband ultrasonic attenuation (BUA) is related to volumetric bone mineral density (vBMD) and mass density. Parallelepipedic samples of bovine cortical bone were cut from three specimens and tested in the axial, radial and tangential directions using an ultrasonic transmission device. BUA was evaluated over a 1-MHz wide bandwidth around 4MHz. In addition, the microstructure of each sample was determined using an optical microscope. BUA values measured in porotic microstructure are significantly higher than in Haversian microstructure. The lowest BUA values are obtained for plexiform microstructure. For all structures, BUA in the axial direction is significantly smaller than in the radial and tangential directions. Moreover, BUA is correlated with both vBMD and density (determination coefficient (R2) equal to 0.44 and 0.65, respectively, in the axial direction). BUA variations can be explained by scattering and viscoelastic mechanisms. This study suggests that BUA measurements have the potential to discriminate among different cortical bone microstructures in addition to providing material properties.     

Size effects in the elasticity and viscoelasticity of bone

 Size effects of large magnitude are observed in the torsional shear modulus and damping of bovine plexiform bone. Damping increases and stiffness decreases with specimen size over all sizes studied. Measurements were conducted in torsion using a laser-based micromechanics apparatus capable of viscoelastic studies over a range of frequencies up to 100 kHz, upon samples of various size, with no parasitic friction or other errors that could mimic any size effect. Torsional tan δ at 1 Hz varies by about a factor of five over the size range 2.8–6.2 mm thick, and is more dependent on specimen thickness at 1 Hz than it is at higher frequency. The size effects are attributed to compliance and viscoelasticity of the interfaces between laminae. These laminae must be substantially stiffer than whole bone. Observed size effects are likely to play a role in understanding scaling laws of bones in living organisms. Received: 12 February 2002 / Accepted: 22 November 2002 R.S. Lakes is also at the Department of Biomedical Engineering, Department of Engineering Physics, Materials Science Program and Rheology Research Center all at the University of Wisconsin. We thank the Whitaker foundation for their support of this work.     

Microdamage evaluation in human trabecular bone based on nonlinear ultrasound vibro-modulation (NUVM)

The primary aim of this work is to investigate the potential of nonlinear ultrasound for microdamage detection in human bone. Microdamage evaluation in human bone is of great importance, because it is considered a significant parameter for characterizing fracture risk. Experiments employing nonlinear acoustic vibro-modulation were carried out in human femoral trabecular specimens removed during surgery. A frequency mixing (inter-modulation) was observed between an ultrasound wave, propagating in the bone, and a low-frequency vibration applied directly to the bone specimens. The appearance of side frequencies, which are related to the vibrational excitation, around the fundamental ultrasound frequency manifests the modulation nonlinear phenomenon. Instead of inducing microdamage by mechanical fatigue loading, specimens with different degree of osteoporosis were used. The experiments demonstrated that osteoporotic bone exhibits stronger nonlinearity compared to healthy bone presenting significant increase of the modulation amplitude with increasing degree of osteoporosis. The obtained results indicate that, in contrast to conventional hysteretic nonlinearity, dissipative acoustic nonlinearity can be of significance in the generation of nonlinear modulation effects. In the proposed technique the size and the shape of samples are not crucial compared to nonlinear resonant ultrasound spectroscopy (NRUS). Furthermore, the method is sensitive to the presence of microdamage, non-invasive, easy to implement and most important, it can be proved valuable tool for in vivo bone damage characterization.